Rubinstein2005-page82

Rubinstein2005-page82 - 0 ε 0 w A clear conclusion can be...

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October 21, 2005 12:18 master Sheet number 80 Page number 64 64 Lecture Five Now, take two prices, H > 1 and L < 1, such that a consumer with utility function u consumes more of the Frst commodity when facing the budget set (( H ,2 ) ,1 ) than when facing the budget set (( L ,2 ) ,1 ) (that is, 1 / H > 1 /( 2 L ) ). When δ is close enough to 0, the demand induced from u δ at B (( H ,2 ) ,1 ) is ( 1 / H ,0 ) . Choose ² such that 1 /( 2 L ) + ²< 1 / H . ±or δ close enough to 0, the bundle in the budget set of B (( L ,2 ) ,1 ) with x 1 = 1 /( 2 L ) + ² is preferred (according to u δ ) over any other bundle in B (( L ,2 ) ,1 ) with a higher quantity of x 1 . Thus, for small enough δ , the induced demand for the Frst commodity at the lower price is at most 1 /( 2 L ) + ² , and is thus lower than the demand at the higher price. “The Law of Demand” We are interested in comparing demand in different environments. We have just seen that the classic assumptions about the consumer do not allow us to draw a clear conclusion regarding the relation between a consumer’s demand when facing B ( p , w ) and his demand when facing B ( p
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Unformatted text preview: + ( 0, . . . , ε , . . . , 0 ) , w ) . A clear conclusion can be drawn when we compare the consumer’s demand when he faces the budget set B ( p , w ) to his demand when facing B ( p , x ( p , w ) p ) . In this comparison we imagine the price vec-tor changing from p to an arbitrary p and wealth changing in such a way that the consumer has exactly the resources allowing him to consume the same bundle he consumed at ( p , w ) . (See Fg. 5.4.) Claim: Let x be a demand function satisfying Walras’s law and WA. If w = p x ( p , w ) , then either x ( p , w ) = x ( p , w ) or [ p − p ][ x ( p , w ) − x ( p , w ) ] < 0. Proof: Assume that x ( p , w ) 6= x ( p , w ) . Then, [ p − p ][ x ( p , w ) − x ( p , w ) ] = p x ( p , w ) − p x ( p , w ) − px ( p , w ) + px ( p , w ) = w − w − px ( p , w ) + w = w − px ( p , w )...
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This note was uploaded on 12/29/2011 for the course ECO 443 taught by Professor Aswa during the Fall '10 term at SUNY Stony Brook.

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