This document was generated at 5:08 PM, 09/24/13 Copyright 2013 Richard T. Woodward

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1 his documen was generaed a 5:8 PM, 9/24/13 Copyrigh 213 Richard. Woodward 6. Lessons in he opimal use of naural resource from opimal conrol heory AGEC I. he model of Hoelling 1931 Hoelling's 1931 aricle, he Economics of Exhausible Resources is a classic ha provides very imporan inuiion ha applies no only o naural resources, bu any form of depleable asse. Hoelling does no use he mehodology of opimal conrol (since i wasn' discovered ye), bu his mehodology is easily applicable o he problem. A. he basic Hoelling model Hoelling considers he problem of a depleable resource (like oil or minerals) and how migh i be opimally used over ime. Wha are he sae and conrol variables of such a problem? Le x be he sock of he resource remaining a ime and le z be he rae a which he sock is being depleed. For simpliciy, firs assume ha exracion coss are zero, and ha he marke is perfecly compeiive. In his case, he represenaive owner of he resource will receive p z from he exracion of z in period and his will be pure profi or, more accuraely, quasi-rens. efiniions (from hp:// Economic ren: A surplus paid o any facor of producion over is supply price. Economic ren is he difference beween wha a facor of producion is earning (is reurn) and wha i would need o be earning o keep i in is presen use. I is, in oher words, he amoun a facor is earning over and above wha i could be earning in is nex bes alernaive use (is ransfer earnings). Quasi-ren: Shor-erm economic ren arising from a emporary inelasiciy of supply. P CS PS= Quasi Ren u ( x ) = z, z, p( z) dz z ime is equal o he area under he inverse demand curve, i.e., u ( x, z, ) ( ) z We consider he problem of a social planner who wans o maximize he presen value of consumer surplus plus rens (= producer surplus in his case). CS + PS a any insan in = p z dz, where p(z) is he inverse demand curve for exracions of he resource.

2 6-2 he problem is consrained by he fac ha he original supply of he resource is finie, x(=)=x, and any exracion of he resource will reduce he available sock, xɺ = z. We know ha in any period x and simple inuiion assures us ha x =. o you see why x =? A formal saemen of he planner's problem, hen, is as follows: z r r max e u ( x,, ) max ( ) z d e p z dz d z = z s.. xɺ = z x(=)=x x he Hamilonian of his problem is, herefore, H=e -r u( ) +λ(-z ) and he maximizaion crieria are: 1. H z =: e -r u'( ) -λ = e -r p(z ) -λ = 2. H x = λ ɺ : λ ɺ = 3. H λ = xɺ : xɺ = z he ransversaliy condiion in his case is found by he erminal poin condiion, 4. x = Looking a 1 and, using he inuiion developed by orfman, we see ha he marginal benefi of exracion in, e -r p(z ), mus be equal o he marginal cos in erms of foregone fuure ne benefis, λ. From 2 we see ha λ is consan a, say, λ so we can drop he subscrip. his is rue in any dynamic opimizaion problem in which neiher he benefi funcion nor he sae equaion depend on he sae variable. his oo is consisen wih he inuiion of orfman since he sae variable does no give rise o benefis a, is marginal value does no change over ime. Subsiuing λ = λ ino 1, we obain p(z ) =λe r. his is imporan. I shows ha he opimal price will grow a he discoun rae, and his is rue regardless of he demand funcion (as long as we have an inerior soluion). [Noe ha in his example he marginal exracion cos is se a zero so ha he price is equal o he marginal quasi-rens earned by he producer. More generally, he marginal quasirens would be equal o price minus marginal cos, and his would grow a he rae of ineres.]

3 6-3 Anoher hing ha is ineresing in his model is ha he value of λ does no change over ime. ha means ha he marginal incremen o he obecive funcion (he whole inegral) of a uni of he resource sock never changes. In oher words, looking a he enire ime horizon, he planner would be compleely indifferen beween receiving a marginal uni of he resource a ime and he insan before, as long as i is known in advance ha a some poin he uni will be arriving. However, noe ha his is he presen value co-sae variable, λ. Wha would he pah of he curren-value cosae variable look like? How does he economic meaning of differ from ha of λ? If we wan o proceed furher, i is necessary o define a paricular funcional form for our demand equaion. Suppose ha p(z)=e -γz so ha he inverse demand curve looks like he figure above. r Hence, from 1, H z = e e γ z γ z = λ, or e r = λe so ha, γ z = ln λ + r or 5 λ + r z = γ A any poin in ime i will always hold ha x = x + xɺ τ dτ. Hence, from our ransversaliy condiion, 4, x x x dτ = = ɺ. τ τ = From 3 and 5, his can be rewrien Evaluaing he inegral leads o 1 r 2 ln ( λ) = x γ 2 r ln λ = x 2 γ. γ r ln λ = x + 2 Hence, we can solve for he unknown value of λ, γ r x 2 τ = ( λ) ln + r γ zd = x or d = λ = e. In his case we can hen solve explicily for z by subsiuing ino 5, yielding x.

