1. Chapter Twenty-One Cost Curves

2. Types of Cost Curves A total cost curve is the graph of a firms total cost function. A variable cost curve is the graph of a firms variable cost function. An average total cost curve is the graph of a firms average total cost function. 3. Types of Cost Curves An average variable cost curve is the graph of a firms average variable cost function. An average fixed cost curve is the graph of a firms average fixed cost function. A marginal cost curve is the graph of a firms marginal cost function. 4. Types of Cost Curves How are these cost curves related to each other? How are a firms long-run and shortrun cost curves related? 5. Fixed, Variable & Total Cost Functions F is the total cost to a firm of its shortrun fixed inputs. F, the firms fixed cost, does not vary with the firms output level. cv(y) is the total cost to a firm of its variable inputs when producing y output units. cv(y) is the firms variable cost function. cv(y) depends upon the levels of the fixed inputs. 6. Fixed, Variable & Total Cost Functions c(y) is the total cost of all inputs, fixed and variable, when producing y output units. c(y) is the firms total cost function;c( y ) = F + c v ( y ). 7. $F y 8. $cv(y)y 9. $cv(y)F y 10. $ c(y) cv(y)c( y ) = F + c v ( y ) FF y 11. Av. Fixed, Av. Variable & Av. Total Cost Curves The firms total cost function is c( y ) = F + c v ( y ). For y > 0, the firms average total cost function isF cv ( y) AC( y ) = + y y = AFC( y ) + AVC( y ). 12. Av. Fixed, Av. Variable & Av. Total Cost Curves What does an average fixed cost curve look like?F AFC( y ) = yAFC(y) is a rectangular hyperbola so its graph looks like ... 13. $/output unitAFC(y) 0 as y AFC(y) 0y 14. Av. Fixed, Av. Variable & Av. Total Cost Curves In a short-run with a fixed amount of at least one input, the Law of Diminishing (Marginal) Returns must apply, causing the firms average variable cost of production to increase eventually. 15. $/output unitAVC(y)0y 16. $/output unitAVC(y) AFC(y) 0y 17. Av. Fixed, Av. Variable & Av. Total Cost Curves AndATC(y) = AFC(y) + AVC(y) 18. $/output unitATC(y) = AFC(y) + AVC(y) ATC(y) AVC(y) AFC(y) 0y 19. $/output unitAFC(y) = ATC(y) - AVC(y) ATC(y) AFCAVC(y) AFC(y)0y 20. $/output unitSince AFC(y) 0 as y , ATC(y) AVC(y) as y .ATC(y) AFCAVC(y) AFC(y)0y 21. $/output unitSince AFC(y) 0 as y , ATC(y) AVC(y) as y .And since short-run AVC(y) must eventually increase, ATC(y) must eventually increase in a short-run. ATC(y) AVC(y) AFC(y) 0y 22. Marginal Cost Function Marginal cost is the rate-of-change of variable production cost as the output level changes. That is, cv ( y) MC( y ) = . y 23. Marginal Cost Function The firms total cost function is c( y ) = F + c v ( y ) and the fixed cost F does not change with the output level y, so c v ( y ) c( y ) MC( y ) = = .yyMC is the slope of both the variable cost and the total cost functions. 24. Marginal and Variable Cost Functions Since MC(y) is the derivative of c v(y), cv(y) must be the integral of MC(y). That is, MC( y ) = c v ( y )yyc v ( y ) = MC( z)dz. 0 25. Marginal and Variable Cost Functions $/output unityc v ( y ) = MC( z)dz 0MC(y)Area is the variable cost of making y units 0 yy 26. Marginal & Average Cost Functions How is marginal cost related to average variable cost? 27. Marginal & Average Cost Functions cv ( y) AVC( y ) = , Since y AVC( y ) y MC( y ) 1 c v ( y ) = . 2 y y 28. Marginal & Average Cost Functions cv ( y) AVC( y ) = , Since y AVC( y ) y MC( y ) 1 c v ( y ) = . 2 y y Therefore, AVC( y ) > =0 y as y MC( y ) = c v ( y ). < 29. Marginal & Average Cost Functions cv ( y) AVC( y ) = , Since y AVC( y ) y MC( y ) 1 c v ( y ) = . 2 y y Therefore, AVC( y ) > =0 y as y MC( y ) = c v ( y ). < > c ( y) AVC( y ) > = 0 as MC( y ) = v = AVC( y ). y < < y 30. Marginal & Average Cost Functions > AVC( y ) > = 0 as MC( y ) = AVC( y ). y < < 31. $/output unitMC(y)AVC(y)y 32. $/output unit AVC( y ) MC( y ) < AVC( y ) AVC( y ) >0 y MC(y)AVC(y)y 34. $/output unit AVC( y ) MC( y ) = AVC( y ) =0 y MC(y)AVC(y)y 35. $/output unit AVC( y ) MC( y ) = AVC( y ) =0 y The short-run MC curve intersects the short-run AVC curve from MC(y) below at the AVC curves minimum. AVC(y)y 36. Marginal & Average Cost Functions c( y ) , Similarly, since ATC( y ) = y ATC( y ) y MC( y ) 1 c( y ) = . 2 y y 37. Marginal & Average Cost Functions c( y ) , Similarly, since ATC( y ) = y ATC( y ) y MC( y ) 1 c( y ) = . 2 y y Therefore, ATC( y ) > =0 y y MC( y ) = c( y ). < 38. Marginal & Average Cost Functions c( y ) , Similarly, since ATC( y ) = y ATC( y ) y MC( y ) 1 c( y ) = . 2 y y Therefore, ATC( y ) > =0 y as y MC( y ) = c( y ). < > c( y ) ATC( y ) > = 0 as MC( y ) = = ATC( y ). y < < y 39. $/output unit> ATC( y ) > = 0 as MC( y ) = ATC( y ) y < < MC(y) ATC(y)y 40. Marginal & Average Cost Functions The short-run MC curve intersects the short-run AVC curve from below at the AVC curves minimum. And, similarly, the short-run MC curve intersects the short-run ATC curve from below at the ATC curves minimum. 