Solving a three player differential game in resource economics the case of exhaustible resources. Petra Huck. Discussion Paper

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1 Solvng a hree player dfferenal game n resource economcs he case of exhausble resources Pera Huck Dscusson Paper 4-5 Technsche Unversä München Envronmenal Economcs and Agrculural Polcy Group

2 Absrac: Dfferenal games lnk sraegc neracons beween agens and opmzaon concernng me. Pas and curren acons of each player nfluence all fuure sraegy ses and pay offs hrough a ranson law. Due o hgh complexy, s hard o fnd a Nash-equlbrum whn a dfferenal game and s even harder o ge some resuls n comparave sacs. I s he purpose of he paper o descrbe an approxmaon roune for an open-loop Nash equlbrum of a smple dfferenal game n exhausble resources. Excel s appled as s a wld spread ool. JEL-classfcaon: A, C73, Q3

3 Inroducon Recenly, reverse engneerng of ecologcal and economcal models n Excel spreads ou n academc leraure. Especally n case of ner emporal analyss Excel can suppor he undersandng of nerdependences beween varables and me. Insofar s adjacen o apply o dfferenal games. The analycal soluon of dfferenal or dynamc games requres a deep undersandng of conrol heory as well as game heory. Conrol heory s a comparavely young feld whn mahemacs. In he ffes and sxes of he las cenury Calculus of Varaon enhanced more and more; a he same me Bellman equaons and he Maxmum Prncple of Ponryagn were nroduced. Wh he knowledge from hese mehods mahemacans have a srong ool a her fngerps. Bu whou a well-founded mahemacal knowledge, sudens from adjacen research felds have a hard job n conrol heory. The same s rue for game heory. In he fores of he las cenury von Neumann / Morgensern proposed o apply game heory n socal and economc suaons. 3 A few years laer Nash nroduced he dsncon beween cooperave and non-cooperave games and esablshed he Nash equlbrum concep n hs dsseraon 4, non-cooperave games go he mayor fracon of aenon n he followng years. To solve for a Nash equlbrum, one has o look for fxed pons n bes response correspondences. Ths mgh requre advanced mahemacal knowledge, oo. An eneranng overvew over he hsory of conrol heory s gven n Fernández-Cara and Zuazua (), chaper 4. Irrgaon sysems were one of he frs felds for he applcaon of conrol heorec elemens; laer on wndmll regulaon and seam engne regulaon subjoned. A mahemacal formulaon dd no arse before he second half of he 9 h cenury. Progress speeded up as world war II cas s shadow before. Regardng Calculus of Varaon, see for nsance: Kamen and Schwarz (98), Inrlgaor (97), chaper, Chang (99), chapers, 3, and 4. Concernng Bellmann equaons, see for example: Inrlgaor (97), chaper 3, and Fernández-Cara and Zuazua (), appendx. On he subjec of he Maxmum Prncple, see for nsance: Inrlgaor (97), chaper 4, Chang (99), chaper 7 and 8 and Fernández-Cara and Zuazua (), appendx. 3 They publshed her famous book Theory of Games and Economc Behavor n John Nash dd hs docorae 95 wh Non-cooperave Games a Prnceon Unversy. In 994 he, Renhard Selen, and John Harsany won he Nobel Prce n Economc Scence due o her conrbuons o noncooperave game heory. 3

