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Review Questions for Final [Key 10-11]

# Review Questions for Final [Key 10-11] - Find the equation...

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Question 10 10. Find the equation of the Pareto frontier for an economy with 2 units of good x, and 5 units of good y when the utility functions of the two people are 10.1 U A = x A + y A 10.2 U B = log ( x B ) + log ( y B ) If the initial endowments are equally distributed between Mr. A and Mr. B find the equilibrium relative prices. Endownments: Economy total endowments for both goods: ω x = 2 , ω y = 5 Endownment distribution: Mr. A: ω A x = 1 , ω A y = 5 / 2 Mr. B: ω B x = 1 , ω B y = 5 / 2 . Resource Constraints: x A + x B = ω x y A + y B = ω y Part A: Pareto Frontier Social Planner Problem: max x A ,y A ,x B ,y B [ λ ( x A + y A ) + (1 - λ ) (log ( x B ) + log ( y B ))] Subject to : x A + x B = 2 y A + y B = 5 Which can be rewritten as max x A ,y A [ λ ( x A + y A ) + (1 - λ ) (log (2 - x A ) + log (5 - y A ))] Assuming internal solution: x A > 0 , y A > 0 (However, x B > 0 , y B > 0 always) First Order Conditions: x A : λ = (1 - λ ) (2 - x A ) y A : λ = (1 - λ ) (5 - y A ) Clearly: (5 - y A ) = (2 - x A ) y A = 3 + x A . Assuming that there can be corner solutions: x A 0 , y A 0 First Order Conditions: x A : λ - (1 - λ ) (2 - x A ) 0 y A : λ - (1 - λ ) (5 - y A ) 0 With equality if x

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Review Questions for Final [Key 10-11] - Find the equation...

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