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ak6 - Assignment for Week Six Answer Key Anton Cheremukhin...

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Unformatted text preview: Assignment for Week Six: Answer Key Anton Cheremukhin October 30, 2008 An Example With Trade & Pareto-Frontier Utilities: U A = log x A + log y A ; U B = log x B + 2 log y B Total Endowments: ! Ax + ! Bx = 1 ; ! Ay + ! By = 2 Resource Contraints: x A + x B = ! Ax + ! Bx ; x A + x B = ! Ay + ! By Pareto Frontier: max x A ;y A ;x B ;y B f &U A + (1 ¡ & ) U B g s.t. resource constraints To simplify computations we can substitute the resource constraints into utility functions: Pareto Frontier: max x A ;y A ;x B ;y B & & ¡ log x A + log y A ¢ + (1 ¡ & ) ¡ log x B + 2 log y B ¢£ Substitute: x A + x B = 1 ) x B = 1 ¡ x A Substitute: y A + y B = 2 ) y B = 2 ¡ y A Pareto Frontier: max x A ;y A & & ¡ log x A + log y A ¢ + (1 ¡ & ) ¡ log ¡ 1 ¡ x A ¢ + 2 log ¡ 2 ¡ y A ¢¢£ First-order conditions: & x A = 1 & & 1 & x A & y A = 2 1 & & 2 & y A Hence, x A 1 & x A = & 1 & & = 2 y A 2 & y A = 2 2 y A & 1 : Therefore, y A = 2 = ¤ 2 x A 1 & x A + 1 ¥ = 2 x A 2 & x A : This is the Pareto Frontier: y A = 2 x A 2 & x A & Decentralization If we want to decentralize a particular Pareto&optimal point as an equilibrium, we need to write...
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ak6 - Assignment for Week Six Answer Key Anton Cheremukhin...

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