4 Dfferenal games as a combnaon of game heory and conrol heory are mporan for nsance n resource managemen as par of economcs and n modellng mlary conflcs, lke a fgh beween arplanes or beween a submarne and an arcraf n reconnassance msson, jus o gve wo examples. Accordngly o s relevance for modellng conflcs and sraegc soluons, research on conrol heory as well as on game heory was parly drven hrough mlary and relaed nsuons. Dfferenal games are a useful ool n resource managemen as hey ake no accoun susanably houghs. Susanably s an ssue n boh, renewable resources and nonrenewable resources. In renewable resources me becomes relevan hrough he re-growh of he resource under consderaon and game heorec aspecs emerge hrough harves and reforesaon decsons, whch depend o some exen on compeors behavour. In many non-renewable resource models player choose some ype of exracon pahs and he game heorec aspec consss n a common sales marke or facor marke. Furher, an exenson for dfferenal games n resource economcs s mplemened hrough he neracon beween nsuonal seup and resources managemen. Even hough dfferenal games are an approprae measure o model a chan of decsons n successon and relaon o oher players, hey have he dsadvanage of complexy. Whou a profound mahemacal knowledge s hard o denfy her equlbra. The paper shows how o solve hs problem n a game consderng as lnchpn he olgopolsc resource marke descrbed n Wacker & Blank (999). A numercal algorhm becomes appled, hus s no necessary o be famlar wh conrol heory o undersand he fndngs. The followng pars wll presen he 3-player-game n exhausble resources, nvesgaed by Wacker and Blank (999), as well as he layou of he Excel fle employed o fnd he open- 4

5 loop Nash equlbrum 5. Subsequen frs experences wh he approxmaon process are repored. The model As n Wacker & Blank (999) a 3-player dfferenal game for an exhausble resource n analysed. Each player owns a gven nal sock S, =,,3, and faces a me-nvaran lnear demand curve represened by a wllngness o pay for he oal perod oupu, p( R ) = a b R. Toal oupu n perod s he sum of he ndvdual oupus n perod,.e. R R = R + R + R 3. Margnal and average exracon coss are dencal and me-nvaran; hey amoun c. The resource owners seek o maxmze her ndvdual oal dscouned prof generaed on he resdual demand. Each one akes he exracon pah of he compeors no accoun when opmzng s own exracon pah. Thus, he oucome s an open-loop Nash equlbrum wh he exracon pah as sraegc varable: ( ( ) ) r max e p R R c * R d R () R refers o he exracon quany of he compeors, a me. Exracon s lmed due o he well-known consrans: R d = S () R (3) 5 A Nash equlbrum s defned as a sraegy confguraon, where each sraegy s an elemen of he respecve bes response correspondence gven all he oher Nash sraeges. In an open-loop Nash equlbrum each player oblges o hs sraegy a he begnnng of he game gven he sraeges of all he oher players, and does no updae hs decson durng he course of he game. Alernave conceps are feedback-, Markovan-, Markov perfec- or closed-loop equlbra. They conen he acual sae of he world as an addonal argumen of he sraegy n conras o he open-loop equlbrum, where me or he perod ndex, respecvely, s he sole argumen. See Dockner e al. (), p. 59 for furher dscusson. 5

6 S & = R (4) () requres oal exracon no o exceed he nal sock, and (3) exracon no o be negave. The law of moon s expressed by (4). The conrol heorec roune o solve hs ype of opmzaon problem ulzes he curren value Hamlon funcon: H = p R R c R λ R (5) The general opmzaon crera are 6 : H R (6) & H λ (7) r λ = S H λ = R (8) Inserng he wllngness o pay n (6), one has: a br br c λ (9) The equy holds as long as he own quany s posve;.e. (7) and express he condon n a me dscree verson: R. I s easy o reformulae + δ ( a br br c ) = ( a br+ br+ c ) or respecvely ( a br br c ) = ( a br br c + δ + δ ). () 6 See for example: Sydsæer, Srøm, Berck () Chaper 6. An exracon pah, fulfllng hese crera s an opmal pah, due o he concavy feaure of he arge funcon. 6

