Rubinstein2005-page101

# Rubinstein2005-page101 - a producer of the commodities L +...

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October 21, 2005 12:18 master Sheet number 99 Page number 83 Production 83 function (unlike the law of demand, which was applied to the compensated demand function). Proft Function 1. π(λ p ) = λπ( p ) . (Follows from z p ) = z ( p ) .) 2. π is continuous. (Follows from the continuity of the supply function.) 3. π is convex. (For all p , p 0 and λ ,if z maximizes pro±ts with λ p + ( 1 λ) p 0 then π(λ p + ( 1 λ) p 0 ) = λ pz + ( 1 λ) p 0 z λπ( p ) + ( 1 λ)π( p 0 ) .) 4. Hotelling’s lemma : For any vector p , π( p ) pz ( p ) for all p . Therefore, the hyperplane { ( p , π) | π = pz ( p ) } is tangent to { ( p , π) | π = π( p ) } , the graph of function π at the point ( p , π( p )) .I f π is differentiable, then d π/ dp k ( p ) = z k ( p ) . 5. From Hotelling’s lemma it follows that if π is differentiable, then dz j / dp k ( p ) = dz k / dp j ( p ) . The Cost Function If we are only interested in the ±rm’s behavior in the output mar- ket (but not in the input markets), it is suf±cient to specify the costs associated with the production of any combination of out-
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Unformatted text preview: a producer of the commodities L + 1, . . . , K , we de±ne c ( p , y ) to be the minimal cost associated with the production of the combina-tion y ∈ < K − L + given the price vector p ∈ < L ++ of the input commodi-ties 1, . . . , L . If the model’s primitive is a technology Z , we have c ( p , y ) = min a { pa | ( − a , y ) ∈ Z } . (See ±g. 7.3.) Discussion In the conventional economic approach we allow the consumer “general” preferences but restrict producer goals to pro±t maximiza-tion. Thus, a consumer who consumes commodities in order to destroy his health is within the scope of our discussion, while a pro-ducer who cares about the welfare of his workers or has in mind a target other than pro±t maximization is not. This is odd since there...
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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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