Lec 6 Costs - 1 Prot maximization and cost minimization of...

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1 Pro&t maximization and cost min- = max x 1 ;x 2 pf ( x 1 ;x 2 ) w 1 x 1 w 2 x 2 = max q;x 1 ;x 2 pq ( w 1 x 1 + w 2 x 2 ) s.t. q = f ( x 1 ;x 2 ) which may also be rewritten as = max q pq c ( w 1 ;w 2 ;q ) c ( w 1 ;w 2 ;q ) = min x 1 ;x 2 w 1 x 1 w 2 x 2 s.t. q = f ( x 1 ;x 2 ) of output q and input prices w 1 and w 2 , what is the cheapest cost of producing it? This cheapest cost can be
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written as a cost function c ( w 1 ;w 2 ;q ) . Stage 2: Given a cost function c ( w 1 ;w 2 ;q ) and the price of output p , 2 Choosing inputs to minimize cost To minimize cost in producing q min L;K wL + rK s.t. f ( L;K ) q ± Iso cost curve: Combinations of L and K which gives the same cost C . C = wL + rK C = w L + r K 0 = w L + r K K L = ² w r
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What happens to slope of isocost curve as input prices change? Show cost minimizing choice of inputs in producing q : tangency of isocost curve and isoquant. Show what happens to choice of inputs when input prices change. All substitution e/ect. Use more of inputs whose relative price fall. At the optimum, the marginal rate of technical sub- stitution of labor for capital, is w r = MRTS ( L;K ) = MPL ( L;K ) MPK ( L;K ) MPL ( L;K ) w = MPK ( L;K ) r
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2.1 Derive optimality condition Consider an initial choice of K and L , K 0 and L 0 which produces q . Now decrease K 0 by K . Reduction in capital cost C k = ± r K Output will fall by q = MPK ( L 0 ;K 0 K q , it will have to increase labor by: MPL ( L 0 ;K 0 L q L = q MPL ( L 0 ;K 0 ) Increase in labor cost C L = w L = w q MPL ( L 0 ;K 0 )
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K if reduction in capital cost exceeds increase in labor cost: r K + w L <
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Lec 6 Costs - 1 Prot maximization and cost minimization of...

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