1987- Computation Techniques for Inter Temporal Allocation of Natural Resources

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Agricultural & Applied Economics Association

Computation Techniques for Intertemporal Allocation of Natural ResourcesAuthor(s): Duane ChapmanReviewed work(s):Source: American Journal of Agricultural Economics, Vol. 69, No. 1 (Feb., 1987), pp. 134-142

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Computationechniquesf o r Intertemporalllocationo f N a t u r a l ResourcesDuaneChapmanApplication of optimal control theory to applied problems is limited by the difficultyof numerical solutions. Typically, terminal values for the production period, price, orproduction level have been assumed rather than optimized. The use of an objectivefunctional with explicit discounting gives direct solution values for n, y(t), p(t), andrent (or consumer surplus) for continuous or discrete problems. The method is usablefor numerical solutions to problems with linear demand, cost trend, or expropriationrisk. It is illustrated with Fisher's widely used discrete problem and with applicationto parameters representing remaining world oil resources for competitive andmonopolistic assumptions.Key words: control theory, natural resources, oil.

The theory of natural resource use over timehas evolved considerably since Hotelling'spathbreakingwork fifty years ago. Clark, Das-gupta, Gilbert, Fisher, Kamien and Schwartz,Heal, Smith and Solow have been among theleaders in this effort. However, applying thistheory to numerical problems has beendifficult. The purpose here is to describe in aprecise and accessible way some problemsand numerical solutions dealing with oil re-sources which can be used in teaching theoryand analysis. The ambition, then, is small, in-tending to restate in simple form materialwhich has been thoroughly discussed in theliterature.Some interesting implications arise from thenumerical solutions. Dramatically differenttime paths of oil use can have about the sameobjective values. Thus, with the present worldeconomy, an oil cartel could for the presentignore depletion. Similarly, the socially op-timal competitive path is unaffected by deple-tion for many years.Production paths for monopoly and compet-

itive solutions are generally thought to cross,as occurs here. But cumulative production isalways greater in the competitive solution,and the difference grows throughout the com-petitive period.The length of time of the production periodis frequently assumed in optimization prob-lems and is usually viewed as years to deple-tion. In part, this assumption reflects thedifficulty of determining numerical answers tothe problem in a continuous framework. Butsuppose that a monopoly has an n-year con-tract. Will it make radically different decisionsthan a monopoly that can choose n? Or sup-pose the expropriation question is defined as aprobability problem for lost profit. Does thisaffect numerical solutions? Of course, bothanswers are affirmative.

Fisher's Problem and Optimal ControlAs a text for the discussion, Fisher's problemis useful because his Resource and Environ-mental Economics has attained wide usage.He gives this problem (pp. 14-18, 37-39):there are ten barrels of oil in the ground, theprice demand function for oil is $10 - y, costis a constant $2, and the discount rate is 10%.What is optimal use over two years?The objectives are to compare social opti-

DuaneChapmans a professorof resourceeconomics,CornellUniversity.Assistance and criticismhave been offeredby SimonLevin,MargaretMorgan,WilliamPodulka,RichardRand,WilliamTo-mek, WilliamLazarus,Gene Heinze-Fry,Nancy Birn, JosephBaldwin,andPaulFassinger,all atCornellUniversity,andaJour-nal reviewer.Copyright1987AmericanAgriculturalEconomics Association


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Chapman NaturalResourcesAllocation 135Table 1. Fisher's Oil Problem: 10 Barrels, 2Periods

Monopoly CompetitiveObjective Profit SocialOptimumPeriod0 use, barrels 4 5.14Period 1 use, barrels 4 4.86Price,period0 $6 $4.86Price,period1 $6 $5.14Totaloil use 8 10Profitpresentvalue $30.55 $28.57Consumers' urplus,presentvalue $15.27 $23.95Note: Cost is a constant $2 per barrel. Consumers' surplus beforediscounting is fpdy - 2y.

