Oligopoly with exhaustible resource input

Transcription

1 Olgopoly wh exhausble resource npu e, P-Y. 78 Olgopoly wh exhausble resource npu Recebmeno dos orgnas: 25/03/202 Aceação para publcação: 3/0/203 Pu-yan e PhD em Scences pela Chnese Academy of Scence Insução: Insue of Indusral Economcs - Jnan Unversy. Endereço: Guangzhou, 50632, P.R. Chna. E-mal: pyne2005@yahoo.com.cn Absrac Ths sudy examnes he effecs of exhausble resource npus. By esablshng a dynamc game, hs paper argues ha he prce of he end produc ncreases over me, whle he exhausble resource npu decreases over me. Frsly, frms wh lower effcences qu hs ndusry early, enablng researchers o calculae a frm s qung me. Secondly, he exhausble resource npu and profs of all frms decrease wh he number of frms. Fnally, n an asymmerc case, he degree of producon effcency beween frms wh hgher effcency and lower effcency has major effecs on he frms sraeges. Keywords: Exhausble resource. Olgopoly. Qung me.. Inroducon Many ypes of producon requre npus of exhausble resources. Ths exhausble resource affecs frms producs and prces. Wha effec does demand have on hs exhausble resource? How does one deermne he prce of he oupus? These quesons movae hs sudy of olgopoles and exhausble resources. Geju Cy of Yunnan Provnce n Chna, s famous because of s rch n reserves (hp://en.wkpeda.org/wk/geju and hp://en.wkpeda.org/wk/tn). In Geju Cy, he ndusry o explore n heavly depends on he n mneral sock, and he n mneral s a ype of exhausble resource. In recen years, along wh he reducon of he n mneral sock, many frms qu hs ndusry sequenally (hp:// Ths example llusraes ha he exhausble resource has crucal mpac on he ndusry dependng on he exhausble resource. Ths movaes our furher research on hs opc. Ths sudy addresses he effecs of exhausble resource on he producon prce and oupus boh under he symmerc suaon Cusos on lne - v. 9, n. Jan/Mar ISS

2 Olgopoly wh exhausble resource npu e, P-Y. 79 and under he asymmerc one. Qung me s also dscussed n hs sudy. Moreover, frms poson affecs frms sraeges wh he exhausble resource npu. Exhausble resources arac exensve aenon n many felds. Bahel (20) developed an opmal managemen sraegy for sraegc reserves of nonrenewable naural resources. Bems and de Carvalho (20) addressed he curren accoun and precauonary savngs of exhausble resources. Gerlagh and Lsk (20) dscussed he sraeges regardng exhausble resource managemen. Lsk and Monero (20) dscussed he mpac of marke power on exhausble resource npus, ncludng he use of polluon perms. Van der Ploeg (200) nvesgaed he compeon over exhausble resources and argued ha propery rgh of exhausble resources has a major effec on frms sraeges. Some emprcal evdence ndcaes ha exhausble resources have major effecs on he economy. Boyce and Emery (20) confrmed ha he correlaon beween growh and naural resource abundance s negave. In anoher emprcal paper on exhausble resources, Boyce and osbakken (20) examned he exploraon and developmen of ol and gas felds n he Uned Saes over he perod and explaned he economc evoluon of he exhausble resource ndusry. Acemoglue al. (202) esablshed a dynamc model o address echnologcal nnovaon wh exhausble resource npus. Ths paper nroduced a producon funcon wh human capal and an exhausble resource. Because an exhausble resource s connuously reduced n producon, s raonal o model wh a dynamc model, he same mehod used by Lsk and Monero (20). The subjec of exhausble resource npus s a challengng opc because covers he envronmen, resources and regulaon. Moreover, exhausble resources have major effecs on frms sraeges (Gerlagh & Lsk 20). Ths sudy focuses on olgopoles of exhausble resource and esablshes a model wh reference o he neresng dynamc model n Acemoglu e al. (202). Due o he propery of exhausble resources, hs sudy sresses he qung me of frms, whch ses apar from Acemoglu e al. (202). Comparng he quany of exhausble resource npus a each sage yelds some neresng conclusons. Ths sudy lms s scope o a sngle ype of produc, whle Acemoglu e al. (202) dscussed a clean produc and a dry produc. The model of hs paper refers o he neresng paper of Van der Ploeg (200). Ths paper s organzed as follows. The model, whch fully consders a ceran ype of exhausble resource, s esablshed n he nex secon. Secon 3 addresses hs model and he Cusos on lne - v. 9, n. Jan/Mar ISS

