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Chapter Twenty Cost Minimization

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A firm is a cost-minimizer if it produces any given output level y 0 at smallest possible total cost. u c(y) denotes the firm’s smallest possible total cost for producing y units of output. u c(y) is the firm’s total cost function.

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Cost Minimization u 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 u Consider a firm using two inputs to make one output. u The production function is y = f(x 1,x 2 ). Take the output level y 0 as given. u 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.

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The Cost-Minimization Problem u For given w 1, w 2 and y, the firm’s cost-minimization problem is to solve subject to

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The Cost-Minimization Problem u 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. u The (smallest possible) total cost for producing y output units is therefore

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Conditional Input Demands u Given w 1, w 2 and y, how is the least costly input bundle located? u And how is the total cost function computed?

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Iso-cost Lines u A curve that contains all of the input bundles that cost the same amount is an iso-cost curve. u E.g., given w 1 and w 2, the $100 iso- cost line has the equation

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Iso-cost Lines u Generally, given w 1 and w 2, the equation of the $c iso-cost line is i.e. u Slope is - w 1 /w 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” x1x1 x2x2

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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” x1x1 x2x2 Slopes = -w 1 /w 2.

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The y’-Output Unit Isoquant x1x1 x2x2 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 x1x1 x2x2 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 x1x1 x2x2 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 x1x1 x2x2 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 x1x1 x2x2 All input bundles yielding y’ units of output. Which is the cheapest? f(x 1,x 2 ) y’ x1*x1* x2*x2*

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The Cost-Minimization Problem x1x1 x2x2 f(x 1,x 2 ) y’ x1*x1* x2*x2* At an interior cost-min input bundle: (a)

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The Cost-Minimization Problem x1x1 x2x2 f(x 1,x 2 ) y’ x1*x1* x2*x2* At an interior cost-min input bundle: (a) and (b) slope of isocost = slope of isoquant

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The Cost-Minimization Problem x1x1 x2x2 f(x 1,x 2 ) y’ x1*x1* x2*x2* At an interior cost-min input bundle: (a) and (b) slope of isocost = slope of isoquant; i.e.

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A Cobb-Douglas Example of Cost Minimization u A firm’s Cobb-Douglas production function is u Input prices are w 1 and w 2. u What are the firm’s conditional input demand functions?

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A Cobb-Douglas Example of Cost Minimization At the input bundle (x 1 *,x 2 *) which minimizes the cost of producing y output units: (a) (b) and

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A Cobb-Douglas Example of Cost Minimization (a)(b)

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A Cobb-Douglas Example of Cost Minimization (a)(b) From (b),

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A Cobb-Douglas Example of Cost Minimization (a)(b) From (b), Now substitute into (a) to get

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A Cobb-Douglas Example of Cost Minimization (a)(b) From (b), Now substitute into (a) to get

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A Cobb-Douglas Example of Cost Minimization (a)(b) From (b), Now substitute into (a) to get Sois the firm’s conditional demand for input 1.

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A Cobb-Douglas Example of Cost Minimization is the firm’s conditional demand for input 2. Sinceand

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A Cobb-Douglas Example of Cost Minimization So the cheapest input bundle yielding y output units is

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Fixed w 1 and w 2. Conditional Input Demand Curves

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Fixed w 1 and w 2. Conditional Input Demand Curves

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Fixed w 1 and w 2. Conditional Input Demand Curves

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Fixed w 1 and w 2. Conditional Input Demand Curves

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Fixed w 1 and w 2. Conditional Input Demand Curves output expansion path

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Fixed w 1 and w 2. Conditional Input Demand Curves output expansion path Cond. demand for input 2 Cond. demand for input 1

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A Cobb-Douglas Example of Cost Minimization For the production function the cheapest input bundle yielding y output units is

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A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is

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A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is

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A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is

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A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is

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A Perfect Complements Example of Cost Minimization u The firm’s production function is u Input prices w 1 and w 2 are given. u What are the firm’s conditional demands for inputs 1 and 2? u What is the firm’s total cost function?

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A Perfect Complements Example of Cost Minimization x1x1 x2x2 min{4x 1,x 2 } y’ 4x 1 = x 2

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A Perfect Complements Example of Cost Minimization x1x1 x2x2 4x 1 = x 2 min{4x 1,x 2 } y’

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A Perfect Complements Example of Cost Minimization x1x1 x2x2 4x 1 = x 2 min{4x 1,x 2 } y’ Where is the least costly input bundle yielding y’ output units?

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A Perfect Complements Example of Cost Minimization x1x1 x2x2 x 1 * = y/4 x 2 * = y 4x 1 = x 2 min{4x 1,x 2 } y’ Where is the least costly input bundle yielding y’ output units?

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A Perfect Complements Example of Cost Minimization The firm’s production function is and the conditional input demands are and

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A Perfect Complements Example of Cost Minimization The firm’s production function is and the conditional input demands are and So the firm’s total cost function is

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A Perfect Complements Example of Cost Minimization The firm’s production function is and the conditional input demands are and So the firm’s total cost function is

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Average Total Production Costs u For positive output levels y, a firm’s average total cost of producing y units is

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Returns-to-Scale and Av. Total Costs u The returns-to-scale properties of a firm’s technology determine how average production costs change with output level. u Our firm is presently producing y’ output units. u 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 u If a firm’s technology exhibits constant returns-to-scale then doubling its output level from y’ to 2y’ requires doubling all input levels.

