# HW2Spring08answer - Homework 2, Spring 08 (Total 20 points)...

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Homework 2, Spring 08 (Total 20 points) Q1 (2 × 4): Ken’s preference is represented by u ( x 1 ,x 2 ) = 2ln x 1 + ln x 2 . Denote the prices by ( p 1 ,p 2 ) and income by m . (a) Obtain the marginal rate of substitution of good 2 for good 1 (as you did before). (b) Obtain the demand function, x 1 ( p 1 ,p 2 ,m ) and x 2 ( p 1 ,p 2 ,m ). (c) Redo (a)(b) for another representation v ( x 1 ,x 2 ) = x 2 3 1 x 1 3 2 . Answer : (a) MU 1 ( x 1 ,x 2 ) = ∂x 1 (2ln x 1 + ln x 2 ) = 2 x 1 , MU 2 ( x 1 ,x 2 ) = ∂x 2 (2ln x 1 + ln x 2 ) = 1 x 2 . MRS ( x 1 ,x 2 ) = MU 1 ( x 1 ,x 2 ) MU 2 ( x 1 ,x 2 ) = 2 x 1 1 x 2 = 2 x 2 x 1 (b) From the tangency condition MRS ( x 1 ,x 2 ) = p 1 p 2 , we have 2 x 2 x 1 = p 1 p 2 , hence x 2 = p 1 2 p 2 · x 1 . Plug this into the budget equation, then p 1 x 1 + p 2 x 2 = p 1 x 1 + p 2 · p 1 2 p 2 · x 1 = 3 p 1 2 · x 1 = m, which yields x 1 = 2 m 3 p 1 . Plug this to the previous formula, then x 2 = m 3 p 2 Summing up, x 1 ( p 1 ,p 2 ,m ) = 2 m 3 p

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## This note was uploaded on 10/25/2010 for the course ECO 420K taught by Professor D during the Spring '10 term at University of Texas at Austin.

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HW2Spring08answer - Homework 2, Spring 08 (Total 20 points)...

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