7 Equaon () corresponds o he Euler-Equaon 7, and can be used o check he qualy of each eraon run n our numercal approxmaon. 8 The Euler equaon can be nerpreed he n he followng way: The dscouned margnal prof s equal n every perod. Moreover, s posve as long as quany s. In order o acheve a dscouned margnal prof n he far fuure whch s as hgh as n he near fuure, he non-dscouned margnal prof n he far fuure has o be hgher han n he near fuure. Ergo, for a monopols he quany has o decrease by me. Gong back o olgopoly, hs mples: as soon as wo socks are depleed and only one suppler s lef, he remanng segmen of he exracon pah has o be decreasng. Ths s no rue for earler perods wh more han one suppler. As long as here s sraegc neracon, ndvdual exracon pahs may have ncreasng segmens. I s no possble o fnd a general analycal soluon for he dfferenal game, hus one has o focus on numercal soluons. In chaper 6 Wacker & Blank look a hree 9 dfferen characerscs for he players: ) hey dffer n he nal sock ) hey dffer n level of exracon coss 3) hey dffer n he dscoun rae Wacker and Blank solve he hree scenaros numercally, bu presen only he qualave resuls. For dfferen nal socks hey choose he followng parameer consellaon: 3 a = 8; b = 5; c = and δ = 6% for =,,3; S = 5; S = 5; S = 5 The resul s compromsed n he followng pcure : 7 See for example: Sydsæer, Srøm, Berck () Chaper 6. 8 For a dscusson of evaluaon echnques n numercal mehods, see Judd (999), Chaper. 9 Only he frs one of hem wll be reproduced here. Bu wh merely slgh changes n workshee desgn, one can easly rebuld he oher wo examples, oo. Noce ha we changed ndex wo and hree. 7

8 R R Toal supply 3 R R Pcure : exracon pah for a hree player seng wh dfferen nal socks Source: analogue o pc. 6., p. 7, n Wacker and Blank The pcure gves an mpresson of he qualave behavour of he model, bu no dea of he me horzon or he magnude of exracon a an arbrary perod, due o he mssng scalng. Wh Excel and s Solver one can reconsruc he pcures and he exracon pahs and answer he queson abou me horzon and magnude. The work can be acceleraed by ulzaon of Macros, bu s n no way necessary o rely on he macro recorder (or Vsual Basc). The procedure can be used manual as well. 3 Excel workshee desgn The followng able dsplays our desgn for he workshee used o reproduce he example wh dfferen nal socks: In he upper par of able, he area A:B9, conans he parameer values; he followng cells go names: B a; B b; B3 r; B4 c_ B5 c_ B6 c_3; B7 S_; B8 S_ ; B9 S_3 The names recur n he formulas n he mddle and he lower par of he able. The mddle par of able, area A:K4, dsplays he frs four rows of he area used o calculae he resuls 8

9 of Wacker and Blank. These rows show he perod ndex, he prce, he exracon quany of player,, and 3 (x, x and x3); he non-dscouned prof of player, and 3 (Π, Π, and Π3) as well as he dscouned prof of player, and 3. These four rows represen he frs four perods; s omed o fll up able wh all oher rows of he Excel workshee, because hey follow jus by copy and pase of he formulas gven n A:K. The lower par of able conans jus he las hree perods and n s las row some sum-formulas: n C, D and E here are he sum of exracon quanes for player, and 3; n I, J, and K he sum of he dscouned profs,.e. for each player he value he wans o maxmze. Table pus on vew pars of he nal exracon pahs (column C, D, E wh x, x, and x3): a unform dsrbuon of he nal socks over a me horzon of perods was choosen. Alhough, hs s surely no a good nal guess, allows o reproduce he resuls of pcure hrough 4 smulaon runs. 9

