Sesion 4_2

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Chapter 10 Applications to Natural ResourcesObjective:

Optimal management and utilization of naturalresources.

Two kinds of natural resource models:

(i)renewable resources such as fish, food, timber,etc., Section 1 .!: an optimal forest thinning model.(ii)nonrenewable or e"haustible resources such as petroleum, minerals, etc. Section 1 .#: an e"haustible resource model.

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10.1 The Sole Owner Fishery Resource o!el1 .1.1 The $%namics of &isher% 'odel

otation and terminolog% is due to lark (1*+ ): ρ - the discount rate,

x (t ) - the biomass of fish population at time t ,

g ( x ) - the natural growth function,u(t ) - the rate of fishing effort at time t ≤ u ≤ U, q = the catchabilit% coefficient, p - the unit price of landed fish, c - the unit cost of effort.

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/ssume growth function g is differentiable andconca0e,

where X denotes the carrying capacity , i.e., thema"imum sustainable fish biomass.The model e uation due to 2ordon(1*34) and

Schaefer(1*3+) is

The instantaneous profit rate is

&rom (1 .1) and (1 .!), it follows that x will sta% in theclosed inter0al ≤ x ≤ X pro0ided x 0 is in the sameinter0al.

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re0enues e"ceed total costs, thereb% attractingadditional fishermen, and increasing the fishing effort.

10.1." The Sole Owner o!elThe bionomic e uilibrium solution obtained from theopen access fisher% model usuall% implies se0erebiological o0erfishing. Suppose a fishing regularagenc% is established to impro0e the operation of thefishing industr%. The ob8ecti0e of the agenc% is:

sub8ect to (1 .!).

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10.1.# Solution $y %reen&s Theore'Sol0ing (1 .!) for u we obtain

Substitute into (1 .#), to obtain

where

where B is a sate tra8ector% in ( x,t ) space, t ∈ 9 ,∞ ).

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et denote a simple closed cur0e in the ( x,t ) spacesurrounding a region R in the space. Then,

let

rewrite (1 .11) as

/s in Section +.!.! and +.!.4, the turnpike le0el isgi0en b%

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The re uired second6order condition is

et be the uni ue solution to (1 .1!) satisf%ing thesecond6order condition.The corresponding 0alue of the control which wouldmaintain the fish stock le0el at is . 5n;"ercise 1 .! %ou are asked to show thatand also that . 5n &igure 1 .1 optimaltra8ectories are shown for two different initial 0alues:

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http://slidepdf.com/reader/full/sesion-42-56d9f0a15ff36 9/38Fi(ure 10.1: Opti'al )olicy *or the Sole OwnerFishery o!el

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+cono'ic interpretation:

where

The interpretation of π ( x ) is that it is the sustainable economic rent at fish stock le0el x .This can be seen b%

substituting into (1 .#), whereobtained using (1 .1!), is the fishing effort re uired tomaintain the fish stock at le0el x . Suppose we ha0eattained the e uilibrium le0el gi0en b% (1 .!), and

suppose we reduce this le0el to b% using fishingeffort of .

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The immediate marginal re0enue, MR , from this actionis

<owe0er, this causes a decrease in the sustainableeconomic rent which e uals

O0er the infinite future, the present 0alue of thisstream, i.e., the marginal cost MC , is

; uating MR and MC , we obtain (1 .1#), which is also(1 .1!).

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7hen the discount rate is zero, e uation (1 .1#)reduces to

so that it will gi0e the e uilibrium fish stock le0elfor ρ - , which ma"imizes the instantaneous profit rate

π ( x ) . This is called in economics the golden rule le el .

7hen ρ - ∞ , we can assume that π ! ( x ) is bounded.&rom (1 .1#) we ha0e pqx" c - , which gi0es

The latter is the bionomic e uilibrium attained in theopen access fisher% solution see (1 .4).

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The sole owner solution satisfies . 5fwe regard a go0ernment regulator% agenc% as the soleowner responsible for operating the fisher% at le0el ,then it can impose restrictions, such as gearregulations, catch limitations,etc., which increase thefishing cost c .5fc is increased to the le0el , then the fisher% canbe turned into an open access fisher% sub8ect to thoseregulations, and it will attain the bionomic e uilibrium

at le0el .

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10." An Opti'al Forest Thinnin( o!el10.".1 The Forestry o!el t 0 - the initial age of the forest,

ρ - the discount rate, x (t )- the 0olume of usable timber in the forest at time t ,u(t ) - the rate of thinning at time t , p - the constant price per unit 0olume of timber, c - the constant cost per unit 0olume of thinning,f ( x ) - the growth function, which is positi0e, conca0e, and has a uni ue ma"imum at x m we assume f ( )- ,g (t ) - the growth coefficient which is positi0e,

decreasing function of time.

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&orm for the forest growth is

where α is a positi0e constant. f is conca0e in the

rele0ant range and that . The% use the growthcoefficient of the form

where a and b are positi0e constants.The forest growth e uation is

Ob8ecti0e function is

The state and control constrains are

(1 .1 ) implies no replanting.

