1 Sterile Resources. This is the simplest case of exhaustion of a finite resource. We will use the terminology

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1 Cmbridge Universiy Press Susinble Nurl Resource Mngemen for Scieniss nd Engineers Excerp More informion Serile Resources In his chper, we inroduce he simple concepion of scrce resource, loclly owned nd globlly rded. I is he clssic Qudrn resource, vluble nd scrce. Producion mouns o mking i vilble for sle ino n economic mrke in which is scrciy ffecs is price. Accordingly, price is endogenous o he resource sysem. Producion is he sme s consumpion, which is nmoun o desrucion: irreversible conversion o oher chemicl forms wih no recycling. The owner s bsic decision is how fs o produce. Th he resource is finie is firs principle. The fc h some porion of he resource is undiscovered ny poin in ime does no chnge is finieness. Wh does chnge, over ime, is he improving se of knowledge bou how much of he resource here is. Decisions bou how fs o produce re lwys reched wihin n environmen of imperfec knowledge nd speculion bou fuure discoveries. There is need o mke decisions in his uncerin environmen nd need o djus coninully s new informion becomes vilble. Explorion reduces, bu does no elimine, unceriny. This cse is exreme in is simpliciy. I is elbored below; he exmple of peroleum is used hroughou. Mny criicl conceps of resource economics re inroduced nd crried forwrd ino subsequen chpers.. COSTLESS PRODUCTION OF A STERILE RESOURCE.. Bse Cse This is he simples cse of exhusion of finie resource. We will use he erminology S() = moun of he resource remining o be produced nd sold X () = producion re P() = mrke price per uni of producion Cmbridge Universiy Press

2 Cmbridge Universiy Press Susinble Nurl Resource Mngemen for Scieniss nd Engineers Excerp More informion Serile Resources I is ssumed h he resource is owned unmbiguously, h i coss nohing o produce i, nd h S, X, nd P re known wih perfec ceriny. Three relions govern: Mss blnce: Price-sensiive demnd: ds d Opiml economic decision mking: = X (.) X = P β (.) dp d = rp (.) The decision equion is reched by considering rdeoff beween uni of resource produced nd sold ody versus wiing nd doing he sme ler. If P grows fser hn r, he ineres re vilble for invesmen of money, hen conserving he resource for ler sle is profible nd producers will do so he vlue of he resource grows fser hn money. If, on he oher hnd, P grows slower hn r, hen conservion is bd invesmen nd selling now is preferble money grows fser hn he vlue of he resource, nd resource owner would prefer o produce now nd inves he proceeds re r. The price equion expresses he poin of indifference beween hese wo opions; i would be relized in siuion of compeiion mong mny producers. (This is Hoelling s Rule [4]. There will be more o sy bou his ler.) The soluion for P is P = P e r (.4) nd hus we hve he producion re X, from he demnd funcion nd he iniil producion re is X = β P e βr (.5) X = P β (.6) Since ds/d = X, we hve S() = S Xd = S [ e βr ] P β (.7) βr We require wo condiions o close he sysem: S, he presen moun of he resource, nd P, he iniil price. Cmbridge Universiy Press

3 Cmbridge Universiy Press Susinble Nurl Resource Mngemen for Scieniss nd Engineers Excerp More informion. Cosless Producion of Serile Resource P Sorge, S 5 5 Producion, X P Figure.. Five differen depleion hisories, idenicl excep for iniil price. P increses by fcors of in he direcion of he rrows. S is presumed known; P is no. If P is se oo high, he demnd will be suned nd he resource will go unuilized. If P is se oo low, he demnd will be oo lrge nd he resource will be depleed premurely, leving our mhemics of decision mking invlid (Figure.). The sysem is closed by invoking he Terminl Condiion (TC): complee resource exhusion s ime goes o infiniy: S s (.8) Thus, S = P β βr (.9) The iniil price is hus P = [ ] β (.) S βr nd he iniil producion re is X = β P = βrs (.) If P is oo high (X oo low), hen S is never exhused. If P is oo low (X oo high), hen he resource is exhused premurely. In eiher cse, producion would be djused o sisfy he TC (Figure.). Ren is he inegred presen worh of ll ne revenues: R = Since P = P e r, we hve e r X ()P()d (.) R = P X ()d = P S (.) Cmbridge Universiy Press