4 6-4 ln λ + r z = γ ln z = γ r x 2 e + γ r γ r r z = x + γ 2γ γ x r r x r z = + = + 2γ γ γ 2 o verify ha his is correc, check he inegral of his, from o x r 2 r 2 z x d = + =. 2γ 2γ 6. Homework ip: Noice ha a his poin z, is a funcion of only known parameers: x, which we know because we can observe his before saring he problem,, which is assumed o be known in advance, and γ, which is a known parameer. If, when solving an opimal conrol problem you sill have unknown variables in your resul, e.g., λ, hen you are no finished. Frequenly, you need o use he ransversaliy condiion a his poin Looking a 6, we see ha he rae of consumpion a any poin in ime is deermined by wo pars: a consan porion of he oal sock, x, plus a porion ha declines linearly r over ime. his second porion is greaer han zero unil γ 2 less han zero for he remainder of he period. Noe ha, ha z < if = 2, and is hen 2γ x 7. >. r So ha if his inequaliy is saisfied, i.e., if he ime horizon is long enough, hen along he opimal pah ha resuls from choices made following 6, z will become negaive, implying ha he resource sock is being rebuil as we approach. his means ha x is negaive over some range, violaing he consrain x. Hence, if 7 holds, he soluion violaes he consrain and we will need o re-solve he problem wih an explici consrain on he opimizaion problem. We will evaluae such consrained problems in lecure 13.

5 6-5 B. Some variaions on he heme and oher resuls Hoelling's analysis cerainly doesn' end here. Q: Consider again he quesion, Wha would happen if we used he curren-value insead of he presen-value Hamilonian? A: Well, you can be sure ha he curren value co-sae variable,, would no be consan over ime how would he change in he shadow price of capial evolve? Wha s he economic inerpreaion of? Q: Wha if here are coss o exracion c(z ) so ha he planner's problem is o maximize he area under he demand curve minus he area under he marginal cos curve? z A: Firs recognize ha if we define ( ) u~ = p( z) c' ( z) dz, where c' is he marginal cos funcion, hen he general resuls will be exacly he same as in he original case afer subsiuing marginal quasi rens for price. ha is, along he opimal pah he marginal surplus will rise a he rae of ineres. Obviously geing a nice clean closed-for soluion z * will be more difficul he more complex and realisic you make c( ), bu he economic inuiion does no change. his economic principle is a cenral o a wide body of economic analysis. Q: Would he social opimum be achieved in a compeiive marke? A: Firs, assuming ha boh consumers and producers are ineresed in maximizing he presen value of heir respecive welfare, hen we've maximized oal surplus, i.e., i is a Pareo Efficien oucome. So we can hen ask, o he assumpions of he 2nd Welfare heorem hold? If hey do, hen wha does ha ell us abou he social opimum? If hese hold, hen for a Pareo efficien here exiss a price vecor for which any Pareo efficien allocaion will be a compeiive equilibrium. Finding he Pareo opimal allocaion also gives a compeiive equilibrium. Hence, our findings are no only normaive, bu more imporanly, hey re posiive; i.e. a predicion of wha choices would acually occur in a perfecly compeiive economy. Now, le's look a his quesion a lile more inuiively. We know ha one of he basic resuls is ha he price (or marginal quasi rens) grow a he rae of ineres? Is his likely o occur in a compeiive economy as well? In he words of Hoelling, i is a maer of indifference o he owner of a mine wheher he receives for a uni of his produc a price p now or a price p e γ afer ime (p. 14). ha is, price akers will look a he fuure and decide o exrac oday, or a uni omorrow a a higher price. he price mus increase by a leas he rae of ineres in his simple model because, if no, he marke would face a glu oday. If he price rose faser han he rae of ineres, hen he owners would choose o exrac none oday. Assuming ha he inverse-demand curve is downward sloping, supply and demand can be equal only if each individual is compleely indifferen as o when he or she exracs which also explains he consancy of λ. his also ges a an imporan difference beween profi and rens. We all know ha in a perfecly compeiive economy wih free enry, profis are pushed o zero -- so why do he holders of he resource sill make money in his case? Because here is no free enry. he oal resource endowmen is fixed a x. An owner of a porion of ha sock is able

