Chapter Twenty Cost Minimization Cost Minimization u A
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Chapter Twenty Cost Minimization
Cost Minimization u. A firm is a cost-minimizer if it produces any given output level y ³ 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.
Cost Minimization u When the firm faces given input prices w = (w 1, w 2, …, wn) the total cost function will be written as c(w 1, …, wn, y).
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). u Take the output level y ³ 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.
The Cost-Minimization Problem u For given w 1, w 2 and y, the firm’s cost-minimization problem is to solve subject to
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
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?
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 isocost line has the equation
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.
Iso-cost Lines x 2 c” º w 1 x 1+w 2 x 2 c’ < c” x 1
Iso-cost Lines x 2 Slopes = -w 1/w 2. c” º w 1 x 1+w 2 x 2 c’ < c” x 1
The y’-Output Unit Isoquant x 2 All input bundles yielding y’ units of output. Which is the cheapest? f(x 1, x 2) º y’ x 1
The Cost-Minimization Problem x 2 All input bundles yielding y’ units of output. Which is the cheapest? f(x 1, x 2) º y’ x 1
The Cost-Minimization Problem x 2 All input bundles yielding y’ units of output. Which is the cheapest? f(x 1, x 2) º y’ x 1
The Cost-Minimization Problem x 2 All input bundles yielding y’ units of output. Which is the cheapest? f(x 1, x 2) º y’ x 1
The Cost-Minimization Problem x 2 All input bundles yielding y’ units of output. Which is the cheapest? x 2* f(x 1, x 2) º y’ x 1* x 1
The Cost-Minimization Problem x 2 At an interior cost-min input bundle: (a) x 2* f(x 1, x 2) º y’ x 1* x 1
The Cost-Minimization Problem x 2 At an interior cost-min input bundle: (a) and (b) slope of isocost = slope of isoquant x 2* f(x 1, x 2) º y’ x 1* x 1
The Cost-Minimization Problem x 2 At an interior cost-min input bundle: (a) and (b) slope of isocost = slope of isoquant; i. e. x 2* f(x 1, x 2) º y’ x 1* x 1
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?
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) and (b)
A Cobb-Douglas Example of Cost Minimization )a( )b(
A Cobb-Douglas Example of Cost Minimization )a( From (b), )b(
A Cobb-Douglas Example of Cost Minimization )a( )b( From (b), Now substitute into (a) to get
A Cobb-Douglas Example of Cost Minimization )a( )b( From (b), Now substitute into (a) to get
A Cobb-Douglas Example of Cost Minimization )a( )b( From (b), Now substitute into (a) to get So is the firm’s conditional demand for input 1.
A Cobb-Douglas Example of Cost Minimization Since and is the firm’s conditional demand for input 2.
A Cobb-Douglas Example of Cost Minimization So the cheapest input bundle yielding y output units is
Conditional Input Demand Curves Fixed w 1 and w 2.
Conditional Input Demand Curves Fixed w 1 and w 2.
Conditional Input Demand Curves Fixed w 1 and w 2.
Conditional Input Demand Curves Fixed w 1 and w 2.
Conditional Input Demand Curves Fixed w 1 and w 2. output expansion path
Conditional Input Demand Curves Fixed w 1 and w 2. Cond. demand for input 2 output expansion path Cond. demand for input 1
A Cobb-Douglas Example of Cost Minimization For the production function the cheapest input bundle yielding y output units is
A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is
A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is
A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is
A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is
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?
A Perfect Complements Example of Cost Minimization x 2 4 x 1 = x 2 min{4 x 1, x 2} º y’ x 1
A Perfect Complements Example of Cost Minimization x 2 4 x 1 = x 2 min{4 x 1, x 2} º y’ x 1
A Perfect Complements Example of Cost Minimization x 2 4 x 1 = x 2 Where is the least costly input bundle yielding y’ output units? min{4 x 1, x 2} º y’ x 1
A Perfect Complements Example of Cost Minimization x 2 4 x 1 = x 2 Where is the least costly input bundle yielding y’ output units? min{4 x 1, x 2} º y’ x 2* = y x 1* = y/4 x 1
A Perfect Complements Example of Cost Minimization The firm’s production function is and the conditional input demands are and
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
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
Average Total Production Costs u For positive output levels y, a firm’s average total cost of producing y units is
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 2 y’ units of output?
