20s - © 2010 W. W. Norton & Company, Inc. 20...

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Unformatted text preview: © 2010 W. W. Norton & Company, Inc. 20 Cost Minimization © 2010 W. W. Norton & Company, Inc. 2 Cost Minimization A firm is a cost-minimizer if it produces any given output level y 0 at smallest possible total cost. c(y) denotes the firm’s smallest possible total cost for producing y units of output. c(y) is the firm’s total cost function. © 2010 W. W. Norton & Company, Inc. 3 Cost Minimization When the firm faces given input prices w = (w 1 ,w 2 ,…,w n ) the total cost function will be written as c(w 1 ,…,w n ,y). © 2010 W. W. Norton & Company, Inc. 4 The Cost-Minimization Problem Consider a firm using two inputs to make one output. The production function is y = f(x 1 ,x 2 ). Take the output level y 0 as given. Given the input prices w 1 and w 2 , the cost of an input bundle (x 1 ,x 2 ) is w 1 x 1 + w 2 x 2 . © 2010 W. W. Norton & Company, Inc. 5 The Cost-Minimization Problem For given w 1 , w 2 and y, the firm’s cost-minimization problem is to solve min , x x w x w x 1 2 1 1 2 2 subject to f x x y ( , ) . 1 2 © 2010 W. W. Norton & Company, Inc. 6 The Cost-Minimization Problem The levels x 1 *(w 1 ,w 2 ,y) and x 1 *(w 1 ,w 2 ,y) in the least-costly input bundle are the firm’s conditional demands for inputs 1 and 2. The (smallest possible) total cost for producing y output units is therefore c w w y w x w w y w x w w y ( , , ) ( , , ) ( , , ). * * 1 2 1 1 1 2 2 2 1 2 © 2010 W. W. Norton & Company, Inc. 7 Conditional Input Demands Given w 1 , w 2 and y, how is the least costly input bundle located? And how is the total cost function computed? © 2010 W. W. Norton & Company, Inc. 8 Iso-cost Lines A curve that contains all of the input bundles that cost the same amount is an iso-cost curve. E.g., given w 1 and w 2 , the $100 iso- cost line has the equation w x w x 1 1 2 2 100 . © 2010 W. W. Norton & Company, Inc. 9 Iso-cost Lines Generally, given w 1 and w 2 , the equation of the $c iso-cost line is i.e. Slope is - w 1 /w 2 . x w w x c w 2 1 2 1 2 . w x w x c 1 1 2 2 © 2010 W. W. Norton & Company, Inc. 10 Iso-cost Lines c’ w 1 x 1 +w 2 x 2 c” w 1 x 1 +w 2 x 2 c’ < c” x 1 x 2 Slopes = -w 1 /w 2 . © 2010 W. W. Norton & Company, Inc. 11 The Cost-Minimization Problem x 1 x 2 All input bundles yielding y’ units of output. Which is the cheapest? f(x 1 ,x 2 ) y’ © 2010 W. W. Norton & Company, Inc. 12 The Cost-Minimization Problem x 1 x 2 All input bundles yielding y’ units of output. Which is the cheapest? f(x 1 ,x 2 ) y’ © 2010 W. W. Norton & Company, Inc. 13 The Cost-Minimization Problem x 1 x 2 All input bundles yielding y’ units of output. Which is the cheapest?...
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20s - © 2010 W. W. Norton & Company, Inc. 20...

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