Review Article A New Theory to Forecast the Price of...

Review ArticleA New Theory to Forecast the Price ofNonrenewable Energy Resources with Mass andEnergy-Capital Conservation Equations

Fabio Gori

Department of Industrial Engineering University of Rome ldquoTor Vergatardquo Via del Politecnico 1 00133 Rome Italy

Correspondence should be addressed to Fabio Gori fammannatiyahoocom

Received 31 December 2013 Accepted 1 April 2014 Published 5 June 2014

Academic Editors K Abhary and Y Demirel

Copyright copy 2014 Fabio Gori This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The mass and energy-capital conservation equations are employed to study the time evolution of mass and price of nonrenewableenergy resources extracted and sold to the market in case of no-accumulation and no-depletion that is when the resources areextracted and sold to the market at the same mass flow rate The Hotelling rule for nonrenewable resources that is an exponentialincrease of the price at the rate of the current interest multiplied the time is shown to be a special case of the general energy-capitalconservation equation when the mass flow rate of extracted resources is unityThemass and energy-capital conservation equationsare solved jointly to investigate the time evolution of the extracted resources

1 Introduction

The price evolution of nonrenewable energy resources isa very important problem in an economy based on non-renewable energy resources The economics of exhaustibleresources was investigated by Hotelling [1] and reviewed in[2] with the conclusion that the price of a resource increasesexponentially with the product of time and interest rate ofcapital

After the oil crisis of 1973 the problem of the energyresources became a very important issue and several energymodels and forecasts have been developed and employedDuring the decade of 1970s among the others McCall [3]presented a linear program for the world oil industry whichwas employed by an oil company Tewksbury [4] describeda practical general approach to forecast foreign oil pricesto the year 2000 Pindyck [5] described a new version ofan econometric policy model of the natural gas industryAdelman [6] forecasted a decade of rising real oil prices forthe US DuMoulin and Newland [7] came to the conclusionthat a rapid take-off in demand is likely to be followed by arapid increase in the price of oil

During the 1980s among the others Pearce [8] presentedan overall look at the world energy demand forecasts with

the conclusion that the judgment rather than explicit mod-elling is used to suggest a world crude oil scenario in whichoil prices rise at 10 per year French [9] discussed the mar-kets of world oil natural gas coal electricity and the energydemand with the conclusion that from mid-decade onwardenergy prices will most likely increase more rapidly than theoverall price level Roberts [10] concluded his analysis thatthrough the mid-1980s real crude prices should be flat todown and thereafter increase at a rate equal to or less thaninflation Campbell and Hubbard [11] showed that forecastsof pricesmdashincluding oil natural gas equipment and drillingcosts and moneymdashaffect the evaluation of all projects Clark[12] focused on generalized price formation in the US gasindustry David and Carson [13] focused on the banks whichuse the long term oil supply and demand situation as theunderlying support for making oil and gas price projectionsLorentsen andRoland [14] outlined a very openmodel whichcan easily impose different assumptions about oil supply anddemand in order to indicate a range of feasible projectionsfor oil prices Curlee [15] concluded that overall assessmentof forecasts and recent oil market trends suggests that priceswill remain constant in real terms for the remainder of the1980s to increase by 2-3 during the 1990s and beyondAnon [16] studied the US demand for petroleum products

Hindawi Publishing CorporationISRN Mechanical EngineeringVolume 2014 Article ID 529748 37 pageshttpdxdoiorg1011552014529748

2 ISRNMechanical Engineering

in 1987 based on the assumption that world oil prices willaverage 13US $bbl through the third quarter of 1986 andrise gradually to 18US $bbl by the end of 1987 Dielwart andColes [17] discussed the current market uncertainties andtheir impact on Canadian oil and natural gas prices Yohe[18] presented an analytical technique based on an adaptiveexpectations model of incorporating current informationinto long-term forecast Sekera [19] presented three priceforecasting analyses with forecasts under each theory andthe recommendation that all long term forecasts be based onfundamentaleconomic analysis while short term forecastingneeds to utilize other methods Gately [20] reviewed recentand current opinions about prospects for prices in the worldmarket for crude oil with two alternative views Holden et al[21] examined the accuracy and properties of forecasts bythe OECD for 24 countries and 8 variables Dougherty [22]presented some rules of thumb that may help to understandthe evolution of oil prices to identify factors to considerwhendeciding on the credibility of an oil price forecast ratherthan to prophesize future oil price Amano [23] developedan annual small-scale econometric model of the world oilmarket to analyze oil market conditions and oil prices for theperiod 1986ndash1991

During the 1990s among the others Garces [24] exploredthe effects of natural gas storage on short-term gas priceforecasts with preliminary results which indicate that byusing natural gas storage as a causal variable the finalgas price forecasts can decrease by as much as 5 overthe forecast period Angelier [25] concluded his analyticalframework with the forecast that in the year 2000 oil priceswill not be significantly different from those of 1990s Boppand Lady [26] concluded their analysis that when actualprices are employed future prices did correctly anticipatethe observed seasonal pattern Abramson and Finizza [27]developed a system that forecasts crude oil prices via MonteCarlo analyses of the network Charleson and Weber [28]forecasted energy consumption in Western Australia withBayesian vector autoregressions to 2010 Fesharaki [29]believed that for the next three to five years oil prices willremain at lower levels than generally predicted because priceincreases even when they occur will not be sustainablefor very long Huntington [30] reviewed forecasts of oilprices over the 1980s that were made in 1980 identifyingthe sources of errors due to such factors as exogenous GNPassumptions resource supply conditions outside the carteland demand adjustments to price changes Moosa and Al-Loughani [31] on the basis of monthly observations on spotand future prices of theWest Texas Intermediate (WTI) crudeoil suggested that future prices are neither unbiased norefficient forecasters of spot prices Abramson [32] discusseda knowledge-based system that models the crude oil marketas a belief network and uses scenario generation and Monte-Carlo analysis to forecast oil prices Skov [33] reviewed someof the more significant forecasts of energy supply demandand oil prices made in the last 25 years and identifies thebasic premises used and why they led to incorrect forecastsAbramson and Finizza [34] described a case study in theuse of inherently probabilistic belief network models toproduce probabilistic forecasts of average annual oil prices

Santini [35] investigated the statistical model of oil pricesexamining issues and trends related to both the US andworld oil supply Williams [36] in order to forecast thefuture examined historical trends and presents the newidea that there is an inverse relationship between crudeoil prices volatility and the strength of the relationshipbetween the owners of crude oil (host governments) andthe oil companies that develop refine market and sell theproducts Chaudhuri and Daniel [37] demonstrated that thenonstationary behavior of US dollar real exchange rates overthe post-Bretton Woods era is due to the nonstationarybehavior of real oil prices Pindyck [38] discussed how priceforecasts also influence investment decisions and the choiceof products to produce Silvapulle and Moosa [39] examinedthe relationship between the spot and futures prices of WTIcrude oil using a sample of daily data with the result thatboth spot and futures markets react simultaneously to newinformation

During the 2000s among the others Alba and Bourdaire[40] consider the actual situation as a ldquocohabitationrdquo betweenoil and the other energieswith the oil price extremely volatilereflecting the trial and error adjustment of the market shareleft to the other energies The Centre for Global EnergyStudies [41] covers several topics including oilfields set tobegin production in 2000 non-OPECproduction demand inwestern OECD countries and the Asia Pacific region and oilprice forecasts According to the US energy information [42]global crude oil prices in 2000 will rise to 24US $bbl up by2US $bbl from its earlier estimate while spot prices forWestTexas Intermediate crude will average more than 22US $bblthrough 2001 Hare [43] reported that according to forecastsby the Energy Information Administration increases innatural gas prices will continue all the way to the summerand winter of 2000 and that high oil prices will still beseen in 2001 According to the managing director of Shellrsquosexploration and production [44] crude oil prices will averagein the mid-teens of US $bbl over the next 10 years TheEnergy Department of Energy Information Administration[45] forecasts themonthly average price of crude oil importedto the USA to stay above 28US $bbl for the rest of 2000Walde and Dos Santos [46] identified factors that have to beconsidered in speculating about the future evolution of oilprices Birky et al [47] examined the potentiality of uncon-ventional natural gas resources with all other fossil fuelscombined with the conclusion of a widespread productionand use of these unconventional reserves The trend of worldcrude oil price in 1999 has been reviewed and analyzed byYang [48] and the prospects on world crude oil price in 2000were predicted MacAvoy and Moshkin [49] used a modelframework consisting of a simultaneous equations system forproduction from reserves and for demands for productionin the residential commercial and industrial sectors with asignificant negative trend in the long-term price of naturalgas Morana [50] showed how the GARCH properties ofoil price changes can be employed to forecast the oil pricedistribution over short-term horizons with a semiparametricmethodology based on the bootstrap approach Tang andHammoudeh [51] investigated the behavior of the nonlinearmodel based on the target zone theory which has very

ISRNMechanical Engineering 3

good forecasting ability when the oil price approaches theupper or the lower limit of the band Alvarez-Ramirezet al [52] studied daily records of international crude oilprices using multifractal analysis methods where rescaledrange Hurst analysis provides evidence that the crude oilmarket is a persistent process with long-run memory effectsdemonstrating that the crude oil market is consistent withthe random-walk assumption only at time scales on theorder of days to weeks Ye et al [53] presented a short-term monthly forecasting model of West Texas Intermediatecrude oil spot price using OECD petroleum inventory levelswhich are a measure of the balance or imbalance betweenpetroleum production and demand and thus provide a goodmarket barometer of crude oil price change Sadorsky [54]used an ARMAX-ARCH model to estimate the conditionalexpected returns of petroleum futures prices under time-varying risk with results from a small forecasting experimentwhich indicates that the out-of-sample forecasts from anARMAX-ARCHmodel generally outperform a randomwalkfor all forecast horizons Fong and See [55] examined thetemporal behaviour of volatility of daily returns on crudeoil futures using a generalised regime switching model thatallows for abrupt changes in mean and variance GARCHdynamics basis-driven time-varying transition probabilitiesand conditional leptokurtosis Taal et al [56] gave a summaryof the most common methods used for cost estimation ofheat exchange equipment in the process industry and thesources of energy price projections showing the relevance ofthe choice of the right method and themost reliable source ofenergy price forecast usedwhen choosing between alternativeretrofit projects or when trying to determine the viabilityof a retrofit project Cortazar and Schwartz [57] developeda parsimonious three-factor model of the term structure ofoil future prices that can be easily estimated from availablefuture price data Cabedo and Moya [58] proposed to usevalue at risk (VaR) for oil price risk quantification providingan estimation for the maximum oil price change associatedwith a likelihood level which fits the continuous oil pricemovements well and provides an efficient risk quantificationBurg et al [59] developed an econometric modeling of theworld oil market suggesting that world oil prices are likely tofall in the latter part of 2004 and in 2005 as global demandpressures ease Burg et al [60] made a market forecastfor various energy sources including oil and gas and coalwith the conclusion that prices are forecasted to decline in2005 to average 38US $bbl WTI Mirmirani and Li [61]applied VAR and ANN techniques to make ex-post forecastof US oil price movements Abosedra and Bagesthani [62]evaluated the predictive accuracy of 1- 3- 6- 9- and 12-month-ahead crude oil future prices for the period fromJanuary 1991 until December 2001 Gori and Takanen [63]forecasted the energy demand in a specific country wherethe analysis of the electricity demand was focused for thefirst time on the energy consumption and the possiblesubstitution among the different energy resources by usinga modified form of the econometric model EDM (EnergyDemand Model) Feng et al [64] predicted the oil pricefluctuation byARFIMAmodel which takes the longmemoryfeature into consideration showing that ARFIMA model

is better than ARMA model Movassagh and Modjtahedi[65] tested the fair-game efficient-markets hypothesis forthe natural gas future prices over the period 1990 through2003 Ye et al [66] presented a short-term forecasting modelof monthly West Texas Intermediate crude oil spot pricesusing readily available OECD industrial petroleum inventorylevels Yousefi et al [67] illustrated an application of waveletsas a possible vehicle for investigating the issue of marketefficiency in futuremarkets for oil Sun and Lai [68] presenteda forecast model of time series oil price about NYMEXwith BP neural networks analyzing the oil price fluctuationtrend Sadorsky [69] used several different univariate andmultivariate statistical models to estimate forecasts of dailyvolatility in petroleum future price returns Ye et al [70]showed the effect that surplus crude oil production capacityhas on short-term crude oil prices Moshiri and Foroutan[71] modelled and forecasted daily crude oil future pricesfrom 1983 to 2003 listed in NYMEX applying ARIMAand GARCH models tested for chaos using embeddingdimension BDS(L) Lyapunov exponent and neural net-works tests and set up a nonlinear and flexible ANN modelto forecast the series The United States Energy InformationAdministration (EIA) [72] expects worldwide oil demand toincrease from80million bblday in 2003 to 98million bbldayin 2015 and to 118 million bblday in 2030 The latest forecastreference case calls for crude oil prices to climb from 31US$bbl in 2003 to 57US $bbl in 2030 Ghouri [73] analyzedqualitatively and quantitatively the relationship between USmonthly ending oil stocks position with that of West TexasIntermediate (WTI) oil prices from February 1995 to July2004 concluding that if other things are held constant WTIis inversely related to the petroleum products (PPP) com-bined petroleum products and crude oil (CPPP) crude oilalone (crude) total oil stocks including petroleum productscrude oil and strategic petroleum reserves SPR (total) totalgasoline (TGO) and total distillate (TDO) Ye et al [74]incorporated low- and high-inventory variables in a singleequation model to forecast short-run WTI crude oil pricesenhancing the model fit and forecast ability The approachof using mass and energy-capital conservation equations toinvestigate the price evolution with time throughout the useof economic parameters was proposed in [75 76] by thepresent author The Hotelling rule was generalized in [75]with the conclusion that the price of the extracted resourcesincreases exponentially with the product of the time and thedifference between the inflation rate and the extraction rateof the resources PIFE ldquoprice increase factor of extractedresourcesrdquo A further generalization was done in [76] withthe introduction of the price of the selling resources whichdepends onPIFE and the newparameter PIFS ldquoprice increasefactor of selling resourcesrdquo that is the difference betweenthe prime rate of interest and the extraction rate of theresources The political events and many complicated factorswhich happened in the last decades have made oil priceshighly nonlinear and even chaotic with irregularities andrandom oscillations linked to the day-to-day events givingreliability to forecast oil price and consumption in reallyshort terms only by using adaptive neural fuzzy inferencesystems (ANFIS) [77] P K Narayan and S Narayan [78]


examined the volatility of the crude oil price using dailydata for the period from 1991 to 2006 in various subsamplesin order to judge the robustness of the results implyingthat the behaviour of the oil prices tend to change overshort periods of time Knetsch [79] developed an oil priceforecasting technique based on the present value model ofrational commodity pricing suggesting the shifting of theforecasting problem to themarginal convenience yield whichcan be derived from the cost-of-carry relationship DeutscheBank [80] reported that light sweet crude oil prices willaverage 80US $bbl on the New YorkMercantile Exchange in2008 while natural gas prices will average 775US $MMbtuand perhaps stay put there It also reported that crude oil inthis decade is likely to average nearly 55US $bbl It seemsthat crude oil markets are following the reciprocal of thedeclining value of the US dollar and ignoring deterioratingoil fundamentals MacAskie and Jablonowski [81] developeda decision-analytic model to value commodity price forecastsin the presence of futures markets and applied the methodto a data set on crude oil prices Askari and Krichene [82]observed markets expecting oil prices to remain volatile andjumpy and with higher probabilities to rise rather than fallabove the expected mean Xu et al [83] investigated with ILSapproach the forecast of the relationship between commodityinventory levels and crude oil spot prices effectively withthe conclusion that their empirical study suggests that boththe ILS method and the confidence interval method canproduce comparable quality forecasts Yu et al [84] proposedempirical mode decomposition (EMD) based on neuralnetwork ensemble learning paradigm for world crude oilspot price forecasting decomposing the original crude oilspot price series into a finite and often small number ofintrinsic mode functions (IMFs) Fan et al [85] appliedpattern matching technique to multistep prediction of crudeoil prices and proposed a new approach generalized patternmatching based on genetic algorithm (GPMGA) whichcan be used to forecast future crude oil price based onhistorical observations This approach can detect the mostsimilar pattern in contemporary crude oil prices from thehistorical data Kang et al [86] investigated the efficacyof a volatility model for three crude oil marketsmdashBrentDubai and West Texas Intermediate (WTI)mdashwith regardto its ability to forecast and identify volatility stylized factsin particular volatility persistence or long memory with theconclusion that CGARCH and FIGARCH models are usefulfor modeling and forecasting persistence in the volatilityof crude oil prices Hamilton [87] examined the factorsresponsible for changes in crude oil prices reviewing thestatistical behavior of oil prices in relation to the predictionsof theory and looking in detail at key features of petroleumdemand and supply discussing the role of commodity spec-ulation OPEC and resource depletion Cuaresma et al [88]using a simple unobserved components model showed thatexplicitly modelling asymmetric cycles on crude oil pricesimproves the forecast ability of univariate time series modelsof the oil price Ye et al [89] predicted crude oil prices from1992 through early 2004 by using OECDrsquos relative inventoriesand OPECrsquos excess production capacity improving forecastsfor the post-Gulf War I time period over models without

the ratchet mechanism Cheong [90] investigated the time-varying volatility of two major crude oil markets the WestTexas Intermediate (WTI) and the Europe Brent with a flex-ible autoregressive conditional heteroskedasticity (ARCH)model to take into account the stylized volatility facts suchas clustering volatility asymmetric news impact and longmemory volatility Ghaffari andZare [91] developed amethodbased on soft computing approaches to predict the dailyvariation of the crude oil price of theWest Texas Intermediate(WTI) with comparison to the actual daily variation of theoil price and the difference is implemented to activate thelearning algorithms The energy supply curve that is priceversus consumption of nonrenewable energy resources wasconstructed in [92]where newparameterswere introduced toforecast the price evolution of nonrenewable resources Thepresent theory has been reviewed and applied to the periodfrom 1966 until 2006 in [93]

During the 2010s among the others de Souza e Silva et al[94] investigated the usefulness of a nonlinear time seriesmodel known as hidden Markov model (HMM) to predictfuture crude oil price movements developing a forecastingmethodology that consists of employing wavelet analysis toremove high frequency price movements assumed as noiseand using the probability distribution of the price returnaccumulated over the next days to infer future price trendsHe et al [95] presented two forecasting models one based ona vector error correctionmechanism and the other based on atransfer function framework with the range taken as a drivervariable for forecasting the daily highs and lows showingthat both of these models offer significant advantages overthe naıve random walk and univariate ARIMA models interms of out-of-sample forecast accuracy Miller and Ni [96]examined how future real GDP growth relates to changesin the forecasted long-term average of discounted real oilprices and to changes in unanticipated fluctuations of real oilprices around the forecasts Forecasts were conducted using astate-space oil market model in which global real economicactivity and real oil prices share a common stochastic trendYaziz et al [97] obtained the daily West Texas Intermedi-ate (WTI) crude oil prices data from Energy InformationAdministration (EIA) from the 2nd of January 1986 to the30th of September 2009 by using the Box-Jenkinsmethodol-ogy and generalized autoregressive conditional heteroscedas-ticity (GARCH) approach in analyzing the crude oil priceswhich is able to capture the volatility by the nonconstantof conditional variance Prat and Uctum [98] suggested amixed expectation model defined as a linear combinationof these traditional processes interpreted as the aggregationof individual mixing behavior and of heterogeneous groupsof agents using these simple processes consistent with theeconomically rational expectations theory It was shown thatthe target oil price included in the regressive componentof this model depends on the long-run marginal cost ofcrude oil production and on the short term macroeconomicfundamentals whose effects are subject to structural changesJammazi and Aloui [99] combined the dynamic propertiesof the multilayer back propagation neural network and therecent Harr A trous wavelet decomposition implementing ahybridmodel HTW-MPNN to achieve prominent prediction


of crude oil price by using three variants of activationfunction namely sigmoid bipolar sigmoid and hyperbolictangent in order to test the modelrsquos flexibility Mingmingand Jinliang [100] constructed a multiple wavelet recurrentneural network (MWRNN) simulationmodel inwhich trendand random components of crude oil and gold prices wereconsidered to capture multiscale data characteristics whilea real neural network (RNN) was utilized to forecast crudeoil prices at different scales showing that the model has highprediction accuracy Baumeister and Kilian [101] constructeda monthly real-time dataset consisting of vintages for theperiod from January 1991 to December 2010 that is suitablefor generating forecasts of the real price of oil from a variety ofmodels and documenting that revisions of the data typicallyrepresent news and introducing backcasting and nowcastingtechniques to fill gaps in the real-time data It was shownthat the real-time forecasts of the real oil price can be moreaccurate than the nochange forecast at horizons up to 1 yearAzadeh [102] presented a flexible algorithmbased on artificialneural network (ANN) and fuzzy regression (FR) to copewith optimum long-term oil price forecasting in noisy uncer-tain and complex environments incorporating the oil supplycrude oil distillation capacity oil consumption of non-OECDUS refinery capacity and surplus capacity as economicindicators Wang et al [103] investigated the interactingimpact between the crude oil prices and the stock marketindices in China and analyzed the corresponding statisticalbehaviors introducing a jump stochastic time effective neuralnetwork model and applying it to forecast the fluctuationsof the time series for the crude oil prices and the stockindices and studying the corresponding statistical propertiesby comparison Hu et al [104] attempted to accuratelyforecast prices of the crude oil futures by adopting threepopular neural networks methods including the multilayerperceptron the Elman recurrent neural network (ERNN)and the recurrent fuzzy neural network (RFNN) with theconclusion that learning performance can be improved byincreasing the training time and that the RFNN has the bestpredictive power and the MLP has the worst one amongthe three underlying neural networks Ruelke et al [105]derived internal consistency restrictions on short mediumand long-term oil price forecasts by analysing whether oilprice forecasts extracted from the Survey of ProfessionalForecasters (SPF) conducted by the European Central Bank(ECB) satisfy these internal consistency restrictions findingthat neither short-term forecast is consistent with medium-term forecasts nor that medium-term forecasts are consistentwith long-term forecasts A new category of cases that isthe negative inflation rate has been introduced within thepresent theory in [106] where some preliminary results havebeen presented Pierdzioch et al [107] found that the lossfunction of a sample of oil price forecasters is asymmetric inthe forecast error indicating that the loss oil price forecastersincurred when their forecasts exceeded the price of oiltended to be larger than the loss they incurred when theirforecast fell short of the price of oil Accounting for theasymmetry of the loss function does not necessarily makeforecasts look rational Shin et al [108] proposed a study toexploit the method of representing the network between the

time-series entities and to then employ SSL to forecast theupward and downward movement of oil prices by using one-month-ahead monthly crude oil price predictions betweenJanuary 1992 and June 2008 Alquist [109] addressed someof the key questions that arise in forecasting the price ofcrude oil Xiong et al [110] proposed a revised hybridmodel built upon empirical mode decomposition (EMD)based on the feed-forward neural network (FNN) modelingframework incorporating the slope-based method (SBM)which is capable of capturing the complex dynamic of crudeoil prices The results obtained in this study indicate that theproposed EMD-SBM-FNNmodel using theMIMO strategyis the best in terms of prediction accuracy with accreditedcomputational load Chang and Lai [111] attempted to developan integrated model to forecast the cycle of energy priceswith respect to the level of economic activity showing thatthe oil price cycle and economic activities have bidirectionalcausality in the short run and that the upwards (downwards)cycle of oil prices is accompanied by expansion (contraction)of economic activities and vice versa with a comovementtrend in the long run Conceptually the model developedin this work is useful with regard to forecasting the levelof economic activities using the oil price cycle as mosteconomic activities depend on energy Salehnia et al [112]used the gamma test for the first time as a mathematicallynonparametric nonlinear smooth modeling tool to choosethe best input combination before calibrating and testingmodels and developing several nonlinear models with the aidof the gamma test including regression models local linearregression (LLR) dynamic local linear regression (DLLR)and artificial neural networks (ANN)models Shin et al [113]exploited the method of representing the network betweenthe time-series entities to employ semisupervised learning(SSL) to forecast the upward and downward movement of oilprices by using one-month-ahead monthly crude oil pricepredictions between January 1992 and June 2008 Yang et al[114] forecasted the international crude oil price by usingthe grey system theory and creating a MATLAB program toachieve it The present theory has been confirmed in caseof negative inflation rate [115] projecting the forecast tothe following five months of 2013 [116] with a very recentverification [117]

2 Mass Conservation Equation ofExtracted Resources

Assume the mass 119872 of nonrenewable energy resources isin the reservoir 119862 reported in Figure 1 The mass flowrate of extraction from the reservoir is 119866 with dimensions(massannum)

The mass conservation equation of the extractedresources is

119889119872

119889119905= minus119866 (1)

which after integration becomes

119872 = 1198720minus int

119905

0

119866119889119905 (2)


Nonrenewable energy resources

GC

Gk

Figure 1 Nonrenewable energy resources in the reservoir 119862

G

Go

120572 gt 0

120572 = 0

120572 lt 0

t

(t0 + infin)

(t0 + 0)

(constant)

Figure 2 Time evolution of the mass flow rate of extraction 119866

where1198720is themass of resources in the reservoir at the initial

time 119905 = 0Assuming the extraction rate 120572 at the time 119905 given by

1

119866

119889119866

119889119905= 120572 (3)

the mass flow rate of extraction with constant 120572 in the timeinterval 0 minus 119905 is

119866 = 1198660exp (120572119905) (4)

The evolution of the mass flow rate of extraction givenby (4) and presented in Figure 2 depends on the value of 120572119866 is constant with time if 120572 = 0 119866 increases if 120572 gt 0 and 119866decreases if 120572 lt 0

The mass of nonrenewable resources in the reservoir 119862becomes dividing by 119866

0

119872

1198660

=1198720

1198660

+[1 minus exp (120572119905)]

120572 (5)

and its time evolution is presented in Figure 3In case of a constant extraction rate119866 = 119866

0 that is120572 = 0

(1) gives

119872 = 1198720minus 1198660sdot 119905 (6)

that is the mass of resources in the reservoir 119862 decreaseslinearly with time Equation (6) gives the time 119905

119891= 11987201198660

when 119872 = 0 that is the time of exhaustion of thenonrenewable resources

t

0

120572 gt 0 120572 = 0

(t0 + 0)

120572c lt 120572 lt 0

120572 lt 120572c lt 0

120572 = 120572c lt 0

MG0

M0G0

M0G0

M(infin)G0 =M0G0 minus 1|120572|

Figure 3 Time evolution of1198721198660(years) at the extraction rate 120572

If for example the ratio11987201198660is equal to 50 years the

time to exhaust the resources is 119905119891= 50 years If 120572 gt 0 the

mass of resources in the reservoir 119862 is exhausted faster thanfor 120572 = 0 that is before 50 years The time 119905

119891 which gives

119872 = 0 is given by (5) as

119905119891=1

120572ln [1 + 120572

1198720

1198660

] (7)

Assuming 11987201198660= 50 years the number of years for the

exhaustion of the resources depends on the value of 120572 From(7) it can be found that

(i) 119905119891= 4055 years for 120572 = 001 (1annum)

(ii) 119905119891= 1792 years for 120572 = 01 (1annum)

If 120572 lt 0 (5) can be written as

119872

1198660

=1198720

1198660

minus[1 minus exp (|120572| 119905)]

|120572| (8)

which for an infinite time 119905 rarr infin gives

119872(infin)

1198660

=1198720

1198660

minus1

|120572| (9)

The extraction policy to exhaust the nonrenewable resourcesin an infinite time 119905 rarr infin is to have

119872(infin)

1198660

=1198720

1198660

minus1

|120572|= 0 (10)

which gives the critical extraction rate 120572119888

10038161003816100381610038161205721198881003816100381610038161003816 =

1198660

1198720

(11)

For11987201198660= 50 years the critical extraction rate is120572

119888= minus002

(1annum)If 120572119888lt 120572 lt 0 the time to exhaust the resources is given

by

119905119891= minus

1

|120572|ln [1 minus |120572|

1198720

1198660

] (12)


which depends on 120572 For11987201198660= 50 years the time 119905

119891is

(i) 119905119891= 6931 years for 120572 = minus001 (1annum)

(ii) 119905119891= 5129 years for 120572 = minus0001 (1annum)

The owner of the nonrenewable resources does not findconvenient to have

119872(infin)

1198660

=1198720

1198660

minus1

|120572|gt 0 (13)

at an infinite time 119905 rarr infin because the resources couldremain nonextracted

If 120572 lt 120572119888

lt 0 the nonrenewable resources willnever be extracted and a certain amount of resources willremain in the reservoir as nonextracted For example if 120572 =minus004 (1annum) the amount of resources which will remainnonextracted is119872(infin)119866

0= 25 years

3 Hotelling Rule

Assume the nonrenewable resource can be extracted withoutcost Denote 119901

119900the initial price per unit of resource at the

initial time 119905 = 0 and suppose the interest rate 119903 is constantin the time interval from 119905 = 0 to 119905The owner of the resourcesunit has two options to keep the unit of resources in thereservoir for the time interval 119905 with the real expectation thatthe price at 119905 will be higher saying 119901(119905) or to sell the unitof resources and put the money obtained from the sale ina bank for the same time interval 119905 to get an interest returnequal to 119901(119905)119905119903 Under competitive conditions the owner isindifferent between these two options then the price of theunit at 119905 119901(119905) must be equal to the price 119901

0at the time 119905

0

increased of the interest return 119901(119905)119905119903 In other words

119901 (119905) = 1199010+ 119901 (119905) 119905119903 (14)

or taking the limit as 119905 rarr 0 it can be obtained

119889119901

119889119905= 119901 sdot 119903 (15)

After integration (15) becomes

119901 (119905) = 1199010exp (119903119905) (16)

also called Hotelling rule [2] in honour of Hotelling [1]where 119903 the interest rate of capital is assumed constant inthe time period 0 minus 119905

Equation (16) states that the price of nonrenewableresources can only increase exponentially with time if theinterest rate 119903 is positive while only decreasing in case of neg-ative inflation rate The price evolution is independent of anyother variable present in the energy game of nonrenewableresources

4 Energy-Capital Conservation Equation ofExtracted Resources

The energy-capital conservation equation can be written inunsteady state similarly to the energy conservation equation

of thermal engineering for the nonextracted resources of thereservoir 119862 Figure 1 with interest rate 119903

119873(1annum) as

119889119870

119889119905= 119866119896 (17)

The left term expresses the time variation of the capital 119870due to the resources of the reservoir 119862 while 119866

119896is the capital

flow rate entering into the reservoir119862 which derives from themass flow rate of extraction 119866

The capital119870 is written as

119870 = 119866 sdot 119901 (18)

where 119901 is the current price of a unity of extracted resourcesand 119866 is the mass flow rate of extraction

The right termof (17) that is the capital flow rate enteringinto the reservoir is written as

119866119896= 119870 sdot 119903

119873= 119866 sdot 119901 sdot 119903

119873 (19)

where 119903119873is the interest rate of the nonextracted resources

Equation (17) then becomes

119889 (119866 sdot 119901)

119889119905= 119866 sdot 119901 sdot 119903

119873 (20)

which for a mass flow rate of extraction being equal to 1that is 119866 = 1 it gives (15) that is the Hotelling rule beforeintegration In conclusion the Hotelling rule can be obtainedfrom the energy-capital conservation equation with a massflow rate of extraction being equal to 119866 = 1

Differentiation of the capital conservation equation (20)and division by the capital119870 gives

1

119901

119889119901

119889119905+1

119866

119889119866

119889119905= 119903119873 (21)

Equation (21) can be used with (3) to study the priceevolution of the extracted resources with interest rate 119903

119873

Equation (21) becomes indicating 119901 as 119901119864for the extracted

resources and using (3)

1

119901119864

119889119901119864

119889119905= 119903119873minus 120572 (22)

which after integration gives

119901119864= 1199011198640sdot exp (120573 sdot 119905) (23)

where

120573 = 119903119873minus 120572 (24)

that is the difference between the interest rate of nonex-tracted resources 119903

119873 and the extraction rate 120572 is a

new parameter called ldquoprice increase factor of extractedresourcesrdquo PIFE The price evolution with time of theextracted resources given by (23) is shown in Figure 4

The price of extracted resources evolves with time accord-ing to the value of PIFE 120573 If the extraction rate 120572 is equalto the interest rate of nonextracted resources 119903

119873 the price


t

(t0 + infin)

pE

pE0

120573 gt 0 (120572 lt r)

120573 = 0 (120572 = r)

120573 lt 0 (120572 gt r)

(t0 + 0)

Figure 4 Time evolution of the price of extracted resources

increase factor PIFE is equal to zero 120573 = 0 and the price 119901119864

remains constant with time If the extraction rate 120572 is higherthan 119903

119873 the price increase factor PIFE is negative 120573 lt 0

and the price 119901119864decreases with time If the extraction rate 120572

is lower than 119903119873 the price increase factor PIFE is positive

120573 gt 0 and the price 119901119864increases with time In case of a

conservative policy of extraction that is 120572 lt 0 the priceincrease factor is greater than zero 120573 gt 0 and the price 119901

119864

increases with time

5 Mass and Energy-Capital ConservationEquations of Selling Resources

Assume that nonextracted and extracted energy resources arein the two reservoirs 119862

119873and 119862

119864 of Figure 5

The nonextracted resources are in the reservoir 119862119873with

mass119872119873and interest rate 119903

119873Themass flow rate of extracted

resources from the reservoir 119862119873is 119866119864while the capital flow

rate relative to the mass flow rate 119866119864 is 119866119870119864

The resourcesof the reservoir 119862

119873are extracted at the price of the extracted

resources 119901119864 and allocated in the reservoir 119862

119864 where119872

119864is

themass of extracted resourcesThe resources of the reservoir119862119864are sold at the price 119901

119878 The mass flow rate of the selling



The mass conservation equation of the extractedresources in the reservoir 119862

119864is

119889119872119864

119889119905= 119866119864minus 119866119878 (25)

while the relative energy-capital conservation equation is

119889

119889119905(119870119864) = 119866119870119878minus 119866119870119864 (26)

Nonextractedenergy resources

Extracted energy resources

CN CE

MN rN ME rE

GE (extracted)pE

GS (selling)pS

GKE(pE) GKS(pS)

