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Unformatted text preview: Problem Set 1 Due Thursday, Sept. 10th, by 10pm Solutions 1. Suppose Joe’s utility function was given by u ( x 1 ,x 2 ) = x 1 / 2 1 + 2 x 1 / 2 2 . (a) Show whether or not Joe’s tastes are homothetic. To show that the tastes are homothetic we need to show that the marginal rate of substi tution is invariant to a proportional change for any bundle of goods (i.e. MRS ( x 1 ,x 2 ) = MRS ( tx 1 ,tx 2 )). MRS = ∂u/∂x 1 ∂u/∂x 2 = 1 2 x 1 2 1 x 1 2 2 = 1 2 x 1 x 2 1 2 = 1 2 tx 1 tx 2 1 2 Since t cancels out, the tastes are homothetic. (b) Calculate Joe’s elasticity of substitution. From the class notes, we saw that the elasticity of substitution ( σ ) is defined as: σ = dln ( x 2 /x 1 ) dln  MRS  The equation for the MRS from part (a) implies that the following holds:  MRS  = 1 2 x 1 x 2 1 2  MRS  = 1 2 x 2 x 1 1 2 4  MRS  2 = x 2 x 1 ln(4) + 2ln  MRS  = ln x 2 x 1 Taking derivatives yields: 2 d ln  MRS  = d ln x 2 x 1 implying that the elasticity of substitution is 2. (c) Suppose the p 1 = 1 , p 2 = 4 , and Joe’s income is $120. How much of x 1 and x 2 does Joe consume? To solve for the demands, set up the lagrangian and taking first order conditions: L = x 1 / 2 1 + 2 x 1 / 2 2 + λ ( I p 1 x 1 p 2 x 2 ) 1 ∂ L ∂x 1 = 1 2 x 1 / 2 1 λp 1...
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This note was uploaded on 05/13/2010 for the course ECON 105D taught by Professor Cur during the Fall '09 term at Duke.
 Fall '09
 CUR
 Microeconomics, Utility

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