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cost_func - 5 Cost Function c(w y = min wx x s.t f(x = y...

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5 Cost Function c ( w , y ) = min x wx s.t. f ( x ) = y. 5.1 Properties 1. Nondecreasing in w . For w 0 w , c ( w 0 , y ) c ( w , y ) . 2. Homogeneous of degree 1 in w : for t > 0, c ( t w , y ) = tc ( w , y ) . 3. Concave in w : for w 00 = t w +(1 t ) w 0 , t [0 , 1], c ( w 00 , y ) tc ( w , y ) + (1 t ) c ( w 0 , y ) . 4. Continuous in w for w À 0 (when de fi ned).
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5.2 Conditional Demand Sheppard’s Lemma: For all i = 1 , . . . , n , for w i > 0, x i ( w ) = ∂c ( w , y ) ∂w i . Proof: Special case of Envelope Theorem. Envelope Theorem (constrained). For the problem M ( a ) = max z f ( z , a ) , s.t. h ( z , a ) = 0 . we have L = f ( z , a ) λh ( z , a ) , dM ( a ) da = L ∂a ¯ ¯ ¯ ¯ z = z ( a ) = ∂f ( x, a ) ∂a ¯ ¯ ¯ ¯ ¯ x = x ( a ) λ ∂h ( x, a ) ∂a ¯ ¯ ¯ ¯ ¯ x = x ( a ) . Application: ∂c ( w , y ) ∂y = λ – shadow cost. 5.3 Comparative statics. (revisited ? ) 1. c ( w , y ) is nondecreasing in w , x i ( w ) = ∂c ( w , y ) ∂w i 0 . 2. x i ( w ) are homogeneous of degree 0. 3. c ( w , y ) is concave, so à ∂x i ( w , y ) ∂w j ! = à 2 c ( w , y ) ∂w i ∂w j ! –symm., neg.-def.
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5.4 Average and Marginal costs. Example: Boston-Chicago
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