UNIVERSITY OF NOTTINGHAM SCHOOL OF ECONOMICS DISCUSSION PAPER 99/28 Welfare Analyss n a Cournot Game wth a Publc Good by Indraneel Dasgupta School of Economcs, Unversty of Nottngham, Nottngham NG7 2RD, UK. Phone: 44(0115)9514297 Fax: 44(0115)9514159. E-mal: ndraneel.dasgupta@nottngham.ac.uk Abstract Ths note develops a method of recoverng ndvdual preferences, and of obtanng money-metrc ndvdual welfare comparsons, from demand functons generated by Cournot-Nash equlbra n games wth publc goods. I thank Prasanta K. Pattanak, R. Robert Russell and Kunal Sengupta for ther comments on earler versons of ths paper.
I. Introducton A number of dfferent economc contexts have been modeled as Cournot games between two or more agents wth one or more publc goods 1. In order to assess the welfare mplcatons of alternatve state polces n such contexts, t s necessary to derve a measure of the changes n each agent s welfare brought about by the relevant polcy. However, the demand functons n these models are generated by Cournot- Nash equlbra, and therefore do not express any one agent s preferences completely. Hence, the standard method, whch proceeds by reconstructng an ndvdual s preferences from her observed demand functons, can no longer be used. Can one generalze the standard approach to take nto account strategc nteracton amongst agents, and thereby derve a money-metrc measure of the welfare mpact of dstortonary nterventons, n such contexts? The purpose of ths note s to address ths ssue. Changng the extent of dstorton n a mxed subsdy-transfer scheme, whle keepng total expendture by the state constant, can be expected to change equlbrum consumpton and nduce welfare gans or losses for each ndvdual. Intutvely, t s clear that the extent of welfare change wll depend on the nteracton between the change n the extent of the deadweght loss and the change n other agents' expendtures on the publc good. To assess the deadweght loss, however, we need to know the agent's utlty functon. How can the polcy-maker recover an agent's utlty functon from demand functons generated by Cournot-Nash equlbra? Second, how can the polcy-maker derve a money-metrc measure of the welfare loss/gan by modfyng the standard noton of equvalent varaton to our strategc context? These are our questons. In Secton II, we frst develop a smple method of recoverng the utlty functon of an agent who makes a postve contrbuton to the publc good from Cournot-Nash demand functons. We then establsh the followng: a modfed noton of equvalent varaton can be used to derve a complete money-metrc rankng of schemes n terms of an agent s ndvdual welfare from the Cournot-Nash demand functons when that agent makes a postve contrbuton to the publc good. II. The Result Consder two agents and, each of whom derves utlty from the consumpton of some prvate goods and a publc good. Purely for notatonal smplcty we assume that the prvate goods are dentcal 1 Applcatons of ths model to the problem of ntra-household resource allocaton nclude Kanbur (1995), Lundberg and Pollack (1993), Ulph (1988) and Woolley (1988). Applcatons to prvate charty are much more extensve, the semnal contrbuton beng Roberts (1984). Kemp (1984) analyzes nternatonal transfers. 1
for the two agents. Therefore, holdng ther prces fxed, they can be treated as a sngle Hcksan composte commodty wth a normalzed prce of 1. Agent s characterzed by a strctly quas-concave utlty functon U U ( x, y) = and ndvdual ncome I, where x s the amount of ths composte prvate good consumed by agent and y s the total amount of the publc good. Smlarly, agent has a strctly quasconcave utlty functon U = W ( x, y) and ncome I, where x s the amount of the prvate good consumed by. Let I = I + I. The prce of the publc good s p. The two agents take the prces as gven and play a Cournot game wth respect to the publc good. Quanttes of the publc good purchased by agents and are, respectvely, y and y. A contrbutory agent n any Nash equlbrum s one who purchases a postve amount of the publc good at that equlbrum. We shall assume that a unque Nash equlbrum exsts n ths game. Proposton 2.1: Consder any Nash equlbrum where the agent has personal expendture x, total consumpton of the publc good s U = U ( x, y ). Then ( V p I x ) = y, and the contrbutory agent has a utlty level U,, where V s s ndrect utlty functon. Proof: Let y be the amount of the publc good provded by agent n the ntal Nash equlbrum. Startng from ths ntal Nash equlbrum, suppose agent s gven a transfer from equal to py. Then her personal ncome s I = I. Snce s contrbutory, the consumpton bundles reman x unchanged n the new Nash equlbrum, the contrbutons to the publc good beng now = y, y y = 0. 2 If, alternatvely, ths agent was maxmzng utlty, n the absence of, and wth ncome I, s maxmzaton problem would be dentcal to that n the two-person Cournot game when she has the same ncome I and s non-contrbutory. Therefore, f s non-contrbutory, 's consumpton would be dentcal. In the Nash equlbrum after the ncome redstrbuton, by the neutralty property, would ndeed be non-contrbutory. The utlty of n the ndvdual maxmzaton case s gven by ( p I x ) V,, and snce has the same consumpton n the orgnal Nash equlbrum, V ( p, I x ) = U. Proposton 2.1 mples that the demand functons y ( p I, I ), x ( p, I, I ), generated by Cournot-Nash equlbra can also be generated by a standard sngle-consumer utlty maxmzaton exercse 2