3 Olgopoly wh exhausble resource npu e, P-Y. 80 concep of qung me. Secon 4 analyzes he model n he conex of he number of frms n ha ndusry. Some concludng remarks are presened n he fnal secon. 2. Model There are frms n an ndusry ha produce dencal producs. The number of frms s denoed by Ν = {,2, L }. In hs ndusry, o produce he correspondng producs requres a ceran exhausble resource. Ths ndusry produces a unque oupu. Gven he prce demand s gven by D = A p, where A s a consan. A lnear demand funcon s employed o smplfy hs model. The producon funcon of frm a me s gven as follows. p, q = θ ( er ) α, () where θ > 0 sands for he producon effcency of frm. Whou he loss of generaly, we assume ha θ θ2 L θ. The noaon er 0 descrbes he exhausble resource npu a me for he frm, and 0 < α s a consan. There are oher npus nvolved n producon, bu hese npus are gnored n hs sudy for he purpose of focusng on he exhausble resource npu. Therefore, he producon funcon enrely depends on he exhausble resource. The sock of he exhausble resource a me s S. A he nal sage, he sock of ds hs exhausble resource s S 0. Obvously, S = S0 ( er ) 0 d or = er for 0. Ths = d = formulaon of he sock of exhausble resources a each sage also appears n Lsk and Monero (20) and Acemoglu e al. (202). The margnal coss of he exhausble resource a me are c( S ). The margnal coss c( S ) ncrease as he sock of he exhausble resource s depleed. Moreover, " c > 0. ( ) ' c < 0 and c S s exogenously deermned by mulple facors. Ths assumpon of margnal coss s conssen wh he emprcal resuls n Boyce and osbakken (20) and Acemoglu e al. (202). Cusos on lne - v. 9, n. Jan/Mar ISS

4 Olgopoly wh exhausble resource npu e, P-Y. 8 Marke clearng condons mply he relaonshp D = q. The game sops a me = T f all frms qu hs ndusry. We noe ha T s addressed n Secon 3. For =,2, L,, he oal profs of frm are T T α α = 0 = 0 =. (2) π [ p q c( S ) er ] d {[ A θ ( er ) ] θ ( er ) c( S ) er } d The dscounng facor for all frms s assumed o be o smplfy he model. The un cos of he exhausble resource vares wh he sock of exhausble resource; namely, decreases as he sock of exhausble resource decreases. The sequence of he game s oulned as follows. A he nal sage, all frms know he sock of he exhausble resource, s margnal prce, he producon effcency of all frms and he demand n hs ndusry. All frms deermne her exhausble resource npus for all. When p q c( S ) er 0 for frm a me ( =, 2, L, ), frm qus hs ndusry. Fnally, all frms qu hs ndusry a me T, hereby endng he game. Ths paper consders fne horzon, whch s dfferen from ha of Van der Ploeg (200). Moreover, he dscoun facor s assumed o be o smplfy he problem. Frsly, n hs work, frms qu hs ndusry f he resource s very scarce. A he las sage, for he seady sae, all frms qung hs ndusry. Moreover, hs paper consders he producon coss whle Van der Ploeg (200) dd no care abou he producon coss. Therefore, s raonal o employ he fne horzon. Secondly, compared wh he neresng paper of Van der Ploeg (200), hs paper consders he common asse. The propery rgh of exhausble resources s no focused by hs work. Fnally, n he fne horzon, under he common asse, s raonal o assume he dscoun facor o be because he dscoun facor has no effecs on he equlbrum. 3. Resuls Ths secon analyzes he above model, resaed as follows: Cusos on lne - v. 9, n. Jan/Mar ISS

5 Olgopoly wh exhausble resource npu e, P-Y. 82 er 0 T T α α = 0 = 0 = Max π [ p q c( S ) er ] d {[ A θ ( er ) ] θ ( er ) c( S ) er } d ds S.. = er. d = (3) α α Denoe he prof of frm a sage o be π = [ A θ ( er ) ] θ ( er ) c( S ) er. Wh = model (3) now characerzed, he exsence of a unque soluon s obaned below. Proposon : There exss a unque soluon o (3). Proof: Ths concluson s mmedaely acheved by drecly calculang he second order dervave under 0 < α. Ths secon addresses he model furher by frs characerzng he equlbrum and hen dscussng qung me. 3.. Equlbrum Model (3) s a ype of fxed endpon problems, whch s also a ype of Euler equaon (Kamen & Schwarz 99, p. 47). If he soluon s a srcly neror pon, he opmal condons of (3) are oulned as follows. The proof n deal s deleed, whch refers o he neresng monograph (Kamen & Schwarz 99). α α {[ A θ ( er ) ] θ ( er ) c( S ) er } = λ α α α ( er ) [ A ( er ) ( er ) ] c( S ) 0. = = αθ θ θ λ = (4) The Lagrangan mulpler λ 0 for all s gven by he followng equaon: α α {[ A θ ( er ) ] θ ( er ) c( S ) er } λ = c S = = er d S S d ( ). (5) Cusos on lne - v. 9, n. Jan/Mar ISS