mality with competitive and monopoly solu-tions. If allof the oil must be used, the monop-olist uses 4.95 barrels n the firstyearandtheremaining5.05 barrels n the secondyear. Butthe competitive solution is socially optimal,and those values are 5.14 barrels in the firstperiodand 4.86 barrels n the second.If the monopolistneed not produceall theoil, the profit-maximizing mounts are fourbarrelsper year in each period, leaving twobarrelsunused.The socially optimal,competi-tive solutionwould, if unconstrainedby a re-source limit, use marginalcost pricing, andseek to use eightbarrelsper period.Since thisis not possible, the constrainedvalues for twoyears (i.e., 5.14 and4.86) arecorrect.Table1shows these values.Fisherhas specifiedthe numberof timepe-riods, the demand unction as linear,andcostas constant. These assumptions rather thanthe specificvalues (i.e., 10 barrelsavailable,2periods)are the key to easy solution.Butif thenumberof time periods, n, is made a controlvariable, the problem is more complex. Infact, for the monopolythe optimalnumberofperiods is eight years. In what follows, amethodof selectingn is developedthat is ap-plicable for finding numerical solutions forgeneralcontinuousproblems involvingrepre-sentativeworld oil data and alternativeobjec-tives.Moreformally,the problem s to maximizethe presentvalue of economic rentby findingthe optimal time path of productionand theoptimal length of time for production.' Thetime path or trajectoryof production s con-

trolledby the decision maker to optimizethedesiredvalue.Price is then determinedby pro-duction,as is economic rent. Productions al-ways the rate of changein cumulativeproduc-tion. Admissiblevalues of productionmustbesuch that price, production,and rent are al-ways nonnegative.If all the resourcemust be used, thencumu-lative productionreachesthe originalstock atthe end of the optimal time. But if resourceexhaustion is not required, then cumulativeproductionmust not exceed the original tock,and optimizationmay leave part of the re-source unused when productionceases. Thisis the basicproblem oranyfiniteresource,onany scale whethermicro(e.g., a smalldeposit)or macro(e.g., globaloil).First, the continuousproblem s stated,andthen Fisher's discreteproblem s considered.In a later section, the monopoly profitobjec-tive is replacedby social welfareandcompeti-tive objectives, and numericalresultsfollow.The continuousproblem,initiallyfor a mo-nopoly, is to maximize the present value ofprofits, V, withrespect to n, the lengthof theproductionperiod, and y(t), the productionpath.

maximize V = P(Y)Y - rydt;{y(t), n) e

where(1) p(y) = P2 - PIY,

dX(t)y(t) dX( ,tydt = X(n)

-S,

p,y -0,py - ay ? 0.Table 2 summarizes he definitionsfor equa-tion(1). Forease of solution,p(y) is linearandcost cx s constant for now.Cumulativeproductionat anytimeis X(t), avariablereflectingthe state of remainingre-sources. Production, y(t), whatever form ittakes, is the rate of changein cumulativepro-duction.Original esourcestockis S; it cannotbe exceeded. However, the not-more-thanconstraintcreates modest difficulties or solu-tions. Will the optimal y(t) use up all of theresourceby n, or willuse cease withoil stocksstill available?Finally, prices, quantities,andThis definition of the problem and first-order canonical condi-tions is based upon Intriligator's discussion, chaps. 11, 14.


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136 February1987 Amer. J. Agr. Econ.Table 2. Definitions of VariablesVariable Definitiona Averagecost perunit3o Optimalmonopolyproduction,uncon-strained13, 32 Parameters n pricedemand unctionP3 Competitive ocially optimalproduction,unconstrainedCV Competitive ndustry,presentvalueofprofitDV DiscreteV;see V belowH Hamiltonian unctionh Dynamic agrangemultiplier, ostatevari-ableSAccumulation factorfor regularnvestmentat interestn Lengthof timeperiodp Priceor pricedemand unctionr Interestor discountrateS Original esource stockSV Socialvalue;presentvalue of consumers'surplust TimeV Presentvalue of monopolyprofitX Cumulativeproductiony Productionor production imepath

profitmust always be nonnegative.The max-imummonopolyprofit s foundby determiningthefunctional orm of y(t), its actualnumericalvalues, and the lengthof time n.The best optimal control techniqueto usewith this problemis the maximumprinciple.This principleasserts that study of a simplerfunction based upon the relationshipsin (1)can provide nformationusefulin findinga so-lution. By examining first-orderconditions,the natureof the solutionmaybe determined.2The Hamiltonianfunction in this contexthas two components. The firstpart is the in-tegrandwhich is being maximizedover time.This is the discountedprofitfunctionhere:

p(y)y - iy(2) H = eY -rt --Xy.The second part of the Hamiltonian is theproductof the co-state variableh (similar o alagrangianmultiplier)and y, the productionrate which is the rate of change in the statevariableX. In (2), h gives the change in dis-countedprofitassociatedwith a small ncreasein resource endowment.Approximately,XisdV/dS.

