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HW06-Solutions

# HW06-Solutions - Economics 11 Solutions to Homework 6...

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Economics 11 Solutions to Homework 6 Question 1 Endowments and utility functions of consumers are: e A = (300 ; 600) u A ( x; y ) = x 1 2 y 1 2 e B = (600 ; 900) u B ( x; y ) = x 1 2 y 1 2 1. Find A and B°s demand for x and y x A = I A 2 p x = 300 p x +600 p y 2 p x = 150 + 300 p y p x y A = I A 2 p y = 300 p x +600 p y 2 p y = 150 p x p y + 300 x B = I B 2 p x = 600 p x +900 p y 2 p x = 300 + 450 p y p x y B = I B 2 p y = 600 p x +900 p y 2 p y = 300 p x p y + 450 2. Find total supply of x and y Total supply of x = 900 Total supply of y = 1500 3. Set Demand = Supply x A + x B = 900 ! 150 + 300 p y p x + 300 + 450 p y p x = 900 4. Solve for the equilibrium price ratio p y p x = 3 5 \$ p x p y = 5 3 5. Substitute into demand functions to ±nd equilibrium choices x A = 330 y A = 550 x B = 570 y B = 950 6. Solve for the pareto set The pareto set consists of all the points for which MRS A = MRS B MRS A = y A x A MRS B = y B x B Notice that the initial allocations cannot be optimal since MRS A = 2 6 = MRS B = 2 3 . Also using the fact that demand equals supply in equilibrium, 1

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x A + x B = 900 ! x B = 900 ° x A y A + y B = 1500 ! y B = 1500 ° y A Substituting for x B and y B : y A x A = y B x B = 1500 ° y A 900 ° x A Solving for y A , the pareto set is: y A = 5 3 x A Along this line, the marginal rates of substitution for both A and B are equal. Notice that the contract curve for B (if we were to draw it from B°s origin) is completely symmetric: y B = 5 3 x B The individually rational set for A, IR A is the set of points that lies above A°s indi/erence curve that passes through her endowment point e A . Likewise, the
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HW06-Solutions - Economics 11 Solutions to Homework 6...

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