k v w q w q 3 k vk I With variable k C v w q 2 q 3 2 p vw I Note C v w q \u00b3 SC k

# K v w q w q 3 k vk i with variable k c v w q 2 q 3 2

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k ( v , w , q ) = w q 3 k + vk I With variable k , C ( v , w , q ) = 2 q 3 2 p vw I Note C ( v , w , q ) ³ SC k ( v , w , q ) for all q with equality only if k = k ± ( v , w , q ) = q 3 2 ³ w v ´ 1 2
Back to Cobb Douglas Example (1) 4 . pdf q 0 1 2 3 4 5 Cost 0 2 4 6 8 10 12 14 16 18 20 22 o o o C(q) SC 1 (q) SC 8 (q)
Algebraic Example with U-Shaped Costs (2) 5 . pdf q Cost C SC q AC, MC AC MC
Properties of Cost Function: holding q °xed I How does C ( v , w , q ) behave when we vary ( v , w ) holding q constant? I Properties similar to consumer±s expenditure function 1. Homogeneous of degree 1 2. Concave 3. Shepherd±s Lemma: cost-minimizing input choices are given by l ± ( v , w , q ) = c / w , k ± ( v , w , q ) = c / v
Properties of Cost Function: holding q °xed (4) . pdf vk a + wl a C(v,w a ,q a ) C w w a