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10."." ,eter'ination o* Opti'al Thinnin(

&orestr% problem has a natural ending at a time # forwhich x (# )- .To get the singular control solution triple , wemust obser0e that and will be functions of time.&rom (1 .1*), we ha0e

which is constant so that . &rom (1 .1=),

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Fi(ure 10.": Sin(ular -sable Ti'ber olu'e

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Fi(ure 10.#: Opti'al )olicy *or the Forest Thinnin(o!el when

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10.".# A Chain o* Forests o!el

Similar to the chain of machines model of Section *.#.7e shall assume that successi0e plantings, sometimescalled forest rotations, take place at e ual inter0als.This is similar to the assumption (*.#*) emplo%ed in themachine replacement problem treated in Sethi (1*+#b).

et # be the rotation period, during the nth rotation, thed%namics of the forest is gi0en b% (1 .1#) witht ∈ 9(n61)# ,n# > and x 9(n61)# >= .

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Fi(ure 10./: Opti'al )olicy *or the Chain o*Forests o!el when

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Case 1:

ote that in the second integral is an

impulse control bringing the forest from 0alue to b% a clearcutting operation differentiate (1 .!3) withrespect to # , e uate the result to zero,

5f the solution # lies in , keep it otherwise set .

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Case ":5n the ?idale67olfe ad0ertising model of hapter +, asimilar case occurs when # is small the solution for

x (# ) is obtained b% integrating (1 .1#) with u - and x 0 - . et this solution be denoted as x%&t ). <ere(1 .!4) becomes

differentiate (1 .!+) and e uate to zero, we get

5f the solution lies in the inter0al keep it otherwiseset The optimal 0alue #% can be obtained b%computing ' @(# ) from both cases and selectingwhiche0er is larger.

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Fi(ure 10. : Opti'al )olicy *or the Chain o* Forestso!el when

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10.# An + haustible Resource o!el7e discuss a simple model taken from Sethi(1*+*a).This paper anal%zes optimal depletion rates b%ma"imizing a social welfare function which in0ol0esconsumersA surplus and producersA surplus with0arious weights. <ere we select a model ha0ing thee uall% weighted criterion function.

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10.#.1 For'ulation o* the o!el /ssume that at a high enough price, sa% p , asubstitute, preferabl% renewable, will become a0ailable.

p (t ) - the price of the resource at time t , q = f ( p ) is the demand function,i.e., the uantit%

demanded at price p and

where is the price at which the substitute completel% replaces the resource. /

t%pical graph of the demand function is shown in&igure 1 . ,

c - ( (q) is the cost function ( (0 )- , ( (q)B for q B , ($)0, and (* ≥ 0 for q ≥ 0 , and ($ (0 )C +the latter

assumption makes it possible for the producers to

make positi0e profit at a price p below ,

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(t ) - the a0ailable stock or reser0e of the resource at time t , ρ - the social discount rate, ρ B , # - the horizon time, which is the latest time at which the substitute will become a0ailable regardless of the price of the natural resource, # B .

et

for which it is ob0ious that

et

denote the profit function of the producers, i.e, the producers$ surplus+

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et be the smallest price at which is nonnegati0e. /ssume further that is a conca0e function in therange as shown in &igure 1 .+. 5n the figure thepoint p m indicates the price which ma"imizes .7e also define

as the consumers$ surplus , i.e., the area shown shadedin &igure 1 . . This uantit% represents the total e"cessamount consumers would be willing to pa%,i.e,

consumers pa% pf ( p ), while the% would be willing to pa%

ote pdq - pf! ( p )dp

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Fi(ure 10.2: The ,e'an! Function

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The instantaneous rate of consumersA surplus andproducersA surplus is the sum . et denotethe ma"imum of this sum, i.e., sol0es

5n ;"ercise 1 .14 %ou will be asked to show thatThe optimal control problem is:

sub8ect to

and . Decall that the sum isconca0e in p .

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Fi(ure 10.3: The )ro*it Function

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10.#." Solution by the a i'u' )rinciple

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The right6hand side of (1 .41) is strictl% negati0ebecause f$ C , and (* ≥ b% assumption. 7e remark

that using (1 .#!) and (1 .#*), and hence thesecond6order condition for of (1 .#!) to gi0e thema"imum of - is 0erified.Case 1 The constraint (# ) ≥ is not binding:λ (t ) ≡ so that

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Case " To obtain the solution re uires finding a 0alue of λ (# )

such that

where

The time t% , if it is less than # , is the time at which

which, when sol0ed for t%, gi0es the second argumentof (1 .43).

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One method to obtain the optimal solution is to define as the longest time horizon during which the

resource can be optimall% used. Such a must satisf%

and therefore,

Subcase "a The optimal control is

learl% in this subcase, t% = and

in &igure 1 .=.

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Subcase "b <ere the optimal price tra8ector% is

where λ (# ) is to be obtained from the transcendentale uation

in &igure 1 .*

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Fi(ure 10.4: Opti'al )rice Trajectory *or

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Sesion 4_2

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