Question

Consider an exchange economy consisting of two people, A and B, endowed with two goods, 1 and 2. Person A is initially endowed with ω A = (0, 3) and person B is initially endowed with ω B = (6, 0). They have identical preferences, which are given by U A(x1, x2) = U B(x1, x2) = x² 1x2.

(A) Write the equation of the contract curve (express xA2 as a function of xA1 ).

(B) Let p2 = 1. Find the competitive equilibrium price, p1, and allocations, x A = (xA1 , xA2 ) and x B = (xB1 , xB2 ).

(C) Now suppose that before A and B ever get to trade, some of B’s endowment of good 1 is destroyed, so she instead has an initial endowment of ω B = (3, 0) (and everything else is the same as in part (a), including A’s endowment and both players’ preferences. Calculate the new equilibrium price, p1, and the new allocations, xA = (xA1 , xA2 ) and xB = (xB1 , xB2 ) (You may assume that p2 = 1.)

(D) Calculate and compare the utility level for person A in equilibrium you found in part b versus part c. Which is higher?

(E) Give a brief, intuitive explanation for why person A’s utility is affected in this way, when we only changed the endowment for person B?

Answer 1

In an exchange economy with two individuals, A and B, and two goods, 1 and 2, we can find the contract curve equation by maximizing the utility function. The competitive equilibrium price and allocations can be found by setting the demand equal to the initial endowments. By modifying the endowment of person B, we can find a new equilibrium price and allocations. Comparing the utility levels, person A has a higher utility in the original equilibrium compared to the modified one. The change in person B's endowment affects person A's utility due to the redistribution of goods and changes in relative prices.

Step-by-step explanation:

(A) To find the equation of the contract curve, we need to find the allocation of goods that maximizes the utility of both individuals A and B. Since both individuals have identical preferences, we can write their utility function as U(x1, x2) = x1^2 * x2. We can use the Lagrange method to find the allocation that maximizes their utility, subject to their endowments and the budget constraint. Solving the Lagrange equations, we get:

xA1 = ωA1 - λ * p1

xA2 = (4λ * ωA1 + 3ωB1) / (8λ)

(B) Assuming p2 = 1, we can find the competitive equilibrium price p1 by setting the demand of the two individuals equal to their initial endowments. Solving the equations:

3 - p1 = 0

6 - 2p1 = 0

we find p1 = 3/2, xA = (1, 9/4), and xB = (3/2, 3/4).

(C) With ωB = (3, 0), the new equilibrium price p1 can be found by setting the demand of the two individuals equal to their modified endowments. Solving the equations:

3 - p1 = 0

6 - 2p1 = 0

we find p1 = 3/2, xA = (1, 3/4), and xB = (3/2, 0).

(D) To compare the utility levels for person A, we can plug the allocations from parts (B) and (C) into their utility function U(x1, x2) = x1^2 * x2. Evaluating the function, we find that person A has a higher utility level in equilibrium in part (B) compared to part (C).

(E) The change in person B's endowment affects person A's utility level because it alters the relative prices and allocation of goods. When person B's endowment of good 1 is decreased, the equilibrium price of good 1 increases, leading to a redistribution of goods between person A and B. This results in a lower allocation of good 1 for person A, reducing their utility level.