Figure 5 Nonextracted and extracted energy resources

which becomes

119889

119889119905(119866119878sdot 119901119878) = 119866119878sdot 119901119878sdot 119903119864minus 119866119864sdot 119901119864sdot 119903119873 (27)

and finally

1

119901119878

sdot119889119901119878

119889119905+1

119866119878

sdot119889119866119878

119889119905= 119903119864minus119866119864sdot 119901119864

119866119878sdot 119901119878

sdot 119903119873 (28)

In case of no-accumulation and no-depletion of theextracted resources 119866

119864= 119866119878 the right term of (25) is zero

that is the mass 119872119864of extracted resources in the reservoir

119862119864is constant with time The mass flow rates of extracted

and selling resources have the same extraction rate 120572 fromthe reservoirs 119862

119873and 119862

119864 that is

1

119866119864

119889119866119864

119889119905=1

119866119878

119889119866119878

119889119905= 120572 (29)

which gives after integration

119866119864= 119866119878= 1198661198640sdot exp sdot (120572 sdot 119905) = 119866

1198780sdot exp sdot (120572 sdot 119905) (30)

and the time evolution is similar to that of Figure 2Equation (28) becomes

1

119901119878

119889119901119878

119889119905+ 120572 = 119903

119864minus119901119864

119901119878

119903119873 (31)

or

119889119901119878

119889119905= 119901119878(119903119864minus 120572) minus 119901

119864sdot 119903119873 (32)

The substitution of (23) into (32) and the solution of therelative differential equation give the time evolution of theprice of sold resources

119901119878= 119901lowast

1198780exp (120573 sdot 119905) + [119901

1198780minus 119901lowast

1198780] exp (1205731015840 sdot 119905) (33)

where

1205731015840= 119903119864minus 120572 (34)

is called the ldquoPrice Increase Factor of Selling resourcesrdquo PIFSand

119901lowast

1198780=119903119873sdot 1199011198640

119903119864minus 119903119873

=119903119873sdot 1199011198640

1205731015840 minus 120573 (35)

the ldquocritical initial price of selling resourcesrdquo CIPS


The price of selling resources (33) has an extreme(maximum or minimum) for the time 119905

119898given by

119905119898=

1

(120573 minus 1205731015840)ln[

1205731015840sdot (119901lowast

1198780minus 1199011198780)

120573 sdot 119901lowast

1198780

] (36)

The extreme is a maximum if

1205731015840(1205731015840minus 120573) (119901


1198780) lt 0 (37)

otherwise it is a minimumThe time 119905

119898of the extreme is zero if 119901

1198780is equal to

119901lowastlowast

1198780=

119901lowast

1198780(1205731015840minus 120573)

1205731015840=1199031198731199011198640

1205731015840 (38)

which is called ldquocritical initial price extreme of sellingresourcesrdquo CIPES

The time 119905119898of the maximum is greater than zero if

1199011198780gt 119901lowastlowast

1198780for 1205731015840 gt 120573 (ie 119903

119864gt 119903119873) (39)

or

1199011198780lt 119901lowastlowast

1198780for 1205731015840 lt 120573 (ie 119903

119864lt 119903119873) (40)

For 1199011198780= 119901lowast

1198780the time of the maximum is

119905119898= +infin (41)

for 1205731015840 gt 120573 or

119905119898= minusinfin (42)

for 1205731015840 lt 120573For 119903119873= 119903119864 that is 1205731015840 = 120573 119901

119878is given by

119901119878= (1199011198780minus 119901lowastlowast

1198780120573119905) exp (120573119905) (43)

where CIPS that is 119901lowast1198780 is not defined

The price of selling resources has an extreme for the time119905119898given by

119905119898=(1199011198780minus 119901lowastlowast

1198780)

119901lowastlowast

11987801205731015840

(44)

The extreme is a maximum if 119901lowastlowast1198780

gt 0 and a minimumif 119901lowastlowast1198780lt 0 The time 119905

119898of the maximum is zero if the initial

price 1199011198780is

1199011198780= 119901lowastlowast

1198780=119903119873sdot 1199011198640

1205731015840 (45)

The possible cases can be classified into six categories

(i) Category 0mdash119903119873= 0

(ii) Category 1 119903119873gt 120572 that is 120573 = PIFE gt 0

(iii) Category 2 120572 gt 119903119873 that is 120573 = PIFE lt 0

(iv) Category 3 120572 = 119903119873 that is 120573 = PIFE = 0

(v) Category 4 119903119873= 119903119864 that is 1205731015840 = 120573 or PIFE = PIFS

(vi) Category 5 119903119873lt 0

t

0

pS120572 lt 0 120573 gt 0

(t0 + infin)(t0 + infin)

(t0 + infin)

120572 = 0 120573 = 0

0 lt 120572 lt rE 120573 lt 0

(constant)

(t0 + 0)

Case 0-A 120572 lt rE 120573998400 gt 0

Case 0-B 120572 = rE 120573998400 = 0 120573 lt 0

Case 0-C 120572 gt rE 120573998400 lt 0 120573 lt 0

Category 0 rN = 0

pS0

Figure 6 Price evolution of selling resources for Category 0

On their side each category presents the following possiblecases

Category 0 (119903119873

= 0) If the interest rate of nonextractedresources is 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast

1198780= 0) the price evolution

of selling resources is

119901119878= 1199011198780exp (1205731015840119905) (46)

where 1205731015840 is the PIF of sold resourcesFigure 6 presents the price evolution of selling resources

according to the relation between extraction rate 120572 andinterest rate of extracted resources 119903

119864

Case 0-A (120572 lt 119903119864 (ie 1205731015840 gt 0)) The price 119901

119878increases with

time at different rates according to the values of 120572 The rateof increase is higher for 120572 lt 0 120573 gt 0 intermediate for 120572 = 0120573 = 0 and lower for 120572 lt 119903

119864 120573 lt 0

Case 0-B (120572 = 119903119864 (ie 1205731015840 = 0 120573 lt 0)) The price 119901

119878is

constant with time at 1199011198780

Case 0-C (120572 gt 119903119864 (ie 1205731015840 lt 0 120573 lt 0)) The price 119901

119878is

decreasing with time towards +0In conclusion for 119903

119873= 0 the criterium to evaluate the

price evolution of selling resources 119901119878 is 1205731015840 If 1205731015840 gt 0 the

price 119901119878increases if 1205731015840 = 0 the price 119901

119878remains constant

and if 1205731015840 lt 0 the price 119901119878decreases towards +0

Category 1 (119903119873

gt 120572 that is 120573 = 119875119868119865119864 gt 0) The priceevolutions of 119901

119878are reported in Figure 7 according to the

relation between 119903119873and 119903119864

Case 1-A (119903119864gt 119903119873gt 120572 that is 1205731015840 gt 120573 gt 0) The price of

selling resource increases with time if the initial price 1199011198780

is1199011198780ge 119901lowast


1198780gt 0 that is equal to or greater than CIPS gt

CIPES gt 0

Case 1-B (119903119864gt 119903119873gt 120572 and 119901lowast

1198780gt 1199011198780gt 119901lowastlowast

1198780gt 0 or 119903

119873gt 119903119864gt

120572 and 1199011198780gt 119901lowastlowast

1198780gt 0 gt 119901

lowast

1198780) The price of selling resource

increases temporarily with time up to 119905119898 given by (19)


Category 1 120572 lt rN 120573 gt 0

(t0 + infin)

(t0 + infin)

Case 1-A rE gt rN gt 120572 120573998400 gt 120573 gt 0

Case 1-ApS0 gt (plowastS0)a

Case 1-ApS0 = (plowastS0)a

Case 1-B rE gt rN gt 120572 and plowastS0 gt pS0 gt plowastlowast

S0 gt 0

or rN gt rE gt 120572 and pS0 gt plowastlowastS0 gt 0 gt plowast

S0

pS

pS0

(plowastS0)apS0

(plowastlowastS0)abpS0

0

tmax

Case 1-B rE gt rN gt 120572 andplowastS0 gt pS0 gt plowastlowast

S0 gt 0

(t0 minus infin) (t0 minus infin)

Case 1-C pS0 lt (plowastlowastS0)a lt (plowast

S0)a

Case 1-C pS0 lt (plowastlowastS0)b

Case 1-C (pS0 = plowastlowastS0 )ab

Case 1-C rE gt rN gt 120572 and plowastS0 gt plowastlowast

S0 gt or rN gt rE gt 120572 and plowastlowast

S0 gt pS0 gt 0 gt plowastS0

or rN gt 120572 gt rE

Case 1-B rN gt rE gt 120572 and pS0 gt plowastlowastS0 gt 0 gt plowast

S0

pS0

t


and then decreases for 119903119864gt 119903119873

gt 120572 if the initial price1199011198780has a value comprised between CIPS and CIPES that is

119901lowast


1198780gt 0 or for 119903

119873gt 119903119864gt 120572 if the initial price

1199011198780is greater than CIPES that is 119901


1198780gt 0 gt 119901

lowast

1198780

Case 1-C (119903119864gt 119903119873gt 120572 and 119901lowast


1198780gt 1199011198780 or 119903119873gt 119903119864gt

120572 and 119901lowastlowast1198780

gt 1199011198780gt 0 gt 119901

lowast

1198780 or 119903119873gt 120572 gt 119903

119864) The price of

selling resource decreases with time for 119903119864gt 119903119873gt 120572 if the

initial price 1199011198780is smaller than CIPS and CIPES that is 119901lowast

1198780gt

119901lowastlowast

1198780gt 1199011198780 or for 119903

119873gt 119903119864gt 120572 if the initial price119901

1198780is smaller

than CIPES that is 119901lowastlowast1198780gt 1199011198780gt 0 gt 119901

lowast

1198780 or for 119903

119873gt 120572 gt 119903

119864

for every initial price 1199011198780because 119901

1198780gt 0 gt 119901

lowast


1198780

Category 2 (120572 gt 119903119873 that is 120573 = PIFE lt 0) The time

evolution is presented in Figure 8 according to 120573 and 1205731015840 orthe relation between 119903

119873and 119903119864

Case 2-A (119903119864gt 120572 gt 119903

119873 that is 1205731015840 gt 120573) The price of selling

resource increases with time if the initial price 1199011198780

is 1199011198780ge

119901lowastlowast

1198780gt 119901lowast

1198780 that is equal to or greater than CIPES gt CIPS

119901lowastlowast

1198780gt 119901lowast

1198780

Case 2-B (119903119864gt 120572 gt 119903


resource decreases temporarily with time up to 119905119898 given by

(19) and then increases if the initial price 1199011198780 has a value

comprised between CIPS and CIPES that is 119901lowast1198780gt 1199011198780gt

119901lowastlowast

1198780gt 0

Case 2-C (120572 gt 119903119873 that is 120573 lt 0) In all other cases the price

of selling resource decreases with time including 119903119864= 120572 gt 119903

119873

or 120572 gt 119903119873gt 119903119864 or 120572 gt 119903

119873gt 119903119864

Category 3 (120572 = 119903119873 that is 120573 = 119875119868119865119864 = 0) The price

evolution of selling resources119901119878 is dependent on the relation

between 119903119873and 119903119864 as reported in Figure 9

Case 3-A (119903119864gt 119903119873= 120572 that is 1205731015840 gt 120573 = 0) The price

of selling resource increases with time if the initial price 1199011198780

is 1199011198780

gt 119901lowastlowast

1198780= 119901lowast

1198780 that is greater than CIPS = CIPES

119901lowastlowast

1198780= 119901lowast

1198780

Case 3-B (119903119864gt 119903119873= 120572 that is 1205731015840 gt 120573 = 0) The price of

selling resource remains constant with time if the initial price1199011198780is 1199011198780= 119901lowastlowast

1198780= 119901lowast

1198780 that is equal to CIPS = CIPES 119901lowastlowast

1198780=

119901lowast

1198780

Case 3-C (119903119864gt 119903119873= 120572 that is 1205731015840 gt 120573 = 0 or 119903

119873= 120572 gt

119903119864 that is 1205731015840 lt 120573 = 0) The price of selling resource for 119903

119864gt

119903119873= 120572 decreases with time if the initial price 119901

1198780is 1199011198780lt

119901lowastlowast

1198780= 119901lowast

1198780 that is smaller than CIPS and CIPES 119901lowastlowast

1198780= 119901lowast

1198780

In the other cases the price of the selling resource decreaseswith time for every value of 119901

1198780

Category 4 (119903119873= 119903119864 that is 1205731015840 = 120573) The time evolutions are

presented in Figure 10

Case 4-A (119903119873= 119903119864gt 120572 that is 1205731015840 = 120573 gt 0) The price cannot

increase with time

Case 4-B (119903119873= 119903119864gt 120572 that is 1205731015840 = 120573 gt 0) The price

of selling resource increases temporarily with time up to 119905119898

given by (2) and then decreases if the initial price 1199011198780

is1199011198780gt 119901lowastlowast

1198780gt 0 that is greater than CIPES gt 0

Case 4-C (119903119873= 119903119864gt 120572 that is 1205731015840 = 120573 gt 0 or 119903

119873= 119903119864lt

120572 that is 1205731015840 = 120573 lt 0 or 119903119873= 119903119864= 120572 that is 1205731015840 = 120573 = 0)

The price of selling resource decreases with time if the initialprice 119901

1198780is 119901lowastlowast1198780

ge 1199011198780gt 0 that is equal to or smaller than

CIPES gt 0 In all the other cases the price of selling resourcedecreases with time for every value of 119901

1198780

Category 5 (119903119873

lt 0) A new category of cases has beenintroduced for negative inflation rate 119903

119873lt 0 which was

registered from March to October 2009 after the economiccrisis of 2008 This new category is interesting becausethe Hotelling rule forecasts a decrease of the oil price if


t0

pS

Category 2 120572 gt rN ie 120573 = PIFE lt 0

(t0 + infin)

pS0

(plowastlowastS0)apS0

(plowastS0)ab

pS0

Case 2-ApS0 gt (plowastlowastS0 )a

(plowastlowastS0 )a

(plowastlowastS0 )a

Case 2-ApS0 =

Case 2-B (plowastlowastS0 )a gt ps0 gt

Case 2-A rE gt 120572 gt rN ie 120573998400 gt 120573

Case 2-B rE gt 120572 gt rN ie 120573998400 gt 120573

Case 2-C pS0 gt (plowastS0)b

t0(pS0 minus (plowastS0)b)

Case 2-C pS0 = (plowastS0)a

Case 2-C pS0 = (plowastS0)b

Case 2-C (plowastS0)c

lt0Case 2-C (plowast

S0)d gt 0(t0 + 0)

(t0 minus 0)

Case 2-C pS0 lt (plowastS0)b

(t0 minus infin)

Case 2-C pS0 lt (plowastS0)a

Case 2-C 120572 gt rN ie 120573 lt 0

pS0 minus (plowastS0)b

pSo minus (plowastS0)b


the interest rate is assumed to be equal to the inflationrate that is negative while the conclusions of the presentapproach are different

The time evolutions are presented in Figure 11

Case 5-A (119903119864gt 120572 gt 0 gt 119903

119873 that is 1205731015840 gt 0 gt 120573) The price

of selling resource increases with time for every value of theinitial price 119901

1198780because 119901

1198780gt 0 gt 119901

lowast


1198780 since CIPS and

CIPES are negative or 0 gt 119901lowast1198780gt 119901lowastlowast

1198780

Case 5-B (120572 gt 119903119864gt 0 gt 119903

119873 that is 0 gt 1205731015840 gt 120573) The price of

selling resource increases temporarily with time up to 119905119898and

then decreases if the initial price 1199011198780 is 119901lowastlowast1198780gt 1199011198780gt 0 gt 119901

lowast

1198780

that is smaller than CIPES gt 0 gt CIPS

Case 5-C (120572 gt 119903119864gt 0 gt 119903


selling resource decreases with time if the initial price 1199011198780 is


1198780gt 0 gt 119901

lowast

1198780 that is greater than CIPES gt 0 gt CIPS

6 Price Difference between Selling andExtracted Resources

The difference between selling and extracted resources Δ119901 =(119901119878minus 119901119864) is defined as

Δ119901 = (119901119878minus 119901119864)

= (1199011198780minus 119901lowast

1198780) exp (1205731015840119905) minus (119901


1198780) exp (120573119905)

= (Δ1199010minus Δlowast1199010) exp (1205731015840119905) + Δlowast119901

0exp (120573119905)

(47)

where

Δlowast1199010= 119901lowast

1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (48)

is the critical initial price difference CIPD Δlowast1199010

The price difference Δ119901 has an extreme (maximum orminimum) for

119905119898=

1

(1205731015840 minus 120573)ln[

120573 sdot (119901lowast

1198780minus 1199011198640)

1205731015840 sdot (119901lowast

1198780minus 1199011198780)]

=1

(1205731015840 minus 120573)ln[

120573 sdot Δlowast1199010

1205731015840 sdot (Δlowast1199010minus Δ1199010)]

=1

(120573 minus 1205731015840)ln[

1205731015840sdot (Δlowast1199010minus Δ1199010)

1205731015840 sdot Δlowast1199010

]

(49)


1205731015840(1205731015840minus 120573) (Δ119901

0minus Δlowast1199010) lt 0 (50)


119898of the extreme is zero if Δ119901

0is equal to

Δlowastlowast1199010= Δlowast1199010[

(1205731015840minus 120573)

1205731015840] = 119901

1198640

(2119903119873minus 119903119864)

1205731015840 (51)


t0

pS

pS0

pS0

(plowastS0)a

(plowastS0)b

Category 3 120572 = rN ie 120573 = PIFE = 0

Case 3-A rE gt rN = 120572 ie 120573998400 gt 0


Case 3-B rE gt rN = 120572 ie 120573998400 gt 120573 = 0

Case 3-B pS0 = (plowastS0)a

Case 3-C rE gt rN = 120572 ie 120573998400 gt 120573 = 0 or rN = 120572 gt rE ie 120573998400 lt 120573 = 0

(t0plowastS0)b

Case 3-C

(t0 + infin)

(t0 minus infin)



t0

pS

pS0

pS0

pS0

(plowastlowastS0 )a

(plowastlowastS0)b

Category 4 rN = rE ie 120573998400 = 120573


(t0 minus infin)

Linear

tmax

tmin

t0

(t0 minus 0)

Case 4-B rN = rE gt 120572 ie 120573998400 = 120573 gt 0

Case 4-B pS0 gt (plowastlowastS0)a

Case 4-C pS0 gt (plowastlowastS0)a

Case 4-C pS0 lt (plowastlowastS0 )a

Case 4-C pS0 lt (plowastlowastS0)a Case 4-C pS0 = (plowastlowast

S0)a Case 4-C rN = rE gt 120572 ie 120573998400 = 120573 gt 0

or rN = rE lt 120572 ie 120573998400 = 120573 lt 0

or rN = rE = 120572 ie 120573998400 = 120573 = 0


where Δlowastlowast1199010is the critical initial extreme of the initial price

difference CIEIPDThe time 119905

119898of the extreme is greater than zero if

Δ1199010gt Δlowastlowast1199010

for 1205731015840 gt 120573Δ1199010lt Δlowastlowast1199010

for 1205731015840 lt 120573(52)

For Δ1199011198780= Δ119901lowast

1198780the time of the extreme is

119905119898= +infin (53)

for 1205731015840 gt 120573 and119905119898= minusinfin (54)

for 1205731015840 lt 120573


t

0

pS

pS0

pS0

pS0

(plowastlowastS0)a

(plowastS0)a

Category 5 rN lt 0

Case 5-A rE gt 120572 gt 0 gt rN ie 120573998400 gt 0 gt 120573

(t0 + infin)

Case 5-ApS0 gt 0

Case 5-B 120572 gt rE gt 0 gt rN ie 0 gt 120573998400 gt 120573

tmax

Case 5-B (plowastS0)a lt pS0 lt (plowastlowast

S0)a

Case 5-C 120572 gt rE gt 0 gt rN ie 0 gt 120573998400 gt 120573

(t0 minus infin)

Case 5-C pS0 gt plowastlowastS0


The time 1199050is

1199050=

1

(1205731015840 minus 120573)ln[

Δlowast1199010

(Δlowast1199010minus Δ1199010)] (55)

when Δ119875 = 0The possible cases can be classified in five categories

Category 0D 119903119873= 0

Category 1D 119903119873gt 120572 that is 120573 = PIFE gt 0

Category 2D 120572 gt 119903119873 that is 120573 = PIFE lt 0

Category 3D 120572 = 119903119873 that is 120573 = PIFE = 0

Category 4D 119903119873= 119903119864 that is 1205731015840 = 120573 or PIFE = PIFS


Category 0D (119903119873= 0) The difference between selling and

extracted resources Δ119901 is function of the initial differenceΔ1199010= (1199011198780minus 1199011198640) and the extraction rate 120572 as compared

to 119903119864 It can be remarked that for 119903

119873= 0 the critical initial

price and differences of sold resources are 119901lowast1198780= 119901lowastlowast

1198780= 0

Δlowast1199010= minus1199011198640 and Δlowastlowast119901

0= minus11990311986411990111986401205731015840

The price difference is then given by

Δ119901 = (119901119878minus 119901119864) = 1199011198780exp (1205731015840119905) minus 119901

1198640exp (120573119905)


0exp (120573119905)

(56)

The time 119905119898of the extreme is given by

119905119898=

1

(1205731015840 minus 120573)ln[

120573 sdot 1199011198640

1205731015840 sdot 1199011198780

] (57)

which is obtained for 119901lowast1198780= 0 and the extreme is a maximum

for

1205731015840(1205731015840minus 120573) 119901

1198780lt 0 (58)

The price difference is Δ119901 = 0 at the time 1199050

1199050=

1

(1205731015840 minus 120573)ln [

1199011198640

1199011198780

] (59)

Four initial conditions are investigated

(i) if the initial price difference Δ1199010is Δ1199010ge 0 or (119901

1198780ge

1199011198640)

(ii) if the initial price difference Δ1199010is 0 gt Δ119901

0ge Δlowastlowast1199010

(iii) if the initial price difference Δ1199010is Δlowastlowast119901

0gt Δ119901

0gt

Δlowast1199010

(iv) if the initial price difference Δ1199010is Δlowast119901

0= Δ119901

0or

1199011198780= 119901lowast

1198780


Case 0D-A (120572 le 0 lt 119903119864 (ie 1205731015840 gt 0 1205731015840 gt 120573 (Δlowastlowast119901

0)119886lt 0))

Case 0D-A1 If the initialΔ1199010ge 0Δ119901 increases exponentially

towards +infin with a higher rate for 120572 lt 0 (120573 gt 0) anintermediate one for 120572 = 0 (120573 = 0) and a smaller one for119903119864gt 120572 gt 0 (120573 lt 0)

Case 0D-A2 If the initial Δ1199010is 0 gt Δ119901

0gt Δlowastlowast1199010the

evolution is similar to the previous case but nowΔ119901 increasesfrom a negative value Δ119901

0and all the curves pass through


Category 0D rN = 0pS minus pE

(Δlowastlowastp0)cpS0

Δp0

0

pS0

0

Δp0pS0

(Δlowastlowastp0)a

Δp0

Δlowastp0

0D-A1120572 lt 0 120573 gt 0

0D-A1120572 = 0 120573 = 0

Δp0

(t0 + infin)

(t0 + infin)

Case 0D-A rE gt 120572 gt 0 0 gt 120573

0D-A1 120572 gt 0 0 gt 120573

0D-B1

0D-B1

0D-C1tmax

tmax (t0 + 0)

(t0 + 0)

(t0 + 0)

t

t

t

0D-A2

120572 lt 0 120573 gt 0

(t0 + infin) 120572 = 0 120573 = 0

0D-A20D-A2120572 gt 0 0 gt 120573

(t0pS0)

(t0pS0)

(t0pS0)

t0

t0

0D-C2

0D-A3120572 lt 0 120573 gt 0 Case 0D-B 120572 = rE 120573

998400 = 0 120573 lt 0

Case 0D-C 120572 gt rE 0 gt 120573998400 gt 120573

tmin

0D-C3

0D-C4

0D-B1

(t0 minus infin)

Figure 12 Price difference evolution for Category 0D

the same point Δ119901 = 0 at 1199050 If the initial Δ119901

0= Δlowastlowast1199010 Δ119901

increases exponentially with the minimum at 119905min = 0

Case 0D-A3 If the initial Δ1199010is Δlowastlowast119901

0gt Δ119901

0gt Δlowast1199010

and 120573 gt 0 Δ119901 decreases initially to a minimum and thenincreases exponentially as in the previous cases

Case 0D-B (120572 = 119903119864 (ie 1205731015840 = 0 gt 120573 119901lowastlowast

1198780is not defined and

Δlowastlowast1199010= minusinfin lt Δ

lowast1199010lt 0))

Case 0D-B1 If the initialΔ1199010ge Δlowast1199010Δ119901 tends towards119901

1198780=

Δ1199010minus Δlowast1199010as asymptote

Case 0D-C (120572 gt 119903119864 (ie 120573 lt 120573

1015840lt 0 (Δlowastlowast119901

0)119888gt 0 gt

(Δlowast1199010)119888))

Case 0D-C1 If the initial value Δ1199010ge Δlowastlowast1199010 the difference

Δ119901 decreases towards +0 with the maximum at 119905 = 0 forΔ1199010= Δlowastlowast1199010

Case 0D-C2 If the initial value Δlowast1199010lt Δ119901

0lt Δlowastlowast1199010 the

difference Δ119901 increases up to the time 119905max when Δ119901 reachesa maximum value Δ119901max and then decreases towards +0

Case 0D-C3 If the initial Δ1199010= Δlowast1199010 Δ119901 increases towards

0

Case 0D-C4 If the initial value Δ1199010= Δlowast1199010lt Δlowastlowast1199010 Δ119901

decreases towards minusinfin for 120573 gt 0 tends to 0 for 120573 lt 0 andremains constant for 120573 = 0

In conclusion for 119903119873= 0 the criteria to evaluate the

evolution of Δ119901 are the extraction rate 120572 and the initial valueΔ1199010 If 120572 lt 119903

119864 Δ119901 increases towards +infin if the initial Δ119901

0

is Δ1199010gt Δlowast1199010 If 120572 = 119903

119864 Δ119901 tends towards the asymptotic

value 1199011198780= Δ1199010minus Δlowast1199010 for Δ119901

0ge Δlowast1199010 If 120572 gt 119903

119864 Δ119901 tends

towards +0 if Δ1199010ge Δlowast1199010

Category 1D (119903119873

gt 120572 that is 120573 = 119875119868119865119864 gt 0) The priceevolution of the difference between selling and extractedresources Δ119901 is investigated according to the initial valueΔ1199010= (1199011198780minus 1199011198640) and the extraction rate 120572 as compared

to 119903119864

Case 1D-A (120572 le 0 lt 119903119873lt 119903119864 (ie 1205731015840 gt 120573 gt 0 119901lowast

1198780ge 0))

Three relations between 119903119864and 119903119873can be investigated

(i) 119903119864gt 2119903119873 that is (Δlowast119901

0)119886lt (Δlowastlowast1199010)119886lt 0

(ii) 119903119864= 2119903119873 that is (Δlowast119901

0)119886= (Δlowastlowast1199010)119886= 0

(iii) 119903119864lt 2119903119873 that is (Δlowast119901

0)119886gt (Δlowastlowast1199010)119886gt 0


pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0

Category 1D 120572 lt rN 120573 gt 01D-A1 120572 lt 0 1D-A1 120572 = 0 (t0 + infin)

1D-A1 120572 gt 0

Case 1D-A 0 lt 120572 le rN lt rE

(ie 120573998400 gt 120573 gt 0 plowastS0 gt 0)

Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0

Case 1D-C1 rN gt 120572 = rE 120573998400 = 0 120573 gt 0 (plowast

S0)b lt 0 (Δlowastp0)b lt 0

Case 1D-C2 rN gt 120572 gt rE 120573 gt 0 gt 120573998400 (plowastS0)c lt 0 (Δlowastlowastp0)c lt (Δlowastp0)c lt 0

t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)


If 119903119864gt 2119903119873 that is Δlowast119901

0lt Δlowastlowast1199010lt 0 the following four

initial conditions are investigated(i) the initial price difference Δ119901

0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)

(ii) the initial price difference Δ1199010is Δlowastlowast119901

0le Δ1199010lt 0

(iii) the initial price difference Δ1199010is Δlowastlowast119901

0gt Δ119901

0gt

Δlowast1199010

(iv) the initial price differenceΔ1199010isΔ1199010le Δlowast1199010 or1199011198780le

119901lowast

1198780

and the price evolutions are presented in Figure 13Case 1D-A1 If Δ119901

0ge 0 the difference Δ119901 increases expo-

nentially towards +infin with a higher rate for 120572 lt 0 anintermediate one for 120572 = 0 and a smaller one for 0 lt 120572Case 1D-A2 If 0 gt Δ119901

0gt Δlowastlowast1199010the difference Δ119901 increases

exponentially towards +infin as in the previous case and thecurves pass through the same point Δ119901 = 0 at 119905

0 If Δ119901

0=

Δlowastlowast1199010the difference Δ119901 increases exponentially with the

minimum at 119905 = 0Case 1D-A3 If Δlowastlowast119901

0gt Δ1199010gt Δlowast1199010 Δ119901 decreases initially

down to a minimum and then increases exponentially as inthe previous cases

If 119903119864= 2119903119873 that isΔlowast119901

0= Δlowastlowast1199010= 0 the following cases

are discussed

Case 1D-A1 If Δ1199010gt Δlowast1199010= 0 the difference Δ119901 increases

exponentially towards +infin with a higher rate for 120572 lt 0 anintermediate one for 120572 = 0 and a smaller one for 0 lt 120572

If 119903119864lt 2119903119873 that isΔlowast119901

0gt Δlowastlowast1199010gt 0 the following cases

are discussed

Case 1D-A1 If Δ1199010ge Δlowast1199010gt 0 the difference Δ119901 increases


Case 1D-B (Δ1199010= Δlowast1199010= 0) Δ119901 remains constant at

Δlowast1199010= 0 with time

Case 1D-C1 (120572 = 119903119864lt 119903119873 (ie 1205731015840 = 0 120573 gt 0 (119901lowast

1198780)119887lt 0 lt

(119901lowastlowast

1198780)119887= infin and (Δlowast119901

0)119887lt 0 lt (Δ

lowastlowast1199010)119887= infin)) If Δ119901

0

is minus1199011198640lt Δ1199010lt Δlowast1199010 Δp decreases towards minusinfin If Δ119901

0is

Δlowast1199010gt Δ1199010gt Δlowastlowast1199010 Δ119901 increases up to a maximum and

then decreases towards minusinfin If Δ1199010le Δlowastlowast1199010 Δ119901 decreases

towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0

(Δlowastlowast1199010)119888lt (Δ

lowast1199010)119888lt 0)) The difference Δ119901 decreases

towards minusinfin for every initial Δ1199010

In conclusion for 120572 lt 119903119873 (ie 120573 gt 0) the criteria

to evaluate the price difference evolution are the extractionrate 120572 the initial condition and the relation between 119903

119864and

119903119873 For 1205731015840 gt 0 and 119903

119864gt 2119903119873 Δ119901 increases towards +infin

if Δ1199010gt Δlowast1199010 For 1205731015840 gt 0 and 119903

119864= 2119903119873 Δ119901 increases

towards +infin if Δ1199010gt Δlowast1199010 For 1205731015840 gt 0 and 119903

119864lt 2119903119873 Δ119901

increases towards +infin if Δ1199010ge Δlowast1199010 while Δ119901 increases up

to a maximum and then decreases if Δlowast1199010gt Δ1199010gt Δlowastlowast1199010

For 1205731015840 le 0 Δ119901 decreases towards minusinfin


Category 2D (120572 gt 119903119873 that is 120573 = 119875119868119865119864 lt 0) The price

evolution of the difference Δ119901 is investigated for 120573 lt 0

according to the initial value Δ1199010and the relation between

119903119873and 119903119864

The price evolutions are presented in Figure 14

Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that

is Δlowastlowast1199010lt Δlowast1199010lt 0 the following situations are discussed

Case 2D-A1 If Δ1199010gt Δlowast1199010the difference Δ119901 increases

towards +infinIf 2119903119873gt 119903119864 that is Δlowastlowast119901

0gt Δlowast1199010gt 0 the following

situations are discussed

Case 2D-A1 If Δ1199010ge Δlowastlowast1199010the difference Δ119901 increases

towards +infin If Δ1199010is Δlowastlowast119901

0gt Δ119901

0gt Δlowast1199010gt 0 the

differenceΔ119901 increases towards +infin after a minimum at 119905minIf 2119903119873= 119903119864 that is Δlowastlowast119901

0= Δlowast1199010= 0 the following



towards +infin

Case 2D-B (120572 = 119903119864gt 119903119873 (ie 1205731015840 = 0 gt 120573 Δlowastlowast119901

0= infin

Δlowast1199010gt 0 for 2119903

119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))

Case 2D-B1 If Δ1199010= Δlowast1199010= 0 the difference Δ119901 remains

constant atΔ1199010= Δlowast1199010= 0 IfΔ119901

0= Δlowast1199010gt 0 the difference

Δ119901 tends to +0

Case 2D-B2 If Δ1199010gt Δlowast1199010 the difference Δ119901 decreases

towards (Δ1199010minus Δlowast1199010) For 119903

119864= 2119903119873 Δ119901 remains constant

at (Δ1199010minus Δlowast1199010)

Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt

0 gt Δlowastlowast1199010for 2119903119873gt 119903119864 Δlowast119901

0lt 0 lt Δ

lowastlowast1199010for 2119903119873lt

119903119864 and Δlowastlowast119901

0= Δlowast1199010= 0 for 2119903

119873= 119903119864))

Case 2D-C1 IfΔ1199010isΔlowastlowast119901

0lt Δ1199010lt Δlowast1199010lt 0 the difference

Δ119901 decreases towards minusinfin after a maximum at 119905max

Case 2D-C2 If Δ1199010lt Δlowast1199010= 0 the difference Δ119901 decreases

towards minusinfin If Δ1199010le Δlowastlowast1199010the difference Δ119901 decreases

towards minusinfin If Δ1199010lt Δlowast1199010the difference Δ119901 decreases

towards minusinfinFor 119903119864gt 2119903119873 Δ119901 tends towards 0 if Δ119901

0ge Δlowastlowast1199010gt 0

Δ119901 increases up to a maximum and then tends towards +0if Δlowastlowast119901