6 Olgopoly wh exhausble resource npu e, P-Y. 83 In economcs, for (4), λ 0 means he margnal profs of frm a me. Apparenly, (5) ndcaes ha he margnal profs decrease wh me. Furher denoe f f = αθ er A θ er θ er c S λ =. Obvously, < 0 α α α ( ) [ ( ) ( ) ] ( ) 0 = f and > 0 c( S ) s large enough. Because s very dffcul o acheve he explc soluon, we analyze dynamc resuls wh mplc funcon heorem or oal dfferenal mehod. From (4) and (5), by he mplc funcon heorem or oal dfferenal mehod o (4), for =, 2, L,, we have he relaonshp f = f > 0 and f = α > 0 α f for a suffcenly small sock of he exhausble resource. Hgher effcency frms are more aggressve n compeon, and frms wh lower effcency are more conservave n compeon. Ths means ha hgher effcency frms requre more exhausble resource npus. Ths s raonal and ndcaes ha he frms wh hgher effcency have larger oupus. When he sock of he exhausble resource s large enough, c( S ) and f f enough. The concluson may be dfferen. Obvously, < 0 and < 0 f λ are all small f c( S ) s small enough. Therefore, for =, 2, L,, we have f = < 0 f and f = α > 0 α f f c( S ) and λ are all very small. Frms are more compeve under ha large sock of he exhausble resource han ha when he exhausble resource s scarce. In hs case, he exhausble resource has no major effec on he producon of frms. Moreover, by a smlar mehod, we mmedaely have < 0 j for, j =,2, L, and j. Ths means ha he hgher producon effcency of one frm reduces s rval s exhausble resource npu and correspondng oupu of producs. Obvously, (4) mples lm c( S ) = + and S 0 lm er = 0. All frms fnally qu hs ndusry because of ncreasng margnal cos. Accordng o Proposon, he equlbrum s enrely deermned by (4) and (5). From (4) and (5), we have: S 0 Cusos on lne - v. 9, n. Jan/Mar ISS

7 Olgopoly wh exhausble resource npu e, P-Y. 84 Proposon 2: Prce p monooncally ncreases n, whle exhausble resource npu er monooncally decreases n for all =, 2, L,. Correspondngly, π monooncally decreases n for all =, 2, L,. Proof: See Appendx. Remarks: Ths concluson mples ha he prce of hese producs connuously ncreases, whle he exhausble resource npu connuously decreases. Ths resul s due o he naure of exhausble resources and dffers from a general suaon whou exhausble resource consrans. Ths concluson s conssen wh some socal phenomena. In Chna, for example, gasolne, whch s a ype of exhausble resource, has been subjec o connuous prce hkes n recen years. The prof funcon under equlbrum mus be addressed. By he envelop heorem, combnng he above conclusons > 0 and < 0, we mmedaely have he relaonshp j π π > 0 and > 0. Takng exhausble resources no accoun, frms benef from hgher α producon effcency. Ths concluson s conssen wh cases even whou exhausble resource npus Qung Because of he exhausble resource, he margnal coss connuously rse wh me. Ths makes he qung me of frms an mporan concep because he varous producon effcences of frms yeld dfferen qung mes. By proposon 2, he prce of he producs ncreases, whle he exhausble resource npu and profs ( π ) decrease wh me. Accordng π o he proof of proposon 2, for me and frm we have he relaonshp > 0. Because s profs a me are less han zero, when p q c( S ) er 0, frm qus hs ndusry. In hs case, frms qung hs ndusry wll no reener, accordng o Proposon 2. Ths case s also equvalen o solvng he maxmzaon problem for frm a me lyng a he corner. Cusos on lne - v. 9, n. Jan/Mar ISS