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 2 y’ requires doubling all input levels.
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 2 y’ requires doubling all input levels. u Total production cost doubles.
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 2 y’ requires doubling all input levels. u Total production cost doubles. u Average production cost does not change.
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 2 y’ requires more than doubling all input levels.
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 2 y’ requires more than doubling all input levels. u Total production cost more than doubles.
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 2 y’ requires more than doubling all input levels. u Total production cost more than doubles. u Average production cost increases.
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 2 y’ requires less than doubling all input levels.
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 2 y’ requires less than doubling all input levels. u Total production cost less than doubles.
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 2 y’ requires less than doubling all input levels. u Total production cost less than doubles. u Average production cost decreases.
Returns-to-Scale and Av. Total Costs /$output unit AC(y) decreasing r. t. s. constant r. t. s. increasing r. t. s. y
Returns-to-Scale and Total Costs u What does this imply for the shapes of total cost functions?
Returns-to-Scale and Total Costs $ Av. cost increases with y if the firm’s technology exhibits decreasing r. t. s. c(2 y’) Slope = c(2 y’)/2 y’ = AC(2 y’). Slope = c(y’)/y’ = AC(y’). c(y’) y’ 2 y’ y
Returns-to-Scale and Total Costs $ Av. cost increases with y if the firm’s technology exhibits decreasing r. t. s. c(y) c(2 y’) Slope = c(2 y’)/2 y’ = AC(2 y’). Slope = c(y’)/y’ = AC(y’). c(y’) y’ 2 y’ y
Returns-to-Scale and Total Costs $ c(2 y’) Av. cost decreases with y if the firm’s technology exhibits increasing r. t. s. Slope = c(2 y’)/2 y’ = AC(2 y’). Slope = c(y’)/y’ = AC(y’). c(y’) y’ 2 y’ y
Returns-to-Scale and Total Costs $ c(2 y’) Av. cost decreases with y if the firm’s technology exhibits increasing r. t. s. c(y) Slope = c(2 y’)/2 y’ = AC(2 y’). Slope = c(y’)/y’ = AC(y’). c(y’) y’ 2 y’ y
Returns-to-Scale and Total Costs c(2 y(’ (’ 2 c(y= $ Av. cost is constant when the firm’s technology exhibits constant r. t. s. c(y) Slope = c(2 y’)/2 y’ = 2 c(y’)/2 y’ = c(y’)/y’ so AC(y’) = AC(2 y’). c(y’) y’ 2 y’ y
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?
Short-Run & Long-Run Total Costs u The long-run cost-minimization problem is subject to u The short-run cost-minimization problem is subject to
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.
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”. u But, if the long-run choice for x 2 ¹ 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.
Short-Run & Long-Run Total Costs x 2 Consider three output levels. x 1
Short-Run & Long-Run Total Costs x 2 In the long-run when the firm is free to choose both x 1 and x 2, the least-costly input bundles are. . . x 1
Short-Run & Long-Run Total Costs x 2 Long-run output expansion path x 1
Short-Run & Long-Run Total Costs Long-run costs are: x 2 Long-run output expansion path x 1
Short-Run & Long-Run Total Costs u Now suppose the firm becomes subject to the short-run constraint that x 2 = x 2”.
Short-Run & Long-Run Total Costs x 2 Short-run output expansion path Long-run costs are: x 1
Short-Run & Long-Run Total Costs x 2 Short-run output expansion path Long-run costs are: x 1
Short-Run & Long-Run Total Costs x 2 Short-run output expansion path Long-run costs are: Short-run costs are: x 1
Short-Run & Long-Run Total Costs x 2 Short-run output expansion path Long-run costs are: Short-run costs are: x 1
Short-Run & Long-Run Total Costs x 2 Short-run output expansion path Long-run costs are: Short-run costs are: x 1
Short-Run & Long-Run Total Costs x 2 Short-run output expansion path Long-run costs are: Short-run costs are: x 1
Short-Run & Long-Run Total Costs u Short-run total cost exceeds longrun 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 shortrun total cost curve.
Short-Run & Long-Run Total Costs 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. cs(y) c(y) y
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