Course: ECO 3553

Imagine an economy composed of I consumers, J firms, and in which there are L goods and services. Each consumer i has an initial endowment wi=(wil,wi2,...wil)

Each consumer i owns a shareTij0 of the firm j, i=0ITij=1. We are in a competitive economy. There is therefore a price vector p = (p1, p2, ..., p_L). The budget constraint of the consumer is, therefore,pxipwi+j=1JTijpyj(p)=Bi(p), where y_j(p) is the profit-maximizing production choice of firm j given the price vector p and its technology constraint. The excess demand vector of this economy isz(p)=i=1Ixi(p,Bi(p))i=1Jwij=1Jyj(p)

The economy is in competitive equilibrium when this excess demand vector is non-positive, i.e.z(p)0

a) Demonstrate that Walrasian general equilibrium exists in this economy, i.e. that there exists a psuch as z(p)0. Please explain in detail each of your steps.

b) Show that the Walrasian general equilibrium of this economy is a Pareto optimum. Please explain in detail each of your steps