The Hamiltonianprovides the first-orderconditionsto solve (1): EitheraH(3a) - 0, oray

(3b) - 0, andat(3c) at - aXIt is still the case thaty = dX/dt.Finally, the second-orderconditionis

d2H(4) 2 0.ay2Beforeapplying hese relationships, he dis-crete problemis introduced n orderto showthat eight years is the optimalmonopolype-riod andhowy, is madeexplicit. The discretegeneral analogue is summarizedwith slightmodificationof equations(1)-(4):

(5a)n-1maximize DV = ( -+ y{y,,n} = (1 + r)tn-1

(5b) y, = X(n - 1) S,t=0(5c) H = p(Y)Y - ay Xy.(1 + r)'

DV is the discrete version of net presentvalue of monopoly profit. The other con-straints, conditions, and definitions n table 2still hold. The logical interpretationof theserelationshipscan be difficultbecause the trueoptimum may fall between the integer timeunits.The Fisher problem values are, as noted,p(y) = $10 - y, P2 = 10, P1 = 1, S = 10barrelsoriginalendowment,r = 10%per yeardiscountrate, and a = $2 per barrelconstantcost.First, considerthe solutionwhen all the re-source is used so X(n - 1) = S. Applying thefirst-order onditions,beginningwith (3a):(6a)

3pOH p + - y - otaH _ y - h = 0,y (1 + r)t2 All of the functions are continuous and differentiable, and thefunctionbeingmaximizeddiscountedprofit n this section) s ap-propriatelyoncave.


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Chapman Natural Resources Allocation 137P2 - 2P1y -

ao6b) (1 + r)' ,- (1 + r)' (P2 - Or)(6c) y = 2 + 2

Apparently, the co-state multiplier does notchange over time, since applying (3c),aH ah(7) - 0

Since h is constant, y in (6c) can be used in the(5b) definition of cumulative production usingall the resource.n--I

(8) : \(2(1+ r) + o S=where o 2 -2r1

n--1(9 n + (1 + r)' = S.(9) n30 2+ >311t=SThe summation of the interest terms is thefamiliar [(1 + r)" - 1]/r, the accumulationfactor for n years of regular investment at in-terest. Define this as gi(n); n has to be deter-mined. Consequently, solving for Kin (9),(10) K = -211(S - nI3o) andp(n)(11) y = o + (S - nI3o)(1+ r)g(n)

Note that {o, n, and S determine whether y,increases or decreases over time. The 1o termdefined in (8) is simply optimal y in the ab-sence of a resource constraint. It is the usualresult of maximizing py - ay. Equation (11),then, says that optimal production in the earlyyears will be closest to the unconstrainedlevel. As time passes, y, diverges from 3o.Thesign and rate of change of this divergence (i.e.,(yt - Bo)/at) depend upon the numericalvalues of resource endowment, the optimalproduction period, and the interest rate. Gen-erally, if the resource constraint applies and nis properly chosen, y, is expected to declineover time.Using Fisher's values given above, the de-mand function is

(12) Y, - 4 + (10 - 4n)(1.1)t(12) y, = 4 + j(n)(t = 0,1,2,...,n).

Equation (12) shows optimal y when n isknown. This can be put back into (5a) to findthe optimal n:(13)

n--ImaximizeV (10 - y,)yt- 2y,

maximize DV={n} t=o (1.1)'where Yt is given in (12).This is easily programmed, and the max-imum discounted present value is $51.57, theoptimal time period is eight years (i.e., n = 7),and oil use declines from 2.08 barrels in theinitial period to 0.25 barrels in the last period.Price rises from $7.92 per barrel in the initialyear to $9.75 in the last year. The value of Xfrom (10) is + $3.85. The interpretation of Xinthis discrete format means that, given n = 7,dDV/dS = $3.85. (Remember that the initialperiod has t = 0, and n = 7 means an eight-year period.)