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Constant Returns-to-Scale and Average Total Costs u If a firm’s technology exhibits constant returns-to-scale then doubling its output level from y’ to 2y’ requires doubling all input levels. u Total production cost doubles.

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Constant Returns-to-Scale and Average Total Costs u If a firm’s technology exhibits constant returns-to-scale then doubling its output level from y’ to 2y’ requires doubling all input levels. u Total production cost doubles. u Average production cost does not change.

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Decreasing Returns-to-Scale and Average Total Costs u 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.

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Decreasing Returns-to-Scale and Average Total Costs u 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. u Total production cost more than doubles.

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Decreasing Returns-to-Scale and Average Total Costs u 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. u Total production cost more than doubles. u Average production cost increases.

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Increasing Returns-to-Scale and Average Total Costs u 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.

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Increasing Returns-to-Scale and Average Total Costs u 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. u Total production cost less than doubles.

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Increasing Returns-to-Scale and Average Total Costs u 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. u Total production cost less than doubles. u Average production cost decreases.

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Returns-to-Scale and Av. Total Costs y $/output unit constant r.t.s. decreasing r.t.s. increasing r.t.s. AC(y)

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Returns-to-Scale and Total Costs u What does this imply for the shapes of total cost functions?

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Returns-to-Scale and Total Costs y $ y’ 2y’ c(y’) c(2y’) Slope = c(2y’)/2y’ = AC(2y’). Slope = c(y’)/y’ = AC(y’). Av. cost increases with y if the firm’s technology exhibits decreasing r.t.s.

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Returns-to-Scale and Total Costs y $ c(y) y’ 2y’ c(y’) c(2y’) Slope = c(2y’)/2y’ = AC(2y’). Slope = c(y’)/y’ = AC(y’). Av. cost increases with y if the firm’s technology exhibits decreasing r.t.s.

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Returns-to-Scale and Total Costs y $ y’ 2y’ c(y’) c(2y’) Slope = c(2y’)/2y’ = AC(2y’). Slope = c(y’)/y’ = AC(y’). Av. cost decreases with y if the firm’s technology exhibits increasing r.t.s.

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Returns-to-Scale and Total Costs y $ c(y) y’ 2y’ c(y’) c(2y’) Slope = c(2y’)/2y’ = AC(2y’). Slope = c(y’)/y’ = AC(y’). Av. cost decreases with y if the firm’s technology exhibits increasing r.t.s.

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Returns-to-Scale 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’). Av. cost is constant when the firm’s technology exhibits constant r.t.s.

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Short-Run & Long-Run Total Costs u In the long-run a firm can vary all of its input levels. u Consider a firm that cannot change its input 2 level from x 2 ’ units. u 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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Short-Run & Long-Run Total Costs u The long-run cost-minimization problem is u The short-run cost-minimization problem is subject to

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Short-Run & Long-Run Total Costs u The short-run cost-min. problem is the long-run problem subject to the extra constraint that x 2 = x 2 ’. u If the long-run choice for x 2 was x 2 ’ then the extra constraint x 2 = x 2 ’ is not really a constraint at all and so the long-run and short-run total costs of producing y output units are the same.

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Short-Run & Long-Run Total Costs u The short-run cost-min. problem is therefore the long-run problem subject to the extra constraint that x 2 = x 2 ”. But, if the long-run choice for x 2 x 2 ” then the extra constraint x 2 = x 2 ” prevents the firm in this short-run from achieving its long-run production cost, causing the short-run total cost to exceed the long-run total cost of producing y output units.

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Short-Run & Long-Run Total Costs x1x1 x2x2 Consider three output levels.

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Short-Run & Long-Run Total Costs x1x1 x2x2 In the long-run when the firm is free to choose both x 1 and x 2, the least-costly input bundles are...

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Short-Run & Long-Run Total Costs x1x1 x2x2 Long-run output expansion path

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Short-Run & Long-Run Total Costs x1x1 x2x2 Long-run output expansion path Long-run costs are:

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Short-Run & Long-Run Total Costs u Now suppose the firm becomes subject to the short-run constraint that x 2 = x 2 ”.

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Short-Run & Long-Run Total Costs x1x1 x2x2 Short-run output expansion path Long-run costs are:

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Short-Run & Long-Run Total Costs x1x1 x2x2 Short-run output expansion path Long-run costs are:

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Short-Run & Long-Run Total Costs x1x1 x2x2 Short-run output expansion path Long-run costs are: Short-run costs are:

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Short-Run & Long-Run Total Costs x1x1 x2x2 Short-run output expansion path Long-run costs are: Short-run costs are:

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Short-Run & Long-Run Total Costs x1x1 x2x2 Short-run output expansion path Long-run costs are: Short-run costs are:

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Short-Run & Long-Run Total Costs x1x1 x2x2 Short-run output expansion path Long-run costs are: Short-run costs are:

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Short-Run & Long-Run Total Costs u Short-run total cost exceeds long-run total cost except for the output level where the short-run input level restriction is the long-run input level choice. u This says that the long-run total cost curve always has one point in common with any particular short- run total cost curve.

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Short-Run & Long-Run Total Costs y $ c(y) c s (y) A short-run total cost curve always has one point in common with the long-run total cost curve, and is elsewhere higher than the long-run total cost curve.

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