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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**Extra resources for Advances in Mathematical Economics**

**Example text**

In Section 3 we shall analyze some particular exampIes in which uj is Cobb-Douglas. In this case, we obtain uj(xo,Xj) = log(xo) + log(xj) + fj and the random vector f = (f1,'" , fn ) enters additively (additive random utility model). Of course, firms don't have to know the underlying stochastic model, but they are supposed to correctly anticipate their demand. It is worth noticing that CES demand functions, which are often used in models of product differentiation, can be obtained as a particular case if the distribution of fis specified as follows [cf.

We are now going to investigate this phcnomenon in more detail by analyzing the behavior of marginal profits Gilli = (Pi - Ci)a;ji + fi for i = 1,2. P2 of the price of the strong firm. 12 I in firm 2's demand. 12 I is very small, since consumers consider both products as roughly identical and very few of firm 2's customers at prices P 10 We would like to remark that example 1 ean be transformed into an example with a symmetrie CES function and asymmetrie eosts. Put Xl = (1/2)1 / /l X1 and X2 = (3/2)I//lX2.

3. (3) ===? (4). Let (x n ) be a bounded sequence in X and M = sUPn Ilxnll. Put H = {l Q x n : n 2': I}. Then H is a bounded subset of L3c(J1). 1 ((2) ===? (3) ===? (5) and foot note (*)), there is a subsequence (lQx nk hof (lQx n ) such that (lQx~Jk is (J(L 1 , L=) convergent. Remark that if k > M, then lQx~k = 1Qx nk . So (lQx nk h is (J(L 1 , L=) convergent and Convergences in L~(JJ) 31 therefore (xnkh is a(X,X*) convergent. Whence X is reflexive. (4)~(1). We can suppose that X is separable.

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

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