Chapter 21 - Cost functions for the common technologies...

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Cost functions for the common technologies Perfect substitutes (1 to 1) – min{w, r}y Perfect complements (1 to 1) – (w+r)y Cobb-Douglas – derive for a+b=1
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Average Production Costs For positive output levels y, a firm’s average cost of producing y units is AC w w y c w w y y ( , , ) ( , , ) . 1 2 1 2 =
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Returns-to-Scale and Average Costs The returns-to-scale properties of a firm’s technology determine how average production costs change with output level. Our firm is presently producing y’ output units. How does the firm’s average production cost change if it instead produces 2y’ units of output?
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Constant Returns-to-Scale and Average Total Costs If a firm’s technology exhibits constant returns-to-scale then doubling its output level from y’ to 2y’ requires doubling all input levels. Total production cost doubles. Average production cost does not change.
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Decreasing Returns-to-Scale and Average Costs If a firm’s technology exhibits decreasing returns-to-scale then doubling its output level from y’ to 2y’ requires more than doubling all input levels. Total production cost more than doubles. Average production cost increases.
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Increasing Returns-to-Scale and Average Total Costs If a firm’s technology exhibits increasing returns-to-scale then doubling its output level from y’ to 2y’ requires less than doubling all input levels. Total production cost less than doubles. Average production cost decreases.
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Returns-to-Scale and Average Costs y $/output unit constant returns decreasing returns increasing returns AC(y)
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R.T.S. and Total Costs What does this imply for the shapes of total cost functions?
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R.T.S. and Total Costs y $ c(y) y’ 2y’ c(y’) c(2y’) Slope = c(2y’)/2y’ = AC(2y’). Slope = c(y’)/y’ = AC(y’). decreasing returns
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R.T.S. and Total Costs y $ c(y) y’ 2y’ c(y’) c(2y’) Slope = c(2y’)/2y’ = AC(2y’). Slope = c(y’)/y’ = AC(y’). increasing returns
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R.T.S. and Total Costs y $ c(y) y’ 2y’ c(y’) c(2y’) =2c(y’) Slope = c(2y’)/2y’ = 2c(y’)/2y’ = c(y’)/y’ so AC(y’) = AC(2y’). constant returns
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Short-Run & Long-Run Total Costs In the long-run a firm can vary all of its input levels. Consider a firm that cannot change its input 2 level from x 2 ’ units. How does the short-run total cost of producing y output units compare to the long-run total cost of producing y units of output?
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The long-run cost-minimization problem is The short-run cost-minimization problem is min , x x w x w x 1 2 0 1 1 2 2 + subject to f x x y ( , ) . 1
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Chapter 21 - Cost functions for the common technologies...

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