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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 06/30/2009 for the course ECON 102 taught by Professor Serra during the Fall '08 term at UCLA.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online