Presentation is loading. Please wait.

Presentation is loading. Please wait.

Chapter Twenty Cost Minimization.  A firm is a cost-minimizer if it produces any given output level y  0 at smallest possible total cost. u c(y) denotes.

Similar presentations


Presentation on theme: "Chapter Twenty Cost Minimization.  A firm is a cost-minimizer if it produces any given output level y  0 at smallest possible total cost. u c(y) denotes."— Presentation transcript:

1 Chapter Twenty Cost Minimization

2  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.

3 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).

4 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.

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

6 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

7 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?

8 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

9 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.

10 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

11 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.

12 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’

13 The Cost-Minimization Problem x1x1 x2x2 All input bundles yielding y’ units of output. Which is the cheapest? f(x 1,x 2 )  y’

14 The Cost-Minimization Problem x1x1 x2x2 All input bundles yielding y’ units of output. Which is the cheapest? f(x 1,x 2 )  y’

15 The Cost-Minimization Problem x1x1 x2x2 All input bundles yielding y’ units of output. Which is the cheapest? f(x 1,x 2 )  y’

16 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*

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

18 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

19 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.

20 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?

21 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

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

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

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

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

26 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.

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

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

29 Fixed w 1 and w 2. Conditional Input Demand Curves

30 Fixed w 1 and w 2. Conditional Input Demand Curves

31 Fixed w 1 and w 2. Conditional Input Demand Curves

32 Fixed w 1 and w 2. Conditional Input Demand Curves

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

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

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

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

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

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

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

40 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?

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

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

43 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?

44 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?

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

46 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

47 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

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

49 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?

50 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.

51 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.

52 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.

53 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.

54 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.

55 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.

56 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.

57 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.

58 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.

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

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

61 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.

62 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.

63 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.

64 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.

65 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.

66 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?

67 Short-Run & Long-Run Total Costs u The long-run cost-minimization problem is u The short-run cost-minimization problem is subject to

68 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.

69 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.

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

71 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...

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

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

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

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

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

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

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

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

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

81 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.

82 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.


Download ppt "Chapter Twenty Cost Minimization.  A firm is a cost-minimizer if it produces any given output level y  0 at smallest possible total cost. u c(y) denotes."

Similar presentations


Ads by Google