Question

3. Consider a general equilibrium model with two individuals (A and B) and two goods (zi and x2). Consumer A has utility function given by Consumer B has utility function given by Consumer A has endowment: 5and and wr = 2. 6, and Consumer B has endowment: (a) Draw the Edgeworth Box for this economy, where the origin for Consumer A is on the bottom-left corner (QA) and the origin for Consumer B is on the top-right corner (Op). Label carefully, and clearly indicate the endowment (E) (b) Consider allocations where each consumer consumes positive amount of each good (that is, all the allocations that are strictly inside the Edgeworth Box). Find all the Pareto- optimal allocations, expressed in terms of a and r (No need to solve for the equation.) (c) Consider the competitive markets. Suppose that the price of i is $1 and the price of r2 is $1, as well, then does this economy reach a competitive equilibrium? Explain your answer. If not, what happens to the prices?

Answer 1

u^A = (x^1/4_1 middot x^3/4_2)A, u^B = (x^3/4_1 middot x^1/4_2)B w^A = (5, 6), w^B = (5, 2) thus w^A_1 + w^B_1 = 5 + 5 = x^A_1 + x^B_1 = 10 w^A_2 + w^B_2 = 6 + 2 = x^A_2 + X^B_2 = 8} (b) now at to efficiency rightarrow MRS L_2^B = MRS L_112^B MRS = partial differential u Vertical-bar partial differential x_1/partial differential u Vertical-bar partial differential x_2 so at PO {1/4 (x_1)^-3/4 (x_2)^3/4/3/4 (x_1)^1/4 (x_2)^-1/4}^A = {3/4 (x_1)^-1/4 x_2^1/4/1/4 (x_1)^3/4 (x_2)^-3/4}^B 1/3 (X_2/x_1)^A = 3(x_2/x_1)^B = 3[8 - x_2^A/10 - x_1^A] implies x_2^A/3 X_1^A = 24 - 3 x_2^A/10 - x_1^A 10 x_2^A - x_1^A middot x_2^A = 72 x_1^A - 9 x_1^A middot x_2^A 10 x_2^A + 8 x_1^A middot x_2^A = 72 x_1^A x_2^A = 72 x_1^A/10 + 8 x_1^A \r\n now since utility function rightarrow x_1^A = 1/4 M_A/P_1 = M_A/4 P_1 M_A = P_1 w_1^A + P_2 w_2^A rightarrow endowment income M_A = 5 + 6 = 11,
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