Munich Personal RePEc Archive Bertrand-Edgeworth equilibrium with a large number of firms Roy Chowdhury, Prabal Indian Statistical Institute, Delhi Center April 2007 Online at https://mpra.ub.uni-muenchen.de/3353/ MPRA Paper No. 3353, posted 30 May 2007 UTC Bertrand-Edgeworth equilibrium with a large number of firms Prabal Roy Chowdhury (Indian Statistical Institute) Abstract We examine a model of price competition with strictly convex costs where the firms simultaneously decide on both price and quantity, are free to supply less than the quantity demanded, and there is discrete pricing. If firms are symmetric then, for a large class of residual de- mand functions, there is a unique equilibrium in pure strategies when- ever, for a fixed grid size, the number of firms is sufficiently large. Moreover, this equilibrium price is within a grid-unit of the competi- tive price. The results go through to a large extent when the firms are asymmetric, or they are symmetric but play a two stage game and the tie-breaking rule is ‘weakly manipulable’. JEL Classification Number: D43, D41, L13. Key words: Bertrand equilibrium, Edgeworth paradox, tie-breaking rule, rationing rule, folk theorem of perfect competition. Address for Correspondence: Indian Statistical Institute, Delhi Center, 7 - S.J.S. Sansanwal Marg, New Delhi - 110016, INDIA. Phone: 91-11-41493930. Fax: 91-11-41493981. e-mail:
[email protected] 1 Introduction Let us consider a Bertrand duopoly where the firms decide on both their price and output levels and the firms are free to supply less than the quantity demanded. Edgeworth (1897) argues that in such models equilibria in pure strategies may not exist (see Dixon (1987), or Friedman (1988) for formal statements of the problem).