{p*y/p*e)e - y, then p-v = 0. Thus, v* + v satisfies the budget constraint. Also, we have y* + y € Y because Y is a convex cone. Hence if the choice v* v and the production plan y* + y are combined, then the resulting consumption is •

Since (v*,y*) is maximal for >-., we must have x? x* + (p-y/p-e)e. By the ii i i r monotonicity, p - y s o.

The profit maximization condition of a Wairasian equilibrium is therefore established. To establish the utility maximization condition, note that, since p'y £ 0 for every y e Y,

(v. + y. e [R : p*v. < p*and y. € Y> - {x. e OxS p-x. s p-u.}.

Hence p-x* ^ p-u. and x* is maximal for >-. in {x. e 0?H p-x. ^ p-w.}.

A possible interpretation of this implication is that if a single convex, constant returns to scale (in fact, additivity suffices) production set Y is accessible to every consumer, then the profit maximization is a consequence of their utility maximization. This .result thus gives a justification for the assumption of profit maximization (to the extent that utility maximization is justified).

17.CM Let p e A, q' e /(p), q" e f(p)t and A e [0,1], We consider two cases, p € Interior A and p € Boundary A, separately. Let p e Interior A, then, for every q e A, zip) • ((1 - A)q* + Aq") = (1 - A)z(p)-q' + Xz(p)-qM £ (1 - A)z(p)-q + Az(p)♦ q = z(p)-q. Hence (1 - A)q' + Aq" € fip).

Let p e Boundary A, If p, > 0, then q' - q" s= 0 and hence (1 - A)ql +

!7,C*2 For each positive integer nr we let An = (p € A: p^ I/n>.

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