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ps2solutions - Problem Set 2 Due Thursday Sept 17th by 10pm...

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Problem Set 2 Due Thursday, Sept. 17th, by 10pm 1. Terri likes to consume chocolate ( x 1 ) and movies ( x 2 ). Terri’s utility function was given by u ( x 1 , x 2 ) = ( x 1 - γ ) α ( x 2 ) 1 - α where γ > 0 . (a) Find Terri’s demands for x 1 and x 2 . Are Terri’s demands homothetic? L = ( x 1 - γ ) α ( x 2 ) 1 - α + λ ( I - p 1 x 1 + p 2 x 2 ) Taking first order conditions and setting them equal to 0: L ∂x 1 = α x 2 x 1 - γ 1 - α - λp 1 = 0 L ∂x 2 = (1 - α ) x 2 x 1 - γ - α - λp 2 = 0 L ∂λ = I - p 1 x 1 + p 2 x 2 = 0 Solving this system gives the marshallian demand: x * 1 ( p 1 , p 2 , I ) = α I p 1 + (1 - α ) γ x * 2 ( p 1 , p 2 , I ) = (1 - α ) p 1 p 2 I p 1 - γ Since MRS is not invariant to a proportional change in the ratio x 1 x 2 the utility function is not homothetic. MRS = - α 1 - α x 2 x 1 - γ (b) Derive Terri’s indirect utility function. The indirect utility function is v ( x * 1 ( p 1 , p 2 , I ) , x * 2 ( p 1 , p 2 , I )) = α I p 1 - αγ α (1 - α ) p 1 p 2 I p 1 - γ 1 - α (c) Derive Terri’s expenditure fuction.
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