Lec 6 Costs

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

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1 Pro°t maximization and cost min- imization of the competitive °rm Competitive °rm is a price taker. What does this mean? Problem of the °rm: ° = 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 ) So we have rewritten the problem of the pro°t maximizing °rm as a two stage problem. Stage 1: For any level 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 , how much output should the °rm produce and sell? 2 Choosing inputs to minimize cost To minimize cost in producing q units, °rm solves: min L;K wL + rK s.t. f ( L; K ) ± q ² 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 = ° w r
² What happens to slope of isocost curve as input prices change? ² Show cost minimizing choice of inputs in producing q : tangency of isocost curve and isoquant. ² Show what happens to choice of inputs when input prices change. All substitution e/ect. Use more of inputs whose relative price fall. ² 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 ² 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 Since °rm has to produce 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 )
So °rm will increase K if reduction in capital cost exceeds increase in labor cost: ° r ° K + w ° L < 0 ° r ° q MPK ( L 0 ; K 0 ) + w ° q MPL ( L 0 ; K 0 ) < 0 wMPK ( L 0 ; K 0 ) ° rMPL ( L 0 ; K 0 ) < 0 Indi/erent if wMPK ( L 0 ; K 0 ) ° rMPL ( L 0 ; K 0 ) = 0 MPK ( L 0 ; K 0 ) r = MPL ( L 0 ; K 0 ) w ²

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