0gt Δ119901

0gt Δlowast1199010gt 0 and Δ119901 tends towards +0

if Δ1199010le Δlowast1199010lt 0 For 119903

119864= 2119903119873 Δ119901 decreases to +0 if

Δ1199010gt Δlowastlowast1199010 remains constant at Δ119901

0if Δ1199010= Δlowastlowast1199010 and

increases towards minus0 if Δ1199010lt Δlowastlowast1199010 For 119903

119864lt 2119903119873 Δ119901

decreases towards 0 for every Δ1199010

Case 2D-D (120572 gt 119903119873gt 119903119864 (ie 1205731015840 lt 120573 Δlowast119901

0lt Δ119901

lowastlowast

0lt

0)) For every Δ1199010the difference Δ119901 decreases down to a

minimum and then increases towards minus0In conclusion for 120572 gt 119903

119873 (ie 120573 lt 0) the criteria to

evaluate the evolution of Δ119901 are the extraction rate 120572 and theinitial condition For 120572 lt 119903

119864 Δ119901 increases towards +infin if

Δ1199010gt Δlowast1199010le 0 or Δ119901

0gt Δlowastlowast1199010gt Δlowast1199010gt 0 or Δlowastlowast119901

0gt

Δ1199010gt Δlowast1199010 Δ119901 decreases towards +0 if Δ119901

0= Δlowast1199010gt

or lt 0 For 120572 = 119903119864 Δ119901 tends towards (Δ119901

0minus Δlowast1199010) If 120572 gt 119903

119864

Δ119901 tends towards +0 for every initial Δ1199010except for 119903

119864gt 2119903119873

and Δlowastlowast1199010gt Δ1199010gt Δlowast1199010when Δ119901 tends towards 0 after a

maximum

Category 3D (120572 = 119903119873 that is 120573 = 119875119868119865119864 = 0) The price

evolution of the difference Δ119901 is investigated for 120573 = 0 (ieΔlowast1199010= Δlowastlowast1199010) according to the initial value Δ119901

0and the

relation between 120572 = 119903119873and 119903119864

The price evolutions are reported in Figure 15

Case 3D-1 (119903119864gt 119903119873= 120572 that is 1205731015840 gt 120573 = 0)

Case 3D-1A If Δ1199010gt Δlowast1199010the difference Δ119901 increases

exponentially with time towards +infin

Case 3D-1B If Δ1199010

= Δlowast1199010the difference Δ119901 remains

constant at the initial value Δ1199010

Case 3D-1C If Δ1199010lt Δlowast1199010the difference Δ119901 decreases

towards minusinfin

Case 3D-2 (120572 = 119903119873gt 119903119864 (ie 1205731015840 lt 120573 = 0)) The difference

Δ119901 decreases towards Δlowast1199010for every initial Δ119901

0

In conclusion for 120573 = 0 the criteria to evaluate theevolution of Δ119901 are the relation between 119903

119873and 119903119864and the

initial condition Δ1199010 For 119903

119873lt 119903119864 Δ119901 increases towards +infin

if Δ1199010gt Δlowast1199010 Δ119901 remains constant if Δ119901

0= Δlowast1199010 and Δ119901

decreases towards minusinfin if Δ1199010lt Δlowast1199010 For 119903

119873gt 119903119864 Δ119901 tends

towards the asymptotic value Δlowast1199010

Category 4D (119903119873= 119903119864 that is 1205731015840 = 120573) The price difference

Δ119901 is given by

Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)

= (Δ1199010minus 119901lowastlowast

11987801205731015840119905) exp (1205731015840119905)

(60)

The critical initial price difference Δlowast1199010and the critical

initial extreme price difference Δlowastlowast1199010are not defined and the

only critical price defined is the critical initial price extremeof the sold resources 119901lowastlowast

1198780 already defined in the second part

of (38) or (45)The price difference Δ119901 has an extreme for the time 119905

119898

given by

119905119898=(Δ1199010minus 119901lowastlowast

1198780)

1205731015840 sdot 119901lowastlowast

1198780

(61)

The extreme is amaximum if1205731015840 gt 0 and aminimum if1205731015840 lt 0The time of maximum 119905max is zero if

Δ1199010= 119901lowastlowast

1198780=1199031198641199011198640

1205731015840 (62)


Category 2D 120573 lt 0 120572 gt rN gt 0

2D-A1 Δp0 gt Δlowastp0

2D-A1 Δp0 ge Δlowastlowastp0(t0 + infin)

2D-A1 Δlowastlowastp0 gt Δp0 gt Δlowastp0 gt 0

Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573

2D-B2 Δp0 gt Δlowastp0

Case 2D-A rE gt 120572 gt rN 120573998400 gt 0 gt 120573

(t0Δp0 minus Δlowastp0)

2D-C2

Case 2D-C 120572 gt rE gt rN rE lt 2rN0 gt 120573998400 gt 120573

2D-B1 Δp0 = Δlowastp0 = 0

(t0 + 0)2D-B1 Δp0 = Δlowastp0 gt 0

t(t0 minus 0)

2D-D

2D-B2 Δp0 lt Δlowastp0

2D-C2 Δp0 = Δlowastp0 lt 0

Case 2D-D 120572 gt rN gt rE 120573998400 lt 120573 lt 0

2D-C1 Δlowastlowastp0 lt Δp0 lt Δlowastp0 lt 0

(t0 minus infin)

2D-C2 Δp0 lt Δlowastlowastp0 lt Δlowastp0 lt 0 or Δp0 lt Δlowastp0 ge 0

pS minus pE

Δp0

Δlowastlowastp0

Δp0 minus Δlowastp0

Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0


Category 3D 120573 = 0 120572 = rN gt 0(t0 + infin)

Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0

3D-1A Δp0 gt Δlowastp0

t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b

3D-1C Δp0 lt Δlowastp0(t0 minus infin)

3D-1B Δp0 = Δlowastp0



and 119905max is greater than zero if

Δ1199010gt 119901lowastlowast

1198780 119901

lowastlowast

11987801205731015840gt 0

Δ1199010lt 119901lowastlowast

1198780 119901

lowastlowast

11987801205731015840lt 0

(63)

The difference Δ119901 is zero for the time 1199050equal to

1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)

The price evolution of the difference Δ119901 is investigatedaccording to the initial value Δ119901

0= (1199011198780minus 1199011198640) and the

extraction rate 120572The price evolutions are reported in Figure 16

Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))

Case 4D-1A The price cannot increase with time

Case 4D-1B IfΔ1199010gt 119901lowastlowast

1198780the differenceΔ119901 initially increases

up to the maximum time 119905max and then decreases towardsminusinfin

Case 4D-1C If Δ1199010= 119901lowastlowast

1198780the difference Δ119901 decreases

towards minusinfin with the maximum at 119905 = 0 If Δ1199010lt 119901lowastlowast

1198780 the

difference Δ119901 decreases towards minusinfin

Case 4D-2 (120572 gt 119903119873 (ie 1205731015840 = 120573 lt 0 119901lowastlowast

1198780lt 0))


Case 4D-2B If Δ1199010gt 119901lowastlowast

1198780 the difference Δ119901 decreases with

time reaches a minimum at 119905min and then increases towardsminus0 If Δ119901

0le 119901lowastlowast

1198780the difference Δ119901 increases towards 0

Case 4D-3 (120572 = 119903119864= 119903119873 (ie 1205731015840 = 120573 = 0 119901lowastlowast

1198780= infin)) The

difference Δ119901 decreases linearly towards minusinfin for every initialΔ1199010

Discussion The discussion on the price evolutions of soldresources 119901

119878 and price difference Δ119901 = (119901

119878minus 119901119864) in

case of no-accumulation and no-depletion of the extractedresources is carried out summarizing the results in thefollowing Tables 1ndash8

Table 1 presents the price trends of 119901119878 119901119864 and Δ119901 for

nonrenewable resources without interest rate 119903119873= 0 that is

1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901

0= minus1199011198640 andΔlowastlowast119901

0= minus11990311986411990111986401205731015840

with the following conclusions

(i) the price 119901119864of extracted resources increases towards

+infin if 120572 lt 0 (ie 120573 gt 0) 119901119864remains constant at 119901

1198640

if 120572 = 0 (ie 120573 = 0) and 119901119864decreases towards +0 if

120573 lt 0(ii) the price 119901

119878of sold resources increases towards +infin

if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901

119878remains constant at 119901

1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901

119878decreases towards +0 if

120572 gt 119903119864 (ie 1205731015840 lt 0)

(iii) if the extraction rate 120572 lt 0 (ie 1205731015840 gt 120573 gt 0) thetrends of the price differenceΔ119901 depend on the initial

value Δ1199010 If Δ119901

0ge Δlowastlowast1199010= Δlowast119901max Δ119901 increases

towards +infin if Δlowast119901min = Δlowast1199010lt Δ119901

0lt Δlowastlowast1199010=

Δlowast119901max Δ119901 increases towards +infin after a minimum

and if Δ1199010= Δlowast1199010= Δlowast119901min Δ119901 decreases towards

minusinfin

(iv) if the extraction rate is 120572 = 0 (ie 1205731015840 gt 120573 = 0) theprice difference Δ119901 increases towards +infin if Δ119901

0gt

Δlowastlowast1199010= Δlowast1199010= Δlowast119901min = Δ

lowast119901max = minus119901

1198640while

tending towards Δ1199010if Δ1199010= Δlowast119901min It can be

noticed thatΔ1199010cannot be smaller thanΔlowast119901

0because

1199011198780cannot be smaller than 0

(v) if the extraction rate 120572 lt 119903119864 (ie 1205731015840 gt 0 gt 120573) the

price difference Δ119901 increases towards +infin if Δ1199010gt

Δlowast1199010= Δlowast119901max = minus119901

1198640while tends towards 0 if

Δlowast1199010= Δlowast119901min

(vi) if the extraction rate 120572 = 119903119864 (ie 1205731015840 = 0 gt 120573) the

price difference Δ119901 tends towards 1199011198780= Δ1199010minus Δlowast1199010

(vii) if the extraction rate 120572 gt 119903119864 (ie 0 gt 120573

1015840gt 120573) the

price difference Δ119901 tends towards +0 for any Δ1199010but

with a maximum at 119905max if Δlowast119901max gt Δ1199010 gt Δ

lowast119901min


120572 lt 119903119873 that is 120573 gt 0 with the following conclusions

(i) the price 119901119864increases towards +infin because 120573 gt 0

(ii) the price 119901119878increases towards +infin if 119903

119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max

(iii) the price 119901119878initially increases reaches a maximum at

119905max and then decreases towards minusinfin in the followingcases 119903

119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901

1198780ge 119901lowastlowast

1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max

(iv) the price 119901119878 decreases towards minusinfin in all the other

cases

(v) the price difference Δ119901 has different trends for 119903119864gt

119903119873according to the relation between 2119903

119873and 119903119864 If

119903119864gt 2119903119873 the price difference Δ119901 increases towards

+infin if Δ1199010ge Δlowast119901max = Δ

lowastlowast1199010 Δ119901 increases towards

+infin after a minimum if Δlowastlowast1199010gt Δ119901

0gt Δlowast1199010

and Δ119901 decreases towards minusinfin if Δ1199010le Δlowast119901min If

119903119864= 2119903119873 the price difference Δ119901 increases towards

+infin if Δ1199010gt Δlowast119901max = Δ

lowast1199010= 0 Δ119901 remains

constant at Δlowast1199010= 0 if Δ119901

0= Δlowast1199010= 0 and Δ119901

decreases to minusinfin if Δ1199010lt Δlowast1199010= 0 If 119903

119864lt 2119903119873 the

price difference Δ119901 increases towards +infin if Δ1199010ge

Δlowast119901max = Δ

lowast1199010 Δ119901 decreases towards minusinfin after a

maximum ifΔlowast1199010gt Δ1199010gt Δlowastlowast1199010 andΔ119901 decreases

to minusinfin if Δ1199010le Δlowastlowast1199010= Δlowast119901min

(vi) the price difference Δ119901 for 119903119873= 119903119864initially increases

up to a maximum at 119905max and then decreases towardsminusinfin if Δ119901


1198780

(vii) the price difference Δ119901 decreases towards minusinfin in allother cases


Table 1 Price trends to 119905 rarr infin of selling resources 119901119878 extracted resources 119901

119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901

0= minus1199011198640 and Δlowastlowast119901

0= minus11990311986411990111986401205731015840)

Extraction rate 120572 1205731015840 = 119903119864minus 120572120573 = 119903

119873minus 120572

Initial condition 1199011198640 1199011198780 or

Δ1199010

Sellingresources 119901

119878

(infin)

Extractedresources 119901

119864

(infin)

Price differenceΔ119901 (infin)

120572 lt 0 1205731015840gt 120573 gt 0

(Δlowast1199010= Δlowast119901min lt Δ

lowast119901max = Δ

lowastlowast1199010lt 0)

1199010gt 119901lowast

1198780+infin +infin

Δ1199010ge Δlowast119901max +infin

Δlowast1199010lt Δ1199010lt Δlowastlowast1199010

+infin (Min)Δ1199010= Δlowast119901min minusinfin

120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ

lowast1199010= Δlowast119901min = Δ

lowast119901max =

Δlowastlowast1199010lt 0)

1199010gt 119901lowast

1198780+infin 119901

1198640

Δ1199010gt Δlowast119901max = Δ

lowast1199010

+infinΔ1199010= Δlowast119901min = Δ

lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ

lowastlowast1199010= Δlowast119901min lt Δ

lowast119901max =

Δlowast1199010lt 0)

1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined


lowast1199010= Δlowast119901max lt 0)

1199010gt 119901lowast

11987801199011198780

0Δ1199010ge Δlowast119901max = Δ

lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ


lowastlowast1199010gt 0 gt Δ

lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0

Δ1199010ge Δlowast119901max 0


0 (Max)Δ1199010= Δlowast119901min 0


119864 and difference Δ119901 = (119901

119878minus 119901119864) in a conservative policy of

extraction 120572 lt 119903119873 120573 gt 0

Extraction rate 120572 le 0 lt 119903119873 120573 gt 0 1205731015840 = 119903

119864minus 120572

120573 = 119903119873minus 120572

Initial condition 1199011198780or Δ119901

0


119878

(infin)


119864

(infin)

Pricedifference Δ119901

(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast

119878min = 119901lowastlowast

1198780gt 0)

1199011198780ge 119901lowast

119878max +infin +infin119901lowast


1198780minusinfin (Max) +infin

0 lt 1199011198780le 119901lowast

119878min minusinfin +infin

2119903119873lt 119903119864 1205731015840 gt 120573




Δ1199010ge Δlowast119901max +infin +infin

Δlowastlowast1199010gt Δ1199010gt Δlowast1199010

+infin +infin (Min)Δ1199010le Δlowast119901min +infin minusinfin

2119903119873= 119903119864 1205731015840 gt 120573

(Δlowast1199010= Δlowast119901min = Δ


lowastlowast1199010= 0)

Δ1199010gt Δlowast119901max +infin +infin

Δ1199010= Δlowast119901min +infin Δ

lowast1199010

Δ1199010lt Δlowast119901min +infin minusinfin

119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573

(Δlowast1199010= Δlowast119901max gt Δ

lowast119901min = Δ

lowastlowast1199010gt 0)

Δ1199010ge Δlowast119901max +infin minusinfin

Δlowast1199010gt Δ1199010gt Δlowast1199010

+infin minusinfin (Max)Δ1199010le Δlowast119901min +infin minusinfin

119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)



0 lt 1199011198780le 119901lowastlowast

1198780minusinfin +infin


1198780+infin minusinfin (Max)

Δ1199010le 119901lowastlowast

1198780+infin minusinfin

119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast

119878max = 119901lowastlowast

1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)

(0 gt Δlowast119901max = Δ

lowast1199010gt Δlowastlowast1199010= Δlowast119901min)



0 lt 1199011198780lt 119901lowastlowast


Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast

119878min lt 0 lt 119901lowastlowast

1198780= 119901lowast

119878max = infin)

(Δlowast1199010= Δlowast119901min lt 0 lt Δ

lowastlowast1199010= Δlowast119901max = infin)

1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)

(t0 minus infin)(t0 minus infin)

Category 4D 120573 = 120573998400 rN = rE

Case 4D-1 120572 le 0 lt rN 120573998400 = 120573 gt 0 (plowastlowast

S0)a gt 0

Case 4D-2 120572 gt rN 120573998400 = 120573 lt 0 (plowastlowast

S0)b lt 0

Case 4D-3 120572 = rN = rE 120573998400 = 120573 = 0 (plowastlowast

S0)c = infin4D-1B Δp0 gt plowastlowast

S0

tmaxtmin

Case 4D-2B Δp0 gt plowastlowastS0

(t0 + 0)

Case 4D-2B Δp0 le plowastlowastS0

4D-1C Δp0 lt plowastlowastS0

4D-1C Δp0 = plowastlowastS0

t0

Case 4D-3



119864 and difference Δ119901 = (119901

119878minus 119901119864) for a rate of extraction

0 lt 120572 = 119903119873 (120573 = 0 119901lowast


1198780 and Δlowast119901

0= Δlowastlowast1199010)

Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)

(Δlowast119901min = Δlowast1199010= Δlowastlowast1199010= Δlowast119901maxgt or = or lt 0)

1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast

119878min 119901lowast

11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901

1198780gt 0 minusinfin (Lin) 119901

1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast

119878max lt 0)(Δlowast119901min = Δ

lowast1199010= Δlowastlowast1199010= Δlowast119901max lt 0)

1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010


120572 = 119903119873 that is 120573 = 0 with the following conclusions

(i) the price 119901119864is constant at 119901

1198640 because 120573 = 0

(ii) the price 119901119878increases towards +infin for 119903

119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min

(iii) the price 119901119878is constant at 119901lowast

1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast

119878min or for 119903119873 gt 119903119864 and every 1199011198780gt 0

(iv) the price 119901119878decreases towards minusinfin for 119903

119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast

119878max or decreases linearly towardsminusinfin for 119903

119873= 119903119864and every 119901

1198780gt 0

(v) the price differenceΔ119901 increases towards+infin for 119903119873lt

119903119864 and Δ119901

0gt Δlowast1199010= Δlowast119901max

(vi) the price difference Δ119901 is constant at Δlowast1199010 for 119903

119873lt

119903119864 and Δ119901

0= Δlowast1199010or for 119903

119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0

Extraction rate120572 gt 119903119873gt 0 120573 lt 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572

Initial condition1199011198780or Δ119901

0

Selling resources119901119878(infin)


119864

(infin)Price

differenceΔ119901 (infin)+0

119903119864gt 120572 gt 119903

119873 1205731015840 gt 0 gt 120573 (119901lowastlowast

1198780= 119901lowast

119878max gt 119901lowast

119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast

119878max +infin119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast

119878min minusinfin

119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573

(0 gt Δlowast1199010= Δlowast119901max gt Δ


lowastlowast1199010)

Δ1199010gt Δlowast119901max +infin

Δ1199010= Δlowast119901max 0

Δlowastlowast1199010lt Δ1199010lt Δlowast1199010

minusinfin (Max)Δ1199010le Δlowast119901min minusinfin

119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901min Δ

lowast1199010

Δ1199010lt Δlowast119901min minusinfin

119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573

(0 lt Δlowast1199010= Δlowast119901min lt Δ





+infin (Min)Δ1199010= Δlowast119901min 0


120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast

119878min gt 0)(Δlowast119901min = Δ

lowast1199010gt or = or lt 0 Δlowast119901max = Δ

lowastlowast1199010= infin)

119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)


Δ1199010minus Δlowast1199010

Δ1199010= Δlowast1199010

Δ1199010minusΔlowast1199010= 0

Δ1199010lt Δlowast1199010


120572 gt 119903119873= 119903119864 1205731015840= 120573 lt 0 (119901lowastlowast

1198780lt 0)

Every 1199011198780

0 (Min)Every Δ119901

00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast

119878max gt 0 gt 119901lowastlowast

1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0






0 (Max)Δ1199010le Δlowast119901min 0

120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573

(Δlowastlowast1199010= Δlowast119901min = Δ

lowast1199010= Δlowast119901max = 0)

Δ1199010gt Δlowast119901min 0

Δ1199010= Δlowast119901min Δ119901

0

Δ1199010lt Δlowast119901min 0

120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573

(Δlowast1199010= Δlowast119901max gt 0 gt Δ



Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast

119878min lt 119901lowastlowast

1198780= 119901lowast

119878max lt 0)


lowastlowast1199010= Δlowast119901max lt 0)

1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)

(vii) the price difference Δ119901 decreases towards minusinfin for119903119873

lt 119903119864 and Δ119901

0lt Δlowast1199010or linearly to minusinfin for

120572 = 119903119873= 119903119864and every Δ119901

0

Table 4 presents the price trends of 119901119878 119901119864 andΔ119901 for 120572 gt

119903119873gt 0 that is 120573 lt 0 with the following conclusions(i) the price 119901

119864decreases towards +0 because 120573 lt 0


119864gt 120572 gt 119903

119873

(ie 1205731015840 gt 0 gt 120573) and 1199011198780gt 119901lowastlowast

1198780= 119901lowast

119878max or after aminimum for 119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780

(iii) the price 119901119878tends to (119901


1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast

119878min(v) the price 119901

119878decreases towards 0 without or with a

minimum in the other two cases(vi) the price differenceΔ119901 increases towards+infin for 119903

119864gt

120572 gt 119903119873and Δ119901

0gt Δlowast119901max if 119903119864 gt 2119903119873 if 119903119864 = 2119903119873 if

119903119864lt 2119903119873and Δ119901

0= Δlowast119901max If 119903119864 lt 2119903119873 a minimum

is present if Δlowastlowast1199010gt Δ1199010gt Δlowast1199010


Table 5 Final conclusions of the price trends at 119905 rarr infin of sellingresources 119901

119878 for 119903

119873= 0 119901lowast


1198780= 0

Relationbetween1205731015840 and 120573

120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0

(vii) the price difference Δ119901 tends to (Δ1199010minus Δlowast1199010) if 119903119864=

120572 gt 119903119873for every Δ119901

0

(viii) the price differenceΔ119901decreases towardsminusinfin for 119903119864gt

120572 gt 119903119873and Δ119901

0lt Δlowast119901min or Δ1199010 = Δ

lowast119901min for 119903119864 gt

2119903119873while Δ119901 passes through a maximum if Δlowast119901

0gt

Δ1199010gt Δlowastlowast1199010and for 119903

119864gt 2119903119873

(ix) the price difference Δ119901 remains constant at Δlowast1199010for

119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010

(x) the price difference Δ119901 tends to 0 in the other threecases

Table 5 resumes the trends of119901119878for 119903119873= 0 that is1205731015840 gt 120573

119901lowast

1198780= 119901lowast

1198780= 0

The price 119901119878increases towards +infin if 1205731015840 gt 0 remains

constant at 1199011198780if 1205731015840 = 0 and decreases to +0 if 1205731015840 lt 0

Table 6 resumes the trends of Δ119901 for 119903119873

= 0 that is1205731015840gt 120573 The trends can be divided according to the initial

conditions which are grouped into four categoriesThe greater value between Δlowast119901

0and Δlowastlowast119901

0is indicated

as Δlowast119901max while the smaller one as Δlowast119901min Consider thefollowing

(i) for Δ1199010gt Δlowast119901max Δ119901 increases to +infin if 1205731015840 gt 0 Δ119901

tends to 1199011198780= Δ1199010minus Δlowast1199010if 1205731015840 = 0 and Δ119901 tends to

+0 if 1205731015840 lt 0(ii) for Δ119901

0= Δlowast119901max Δ119901 increases to +infin if 1205731015840 gt 120573 gt 0

Δ119901 remains constant at Δ1199010if 1205731015840 gt 120573 = 0 Δ119901 tends

to 0 if 1205731015840 gt 0 gt 120573 and 1205731015840 lt 0 Δ119901 tends to 1199011198780=

Δ1199010minus Δlowast1199010if 1205731015840 = 0

(iii) for Δlowast1199010= Δlowast119901min lt Δ119901

0lt Δlowast119901max = Δ

lowastlowast1199010 Δ119901

increases to +infin after a minimum if 1205731015840 gt 0 and Δ119901tends to +0 after a maximum if 1205731015840 lt 0

(iv) for Δlowast1199010= Δlowast119901min gt Δ119901

0 Δ119901 decreases to minusinfin if

1205731015840gt 120573 gt 0 Δ119901 remains constant at Δ119901

0if 1205731015840 gt 120573 = 0

and Δ119901 tends to 0 if 1205731015840 lt 0

Table 7 resumes the trends of 119901119878for 119903119873

different from0 The trends can be divided into four classes A-B-C-Daccording to the relation between 1205731015840 and 120573 and 1205731015840 = 0 Ineach class the trends are depending on the value of 120573 or 1205731015840and the initial value 119901

1198780 The greater between 119901lowast

1198780and 119901lowastlowast

1198780is

indicated as 119901lowast119878max and the smaller one as 119901lowast

119878minIn class A 1205731015840 gt 120573 the trends of 119901

119878are to increase towards

+infin for 1205731015840 gt 120573 gt 0 if 1199011198780ge 119901lowast

119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast

119878max and for 1205731015840 gt 0 gt 120573 if 1199011198780ge 119901lowast

119878maxand if 119901lowast

1198780gt 1199011198780


1198780after a minimum A temporary

increase followed by a decrease towards minusinfin is present if119901lowast

1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast


1198780and 120573 gt 0 In all the other

cases 119901119878tends towards asymptotes equal to 119901

1198780 0 1199011198780minus 119901lowast

1198780

or decreases towards minusinfinIn class B (1205731015840 = 120573) for 120573 gt 0 a temporary increase

followed by a decrease towards minusinfin is present if 1199011198780ge 119901lowastlowast

1198780

In all the other cases 119901119878tends towards an asymptote equal to

+0 or decreases towards minusinfinIn class C (1205731015840 lt 120573) for 120573 gt 0 a temporary increase

followed by a decrease towards minusinfin is present if 1199011198780

ge

119901lowast


1198780 In all the other cases 119901

119878tends towards

asymptotes equal to 119901lowast1198780 0 or decreases towards minusinfin

In class D (1205731015840 = 0) 119901119878decreases towards minusinfin

Table 8 resumes the trends of Δ119901 for 119903119873different from

0 The trends can be divided into four classes A-B-C-Daccording to the relation between 1205731015840 and 120573 and 1205731015840 = 0 Ineach class the trends depend on the value of 120573 or 1205731015840 andthe initial value Δ119901

0 The greater between Δlowast119901


0

is indicated as Δlowast119901max and the smaller one as Δlowast119901minIn class A (1205731015840 gt 120573) the trends of Δ119901 are to increase

towards +infin in the following cases

(i) for 1205731015840 gt 0 and Δ1199010gt Δlowast119901max

(ii) for 1205731015840 gt 120573 gt 0 and Δ1199010= Δlowast119901max = 0

(iii) for 1205731015840 gt 0 gt 120573 and Δ1199010= Δlowast119901max gt 0

(iv) for 1205731015840 gt 120573 gt 0 1205731015840 gt 0 gt 120573 and Δlowastlowast1199010gt Δ1199010gt Δlowast1199010

after a minimum

A temporary increase followed by a decrease towardsminusinfin is present for 1205731015840 gt 120573 gt 0 and 1205731015840 gt 0 gt 120573 if Δlowast119901

0=

Δlowast119901max gt Δ1199010 gt Δ


lowastlowast1199010 In all the other cases Δ119901

decreases towards asymptotes equal to Δlowast1199010 0 or decreases

towards minusinfinIn class B (1205731015840 = 120573) for 120573 gt 0 a temporary increase

followed by a decrease towards minusinfin is present if Δ1199010gt 119901lowastlowast

1198780

In all the other cases Δ119901 decreases towards the asymptoteequal to 0 or decreases towards minusinfin

In class C (1205731015840 lt 120573) Δ119901 decreases towards asymptotesequal to Δlowast119901

0 +0 or decreases towards minusinfin

In class D (1205731015840 = 0)Δ119901 decreases towards the asymptotesequal to (Δ119901

0minus Δlowast1199010) or decreases towards minusinfin

7 Energy Supply Curve

71 Supply Curve of Extracted Resources The mass conser-vation equations of extracted and sold resources under thehypothesis of no accumulation nor depletion of the resourcescan be written in dimensionless form as

1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)

The price evolution with time of extracted resources is indimensionless form

1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced

ifferenceΔ119901for119903119873=0Δlowast1199010=minus1199011198640lt0Δlowastlowast1199010=minus1199011198640(1205731015840minus120573)1205731015840

Relatio

nbetween1205731015840

and120573

120573Δ1199010gtΔlowast119901max

Δ1199010=Δlowast119901max

Δlowast119901minltΔ1199010ltΔlowast119901max

Δ1199010=Δlowast119901min

A1205731015840gt120573

A1

1205731015840gt120573gt0(0gtΔlowast119901max=Δlowastlowast1199010gtΔlowast1199010=Δlowast119901min)

+infin+infin

+infin(M

in)

minusinfin

A2

1205731015840gt120573=0(0gtΔlowast119901max=Δlowastlowast1199010=Δlowast1199010=Δlowast119901min)

+infinΔ1199010

Δ1199010

A3

1205731015840gt0gt120573(0gtΔlowast119901max=Δlowast1199010gtΔlowastlowast1199010=Δlowast119901min)

+infin0

A4

1205731015840=0gt120573(0gtΔlowast1199010=Δlowast119901maxgtΔlowastlowast1199010=Δlowast119901min=minusinfin)

1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5

0gt1205731015840gt120573(Δlowast119901max=Δlowastlowast1199010gt0gtΔlowast1199010=Δlowast119901min)

00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780

119901lowast 1198780gt1199011198780gt119901lowastlowast

1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1

1205731015840gt120573gt0(119901lowast 119878max=119901lowast 1198780gt119901lowastlowast

1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2

1205731015840gt120573=0(119901lowast 119878min=119901lowast 1198780=119901lowastlowast

1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3

1205731015840gt0gt120573(119901lowast 119878max=119901lowastlowast

1198780gt119901lowast 1198780=119901lowast 119878mingt0)

+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin

gt119901lowast 119878min=119901lowast 1198780)

1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5

0gt1205731015840gt120573(119901lowast 119878max=119901lowast 1198780gt0gt119901lowastlowast

1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1

120573gt0(119901lowast 119878min=119901lowastlowast

1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2

120573=0(119901lowast 119878min=119901lowastlowast

1198780=infin)

minusinfin

(Lin)

B3

120573lt0(119901lowast 119878min=119901lowastlowast

1198780lt0)

0(M

in)

C1205731015840lt120573

C1

120573gt0(119901lowast 119878max=119901lowastlowast

1198780gt0gt119901lowast 1198780=119901lowast 119878min)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2

120573=0(119901lowast 119878min=119901lowast 1198780=119901lowastlowast

1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3


1198780lt0lt119901lowast 1198780=119901lowast 119878max)

0(M

in)

D1205731015840=0

D1

120573gt0(119901lowast 119878min=119901lowast 1198780lt0lt119901lowastlowast

1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced

ifferenceΔ119901for119903119873

=0



Δlowast119901min=Δlowast1199010lt

Δ1199010ltΔlowast119901max=Δlowastlowast1199010

Δlowast119901max=Δlowast1199010gt

Δ1199010gtΔlowast119901min=Δlowastlowast1199010


Δ1199010ltΔlowast119901min

A1205731015840gt120573

A1

1205731015840gt120573gt0(0gtΔlowastlowast1199010=Δlowast119901maxgtΔlowast1199010=Δlowast119901min)

+infin+infin

+infin(M

in)

minusinfin

minusinfin

1205731015840gt120573gt0(Δlowast1199010=Δlowast119901maxgtΔlowastlowast1199010=Δlowast119901mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

1205731015840gt120573gt0(Δlowast119901max=Δlowast119901min=Δlowast1199010=Δ119901lowastlowast

0=0)

+infinΔlowast1199010

minusinfin

A2

1205731015840gt120573=0

(Δlowast119901max=Δlowast119901min=Δlowastlowast1199010=Δlowast1199010gtor=orlt0)


minusinfin

A3

1205731015840gt0gt120573(0ltΔlowast119901min=Δlowast1199010ltΔ119901lowastlowast

0=Δlowast119901max)

+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin

1205731015840gt0gt120573(Δlowast119901max=Δlowast1199010=Δlowastlowast1199010=Δlowast119901min=0)


minusinfin

A4

0gt1205731015840gt120573(Δlowast119901min=Δlowast1199010lt0ltΔlowast119901max=Δlowastlowast1199010)

00

0(M

ax)

00

A5

0gt1205731015840gt120573(Δlowast119901min=Δlowast1199010gt0gtΔlowast119901max=Δlowastlowast1199010)

00

00

0

B1205731015840=120573

B1

120573gt0 (Δ119901lowast min=119901lowastlowast

1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2

120573=0(Δlowast119901min=119901lowastlowast

1198780=infin)

minusinfin

(Lin)

B3

120573lt0(Δlowast119901min=119901lowastlowast

1198780lt0)

00

0

C1205731015840lt120573

C1

120573gt0(Δlowast1199010=Δlowast119901minltΔlowastlowast1199010=Δlowast119901maxlt0)

minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2

120573=0 (Δlowast119901max=Δlowast1199010=Δlowast119901min=Δlowastlowast1199010lt0)

Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3

120573lt0(Δlowast1199010=Δlowast119901minltΔlowastlowast1199010=Δlowast119901maxlt0)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1

120573gt0(Δlowast119901min=Δlowast1199010lt0ltΔlowastlowast1199010=infin)

minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3

120573lt0 (Δlowast1199010=Δlowast119901mingtor=orlt0Δlowastlowast1199010=infin)

Δ1199010minusΔlowast1199010




The extraction rate 120572 can be obtained from (65) on thebase of the mass flow rates of extraction in two successiveyears

120572 =ln1198661015840119878

119905 (67)

The elimination of the time variable 119905 between (65) and(66) allows obtaining the relation called supply curve ofextracted resources between the dimensionless price1199011015840

119864 and

the dimensionless mass flow rate 1198661015840119864 of extracted resources

1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)

where the new variable 119910

119910 =119903119873

120572 (69)

is called rate of interest of nonextracted resources on theextraction rate RINE

The variation of 1199011015840119864with 119866

1015840

119864is presented in Figure 17

where the only variable affecting the evolution is 119910 Considerthe following

(i) for 119910 = 0 (ie 119903119873

= 0 or 120572 = plusmninfin) (68) is anequilateral hyperbole 1199011015840

119864decreases with the increase

of 1198661015840119864or increases with the decrease of 1198661015840

119864

(ii) for 0 lt 119910 lt 1 1199011015840119864decreases with the increase of 1198661015840

119864

(iii) for 119910 = 1 (ie 120572 = 119903119873) 1199011015840119864is constant with the

increase of 1198661015840119864

(iv) for 119910 gt 1 1199011015840119864increases with the increase of 1198661015840

