Question

2. Suppose there are two consumers, A and B, and two goods, X and Y. Consumer A is given an initial endowment of 2 units of good X and 3 units of good Y. Consumer B is given an initial endowment of 6 units of good X and 5 units of good Y. Consumer A’s utility function is given by:

U_A(X,Y) = X^1/2*Y^1/2,

And consumer B’s utility function is given by

U_B(X,Y) = X^1/4*Y^3/4.

Therefore, consumer A’s marginal utilities for each good are given by:

MU_X = (1/2)X^-1/2Y^1/2

MU_Y = (1/2)X^1/2Y^-1/2

Also, consumer B’s marginal utilities for each good are given by:

MU_X = (1/4)X^-3/4Y^3/4

MU_Y = (3/4)X^1/4Y^-1/4

a) Suppose the price of good Y is equal to one. Calculate the price of good X that will lead to a competitive equilibrium.

b) How much of each good does each consumer demand in equilibrium?

c) What is the marginal rate of substitution for consumer A at the competitive equilibrium?

d) Illustrate the situation in an Edgeworth Box. Be sure to label your box carefully and accurately. Identify the initial endowment and label it W. Identify the competitive equilibrium and label it D. Draw the budget constraint that each consumer faces and identify the values where it intercepts the perimeter of the Edgeworth Box (there are two different intercepts to identify).