Global, Discrete Oil Use: Monopoly SolutionThere are 1.189 trillion barrels of oil remain-ing, more or less, from the earth's original en-dowment of 2 trillion barrels. This includesproved reserves as well as oil remaining to bediscovered.3 If the demand elasticity is -0.5at current annual world oil use of 20 billionbarrels, then for linear demand, p = $75 -2.5y.4 Figuring average world production costat $20 per barrel, parameters for a global oilproblem arep(y) = $75 - 2.5y, P2 = 75, 13=2.5; S = 1,189 billion barrels remaining oilresource; r = 10%per year discount rate; a =$20 per barrel constant cost; and 1o = 11.Using equation (11), oil production in year tis(14) Yt = ll + (1189 - lln)(1.1)'14) yr = 11 + gnix(n)

Searching for the optimal n is more timeconsuming with these values. It is helpful tofind an initial value which may be close to theoptimum. Suppose expropriation is not aproblem and y declines to near zero as is often

3 From USGS, Masters et al., 1983 resource estimate, modifiedfor 1983-84 actual use. Other accessible discussions of total oilresources are the Oil and Gas Journal; Exxon; Chapman; andDaly, Griffin, and Steele.4 Long-run price elasticity values include Daly, Griffin, andSteele's -0.73; Pindyck's linear demand values which include-0.33 and -0.90; and -0.30 from Adams and Marquez. Kourissummarizes several studies; for retail gasoline in the U.S., valuesrange from -0.36 to -1.02.


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138 February 1987 Amer. J. Agr. Econ.assumed. Then, by setting y(t = n) = 0, aninitialn canbe found.Using (12),andthentheglobal values,(15)y(t = n) = o30+ r(+r)= 0,(1 + r)n - 1

(16) S 1 _ (1 + r)-"(16) n +--1o r rwhich gives(17) n = 118.09- (.1'1

Following the simple iteration first used byHotelling (p. 142), n = 118 is the closest inte-ger value. Below, this method is shown todefine the unique optimal n for the continuousproblem.The present value of discretely discountedprofit is $3.3274 trillion. Oil production andconsumption would decline from 11 billionbarrels in year "O" to 1.09 billion barrels inyear 117. Price would rise from $47.50 per bar-rel in the initial period to $74.77 in the finalperiod. Given oil use of about 20 billion bar-rels annually and a price of $27 per barrel in1985, monopoly profit maximization is not at-tained, at least based on these assumptions.5

Continuous Solutions: Competition,Monopoly, and Social OptimaAlthough continuous relationships are moreeasily interpreted, they have not attained fre-quent use in numerical problems. In part, thisis because evaluation of present value V in (1)can be very difficult. However, new programssuch as MACSYMA now permit resolution ofthis difficulty.6 Explicit answers to the prob-lem in (1) for the global numerical assumptionsfollow.Beginning with equations (1)-(4) for the mo-nopoly case, the first-order (canonical) condi-tions are

dpp+ y-pHy -(18) H = - = ,y ert- herr (P2 - a)19) y

-+2P1 2P1

As above, since (3c) leads to - aHIaX = 0,then ah/lat= 0, and h is constant for a given n.Consequently, if the resource is exhausted,then production over the period is(O

'-- kert )(20a) + Bo0dt = S,2pB2where 3o = 21 , Or2131-- (er"- 1) + n[o = S.(20b) l r~P1 rSo, with the accumulation factor, now

g(n) = (er - 1)/r,S2P1(S - niPo)(21) - x = 23( - n3)I,(n)

(22) y(t) = (S - n3o)ert + 3o, t = 0, n.px(n)These expressions for y(t) and X are identi-cal to the discrete forms for y, and h, [(10) and(11) above]. Clark (p. 146)outlined part of thisapproach ten years ago, noting, "Determina-tion of T [the optimal exhaustion date], how-ever, is a non-trivial matter." He suggestediterating X.However, an explicit solution for the op-timal present value of monopoly profit is de-rived by using y(t) in (1),7 and

