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

ISBN-10: 4431658955

ISBN-13: 9784431658955

ISBN-10: 4431658971

ISBN-13: 9784431658979

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**Additional info for Advances in Mathematical Economics**

**Sample text**

Since w. 1 and w. 2 are convex and closed in measure, 1 E w. 1 and 9 E w. 2 . Thus u=l+gEW.. 0 The following is a minimisation result. 3. 11 w. 1 and w. ) which are closed in measure and il w. 1 is bounded, then there exist 1 E w. 1 and 9 E w. 2 such that Praof. Let us set m := inf{II11-vl Ir : 11 E w. 1 , v E W. 2 }. There exist a sequence (fn) in w. 1 and a sequence (gn) in w. 2 such that m = limn-too Illn - 9nllr· Since w. e. to a function 1 E L~(p,) (see, [9, 10]). Each 1~ has the form 1~ = :E~~nAifi with Ai ~ 0, :E~~nAi = 1.

If the degree t of product differentiation is very elose to zero, then the corresponding CES indifference curves are rat her Hat with normal vector elose to (1,3) (apart from a very small area elose to the boundary of IR~ where the indifferenee curves bend towards the horizontal and vertieal coordinate axis, respeetively). As a eonsequenee, the weak firm's priee PI is nearly equal to its unit eosts c = 1 and the strong firm eharges approximately three times as mueh. The weak firm has a negligible market share and the strong firm is nearly a monopolist restricted, though, to some extent by the presenee of the weak firm.

Iii) How does the degree of product differentiation affect the existence of Nash equilibria if costs are strictly concave? A natural framework for the comparison of Nash equilibria, which lies at the heart of the result by Perloff and Salop, is provided by the theory of supermodular games that has been developed in recent years. This theory is based on seminal results by Tarski (1955) and by Topkis (1978, 1979), who also pointed out explicitly the usefulness of thc concept of supermodularity 1 A major part of the literature allowing for increasing returns to scale does so by adding a fixed cost component to a linear variable cost function.

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

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