4 Cmbridge Universiy Press Susinble Nurl Resource Mngemen for Scieniss nd Engineers Excerp More informion 4 Serile Resources Sorge, S 6 4 Producion, X Price, P Revenue, PX Figure.. Exhusion hisory h mches he TC. Demnd is X = /P β, wih (, β) = (,.5). For his simple cse, he presen worh of ll fuure producion is equl o ody s price imes ody s ol supply. Progrm Oil illusres he exhusion hisory under hese condiions. The ODE s re inegred forwrd in ime wih n explici (Euler) forwrd-difference mehod. The iniil price P needs o be djused mnully o sisfy he TC. Becuse numericl inegrion is no perfec, he relions developed bove using he clculus correspond only pproximely o he Oil simulion; he discrepncies vnish s he numericl imesep becomes infiniesimlly smll... Finie Demnd Nex, dd ceiling price P, which limis demnd (Figure.). Above his price, cusomers purchse subsiue produc. The previous soluion, in which P rises wihou bound, is invlid. Equions.. sill govern, bu he TC needs o be lered. The correc TC in his cse is S sp P (.4) Cmbridge Universiy Press

5 Cmbridge Universiy Press Susinble Nurl Resource Mngemen for Scieniss nd Engineers Excerp More informion. Cosless Producion of Serile Resource 5 () 5 Producion re, X (b) (c) 5 4 Producion re, X (d).5.5 Price, P.5.5 Price, P Figure.. Four differen demnd funcions X (P ). () bse cse X = P β ; (b) bse cse wih P P = ; (c) liner demnd X c = X ( P ); (d) bse cse shifed, X d + = P β. The dsh lines indice he coninuion of he bse cse curve beyond X. Cses b, c, nd d hve finie demnd. which leds o exhusion finie ime T. From Equions. nd., we hve from which we obin he finl resuls These relions reduce o he previous ones s P. P = P e rt (.5) S = [ P β e βrt βr (.6) [ ] β P = βrs + (.7) P β X = βrs + P β (.8) [ T = βr ln βrs P β ] + (.9) The bove relions mus govern ny ime during he exrcion hisory, else he rjecory would no be opiml nd i would be lered, conrry o hypohesis. Cmbridge Universiy Press

6 Cmbridge Universiy Press Susinble Nurl Resource Mngemen for Scieniss nd Engineers Excerp More informion 6 Serile Resources Hence, we my drop he subscrips nd i is lwys rue h [ ] β P = βrs + (.) P β X = βrs + P β (.) [ T = βr ln βrsp β ] + (.) wih S he remining unexploied resource ny ime nd P, X, T he conemporry price, producion re, nd remining ime o exhusion. Equions.,., nd. compleely chrcerize he soluion o Equions.,., nd., subjec o he TC of exhusion s price reches he ceiling P. Progrm Oil illusres hese relionships. Figure.4 displys simulion resuls for finie P nd T, chieved wih he decision rule X = X (S) (Equion.). Ren is, s bove, he inegred presen worh of ll fuure producion: R = T PXe r d = P Xd = P S (.) Sorge, S Producion, X Price, P 6 4 Revenue, PX Figure.4. Exrcion hisory wih finie demnd, X = P β wih P = 9 (cse (b) in Figure.). This leds o exhusion finie ime s shown. Cmbridge Universiy Press