(23) V = p(n)P13o2 _ 13(S - nIo)2ern (n)Also, note that maximizing V for n gives aunique solution to the problem of optimal n:

S 1 e-rn(24) n = + er1o r rThis of course provides the basis for theHotelling iteration for equations (15)-(17). Forthis problem, (24) is the single unique op-timum length of time for production. TheSThe $27 per barrel figure represents an average price for all of1985. See Weekly Petroleum Status Report, recent issues (U.S.Department of Energy).6 MACSYMA is a computer program for algebra and appliedmathematics which, because of its calculus capabilities, is particu-larly useful for solving optimal control values. Rand offers a goodprogram guide, and complete documentation is published by theMIT Mathlab Group and Symbolics, Inc. Microcomputers can usesimilar packages.

7 Equations (23) and (35) and the proof of their maximization by(24) and (33) are contributions by Margaret Morgan and WilliamPodulka in a Cornell graduate seminar, "Economics of ResourceUse."


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Chapman Natural Resources Allocation 139value for n is found by simple iteration or byNewton's method and is 118.09.8The present value of monopoly profit is$3.025 trillion. This is reassuringly close to thediscrete solution. The second-order condition[from equations (18) and (4)] is

a2H - 2P1 0,dtP

(39) dt rcert > 0.Here hm is .04 of 1? and hK s 9?. The deple-tion periods are considerably different. Theseresults imply that an increment to world oilresources is more valuable to a competitiveworld economy than to a world cartel.

The Numerical Paradox: Impatience andMyopia Are Almost OptimalProduction paths that diverge sharply from theoptimal can give essentially equivalent profit.This is equally applicable to solutions for boththe social optimum and monopoly problems.Define a myopic planner as setting y at a8 Conrad and Clark (chap. 1.6) show applications of Newton'ssolution method to optimal control resource problems.


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140 February 1987 Amer. J. Agr. Econ.50 50 Ya: impatient paion

yb: myopic paion-- Y: optimal plan

40 -.5.74

T0..

S ~ ~ ~ ~ ~ I : f ~ i i i i l ii i i i it i i i i ii i i i ii i i il ~ i i r ~ i i i i i i ii l i i- 30- Y.0S .0itiiijijiiiiiiiiitiijritiii~iiit~i~iiiiirii

.....-. ..........."..'.'.'. - ... ''- :' '_',,,i,,;_'::::::::::: : :2

: : : : ::::::i::::::: y20o

.0 I (s 6.03 T

I0-

0 8 16 24 32 40 48 56 64Years" 1985 to 2049Figure 1. Petroleum use profiles and net social value

constant 33 level until the resource is ex-hausted afterS/33 years. (Recall 33 is the so-cially optimal level of productionin the ab-sence of a finite resource constraint.) Themyopic planner's 22 billion barrels per yearlasts 54 years, earninga discountednet socialvalue of $6.023 trillion,rounded to $6.02 tril-lion for path yb in figure1. The trueoptimum,Y*in figure1, has productionwhich exhauststhe resource after 64 years, attaining$6.030trillion.In this instance, full optimization m-proves presentvalue by a mere $7 billion.Define an impatient planneras wishing toinitiateproductionat the 33 level, gain a netsocial value comparableto the true optimum$6.03 trillion,and leave the oil business. Thiscan be accomplishedas follows. First, use aform ya(t) = 33 + 34ert. Second, define com-parable social value (or monopoly profit)as95%of the optimumvalue:

SVa{ya(t), na} = .95 SV*{y*(t), n*}, orVa{ya(t), na} = .95 V*{y*(t), n*}.