8 Olgopoly wh exhausble resource npu e, P-Y. 85 π > 0 and Proposon 2 ndcae he followng concluson: Frm frs qus hs ndusry. Then, frm qus. Evenually, he las frm qus hs ndusry. When he frs α α frm qus hs ndusry, should sasfy he relaonshp αθ ( er ) [ A 2 θ ( er ) ] c( S ) = 0 T T T and Aθ ( er ) θ ( er ) c( S ) er 0 whle α 2 2α T T T T αθ ( er ) [ A 2 θ ( er ) ] c( S ) = 0 and α α Aθ ( er ) θ ( er ) c( S ) er 0 for all < T hold. α 2 2α Remarks: Accordng o he above concluson, frms qu hs ndusry n order accordng o her producon effcency. Frms wh lower effcency qu earler. A he las sage, all frms qu hs ndusry. As some frms qu hs ndusry, compeve pressure s reduced, whle producon coss rse. When frm deermnes o qu hs ndusry, we have p q c( S ) er = 0. By he connuy of he correspondng prof funcon, here exss a unque soluon o hs equaon. Ths s rewren as follows: j α [ A θ j ( er ) ] q c( S ) er 0. j= er, j =,2, L, s deermned by (4) and (5) wh j er = er = L = er = 0. The exac qung me s closely relaed o he margnal cos of he exhausble resource npu and he producon effcency of each frm. 4. Furher dscusson Ths secon llusraes he relaonshp beween he number of frms and he exhausble resource npu, ncludng a dscusson of a herarchy of frms. 4.. Symmerc cases Under symmerc cases, θ = θ2 = L = θ. Under equlbrum, he exhausble resource npu s dencal for all frms. Alernavely, er = er = L = er 2 Accordng o (4) and (5), combnng θ = θ2 = L = θ, we have and λ = λ = L = λ = λ. 2 Cusos on lne - v. 9, n. Jan/Mar ISS

9 Olgopoly wh exhausble resource npu e, P-Y. 86 {[ A θ ( er ) ] θ ( er ) c( S ) er } λ α α = αθ ( er ) [ A ( + ) θ ( er ) ] c( S ) λ = 0. α α (6) λ 0 for all s gven by (5). Gven he sock of exhausble resource S, by (5) and (6) we have he followng concluson: Proposon 3: The exhausble resource npu er and prof of each frm a any me monooncally decrease n, and er monooncally ncreases n. = Remarks: Proposon 3 means ha he exhausble resource npus and frms profs monooncally decrease n, whle he oal exhausble resource npus monooncally ncreases n. The exhausble resource s exhaused sooner wh more frms n hs ndusry. Ths concluson s hghly conssen wh he Couran model whou exhausble resources. Under symmerc cases, he exhausble resource npus and profs are all capured. Ths concluson provdes heorecal suppors for governmen decsons o proec exhausble resources. In pracce, many ndusres wh exhausble resource npus are regulaed o resrc he number of frms so ha he exhausble resource s avalable longer. Ths s an neresng applcaon of he above concluson. = er 4.2. Leader-follower poson Ths secon analyzes frms sraeges accordng o he hypohess ha θ > θ = L = θ. In hs case, he frs frm has he hghes producon effcency and assumes 2 he leadng poson. The frms wh lower producon effcency assume followng posons. Ths sequence s denoed by τ = θ θ2 ; θ2 = θ3 = L = θ mples he relaonshp 2 er = er = L = er and λ = L = λ = λ for all me. The frs opmal condons are resaed 2 3 as follows. For he frs frm, we have: Cusos on lne - v. 9, n. Jan/Mar ISS

10 Olgopoly wh exhausble resource npu e, P-Y. 87 αθ ( er ) [ A ( ) θ ( er ) 2 θ ( er ) ] c( S ) λ α 2 α α 2 = αθ ( er ) [ A ( )( θ τ )( er ) 2 θ ( er ) ] c( S ) λ = 0. α 2 α α (7) For oher frms, we have: αθ ( er ) [ A θ ( er ) ( ) θ ( er ) θ ( er ) ] c( S ) λ α α α α = αθ ( er ) [ A θ ( er ) ( ) θ ( er ) ( θ + τ )( er ) ] c( S ) λ = 0. α α α α (8) concluson. λ 0 and λ 0 for all s gven by (5), (7) and (8) drecly mply he followng Proposon 4: If he frs frm has a producon effcency advanage, hen he frs frm s exhausble resource npu ncreases whle hose of oher frms decrease n he degree of producon effcency advanage. Proof: (7) and (8) yeld > 0 and < 0 τ τ concluson s acheved, and he proof s complee. for any me and = 2,3, L,. The Remarks: Ths concluson means ha he degree of producon effcency advanage s posvely relaed o he producon gap beween he more effcen and less effcen frms. Under θ2 = θ3 = L = θ, Proposon 3 also holds. Gven θ2 = θ3 = L = θ, (8) ndcaes er he relaonshp = τ < 0. Alernavely, he oal level of exhausble resource npus s reduced when here s an advanage n producon effcency. Ths means ha a frm wh an advanage n producon effcency decreases he consumng exhausble resources n hs ndusry. 5. Concludng Remarks Ths sudy addresses he effecs of exhausble resource npus on producon and fnds ha he exhausble properes have deep effecs on frms sraeges, ncludng arge oupu Cusos on lne - v. 9, n. Jan/Mar ISS