119864

(v) for 119910 = plusmninfin (or 120572 = 0) (68) is a vertical line 1199011015840119864

increases at constant 1198661015840119864= 1

(vi) for 119910 lt 0 1199011015840119864increases with the decrease of 1198661015840

119864

72 Supply Curve of Sold Resources The price evolution withtime of sold resources becomes in dimensionless form

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) minus 1199011015840lowast1198780[exp (1205731015840119905) minus exp (120573119905)] (70)

where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)

is the dimensionless critical initial price of sold resourcesDCIPS and the new variable 119909

119909 =119903119864

120572 (72)

is the rate of interest of sold resources on the extraction rateRISE

For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E

y = 0 (120572 = minusinfin)

G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)

Figure 17 Dimensionless price of extracted resources 1199011015840119864 versus

1198661015840

119864

An extreme value of 1199011015840119878is present for

1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]

[(119909 minus 1) (1 minus 1199011015840lowast

1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780

The maximum of 1199011015840119878is present at 1198661015840

119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast

1198780=119901lowastlowast

1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)

which is the dimensionless critical initial price extreme ofsold resources DCIPES

721 Supply Curve of Sold Resources for 119903119873= 0 That Is 1199011015840lowast

1198780=

0 For 119903119873= 0 that is 1199011015840lowast

1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)

which combined with (65) gives

1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)


119878is presented in Figure 18 for

119910 = 0 (ie 119903119873= 0)

The only variable affecting the evolution is 119909 Considerthe following

(i) for 119909 = 0 (ie 119903119864= 0 or 120572 = plusmninfin) (78) is an

equilateral hyperbole 1199011015840119878decreases with the increase


119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S

x = 0 (120572 = minusinfin)

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)

Figure 18 Dimensionless price of selling resources 1199011015840119878 versus 1198661015840

119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)

Figure 19 Dimensionless price of selling resources 1199011015840119878versus 1198661015840

119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)

(iii) for 119909 = 1 (ie 120572 = 119903119864) 1199011015840119878is constant with the increase

of 1198661015840119878


119878




119878

722 Supply Curve of Selling Resources for 1199011015840lowast1198780= 1 For 1199011015840lowast

1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)

and the variation of 1199011015840119878is only dependent on 119910 RINE


119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if

1199011198640= 1199011198780= 1) is presented in Figure 19 The only variable

involved in the evolution is 119910 Consider the following(i) for 119910 = 0 (ie 119903

119864= 0 or 120572 = plusmninfin) (79) is an



119878


119878

(iii) for 119910 = 1 1199011015840119878is constant with the increase of 1198661015840

119878


119878 A

special case is119910 = 2when the variation of1199011015840119878is a linear

increase with the increase of 1198661015840119878




119878

723 Supply Curve of Sold Resources for |119909| gt |119910| gt 1 Thevariation of 1199011015840

119878with 1198661015840

119878is presented in Figure 20 for |119909| gt

|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)

The variables involved in the evolution are here two thatis 1199011015840lowast1198780and 1199011015840lowastlowast

1198780 The main role is played by 1199011015840lowast

1198780when 1199011015840lowast

1198780le 1

but when 1199011015840lowast1198780gt 1 it is determinant the second variable 1199011015840lowastlowast

1198780

Consider the following

(i) for 1199011015840lowast1198780

le 1 1199011015840119878increases with the increase or the

decrease of 1198661015840119878

(ii) for 1199011015840lowast1198780gt 1 gt 119901

1015840lowastlowast

1198780 1199011015840119878increases up to a maximum

and then decreases with the increase or the decreaseof 1198661015840119878

(iii) for 1199011015840lowastlowast1198780

= 1 1199011015840119878decreases with the increase or the

decrease of 1198661015840119878and the maximum is at 1198661015840

119878= 1

(iv) for 1199011015840lowast1198780gt 1199011015840lowastlowast

1198780gt 1 1199011015840

119878decreases with the increase or

the decrease of 1198661015840119878

724 Supply Curve of Sold Resources for |119910| gt |119909| gt 1 or 119910 gt1 gt 119909 The variation of 1199011015840

119878with 1198661015840


for |119910| gt |119909| gt 1 (ie 119903119873gt 119903119864gt 120572 and 1199011015840lowast

1198780lt 0 lt 119901

1015840lowastlowast

1198780)

The only variable influencing the evolution is 1199011015840lowastlowast

1198780

because 1199011015840lowast1198780lt 0 Consider the following

(i) for 1199011015840lowastlowast1198780

lt 1 1199011015840119878increases up to a maximum and then

decreases with the decrease or the increase of 1198661015840119878

(ii) for 1199011015840lowastlowast1198780

= 1 1199011015840119878decreases with the decrease or the

increase of 1198661015840119878and the maximum is at 1198661015840

119878= 1


gt 1 1199011015840119878decreases with the increase or the

decrease of 1198661015840119878


119878is presented in Figure 21 also

for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The

price 1199011015840119878decreases with the increase of 1198661015840

119878

The conclusion of Figure 21 is that 1199011015840119878cannot increase

indefinitely for |119910| gt |119909| gt 1 or 119910 gt 1 gt 119909

725 Supply Curve of Sold Resources for 119910 = 1 or 119909 = 1 Thevariation of 1199011015840

119878with 1198661015840

119878is presented in Figure 22 for 119910 = 1

(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10

x lt y lt 0 x gt y gt 1|x| gt |y| gt 1 (120572 lt rN lt rE)

p998400lowastS0 gt p998400lowastlowast

S0 gt 0

p998400lowastS0 = 1

p998400lowastS0 lt 1

p998400lowastS0 = 1

p998400lowastS0 = 1

p998400lowastS0 gt 1 gt p998400lowastlowast

S0


S0 = 1p998400lowastS0 gt 1 gt p998400lowastlowast

S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast

S0 gt p998400lowastlowastS0 = 1


119878 for |119909| gt |119910| gt 1

p998400S

1

1

0

y lt x lt 0 |y| gt |x| gt 1 (120572 lt rE lt rN) p998400lowastS0 lt 0 lt p998400lowastlowast

S0

p998400lowastlowastS0 lt 1


p998400lowastlowastS0 = 1

p998400lowastlowastS0 = 1p998400lowastlowast

S0 gt 1

p998400lowastlowastS0 gt 1

G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

0 gt p998400lowastS0 gt p998400lowastlowast

S0

y gt 1 gt x (rE lt 120572 lt rN)

Figure 21 Dimensionless price of sold resources 1199011015840119878 1198661015840119878 for |119910| gt |119909| gt 1 or 119910 gt 1 gt 119909

Themain variable influencing the evolution is 1199011015840lowastlowast1198780

= 1199011015840lowast

1198780


(i) for 119909 gt 119910 = 1 (ie 120572 = 119903119873lt 119903119864and 1199011015840lowastlowast

1198780= 1199011015840lowast

1198780gt

0) 1199011015840119878increases with the increase of 1198661015840

119878if 1199011015840lowast1198780lt 1 1199011015840

119878

remains constant with1198661015840119878if1199011015840lowast1198780= 11199011015840

119878decreases with

1198661015840

119878if 1199011015840lowast1198780gt 1

(ii) for 119909 lt 119910 = 1 (ie 120572 = 119903119873gt 119903119864and 1199011015840lowastlowast

1198780= 1199011015840lowast

1198780lt 0)

1199011015840

119878decreases asymptotically to 1199011015840lowast

1198780with the increase

of 1198661015840119878

The evolution of 1199011015840119878with 1198661015840


for 119909 = 1 lt 119910 (ie 119903119873gt 119903119864= 120572 and 1199011015840lowast

1198780lt 0 1199011015840lowastlowast

1198780= +infin)

The price 1199011015840119878decreases with the increase of 1198661015840

119878

726 Supply Curve of Sold Resources for 119910 lt 1 The variationof 1199011015840119878with1198661015840

119878is presented in Figure 23 for 119910 lt 1 (ie 120572 gt 119903

119873)

The main variables influencing the evolution are 1199011015840lowast1198780and

1199011015840lowastlowast

1198780 Consider the following

(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)

five cases are possible1199011015840119878increaseswith1198661015840

119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)

p998400lowastS0 gt 1

x gt y = 1 (120572 = rN lt rE) p998400lowastS0 = p998400lowastlowast

S0 gt 0

x = 1 lt y (120572 = rE lt rN) p998400lowastS0 lt 0 p998400lowastlowast

S0 = infin

x lt y = 1 (120572 = rN gt rE) p998400lowastS0 = p998400lowastlowast

S0 lt 0

p998400lowastS0

Figure 22 Dimensionless price of sold resources 1199011015840119878 versus 1198661015840

119878 for 119910 = 1 or 119909 = 1

p998400S

1

01

1 minus p998400lowastS0




p998400lowastS0 le 1


p998400lowastlowastS0 gt 1 gt p998400lowast

S0

G998400S

p998400lowastlowastS0 gt p998400lowast

S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)

1 gt x gt y(120572 gt rE gt rN) p998400lowastS0 gt 0 gt p998400lowastlowast

S0

t0 + 0



0 gt p998400lowastlowastS0 gt p998400lowast

S0


S0 gt 1

x gt 1 gt y (rE gt 120572 gt rN)

x = 1 gt y (rE = 120572 gt rN)

1 gt y gt x (120572 gt rN gt rE)

p998400lowastS0 gt 0 p998400lowastlowast

S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0

p998400lowastlowastS0 lt 1 p998400lowastlowast

S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)

x = y = 1 (120572 = rE = rN) p998400lowastS0 = p998400lowastlowast

S0 = infin

x = y lt 1 (120572 gt rE = rN) p998400lowastlowastS0 lt 0 p998400lowast

S0 = infin

|x| = |y| gt 1 (120572 lt rE = rN) p998400lowastlowastS0 gt 0 p998400lowast

S0 = infin


119878 for 119909 = 119910

1199011015840

119878increases with 119866

1015840

119878 after a minimum if 1199011015840lowastlowast

1198780gt

1 gt 1199011015840lowast

1198780 1199011015840119878decreases asymptotically to +0 with 1198661015840

119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840

119878decreases asymptotically to (1 minus 1199011015840lowast

1198780) with

the increase of 1198661015840119878

(iii) for 1 gt 119909 gt 119910 (ie 120572 gt 119903119864gt 119903119873and 1199011015840lowast

1198780gt 0 gt 119901

1015840lowastlowast

1198780)

two cases are possible 1199011015840119878decreases asymptotically to

+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840

119878decreases asymptotically to

minus0 with 1198661015840119878 after a minimum if 1199011015840lowast

1198780gt 1

(iv) for 1 gt 119910 gt 119909 (ie 120572 gt 119903119873gt 119903119864and 0 gt 119901

1015840lowastlowast

1198780gt

1199011015840lowast

1198780) 1199011015840119878decreases asymptotically to minus0 with 1198661015840

119878after a

minimum

727 Supply Curve of Sold Resources for 119909 = 119910 (ie 119903119873=119903119864)

Theprice evolution of sold resources for 119909 = 119910 (ie 119903119873= 119903119864)

can be written in dimensionless form as

1199011015840

119878=119901119878

1199011198780

= (1 minus 1199011015840lowastlowast

1198780120573119905) exp (120573119905) (80)

Using (65) (80) becomes finally

1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[

1 minus 1199011015840lowastlowast

1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)

The maximum value of 1199011015840119878corresponds to

1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)


The price evolution of sold resources for 119909 = 119910 = 1 (ie119903119873= 119903119864= 120572) becomes in dimensionless form

1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)



119909 = 119910 (ie 119903119873= 119903119864) Consider the following

(i) for |119909| = |119910| gt 1 (ie 119903119864= 119903119873gt 120572 and 1199011015840lowastlowast

1198780gt 0

1199011015840lowast

1198780= +infin) the only variable affecting the evolution

is 1199011015840lowastlowast1198780

gt 0 The price 1199011015840119878decreases with the increase

or the decrease of 1198661015840119878for 1199011015840lowastlowast1198780

ge 1 The price 1199011015840119878

increases up to a maximum and then decreases withthe decrease or the decrease of 1198661015840

119878for 1199011015840lowastlowast1198780

lt 1(ii) for 119909 = 119910 lt 1 (ie 119903

119864= 119903119873lt 120572 and 1199011015840lowastlowast

1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840


minus0 with the increase of 1198661015840119878after a negative minimum


10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0

0 gt Δp0 gt Δlowastlowastp0Δp0 = Δlowastlowastp0

Δlowastlowastp0 gt Δp0 gt Δlowastp00 gt Δp0 gt Δlowastlowastp0

Δp0 = Δlowastlowastp0

Δlowastlowastp0 gt Δp0 gt Δlowastp0

Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y = 0 (rN = 0) p998400lowastS0 = p998400lowastlowast

S0 = 0 Δlowastp0 = minuspE0 lt 0

|x| gt 1 Δlowastlowastp0 lt 0

Δp0 = Δlowastp0Δlowastp0

Δp0 = Δlowastp0

Δlowastlowastp0

Figure 25 Price difference versus 1198661015840119878 for 119910 = 0 and |119909| gt 1

(iii) for119909 = 119910 = 1 (ie 119903119864= 119903119873= 120572 and1199011015840lowast

1198780= 1199011015840lowastlowast

1198780= +infin)

the price 1199011015840119878decreases with the increase of 1198661015840

119878

8 Supply Curve of the Difference betweenSelling and Extracted Resources

The difference Δ119901 between the price of selling and extractedresources is given by

Δ119901 = 119901119878minus 119901119864= (Δ119901

0minus Δlowast1199010) 1198661015840(119909minus1)

119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)

is the critical initial price difference CIPD The price differ-ence Δ119901 has an extreme for

1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]

[(119909 minus 1) (1 minus Δ1199011015840lowast

1198780)]

1(119909minus119910)

(88)

which is equal to 1 for Δ1199010equal to

Δlowastlowast1199010= Δlowast1199010

(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)

which is the critical extreme of the initial price differenceCEIPD The relative evolutions of Δ119901 with 1198661015840

119878are presented

in Figures 25ndash28

81 Supply Curve of Δ119901 for 119910 = 0 and |119909| gt 1 Figure 25presents the price difference Δ119901 for 119910 = 0 (ie 119903

119873= 0) and

|119909| gt 1 (ie 120572 lt 119903119864 Δlowast1199010= minus1199011198640lt 0 and Δlowast119901

0lt Δlowastlowast1199010lt

0) Consider the following(i) for Δ119901

0gt 0 the price difference Δ119901 increases with

the increase or the decrease of 1198661015840119878 that is for 1198661015840

119878lt 1

or 1198661015840119878gt 1 For 1198661015840

119878lt 1 the difference Δ119901 increases

exponentially while the difference Δ119901 increases at alower rate for 1198661015840

119878gt 1

(ii) for Δ1199010in the range from 0 to Δlowastlowast119901

0 that is Δlowastlowast119901

0lt

Δ1199010lt 0 the price difference Δ119901 increases with

the increase or the decrease of 1198661015840119878 The increase is

exponential for 1198661015840119878lt 1 while having a lower rate of

increase for 1198661015840119878gt 1

(iii) for Δ1199010= Δlowastlowast1199010 the price difference Δ119901 increases

with the increase or the decrease of 1198661015840119878 For 1198661015840

119878lt 1

the difference Δ119901 increases exponentially with theminimum at 1198661015840

119878= 1 while the increase has a lower

rate for 1198661015840119878gt 1

(iv) for Δ1199010in the range from Δ

lowast1199010to Δlowastlowast119901

0 that is

Δlowast1199010lt Δ119901

0lt Δlowastlowast1199010 the price difference Δ119901

increases exponentially after a minimum with thedecrease of1198661015840

119878 For1198661015840

119878gt 1 the differenceΔ119901 increases

with a lower rate(v) for Δ119901

0= Δlowast1199010 the price difference Δ119901 increases

asymptotically towards minus0 for 1198661015840119878gt 1 while Δ119901

decreases exponentially for 1198661015840119878lt 1

82 Supply Curve of Δ119901 for 119910 = 0 and 0 lt 119909 le 1 The pricedifference Δ119901 for 119910 = 0 (ie 119903

119873= 0 Δlowast119901

0= minus1199011198640lt 0) and

0 lt 119909 le 1 (ie 120572 ge 119903119864gt 0) has the following trends

(i) for 119909 = 1 (ie 120572 = 119903119864and Δlowastlowast119901

0= minusinfin) the price

difference Δ119901 increases asymptotically towards 1199011198780

for any value of Δ1199010(ie Δ119901

0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)

(ii) for 119909 lt 1 (ie 120572 gt 119903119864gt 0 and Δlowastlowast119901

0gt 0) the price

difference Δ119901 tends to 0 for any value of Δ1199010


1

Δp

0

x lt 0 x gt 0

G998400S

|x| gt 2|y| ge 1 (rE gt 2rN ge 120572) plowastS0 gt 0 Δlowastp0 lt Δlowastlowastp0 lt 0

y gt 1 gt x (rN gt 120572 gt rE) plowastS0 lt 0 Δlowastlowastp0 lt Δlowastp0 lt 0

Δp0 gt 0

Δp0 gt Δlowastlowastp0



Δp0 gt 0 Δp0 gt Δlowastlowastp0



G998400S gt 1 (120572 gt 0)Δlowastlowastp0

Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0

y gt x = 1 (rN gt 120572 = rE) plowastS0 lt

lowastp0 lt 00 Δ

Figure 26 Price difference versus 1198661015840119878 for |119909| gt 2|119910| ge 1 or 119910 gt 119909 = 1 or 119910 gt 1 gt 119909

83 Supply Curve of Δ119901 for |119909| gt 2|119910| ge 1 or 119910 gt 119909 = 1 or119910 gt 1 gt 119909 Figure 26 presents the price difference Δ119901 for|119909| gt 2|119910| ge 1 (ie 119903

119864gt 2119903119873ge 120572 and Δlowast119901

0lt Δlowastlowast1199010lt 0)

119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903

119864and Δlowast119901

0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903


0lt Δlowast1199010lt 0 Consider the following

(i) for |119909| gt 2|119910| ge 1 and Δ1199010gt 0 or Δ119901

0gt Δlowastlowast1199010

the price difference Δ119901 increases exponentially withthe increase or the decrease of 1198661015840

119878 For Δ119901

0= Δlowastlowast1199010

the price differenceΔ119901 increases exponentially witha minimum at 1198661015840

119878= 1 For Δlowastlowast119901

0gt Δ119901

0gt Δlowast1199010

the price difference Δ119901 increases exponentially aftera minimum with the decrease or the increase of 1198661015840

119878

For Δ1199010= Δlowast1199010the price difference Δ119901 decreases

exponentially with the decrease or the increase of 1198661015840119878

Figure 26 presents also the price difference Δ119901 for 119910 gt

119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903


0lt Δlowast1199010lt 0) The price difference Δ119901

decreases exponentially with 1198661015840119878

84 Supply Curve of Δ119901 for |119910| gt 1 and |119909| = 2119910 The pricedifference Δ119901 for |119910| gt 1 and |119909| = 2|119910| (ie 120572 lt 119903

119873 119903119864=

2119903119873 and Δlowast119901

0= Δlowastlowast1199010= 0) has the following trends For

Δ1199010gt 0 the price difference Δ119901 increases exponentially with

the increase or the decrease of 1198661015840119878 For Δ119901

0lt Δlowast1199010the price

differenceΔ119901 decreases exponentially with the increase or thedecrease of 1198661015840

119878

85 Supply Curve of Δ119901 for |119910| gt 1 and |119909| lt 2119910 Figure 27presents the price difference Δ119901 for |119910| gt 1 and |119909| lt 2|119910|

(ie 120572 lt 119903119873 119903119864lt 2119903119873and Δlowast119901

0gt Δlowastlowast1199010gt 0)

For Δ1199010

ge Δlowast1199010the price difference Δ119901 increases

exponentially with the increase or the decrease of 1198661015840119878 For

Δlowast1199010gt Δ1199010gt Δlowastlowast1199010the price differenceΔ119901 increases up to a

maximumand then decreases exponentially with the increaseor the decrease of 1198661015840

119878 For Δ119901

0= Δlowastlowast1199010the price difference

Δ119901 decreases exponentially with the increase or the decreaseof 1198661015840119878and the maximum is at 1198661015840

119878= 1 For Δ119901

0lt Δlowastlowast1199010the

price difference Δ119901 decreases exponentially with the increaseor the decrease of 1198661015840

119878

86 Supply Curve ofΔ119901 for 119910 = 1 Theprice differenceΔ119901 for119910 = 1 (ie 120572 = 119903

119873) has the following trends

(i) for 119909 gt 119910 = 1 (ie 0 lt 120572 = 119903119873lt 119903119864and Δlowast119901

0=

Δlowastlowast1199010) three cases are possible For Δ119901

0gt Δlowast1199010

the price difference Δ119901 increases with the increase of1198661015840

119878 for Δ119901

0= Δlowast1199010the price difference Δ119901 remains

constant with the increase of 1198661015840119878 for Δ119901

0lt Δlowast1199010the

price difference Δ119901 decreases with the increase of 1198661015840119878

(ii) for 119909 lt 119910 = 1 (ie 120572 = 119903119873gt 119903119864and Δlowast119901


the price difference Δ119901 decreases with the increase of1198661015840

119878

87 Supply Curve of Δ119901 for 119910 lt 1 The price difference Δ119901 for119910 lt 1 (ie 120572 gt 119903

119873) and Δlowastlowast119901

0gt Δlowast1199010gt 0 has the following

trends For Δ1199010gt Δlowastlowast1199010the price difference Δ119901 increases

with the increase of1198661015840119878 For Δ119901


Δ119901 increases with the increase of 1198661015840119878and the minimum is at

1198661015840


0gt Δ119901

0gt Δlowast1199010the price difference Δ119901

increases with the increase of1198661015840119878after a minimum For Δ119901

0=

Δlowast1199010the price difference Δ119901 decreases asymptotically to +0

with the increase of 1198661015840119878 For Δ119901

0lt Δlowast1199010the price difference

Δ119901 decreases with the increase of 1198661015840119878

The price difference Δ119901 for 119910 lt 1 (ie 120572 gt 119903119873gt 0)

and Δlowastlowast1199010lt Δlowast1199010lt 0 has the following trends For

Δ1199010gt 0 gt Δ

lowast1199010the price difference Δ119901 increases with

the increase of 1198661015840119878 For Δ119901

0= Δlowast1199010the price difference Δ119901

increases asymptotically to minus0 with the increase of 1198661015840119878 For

Δlowast1199010gt Δ119901

0gt Δlowastlowast1199010the price difference Δ119901 increases up

to a maximum and then decreases with the increase of 1198661015840119878


Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0

Δp0 gt Δlowastp0Δp0 gt Δlowastp0

Δp0 = Δlowastp0

Δlowastlowastp0 lt Δp0 lt Δlowastp0Δlowastp0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0) Δlowastlowastp0 lt Δp0 lt Δlowastp0

Δp0 = Δlowastlowastp0 Δp0 lt Δlowastlowastp0 Δp0 = Δlowastlowastp0

1

(rE lt 2rN 120572 lt rN) plowastS0 gt 0 Δlowastp0 gt Δlowastlowastp0 gt 01 lt |x| lt 2|y| |y| gt 1

Figure 27 Price difference versus 1198661015840119878 for |119910| gt 1 and |119909| lt 2|119910|

0

1

Δp

G998400S

x lt 0 (120572 lt 0 lt rE) plowastlowastS0 gt 0 x lt 1 (120572 gt rE) p

lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0

Δp0 gt plowastlowastS0

Δp0 = plowastlowastS0

Δp0 lt plowastlowastS0

Figure 28 Price difference versus 1198661015840119878 for 119909 = 119910

For Δ1199010le Δlowastlowast1199010the price difference Δ119901 decreases with the

increase of 1198661015840119878

The price difference Δ119901 for the other three cases whichare possible according to the relation between 119909 and 119910 (ie 120572and 119903119864) has the following trends

(i) for 1 ge 119909 gt 119910 (ie 120572 ge 119903119864gt 119903119873) the price difference

Δ119901 decreases asymptotically to +0 with the increaseof 1198661015840119878

(ii) for 1 gt 119910 gt 119909 (ie 120572 gt 119903119873gt 119903119864) the price difference

Δ119901 decreases asymptotically to minus0 after a negativeminimum with the increase of 1198661015840

119878

88 Supply Curve of Δ119901 for 119909 = 119910 The price difference Δ119901is in dimensionless form

Δ119901 = 119901119878minus 119901119864= (Δ119901

0minus 119901lowastlowast

11987801205731015840119905) exp (1205731015840119905) (90)

For 119909 = 119910 the critical initial price difference Δlowast1199010and the

critical initial extreme price difference Δlowastlowast1199010are not defined

and the only critical price defined is the critical initial priceextreme of the sold resources 119901lowastlowast

1198780

The price difference Δ119901 as a function of 1198661015840119878 is then

Δ119901 = [Δ1199010minus 119901lowastlowast

1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)


The price difference Δ119901 has an extreme for

1198661015840

119878= exp[

(Δ1199010minus 119901lowastlowast

1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)

which is equal to 1 if Δ1199010is equal to


0 (93)

Figure 28 presents the price difference Δ119901 for 119909 = 119910 (ie119903119873= 119903119864) Consider the following

(i) for 119909 lt 0 (ie 120572 lt 0 lt 119903119864and 119901lowastlowast

0gt 0) three cases

are possible with the decrease of 1198661015840119878 For Δ119901


0

the price difference Δ119901 decreases after a maximumfor Δ119901


0the price difference Δ119901 decreases with

the maximum at 1198661015840119878= 1 for Δ119901


0the price

difference Δ119901 decreases exponentially(ii) for 119909 lt 1 (ie 120572 gt 119903


0lt 0) three cases are

possible with the increase of 1198661015840119878 For Δ119901

0gt 0 gt 119901

lowastlowast

0

the price differenceΔ119901 decreases asymptotically tominus0after a negative minimum for 0 gt Δ119901

0gt or = or lt

119901lowastlowast

0the price difference Δ119901 increases asymptotically

to minus0(iii) for 119909 = 119910 = 1 (ie 120572 = 119903

119864= 119903119873) the price difference

Δ119901 which has the following dependence on 1198661015840119878

Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)

decreases with the increase of 1198661015840119878

Nomenclature

Latin

119866119896= 119866119901119903 Capital flow rate

119866 Mass flow rate119870 = 119866119901 Capital flow119872 Mass119901 Price119901lowast

1198780 Critical initial price of selling

resources CIPS119901lowastlowast

1198780 Critical initial extreme price of

selling resources CIPES1199011015840lowast

1198780 Dimensionless critical initial price

of sold resources DCIPS1199011015840lowastlowast


extreme of sold resources DCIPES119903 Interest rate (annumminus1)119905 Time119909 = 119903119864120572 Rate of interest of sold resources

on extracted rate RISE119910 = 119903119873120572 Rate of interest of nonextracted

resources on extracted rate RINE

Greek120572 = (1119866)(119889119866119889119905) Extraction rate120573 = 119903119873minus 120572 Price-increase factor of extracted

resources PIFE

1205731015840= 119903119864minus 120572 Price-increase factor of selling

resources PIFSΔ119901 = 119901

119878minus 119901119864 Difference between price of soldand extracted resources

Δlowast1199010 Critical initial price difference

CIPDΔlowastlowast1199010 Critical extreme initial price

difference CEIPDSubscript0 Initial119888 Critical119864 Extracted resources119891 Final119896 Capital119898 Maximum or minimummax Maximummin Minimum119873 Nonextracted resources119878 Selling resources

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] H Hotelling ldquoThe economics of exhaustible resourcesrdquo Journalof Political Economy vol 39 no 2 pp 137ndash175 1931

[2] P S Dasgupta andGMHealEconomicTheory and ExhaustibleResources Cambridge University Press 1979

[3] P P McCall ldquoEnergy models and forecasts a userrsquos point ofviewrdquo in Proceedings of the Conference on Energy Model andForecast pp 159ndash165 Berkeley Calif USA June 1974

[4] J G Tewksbury ldquoLong range forecasting of foreign oil pricetechnique and resultsrdquo in Proceedings of the AIME Symposiumon Petroleum Economics and Valuation pp 89ndash96 Society ofPetroleum Engineers Dallas Tex USA February 1977

[5] R S Pindyck ldquoHigher energy prices and the supply of naturalgasrdquo Energy Systems and Policy vol 2 no 2 pp 177ndash209 1978

[6] M A Adelman ldquoInternational oilrdquo Natural Resources Journalvol 18 no 4 pp 725ndash730 1978

[7] H DuMoulin and E V Newland ldquoForetelling the energyfuturerdquo in Proceedings of the National Meeting of the AmericanChemical Society pp 21ndash31 Honolulu Hawaii USA April 1979

[8] D Pearce ldquoWorld energy demand and crude oil prices to theyear 2000rdquo Journal of Agricultural Economics vol 32 no 3 pp341ndash354 1981

[9] M W French ldquoEnergy forecast for 1983ndash1993rdquo Energy Eco-nomics Policy and Management vol 3 no 3 pp 4ndash9 1983

[10] M L Roberts ldquoCrude oil pricesmdashthe directionrdquo Journal ofPetroleum Technology vol 36 no 5 pp 811ndash816 1984

[11] J M Campbell and R A Hubbard ldquoPrice forecasting andproject evaluation in the 1980rsquosrdquo Journal of Petroleum Technol-ogy vol 36 no 5 pp 817ndash825 1984

[12] S H Clark ldquoUS natural gas price forecastingrdquo in Proceedings ofthe American Institute of Chemical Engineers Summer NationalMeeting p 13 Philadelphia Pa USA 1984


[13] A David and G P Carson ldquoPrice forecast for energy lendingrdquoinProceedings of the Eastern RegionalMeeting pp 51ndash55 Societyof Petroleum Engineers of AIME CharlestonWVa USA 1984

[14] L Lorentsen andK Roland ldquoModelling the crude oilmarket oilprices in the long termrdquo in Bridge between Control Science andTechnology Biomedical Applications Water Resources Environ-ment Energy Systems Development Social Effects Proceedingsof the 9th Triennial World Congress of IFAC vol 6 of IFACProceedings Series pp 3309ndash3312 SWIIS Education BudapestHungary 1985

[15] T R Curlee ldquoForecasting world oil prices the evolution ofmodeling methodologies and summary of recent projectionsrdquoMaterials and Society vol 9 no 4 pp 413ndash442 1985

[16] A Anon ldquoDeclining oil prices have not resulted in increaseddemand decreased supplyrdquo Energy vol 11 no 5 pp 4ndash6 1986

[17] J P Dielwart and F C Coles ldquoCanadian petroleum pricesand demandmdashan overviewrdquo in Proceedings of the 37th AnnualTechnical Meeting of the Petroleum Society of CIM vol 2 pp231ndash242 Alberta Canada 1986

[18] G W Yohe ldquoEvaluating the efficiency of long-term forecastswith limited information Revisions in the IEW poll responsesrdquoResources and Energy vol 8 no 4 pp 331ndash339 1986

[19] P Sekera ldquoOil prices 1921 to 2121 (past present and future)rdquoin Proceedings of the 37th Annual Technical Meeting of thePetroleum Society of CIM vol 2 pp 247ndash291 Alberta Canada1986

[20] D Gately ldquoProspects for oil prices revisitedrdquo Annual Review ofEnergy vol 11 pp 513ndash538 1986

[21] K Holden D Peel and B Sandhu ldquoThe accuracy of OECDforecastsrdquo Empirical Economics vol 12 no 3 pp 175ndash186 1987

[22] E L Dougherty ldquoOil prices can we predict where they aregoingrdquo in Proceedings of the SPE Annual Technical Conferenceand Exhibition pp 147ndash157 Society of Petroleum Engineers ofAIME Dallas Tex USA 1987

[23] A Amano ldquoA small forecasting model of the world oil marketrdquoJournal of Policy Modeling vol 9 no 4 pp 615ndash635 1987

[24] J S Garces ldquoEffects of natural gas storage on gas price forecastsrdquoin Proceedings of the ASCE Energy Division Specialty Conferenceon Energy pp 59ndash64 Pittsburgh Pa USA March 1991

[25] J-P Angelier ldquoThe determinants of oil pricesrdquo Energy StudiesReview vol 3 no 3 pp 217ndash226 1991

[26] A E Bopp andGM Lady ldquoA comparison of petroleum futuresversus spot prices as predictors of prices in the futurerdquo EnergyEconomics vol 13 no 4 pp 274ndash282 1991

[27] B Abramson and A Finizza ldquoUsing belief networks to forecastoil pricesrdquo International Journal of Forecasting vol 7 no 3 pp299ndash315 1991

[28] L R Charleson and E J Weber ldquoEnergy forecasts for WesternAustralia 1992ndash2010rdquo Energy Economics vol 15 no 2 pp 111ndash122 1993

[29] F Fesharaki ldquoOil prices in a new lightrdquoHydrocarbon Processingvol 73 article 5 1994

[30] H G Huntington ldquoOil price forecasting in the 1980s what wentwrongrdquo Energy Journal vol 15 no 2 pp 1ndash22 1994

[31] I A Moosa and N E Al-Loughani ldquoUnbiasedness and timevarying risk premia in the crude oil futures marketrdquo EnergyEconomics vol 16 no 2 pp 99ndash105 1994

[32] B Abramson ldquoThe design of belief network-based systems forprice forecastingrdquo Computers amp Electrical Engineering vol 20no 2 pp 163ndash180 1994

[33] AM Skov ldquoAnalysis of forecasts of energy supply demand andoil pricesrdquo inProceedings of the SPEHydrocarbon Economics andEvaluation Symposium pp 239ndash253 Dallas Tex USA March1995

[34] B Abramson andA Finizza ldquoProbabilistic forecasts from prob-abilistic models a case study in the oil marketrdquo InternationalJournal of Forecasting vol 11 no 1 pp 63ndash72 1995

[35] D J Santini ldquoAn assessment of oil supply and its implicationsfor future pricesrdquo Natural Resources Research vol 7 no 2 pp101ndash121 1998

[36] M D Williams ldquoThe future of OPEC and crude oil pricesrdquoInternational Journal of Global Energy Issues vol 10 no 1 pp22ndash37 1998

[37] K Chaudhuri and B C Daniel ldquoLong-run equilibrium realexchange rates and oil pricesrdquo Economics Letters vol 58 no 2pp 231ndash238 1998

[38] R S Pindyck ldquoLong-run evolution of energy pricesrdquo EnergyJournal vol 20 no 2 pp 1ndash27 1999