Now find 34,which, meetingthenonnegativityconstraints,gives the leastyearsof productionna. For figure 1, for example, with the basicparameters for the maximum social valueproblem,the impatientplannerpathis ya(t) =22 + .577e"/. Petroleum use increases over a

36-year period. But the net social value forthis dramaticallydifferentpath is $5.74 tril-lion, 95%of the trueoptimum.Note infigure1the contrastbetween Ya and Y*.Figure 1 raises a problem:optimalplanningmay not always be necessary to gain optimalvalue. Strongly divergent productionpaths insome cases yield comparableobjective func-tion values. This is notalwaysthecase, partic-ularlywhere the resource is very limited. Inthe Fisher problemabove, for example, thiscomparabilitydisappears.A sensitivityanalysisindicatesthat thephe-nomenonmay applyto a broad domainof pa-rameter values applicable to world oil mar-kets. In figure1, the unshadedarearepresentsoil productioncommon to bothdivergentpathYa and optimal path Y*. The shaded area isproductionnot common to both paths. A di-vergence ratio can be the ratio of the sum ofnot-commonproductionto the total resourceproduced on the optimal path. Graphically,this is the ratio of shadedareato the appropri-ate S for each case. Numerically, hisresult s:f (lya(t) - y*(t)l)/Sdt.9

9 Since ya usually accelerates while Y* declines, and the pro-duction period forl is always shorter, a correlation measure ofdivergence is less descriptive.


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Chapman NaturalResources Allocation 141Table 3. Sensitivity Analysis of DivergenceRatios

Competitionor SociallyOptimalCases Monopoly Plan1. Original lobaloil parame-ters .75 .532. But r = 5% .53 .413. r = 10%, but S = 594.5billion bl .53 .414. r = 5% and S = 594.5billion bl .40 .395. Elasticity s -1.2, cost is$10 bl, r = 5%, S =594.5billionbl .39 .40

Table 3 shows a sensitivity analysis wheredivergence ratios are compared for severalmonopoly and social optimum-competitioncases. In each instance, original resources, in-terest rate, cost, or elasticity are varied bysizable decrements for each market structure.Optimal paths and optimal lengths of produc-tion vary accordingly, as does the impatientpath ya for each case. Figures analogous tofigure 1 exist for each of the ten cases. Diver-gence ratios are noticeably large for all cases.A similar sensitivity analysis for the myopicplanner (yb(t) = fo for monopoly, [3 for socialoptimum) is not shown, but in eight of the tencases the value attained or exceeded 95% ofthe true optimum.The problem is not resolved. The observedcomparability of solution values may be de-pendent upon the functional specification orparametric values used here.

Concluding CommentsIn mid-1986, average world contract and spotprices were fluctuating around $15 per barrel.This compares to a mid-1985 value of $27 perbarrel. U.S. and world petroleum use was in-creasing as a consequence of lower prices. Ageneral conclusion is that world oil marketsmoved from a mixed competition/cartel com-bination in 1985 to a competitive market in1986. The presence or absence of an effectiveworld monopoly is probably much more im-portant in determining annual use, prices, andthe years to exhaustion than are specific pa-rameter values.

There is no intentionhere to determine heempiricalparametersof worldoil marketsandthe natureor durationof the current ransitionto a primarily ompetitivemarket.Griffin's e-cent article on the Organization f PetroleumExportingCountries addresses these points.Solow observedtwelve yearsagothatbecauseoptimaln is longerfor monopolythanfor thecompetitivesocial optimum,"the monopolistis the conservationist'sfriend" (p. 8). How-ever, the numbersshow an often-obscured e-lationship.Potentialmonopoly profitwith ba-sic parameters s $3 trillion, one-half of thepotentialnet social value of $6 trillion.The general point here is that numericalanalysis with optimal control gives differenteconomicinterpretationshan would be possi-ble with theory alone, and that dramaticallynonoptimalpathscan have almostoptimalob-jective function values.To recapitulate, indingnumericalvaluesforthe optimallengthof a productionperiodhasbeendifficult,andfindingnumericalvaluesforeconomic rent and social value functionshasbeen nearly impossible. With computer-assisted algebra and analytic solutions, theprocess can be shortened significantly.Thesteps are (a) define the first-ordercanonicalconditions with discounting explicit; (b) ex-pressy, the equilibrium uantity,as a functionof the parameters or demand, cost, and theco-state multiplier; c) solve for the co-statemultiplier X as a function of the resourcestock, the demandand cost parameters,andthe interestaccumulation actor;(d) solve fory, now anexplicitfunctionof n, t, the resourcestock, and the demandand cost parameters;(e) find optimaln; (f) evaluate the objectivefunction; and (g) evaluate the second-orderconditions.The general technique should be followedfor possible optimawhen all of the resourcesneed not be used. In both cases only permis-sable values of quantities, prices, and profitshould be examined.Compareas appropriatethe constrainedoptimum path and length oftime periodin which all of the resource mustbe used with the unconstrainedoptimum.These steps lead quickly to numerical an-swers for different decision goals, whetherthey be monopolyprofit,consumers'surplus,or competitive equilibria. Both continuousand discrete formulationsare easily solved,and the method is easily extended to includethe risk of expropriation and a technicalchange/environmentalcost continuous shift(Chapman1986, pp. 21-23). With these im-


