the economy consists of two consumers a and b

Question

The economy consists of two consumers, A and B. Both consumers are endowed with one unit of good 1 and one unit of good 2. Consumer A is entirely indi?erent between all consumption plans. Consumer B has the utility function u(x₁^B; x₂^B) = x₁^B .x₂^B
(i) Find a competitive equilibrium for this economy. (Hint: guess an equilibrium price and then check that it works.)

(ii) Find a second competitive equilibrium (different from the one you found in part (i)).

(iii) Show that the equilibrium is not effcient.

Problem 4

In problem 3 we found an example that shows that if the utility functions are not increasing then the competitive equilibrium may not be effcient. Here is a weaker de?nition of effciency: A feasible consumption plan is weakly e?cient if there is no other feasible consumption plan that increases every consumer's utility.

Prove that every competitive equilibrium of an exchange economy is weakly effcient.