[39] P Silvapulle and I A Moosa ldquoThe relationship between spotand futures prices evidence from the crude oil marketrdquo Journalof Futures Markets vol 19 no 2 pp 175ndash193 1999

[40] P P Alba and J-M Bourdaire ldquoTheoil pricerdquoRevue de lEnergieno 516 pp 209ndash217 2000

[41] Centre for Global Energy Studies (CGES) ldquoAnnual oil marketforecast and reviewrdquo Middle East Economic Survey vol 43 no5 pp A17ndashA18 2000

[42] ldquoEIA revises average oil price forecast for 2000 upward to $24brdquoPetrostrategies vol 14 no 667 pp 8ndash9

[43] A Hare ldquoGas to follow oil price rise through 2001 EIAforecastsrdquo Natural Gas Week vol 16 no 11 p 20 2000

[44] ldquoShell forecasts mid-teens average price for next 10 yearsrdquoMiddle East Economic Survey vol 43 no 22 p A12 2000

[45] ldquoUS Energy Department predicts WTI crude price of $30B-plus for remainder of yearrdquo Middle East Economic Survey vol43 no 38 pp A11ndashA12 2000

[46] T Walde and E M dos Santos ldquoComment on the ups (and pastand future downs) of the oil pricerdquo Revue de lEnergie no 520pp 453ndash459 2000

[47] A K Birky J D Maples J S Moore Jr and P D PattersonldquoFuture world oil prices and the potential for new transporta-tion fuelsrdquo Transportation Research Record no 1738 pp 94ndash992000

[48] B Yang ldquoPrediction of world crude oil price in 2000rdquo PetroleumProcessing and Petrochemicals vol 31 no 3 pp 63ndash65 2000

[49] P W MacAvoy and N V Moshkin ldquoThe new trend in the long-term price of natural gasrdquo Resource and Energy Economics vol22 no 4 pp 315ndash338 2000

[50] C Morana ldquoA semiparametric approach to short-term oil priceforecastingrdquo Energy Economics vol 23 no 3 pp 325ndash338 2001

[51] L Tang and S Hammoudeh ldquoAn empirical exploration of theworld oil price under the target zonemodelrdquo Energy Economicsvol 24 no 6 pp 577ndash596 2002

[52] J Alvarez-Ramirez M Cisneros C Ibarra-Valdez and ASoriano ldquoMultifractalHurst analysis of crude oil pricesrdquoPhysicaA StatisticalMechanics and Its Applications vol 313 no 3-4 pp651ndash670 2002

[53] M Ye J Zyren and J Shore ldquoForecasting crude oil spotprice using OECD petroleum inventory levelsrdquo InternationalAdvances in Economic Research vol 8 no 4 pp 324ndash333 2002

[54] P Sadorsky ldquoTime-varying risk premiums in petroleum futurespricesrdquo Energy Economics vol 24 no 6 pp 539ndash556 2002


[55] W M Fong and K H See ldquoA Markov switching model ofthe conditional volatility of crude oil futures pricesrdquo EnergyEconomics vol 24 no 1 pp 71ndash95 2002

[56] M Taal I Bulatov J Klemes and P Stehlık ldquoCost estimationand energy price forecasts for economic evaluation of retrofitprojectsrdquo AppliedThermal Engineering vol 23 no 14 pp 1819ndash1835 2003

[57] G Cortazar and E S Schwartz ldquoImplementing a stochasticmodel for oil futures pricesrdquo Energy Economics vol 25 no 3pp 215ndash238 2003

[58] J D Cabedo and I Moya ldquoEstimating oil price ldquovalue at riskrdquousing the historical simulation approachrdquo Energy Economicsvol 25 no 3 pp 239ndash253 2003

[59] G Burg S Richmond I Haine A Maurer and T WellsldquoEnergy prices to remain high in 2004rdquo Australian Commodi-ties vol 11 no 2 pp 275ndash283 2004

[60] G Burg S Richmond I Haine and A Maurer ldquoEnergy pricesremain high in 2004rdquo Australian Commodities vol 11 no 3 pp419ndash429 2004

[61] S Mirmirani and H C Li ldquoA comparison of VAR and neuralnetworks with genetic algorithm in forecasting price of oilrdquoAdvances in Econometrics vol 19 pp 203ndash223 2004

[62] S Abosedra and H Baghestani ldquoOn the predictive accuracy ofcrude oil futures pricesrdquo Energy Policy vol 32 no 12 pp 1389ndash1393 2004

[63] F Gori and C Takanen ldquoForecast of energy consumptionof industry and household amp services in Italyrdquo InternationalJournal of Heat and Technology vol 22 no 2 pp 115ndash121 2004

[64] C-S Feng J-C Wu and F Jiang ldquoForecasting the oil price byARFIMA modelrdquo Journal of University of Shanghai for Scienceand Technology vol 27 no 6 pp 539ndash542 2005

[65] N Movassagh and B Modjtahedi ldquoBias and backwardation innatural gas futures pricesrdquo Journal of Futures Markets vol 25no 3 pp 281ndash308 2005

[66] M Ye J Zyren and J Shore ldquoA monthly crude oil spotprice forecastingmodel using relative inventoriesrdquo InternationalJournal of Forecasting vol 21 no 3 pp 491ndash501 2005

[67] S Yousefi IWeinreich and D Reinarz ldquoWavelet-based predic-tion of oil pricesrdquo Chaos Solitons and Fractals vol 25 no 2 pp265ndash275 2005

[68] D-L Sun and J Lai ldquoOil price forecasting using neuralnetworks methodrdquo Journal of Petrochemical Universities vol 19no 2 pp 89ndash92 2006

[69] P Sadorsky ldquoModeling and forecasting petroleum futuresvolatilityrdquo Energy Economics vol 28 no 4 pp 467ndash488 2006

[70] M Ye J Zyren and J Shore ldquoShort-run crude oil priceand surplus production capacityrdquo International Advances inEconomic Research vol 12 no 3 pp 390ndash394 2006

[71] S Moshiri and F Foroutan ldquoForecasting nonlinear crude oilfutures pricesrdquo Energy Journal vol 27 no 4 pp 81ndash95 2006

[72] N Snow ldquoEIA world energy forecast reflects higher oil pricesrdquoOil amp Gas Journal vol 104 no 25 pp 34ndash35 2006

[73] S S Ghouri ldquoAssessment of the relationship between oil pricesand US oil stocksrdquo Energy Policy vol 34 no 17 pp 3327ndash33332006

[74] M Ye J Zyren and J Shore ldquoForecasting short-run crude oilprice using high- and low-inventory variablesrdquo Energy Policyvol 34 no 17 pp 2736ndash2743 2006

[75] F Gori ldquoMass and energy-capital conservation equations tostudy the price evolution of non-renewable energy resourcesmdashpart I generalization of the Hotelling rulerdquo Applied ThermalEngineering vol 26 no 14-15 pp 1746ndash1750 2006

[76] F Gori ldquoMass and energy-capital conservation equations tostudy the price evolution of non-renewable energy resourcesmdashpart II extension to resources sold to the marketrdquo AppliedThermal Engineering vol 26 no 14-15 pp 1751ndash1770 2006

[77] F Gori D Ludovisi and P F Cerritelli ldquoForecast of oil price andconsumption in the short termunder three scenarios paraboliclinear and chaotic behaviourrdquo Energy vol 32 no 7 pp 1291ndash1296 2007

[78] P K Narayan and S Narayan ldquoModelling oil price volatilityrdquoEnergy Policy vol 35 no 12 pp 6549ndash6553 2007

[79] T A Knetsch ldquoForecasting the price of crude oil via conve-nience yield predictionsrdquo Journal of Forecasting vol 26 no 7pp 527ndash549 2007

[80] P Dittrick ldquoDeloitte Sieminski forecasts $80bbl oil in 2008rdquoOil amp Gas Journal vol 105 no 48 pp 29ndash30 2007

[81] R MacAskie and C Jablonowski ldquoThe value of oil and gasprice forecasts in the presence of futuresrdquo Energy Explorationamp Exploitation vol 26 no 3 pp 157ndash174 2008

[82] H Askari and N Krichene ldquoOil price dynamics (2002ndash2006)rdquoEnergy Economics vol 30 no 5 pp 2134ndash2153 2008

[83] S Xu X Chen and A Han ldquoInterval forecasting of crude oilpricerdquo Advances in Soft Computing vol 46 pp 353ndash363 2008

[84] L Yu S Wang and K K Lai ldquoForecasting crude oil price withan EMD-based neural network ensemble learning paradigmrdquoEnergy Economics vol 30 no 5 pp 2623ndash2635 2008

[85] Y Fan Q Liang and Y-M Wei ldquoA generalized patternmatching approach for multi-step prediction of crude oil pricerdquoEnergy Economics vol 30 no 3 pp 889ndash904 2008

[86] S H Kang S-M Kang and S-M Yoon ldquoForecasting volatilityof crude oil marketsrdquo Energy Economics vol 31 no 1 pp 119ndash125 2009

[87] J D Hamilton ldquoUnderstanding crude oil pricesrdquo Energy Jour-nal vol 30 no 2 pp 179ndash206 2009

[88] J C Cuaresma A Jumah and S Karbuz ldquoModelling andforecasting oil prices the role of asymmetric cyclesrdquo EnergyJournal vol 30 no 3 pp 81ndash90 2009

[89] M Ye J Zyren C J Blumberg and J Shore ldquoA short-run crudeoil price forecast model with ratchet effectrdquo Atlantic EconomicJournal vol 37 no 1 pp 37ndash50 2009

[90] C W Cheong ldquoModeling and forecasting crude oil marketsusing ARCH-type modelsrdquo Energy Policy vol 37 no 6 pp2346ndash2355 2009

[91] A Ghaffari and S Zare ldquoA novel algorithm for predictionof crude oil price variation based on soft computingrdquo EnergyEconomics vol 31 no 4 pp 531ndash536 2009

[92] F Gori ldquoMass and energy-capital conservation equations tostudy price evolution of non-renewable energy resourcesmdashpartIII energy supply curverdquo Applied Thermal Engineering vol 29no 11-12 pp 2172ndash2186 2009

[93] F Gori ldquoMass and energy-capital conservation equations forthe time evolution of non-renewable energy resourcesrdquo inThermal Engineering Research Developments J Evgova and OKostadinov Eds chapter 5 pp 245ndash282 Nova Science NewYork NY USA 2009

[94] E G de Souza e Silva L F L Legey and E A de Souza e SilvaldquoForecasting oil price trends using wavelets and hiddenMarkovmodelsrdquo Energy Economics vol 32 no 6 pp 1507ndash1519 2010

[95] A W W He J T K Kwok and A T K Wan ldquoAn empiricalmodel of daily highs and lows ofWest Texas Intermediate crudeoil pricesrdquo Energy Economics vol 32 no 6 pp 1499ndash1506 2010


[96] J I Miller and S Ni ldquoLong-term oil price forecasts a newperspective on oil and the macroeconomyrdquo MacroeconomicDynamics vol 15 supplement 3 pp 396ndash415 2011

[97] S R Yaziz M H Ahmad L C Nian and N MuhammadldquoA comparative study on Box-Jenkins and Garch models inforecasting crude oil pricesrdquo Journal of Applied Sciences vol 11no 7 pp 1129ndash1135 2011

[98] G Prat and R Uctum ldquoModelling oil price expectationsevidence from survey datardquoQuarterly Review of Economics andFinance vol 51 no 3 pp 236ndash247 2011

[99] R Jammazi and C Aloui ldquoCrude oil price forecasting exper-imental evidence from wavelet decomposition and neuralnetwork modelingrdquo Energy Economics vol 34 no 3 pp 828ndash841 2012

[100] T Mingming and Z Jinliang ldquoA multiple adaptive waveletrecurrent neural network model to analyze crude oil pricesrdquoJournal of Economics and Business vol 64 no 4 pp 275ndash2862012

[101] C Baumeister and L Kilian ldquoReal-time forecasts of the realprice of oilrdquo Journal of Business amp Economic Statistics vol 30no 2 pp 326ndash336 2012

[102] A AzadehMMoghaddamMKhakzad andV EbrahimipourldquoA flexible neural network-fuzzy mathematical programmingalgorithm for improvement of oil price estimation and forecast-ingrdquoComputers amp Industrial Engineering vol 62 no 2 pp 421ndash430 2012

[103] J Wang H Pan and F Liu ldquoForecasting crude oil price andstock price by jump stochastic time effective neural networkmodelrdquo Journal of Applied Mathematics vol 2012 Article ID646475 15 pages 2012

[104] J J-W Hu Y-C Hu and R R-W Lin ldquoApplying neural net-works to prices prediction of crude oil futuresrdquo MathematicalProblems in Engineering vol 2012 Article ID 959040 12 pages2012

[105] J C Ruelke C Pierdzioch and G Stadtmann ldquoOn the internalconsistency of short-term medium-term and long-term oilprice forecastsrdquo Applied Economics vol 44 no 21 pp 2757ndash2765 2012

[106] F Gori ldquoForecasting the time evolution of oil price withnegative inflation raterdquo inProceedings of the ASME InternationalMechanical Engineering Conference and Exhibition (IMECE rsquo12)vol 6 pp 1491ndash1496 Houston Tex USA November 2012

[107] C Pierdzioch J-C Rulkeb and G Stadtmann ldquoOil priceforecasting under asymmetric lossrdquo Applied Economics vol 45no 17 pp 2371ndash2379 2013

[108] H Shin T Hou K Park C-K Park and S Choi ldquoPredictionof movement direction in crude oil prices based on semi-supervised learningrdquoDecision Support Systems vol 55 no 1 pp348ndash358 2013

[109] R Alquist L Kilian and R J Vigfusson ldquoForecasting the priceof oilrdquo inHandbook of Economic Forecasting vol 2 pp 427ndash5072013

[110] T Xiong Y Bao and Z Hu ldquoBeyond one-step-ahead fore-casting evaluation of alternative multi-step-ahead forecastingmodels for crude oil pricesrdquo Energy Economics vol 40 pp 405ndash415 2013

[111] C C Chang and T C Lai ldquoThe oil energy price cycle ineconomic activities a stochastic modelrdquo Energy Sources BEconomics Planning and Policy vol 8 no 4 pp 369ndash381 2013

[112] N Salehnia M A Falahi A Seifi and M H M AdelildquoForecasting natural gas spot prices with nonlinear modeling

using gamma test analysisrdquo Journal of Natural Gas Science andEngineering vol 14 pp 238ndash249 2013


[114] X L Yang Q He and F Yang ldquoAn error modified grey theoryfor forecasting international oil pricerdquo Applied Mechanics andMaterials vol 380ndash384 pp 1525ndash1528 2013

[115] F Gori ldquoForecasting the time evolution of the oil price duringmonths of negative inflation raterdquo Journal of Energy ResourcesTechnology vol 135 no 1 Article ID 011602 8 pages 2013

[116] F Gori ldquoMass and energy-capital conservation equations toforecast monthly oil pricerdquo Applied Thermal Engineering vol61 no 2 pp 623ndash632 2013

[117] F Gori ldquoRemarks on the paper lsquoMass and energy-capitalconservation equations to forecast monthly oil pricersquordquo AppliedThermal Engineering vol 68 no 1-2 pp 59ndash61 2014

International Journal of

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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



in 1987 based on the assumption that world oil prices willaverage 13US $bbl through the third quarter of 1986 andrise gradually to 18US $bbl by the end of 1987 Dielwart andColes [17] discussed the current market uncertainties andtheir impact on Canadian oil and natural gas prices Yohe[18] presented an analytical technique based on an adaptiveexpectations model of incorporating current informationinto long-term forecast Sekera [19] presented three priceforecasting analyses with forecasts under each theory andthe recommendation that all long term forecasts be based onfundamentaleconomic analysis while short term forecastingneeds to utilize other methods Gately [20] reviewed recentand current opinions about prospects for prices in the worldmarket for crude oil with two alternative views Holden et al[21] examined the accuracy and properties of forecasts bythe OECD for 24 countries and 8 variables Dougherty [22]presented some rules of thumb that may help to understandthe evolution of oil prices to identify factors to considerwhendeciding on the credibility of an oil price forecast ratherthan to prophesize future oil price Amano [23] developedan annual small-scale econometric model of the world oilmarket to analyze oil market conditions and oil prices for theperiod 1986ndash1991

During the 1990s among the others Garces [24] exploredthe effects of natural gas storage on short-term gas priceforecasts with preliminary results which indicate that byusing natural gas storage as a causal variable the finalgas price forecasts can decrease by as much as 5 overthe forecast period Angelier [25] concluded his analyticalframework with the forecast that in the year 2000 oil priceswill not be significantly different from those of 1990s Boppand Lady [26] concluded their analysis that when actualprices are employed future prices did correctly anticipatethe observed seasonal pattern Abramson and Finizza [27]developed a system that forecasts crude oil prices via MonteCarlo analyses of the network Charleson and Weber [28]forecasted energy consumption in Western Australia withBayesian vector autoregressions to 2010 Fesharaki [29]believed that for the next three to five years oil prices willremain at lower levels than generally predicted because priceincreases even when they occur will not be sustainablefor very long Huntington [30] reviewed forecasts of oilprices over the 1980s that were made in 1980 identifyingthe sources of errors due to such factors as exogenous GNPassumptions resource supply conditions outside the carteland demand adjustments to price changes Moosa and Al-Loughani [31] on the basis of monthly observations on spotand future prices of theWest Texas Intermediate (WTI) crudeoil suggested that future prices are neither unbiased norefficient forecasters of spot prices Abramson [32] discusseda knowledge-based system that models the crude oil marketas a belief network and uses scenario generation and Monte-Carlo analysis to forecast oil prices Skov [33] reviewed someof the more significant forecasts of energy supply demandand oil prices made in the last 25 years and identifies thebasic premises used and why they led to incorrect forecastsAbramson and Finizza [34] described a case study in theuse of inherently probabilistic belief network models toproduce probabilistic forecasts of average annual oil prices

Santini [35] investigated the statistical model of oil pricesexamining issues and trends related to both the US andworld oil supply Williams [36] in order to forecast thefuture examined historical trends and presents the newidea that there is an inverse relationship between crudeoil prices volatility and the strength of the relationshipbetween the owners of crude oil (host governments) andthe oil companies that develop refine market and sell theproducts Chaudhuri and Daniel [37] demonstrated that thenonstationary behavior of US dollar real exchange rates overthe post-Bretton Woods era is due to the nonstationarybehavior of real oil prices Pindyck [38] discussed how priceforecasts also influence investment decisions and the choiceof products to produce Silvapulle and Moosa [39] examinedthe relationship between the spot and futures prices of WTIcrude oil using a sample of daily data with the result thatboth spot and futures markets react simultaneously to newinformation

During the 2000s among the others Alba and Bourdaire[40] consider the actual situation as a ldquocohabitationrdquo betweenoil and the other energieswith the oil price extremely volatilereflecting the trial and error adjustment of the market shareleft to the other energies The Centre for Global EnergyStudies [41] covers several topics including oilfields set tobegin production in 2000 non-OPECproduction demand inwestern OECD countries and the Asia Pacific region and oilprice forecasts According to the US energy information [42]global crude oil prices in 2000 will rise to 24US $bbl up by2US $bbl from its earlier estimate while spot prices forWestTexas Intermediate crude will average more than 22US $bblthrough 2001 Hare [43] reported that according to forecastsby the Energy Information Administration increases innatural gas prices will continue all the way to the summerand winter of 2000 and that high oil prices will still beseen in 2001 According to the managing director of Shellrsquosexploration and production [44] crude oil prices will averagein the mid-teens of US $bbl over the next 10 years TheEnergy Department of Energy Information Administration[45] forecasts themonthly average price of crude oil importedto the USA to stay above 28US $bbl for the rest of 2000Walde and Dos Santos [46] identified factors that have to beconsidered in speculating about the future evolution of oilprices Birky et al [47] examined the potentiality of uncon-ventional natural gas resources with all other fossil fuelscombined with the conclusion of a widespread productionand use of these unconventional reserves The trend of worldcrude oil price in 1999 has been reviewed and analyzed byYang [48] and the prospects on world crude oil price in 2000were predicted MacAvoy and Moshkin [49] used a modelframework consisting of a simultaneous equations system forproduction from reserves and for demands for productionin the residential commercial and industrial sectors with asignificant negative trend in the long-term price of naturalgas Morana [50] showed how the GARCH properties ofoil price changes can be employed to forecast the oil pricedistribution over short-term horizons with a semiparametricmethodology based on the bootstrap approach Tang andHammoudeh [51] investigated the behavior of the nonlinearmodel based on the target zone theory which has very














119889119872

119889119905= minus119866 (1)


119872 = 1198720minus int

119905

0

119866119889119905 (2)



GC

Gk


G

Go

120572 gt 0

120572 = 0

120572 lt 0

t

(t0 + infin)

(t0 + 0)

(constant)




1

119866

119889119866

119889119905= 120572 (3)


119866 = 1198660exp (120572119905) (4)



0

119872

1198660

=1198720

1198660

+[1 minus exp (120572119905)]

120572 (5)


0 that is120572 = 0

(1) gives

119872 = 1198720minus 1198660sdot 119905 (6)


119891= 11987201198660


t

0

120572 gt 0 120572 = 0

(t0 + 0)

120572c lt 120572 lt 0

120572 lt 120572c lt 0

120572 = 120572c lt 0

MG0

M0G0

M0G0






119891 which gives

119872 = 0 is given by (5) as

119905119891=1

120572ln [1 + 120572

1198720

1198660

] (7)



(i) 119905119891= 4055 years for 120572 = 001 (1annum)



119872

1198660

=1198720

1198660

minus[1 minus exp (|120572| 119905)]

|120572| (8)


119872(infin)

1198660

=1198720

1198660

minus1

|120572| (9)


119872(infin)

1198660

=1198720

1198660

minus1

|120572|= 0 (10)


10038161003816100381610038161205721198881003816100381610038161003816 =

1198660

1198720

(11)


119888= minus002


by

119905119891= minus

1

|120572|ln [1 minus |120572|

1198720

1198660

] (12)



119891is




119872(infin)

1198660

=1198720

1198660

minus1

|120572|gt 0 (13)


If 120572 lt 120572119888


0= 25 years

3 Hotelling Rule




0at the time 119905

0


119901 (119905) = 1199010+ 119901 (119905) 119905119903 (14)


119889119901

119889119905= 119901 sdot 119903 (15)


119901 (119905) = 1199010exp (119903119905) (16)






119873(1annum) as

119889119870

119889119905= 119866119896 (17)





119870 = 119866 sdot 119901 (18)



119866119896= 119870 sdot 119903

119873= 119866 sdot 119901 sdot 119903

119873 (19)



119889 (119866 sdot 119901)

119889119905= 119866 sdot 119901 sdot 119903

119873 (20)



1

119901

119889119901

119889119905+1

119866

119889119866

119889119905= 119903119873 (21)


119873



1

119901119864

119889119901119864

119889119905= 119903119873minus 120572 (22)


119901119864= 1199011198640sdot exp (120573 sdot 119905) (23)

where

120573 = 119903119873minus 120572 (24)





119873 the price


t

(t0 + infin)

pE

pE0

120573 gt 0 (120572 lt r)

120573 = 0 (120572 = r)

120573 lt 0 (120572 gt r)

(t0 + 0)









119864

increases with time



119873and 119862

119864 of Figure 5









119864 where119872

119864is






119864is

119889119872119864

119889119905= 119866119864minus 119866119878 (25)


119889

119889119905(119870119864) = 119866119870119878minus 119866119870119864 (26)



CN CE

MN rN ME rE

GE (extracted)pE

GS (selling)pS

GKE(pE) GKS(pS)


which becomes

119889


and finally

1

119901119878

sdot119889119901119878

119889119905+1

119866119878

sdot119889119866119878

119889119905= 119903119864minus119866119864sdot 119901119864

119866119878sdot 119901119878

sdot 119903119873 (28)






119873and 119862

119864 that is

1

119866119864

119889119866119864

119889119905=1

119866119878

119889119866119878

119889119905= 120572 (29)





1

119901119878

119889119901119878

119889119905+ 120572 = 119903

119864minus119901119864

119901119878

119903119873 (31)

or

119889119901119878

119889119905= 119901119878(119903119864minus 120572) minus 119901

119864sdot 119903119873 (32)


119901119878= 119901lowast

1198780exp (120573 sdot 119905) + [119901


1198780] exp (1205731015840 sdot 119905) (33)

where

1205731015840= 119903119864minus 120572 (34)


119901lowast

1198780=119903119873sdot 1199011198640

119903119864minus 119903119873

=119903119873sdot 1199011198640

1205731015840 minus 120573 (35)




119898given by

119905119898=

1

(120573 minus 1205731015840)ln[

1205731015840sdot (119901lowast

1198780minus 1199011198780)


1198780

] (36)


1205731015840(1205731015840minus 120573) (119901


1198780) lt 0 (37)



1198780is equal to

119901lowastlowast

1198780=

119901lowast

1198780(1205731015840minus 120573)

1205731015840=1199031198731199011198640

1205731015840 (38)




1198780for 1205731015840 gt 120573 (ie 119903

119864gt 119903119873) (39)

or


1198780for 1205731015840 lt 120573 (ie 119903

119864lt 119903119873) (40)

For 1199011198780= 119901lowast


119905119898= +infin (41)

for 1205731015840 gt 120573 or

119905119898= minusinfin (42)

for 1205731015840 lt 120573For 119903119873= 119903119864 that is 1205731015840 = 120573 119901

119878is given by


1198780120573119905) exp (120573119905) (43)




1198780)

119901lowastlowast

11987801205731015840

(44)




price 1199011198780is

1199011198780= 119901lowastlowast

1198780=119903119873sdot 1199011198640

1205731015840 (45)







(vi) Category 5 119903119873lt 0

t

0

pS120572 lt 0 120573 gt 0


(t0 + infin)

120572 = 0 120573 = 0

0 lt 120572 lt rE 120573 lt 0

(constant)

(t0 + 0)

Case 0-A 120572 lt rE 120573998400 gt 0

Case 0-B 120572 = rE 120573998400 = 0 120573 lt 0

Case 0-C 120572 gt rE 120573998400 lt 0 120573 lt 0

Category 0 rN = 0

pS0



Category 0 (119903119873


119873= 0 (ie 119901lowast




119901119878= 1199011198780exp (1205731015840119905) (46)



119864




119864 120573 lt 0


119878is



119878is







Category 1 (119903119873






is1199011198780ge 119901lowast



CIPES gt 0



1198780gt 0 or 119903

119873gt 119903119864gt


1198780gt 0 gt 119901

lowast





(t0 + infin)

(t0 + infin)





S0 gt 0


S0

pS

pS0

(plowastS0)apS0


0

tmax


S0 gt 0



S0)a








S0

pS0

t




119901lowast


1198780gt 0 or for 119903




1198780gt 0 gt 119901

lowast

1198780



1198780gt 1199011198780 or 119903119873gt 119903119864gt


gt 1199011198780gt 0 gt 119901

lowast

1198780 or 119903119873gt 120572 gt 119903




1198780gt

119901lowastlowast

1198780gt 1199011198780 or for 119903


1198780is smaller


lowast

1198780 or for 119903

119873gt 120572 gt 119903

119864


1198780gt 0 gt 119901

lowast


1198780



119873and 119903119864

Case 2-A (119903119864gt 120572 gt 119903



is 1199011198780ge

119901lowastlowast

1198780gt 119901lowast


119901lowastlowast

1198780gt 119901lowast

1198780

Case 2-B (119903119864gt 120572 gt 119903





119901lowastlowast

1198780gt 0



119873

or 120572 gt 119903119873gt 119903119864 or 120572 gt 119903

119873gt 119903119864






is 1199011198780


1198780= 119901lowast


119901lowastlowast

1198780= 119901lowast

1198780



1198780= 119901lowast


1198780=

119901lowast

1198780


119873= 120572 gt


119864gt


1198780is 1199011198780lt

119901lowastlowast

1198780= 119901lowast


1198780= 119901lowast

1198780


1198780




increase with time







119873= 119903119864lt






1198780

Category 5 (119903119873





t0

pS


(t0 + infin)

pS0


(plowastS0)ab

pS0


(plowastlowastS0 )a

(plowastlowastS0 )a

Case 2-ApS0 =










S0)d gt 0(t0 + 0)

(t0 minus 0)


(t0 minus infin)


Case 2-C 120572 gt rN ie 120573 lt 0






Case 5-A (119903119864gt 120572 gt 0 gt 119903



1198780because 119901

1198780gt 0 gt 119901

lowast




1198780

Case 5-B (120572 gt 119903119864gt 0 gt 119903




lowast

1198780


Case 5-C (120572 gt 119903119864gt 0 gt 119903




1198780gt 0 gt 119901

lowast




Δ119901 = (119901119878minus 119901119864)

= (1199011198780minus 119901lowast

1198780) exp (1205731015840119905) minus (119901


1198780) exp (120573119905)


0exp (120573119905)

(47)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (48)



119905119898=

1

(1205731015840 minus 120573)ln[


1198780minus 1199011198640)

1205731015840 sdot (119901lowast

1198780minus 1199011198780)]

=1

(1205731015840 minus 120573)ln[



=1

(120573 minus 1205731015840)ln[


1205731015840 sdot Δlowast1199010

]

(49)


1205731015840(1205731015840minus 120573) (Δ119901




0is equal to


(1205731015840minus 120573)

1205731015840] = 119901

1198640

(2119903119873minus 119903119864)

1205731015840 (51)


t0

pS

pS0

pS0

(plowastS0)a

(plowastS0)b







(t0plowastS0)b

Case 3-C

(t0 + infin)

(t0 minus infin)



t0

pS

pS0

pS0

pS0

(plowastlowastS0 )a

(plowastlowastS0)b

Category 4 rN = rE ie 120573998400 = 120573


(t0 minus infin)

Linear

tmax

tmin

t0

(t0 minus 0)







or rN = rE lt 120572 ie 120573998400 = 120573 lt 0

or rN = rE = 120572 ie 120573998400 = 120573 = 0







for 1205731015840 lt 120573(52)

For Δ1199011198780= Δ119901lowast


119905119898= +infin (53)


for 1205731015840 lt 120573


t

0

pS

pS0

pS0

pS0

(plowastlowastS0)a

(plowastS0)a

Category 5 rN lt 0


(t0 + infin)

Case 5-ApS0 gt 0


tmax


S0)a


(t0 minus infin)



The time 1199050is

1199050=

1

(1205731015840 minus 120573)ln[

Δlowast1199010

(Δlowast1199010minus Δ1199010)] (55)


Category 0D 119903119873= 0











1198780= 0


0= minus11990311986411990111986401205731015840


Δ119901 = (119901119878minus 119901119864) = 1199011198780exp (1205731015840119905) minus 119901

1198640exp (120573119905)


0exp (120573119905)

(56)


119905119898=

1

(1205731015840 minus 120573)ln[

120573 sdot 1199011198640

1205731015840 sdot 1199011198780

] (57)


for

1205731015840(1205731015840minus 120573) 119901

1198780lt 0 (58)


1199050=

1

(1205731015840 minus 120573)ln [

1199011198640

1199011198780

] (59)



1198780ge

1199011198640)




0gt Δ119901

0gt

Δlowast1199010


0= Δ119901

0or

1199011198780= 119901lowast

1198780



0)119886lt 0))










Δp0

0

pS0

0

Δp0pS0

(Δlowastlowastp0)a

Δp0

Δlowastp0

0D-A1120572 lt 0 120573 gt 0

0D-A1120572 = 0 120573 = 0

Δp0

(t0 + infin)

(t0 + infin)

Case 0D-A rE gt 120572 gt 0 0 gt 120573

0D-A1 120572 gt 0 0 gt 120573

0D-B1

0D-B1

0D-C1tmax

tmax (t0 + 0)

(t0 + 0)

(t0 + 0)

t

t

t

0D-A2

120572 lt 0 120573 gt 0

(t0 + infin) 120572 = 0 120573 = 0

0D-A20D-A2120572 gt 0 0 gt 120573

(t0pS0)

(t0pS0)

(t0pS0)

t0

t0

0D-C2

0D-A3120572 lt 0 120573 gt 0 Case 0D-B 120572 = rE 120573

998400 = 0 120573 lt 0

Case 0D-C 120572 gt rE 0 gt 120573998400 gt 120573

tmin

0D-C3

0D-C4

0D-B1

(t0 minus infin)






0gt Δ119901

0gt Δlowast1199010





lowast1199010lt 0))


1198780=


Case 0D-C (120572 gt 119903119864 (ie 120573 lt 120573


0)119888gt 0 gt

(Δlowast1199010)119888))







0






0

is Δ1199010gt Δlowast1199010 If 120572 = 119903



0ge Δlowast1199010 If 120572 gt 119903

119864 Δ119901 tends


Category 1D (119903119873


to 119903119864


1198780ge 0))





0)119886= (Δlowastlowast1199010)119886= 0




pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0


1D-A1 120572 gt 0



Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0




t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)





0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)


0le Δ1199010lt 0


0gt Δ119901

0gt

Δlowast1199010


119901lowast

1198780






0 If Δ119901

0=







are discussed





are discussed






1198780)119887lt 0 lt

(119901lowastlowast


0)119887lt 0 lt (Δ


0


0is



towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0






119864and

119903119873 For 1205731015840 gt 0 and 119903



119864= 2119903119873 Δ119901 increases


119864lt 2119903119873 Δ119901








119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















119889119872

119889119905= minus119866 (1)


119872 = 1198720minus int

119905

0

119866119889119905 (2)



GC

Gk


G

Go

120572 gt 0

120572 = 0

120572 lt 0

t

(t0 + infin)

(t0 + 0)

(constant)




1

119866

119889119866

119889119905= 120572 (3)


119866 = 1198660exp (120572119905) (4)



0

119872

1198660

=1198720

1198660

+[1 minus exp (120572119905)]

120572 (5)


0 that is120572 = 0

(1) gives

119872 = 1198720minus 1198660sdot 119905 (6)


119891= 11987201198660


t

0

120572 gt 0 120572 = 0

(t0 + 0)

120572c lt 120572 lt 0

120572 lt 120572c lt 0

120572 = 120572c lt 0

MG0

M0G0

M0G0






119891 which gives

119872 = 0 is given by (5) as

119905119891=1

120572ln [1 + 120572

1198720

1198660

] (7)



(i) 119905119891= 4055 years for 120572 = 001 (1annum)



119872

1198660

=1198720

1198660

minus[1 minus exp (|120572| 119905)]

|120572| (8)


119872(infin)

