Week8 Handout (6pp)(1)

Week8 Handout (6pp)(1) - 9/16/2011 Cost Minimization Cost...

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9/16/2011 1 1 Cost minimisation & cost curves Chapters 20 and 21 Cost Minimization A firm is a cost-minimiser 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. 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). 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 . The Cost-Minimization Problem For given w 1 , w 2 and y, the firm’s cost- minimization problem is to solve subject to: min , x x w x w x 1 2 0 1 1 2 2 f x x y ( , ) . 1 2 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 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?
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9/16/2011 2 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 . Iso-cost Lines Generally, given w 1 and w 2 , the equation of the $c iso-cost line is that is: 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 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 . The y’-Output Unit Isoquant x 1 x 2 All input bundles yielding y’ units of output. Which is the cheapest? f(x 1 ,x 2 ) y’ 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’ 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’ x 1 * x 2 *
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9/16/2011 3 The Cost-Minimization Problem x 1 x 2 f(x 1 ,x 2 ) y’ x 1 * x 2 * At an interior cost-min input bundle: (a) and (b) slope of isocost = slope of isoquant f x x y ( , ) * * 1 2   w w TRS MP MP at x x 1 2 1 2 1 2 ( , ). * * A Cobb-Douglas Example A firm’s Cobb-Douglas production function is Input prices are w 1 and w 2 .
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Week8 Handout (6pp)(1) - 9/16/2011 Cost Minimization Cost...

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