6 6-6 o make resource rens because he or she has access o a resriced profiable inpu. Furher, he owner is able o exploi he radeoffs beween curren and fuure use o make economic gains. his is wha is mean by Hoelling rens. II. Harwick's model of naional accouning and he general inerpreaion of he Hamilonian Harwick (199) has a very nice presenaion of he Hamilonian's inuiive appeal as a measure of welfare in a growh economy. he analogies o microeconomic problems will be considered a he end of his secion. he 199 paper builds on Weizman (1976) and is a generalizaion of Harwick s more ofen cied paper from A. he general case We'll firs presen he general case and hen look a some of Harwick's pariculars. Consider he problem of opimal growh in an economy maximizing ( ) U C e ρ d subec o a sae equaion for a malleable capial sock, x, ha can eiher be consumed or saved for nex period xɺ = g x, z C ( ) and n addiional sae equaions for he n oher asses in he economy (e.g., infrasrucure, human capial, environmenal qualiy, ec.). xɺ = g x,z, i=1,,n. i i ( ) Please excuse he possibly confusing noaion. Here he subscrip is an index of he good and he ime subscrip is suppressed. Le z represen a vecor of conrol variables and C be he numeraire choice variable (hink consumpion). he vecor of sae variables is denoed x. he general curren value Hamilonian of his opimizaion problem is n Hc = U ( C) + ( g ( x, z) C) + g ( x, z ). 1 = 1 his is our firs exposure o he problem of opimal conrol wih muliple sae and conrol variables, bu he maximizaion condiions are he simple analogues of he single variable case: = = for all i [or in general, maximize H wih respec C and all he z i s] C z i 1 Again o wrie more concisely, from his poin furher we will wrie H insead of H c, which we ypically use in hese noes.

7 6-7 = ρ ɺ for all x = x ɺ for all Given he specificaion of uiliy, = U ' = =U'. C (remember, is he cosae variable on he numeraire good, no he cosae variable a =.) Similar o he approach used by orfman, Harwick uses a linear approximaion of curren uiliy, U(C) U' C, and, if we measure consumpion in erms of dollars, U' is he marginal uiliy of income. Using his approximaion, he Hamilonian can be rewrien n = + + = 1 ( ( x z) ) ( x z ) H U ' C g, C g,. ividing boh sides by he marginal uiliy of consumpion and remembering ha =U', he obains he relaionship n H = C + x ɺ + x ɺ U ' = 1 If you look a he RHS of his equaion, you will see ha his is equivalen o ne naional produc (NNP) in a closed economy wihou governmen. NNP is equal o he value of goods and services (C) plus he ne change in he value of he asses of he economy, n xɺ + xɺ. = 1 he firs lesson from his model, herefore, is a general one and, as we will discuss below, i carries over quie nicely o microeconomic problems: maximizing he Hamilonian is equivalen o maximizing NNP, which seems like a prey reasonable goal. he Hamilonian, herefore, is a measure of he benefis o he decision maker ha correcly akes ino accoun boh curren benefis and coss, and fuure consequences. Second, using some simplisic economies, Harwick helps us undersand wha he appropriae shadow prices o be used when measuring changes in an economy's asses:. In he nex secion couple of secions we consider cases where his framework can be helpful in idenifying he righ prices for resource sock. B. he case of a non-renewable resource he firs case o consider is an economy in which here are hree sae variables. Firs, here's he fungible capial sock, x which we will now call. Second, here's a nonrenewable resource or mine, S which falls as he resource is exraced, R, and grows when here are discoveries,. Exracions, R are used in he producion funcion F( ) bu here is a cos o exracion, f(r,s).

8 6-8 iscovery coss rise over ime as a funcion of cumulaive discoveries so ha he marginal cos of finding more of he resource increases over ime. he oal cos of discovery in a period is v(), linearly approximaed as v wih v changing over ime. 2 Harwick also includes labor, L. However, since he economy is always assumed o be a full employmen and he growh rae of labor is exogenous, labor can be reaed as an inermediae variable and can, herefore, be largely ignored. he hree sae equaions are, herefore, Capial sock: ɺ = F, L, R C f R, S v Resource Sock: iscovery Cos: Sɺ = R + vɺ ( ) ( ) ( ) = g and he resuling curren value Hamilonian is H = U C + F, L, R C f R, S v + S R + + g he FOCs w.r.. he choice variables are: ( ) ( ) ( ) [ ] ( ) H C =: U ' = H R =: [ F f ] =. R R S H =: v + + g =. S A linear approximaion of he curren-value Hamilonian can be wrien H = U ' C + ɺ + R + + g ' [ ] S ividing by U'= k, we ge H S C R S = + ɺ + + g U ' S Using he H R and H condiions, i follows ha [ F f ] [ ] R R = and v S v FR f R = or = g g g g Hence, he linear approximaion of he Hamilonian can be rewrien H [ FR fr ] [ FR fr ] v [ FR fr ] = C + ɺ R + + g U ' g g H = C + ɺ [ F R f R ] R + v U ' We know ha in a compeiive economy, he price paid for he resource would equal F R (resources are paid heir marginal value produc). Hence, o arrive a NNP curren 2 his is a refinemen of he specificaion in Harwick (199) as proposed Hamilon (1994).