1198660

=1198720

1198660

minus1

|120572| (9)


119872(infin)

1198660

=1198720

1198660

minus1

|120572|= 0 (10)


10038161003816100381610038161205721198881003816100381610038161003816 =

1198660

1198720

(11)


119888= minus002


by

119905119891= minus

1

|120572|ln [1 minus |120572|

1198720

1198660

] (12)



119891is




119872(infin)

1198660

=1198720

1198660

minus1

|120572|gt 0 (13)


If 120572 lt 120572119888


0= 25 years

3 Hotelling Rule




0at the time 119905

0


119901 (119905) = 1199010+ 119901 (119905) 119905119903 (14)


119889119901

119889119905= 119901 sdot 119903 (15)


119901 (119905) = 1199010exp (119903119905) (16)






119873(1annum) as

119889119870

119889119905= 119866119896 (17)





119870 = 119866 sdot 119901 (18)



119866119896= 119870 sdot 119903

119873= 119866 sdot 119901 sdot 119903

119873 (19)



119889 (119866 sdot 119901)

119889119905= 119866 sdot 119901 sdot 119903

119873 (20)



1

119901

119889119901

119889119905+1

119866

119889119866

119889119905= 119903119873 (21)


119873



1

119901119864

119889119901119864

119889119905= 119903119873minus 120572 (22)


119901119864= 1199011198640sdot exp (120573 sdot 119905) (23)

where

120573 = 119903119873minus 120572 (24)





119873 the price


t

(t0 + infin)

pE

pE0

120573 gt 0 (120572 lt r)

120573 = 0 (120572 = r)

120573 lt 0 (120572 gt r)

(t0 + 0)









119864

increases with time



119873and 119862

119864 of Figure 5









119864 where119872

119864is






119864is

119889119872119864

119889119905= 119866119864minus 119866119878 (25)


119889

119889119905(119870119864) = 119866119870119878minus 119866119870119864 (26)



CN CE

MN rN ME rE

GE (extracted)pE

GS (selling)pS

GKE(pE) GKS(pS)


which becomes

119889


and finally

1

119901119878

sdot119889119901119878

119889119905+1

119866119878

sdot119889119866119878

119889119905= 119903119864minus119866119864sdot 119901119864

119866119878sdot 119901119878

sdot 119903119873 (28)






119873and 119862

119864 that is

1

119866119864

119889119866119864

119889119905=1

119866119878

119889119866119878

119889119905= 120572 (29)





1

119901119878

119889119901119878

119889119905+ 120572 = 119903

119864minus119901119864

119901119878

119903119873 (31)

or

119889119901119878

119889119905= 119901119878(119903119864minus 120572) minus 119901

119864sdot 119903119873 (32)


119901119878= 119901lowast

1198780exp (120573 sdot 119905) + [119901


1198780] exp (1205731015840 sdot 119905) (33)

where

1205731015840= 119903119864minus 120572 (34)


119901lowast

1198780=119903119873sdot 1199011198640

119903119864minus 119903119873

=119903119873sdot 1199011198640

1205731015840 minus 120573 (35)




119898given by

119905119898=

1

(120573 minus 1205731015840)ln[

1205731015840sdot (119901lowast

1198780minus 1199011198780)


1198780

] (36)


1205731015840(1205731015840minus 120573) (119901


1198780) lt 0 (37)



1198780is equal to

119901lowastlowast

1198780=

119901lowast

1198780(1205731015840minus 120573)

1205731015840=1199031198731199011198640

1205731015840 (38)




1198780for 1205731015840 gt 120573 (ie 119903

119864gt 119903119873) (39)

or


1198780for 1205731015840 lt 120573 (ie 119903

119864lt 119903119873) (40)

For 1199011198780= 119901lowast


119905119898= +infin (41)

for 1205731015840 gt 120573 or

119905119898= minusinfin (42)

for 1205731015840 lt 120573For 119903119873= 119903119864 that is 1205731015840 = 120573 119901

119878is given by


1198780120573119905) exp (120573119905) (43)




1198780)

119901lowastlowast

11987801205731015840

(44)




price 1199011198780is

1199011198780= 119901lowastlowast

1198780=119903119873sdot 1199011198640

1205731015840 (45)







(vi) Category 5 119903119873lt 0

t

0

pS120572 lt 0 120573 gt 0


(t0 + infin)

120572 = 0 120573 = 0

0 lt 120572 lt rE 120573 lt 0

(constant)

(t0 + 0)

Case 0-A 120572 lt rE 120573998400 gt 0

Case 0-B 120572 = rE 120573998400 = 0 120573 lt 0

Case 0-C 120572 gt rE 120573998400 lt 0 120573 lt 0

Category 0 rN = 0

pS0



Category 0 (119903119873


119873= 0 (ie 119901lowast




119901119878= 1199011198780exp (1205731015840119905) (46)



119864




119864 120573 lt 0


119878is



119878is







Category 1 (119903119873






is1199011198780ge 119901lowast



CIPES gt 0



1198780gt 0 or 119903

119873gt 119903119864gt


1198780gt 0 gt 119901

lowast





(t0 + infin)

(t0 + infin)





S0 gt 0


S0

pS

pS0

(plowastS0)apS0


0

tmax


S0 gt 0



S0)a








S0

pS0

t




119901lowast


1198780gt 0 or for 119903




1198780gt 0 gt 119901

lowast

1198780



1198780gt 1199011198780 or 119903119873gt 119903119864gt


gt 1199011198780gt 0 gt 119901

lowast

1198780 or 119903119873gt 120572 gt 119903




1198780gt

119901lowastlowast

1198780gt 1199011198780 or for 119903


1198780is smaller


lowast

1198780 or for 119903

119873gt 120572 gt 119903

119864


1198780gt 0 gt 119901

lowast


1198780



119873and 119903119864

Case 2-A (119903119864gt 120572 gt 119903



is 1199011198780ge

119901lowastlowast

1198780gt 119901lowast


119901lowastlowast

1198780gt 119901lowast

1198780

Case 2-B (119903119864gt 120572 gt 119903





119901lowastlowast

1198780gt 0



119873

or 120572 gt 119903119873gt 119903119864 or 120572 gt 119903

119873gt 119903119864






is 1199011198780


1198780= 119901lowast


119901lowastlowast

1198780= 119901lowast

1198780



1198780= 119901lowast


1198780=

119901lowast

1198780


119873= 120572 gt


119864gt


1198780is 1199011198780lt

119901lowastlowast

1198780= 119901lowast


1198780= 119901lowast

1198780


1198780




increase with time







119873= 119903119864lt






1198780

Category 5 (119903119873





t0

pS


(t0 + infin)

pS0


(plowastS0)ab

pS0


(plowastlowastS0 )a

(plowastlowastS0 )a

Case 2-ApS0 =










S0)d gt 0(t0 + 0)

(t0 minus 0)


(t0 minus infin)


Case 2-C 120572 gt rN ie 120573 lt 0






Case 5-A (119903119864gt 120572 gt 0 gt 119903



1198780because 119901

1198780gt 0 gt 119901

lowast




1198780

Case 5-B (120572 gt 119903119864gt 0 gt 119903




lowast

1198780


Case 5-C (120572 gt 119903119864gt 0 gt 119903




1198780gt 0 gt 119901

lowast




Δ119901 = (119901119878minus 119901119864)

= (1199011198780minus 119901lowast

1198780) exp (1205731015840119905) minus (119901


1198780) exp (120573119905)


0exp (120573119905)

(47)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (48)



119905119898=

1

(1205731015840 minus 120573)ln[


1198780minus 1199011198640)

1205731015840 sdot (119901lowast

1198780minus 1199011198780)]

=1

(1205731015840 minus 120573)ln[



=1

(120573 minus 1205731015840)ln[


1205731015840 sdot Δlowast1199010

]

(49)


1205731015840(1205731015840minus 120573) (Δ119901




0is equal to


(1205731015840minus 120573)

1205731015840] = 119901

1198640

(2119903119873minus 119903119864)

1205731015840 (51)


t0

pS

pS0

pS0

(plowastS0)a

(plowastS0)b







(t0plowastS0)b

Case 3-C

(t0 + infin)

(t0 minus infin)



t0

pS

pS0

pS0

pS0

(plowastlowastS0 )a

(plowastlowastS0)b

Category 4 rN = rE ie 120573998400 = 120573


(t0 minus infin)

Linear

tmax

tmin

t0

(t0 minus 0)







or rN = rE lt 120572 ie 120573998400 = 120573 lt 0

or rN = rE = 120572 ie 120573998400 = 120573 = 0







for 1205731015840 lt 120573(52)

For Δ1199011198780= Δ119901lowast


119905119898= +infin (53)


for 1205731015840 lt 120573


t

0

pS

pS0

pS0

pS0

(plowastlowastS0)a

(plowastS0)a

Category 5 rN lt 0


(t0 + infin)

Case 5-ApS0 gt 0


tmax


S0)a


(t0 minus infin)



The time 1199050is

1199050=

1

(1205731015840 minus 120573)ln[

Δlowast1199010

(Δlowast1199010minus Δ1199010)] (55)


Category 0D 119903119873= 0











1198780= 0


0= minus11990311986411990111986401205731015840


Δ119901 = (119901119878minus 119901119864) = 1199011198780exp (1205731015840119905) minus 119901

1198640exp (120573119905)


0exp (120573119905)

(56)


119905119898=

1

(1205731015840 minus 120573)ln[

120573 sdot 1199011198640

1205731015840 sdot 1199011198780

] (57)


for

1205731015840(1205731015840minus 120573) 119901

1198780lt 0 (58)


1199050=

1

(1205731015840 minus 120573)ln [

1199011198640

1199011198780

] (59)



1198780ge

1199011198640)




0gt Δ119901

0gt

Δlowast1199010


0= Δ119901

0or

1199011198780= 119901lowast

1198780



0)119886lt 0))










Δp0

0

pS0

0

Δp0pS0

(Δlowastlowastp0)a

Δp0

Δlowastp0

0D-A1120572 lt 0 120573 gt 0

0D-A1120572 = 0 120573 = 0

Δp0

(t0 + infin)

(t0 + infin)

Case 0D-A rE gt 120572 gt 0 0 gt 120573

0D-A1 120572 gt 0 0 gt 120573

0D-B1

0D-B1

0D-C1tmax

tmax (t0 + 0)

(t0 + 0)

(t0 + 0)

t

t

t

0D-A2

120572 lt 0 120573 gt 0

(t0 + infin) 120572 = 0 120573 = 0

0D-A20D-A2120572 gt 0 0 gt 120573

(t0pS0)

(t0pS0)

(t0pS0)

t0

t0

0D-C2

0D-A3120572 lt 0 120573 gt 0 Case 0D-B 120572 = rE 120573

998400 = 0 120573 lt 0

Case 0D-C 120572 gt rE 0 gt 120573998400 gt 120573

tmin

0D-C3

0D-C4

0D-B1

(t0 minus infin)






0gt Δ119901

0gt Δlowast1199010





lowast1199010lt 0))


1198780=


Case 0D-C (120572 gt 119903119864 (ie 120573 lt 120573


0)119888gt 0 gt

(Δlowast1199010)119888))







0






0

is Δ1199010gt Δlowast1199010 If 120572 = 119903



0ge Δlowast1199010 If 120572 gt 119903

119864 Δ119901 tends


Category 1D (119903119873


to 119903119864


1198780ge 0))





0)119886= (Δlowastlowast1199010)119886= 0




pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0


1D-A1 120572 gt 0



Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0




t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)





0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)


0le Δ1199010lt 0


0gt Δ119901

0gt

Δlowast1199010


119901lowast

1198780






0 If Δ119901

0=







are discussed





are discussed






1198780)119887lt 0 lt

(119901lowastlowast


0)119887lt 0 lt (Δ


0


0is



towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0






119864and

119903119873 For 1205731015840 gt 0 and 119903



119864= 2119903119873 Δ119901 increases


119864lt 2119903119873 Δ119901








119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











GC

Gk


G

Go

120572 gt 0

120572 = 0

120572 lt 0

t

(t0 + infin)

(t0 + 0)

(constant)




1

119866

119889119866

119889119905= 120572 (3)


119866 = 1198660exp (120572119905) (4)



0

119872

1198660

=1198720

1198660

+[1 minus exp (120572119905)]

120572 (5)


0 that is120572 = 0

(1) gives

119872 = 1198720minus 1198660sdot 119905 (6)


119891= 11987201198660


t

0

120572 gt 0 120572 = 0

(t0 + 0)

120572c lt 120572 lt 0

120572 lt 120572c lt 0

120572 = 120572c lt 0

MG0

M0G0

M0G0






119891 which gives

119872 = 0 is given by (5) as

119905119891=1

120572ln [1 + 120572

1198720

1198660

] (7)



(i) 119905119891= 4055 years for 120572 = 001 (1annum)



119872

1198660

=1198720

1198660

minus[1 minus exp (|120572| 119905)]

|120572| (8)


119872(infin)

1198660

=1198720

1198660

minus1

|120572| (9)


119872(infin)

1198660

=1198720

1198660

minus1

|120572|= 0 (10)


10038161003816100381610038161205721198881003816100381610038161003816 =

1198660

1198720

(11)


119888= minus002


by

119905119891= minus

1

|120572|ln [1 minus |120572|

1198720

1198660

] (12)



119891is




119872(infin)

1198660

=1198720

1198660

minus1

|120572|gt 0 (13)


If 120572 lt 120572119888


0= 25 years

3 Hotelling Rule




0at the time 119905

0


119901 (119905) = 1199010+ 119901 (119905) 119905119903 (14)


119889119901

119889119905= 119901 sdot 119903 (15)


119901 (119905) = 1199010exp (119903119905) (16)






119873(1annum) as

119889119870

119889119905= 119866119896 (17)





119870 = 119866 sdot 119901 (18)



119866119896= 119870 sdot 119903

119873= 119866 sdot 119901 sdot 119903

119873 (19)



119889 (119866 sdot 119901)

119889119905= 119866 sdot 119901 sdot 119903

119873 (20)



1

119901

119889119901

119889119905+1

119866

119889119866

119889119905= 119903119873 (21)


119873



1

119901119864

119889119901119864

119889119905= 119903119873minus 120572 (22)


119901119864= 1199011198640sdot exp (120573 sdot 119905) (23)

where

120573 = 119903119873minus 120572 (24)





119873 the price


t

(t0 + infin)

pE

pE0

120573 gt 0 (120572 lt r)

120573 = 0 (120572 = r)

120573 lt 0 (120572 gt r)

(t0 + 0)









119864

increases with time



119873and 119862

119864 of Figure 5









119864 where119872

119864is






119864is

119889119872119864

119889119905= 119866119864minus 119866119878 (25)


119889

119889119905(119870119864) = 119866119870119878minus 119866119870119864 (26)



CN CE

MN rN ME rE

GE (extracted)pE

GS (selling)pS

GKE(pE) GKS(pS)


which becomes

119889


and finally

1

119901119878

sdot119889119901119878

119889119905+1

119866119878

sdot119889119866119878

119889119905= 119903119864minus119866119864sdot 119901119864

119866119878sdot 119901119878

sdot 119903119873 (28)






119873and 119862

119864 that is

1

119866119864

119889119866119864

119889119905=1

119866119878

119889119866119878

119889119905= 120572 (29)





1

119901119878

119889119901119878

119889119905+ 120572 = 119903

119864minus119901119864

119901119878

119903119873 (31)

or

119889119901119878

119889119905= 119901119878(119903119864minus 120572) minus 119901

119864sdot 119903119873 (32)


119901119878= 119901lowast

1198780exp (120573 sdot 119905) + [119901


1198780] exp (1205731015840 sdot 119905) (33)

where

1205731015840= 119903119864minus 120572 (34)


119901lowast

1198780=119903119873sdot 1199011198640

119903119864minus 119903119873

=119903119873sdot 1199011198640

1205731015840 minus 120573 (35)




119898given by

119905119898=

1

(120573 minus 1205731015840)ln[

1205731015840sdot (119901lowast

1198780minus 1199011198780)


1198780

] (36)


1205731015840(1205731015840minus 120573) (119901


1198780) lt 0 (37)



1198780is equal to

119901lowastlowast

1198780=

119901lowast

1198780(1205731015840minus 120573)

1205731015840=1199031198731199011198640

1205731015840 (38)




1198780for 1205731015840 gt 120573 (ie 119903

119864gt 119903119873) (39)

or


1198780for 1205731015840 lt 120573 (ie 119903

119864lt 119903119873) (40)

For 1199011198780= 119901lowast


119905119898= +infin (41)

for 1205731015840 gt 120573 or

119905119898= minusinfin (42)

for 1205731015840 lt 120573For 119903119873= 119903119864 that is 1205731015840 = 120573 119901

119878is given by


1198780120573119905) exp (120573119905) (43)




1198780)

119901lowastlowast

11987801205731015840

(44)




price 1199011198780is

1199011198780= 119901lowastlowast

1198780=119903119873sdot 1199011198640

1205731015840 (45)







(vi) Category 5 119903119873lt 0

t

0

pS120572 lt 0 120573 gt 0


(t0 + infin)

120572 = 0 120573 = 0

0 lt 120572 lt rE 120573 lt 0

(constant)

(t0 + 0)

Case 0-A 120572 lt rE 120573998400 gt 0

Case 0-B 120572 = rE 120573998400 = 0 120573 lt 0

Case 0-C 120572 gt rE 120573998400 lt 0 120573 lt 0

Category 0 rN = 0

pS0



Category 0 (119903119873


119873= 0 (ie 119901lowast




119901119878= 1199011198780exp (1205731015840119905) (46)



119864




119864 120573 lt 0


119878is



119878is







Category 1 (119903119873






is1199011198780ge 119901lowast



CIPES gt 0



1198780gt 0 or 119903

119873gt 119903119864gt


1198780gt 0 gt 119901

lowast





(t0 + infin)

(t0 + infin)





S0 gt 0


S0

pS

pS0

(plowastS0)apS0


0

tmax


S0 gt 0



S0)a








S0

pS0

t




119901lowast


1198780gt 0 or for 119903




1198780gt 0 gt 119901

lowast

1198780



1198780gt 1199011198780 or 119903119873gt 119903119864gt


gt 1199011198780gt 0 gt 119901

lowast

1198780 or 119903119873gt 120572 gt 119903




1198780gt

119901lowastlowast

1198780gt 1199011198780 or for 119903


1198780is smaller


lowast

1198780 or for 119903

119873gt 120572 gt 119903

119864


1198780gt 0 gt 119901

lowast


1198780



119873and 119903119864

Case 2-A (119903119864gt 120572 gt 119903



is 1199011198780ge

119901lowastlowast

1198780gt 119901lowast


119901lowastlowast

1198780gt 119901lowast

1198780

Case 2-B (119903119864gt 120572 gt 119903





119901lowastlowast

1198780gt 0



119873

or 120572 gt 119903119873gt 119903119864 or 120572 gt 119903

119873gt 119903119864






is 1199011198780


1198780= 119901lowast


119901lowastlowast

1198780= 119901lowast

1198780



1198780= 119901lowast


1198780=

119901lowast

1198780


119873= 120572 gt


119864gt


1198780is 1199011198780lt

119901lowastlowast

1198780= 119901lowast


1198780= 119901lowast

1198780


1198780




increase with time







119873= 119903119864lt






1198780

Category 5 (119903119873





t0

pS


(t0 + infin)

pS0


(plowastS0)ab

pS0


(plowastlowastS0 )a

(plowastlowastS0 )a

Case 2-ApS0 =










S0)d gt 0(t0 + 0)

(t0 minus 0)


(t0 minus infin)


Case 2-C 120572 gt rN ie 120573 lt 0






Case 5-A (119903119864gt 120572 gt 0 gt 119903



1198780because 119901

1198780gt 0 gt 119901

lowast




1198780

Case 5-B (120572 gt 119903119864gt 0 gt 119903




lowast

1198780


Case 5-C (120572 gt 119903119864gt 0 gt 119903




1198780gt 0 gt 119901

lowast




Δ119901 = (119901119878minus 119901119864)

= (1199011198780minus 119901lowast

1198780) exp (1205731015840119905) minus (119901


1198780) exp (120573119905)


0exp (120573119905)

(47)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (48)



119905119898=

1

(1205731015840 minus 120573)ln[


1198780minus 1199011198640)

1205731015840 sdot (119901lowast

1198780minus 1199011198780)]

=1

(1205731015840 minus 120573)ln[



=1

(120573 minus 1205731015840)ln[


1205731015840 sdot Δlowast1199010

]

(49)


1205731015840(1205731015840minus 120573) (Δ119901




0is equal to


(1205731015840minus 120573)

1205731015840] = 119901

1198640

(2119903119873minus 119903119864)

1205731015840 (51)


t0

pS

pS0

pS0

(plowastS0)a

(plowastS0)b







(t0plowastS0)b

Case 3-C

(t0 + infin)

(t0 minus infin)



t0

pS

pS0

pS0

pS0

(plowastlowastS0 )a

(plowastlowastS0)b

Category 4 rN = rE ie 120573998400 = 120573


(t0 minus infin)

Linear

tmax

tmin

t0

(t0 minus 0)







or rN = rE lt 120572 ie 120573998400 = 120573 lt 0

or rN = rE = 120572 ie 120573998400 = 120573 = 0







for 1205731015840 lt 120573(52)

For Δ1199011198780= Δ119901lowast


119905119898= +infin (53)


for 1205731015840 lt 120573


t

0

pS

pS0

pS0

pS0

(plowastlowastS0)a

(plowastS0)a

Category 5 rN lt 0


(t0 + infin)

Case 5-ApS0 gt 0


tmax


S0)a


(t0 minus infin)



The time 1199050is

1199050=

1

(1205731015840 minus 120573)ln[

Δlowast1199010

(Δlowast1199010minus Δ1199010)] (55)


Category 0D 119903119873= 0











1198780= 0


0= minus11990311986411990111986401205731015840


Δ119901 = (119901119878minus 119901119864) = 1199011198780exp (1205731015840119905) minus 119901

1198640exp (120573119905)


0exp (120573119905)

(56)


119905119898=

1

(1205731015840 minus 120573)ln[

120573 sdot 1199011198640

1205731015840 sdot 1199011198780

] (57)


for

1205731015840(1205731015840minus 120573) 119901

1198780lt 0 (58)


1199050=

1

(1205731015840 minus 120573)ln [

1199011198640

1199011198780

] (59)



1198780ge

1199011198640)




0gt Δ119901

0gt

Δlowast1199010


0= Δ119901

0or

1199011198780= 119901lowast

1198780



0)119886lt 0))










Δp0

0

pS0

0

Δp0pS0

(Δlowastlowastp0)a

Δp0

Δlowastp0

0D-A1120572 lt 0 120573 gt 0

0D-A1120572 = 0 120573 = 0

Δp0

(t0 + infin)

(t0 + infin)

Case 0D-A rE gt 120572 gt 0 0 gt 120573

0D-A1 120572 gt 0 0 gt 120573

0D-B1

0D-B1

0D-C1tmax

tmax (t0 + 0)

(t0 + 0)

(t0 + 0)

t

t

t

0D-A2

120572 lt 0 120573 gt 0

(t0 + infin) 120572 = 0 120573 = 0

0D-A20D-A2120572 gt 0 0 gt 120573

(t0pS0)

(t0pS0)

(t0pS0)

t0

t0

0D-C2

0D-A3120572 lt 0 120573 gt 0 Case 0D-B 120572 = rE 120573

998400 = 0 120573 lt 0

Case 0D-C 120572 gt rE 0 gt 120573998400 gt 120573

tmin

0D-C3

0D-C4

0D-B1

(t0 minus infin)






0gt Δ119901

0gt Δlowast1199010





lowast1199010lt 0))


1198780=


Case 0D-C (120572 gt 119903119864 (ie 120573 lt 120573


0)119888gt 0 gt

(Δlowast1199010)119888))







0






0

is Δ1199010gt Δlowast1199010 If 120572 = 119903



0ge Δlowast1199010 If 120572 gt 119903

119864 Δ119901 tends


Category 1D (119903119873


to 119903119864


1198780ge 0))





0)119886= (Δlowastlowast1199010)119886= 0




pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0


1D-A1 120572 gt 0



Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0




t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)





0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)


0le Δ1199010lt 0


0gt Δ119901

0gt

Δlowast1199010


119901lowast

1198780






0 If Δ119901

0=







are discussed





are discussed






1198780)119887lt 0 lt

(119901lowastlowast


0)119887lt 0 lt (Δ


0


0is



towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0






119864and

119903119873 For 1205731015840 gt 0 and 119903



119864= 2119903119873 Δ119901 increases


119864lt 2119903119873 Δ119901








119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119891is




119872(infin)

1198660

=1198720

1198660

minus1

|120572|gt 0 (13)


If 120572 lt 120572119888


0= 25 years

3 Hotelling Rule




0at the time 119905

0


119901 (119905) = 1199010+ 119901 (119905) 119905119903 (14)


119889119901

119889119905= 119901 sdot 119903 (15)


119901 (119905) = 1199010exp (119903119905) (16)






119873(1annum) as

119889119870

119889119905= 119866119896 (17)





119870 = 119866 sdot 119901 (18)



119866119896= 119870 sdot 119903

119873= 119866 sdot 119901 sdot 119903

119873 (19)



119889 (119866 sdot 119901)

119889119905= 119866 sdot 119901 sdot 119903

119873 (20)



1

119901

119889119901

119889119905+1

119866

119889119866

119889119905= 119903119873 (21)


119873



1

119901119864

119889119901119864

119889119905= 119903119873minus 120572 (22)


119901119864= 1199011198640sdot exp (120573 sdot 119905) (23)

where

120573 = 119903119873minus 120572 (24)





119873 the price


t

(t0 + infin)

pE

pE0

120573 gt 0 (120572 lt r)

120573 = 0 (120572 = r)

120573 lt 0 (120572 gt r)

(t0 + 0)









119864

increases with time



119873and 119862

119864 of Figure 5









119864 where119872

119864is






119864is

119889119872119864

119889119905= 119866119864minus 119866119878 (25)


119889

119889119905(119870119864) = 119866119870119878minus 119866119870119864 (26)



CN CE

MN rN ME rE

GE (extracted)pE

GS (selling)pS

GKE(pE) GKS(pS)


which becomes

119889


and finally

1

119901119878

sdot119889119901119878

119889119905+1

119866119878

sdot119889119866119878

119889119905= 119903119864minus119866119864sdot 119901119864

119866119878sdot 119901119878

sdot 119903119873 (28)






119873and 119862

119864 that is

1

119866119864

119889119866119864

119889119905=1

119866119878

119889119866119878

119889119905= 120572 (29)





1

119901119878

119889119901119878

119889119905+ 120572 = 119903

119864minus119901119864

119901119878

119903119873 (31)

or

119889119901119878

119889119905= 119901119878(119903119864minus 120572) minus 119901

119864sdot 119903119873 (32)


119901119878= 119901lowast

1198780exp (120573 sdot 119905) + [119901


1198780] exp (1205731015840 sdot 119905) (33)

where

1205731015840= 119903119864minus 120572 (34)


119901lowast

1198780=119903119873sdot 1199011198640

119903119864minus 119903119873

=119903119873sdot 1199011198640

1205731015840 minus 120573 (35)




119898given by

119905119898=

1

(120573 minus 1205731015840)ln[

1205731015840sdot (119901lowast

1198780minus 1199011198780)


1198780

] (36)


1205731015840(1205731015840minus 120573) (119901


1198780) lt 0 (37)



1198780is equal to

119901lowastlowast

1198780=

119901lowast

1198780(1205731015840minus 120573)

1205731015840=1199031198731199011198640

1205731015840 (38)




1198780for 1205731015840 gt 120573 (ie 119903

119864gt 119903119873) (39)

or


1198780for 1205731015840 lt 120573 (ie 119903

119864lt 119903119873) (40)

For 1199011198780= 119901lowast


119905119898= +infin (41)

for 1205731015840 gt 120573 or

119905119898= minusinfin (42)

for 1205731015840 lt 120573For 119903119873= 119903119864 that is 1205731015840 = 120573 119901

119878is given by


1198780120573119905) exp (120573119905) (43)




1198780)

119901lowastlowast

11987801205731015840

(44)




price 1199011198780is

1199011198780= 119901lowastlowast

1198780=119903119873sdot 1199011198640

1205731015840 (45)







(vi) Category 5 119903119873lt 0

t

0

pS120572 lt 0 120573 gt 0


(t0 + infin)

120572 = 0 120573 = 0

0 lt 120572 lt rE 120573 lt 0

(constant)

(t0 + 0)

Case 0-A 120572 lt rE 120573998400 gt 0

Case 0-B 120572 = rE 120573998400 = 0 120573 lt 0

Case 0-C 120572 gt rE 120573998400 lt 0 120573 lt 0

Category 0 rN = 0

pS0



Category 0 (119903119873


119873= 0 (ie 119901lowast




119901119878= 1199011198780exp (1205731015840119905) (46)



119864




119864 120573 lt 0


119878is



119878is







Category 1 (119903119873






is1199011198780ge 119901lowast



CIPES gt 0



1198780gt 0 or 119903

119873gt 119903119864gt


1198780gt 0 gt 119901

lowast





(t0 + infin)

(t0 + infin)





S0 gt 0


S0

pS

pS0

(plowastS0)apS0


0

tmax


S0 gt 0



S0)a








S0

pS0

t




119901lowast


1198780gt 0 or for 119903




1198780gt 0 gt 119901

lowast

1198780



1198780gt 1199011198780 or 119903119873gt 119903119864gt


gt 1199011198780gt 0 gt 119901

lowast

1198780 or 119903119873gt 120572 gt 119903




1198780gt

119901lowastlowast

1198780gt 1199011198780 or for 119903


1198780is smaller


lowast

1198780 or for 119903

119873gt 120572 gt 119903

119864


1198780gt 0 gt 119901

lowast


1198780



119873and 119903119864

Case 2-A (119903119864gt 120572 gt 119903



is 1199011198780ge

119901lowastlowast

1198780gt 119901lowast


119901lowastlowast

1198780gt 119901lowast

1198780

Case 2-B (119903119864gt 120572 gt 119903





119901lowastlowast

1198780gt 0



119873

or 120572 gt 119903119873gt 119903119864 or 120572 gt 119903

119873gt 119903119864






is 1199011198780


1198780= 119901lowast


119901lowastlowast

1198780= 119901lowast

1198780



1198780= 119901lowast


1198780=

119901lowast

1198780


119873= 120572 gt


119864gt


1198780is 1199011198780lt

119901lowastlowast

1198780= 119901lowast


1198780= 119901lowast

1198780


1198780




increase with time







119873= 119903119864lt






1198780

Category 5 (119903119873





t0

pS


(t0 + infin)

pS0


(plowastS0)ab

pS0


(plowastlowastS0 )a

(plowastlowastS0 )a

Case 2-ApS0 =










S0)d gt 0(t0 + 0)

(t0 minus 0)


(t0 minus infin)


Case 2-C 120572 gt rN ie 120573 lt 0






Case 5-A (119903119864gt 120572 gt 0 gt 119903



1198780because 119901

1198780gt 0 gt 119901

lowast




1198780

Case 5-B (120572 gt 119903119864gt 0 gt 119903




lowast

1198780


Case 5-C (120572 gt 119903119864gt 0 gt 119903




1198780gt 0 gt 119901

lowast




Δ119901 = (119901119878minus 119901119864)

= (1199011198780minus 119901lowast

1198780) exp (1205731015840119905) minus (119901


1198780) exp (120573119905)


0exp (120573119905)

(47)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (48)



119905119898=

1

(1205731015840 minus 120573)ln[


1198780minus 1199011198640)

1205731015840 sdot (119901lowast

1198780minus 1199011198780)]

=1

(1205731015840 minus 120573)ln[



=1

(120573 minus 1205731015840)ln[


1205731015840 sdot Δlowast1199010

]

(49)


1205731015840(1205731015840minus 120573) (Δ119901




0is equal to


(1205731015840minus 120573)

1205731015840] = 119901

1198640

(2119903119873minus 119903119864)

1205731015840 (51)


t0

pS

pS0

pS0

(plowastS0)a

(plowastS0)b







(t0plowastS0)b

Case 3-C

(t0 + infin)

(t0 minus infin)



t0

pS

pS0

pS0

pS0

(plowastlowastS0 )a

(plowastlowastS0)b

Category 4 rN = rE ie 120573998400 = 120573


(t0 minus infin)

Linear

tmax

tmin

t0

(t0 minus 0)







or rN = rE lt 120572 ie 120573998400 = 120573 lt 0

or rN = rE = 120572 ie 120573998400 = 120573 = 0







for 1205731015840 lt 120573(52)

For Δ1199011198780= Δ119901lowast


119905119898= +infin (53)


for 1205731015840 lt 120573


t

0

pS

pS0

pS0

pS0

(plowastlowastS0)a

(plowastS0)a

Category 5 rN lt 0


(t0 + infin)

Case 5-ApS0 gt 0


tmax


S0)a


(t0 minus infin)



The time 1199050is

1199050=

1

(1205731015840 minus 120573)ln[

Δlowast1199010

(Δlowast1199010minus Δ1199010)] (55)


Category 0D 119903119873= 0











1198780= 0


0= minus11990311986411990111986401205731015840


Δ119901 = (119901119878minus 119901119864) = 1199011198780exp (1205731015840119905) minus 119901

1198640exp (120573119905)


0exp (120573119905)

(56)


119905119898=

1

(1205731015840 minus 120573)ln[

120573 sdot 1199011198640

1205731015840 sdot 1199011198780

] (57)


for

1205731015840(1205731015840minus 120573) 119901

1198780lt 0 (58)


1199050=

1

(1205731015840 minus 120573)ln [

1199011198640

1199011198780

] (59)



1198780ge

1199011198640)




0gt Δ119901

0gt

Δlowast1199010


0= Δ119901

0or

1199011198780= 119901lowast

1198780



0)119886lt 0))










Δp0

0

pS0

0

Δp0pS0

(Δlowastlowastp0)a

Δp0

Δlowastp0

0D-A1120572 lt 0 120573 gt 0

0D-A1120572 = 0 120573 = 0

Δp0

(t0 + infin)

(t0 + infin)

Case 0D-A rE gt 120572 gt 0 0 gt 120573

0D-A1 120572 gt 0 0 gt 120573

0D-B1

0D-B1

0D-C1tmax

tmax (t0 + 0)

(t0 + 0)

(t0 + 0)

t

t

t

0D-A2

120572 lt 0 120573 gt 0

(t0 + infin) 120572 = 0 120573 = 0

0D-A20D-A2120572 gt 0 0 gt 120573

(t0pS0)

