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Rubinstein2005-page89

# Rubinstein2005-page89 - v/∂ p k p ∗ w ∗ ∂ v/∂ w p...

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October 21, 2005 12:18 master Sheet number 87 Page number 71 Choice over Budget Sets and the Dual Problem 71 Claim: Assume that the demand function satisfies Walras’s law. Let H = { ( p , w ) | ( x ( p , w ) , 1 )( p , w ) = 0 } for some ( p , w ) . The hyperplane H is tangent to the indifference curve through ( p , w ) . Proof: Of course ( p , w ) H . For any ( p , w ) H , the bundle x ( p , w ) B ( p , w ) . Hence x ( p , w ) x ( p , w ) , and thus ( p , w ) ( p , w ) . In the case in which is represented by differentiable v , H = { ( p , w ) | (∂ v /∂ p 1 ( p , w ) , . . . , v /∂ p K ( p , w ) , v /∂ w ( p , w ))( p p , w w ) = 0 } . From the above claim and since w = p x ( p , w ) we have also H = { ( p , w ) | ( x ( p , w ) , 1 )( p p , w w ) = 0 } . Therefore, the vector (∂ v /∂ p 1 ( p , w ) , . . . , v /∂ p K ( p , w ) , v /∂ w ( p , w )) is proportional to the vector ( x 1 ( p , w ) , . . . , x K ( p , w ) , 1 ) , and thus, −[ v
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Unformatted text preview: v /∂ p k ( p ∗ , w ∗ ) ] / [ ∂ v /∂ w ( p ∗ , w ∗ ) ] = x k ( p ∗ , w ∗ ). Dual Problems In normal discourse, we consider the following two statements to be equivalent: 1. The maximal distance a turtle can travel in 1 day is 1 km. 2. The minimal time it takes a turtle to travel 1 km is 1 day. This equivalence actually relies on two “hidden” assumptions: a. ±or (1) to imply (2) we need to assume the turtle travels a posi-tive distance in any period of time. Contrast this with the case in which the turtle’s speed is 2 km/day but, after half a day,...
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