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by: Dwight Marquardt IV


Dwight Marquardt IV
GPA 3.65

Aaron Benson

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Aaron Benson
Class Notes
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This 17 page Class Notes was uploaded by Dwight Marquardt IV on Thursday October 22, 2015. The Class Notes belongs to AAEC 5308 at Texas Tech University taught by Aaron Benson in Fall. Since its upload, it has received 25 views. For similar materials see /class/226379/aaec-5308-texas-tech-university in Agriculture Education at Texas Tech University.

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Date Created: 10/22/15
AAEC 56308 Lecture 10 Page 1 Lecture 10 7 Optimal Control Theory and Nonrenewable Resource Use 1 Optimal control theory 11 To determine optimal extraction of a nonrenewable resource we need new tools 12 We need to determine optimal response at every point in time in response to changing state conditions 121 What does that mean 122 Our economic agent is faced with choosing some response u at each point in time in the presence of a changing state x subject to fx t ut0 t T 123 We restrict u to some class of admissible controls U which is the class of all piecewisecontinuous real functions ut de ned for 0 StST and satisfying u H6 U I where U is a given interval called the control set 13 Given the changing state we need an objective function to maximize by choosing the response ut for every t 131 To do so we introduce the objective functional T Juj gxttutldt 0 where gx t u is a given continuously differentiable function and xt denotes the response to ut 132 The fundamental problem in optimal control theory is to determine a feasible control ut that maximizes J u 1321 Note that xt will evolve according to the 56 equation given above AAEC 56308 Lecture 10 Page 2 133 Such a control if it exists is called an optimal control 14 The maximum principle gives certain necessary conditions that must be satis ed by an optimal control 141 To motivate these necessary conditions we will consider a rm which owns a shery 142 The rm possesses an initial stock of sh x0 and will maximize pro ts from time t0totT 143 The population of sh grows according to the function 5cfxt ut t 144 Pro t is a function of shing effort ut the stock xt and a discount factor 639quot so total discounted pro t is T Wxufrrux re Td r 145 The rm will choose ut for every 0 St 5 T7 this is called the timepath for the control variable 1451 The time path of the control will determine the time path of the state 7 using the state equation of motion above 146 Now divide W into two parts pro t earned during an initial period of duration A and profit from the remaining time t A to T T Wx0utrruxtAJ TruxTdT z A Or more concisely wx0utTruxtAWx AutA AAEC 56308 Lecture 10 Page 3 147 The value from an optimal choice of u is given by J xi tmaXW xi ut and when divided between an initial period and all future periods J x tmaxlrru x t AJxZ tAl 148 Differentiate the above by the decision variable ut and set equal to zero this gives Aaniimx arm Buff But 149 The second term above is expanded by employing the chain rule 393Jx AIA70Jx AtA 3x A But 3x A But 1410 To clarify the above expression we define the costate variable i 393Jx t 7 3x Mt 1411 and we note that xtAxtAxt 1412 Since J39Cfxt I we can combine the equations under 1410 and 1411 to restate 149 as 3 tA ltx A 2tAA But 39Bu 1413 Equivalently AtAAtA2 1414 Inserting 1413 into 1412 gives AAEC 56308 Lecture 10 Page4 6 tA ltx A AIA A2A But 39Bu Bu 1415 Combining this result with 148 and canceling out the common factor A gives an af af 2 2A 0 Eu taut BuZ 1416 If we let A go to zero the above expression becomes 14161 This expression has the interpretation that the marginal pro t of harvesting sh equals the marginal value of the stock of sh 14162 This is a necessary condition for a maximum 14163 Now we need to know how the shadow price of the stock 7 changes through time 1417 We can go through the same process by differentiating the expression in 147 with respect to x and simplifying to get any 6x 6x 14171 The term on the left hand side gives the rate at which the present value of the shadow price changes 14172 Equation 1417 ensures that given the shadow price 7 the rm is indifferent between holding the last unit of stock or selling the stock and investing in the numeraire asset 1418 The necessary conditions above 1416 and 1417 can be represented more AAEC 56308 Lecture 10 Page 5 succinctly by introducing the Hamiltonian function H Hux Atrrux t2fxu t 14181 The Hamiltonian is equal to the pro t plus the change in the stock valued by its