11 Olgopoly wh exhausble resource npu e, P-Y. 88 quany and qung me. Ths arcle shows ha ha prce of he end producs ncreases as he exhausble resource npu decreases wh me. The qung me s dscussed. The number of frms n he ndusry has major effecs on he producon schedule and he me a whch frms qu he ndusry. The heorecal conclusons and some hypoheses regardng he model are conssen wh he emprcal evdence of Boyce and osbakken (20), such as he cos of exhausble resources and producon sraeges of frms. The resuls acheved n hs paper are also conssen wh hose n Van der Ploeg (200). Exhausble resource npus creae many complcaed economc phenomena. Ths sudy only addresses one oupu; when mulple oupus are consdered, he suaon becomes more complcaed, whch s a opc for furher research. 6. References ACEMOGLU, D; AGHIO, P; BURSZTY L, HEMOUS, D. The envronmen and dreced echncal change. Amercan Economc Revew. 02(), BAHEL, E. Opmal managemen of sraegc reserves of nonrenewable naural resources. Journal of Envronmenal Economcs and Managemen. 6(3), BEMS, R; CARVALHO, I. de. The curren accoun and precauonary savngs for exporers of exhausble resources. Journal of Inernaonal Economcs. 84(), BOYCE, J. R; EMERY, J.C.H. Is a negave correlaon beween resource abundance and growh suffcen evdence ha here s a "resource curse"? Resources. 36(), BOYCE, J.R; OSTBAKKE, L. Exploraon and developmen of US ol and gas felds, Journal of Economc Dynamcs & Conrol. 35(6), GERLAGH, R; LISKI, M. Sraegc resource dependence. Journal of Economc Theory. 46(2), Cusos on lne - v. 9, n. Jan/Mar ISS

12 Olgopoly wh exhausble resource npu e, P-Y. 89 KAMIE, M.I; SCHWARTZ,.L. Dynamc Opmzaon: The Calculus of Varaons and Opmzaon Conrol n Economcs and Managemen, Elesver Scence Publshng Co. Inc, ew York. 99. LISKI, M; MOTERO, J.P. Marke power n an exhausble resource marke: The case of sorable polluon perms. Economc Journal. 2(March), VA DER PLOEG, F. Voracous ransformaon of a common naural resource no producve capal. Inernaonal Economc Revew. 5 (2), hp://en.wkpeda.org/wk/geju hp://en.wkpeda.org/wk/tn hp:// Cusos on lne - v. 9, n. Jan/Mar ISS

13 Olgopoly wh exhausble resource npu e, P-Y. 90 Appendx Proof of Proposon 2 Because dc( S ) c( S ) = er, (4) and (5) mply ha c( S ) λ s decreased n. Model (4) d S = α α α mples ha he erm αθ ( er ) [ A θ ( er ) θ ( er ) ] s ncreased n. On he oher hand, = (4) mples ha α α α ( er ) [ A ( er ) ( er ) ] > 0 = αθ θ θ. The concavy of he prof funcon ndcaes α α α { αθ ( er ) [ A θ ( er ) θ ( er ) ]} = < 0 or α α α ( er ) [ A ( er ) ( er ) ] = αθ θ θ decreases wh α α α { αθ ( er ) [ A θ ( er ) θ ( er ) ]} = er. The formulaons < 0 and α α α α α α { αθ ( er ) [ A θ ( er ) θ ( er ) ]} { αθ ( er ) [ A θ ( er ) θ ( er ) ]} = = der d = > 0jonly mply ha he exhausble resource npu Because he exhausble resource npu er monooncally decreases n for all =,2, L,. er monooncally decreases n for all =, 2, L,, n θ = p = A ( er ) α shows ha he prce p monooncally ncreases n. By he envelop heorem, because c( S ) λ s decreased n, we see ha π monooncally decreases n for all =, 2, L,. Conclusons are acheved, and he proof s complee. Acknowledgemen: Sncere hanks o he anonymous revewers and he edor for her excellen job. Ths work s parally suppored by Fundamenal Research Funds for he Cenral Unverses, GDUPS(202) and aonal aural Scence Foundaon of PRC (72700). Cusos on lne - v. 9, n. Jan/Mar ISS