Varian_Chapter20_Cost_Minimization

# Varian_Chapter20_Cost_Minimization - Chapter Twenty Cost...

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Chapter Twenty Cost Minimization

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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 .
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).

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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 min , xx wx w x 12 0 11 2 2 + subject to fx x y (, ) . =

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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 cw w y wx w w y w w y (, , ) , ) , ) . * * 12 1 112 22 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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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 wx w x 11 2 2 100 + = .
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 =− + . wx w x c 11 2 2 + =

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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
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 .

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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’

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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’

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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 *
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) fx x y (, ) ** 12 =

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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 fx x y (, ) ** 12 =
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; i.e.

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Varian_Chapter20_Cost_Minimization - Chapter Twenty Cost...

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