shadow price 14182 The Hamiltonian allows a convenient representation of the necessary conditions which comprise the maximum principle 14 1821 First differentiate H with respect to u 63H 0 393 u which gives equation 1416 141822 Equation 1417 is given by 63H E 7A which we call the co state condition 14 1823 The two necessary conditions plus the equation of motion 5c f x u I de ne a system of differential equations which if they satisfy sufficiency conditions define an optimal solution 15 So in general necessary conditions for solving the following problem T max g xT T uTldT 14 In subject to are found by 151 Creating the Hamiltonian AAEC 56308 Lecture 10 Page 6 HgHQLLuQHAthLLuUH 152 Differentiating H with respect to u and setting to zero 393 H 0 393 u 153 Differentiating H with respect to x and setting equal to the negative time derivative of the shadow value of the state variable 5 H 3x 7A 154 And restating the equation of motion as a necessary condition 16 It may occasionally be advantageous to describe a Hamiltonian in a currentvalue form rather than presentvalue form 161 The currentvalue Hamiltonian Xis given by 2 He 162 fis equal to the presentvalue Hamiltonian compounded xm tgx u tufx ut where u he 163 The rstorder condition is still 6 620 393 u 164 and the costate condition becomes 0 6x ery 17 Transversality conditions are conditions placed on the nal state of the statecontrol system and are required to be met to ensure that a maximum has been reached AAEC 56308 Lecture 10 Page 7 18 There are three essential attributes of transversality conditions 181 Constraints on the state variable 1811 If terminal time is xed and stock value is unconstrained it has a zero value at the end of the planning horizon 181117 T 0 Tmay be nite or in nite 18112 If otherwise it would imply that pro t could be increased by further exploitation of the stock 1812 If stock value is constrained with a weak inequality ie xT 2 x7 the above is modi ed as a KuhnTucker condition 18121 AT20andlxT7leAT0 182 Constraints on the terminal time 1821 The terminal time of the optimization problem need not be xed in the problem statement 1822 A resource owner may not know when the optimal time is to end extraction of the resource 1823 The terminal time T can become an endogenous variable in the maximization problem and solved for after optimal control and state variable time paths are determined 1824 For example in the case of a mine operator it can be shown that optimal T is the value at which the value of the Hamiltonian equals zero 183 The presence of a 39scrapvalue function 1831 In resource management problems where the terminal time is nite there AAEC 56308 Lecture 10 Page 8 may be good reason to include some condition based on some value of the stock at the terminal time BxT 18311 For example a rm that rents a resource like a shery for a period of time and is compensated based on the stock size at when the lease expires 1832 The transversality condition in this case is BB EMT 18321 MT 18322 In other words the costate value equals the marginal scrap value of stock in the terminal period 2 Nonrenewable resources 21 These are resources that exhibit no growth or regenerative processes 211 ie Rtqt 212 Later we will include the possibility of new reserves being discovered through exploratory effort wt and discovery where Xt is cumulative discoveries 22 Hotelling model 1931 221 What Hotelling described as the simplest meaningful model of exhaustible resource exploitation 222 Per unit price of ore is pt and is a function of the form pt p0e 2221 Which means that the mine owner is indifferent between receiving price p0 now or p0e after time t AAEC 56308 Lecture 10 Page 9 223 The ore producer or mine owner in this model is perfectly competitive so extraction at time tis determined by the demand function 90 DPt 224 If there are no costs of extraction the initial reserves R will be exhausted qtdtR oh 225 AttTqT0and qltTgtDplt0gte 0 2251 That is that the choke price for ore is reached at terminal time T 226 The three equations above determine p0 T and the entire timepath of extraction 2261 Suppose that D is linear qtDlptJabpt 2262 Then qtae 13100 2263 and qTa7bp0eM0 2264 Which implies that plt0e 2265 and therefore qtaliem D 2266 Since initial reserves will be exhausted AAEC 56308 Lecture 10 Page 10 lie dtR a oh 2266l Integrating the left hand side of 2266 gives aTiaLlie 5T5R 22662 And T can be solved for numerically and when substituted into 2265 gives the complete time path of extraction 2267 So ifa 100 3 05 andR 500 22661 500 at T 16 and the time path of extraction becomes qt100lliem 16 23 Now consider a monopolistic producer 231 