MWG 3G7 Let us consider the indirect demand function g x ...

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Econ 201A Fall 2013 Problem Set 5 Suggested Solutions 1. MWG 3.G.7 Let us consider the indirect demand function g ( x ) given as the inverse of x * ( p, 1), i.e. x = x * ( g ( x ) , 1) (assuming that x * and g are well-defined, smooth functions). (a) Prove that g ( x ) = 1 x · ∇ u ( x ) u ( x ) .
(b) Deduce from Roy’s identity that x * ( p, 1) = 1 p · ∇ v ( p, 1) v ( p, 1) .
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Econ 201A Fall 2013 Problem Set 5 Suggested Solutions constraint is binding (i.e. Walras’ Law holds), then we can use the Lagrangian approach and get v ( p, w ) = max x,λ u ( x ) - λ · ( p · x - w ) . Hence, by the Envelope theorem, we get p v ( p, w ) = - λx * ( p, w ) and ∂w v ( p, w ) = λ, so p v ( p, w ) = - ∂w v ( p, w ) · x * ( p, w ). Finally, we can conclude since we assumed that Walras’ Law holds, that p v ( p, 1) · p = - ∂w v ( p, w ). 2. MWG 3.G.16 Consider the expenditure function e ( p, v ) = exp ( X l α l log p l + Y l p β l l ! v ) . (1) (a) What restrictions on α 1 , ..., α n , β 1 , ..., β n are necessary for this to be derivable from ex- penditure minimization (assuming a continuous, locally nonsatiated and strictly quasi- concave utility function)?

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