10 A B C D E F G H I J K a 8 b 5 r,6 c c c3 S 5 S 5 S3 5 p x x x3 Π Π Π 3 dsc. Π dsc. Π dsc. Π3 a-b*(c+d+e),5,5 5 (B-c_)*C (B-c_)*D (B-c_3)*E ((/(+r))^a)*f ((/(+r))^a)*g ((/(+r))^a)*h a-b*(c+d+e),5,5 5 (B-c_)*C (B-c_)*D (B-c_3)*E ((/(+r))^a)*f ((/(+r))^a)*g ((/(+r))^a)*h 3 a-b*(c3+d3+e3),5,5 5 (B3-c_)*C3 (B3-c_)*D3 (B3-c_3)*E3 ((/(+r))^a3)*f3 ((/(+r))^a3)*g3 ((/(+r))^a3)*h3 4 a-b*(c4+d4+e4),5,5 5 (B4-c_)*C4 (B4-c_)*D4 (B4-c_3)*E4 ((/(+r))^a4)*f4 ((/(+r))^a4)*g4 ((/(+r))^a4)*h4 98 a-b*(c8+d8+e8),5,5 5 (B8-c_)*C8 (B8-c_)*D8 (B8-c_3)*E8 ((/(+r))^a8)*f8 ((/(+r))^a8)*g8 ((/(+r))^a8)*h8 99 a-b*(c9+d9+e9),5,5 5 (B9-c_)*C9 (B9-c_)*D9 (B9-c_3)*E9 ((/(+r))^a9)*f9 ((/(+r))^a9)*g9 ((/(+r))^a9)*h9 a-b*(c+d+e),5,5 5 (B-c_)*C (B-c_)*D (B-c_3)*E ((/(+r))^a)*f ((/(+r))^a)*g ((/(+r))^a)*h SUMME (C:C) SUMME (D:D) SUMME (E:E) SUMME(I:I) SUMME(J:J) SUMME(K:K) Table : desgn of calculaon workshee

11 Roaonal opmzaon of dscouned profs n 4 smulaon runs reurns accepable approxmaons for he sough-afer depleon pahs. The opmzaon based on Excels Solver. As he opmzaon problem s non-lnear, s necessary o change some of he solver opons o he one shown n he followng Screensho: Screensho : opons for SOLVER To accoun for he non-lneary, he check mark for assume lnear s deacvaed and quadrac nsead of lnear as esmaes opon 3 s chosen. Furher, he preseng of max me 4 and eraons are boh ncreased by facor 5. The hree followng pcures llusrae he aonemen process for each of he players and anoher pcure (pcure 5) dsplays he oal supply developmen a he end of round 4. The smlary o pcure s hghly vsble. A check of he Euler-Equaon provded he bass of nformaon abou convergence of he procedure. Each me a player opmzes hs dscouned prof, dscouned margnal prof s he same for all perods wh posve exracon (see equaon ()). Bu as soon as he nex player sars opmzaon, equaon () no longer holds for player. In rackng he developmen of dfferences beween he hghes and he lowes dscouned margnal prof for hose players how are n a wang poson, one ges an esmae abou he qualy of he curren smulaon run. Lneares Modell voraussezen 3 Schäzung 4 Höchsze

12 Taonemen_Process for player (bes reply pah a acual round) x 4 3,5 3,5,5 Round Round Round Round 3 Round 4, Pcure : player s exracon pah n a aonemen process Taonemen_Process for player (bes reply pah a acual round) x Round Round Round Round 3 Round Pcure 3: player s exracon pah n a aonemen process

13 Taonemen_Process for player 3 (bes reply pah a acual round) 4,5 x3 4 3,5 3,5 Round Round Round Round 3 Round 4,5, Pcure 4: player 3 s exracon pah n a aonemen process exracon pahs for he players, and 3 x+x+x x x3 x Pcure 5: oal exracon and s spl beween players 3

14 The smulaon runs show ha he exhausble resources are used up oally n perod 95; depleon of he smalles sock runs ou before perod 3, and of he mddle sock wh perod 5. A he begnnng oal exracon amouns beween and uns; n perod 3 we have jus under uns, and so on. 4 Experences wh he aonemen process The speed and workng of he aonemen process depends very much on he nal guess abou exracon pahs. If he choce s bad, he algorhm wll ge suck. Neverheless subopmal exracon pahs usually can be denfed by jerng curves. Furher, a leas for one player a check of he Euler-equaon dsplays he problem. Whenever he algorhm ges suck, one way ou may be o choose a new parameer consellaon near he orgnal one, apply he algorhm o, and use he resul as nal guess for he orgnal problem. 5 Bu as long as he nal guess s no oo bad and concavy requremens are fulflled, he algorhm descrbed approxmaes he Nash equlbrum. I s a mehod o ge resuls quckly when me s rare o become acquaned wh he mahemacal ools, even hough proper knowledge s desrable. 5 Dscusson In order o demonsrae dfferenal games n resource economcs - whou nvesmen n mahemacal ranng one can develop a smple procedure n Excel. Provded ha concavy requremens are fulflled, exracon sops whn accepable perods and he nal guess abou he exracon pahs s no oo bad, he procedure proposed n he paper a hand approxmaes he Nash equlbrum. Furhermore, havng an easy way of generaon of equlbra, comparave sacs becomes vable. Therefore, comparave sacs consue an neresng opporuny for furher work. 5 For dealed descrpon, see Judd (999), p. 9 concernng ho sars. Furher a rearrangemen of opmzaon sequence beween players may be helpful; Judd (999), p.7. 4

