Download Advances in Mathematical Economics by Gérard Debreu (auth.), Shigeo Kusuoka, Toru Maruyama (eds.) PDF

By Gérard Debreu (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)

ISBN-10: 4431658971

ISBN-13: 9784431658979

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Finally, both firms become more powerful if t gets suffieiently large. The latter ease eorresponds to the intuition expressed by Anderson et al. (1992) as cited in the introduction. Their discussion foeuses on the market power all firms together exert over the eonsumers, but it abstracts from the question of how the strategie position of an individual firm in relation to its eompetitors is affected by the degree of product differentiation. Produet differentiation and market power 49 Example 2 (small cost asymmetr'Y): The above discussion suggests that counterintuitive behavior for sufficiently low degrees t of product differentiation can occur independently of the size and of the source of the underlying asymmetry.

Suppose f(s, t) : S x T -t lR is supermodular in s for given t and has inereasing differences in sand t. Suppose that t 2: t' and that sEM = argmaxf(8,t) and s' E M' = argmaxf(s,t'). \s' E M' and8Vs' E M. In particular (when t = t'), the set of maximizers of f is a sublattice. We shall consider smooth supermodular games of the following type. Every player i = 1,'" ,n has a strategy set Si that is a compact interLet S = Si be the (complete) lattice of strategy profiles. val in lR The family of games is parameterized by an interval T c R The payoff fi : S x T -t lR of every player i is C 2 and satisfies the supermodularity condition Ö/Jdi(S; t) 2: 0 where thc mixed partial derivatives are taken with respect to strategics Si, 8j, j i= i.

Ey [9,10] we know that Rw(X)-tight==}WTP. ) be a bounded sequence in Sf. Let us consider a dense sequence (e;)pEN for the Mackey topology T(X*, X). 1) . 2) for each subsequence (wt) = (vm[). 1) we see that (~~j=lWj('))n is pointwise bounded and by hypothesis ~~j=lWj(W) E r(w) for almost all wEn and for all n E N*. )II. ~j=lWj(W)) n = (e*,u(w)) for each subsequence thus showing that (u n ) has the WKP. D Remarks. 1. If cwk(X) denotes the collection of non empty convex weakly compact subsets of X, Saadoune [31, Theorem 1] introduced the following tighness condition, namely for every sequence (u n ) in the set into consideration, there is a sequence (u n ) satisfying: (i) un E Co{ Um : m ~ n} for every n, (ii) for every c > 0, there is a convex cwk(X)-valued measurable multifunction r e : n --+ X such that SUP/L({w E n: un(w) ~ n r e (w)}) :::; c.

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