Question

I need help solving this exercise from a course called Advanced Microeconomics in MSc in Economics. Thank you in advance

Exercise 4 (General Equilibrium). Two agents have identical quasilinear preferences U(x,y) -u(x) +y, where u(x) Agent 1's endowment is (3/2, 1/2) and agent 2's endowment is (1/2,3/2). Normalize so that the price of good 2 is 1. 1. Calculate a Walrasian equilibrium at which the price of good 1 is greater than 1/2. Are there other Walrasian equilibria? 2. Draw an Edgeworth box for this economy. Draw and label the following elements: (a) The Walrasian equilibrium you just found (including the price line, the allocation, and each agent's indifference curve through the allocation) (b) the entire Pareto Efficient set (c) the initial endowment point. (d) the core 3. Suppose agent 1's endowment were (2. Уг). Find a Walrasian equilibrium for this economy. Note that agent 1 is actually worse off, despite having a higher endowment. Briefly explain, in terms of supply and demand, why this happened

Answer 1

U (x, y) = u(x) + y, where u(x) = {x, x elementof [0,1] x - 1/2 + 1, x > 1 (w_1) Agent 1's endowment = [3/2, 1/2] (w_2) Agent 2's endowment = [1/2, 3/2] (p_1) price of good 2 = 1 (p_2) Price of good 1 > 1/2 Budget set of Agent 1 p_1 x_1^1 + p_2 x_2^1 lessthanorequalto p_1 w_1^1 + p_2 w_2^1 Budget set of Agent 2 p_1 x_1^2 + p_2 x_2^2 lessthanorequalto p_1 w_1^2 + p_2 w_2^2 condition for walrasian Equilibrium] rightarrow Each consumer maximizes utility subject to budget constraint In the above equation, assume x_i^1 + x_i^2 = w_i or x_1^1 + x_1^2 = w_1 Similarly x_2^1 + x_2^2 w_2 x_1^2 = w_1 - x_1^1 therefore P_1 X_1^1 + P_2 X_2^1 greaterthanorequalto p_1 w_1^1 + p_2 w_2^1 \r\n w_1 = (3/2, 1/2) U(x, y) = u(x) + y = x - 1/2 + 1 + y = 3/2 - 1/2 + 1 + 1/2 = 1/2/2 + 1 +..