Consumer's surplus in monopolistic competition a la Dixit and Stiglitz.
Dixit and Stiglitz's ["Monopolistic Competition and Optimum Product Diversity," AER, 1977] model of monopolistic competition has been used by a number of economists in recent years to gain insight into phenomena ranging from monetary policy to intra-industry trade. In this model, there are n (i = 1,...n) monopolistically competitive (MC) firms facing demand given by [q.sub.i] = [k.sub.i] [([P.sub.i]/P).sup.[epsilon]] where [k.sub.i] is a firm specific constant, [P.sub.i] is firm i's nominal price, and n is large so that a small change in [P.sub.i] has little impact on the industry price index, P. The elasticity of demand for firms in the industry, [epsilon], is therefore constant. A lower value of [epsilon] indicates greater product differentiation and (assuming constant returns to technology) a higher price-cost margin ( = 1 / [epsilon]). Conversely, a higher value of [epsilon] indicates greater product homogeneity and a lower margin. As [epsilon] [arrow right] [infinity], the structure of an industry approaches that of perfect competition.
This paper reports an interesting property of constant elasticity demand. It appears to be singularly partial to consumers in the division of gross surplus.
Proposition 1: Assume that the production technology exhibits constant returns and the reservation prices are sufficiently high. Then consumer's surplus, due to a MC firm facing constant elasticity demand, will be at least as large as the firm's gross profit.
Proof: At any instant, consumer's surplus due to firm i is:
(1) [Mathematical Expression Omitted]
where [P.sub.i] is the reservation price, which is assumed to be high. Constant elasticity of demand implies homothetic preferences, which allows one to use the integral in (1) even when all prices change simultaneously [Novshek, Mathematics for Economists, 1993, p. 191]. Firm i's gross profit before depreciation is [q.sub.i] [P.sub.i] / [epsilon] <- Csi. Q.E.D.
An even stronger result is available if one assumes that the expected value of producer's surplus net of cost of entry is zero in every industry due to entry by new firms.
Proposition 2: The expected values of total and consumer's surplus produced by a MC firm will always be at least as high as those produced by a perfectly competitive firm and will usually be higher.
Proof: Since the expected value of producer's surplus net of cost of entry is zero, the expected value of total surplus equals that of consumer's surplus. In perfect competition [epsilon] = [infinity] and, therefore, from (1), [Mathematical Expression Omitted]. For a MC firm, [epsilon] < [infinity] and [Mathematical Expression Omitted]. Q.E.D. Note, however, that although MC firms outperform perfect competitors in terms of consumer's surplus, it does not mean that a [derivatives] (CSi)/[derivatives][epsilon] < 0. The results should not be interpreted to mean that in the case of monopoly, more is better than less. They serve the purpose of highlighting the extreme nature of the assumptions underlying perfect competition (that is, E = oo), which make it a vulnerable straw-man. Researchers should, therefore, be careful in using perfect competition as the standard against which to judge welfare performance of economic models.
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|Author:||Sinha, Deepak K.|
|Publication:||Atlantic Economic Journal|
|Date:||Dec 1, 1995|
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