by the contrbutory agent when ths agent s endowed wth a varable personal ncome ( p, I I ) I = I x,. Snce ths transformaton allows one to derve the ndvdual Marshallan demand functons of from the market demand functons f prvate demand functons and ndvdual ncomes are known, t also allows the utlty functon of to be recovered n the standard way when s a contrbutory agent. Proposton 2.1 can be extended n a straghtforward fashon to cover the case of multple publc goods by usng the correspondng neutralty property 3. Suppose now that the state has a budgetary surplus (defct) whch t can transfer to (rase from) agent ether as a cash lump-sum or as subsdy (tax) on the publc good. 4 It has a gven budgetary support B and can choose alternatve combnatons of cash transfer T and subsdy rate t. The post-nterventon demand for the publc good s gven by = y p( t) ( 1, I + T I ) y,, and the expendture on the subsdy s S = pt y = B T. (2.1) Let the prvate expendture of agent n the post-nterventon equlbrum be x. The utlty derved by agent n the post-nterventon equlbrum s U = U x, y. Let the mnmum expendture requred at orgnal prce p to purchase a consumpton bundle that yelds the same utlty to agent be e, whle the actual cost of the post-subsdy consumpton bundle x, y at the orgnal prce s e + L, where L 0. Thus, U = V ( p, e), where V s 's ndrect utlty functon, and L s the (-specfc) money-metrc deadweght loss assocated wth the subsdy. Gven B and the budget constrant for the state (2.1), consder any two alternatve schemes 1 and 2. Denote the relevant magntudes for each scheme by ts correspondng number. Corollary 2.2: The contrbutory agent prefers scheme 1 to scheme 2 f and only f [ x$ 1 L ] [ x$ 2 L 1 2 ] + < +. 2 For analyss of ths neutralty property of Cournot games wth publc goods, see Bergstrom et al. (1986). 3 Kemp (1984) and Bergstrom et. al. (1986) nvestgate the neutralty property wth multple publc goods. 4 The treatment can be easly generalzed to allow a tax or subsdy on one or both of the prvate goods. 3
Proof: Consder the general case for any such scheme. Snce s a contrbutory agent, by Proposton 2.1, we get ( ) U = V p 1 t, E, where her effectve ncome Ê s gven by E = I + T x = I + B S xˆ (usng (2.1)). (2.2) However, e e ( p U ) =, $ s the mnmum expendture requred at orgnal prce p to provde the postnterventon utlty level to. Thus, the subsdy component transfers to the equvalent varaton R = e E = S L.. (2.3) Equatons (2.2) - (2.3) together yeld ( x L) e = I + B ˆ +. (2.4) Snce U = V ( p, e) and V s strctly ncreasng n expendture, the result follows from (2.4). Equaton (2.4) completely specfes each contrbutory agent s cardnal rankng over all possble nterventon schemes nvolvng lump-sum transfers to that agent and/or subsdy on the publc good. The deadweght loss can be measured by recoverng the utlty functon from the Cournot-Nash demand functons, as dscussed earler. Our results thus provde a smple method of dervng a money-metrc measure of the ndvdual-specfc welfare mpact of alternatve dstortonary nterventons n contexts that may be modeled as Cournot games between two or more agents wth one or more publc goods. Aggregate welfare calculatons can then proceed along standard lnes. It s clear from Corollary 2.2 that, f a subsdy ncreases the spendng by the other agent on the publc good, then the standard result regardng the superorty of lump-sum transfers vs-à-vs subsdes may get reversed n our strategc context. 4
REFERENCES Bergstrom, T., L. Blume, and H. Varan (1986): On the Prvate Provson of Publc Goods. Journal of Publc Economcs, 29: 25-49. Kanbur, R. (1995): Chldren and Intra-Household Inequalty: A Theoretcal Analyss. n K. Basu, P. K. Pattanak and K. Suzumura (eds.) Choce, Welfare and Development: A Festschrft n Honour of Amartya K. Sen (New York: Oxford Unversty Press). Kemp, M. (1984): "A Note on the Theory of Internatonal Transfers." Economcs Letters, 14: 259-262. Lundberg, S. and R. A. Pollack (1993): Separate Spheres Barganng and the Marrage Market. Journal of Poltcal Economy, 101(6, December): 988-1010. Roberts, R. (1984): A Postve Model of Prvate Charty and Wealth Transfers. Journal of Poltcal Economy, 92 : 136-148. Ulph, D. (1988): General Noncooperatve Nash Model of Household Consumpton Behavour. Department of Economcs, Unversty of Brstol, UK. Mmeo, Woolley, F. (1988): A Non-Cooperatve Model of Famly Decson Makng. Workng Paper, No.TIDI /125, London School of Economcs, UK. 5
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