By Professor Charalambos D. Aliprantis, Professor Donald J. Brown, Professor Owen Burkinshaw (auth.)

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Next, fix some constant c > 0 such that for each coalition 5 and each x E V(5) we have x; < c for all i E 5, and then consider the set W = [ U V ( 5)] n(-oo, cr . SEN Clearly, W is a closed, comprehensive from below (i. e. , W - R+ = W holds), bounded from above in nn by c = (c, c, ... 5-2. In particular, the boundary oW of lV is contained in W, i. , oW~ W holds. Fig. 5-2 The following property ( *) of the set W will be employed in the proof. If X E oW and Xr =0 for some r' then X; =c also holds for some z.

For this exercise ~ will denote a continuous and convex preference relation on R~ having an extremely desirable bundle. l fixed vector. :y holdsforall yEBw(P)}, pEint(R~). :::: is strictly convex, then the demand correspondence coincides with the demand function. Establish the following properties for the demand correspondence. , the demand function is non-empty, convex-valued and compact-valued. b) For each y E x(p ), we have p · y = p · w. A > 0. d) If {Pn} ~ Int(R~) satisfies Pn - - f p ~ 0, then there exists a bounded subset B of R~ such that x(pn) ~ B holds for each n.

THE ARROW-DEBREU MODEL 36 [Chap. I Fig. 4-2 EXERCISES 1. } having three consumers with the following characteristics: Consumer 1. Initial endowment (1, 2) and utility function u 1 (x, y) =-/X+ Vfj. Consumer 2. Initial endowment (3, 4) and utility function u 2 (x, y) =min{ x, y }. Consumer 3. Initial endowment ( 1, 1) and utility function u 3 ( x, y) = yex. Find the excess demand function and the equilibrium prices for this ex2 tt-2) ~"( p ) -_ ( 3t 2Itt tt-2 , - 3tt(Itt) h c ange economy. A nswer ..