(t0pS0)

(t0pS0)

t0

t0

0D-C2

0D-A3120572 lt 0 120573 gt 0 Case 0D-B 120572 = rE 120573

998400 = 0 120573 lt 0

Case 0D-C 120572 gt rE 0 gt 120573998400 gt 120573

tmin

0D-C3

0D-C4

0D-B1

(t0 minus infin)






0gt Δ119901

0gt Δlowast1199010





lowast1199010lt 0))


1198780=


Case 0D-C (120572 gt 119903119864 (ie 120573 lt 120573


0)119888gt 0 gt

(Δlowast1199010)119888))







0






0

is Δ1199010gt Δlowast1199010 If 120572 = 119903



0ge Δlowast1199010 If 120572 gt 119903

119864 Δ119901 tends


Category 1D (119903119873


to 119903119864


1198780ge 0))





0)119886= (Δlowastlowast1199010)119886= 0




pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0


1D-A1 120572 gt 0



Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0




t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)





0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)


0le Δ1199010lt 0


0gt Δ119901

0gt

Δlowast1199010


119901lowast

1198780






0 If Δ119901

0=







are discussed





are discussed






1198780)119887lt 0 lt

(119901lowastlowast


0)119887lt 0 lt (Δ


0


0is



towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0






119864and

119903119873 For 1205731015840 gt 0 and 119903



119864= 2119903119873 Δ119901 increases


119864lt 2119903119873 Δ119901








119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










t

(t0 + infin)

pE

pE0

120573 gt 0 (120572 lt r)

120573 = 0 (120572 = r)

120573 lt 0 (120572 gt r)

(t0 + 0)









119864

increases with time



119873and 119862

119864 of Figure 5









119864 where119872

119864is






119864is

119889119872119864

119889119905= 119866119864minus 119866119878 (25)


119889

119889119905(119870119864) = 119866119870119878minus 119866119870119864 (26)



CN CE

MN rN ME rE

GE (extracted)pE

GS (selling)pS

GKE(pE) GKS(pS)


which becomes

119889


and finally

1

119901119878

sdot119889119901119878

119889119905+1

119866119878

sdot119889119866119878

119889119905= 119903119864minus119866119864sdot 119901119864

119866119878sdot 119901119878

sdot 119903119873 (28)






119873and 119862

119864 that is

1

119866119864

119889119866119864

119889119905=1

119866119878

119889119866119878

119889119905= 120572 (29)





1

119901119878

119889119901119878

119889119905+ 120572 = 119903

119864minus119901119864

119901119878

119903119873 (31)

or

119889119901119878

119889119905= 119901119878(119903119864minus 120572) minus 119901

119864sdot 119903119873 (32)


119901119878= 119901lowast

1198780exp (120573 sdot 119905) + [119901


1198780] exp (1205731015840 sdot 119905) (33)

where

1205731015840= 119903119864minus 120572 (34)


119901lowast

1198780=119903119873sdot 1199011198640

119903119864minus 119903119873

=119903119873sdot 1199011198640

1205731015840 minus 120573 (35)




119898given by

119905119898=

1

(120573 minus 1205731015840)ln[

1205731015840sdot (119901lowast

1198780minus 1199011198780)


1198780

] (36)


1205731015840(1205731015840minus 120573) (119901


1198780) lt 0 (37)



1198780is equal to

119901lowastlowast

1198780=

119901lowast

1198780(1205731015840minus 120573)

1205731015840=1199031198731199011198640

1205731015840 (38)




1198780for 1205731015840 gt 120573 (ie 119903

119864gt 119903119873) (39)

or


1198780for 1205731015840 lt 120573 (ie 119903

119864lt 119903119873) (40)

For 1199011198780= 119901lowast


119905119898= +infin (41)

for 1205731015840 gt 120573 or

119905119898= minusinfin (42)

for 1205731015840 lt 120573For 119903119873= 119903119864 that is 1205731015840 = 120573 119901

119878is given by


1198780120573119905) exp (120573119905) (43)




1198780)

119901lowastlowast

11987801205731015840

(44)




price 1199011198780is

1199011198780= 119901lowastlowast

1198780=119903119873sdot 1199011198640

1205731015840 (45)







(vi) Category 5 119903119873lt 0

t

0

pS120572 lt 0 120573 gt 0


(t0 + infin)

120572 = 0 120573 = 0

0 lt 120572 lt rE 120573 lt 0

(constant)

(t0 + 0)

Case 0-A 120572 lt rE 120573998400 gt 0

Case 0-B 120572 = rE 120573998400 = 0 120573 lt 0

Case 0-C 120572 gt rE 120573998400 lt 0 120573 lt 0

Category 0 rN = 0

pS0



Category 0 (119903119873


119873= 0 (ie 119901lowast




119901119878= 1199011198780exp (1205731015840119905) (46)



119864




119864 120573 lt 0


119878is



119878is







Category 1 (119903119873






is1199011198780ge 119901lowast



CIPES gt 0



1198780gt 0 or 119903

119873gt 119903119864gt


1198780gt 0 gt 119901

lowast





(t0 + infin)

(t0 + infin)





S0 gt 0


S0

pS

pS0

(plowastS0)apS0


0

tmax


S0 gt 0



S0)a








S0

pS0

t




119901lowast


1198780gt 0 or for 119903




1198780gt 0 gt 119901

lowast

1198780



1198780gt 1199011198780 or 119903119873gt 119903119864gt


gt 1199011198780gt 0 gt 119901

lowast

1198780 or 119903119873gt 120572 gt 119903




1198780gt

119901lowastlowast

1198780gt 1199011198780 or for 119903


1198780is smaller


lowast

1198780 or for 119903

119873gt 120572 gt 119903

119864


1198780gt 0 gt 119901

lowast


1198780



119873and 119903119864

Case 2-A (119903119864gt 120572 gt 119903



is 1199011198780ge

119901lowastlowast

1198780gt 119901lowast


119901lowastlowast

1198780gt 119901lowast

1198780

Case 2-B (119903119864gt 120572 gt 119903





119901lowastlowast

1198780gt 0



119873

or 120572 gt 119903119873gt 119903119864 or 120572 gt 119903

119873gt 119903119864






is 1199011198780


1198780= 119901lowast


119901lowastlowast

1198780= 119901lowast

1198780



1198780= 119901lowast


1198780=

119901lowast

1198780


119873= 120572 gt


119864gt


1198780is 1199011198780lt

119901lowastlowast

1198780= 119901lowast


1198780= 119901lowast

1198780


1198780




increase with time







119873= 119903119864lt






1198780

Category 5 (119903119873





t0

pS


(t0 + infin)

pS0


(plowastS0)ab

pS0


(plowastlowastS0 )a

(plowastlowastS0 )a

Case 2-ApS0 =










S0)d gt 0(t0 + 0)

(t0 minus 0)


(t0 minus infin)


Case 2-C 120572 gt rN ie 120573 lt 0






Case 5-A (119903119864gt 120572 gt 0 gt 119903



1198780because 119901

1198780gt 0 gt 119901

lowast




1198780

Case 5-B (120572 gt 119903119864gt 0 gt 119903




lowast

1198780


Case 5-C (120572 gt 119903119864gt 0 gt 119903




1198780gt 0 gt 119901

lowast




Δ119901 = (119901119878minus 119901119864)

= (1199011198780minus 119901lowast

1198780) exp (1205731015840119905) minus (119901


1198780) exp (120573119905)


0exp (120573119905)

(47)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (48)



119905119898=

1

(1205731015840 minus 120573)ln[


1198780minus 1199011198640)

1205731015840 sdot (119901lowast

1198780minus 1199011198780)]

=1

(1205731015840 minus 120573)ln[



=1

(120573 minus 1205731015840)ln[


1205731015840 sdot Δlowast1199010

]

(49)


1205731015840(1205731015840minus 120573) (Δ119901




0is equal to


(1205731015840minus 120573)

1205731015840] = 119901

1198640

(2119903119873minus 119903119864)

1205731015840 (51)


t0

pS

pS0

pS0

(plowastS0)a

(plowastS0)b







(t0plowastS0)b

Case 3-C

(t0 + infin)

(t0 minus infin)



t0

pS

pS0

pS0

pS0

(plowastlowastS0 )a

(plowastlowastS0)b

Category 4 rN = rE ie 120573998400 = 120573


(t0 minus infin)

Linear

tmax

tmin

t0

(t0 minus 0)







or rN = rE lt 120572 ie 120573998400 = 120573 lt 0

or rN = rE = 120572 ie 120573998400 = 120573 = 0







for 1205731015840 lt 120573(52)

For Δ1199011198780= Δ119901lowast


119905119898= +infin (53)


for 1205731015840 lt 120573


t

0

pS

pS0

pS0

pS0

(plowastlowastS0)a

(plowastS0)a

Category 5 rN lt 0


(t0 + infin)

Case 5-ApS0 gt 0


tmax


S0)a


(t0 minus infin)



The time 1199050is

1199050=

1

(1205731015840 minus 120573)ln[

Δlowast1199010

(Δlowast1199010minus Δ1199010)] (55)


Category 0D 119903119873= 0











1198780= 0


0= minus11990311986411990111986401205731015840


Δ119901 = (119901119878minus 119901119864) = 1199011198780exp (1205731015840119905) minus 119901

1198640exp (120573119905)


0exp (120573119905)

(56)


119905119898=

1

(1205731015840 minus 120573)ln[

120573 sdot 1199011198640

1205731015840 sdot 1199011198780

] (57)


for

1205731015840(1205731015840minus 120573) 119901

1198780lt 0 (58)


1199050=

1

(1205731015840 minus 120573)ln [

1199011198640

1199011198780

] (59)



1198780ge

1199011198640)




0gt Δ119901

0gt

Δlowast1199010


0= Δ119901

0or

1199011198780= 119901lowast

1198780



0)119886lt 0))










Δp0

0

pS0

0

Δp0pS0

(Δlowastlowastp0)a

Δp0

Δlowastp0

0D-A1120572 lt 0 120573 gt 0

0D-A1120572 = 0 120573 = 0

Δp0

(t0 + infin)

(t0 + infin)

Case 0D-A rE gt 120572 gt 0 0 gt 120573

0D-A1 120572 gt 0 0 gt 120573

0D-B1

0D-B1

0D-C1tmax

tmax (t0 + 0)

(t0 + 0)

(t0 + 0)

t

t

t

0D-A2

120572 lt 0 120573 gt 0

(t0 + infin) 120572 = 0 120573 = 0

0D-A20D-A2120572 gt 0 0 gt 120573

(t0pS0)

(t0pS0)

(t0pS0)

t0

t0

0D-C2

0D-A3120572 lt 0 120573 gt 0 Case 0D-B 120572 = rE 120573

998400 = 0 120573 lt 0

Case 0D-C 120572 gt rE 0 gt 120573998400 gt 120573

tmin

0D-C3

0D-C4

0D-B1

(t0 minus infin)






0gt Δ119901

0gt Δlowast1199010





lowast1199010lt 0))


1198780=


Case 0D-C (120572 gt 119903119864 (ie 120573 lt 120573


0)119888gt 0 gt

(Δlowast1199010)119888))







0






0

is Δ1199010gt Δlowast1199010 If 120572 = 119903



0ge Δlowast1199010 If 120572 gt 119903

119864 Δ119901 tends


Category 1D (119903119873


to 119903119864


1198780ge 0))





0)119886= (Δlowastlowast1199010)119886= 0




pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0


1D-A1 120572 gt 0



Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0




t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)





0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)


0le Δ1199010lt 0


0gt Δ119901

0gt

Δlowast1199010


119901lowast

1198780






0 If Δ119901

0=







are discussed





are discussed






1198780)119887lt 0 lt

(119901lowastlowast


0)119887lt 0 lt (Δ


0


0is



towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0






119864and

119903119873 For 1205731015840 gt 0 and 119903



119864= 2119903119873 Δ119901 increases


119864lt 2119903119873 Δ119901








119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119898given by

119905119898=

1

(120573 minus 1205731015840)ln[

1205731015840sdot (119901lowast

1198780minus 1199011198780)


1198780

] (36)


1205731015840(1205731015840minus 120573) (119901


1198780) lt 0 (37)



1198780is equal to

119901lowastlowast

1198780=

119901lowast

1198780(1205731015840minus 120573)

1205731015840=1199031198731199011198640

1205731015840 (38)




1198780for 1205731015840 gt 120573 (ie 119903

119864gt 119903119873) (39)

or


1198780for 1205731015840 lt 120573 (ie 119903

119864lt 119903119873) (40)

For 1199011198780= 119901lowast


119905119898= +infin (41)

for 1205731015840 gt 120573 or

119905119898= minusinfin (42)

for 1205731015840 lt 120573For 119903119873= 119903119864 that is 1205731015840 = 120573 119901

119878is given by


1198780120573119905) exp (120573119905) (43)




1198780)

119901lowastlowast

11987801205731015840

(44)




price 1199011198780is

1199011198780= 119901lowastlowast

1198780=119903119873sdot 1199011198640

1205731015840 (45)







(vi) Category 5 119903119873lt 0

t

0

pS120572 lt 0 120573 gt 0


(t0 + infin)

120572 = 0 120573 = 0

0 lt 120572 lt rE 120573 lt 0

(constant)

(t0 + 0)

Case 0-A 120572 lt rE 120573998400 gt 0

Case 0-B 120572 = rE 120573998400 = 0 120573 lt 0

Case 0-C 120572 gt rE 120573998400 lt 0 120573 lt 0

Category 0 rN = 0

pS0



Category 0 (119903119873


119873= 0 (ie 119901lowast




119901119878= 1199011198780exp (1205731015840119905) (46)



119864




119864 120573 lt 0


119878is



119878is







Category 1 (119903119873






is1199011198780ge 119901lowast



CIPES gt 0



1198780gt 0 or 119903

119873gt 119903119864gt


1198780gt 0 gt 119901

lowast





(t0 + infin)

(t0 + infin)





S0 gt 0


S0

pS

pS0

(plowastS0)apS0


0

tmax


S0 gt 0



S0)a








S0

pS0

t




119901lowast


1198780gt 0 or for 119903




1198780gt 0 gt 119901

lowast

1198780



1198780gt 1199011198780 or 119903119873gt 119903119864gt


gt 1199011198780gt 0 gt 119901

lowast

1198780 or 119903119873gt 120572 gt 119903




1198780gt

119901lowastlowast

1198780gt 1199011198780 or for 119903


1198780is smaller


lowast

1198780 or for 119903

119873gt 120572 gt 119903

119864


1198780gt 0 gt 119901

lowast


1198780



119873and 119903119864

Case 2-A (119903119864gt 120572 gt 119903



is 1199011198780ge

119901lowastlowast

1198780gt 119901lowast


119901lowastlowast

1198780gt 119901lowast

1198780

Case 2-B (119903119864gt 120572 gt 119903





119901lowastlowast

1198780gt 0



119873

or 120572 gt 119903119873gt 119903119864 or 120572 gt 119903

119873gt 119903119864






is 1199011198780


1198780= 119901lowast


119901lowastlowast

1198780= 119901lowast

1198780



1198780= 119901lowast


1198780=

119901lowast

1198780


119873= 120572 gt


119864gt


1198780is 1199011198780lt

119901lowastlowast

1198780= 119901lowast


1198780= 119901lowast

1198780


1198780




increase with time







119873= 119903119864lt






1198780

Category 5 (119903119873





t0

pS


(t0 + infin)

pS0


(plowastS0)ab

pS0


(plowastlowastS0 )a

(plowastlowastS0 )a

Case 2-ApS0 =










S0)d gt 0(t0 + 0)

(t0 minus 0)


(t0 minus infin)


Case 2-C 120572 gt rN ie 120573 lt 0






Case 5-A (119903119864gt 120572 gt 0 gt 119903



1198780because 119901

1198780gt 0 gt 119901

lowast




1198780

Case 5-B (120572 gt 119903119864gt 0 gt 119903




lowast

1198780


Case 5-C (120572 gt 119903119864gt 0 gt 119903




1198780gt 0 gt 119901

lowast




Δ119901 = (119901119878minus 119901119864)

= (1199011198780minus 119901lowast

1198780) exp (1205731015840119905) minus (119901


1198780) exp (120573119905)


0exp (120573119905)

(47)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (48)



119905119898=

1

(1205731015840 minus 120573)ln[


1198780minus 1199011198640)

1205731015840 sdot (119901lowast

1198780minus 1199011198780)]

=1

(1205731015840 minus 120573)ln[



=1

(120573 minus 1205731015840)ln[


1205731015840 sdot Δlowast1199010

]

(49)


1205731015840(1205731015840minus 120573) (Δ119901




0is equal to


(1205731015840minus 120573)

1205731015840] = 119901

1198640

(2119903119873minus 119903119864)

1205731015840 (51)


t0

pS

pS0

pS0

(plowastS0)a

(plowastS0)b







(t0plowastS0)b

Case 3-C

(t0 + infin)

(t0 minus infin)



t0

pS

pS0

pS0

pS0

(plowastlowastS0 )a

(plowastlowastS0)b

Category 4 rN = rE ie 120573998400 = 120573


(t0 minus infin)

Linear

tmax

tmin

t0

(t0 minus 0)







or rN = rE lt 120572 ie 120573998400 = 120573 lt 0

or rN = rE = 120572 ie 120573998400 = 120573 = 0







for 1205731015840 lt 120573(52)

For Δ1199011198780= Δ119901lowast


119905119898= +infin (53)


for 1205731015840 lt 120573


t

0

pS

pS0

pS0

pS0

(plowastlowastS0)a

(plowastS0)a

Category 5 rN lt 0


(t0 + infin)

Case 5-ApS0 gt 0


tmax


S0)a


(t0 minus infin)



The time 1199050is

1199050=

1

(1205731015840 minus 120573)ln[

Δlowast1199010

(Δlowast1199010minus Δ1199010)] (55)


Category 0D 119903119873= 0











1198780= 0


0= minus11990311986411990111986401205731015840


Δ119901 = (119901119878minus 119901119864) = 1199011198780exp (1205731015840119905) minus 119901

1198640exp (120573119905)


0exp (120573119905)

(56)


119905119898=

1

(1205731015840 minus 120573)ln[

120573 sdot 1199011198640

1205731015840 sdot 1199011198780

] (57)


for

1205731015840(1205731015840minus 120573) 119901

1198780lt 0 (58)


1199050=

1

(1205731015840 minus 120573)ln [

1199011198640

1199011198780

] (59)



1198780ge

1199011198640)




0gt Δ119901

0gt

Δlowast1199010


0= Δ119901

0or

1199011198780= 119901lowast

1198780



0)119886lt 0))










Δp0

0

pS0

0

Δp0pS0

(Δlowastlowastp0)a

Δp0

Δlowastp0

0D-A1120572 lt 0 120573 gt 0

0D-A1120572 = 0 120573 = 0

Δp0

(t0 + infin)

(t0 + infin)

Case 0D-A rE gt 120572 gt 0 0 gt 120573

0D-A1 120572 gt 0 0 gt 120573

0D-B1

0D-B1

0D-C1tmax

tmax (t0 + 0)

(t0 + 0)

(t0 + 0)

t

t

t

0D-A2

120572 lt 0 120573 gt 0

(t0 + infin) 120572 = 0 120573 = 0

0D-A20D-A2120572 gt 0 0 gt 120573

(t0pS0)

(t0pS0)

(t0pS0)

t0

t0

0D-C2

0D-A3120572 lt 0 120573 gt 0 Case 0D-B 120572 = rE 120573

998400 = 0 120573 lt 0

Case 0D-C 120572 gt rE 0 gt 120573998400 gt 120573

tmin

0D-C3

0D-C4

0D-B1

(t0 minus infin)






0gt Δ119901

0gt Δlowast1199010





lowast1199010lt 0))


1198780=


Case 0D-C (120572 gt 119903119864 (ie 120573 lt 120573


0)119888gt 0 gt

(Δlowast1199010)119888))







0






0

is Δ1199010gt Δlowast1199010 If 120572 = 119903



0ge Δlowast1199010 If 120572 gt 119903

119864 Δ119901 tends


Category 1D (119903119873


to 119903119864


1198780ge 0))





0)119886= (Δlowastlowast1199010)119886= 0




pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0


1D-A1 120572 gt 0



Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0




t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)





0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)


0le Δ1199010lt 0


0gt Δ119901

0gt

Δlowast1199010


119901lowast

1198780






0 If Δ119901

0=







are discussed





are discussed






1198780)119887lt 0 lt

(119901lowastlowast


0)119887lt 0 lt (Δ


0


0is



towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0






119864and

119903119873 For 1205731015840 gt 0 and 119903



119864= 2119903119873 Δ119901 increases


119864lt 2119903119873 Δ119901








119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(t0 + infin)

(t0 + infin)





S0 gt 0


S0

pS

pS0

(plowastS0)apS0


0

tmax


S0 gt 0



S0)a








S0

pS0

t




119901lowast


1198780gt 0 or for 119903




1198780gt 0 gt 119901

lowast

1198780



1198780gt 1199011198780 or 119903119873gt 119903119864gt


gt 1199011198780gt 0 gt 119901

lowast

1198780 or 119903119873gt 120572 gt 119903




1198780gt

119901lowastlowast

1198780gt 1199011198780 or for 119903


1198780is smaller


lowast

1198780 or for 119903

119873gt 120572 gt 119903

119864


1198780gt 0 gt 119901

lowast


1198780



119873and 119903119864

Case 2-A (119903119864gt 120572 gt 119903



is 1199011198780ge

119901lowastlowast

1198780gt 119901lowast


119901lowastlowast

1198780gt 119901lowast

1198780

Case 2-B (119903119864gt 120572 gt 119903





119901lowastlowast

1198780gt 0



119873

or 120572 gt 119903119873gt 119903119864 or 120572 gt 119903

119873gt 119903119864






is 1199011198780


1198780= 119901lowast


119901lowastlowast

1198780= 119901lowast

1198780



1198780= 119901lowast


1198780=

119901lowast

1198780


119873= 120572 gt


119864gt


1198780is 1199011198780lt

119901lowastlowast

1198780= 119901lowast


1198780= 119901lowast

1198780


1198780




increase with time







119873= 119903119864lt






1198780

Category 5 (119903119873





t0

pS


(t0 + infin)

pS0


(plowastS0)ab

pS0


(plowastlowastS0 )a

(plowastlowastS0 )a

Case 2-ApS0 =










S0)d gt 0(t0 + 0)

(t0 minus 0)


(t0 minus infin)


Case 2-C 120572 gt rN ie 120573 lt 0






Case 5-A (119903119864gt 120572 gt 0 gt 119903



1198780because 119901

1198780gt 0 gt 119901

lowast




1198780

Case 5-B (120572 gt 119903119864gt 0 gt 119903




lowast

1198780


Case 5-C (120572 gt 119903119864gt 0 gt 119903




1198780gt 0 gt 119901

lowast




Δ119901 = (119901119878minus 119901119864)

= (1199011198780minus 119901lowast

1198780) exp (1205731015840119905) minus (119901


1198780) exp (120573119905)


0exp (120573119905)

(47)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (48)



119905119898=

1

(1205731015840 minus 120573)ln[


1198780minus 1199011198640)

1205731015840 sdot (119901lowast

1198780minus 1199011198780)]

=1

(1205731015840 minus 120573)ln[



=1

(120573 minus 1205731015840)ln[


1205731015840 sdot Δlowast1199010

]

(49)


1205731015840(1205731015840minus 120573) (Δ119901




0is equal to


(1205731015840minus 120573)

1205731015840] = 119901

1198640

(2119903119873minus 119903119864)

1205731015840 (51)


t0

pS

pS0

pS0

(plowastS0)a

(plowastS0)b







(t0plowastS0)b

Case 3-C

(t0 + infin)

(t0 minus infin)



t0

pS

pS0

pS0

pS0

(plowastlowastS0 )a

(plowastlowastS0)b

Category 4 rN = rE ie 120573998400 = 120573


(t0 minus infin)

Linear

tmax

tmin

t0

(t0 minus 0)







or rN = rE lt 120572 ie 120573998400 = 120573 lt 0

or rN = rE = 120572 ie 120573998400 = 120573 = 0







for 1205731015840 lt 120573(52)

For Δ1199011198780= Δ119901lowast


119905119898= +infin (53)


for 1205731015840 lt 120573


t

0

pS

pS0

pS0

pS0

(plowastlowastS0)a

(plowastS0)a

Category 5 rN lt 0


(t0 + infin)

Case 5-ApS0 gt 0


tmax


S0)a


(t0 minus infin)



The time 1199050is

1199050=

1

(1205731015840 minus 120573)ln[

Δlowast1199010

(Δlowast1199010minus Δ1199010)] (55)


Category 0D 119903119873= 0











1198780= 0


0= minus11990311986411990111986401205731015840


Δ119901 = (119901119878minus 119901119864) = 1199011198780exp (1205731015840119905) minus 119901

1198640exp (120573119905)


0exp (120573119905)

(56)


119905119898=

1

(1205731015840 minus 120573)ln[

120573 sdot 1199011198640

1205731015840 sdot 1199011198780

] (57)


for

1205731015840(1205731015840minus 120573) 119901

1198780lt 0 (58)


1199050=

1

(1205731015840 minus 120573)ln [

1199011198640

1199011198780

] (59)



1198780ge

1199011198640)




0gt Δ119901

0gt

Δlowast1199010


0= Δ119901

0or

1199011198780= 119901lowast

1198780



0)119886lt 0))










Δp0

0

pS0

0

Δp0pS0

(Δlowastlowastp0)a

Δp0

Δlowastp0

0D-A1120572 lt 0 120573 gt 0

0D-A1120572 = 0 120573 = 0

Δp0

(t0 + infin)

(t0 + infin)

Case 0D-A rE gt 120572 gt 0 0 gt 120573

0D-A1 120572 gt 0 0 gt 120573

0D-B1

0D-B1

0D-C1tmax

tmax (t0 + 0)

(t0 + 0)

(t0 + 0)

t

t

t

0D-A2

120572 lt 0 120573 gt 0

(t0 + infin) 120572 = 0 120573 = 0

0D-A20D-A2120572 gt 0 0 gt 120573

(t0pS0)

(t0pS0)

(t0pS0)

t0

t0

0D-C2

0D-A3120572 lt 0 120573 gt 0 Case 0D-B 120572 = rE 120573

998400 = 0 120573 lt 0

Case 0D-C 120572 gt rE 0 gt 120573998400 gt 120573

tmin

0D-C3

0D-C4

0D-B1

(t0 minus infin)






0gt Δ119901

0gt Δlowast1199010





lowast1199010lt 0))


1198780=


Case 0D-C (120572 gt 119903119864 (ie 120573 lt 120573


0)119888gt 0 gt

(Δlowast1199010)119888))







0






0

is Δ1199010gt Δlowast1199010 If 120572 = 119903



0ge Δlowast1199010 If 120572 gt 119903

119864 Δ119901 tends


Category 1D (119903119873


to 119903119864


1198780ge 0))





0)119886= (Δlowastlowast1199010)119886= 0




pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0


1D-A1 120572 gt 0



Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0




t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)





0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)


0le Δ1199010lt 0


0gt Δ119901

0gt

Δlowast1199010


119901lowast

1198780






0 If Δ119901

0=







are discussed





are discussed






1198780)119887lt 0 lt

(119901lowastlowast


0)119887lt 0 lt (Δ


0


0is



towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0






119864and

119903119873 For 1205731015840 gt 0 and 119903



119864= 2119903119873 Δ119901 increases


119864lt 2119903119873 Δ119901








119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










t0

pS


(t0 + infin)

pS0


(plowastS0)ab

pS0


(plowastlowastS0 )a

(plowastlowastS0 )a

Case 2-ApS0 =










S0)d gt 0(t0 + 0)

(t0 minus 0)


(t0 minus infin)


Case 2-C 120572 gt rN ie 120573 lt 0






Case 5-A (119903119864gt 120572 gt 0 gt 119903



1198780because 119901

1198780gt 0 gt 119901

lowast




1198780

Case 5-B (120572 gt 119903119864gt 0 gt 119903




lowast

1198780


Case 5-C (120572 gt 119903119864gt 0 gt 119903




1198780gt 0 gt 119901

lowast




Δ119901 = (119901119878minus 119901119864)

= (1199011198780minus 119901lowast

1198780) exp (1205731015840119905) minus (119901


1198780) exp (120573119905)


0exp (120573119905)

(47)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (48)



119905119898=

1

(1205731015840 minus 120573)ln[


1198780minus 1199011198640)

1205731015840 sdot (119901lowast

1198780minus 1199011198780)]

=1

(1205731015840 minus 120573)ln[



=1

(120573 minus 1205731015840)ln[


1205731015840 sdot Δlowast1199010

]

(49)


1205731015840(1205731015840minus 120573) (Δ119901




0is equal to


(1205731015840minus 120573)

1205731015840] = 119901

1198640

(2119903119873minus 119903119864)

1205731015840 (51)


t0

pS

pS0

pS0

(plowastS0)a

(plowastS0)b







(t0plowastS0)b

Case 3-C

(t0 + infin)

(t0 minus infin)



t0

pS

pS0

pS0

pS0

(plowastlowastS0 )a

(plowastlowastS0)b

Category 4 rN = rE ie 120573998400 = 120573


(t0 minus infin)

Linear

tmax

tmin

t0

(t0 minus 0)







or rN = rE lt 120572 ie 120573998400 = 120573 lt 0

or rN = rE = 120572 ie 120573998400 = 120573 = 0







for 1205731015840 lt 120573(52)

For Δ1199011198780= Δ119901lowast


119905119898= +infin (53)


for 1205731015840 lt 120573


t

0

pS

pS0

pS0

pS0

(plowastlowastS0)a

(plowastS0)a

Category 5 rN lt 0


(t0 + infin)

Case 5-ApS0 gt 0


tmax


S0)a


(t0 minus infin)



The time 1199050is

1199050=

1

(1205731015840 minus 120573)ln[

Δlowast1199010

(Δlowast1199010minus Δ1199010)] (55)


Category 0D 119903119873= 0











1198780= 0


0= minus11990311986411990111986401205731015840


Δ119901 = (119901119878minus 119901119864) = 1199011198780exp (1205731015840119905) minus 119901

1198640exp (120573119905)


0exp (120573119905)

(56)


119905119898=

1

(1205731015840 minus 120573)ln[

120573 sdot 1199011198640

1205731015840 sdot 1199011198780

] (57)


for

1205731015840(1205731015840minus 120573) 119901

1198780lt 0 (58)


1199050=

1

(1205731015840 minus 120573)ln [

1199011198640

1199011198780

] (59)



1198780ge

1199011198640)




0gt Δ119901

0gt

Δlowast1199010


0= Δ119901

0or

1199011198780= 119901lowast

1198780



0)119886lt 0))










Δp0

0

pS0

0

Δp0pS0

(Δlowastlowastp0)a

Δp0

Δlowastp0

0D-A1120572 lt 0 120573 gt 0

0D-A1120572 = 0 120573 = 0

Δp0

(t0 + infin)

(t0 + infin)

Case 0D-A rE gt 120572 gt 0 0 gt 120573

0D-A1 120572 gt 0 0 gt 120573

0D-B1

0D-B1

0D-C1tmax

tmax (t0 + 0)

(t0 + 0)

(t0 + 0)

t

t

t

0D-A2

120572 lt 0 120573 gt 0

(t0 + infin) 120572 = 0 120573 = 0

0D-A20D-A2120572 gt 0 0 gt 120573

(t0pS0)

(t0pS0)

(t0pS0)

t0

t0

0D-C2

0D-A3120572 lt 0 120573 gt 0 Case 0D-B 120572 = rE 120573

998400 = 0 120573 lt 0

Case 0D-C 120572 gt rE 0 gt 120573998400 gt 120573

tmin

0D-C3

0D-C4

0D-B1

(t0 minus infin)






0gt Δ119901

0gt Δlowast1199010





lowast1199010lt 0))


1198780=


Case 0D-C (120572 gt 119903119864 (ie 120573 lt 120573


0)119888gt 0 gt

(Δlowast1199010)119888))







0






0

is Δ1199010gt Δlowast1199010 If 120572 = 119903



0ge Δlowast1199010 If 120572 gt 119903

119864 Δ119901 tends


Category 1D (119903119873


to 119903119864


1198780ge 0))





0)119886= (Δlowastlowast1199010)119886= 0




pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0


1D-A1 120572 gt 0



Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0




t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)





0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)


0le Δ1199010lt 0


0gt Δ119901

0gt

Δlowast1199010


119901lowast

1198780






0 If Δ119901

0=







are discussed





are discussed






1198780)119887lt 0 lt

(119901lowastlowast


0)119887lt 0 lt (Δ


0


0is



towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0






119864and

119903119873 For 1205731015840 gt 0 and 119903



119864= 2119903119873 Δ119901 increases


119864lt 2119903119873 Δ119901








119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










t0

pS

pS0

pS0

(plowastS0)a

(plowastS0)b







(t0plowastS0)b

Case 3-C

(t0 + infin)

(t0 minus infin)



t0

pS

pS0

pS0

pS0

(plowastlowastS0 )a

(plowastlowastS0)b

Category 4 rN = rE ie 120573998400 = 120573


(t0 minus infin)

Linear

tmax

tmin

t0

(t0 minus 0)







or rN = rE lt 120572 ie 120573998400 = 120573 lt 0

or rN = rE = 120572 ie 120573998400 = 120573 = 0







for 1205731015840 lt 120573(52)

For Δ1199011198780= Δ119901lowast


119905119898= +infin (53)


for 1205731015840 lt 120573


t

0

pS

pS0

pS0

pS0

(plowastlowastS0)a

(plowastS0)a

Category 5 rN lt 0


(t0 + infin)

Case 5-ApS0 gt 0


tmax


S0)a


(t0 minus infin)



The time 1199050is

1199050=

1

(1205731015840 minus 120573)ln[

Δlowast1199010

(Δlowast1199010minus Δ1199010)] (55)


Category 0D 119903119873= 0











1198780= 0


0= minus11990311986411990111986401205731015840


Δ119901 = (119901119878minus 119901119864) = 1199011198780exp (1205731015840119905) minus 119901

1198640exp (120573119905)


0exp (120573119905)

(56)


119905119898=

1

(1205731015840 minus 120573)ln[

120573 sdot 1199011198640

1205731015840 sdot 1199011198780

] (57)


for

1205731015840(1205731015840minus 120573) 119901

1198780lt 0 (58)


1199050=

1

(1205731015840 minus 120573)ln [

1199011198640

1199011198780

] (59)