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142 February 1987 Amer. J. Agr. Econ.proved solution techniques, classroom useand appliedresearch can grow substantially.

[Received July 1985;final revision receivedJune 1986.]

ReferencesAdams, F. Gerard, and Jaime Marquez. "Petroleum PriceElasticity, Income Effects, and OPEC's Pricing Pol-icy." Energy J. 5(1984):155-228.Chapman, Duane. Energy Resources and Energy Corpo-rations. Ithaca NY: Cornell University Press, 1983.

. Working on Fisher's Problem with ComputerAlgebra. Cornell Agr. Econ. Staff Pap. No. 86-11,1986.Clark, Colin W. Mathematical Bioeconomics. New York:John Wiley & Sons, 1976.Conrad, Jon M., and Colin W. Clark. Notes and Problemsin Resource Economics. Cambridge: Cambridge Uni-versity Press, forthcoming.Daly, George, James M. Griffin, and Henry B. Steele."Recent Oil Price Escalations: Implications forOPEC Stability." OPEC Behavior and World OilPrices, ed. J. M. Griffinand David J. Teece. Boston:Allen & Unwin, 1982.Exxon Corporation. "Improved Oil Recovery." NewYork, 1982.Fisher, Anthony C. Resource and Environmental Eco-

nomics. Cambridge: Cambridge University Press,1981.Griffin,James M. "OPEC Behavior: A Test of Alternative

Hypotheses." Amer. Econ. Rev. 75(1985):954-63.Hotelling, Harold. "The Economics of Exhaustible Re-sources." J. Polit. Econ. 39(1931):137-75.

Intriligator, Michael D. Mathematical Optimization andEconomic Theory. Englewood Cliffs NJ: Prentice-Hall, 1971.Kouris, George. "Energy Demand Elasticities in Industri-alized Countries." Energy J. 4(1983):73-94.Mathlab Group. MACSYMA Reference Manual, 2 vols.,rev. ed. Cambridge MA: MIT Laboratory for Com-puter Science, and Symbolics, Inc., 1983.Morgan, Margaret. "An Examination of the Effect of aMiscalculation of the Level of Available Oil Re-

sources on the Optimal Rate and Length of Deple-tion." Paper presented in Economics of ResourceUse seminar, Dep. Agr. Econ., Cornell University,1985.Oil and Gas Journal, 31 Dec. 1984.Pindyck, Robert S. "Gains to Producers from the Cartel-ization of Exhaustible Resources." Rev. Econ. and

Statist. 60(1978):238-51.Podulka, William. "Fisher's Problem: Some General Ob-servations and an Example Using Variable InterestRates." Paper presented in Economics of ResourceUse seminar, Dep. Agr. Econ., Cornell University,1985.Rand, R. H. Computer Algebra in Applied Mathematics:An Introduction to MACS YMA. Boston MA: Pitman,1984.Solow, Robert M. "The Economics of Resources orthe Resources of Economics." Amer. Econ. Rev.

64(1974):1-14.U.S. Department of Energy. Weekly Petroleum StatusReport. Washington DC, various issues.U.S. Geological Survey, C. D. Masters et al. "Distribu-tion and Quantitative Assessment of World Crude-OilReserves and Resources." Open-File Rep. 83-728,Reston VA, 1983.-. "World Petroleum Resources-A Perspective."Open-File Rep. 85-248, Reston VA, 1985.

1987- Computation Techniques for Inter Temporal Allocation of Natural Resources

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