HW07-solutions

HW07-solutions - UCLA Economics 11 Microeconomic Theory...

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UCLA Economics 11 Microeconomic Theory Winter 2006 Homework # 7 Answer Key Professor William Zame March 2, 2006 ——————————————————————————————————————— For each of the following two-person, two-good economies, find the equilibrium and the Pareto set, and sketch an Edgeworth box showing the endowments, the Pareto set and the equilibrium price line and consumption choices. 1. e A 100, 200 ; u A x , y x 1 3 y 2 3 ; e B 300, 200 ; u B x , y x 1 2 y 1 2 Equilibrium Prices and Choices Step 1: Calculate the total endowments: E 100 300, 200 200 400, 400 Step 2: Derive the demand functions for each individuals: since the utility functions are Cobb-Douglas functions, the demand functions are as follows: x A I A 3 P x 100 P x 200 P y 3 P x 100 3 200 3 P y P x y A 2 I A 3 P y 2 100 P x 200 P y 3 P y 200 3 P x P y 400 3 x B I B 2 P x 300 P x 200 P y 2 P x 150 100 P y P x y B I B 2 P y 300 P x 200 P y 2 P y 150 P x P y 100 Step 3: Markets clear; therefore demand has to equal supply in equilibrium. Total demand for x x A x B 100 3 200 3 P y P x 150 100 P y P x 550 3 500 3 P y P x Total supply of x 400 In equilibrium, 550 3 500 3 P y P x 400 Solve for P y P x , we obtain P y P x 13 10 Step 4: Plug the equilibrium prices back into the individual demand functions, we can obtain the equilibrium choices:
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x A 100 3 200 3 P y P x 100 3 200 3 13 10 120 y A 200 3 P x P y 400 3 200 3 10 13 400 3 2400 13 x B 150 100 P y P x 150 100 13 10 280 y B 150 P x P y 100 150 10 13 100 2800 13 Pareto Set Step 1: Calculate MRS for both individuals: MRS A MU x A MU y A 1 3 x A 2 3 y A 2 3 2 3 x A 1 3 y A 1 3 y A 2 x A MRS B MU x B MU y B 1 2 x B 1 2 y B 1 2 1 2 x B 1 2 y B 1 2 y B x B Step 2: On the Pareto set, MRS A MRS B : y A 2 x A y B x B We also know x B 400 x A and y B 400 y A ; substitute these back: y A 2 x A 400 y A 2 400 x A Therefore, the function of the Pareto set is as the following, y A 800 x A 400 x A Graph
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2. . e A 100, 700 ; u A
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This note was uploaded on 04/01/2010 for the course ECON Econ 11 taught by Professor Mcdevitt during the Fall '07 term at UCLA.

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HW07-solutions - UCLA Economics 11 Microeconomic Theory...

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