Question

Q1) Consider the fowling economy, with two consumers and two commodities. Consumer 1 has the endowment vector, w1 = (1,1), and the utility function,
u1(x1) = log(x11) + A log(x21)
where A is a positive constant. Consumer 2 has the endowment vector, w = (1,1), and the utility function,
u2(x2) = log(x12) + log(x22)
a) Define competitive equilibrium for this economy.
b) Calculate the competitive equilibrium price vector, as a function of the parameter A.

Answer 1

a. Competitive equilibrium is a condition in which profit-maximizing producers and utility-maximizing consumers in competitive markets with freely determined prices arrive at an equilibrium price

b. w1 = (1,1)

$u1(x1) = log(x11) + A log(x21)$

$u2(x2) = log(x12) + log(x22)$

Alog(x21)=log(x11)

A=log(x11)/log(x22)