7 Cmbridge Universiy Press Susinble Nurl Resource Mngemen for Scieniss nd Engineers Excerp More informion. Cosless Producion of Serile Resource 7 P S X S 4 4 T R 5 5 S 5 5 S Figure.5. P, X, nd T s funions of he ol reserve S ny ime (Equions..). Demnd: X = P β ; = ; β =.5; r =.5; P = (,, ) s indiced by he linesyles. This resul is unchnged by he imposiion of ceiling price nd he resuln finie T. Since P decreses s P decreses, ceiling price hs he effec of diminishing overll ren, in ccord wih inuiion. Equions.. give P, X, nd T s funions of he ol reserve S ny ime, ssuming complee exhusion, X = P β nd P P. Figure.5 plos hese for hree differen vlues of P. Ren peks nd begins o decline wih S high bundnce in his scenrio. Consumers Surplus Consumpion price P indices willingness o py les P h is, he vlue of he consumpion V P. The consumer obins surplus equl o he difference V P. Figure.6 illusres demnd curve mde up of individul uses X, ech wih is own vlue V. If price is se P, hose users wih higher vlue will purchse, nd hose wih lower vlue will no. The consumers surplus (CS) is heir ccumulion: CS = (V P) X (.4) for ll incremens where (V P) >. In he limi, CS(P) = X (P) (V P)dx (.5) Cmbridge Universiy Press

8 Cmbridge Universiy Press Susinble Nurl Resource Mngemen for Scieniss nd Engineers Excerp More informion 8 Serile Resources X X 6 X 5 X 4 X X X Vlue Figure.6. Demnd curve buil up of individul incremens X, ordered by decresing individul vlue, ploed on he horizonl xis. X Ren X Consumers Surplus P Vlue Figure.7. Demnd curve s in Figure.6, dding he cul price P. The re o he righ of he price line is he consumers surplus; h o he lef is he ren rnsferred o he seller. Clerly, CS is funcion of P. Grphiclly, his is illusred in Figure.7 s he re under he demnd curve, o he righ of P. The moun PX shown grphiclly is he ol ren rnsferred o he seller. So rnscions P genere consumers surplus s well s ren. Grphiclly, i is esy o see h n equivlen inegrl is P CS(P) = XdV (.6) An nlogous concep of producers surplus (PS) divides he ren ino producion cos plus surplus: ne ren. When producion is cosly, he producers surplus is he ne ren. P Cmbridge Universiy Press

9 Cmbridge Universiy Press Susinble Nurl Resource Mngemen for Scieniss nd Engineers Excerp More informion. Cosless Producion of Serile Resource 9 Consumers surplus is sic concep; ime is fixed in is consrucion. Clerly in depleion conex, s P rises over ime, CS will decrese: CS(P) = CS(P()). Suppose we hve he bse cse demnd X = P β, wih ceiling price P. Then i is esy o verify h CS my be inegred o obin CS(P) = [ P β P β] (.7) β The presen worh of he consumers surplus is explored in Problem 4... Liner Demnd As n exension of he preceding, consider he lernive demnd funcion P = P bx (.8) We sill hve he requiremen of exponenil price growh P = P e r (.9) nd hus X = P P e r b (.) Inegring ds/d = X gives S() = S + b [ ] P r (er ) P (.) The Terminl Condiion is S(T ) sp(t ) P (.) nd herefore P e rt = P (.) ( ) P rt = ln P (.4) nd S = b [ ] P r (ert ) PT (.5) Cmbridge Universiy Press

10 Cmbridge Universiy Press Susinble Nurl Resource Mngemen for Scieniss nd Engineers Excerp More informion Serile Resources.8 Compeiive.8 P.6.4 X S S 4 T S R S Figure.8. Opiml exrcion relions for he liner demnd cse: r =.5; P = X ; compeiive cse. A lile rerrngemen leds o ( ) T = r ln P S = P br X = P P b P [ ( )] P P + ln P P (.6) (.7) (.8) These ls hree equions rele S, X, nd T o P ; hey comprise implici funcions X (S), T (S), nd P (S). There re no simple closed-form soluions, bu X (S), T (S), nd P (S) cn be evlued numericlly s in Oil6M+C; he plos shown herein re reproduced in Figure.8. They chrcerize his sysem under liner demnd, s did he closed-form Equions.. for he erlier demnd funcion. As before, ren R = P S. I is ineresing o noe here h s S increses, P ulimely decreses owrd he limiing cse P ss. As resul, he R iniilly grows wih S bu ulimely peks nd hen decreses wih incresing S. The poin of mximum ren is found by seing dr/d = ; he resul is ( ) P ln P ( = P ) P (.9) Cmbridge Universiy Press