9 6-9 Hoelling Rens from exracions, namely [ ] FR fr R, should be need ou of GNP, and discoveries, priced a he marginal cos of discovery, should be added back in. 3 So we see ha here are appropriae prices ha could be used o adus NNP o ake ino accoun resource exracion and discoveries. Is his common pracice in naional accouning? No. he depreciaion of naural resource asses is ignored in he sysem of naional accouns. his resuls in a misrepresenaion of naional welfare. One reason for his is he abiliy o acually implemen he necessary accouning pracice. Harwick elaboraes, he principal problem of implemening he accouning rule above is in obaining marginal exracion coss for minerals exraced. C. An economy wih annoying polluion he final example ha Harwick presens is ha of an economy in which here is a disuiliy associaed wih polluion. he case Harwick considers is where naional welfare is affeced by changes in he polluion sock. ha is, if he sock of polluion is increasing, welfare goes down. If he sock of polluion is falling, welfare goes up. In his case we would have U = U( C, Xɺ ), where X ɺ is he change in he polluion sock, wih U < X ɺ. 4 Producion is assumed o be affeced by polluion, i.e., F(,L,X) so, for example, more polluion makes producion more difficul. he polluion sock is assumed o increase wih he producion a he rae γ, and decrease wih choices made regarding he level of cleanup, b, which coss f(b), i.e., Xɺ = bx + γ F (, L, X ) and he evoluion of he ɺ = F, L, R C f b. numeraire capial sock follows ( ) ( ) he curren value Hamilonian wih his sock change incorporaed in he uiliy funcion, herefore, is H = U ( C, bx + γ F(, L, X )) + [ F(, L, X ) C f ( b) ] + X [ bx + γf(, L, X )]. Again, he FOC w.r.. he conrol variables C and b are: = U C = C U x fb X = = U x X fb X X = X b Using he linear approximaion of H, herefore, yields. (8) 3 his resul differs from ha presened in Hamilon (1994). I have no aemped o unravel where he difference comes from. Help is welcome o improve hese noes. 4 I m no paricularly fond of his specificaion, bu i does lead o some ineresing resuls.

10 6-1 H = U C + U X + ɺ + Xɺ ' c x X U H C x X X = + + ɺ + X ɺ U c U c hen, using he FOC wih respec o b, H U x U X fb = C + X + ɺ + X ɺ Uc Uc X Hence, if we wan o correcly calculae NNP, aking ino accoun he changes in he sock of polluion in he calculaion of welfare, he price ha should be placed on hese changes is a funcion no only of he marginal damage of changes in he sock of polluion, bu he marginal cos of clean-up as well.. Implicaions beyond he realm of naional income accouning If you're no paricularly ineresed in he naional income accouns or environmenal and naural resource economics, he above discussion may seem academic. However, clearly, he correc measuremen of income is no an academic pursui limied o he naional income accouns. Hicks' (1939, Value and Capial) defined income as, o paraphrase, he maximum amoun ha an individual can consume in a week wihou diminishing his or her abiliy o consume nex week. Clearly, us as for a naional accoun, farmers and managers also need o be aware of he disincion beween invesmen, capial consumpion, and rue income. Harwick's Hamilonian formulaion of NNP, herefore, wih is useful presenaion of he correc prices for use in he calculaion of income, migh readily be applied o a hos of microeconomic problems of concern o applied economiss. III. References Harwick, John M Inergeneraional Equiy and he Invesing of Rens from Exhausible Resources. American Economic Review 67(5): Harwick, John M. "Naural Resources, Naional Accouning and Economic epreciaion." Journal of Public Economics 43(ecember 199): Hamilon, irk Green Adusmens o GP. Resources Policy 2(3): Hicks, John Richard. Value and capial; an inquiry ino some fundamenal principles of economic heory. Oxford: he Clarendon press, Hoelling, Harold he Economics of Exhausible Resources. he Journal of Poliical Economy 39(2): Weizman, Marin L. "On he Welfare Significance of Naional Produc in a ynamic Economy." Quarerly Journal of Economics 9(Feb 1976):

11 IV. Readings for nex class For fun, read reyfus, S. 22. Richard Bellman on he Birh of ynamic Programming. Operaions Research 5(1): (link o on-line version from noes page). his is us ligh reading ha will give you a glimpse of how he principles of P arose. 6-11