The monopolist wishes to maximize Tm Trf Pqt qte malt 0 where Pqt is the inverse demand function for ore price as a function of quantity supplied 232 We will now formulate the monopolists decision as an optimal control problem by denoting remaining reserves Rt so that dRt dt RQt R0R 233 The monopolist s current value Hamiltonian is 234 And the necessary conditions are 0 PP39qtut0 2341 7 0W Page 1 1 AAEC 56308 Lecture 10 3 2w 1 5HltIm0 2343 R7qt 235 P P39qt is marginal revenue which we denote as MRO from now on 236 Equation 2342 implies 5 u or that the shadow price of the the resource increases as the rate of interest 237 By 2341 the shadow price is equated to marginal revenue so we have MR t 6 MR I so the monopolist extracts ore such that marginal revenues increase at the rate of interest 238 Suppose the monopolist faces the same linear demand curve as in 2261 the inverse of that demand is ptab7qtb 239 Remembering that MR P P39qt we have MRtabi2qtb 2310 One of our transversality conditions is that H Tm 0 23101 Equation 233 with our transversality condition implies that qTm 0 as well 23102 Evaluating 2341 at t Tm implies abuTm 23103 But utu0em and uTmu0eM so u0ae T b AAEC 56308 Lecture 10 Page 12 nally giving utae I mlb 2311 The rst necessary condition requires that MR u so we can equate 23103 with 239 and get ab72qtbae I mlb 2312 Solve the above for qt 5 2 TM qlttllie 2313 To nd Tm we need to consider the condition on total reserves ie aha 317e H 4521 2 rm rm f gable Wait 0 2 0 0T Tm a 3 3T m 7 quotdt 2 1 28 e aTm a 5T M a mo 5T aTm at 5T 7 m m m 1 M R 2 256 8 Se 8 2 2 e 2314 Similarly to the competitive market solution we can solve for Tm 24 Now we can compare the differences between exploitation in competitive markets and monopoly 241 See the example spreadsheet that compares the extraction paths between the two industries AAEC 56308 Lecture 10 Page 13 3 Extraction costs 7 the case of a single mine 31 We complicate the basic model above by adding extraction costs 311 We assume costs depend only on the rate of extraction 3111 Ct Cqt 393 C 02C 3112 Where a gt0 and 2gt0 q 393 q 312 Also we let pt be exogenous and known in advance to the mine owner 32 The mine owner chooses extraction rate according to T maXllPltMUFC39qume d st Rt7qtR0R0Rt20 321 The currentvalue Hamiltonian is MtplttqltteCqlttJiuIMO 322 The rst order conditions are 63 3221 aqltt0gtptiC qut 3222 39 5 0 1 5 7 gt 63R u 323 The above equations can be combined to show d t 7C39 dt p 61 ptC 39q 3231 Or that price net of marginal costs rise at the rate of interest 324 To develop a solution for the mine owner ie to determine the time path of q we will assume that the cost curve is Ushaped as below AAEC 56308 Lecture 10 Page 14 3241 As long as pt gt C39q we will have q lt qt for 0 S IS T 325 The transversality condition for the free terminal time problem ie T is not xed implies that 0 at t T 3251 So by the setting the Hamiltonian equal to zero and solVing for ut we have 3254 So by our above assumption 610 qquot 326 With the result in 3254 the problem would be solved ifwe knew T AAEC 56308 Lecture 10 Page 15 3261 Because with qT known equation 3251 gives T 3262 Equation 3222 indicates that u Ie u0 3263 Which in turn implies that u Te 0 3264 If we divide the above two equations we get HQ 7 m T m4 gt ute uT 3265 and then equation 3221 could be rearranged as and qt could be solved for directly 327 So to solve for T we use the condition that RT 0 3271 Since pt gt C 39q for all t the mine operator will extract in every period so at some point the resource will be exhausted 33 As an example suppose that price pt p and assume that costs are Cq a qu 331 Then lm 3311 Because q is q such that 3312 So given our cost function above 33 we have AAEC 56308 Lecture 10 Page 16 2bqaqbq gt 2bq2abq2 bq2a gt qv im 332 And equations 3264 and 3265 will be used to determine the time path ofq 3321 First to nd uT we evaluate the rstorder condition 3221 at time T and solve for T uTpC39q 3322 Given 3311 and 3312 HTPV 3323 Now foc 3265 also gives C39qtl2bqlttpiultt 3324 So by 3264 and 3322 ultte H Pi 1 3325 So 3323 becomes 2bqtp7e5 I lpiv l 3326 And we have WL 3 l 2bl 10 le pie 3327 Finally T is determined by AAEC 56308 Lecture 10 Page 17 T 139 5739 T 72Vab lie tdtL7L R gm 2b 2b 6 333 By solving numerically for T a time path can be determined for qt 3331 The nonrenewable resources spreadsheet can be used to conduct comparative statics analysis 3332 What happens as costs of extraction increase 3333 What happens when the market price of the resource commodity decreases 4 Readings PMMC Chapter 17 pp 555597


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