1198780ge

1199011198640)




0gt Δ119901

0gt

Δlowast1199010


0= Δ119901

0or

1199011198780= 119901lowast

1198780



0)119886lt 0))










Δp0

0

pS0

0

Δp0pS0

(Δlowastlowastp0)a

Δp0

Δlowastp0

0D-A1120572 lt 0 120573 gt 0

0D-A1120572 = 0 120573 = 0

Δp0

(t0 + infin)

(t0 + infin)

Case 0D-A rE gt 120572 gt 0 0 gt 120573

0D-A1 120572 gt 0 0 gt 120573

0D-B1

0D-B1

0D-C1tmax

tmax (t0 + 0)

(t0 + 0)

(t0 + 0)

t

t

t

0D-A2

120572 lt 0 120573 gt 0

(t0 + infin) 120572 = 0 120573 = 0

0D-A20D-A2120572 gt 0 0 gt 120573

(t0pS0)

(t0pS0)

(t0pS0)

t0

t0

0D-C2

0D-A3120572 lt 0 120573 gt 0 Case 0D-B 120572 = rE 120573

998400 = 0 120573 lt 0

Case 0D-C 120572 gt rE 0 gt 120573998400 gt 120573

tmin

0D-C3

0D-C4

0D-B1

(t0 minus infin)






0gt Δ119901

0gt Δlowast1199010





lowast1199010lt 0))


1198780=


Case 0D-C (120572 gt 119903119864 (ie 120573 lt 120573


0)119888gt 0 gt

(Δlowast1199010)119888))







0






0

is Δ1199010gt Δlowast1199010 If 120572 = 119903



0ge Δlowast1199010 If 120572 gt 119903

119864 Δ119901 tends


Category 1D (119903119873


to 119903119864


1198780ge 0))





0)119886= (Δlowastlowast1199010)119886= 0




pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0


1D-A1 120572 gt 0



Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0




t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)





0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)


0le Δ1199010lt 0


0gt Δ119901

0gt

Δlowast1199010


119901lowast

1198780






0 If Δ119901

0=







are discussed





are discussed






1198780)119887lt 0 lt

(119901lowastlowast


0)119887lt 0 lt (Δ


0


0is



towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0






119864and

119903119873 For 1205731015840 gt 0 and 119903



119864= 2119903119873 Δ119901 increases


119864lt 2119903119873 Δ119901








119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










t

0

pS

pS0

pS0

pS0

(plowastlowastS0)a

(plowastS0)a

Category 5 rN lt 0


(t0 + infin)

Case 5-ApS0 gt 0


tmax


S0)a


(t0 minus infin)



The time 1199050is

1199050=

1

(1205731015840 minus 120573)ln[

Δlowast1199010

(Δlowast1199010minus Δ1199010)] (55)


Category 0D 119903119873= 0











1198780= 0


0= minus11990311986411990111986401205731015840


Δ119901 = (119901119878minus 119901119864) = 1199011198780exp (1205731015840119905) minus 119901

1198640exp (120573119905)


0exp (120573119905)

(56)


119905119898=

1

(1205731015840 minus 120573)ln[

120573 sdot 1199011198640

1205731015840 sdot 1199011198780

] (57)


for

1205731015840(1205731015840minus 120573) 119901

1198780lt 0 (58)


1199050=

1

(1205731015840 minus 120573)ln [

1199011198640

1199011198780

] (59)



1198780ge

1199011198640)




0gt Δ119901

0gt

Δlowast1199010


0= Δ119901

0or

1199011198780= 119901lowast

1198780



0)119886lt 0))










Δp0

0

pS0

0

Δp0pS0

(Δlowastlowastp0)a

Δp0

Δlowastp0

0D-A1120572 lt 0 120573 gt 0

0D-A1120572 = 0 120573 = 0

Δp0

(t0 + infin)

(t0 + infin)

Case 0D-A rE gt 120572 gt 0 0 gt 120573

0D-A1 120572 gt 0 0 gt 120573

0D-B1

0D-B1

0D-C1tmax

tmax (t0 + 0)

(t0 + 0)

(t0 + 0)

t

t

t

0D-A2

120572 lt 0 120573 gt 0

(t0 + infin) 120572 = 0 120573 = 0

0D-A20D-A2120572 gt 0 0 gt 120573

(t0pS0)

(t0pS0)

(t0pS0)

t0

t0

0D-C2

0D-A3120572 lt 0 120573 gt 0 Case 0D-B 120572 = rE 120573

998400 = 0 120573 lt 0

Case 0D-C 120572 gt rE 0 gt 120573998400 gt 120573

tmin

0D-C3

0D-C4

0D-B1

(t0 minus infin)






0gt Δ119901

0gt Δlowast1199010





lowast1199010lt 0))


1198780=


Case 0D-C (120572 gt 119903119864 (ie 120573 lt 120573


0)119888gt 0 gt

(Δlowast1199010)119888))







0






0

is Δ1199010gt Δlowast1199010 If 120572 = 119903



0ge Δlowast1199010 If 120572 gt 119903

119864 Δ119901 tends


Category 1D (119903119873


to 119903119864


1198780ge 0))





0)119886= (Δlowastlowast1199010)119886= 0




pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0


1D-A1 120572 gt 0



Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0




t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)





0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)


0le Δ1199010lt 0


0gt Δ119901

0gt

Δlowast1199010


119901lowast

1198780






0 If Δ119901

0=







are discussed





are discussed






1198780)119887lt 0 lt

(119901lowastlowast


0)119887lt 0 lt (Δ


0


0is



towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0






119864and

119903119873 For 1205731015840 gt 0 and 119903



119864= 2119903119873 Δ119901 increases


119864lt 2119903119873 Δ119901








119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Δp0

0

pS0

0

Δp0pS0

(Δlowastlowastp0)a

Δp0

Δlowastp0

0D-A1120572 lt 0 120573 gt 0

0D-A1120572 = 0 120573 = 0

Δp0

(t0 + infin)

(t0 + infin)

Case 0D-A rE gt 120572 gt 0 0 gt 120573

0D-A1 120572 gt 0 0 gt 120573

0D-B1

0D-B1

0D-C1tmax

tmax (t0 + 0)

(t0 + 0)

(t0 + 0)

t

t

t

0D-A2

120572 lt 0 120573 gt 0

(t0 + infin) 120572 = 0 120573 = 0

0D-A20D-A2120572 gt 0 0 gt 120573

(t0pS0)

(t0pS0)

(t0pS0)

t0

t0

0D-C2

0D-A3120572 lt 0 120573 gt 0 Case 0D-B 120572 = rE 120573

998400 = 0 120573 lt 0

Case 0D-C 120572 gt rE 0 gt 120573998400 gt 120573

tmin

0D-C3

0D-C4

0D-B1

(t0 minus infin)






0gt Δ119901

0gt Δlowast1199010





lowast1199010lt 0))


1198780=


Case 0D-C (120572 gt 119903119864 (ie 120573 lt 120573


0)119888gt 0 gt

(Δlowast1199010)119888))







0






0

is Δ1199010gt Δlowast1199010 If 120572 = 119903



0ge Δlowast1199010 If 120572 gt 119903

119864 Δ119901 tends


Category 1D (119903119873


to 119903119864


1198780ge 0))





0)119886= (Δlowastlowast1199010)119886= 0




pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0


1D-A1 120572 gt 0



Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0




t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)





0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)


0le Δ1199010lt 0


0gt Δ119901

0gt

Δlowast1199010


119901lowast

1198780






0 If Δ119901

0=







are discussed





are discussed






1198780)119887lt 0 lt

(119901lowastlowast


0)119887lt 0 lt (Δ


0


0is



towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0






119864and

119903119873 For 1205731015840 gt 0 and 119903



119864= 2119903119873 Δ119901 increases


119864lt 2119903119873 Δ119901








119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










pS minus pE

Δp0

0

0

Δp0

(Δlowastlowastp0)a

Δp0

Δlowastp0


1D-A1 120572 gt 0



Case 1D-B

1D-A2 120572 lt 01D-A2 120572 = 0

(t0 + infin)1D-A2 120572 gt 0




t

t

t0

1D-A2

(t0 + infin)1D-A3

tmin

1D-C1D-C1D-C

(t0 minus infin)





0is Δ1199010ge 0 (or 119901

1198780ge

1199011198640)


0le Δ1199010lt 0


0gt Δ119901

0gt

Δlowast1199010


119901lowast

1198780






0 If Δ119901

0=







are discussed





are discussed






1198780)119887lt 0 lt

(119901lowastlowast


0)119887lt 0 lt (Δ


0


0is



towards minusinfin

Case 1D-C2 (119903119864lt 120572 lt 119903

119873 (ie 1205731015840 lt 0 lt 120573 (119901lowast

1198780)119888lt 0






119864and

119903119873 For 1205731015840 gt 0 and 119903



119864= 2119903119873 Δ119901 increases


119864lt 2119903119873 Δ119901








119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119903119873and 119903119864


Case 2D-A (119903119864gt 120572 gt 119903

119873 (ie 1205731015840 gt 0 gt 120573)) If 2119903

119873lt 119903119864 that








0gt Δ119901






towards +infin


0= infin


119873gt 119903119864 Δlowast1199010= 0 for 2119903

119873= 119903119864 and Δlowast119901

0lt

0 for 2119903119873lt 119903119864))









Case 2D-C (120572 gt 119903119864gt 119903119873 (ie 0 gt 120573

1015840gt 120573 Δlowast119901

0gt


0lt 0 lt Δ



0= Δlowast1199010= 0 for 2119903

119873= 119903119864))










0gt Δ119901



119864= 2119903119873 Δ119901 decreases to +0 if




119864lt 2119903119873 Δ119901



0lt Δ119901

lowastlowast

0lt








0gt





119864


119864gt 2119903119873


maximum



0and the










towards minusinfin



0


119873and 119903119864and the




0= Δlowast1199010 and Δ119901


119873gt 119903119864 Δ119901 tends




Δ119901 = (119901119878minus 119901119864)

= [(1199011198780minus 1199031198641199011198640119905) minus 119901

1198640] exp (1205731015840119905)

= (Δ1199010minus 1199031198641199011198640119905) exp (1205731015840119905)


11987801205731015840119905) exp (1205731015840119905)

(60)






119898

given by


1198780)


1198780

(61)



1198780=1199031198641199011198640

1205731015840 (62)






Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Case 2D-B 120572 = rE gt rN 120573998400 = 0 gt 120573




2D-C2




t(t0 minus 0)

2D-D





(t0 minus infin)


pS minus pE

Δp0

Δlowastlowastp0


Δlowastp0

Δp0

0


Δlowastp0

Δp0

Δlowastlowastp0



Case 3D-1 rE gt rN 120573998400 lt 120573 = 0

Case 3D-2 rN gt rE 120573998400 lt 120573 = 0


t

3D-2

pS minus pE

Δp0

(Δlowastp0)a

Δp0

(Δlowastp0)b







1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1198780 119901

lowastlowast

11987801205731015840gt 0


1198780 119901

lowastlowast

11987801205731015840lt 0

(63)


1199050=(1199011198780minus 1199011198640)

(1199031198641199011198640)

=Δ1199010

(1199031198641199011198640)=

Δ1199010

(119901lowastlowast

11987801205731015840)

(64)


0= (1199011198780minus 1199011198640) and the


Case 4D-1 (119903119873= 119903119864gt 120572 (ie 1205731015840 = 120573 gt 0))








1198780 the



1198780lt 0))












119878minus 119901119864) in




1205731015840gt 120573 119901lowast


1198780= 0Δlowast119901


0= minus11990311986411990111986401205731015840




1198640


120573 lt 0(ii) the price 119901


if 120572 le 0 lt 119903119864 (ie 1205731015840 gt 0) 119901


1198780

if 120572 = 119903119864 (ie 1205731015840 = 0) and 119901


120572 gt 119903119864 (ie 1205731015840 lt 0)


value Δ1199010 If Δ119901






minusinfin


0gt



1198640while



0because










1015840gt 120573) the



lowast119901min





119864gt 119903119873 (ie

1205731015840gt 120573 gt 0) and 119901

1198780ge 119901lowast

1198780= 119901lowast

119878max



119873lt 119903119864and 119901lowast

119878max = 119901lowast


1198780= 119901lowast

119878min119903119873= 119903119864and 119901


1198780 and 119903

119873gt 119903119864and 119901


1198780=

119901lowast

119878max


cases



119873and 119903119864 If





0gt Δlowast1199010






0= Δlowast1199010= 0 and Δ119901


119864lt 2119903119873 the









1198780




119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119864 and difference Δ119901 = (119901

119878minus 119901119864) for 119903

119873= 0 (ie 119901lowast

1198780= 119901lowastlowast1198780

=0 1205731015840 gt 120573 Δlowast119901


0= minus11990311986411990111986401205731015840)


119873minus 120572


Δ1199010


119878

(infin)


119864

(infin)


120572 lt 0 1205731015840gt 120573 gt 0




1199010gt 119901lowast





120572 = 0 lt 119903119864 1205731015840gt 120573 = 0 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 119901

1198640


lowast1199010


lowast1199010

Δ1199010

0 lt 120572 lt 119903119864 1205731015840gt 0 gt 120573 (Δ


lowast119901max =


1199010gt 119901lowast

1198780+infin 0


lowast1199010


lowast1199010

00 lt 120572 = 119903

119864 1205731015840= 0 gt 120573 (119901

lowastlowast

1198780undefined



1199010gt 119901lowast

11987801199011198780


lowast1199010

1199011198780= Δ1199010minusΔlowast1199010

120572 gt 119903119864 120573 lt 120573

1015840lt 0 (Δ



lowast1199010=

Δlowast119901min)

1199010gt 119901lowast

11987800 0





119864 and difference Δ119901 = (119901


extraction 120572 lt 119903119873 120573 gt 0


119864minus 120572

120573 = 119903119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840gt 120573 (119901lowast

1198780= 119901lowast119878max gt 119901

lowast


1198780gt 0)

1199011198780ge 119901lowast




0 lt 1199011198780le 119901lowast


2119903119873lt 119903119864 1205731015840 gt 120573







2119903119873= 119903119864 1205731015840 gt 120573






lowast1199010


119903119873lt 119903119864 2119903119873gt 119903119864 1205731015840gt 120573







119903119873= 119903119864gt 120572 120573

1015840= 120573 (119901lowastlowast

1198780gt 0)









119903119873gt 119903119864gt 120572 120573

1015840lt 120573 (119901lowast


1198780gt 0 gt 119901

lowast

1198780= 119901lowast

119878min)







Every Δ1199010

+infin minusinfin

119903119864= 120572 lt 119903

119873 1205731015840= 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max = infin)



1199011198780gt 0

Every Δ1199010

minusinfin +infin

+infin minusinfin


t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










t

pS minus pE

Δp0

(plowastlowastS0)a

Δp0

(plowastlowastS0)b

0

(t0 minus infin)


Category 4D 120573 = 120573998400 rN = rE


S0)a gt 0


S0)b lt 0



S0

tmaxtmin


(t0 + 0)




t0

Case 4D-3



119864 and difference Δ119901 = (119901


0 lt 120572 = 119903119873 (120573 = 0 119901lowast




Extraction rate120572 = 119903119873 120573 = 0

1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0


119878

(infin)


119864

(infin)


(infin)

119903119873lt 119903119864 1205731015840 gt 120573 (119901lowast

119878min = 119901lowast

1198780= 119901lowast


1198780gt 0)


1199011198780gt 119901lowast

119878max +infin 1199011198640

1199011198780= 119901lowast


11987801199011198640

0 lt 1199011198780lt 119901lowast

119878min minusinfin 1199011198640

Δ1199010gt Δlowast119901max 119901

1198640+infin

Δ1199010= Δlowast119901min 119901

1198640Δlowast1199010

Δ1199010lt Δlowast119901min 119901

1198640minusinfin

120572 = 119903119873= 119903119864 1205731015840 = 120573 (119901lowastlowast

1198780= infin) Every 119901


1198640

Every Δ1199010

1199011198640

minusinfin (Lin)

119903119873gt 119903119864 1205731015840lt 120573 (119901lowast

119878min = 119901lowast


1198780= 119901lowast



1199011198780gt 0

Every Δ1199010

119901lowast

11987801199011198640

1199011198640 Δ

lowast1199010




1198640 because 120573 = 0


119873lt 119903119864 and

1199011198780gt 119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878min


1198780for 119903119873lt 119903119864and 119901

1198780=

119901lowast

1198780= 119901lowast



119873lt 119903119864and

0 lt 1199011198780lt 119901lowast

1198780= 119901lowast


119873= 119903119864and every 119901

1198780gt 0


119903119864 and Δ119901



119873lt

119903119864 and Δ119901


119873gt 119903119864and every Δ119901

0



119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119864 and difference Δ119901 = (119901


120572 gt 119903119873gt 0 120573 lt 0


1205731015840= 119903119864minus 120572 120573 = 119903

119873minus 120572


0



119864

(infin)Price


119903119864gt 120572 gt 119903


1198780= 119901lowast


119878min = 119901lowast

1198780gt 0)

1199011198780ge 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780+infin (Min)

1199011198780= 119901lowast

119878min 00 lt 119901

1198780lt 119901lowast


119903119864gt 120572 gt 119903

119873 119903119864gt 2119903119873 1205731015840 gt 0 gt 120573





Δ1199010= Δlowast119901max 0



119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873 1205731015840 gt 0 gt 120573






lowast1199010


119903119864gt 120572 gt 119903

119873 2119903119873gt 119903119864 1205731015840 gt 0 gt 120573








120572 = 119903119864gt 119903119873gt 0 1205731015840 = 0 gt 120573

(119901lowast119878max = 119901

lowastlowast

1198780= infin gt 119901

lowast

1198780= 119901lowast




119901lowastlowast

1198780gt 1199011198780gt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)

1199011198780= 119901lowast

1198780(1199011198780minus 119901lowast

1198780) = 0

0 lt 1199011198780lt 119901lowast

1198780(1199011198780minus 119901lowast

1198780)



Δ1199010= Δlowast1199010





1198780lt 0)

Every 1199011198780


00

120572 gt 119903119864gt 119903119873 120573 lt 120573

1015840lt 0 (119901

lowast

1198780= 119901lowast


1198780= 119901lowast

119878min) 1199011198780gt 119901lowast

119878max 0 (Min)

120572 gt 119903119864gt 2119903119873 120573 lt 120573

1015840lt 0







120572 gt 119903119864gt 119903119873 119903119864= 2119903119873 1205731015840= 120573




Δ1199010= Δlowast119901min Δ119901

0


120572 gt 119903119864gt 119903119873 2119903119873gt 119903119864 0 gt 120573

1015840gt 120573




Every Δ1199010

0

120572 gt 119903119873gt 119903119864gt 0 120573

1015840lt 120573 lt 0

(119901lowast

1198780= 119901lowast


1198780= 119901lowast

119878max lt 0)



1199011198780gt 0

Every Δ1199010

0 (Min)

0 (Min)


lt 119903119864 and Δ119901


120572 = 119903119873= 119903119864and every Δ119901

0





119864gt 120572 gt 119903

119873


1198780= 119901lowast


1198780gt 1199011198780gt 119901lowast

1198780



1198780) for 119903

119864= 120572 gt 119903

119873


119864gt 120572 gt 119903

119873

and 1199011198780lt 119901lowast

1198780= 119901lowast




119864gt

120572 gt 119903119873and Δ119901


119903119864lt 2119903119873and Δ119901





119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119878 for 119903

119873= 0 119901lowast


1198780= 0


120573 1199011198780gt 119901lowast


1198780= 0

A 1205731015840gt 120573

A1 1205731015840gt 120573 gt 0 +infin

A2 1205731015840gt 120573 = 0 +infin

A3 1205731015840gt 0 gt 120573 +infin

A4 1205731015840= 0 gt 120573 119901

1198780

A5 0 gt 1205731015840gt 120573 +0


120572 gt 119903119873for every Δ119901

0


120572 gt 119903119873and Δ119901




0gt


119864gt 2119903119873


119903119864gt 120572 gt 119903

119873 119903119864= 2119903119873and Δ119901

0= Δlowast1199010



119901lowast

1198780= 119901lowast

1198780= 0







0is indicated




+0 if 1205731015840 lt 0(ii) for Δ119901












0if 1205731015840 gt 120573 = 0






1198780is





119878max = 119901lowast

1198780 for 1205731015840 gt 120573 = 0

if 1199011198780gt 119901lowast

1198780= 119901lowast



1198780gt 1199011198780




1198780= 119901lowast

119878max gt 1199011198780 gt 119901lowast




1198780 0 1199011198780minus 119901lowast

1198780



1198780




ge

119901lowast



119878tends towards







0







after a minimum


0=







1198780








1198661015840

119864=119866119864

1198661198640

= 1198661015840

119878=119866119878

1198661198780

= exp (120572119905) (65)


1199011015840

119864=119901119864

1199011198640

= exp (120573119905) (66)


Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table6Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

thep

riced


Relatio

nbetween1205731015840

and120573





A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

A2


+infinΔ1199010

Δ1199010

A3


+infin0

A4


1199011198780=Δ1199010minusΔlowast1199010

1199011198780=Δ1199010minusΔlowast1199010

A5


00

0(M

ax)

0


Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table7Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

selling

resources119901119878for119903119873

=0

1205731199011198780gt119901lowast 119878max

1199011198780=119901lowast 119878max

119901lowastlowast

1198780gt1199011198780gt119901lowast 1198780


1198780

1199011198780=119901lowast 119878min

0lt1199011198780lt119901lowast 119878min

A1205731015840gt120573

A1


1198780=119901lowast 119878mingt0)

+infin+infin

minusinfin

(Max)

minusinfin

minusinfin

A2


1198780=119901lowast 119878maxgt0)

+infin1199011198780

minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin

A4

1205731015840=0gt120573(119901lowastlowast

1198780=119901lowast 119878max=infin


1199011198780minus119901lowast 1198780

1199011198780minus119901lowast 1198780=0

1199011198780minus119901lowast 1198780

A5


1198780=119901lowast 119878min)

0(M

in)

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

(Max)

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

0(M

in)

C1205731015840lt120573

C1



minusinfin

(Max)

minusinfin

(Max)

minusinfin

C2


1198780=119901lowast 119878maxlt0)

119901lowast 1198780

C3



0(M

in)

D1205731015840=0

D1


1198780=infin)

minusinfin


Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table8Finalcon

clusio

nsof

thetrend

sto119905rarr

infinof

priced


=0









A1205731015840gt120573

A1


+infin+infin

+infin(M

in)

minusinfin

minusinfin


+infin+infin

minusinfin

(Max)

minusinfin

minusinfin


0=0)


minusinfin

A2

1205731015840gt120573=0



minusinfin

A3



+infin+infin

+infin(M

in)

0minusinfin


+infin0

minusinfin

(Max)

minusinfin

minusinfin



minusinfin

A4


00

0(M

ax)

00

A5


00

00

0

B1205731015840=120573

B1


1198780gt0)

minusinfin

(Max)

minusinfin

minusinfin

B2


1198780=infin)

minusinfin

(Lin)

B3


1198780lt0)

00

0

C1205731015840lt120573

C1


minusinfin

minusinfin

minusinfin

minusinfin

minusinfin

C2


Δlowast1199010

Δlowast1199010

Δlowast1199010

Δlowast1199010

C3


0(M

in)

0(M

in)

0(M

in)

0(M

in)

0(M

in)

D1205731015840=0

D1


minusinfin

minusinfin

minusinfin

D2


1198780=infin)

minusinfin

(Lin)

D3







120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120572 =ln1198661015840119878

119905 (67)


119864 and


1199011015840

119864= 1198661015840(120573120572)

119864= 1198661015840(119910minus1)

119864 (68)


119910 =119903119873

120572 (69)



1015840



(i) for 119910 = 0 (ie 119903119873




119864


119864


increase of 1198661015840119864


119864




119864


1199011015840

119878=119901119878

1199011198780


where

1199011015840lowast

1198780=119901lowast

1198780

1199011198780

=1199031198731199011198640

1199011198780sdot (1205731015840 minus 120573)

=1199011198640sdot 119910

1199011198780(119909 minus 119910)

(71)


119909 =119903119864

120572 (72)


For 119909 = 119910 (ie 119903119873

= 119903119864) (70) combined with (65)

becomes

1199011015840

119878=119901119878

1199011198780

= 1198661015840(1205731015840120572)

119878minus 1199011015840lowast

1198780(1198661015840(1205731015840120572)

119878minus 1198661015840(120573120572)

119878)

= 1198661015840(119909minus1)

119878minus 1199011015840lowast

1198780(1198661015840(119909minus1)

119878minus 1198661015840(119910minus1)

119878)

= 1198661015840(119909minus1)

119878(1 minus 119901

1015840lowast

1198780) + 1199011015840lowast

11987801198661015840(119910minus1)

119878

(73)

p998400E

1

0

y = minus5y = 5

y = 2

y = 1

y = 05

1 G998400E


G998400E lt 1 (120572 lt 0) G998400

E gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


1198661015840

119864


1198661015840

119878=

[119910 (1 minus 119910)]

[(119909 minus 1) (119909 minus 2119910)]

1(119909minus119910)

(74)

when 1199011198640= 1199011198780 and

1198661015840

119878=

[1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(75)

when 1199011198640

= 1199011198780


119878= 1 for 1199011015840

1198780

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780 (119909 minus 1)

(76)



1198780=


1198780= 0 (73) becomes

1199011015840

119878=119901119878

1199011198780

= exp (1205731015840119905) (77)


1199011015840

119878= 1198661015840(1205731015840120572)

119878= 1198661015840(119909minus1)

119878 (78)



119910 = 0 (ie 119903119873= 0)





119878


119878


p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










p998400S

1

0

x = minus5

x = 5

x = 2

x = 1

x = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

x rarr |infin| (120572 = 0)

x = 0 (120572 = +infin)


119878

for 119903119873= 0

p998400S

1

0

y = minus5

y = 5

y = 2

y = 1

y = 05

1 G998400S


G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)

y rarr |infin| (120572 = 0)

y = 0 (120572 = +infin)


119878

for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if 119901

1198780= 1199011198640= 1)


of 1198661015840119878


119878




119878


1198780=

1 (73) reduces to

1199011015840

119878=119901119878

1199011198780

= 1198661015840(119910minus1)

119878 (79)



119878for 1199011015840lowast1198780= 1 (ie 119909 = 2119910 if






119878


119878


119878


119878 A






119878


119878with 1198661015840


|119910| gt 1 (ie 119903119864gt 119903119873gt 120572)



1198780when 1199011015840lowast

1198780le 1


1198780


(i) for 1199011015840lowast1198780


decrease of 1198661015840119878


1015840lowastlowast






119878= 1


1198780gt 1 1199011015840


the decrease of 1198661015840119878


119878with 1198661015840



1198780lt 0 lt 119901

1015840lowastlowast

1198780)


1198780








119878= 1



decrease of 1198661015840119878



for 119910 gt 1 gt 119909 (ie 119903119873gt 120572 gt 119903

119864and 0 gt 119901

1015840lowast


1198780) The


119878




119878with 1198661015840


(ie 120572 = 119903119873gt 0 and 1199011015840lowast

1198780= 1199011015840lowastlowast

1198780)


p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










p998400S

1

10



S0 gt 0

p998400lowastS0 = 1


p998400lowastS0 = 1

p998400lowastS0 = 1


S0



S0


S0 gt 1G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)p998400lowast



119878 for |119909| gt |119910| gt 1

p998400S

1

1

0


S0





S0 gt 1


G998400S

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)


S0




= 1199011015840lowast

1198780



1198780= 1199011015840lowast

1198780gt


119878if 1199011015840lowast1198780lt 1 1199011015840

119878



1198661015840

119878if 1199011015840lowast1198780gt 1


1198780= 1199011015840lowast

1198780lt 0)

1199011015840



of 1198661015840119878




1198780lt 0 1199011015840lowastlowast

1198780= +infin)


119878



119873)




(i) for 119909 gt 1 gt 119910 (ie 119903119864gt 120572 gt 119903


1198780gt 1199011015840lowast

1198780gt 0)


119878if1199011015840lowastlowast1198780

le 1


p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










p998400S

1

10


p998400lowastS0 = 1

G998400S

G998400S gt 1 (120572 gt 0)



S0 gt 0


S0 = infin


S0 lt 0

p998400lowastS0


119878 for 119910 = 1 or 119909 = 1

p998400S

1

01








S0

G998400S


S0 gt 0


S0 = 0

G998400S gt 1 (120572 gt 0)


S0

t0 + 0




S0


S0 gt 1


x = 1 gt y (rE = 120572 gt rN)



S0 = +infin


119878 for 119910 lt 1


p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










p998400S

0 1

G998400S

x = y lt 0


S0 lt 1



G998400S lt 1 (120572 lt 0)



G998400S gt 1 (120572 gt 0)


S0 = infin


S0 = infin


S0 = infin


119878 for 119909 = 119910

1199011015840


1015840


1198780gt

1 gt 1199011015840lowast


119878if


1198780gt 1199011015840lowast

1198780= 1 1199011015840

119878decreases with 1198661015840

119878if 1199011015840lowastlowast1198780

gt 1199011015840lowast

1198780gt

1(ii) for 119909 = 1 gt 119910 (ie 119903

119864= 120572 gt 119903


1198780= +infin

1199011015840lowast

1198780gt 0) 1199011015840


1198780) with

the increase of 1198661015840119878


1198780gt 0 gt 119901

1015840lowastlowast

1198780)


+0 with 1198661015840119878if 1199011015840lowast1198780le 1 1199011015840



1198780gt 1


1015840lowastlowast

1198780gt

1199011015840lowast


119878after a

minimum




1199011015840

119878=119901119878

1199011198780


1198780120573119905) exp (120573119905) (80)


1199011015840

119878=119901119878

1199011198780

=1198661015840(120573120572)

119878[


1198780120573 (ln (1198661015840

119878))

120572]

= 1198661015840(119910minus1)

119878[1 minus 119901

1015840lowastlowast

1198780(119910 minus 1) (ln (1198661015840

119878))]

(81)


1198661015840

119878= exp[

(119909 minus 119910 minus 1)

119910 sdot (119910 minus 1)] (82)

for 1199011198640= 1199011198780 and to

1198661015840

119878= exp[

(11199011015840lowast

1198780minus 1)

(119910 minus 1)] (83)

for 1199011198640

= 1199011198780


119878= 1 for

1199011015840

1198780= 1199011015840lowastlowast


1198780

1199011198780

= 1199011015840lowast

1198780

(1205731015840minus 120573)

1205731015840

=1199031198731199011198640

1199011198780sdot 1205731015840

=1199011198640sdot 119910

1199011198780(119909 minus 1)

(84)



1199011015840

119878=119901119878

1199011198780

= ln (1198661015840119878)1 minus 1199011198640

1199011198780

(85)





1198780gt 0

1199011015840lowast





ge 1 The price 1199011015840119878



lt 1(ii) for 119909 = 119910 lt 1 (ie 119903


1198780lt 0

1199011015840lowast

1198780= +infin) the price 1199011015840




10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10 G998400

S

Δpx lt 0 x gt 1

t0 + infin

t0 + infin

t0 + infin

t0 + infin

t0 minus 0





Δp0 gt 0 Δp0 gt 0

G998400S lt 1 (120572 lt 0) G998400

S gt 1 (120572 gt 0)





Δp0 = Δlowastp0

Δlowastlowastp0



1198780= 1199011015840lowastlowast

1198780= +infin)


119878



Δ119901 = 119901119878minus 119901119864= (Δ119901


119878+ Δlowast11990101198661015840(119910minus1)

119878 (86)

where


1198780minus 1199011198640= 1199011198640

(2119903119873minus 119903119864)

(119903119864minus 119903119873) (87)


1198661015840

119878=

[Δ1199011015840lowast

1198780(1 minus 119910)]


1198780)]

1(119909minus119910)

(88)



(119909 minus 119910)

(119909 minus 1)

= Δlowast1199010

(1205731015840minus 120573)

1205731015840= 1199011198640

(2 sdot 119903119873minus 119903119864)

1205731015840

(89)


119878are presented



119873= 0) and






119878lt 1

or 1198661015840119878gt 1 For 1198661015840



119878gt 1



0lt




increase for 1198661015840119878gt 1



119878lt 1



rate for 1198661015840119878gt 1



0 that is




119878 For1198661015840







119873= 0 Δlowast119901

0= minus1199011198640lt 0) and






0gt 0 0 gt Δ119901

0gt Δlowast1199010

or Δ1199010= Δlowast1199010)


0gt 0) the price



1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

Δp

0

x lt 0 x gt 0

G998400S



Δp0 gt 0








Δlowastp0

G998400S lt 1 (120572 lt 0)

Δp0 = Δlowastp0


lowastp0 lt 00 Δ



119864gt 2119903119873ge 120572 and Δlowast119901


119910 gt 119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903






119878 For Δ119901




0gt Δ119901

0gt Δlowast1199010


119878




119909 = 1 (ie 119903119873gt 120572 = 119903


0lt 0) and for 119910 gt 1 gt 119909 (ie

119903119873gt 120572 gt 119903





119873 119903119864=

2119903119873 and Δlowast119901






119878




For Δ1199010





119878 For Δ119901



119878= 1 For Δ119901



119878




0=


0gt Δlowast1199010


119878 for Δ119901








119878








1198661015840


0gt Δ119901



0=







Δ1199010gt 0 gt Δ









Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Δp

0G998400

S

x lt 0 x gt 0

Δp0 = Δlowastp0


Δp0 = Δlowastp0


G998400S lt 1 (120572 lt 0) G998400



1



0

1

Δp

G998400S


lowastlowastS0 lt 0

x = y (rE = rN)

x = y = 1 (120572 = rE = rN)

plowastlowastS0

plowastlowastS0






increase of 1198661015840119878






119878


Δ119901 = 119901119878minus 119901119864= (Δ119901


11987801205731015840119905) exp (1205731015840119905) (90)




1198780



1198780(119910 minus 1) ln (1198661015840

119878)]1198661015840(119910minus1)

119878 (91)



1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198661015840

119878= exp[


1198780)

119901lowastlowast

1198780sdot (1 minus 119910)

] (92)



0 (93)



0gt 0) three cases



0






0the price





0gt 0 gt 119901

lowastlowast

0


0gt or = or lt

119901lowastlowast





Δ119901 = Δ1199010minus 1199101199011198640ln (1198661015840119878) (94)


Nomenclature

Latin














resources PIFE









References




























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Review Article A New Theory to Forecast the Price of...

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Transcript of Review Article A New Theory to Forecast the Price of...