15 References: Bleymüller, J.; Gehler, G.; Gülcher, H. (988): Sask für Wrschafswssenschafler; Verlag Vahlen, München Buongorno, J., Glles, J.K. (3): Decson Mehods for Fores Resource Managemen, Academc Press, Elsever Scence Chang, A. (99): Elemens of Dynamc Opmzaon, Waveland Press, Inc., Prospec Heghs, Illnos Clark, C.W. (99): Mahemacal Boeconomcs: The Opmal Managemen of Renewable Resources, Wley-Inerscence, John Wley & Sons, New York Conrad, J.M. (999): Resource Economcs, Cambrdge Unversy Press, New York Dockner, Engelber e al.(): Dfferenal games n Economcs and Managemen Scence, Cambrge, Mass. Dorfman, R. (969): An Economc Inerpreaon of Opmal Conrol Theory, Amercan Economc Revew, 59 (5), Fernández-Cara, E.; Zuazua, E. () : Conrol Theory: Hsory, Mahemacal Achevemens and Perspecves, Bol. Sc. Esp. Ma. Apl. n, -63 5

16 Glber, N.; Trozsch, G. (999): Smulaon for he Socal Scens; Open Unversy Press, Buckngham, Phladelpha Inrlgaor, M. (97): Mahemacal Opmzaon and Economc Theory, Prence-Hall, Inc., Englewood Clffs, N.J. Judd, K. L. (999): Numercal Mehods n Economcs, MIT Press; Cambrdge Kamen, M.; Schwarz, N. (98): Dynamc Opmzaon, The calculus of Varaons and Opmal Conrol n Economcs and Managemen; Norh Holland; New York, Oxford Krschke, D., Jelschka, K. (): Angewande Mkroökonome und Wrschafspolk m Excel, Verlag Vahlen, München Krschke, D., Jelschka, K. (3): Inerakve Programmerungsansäze für de Gesalung von Agrar- und Umwelprogrammen, Agrarwrschaf 5 (4), - 7 Kolberg, W.C. (993): Quck and Easy Opmal Approach Pahs for Nonlnear Naural Resource Models, Amercan Journal of Agrculural Economcs 75, Kraukraemer, J. (998): Nonrenewable Resource Scarcy, Journal of Economc Leraure 36 (Dec.), 65 7 Kraukraemer, J.; Toman, M. (3): Fundamenal Economcs of Depleale Energy Supply; Resources for he Fuure; Dscusson Paper 3- (hp:// 6

17 Papadaos, A., Berger, A.M., Pra, J.E., Barbano, D.M. (): A Nonlnear Programmng Opmzaon Model o maxmze Ne Revenue n Cheese Manufacure, Journal of Dary Scence 85, Pezzey, J.; Toman, M. (99): The Economcs of Susanably: A Revew of Journal Arcles; Resources for he Fuure; Dscusson Paper -3 (hp:// Ragsdale, C. (): Spreadshee Modelng and Decson Analyss, Souh-Wesern College Publshng, Cncnna Sydsæer, K., Srøm, A., Berck, P. (): Economss Mahemacal Manual; Sprnger- Verlag, Berln-Hedelberg Wacker, H.; Blank, J. E. (999): Ressourceökonome, Band II: Erschöpfbare naürlche Ressourcen, Oldenbourg, München 7