41. $/output unitMC(y) ATC(y) AVC(y)y 42. Short-Run & Long-Run Total Cost Curves A firm has a different short-run total cost curve for each possible shortrun circumstance. Suppose the firm can be in one of just three short-runs; x 2 = x 2 or x2 = x2 x2 < x2 < x2. or x2 = x2. 43. $F = w2x2cs(y;x2)F 0y 44. $F = w2x2 F = w2x2cs(y;x2) cs(y;x2 )F F 0y 45. $F = w2x2 F = w2x2 A larger amount of the fixed input increases the firms fixed cost.F F 0cs(y;x2) cs(y;x2 )y 46. $F = F2x2 w = w2x2 A larger amount of the fixed input increases the firms fixed cost.cs(y;x2) cs(y;x2 )Why does a larger amount of the fixed input reduce the slope of the firms total cost curve?F F 0y 47. Short-Run & Long-Run Total Cost Curves MP1 is the marginal physical productivity of the variable input 1, so one extra unit of input 1 gives MP1 extra output units. Therefore, the extra amount of input 1 needed for 1 extra output unit is 48. Short-Run & Long-Run Total Cost Curves MP1 is the marginal physical productivity of the variable input 1, so one extra unit of input 1 gives MP1 extra output units. Therefore, the extra amount of input 1 needed for 1 extra output unit is 1 / MP1 units of input 1. 49. Short-Run & Long-Run Total Cost Curves MP1 is the marginal physical productivity of the variable input 1, so one extra unit of input 1 gives MP1 extra output units. Therefore, the extra amount of input 1 needed for 1 extra output unit is 1 / MP1 units of input 1. Each unit of input 1 costs w 1, so the firms extra cost from producing one extra unit of output is 50. Short-Run & Long-Run Total Cost Curves MP1 is the marginal physical productivity of the variable input 1, so one extra unit of input 1 gives MP1 extra output units. Therefore, the extra amount of input 1 needed for 1 extra output unit is 1 / MP1 units of input 1. Each unit of input 1 costs w 1, so the firms extra cost from producing one extra unit w1 of output is MC = . MP1 51. Short-Run & Long-Run Total Cost Curves w1 MC = is the slope of the firms total MP1 cost curve. 52. Short-Run & Long-Run Total Cost Curves w1 MC = is the slope of the firms total MP1 cost curve. If input 2 is a complement to input 1 then MP1 is higher for higher x2. Hence, MC is lower for higher x2. That is, a short-run total cost curve starts higher and has a lower slope if x2 is larger. 53. $F = F2x2 w = F 2 w2x = w2x2 F F F 0cs(y;x2) cs(y;x2 ) cs(y;x2 )y 54. Short-Run & Long-Run Total Cost Curves The firm has three short-run total cost curves. In the long-run the firm is free to choose amongst these three since it is free to select x2 equal to any of x2, x2, or x2. How does the firm make this choice? 55. $For 0 y y, choose x2 = ?cs(y;x2) cs(y;x2 ) cs(y;x2 )F F F 0yyy 56. $For 0 y y, choose x2 = x2. cs(y;x2) cs(y;x2 ) cs(y;x2 )F F F 0yyy 57. $For 0 y y, choose x2 = x2. cs(y;x2) For y y y, choose x2 = ? cs(y;x2 ) cs(y;x2 )F F F 0yyy 58. $For 0 y y, choose x2 = x2. cs(y;x2) For y y y, choose x2 = x2. cs(y;x2 ) cs(y;x2 )F F F 0yyy 59. $For 0 y y, choose x2 = x2. cs(y;x2) For y y y, choose x2 = x2. For y < y, choose x2 = ? cs(y;x2 ) cs(y;x2 F ) F F 0 y y y 60. $For 0 y y, choose x2 = x2. cs(y;x2) For y y y, choose x2 = x2. For y < y, choose x2 = x2. cs(y;x2 ) cs(y;x2 F ) F F 0 y y y 61. $For 0 y y, choose x2 = x2. cs(y;x2) For y y y, choose x2 = x2. For y < y, choose x2 = x2. cs(y;x2 ) cs(y;x2 c(y), the ) F firms long run total F cost curve. F 0 y y y 62. Short-Run & Long-Run Total Cost Curves The firms long-run total cost curve consists of the lowest parts of the short-run total cost curves. The long-run total cost curve is the lower envelope of the short-run total cost curves. 63. Short-Run & Long-Run Total Cost Curves If input 2 is available in continuous amounts then there is an infinity of short-run total cost curves but the long-run total cost curve is still the lower envelope of all of the short-run total cost curves. 64. $cs(y;x2)cs(y;x2 )F F F 0cs(y;x2 ) c(y)y 65. Short-Run & Long-Run Average Total Cost Curves For any output level y, the long-run total cost curve always gives the lowest possible total production cost. Therefore, the long-run av. total cost curve must always give the lowest possible av. total production cost. The long-run av. total cost curve must be the lower envelope of all of the firms short-run av. total cost curves. 66. Short-Run & Long-Run Average Total Cost Curves E.g. suppose again that the firm can be in one of just three short-runs; x 2 = x 2 or x2 = x2 (x2 < x2 < x2) or x2 = x2 then the firms three short-run average total cost curves are ... 67. $/output unitACs(y;x2) ACs(y;x2) ACs(y;x2) y 68. Short-Run & Long-Run Average Total Cost Curves The firms long-run average total cost curve is the lower envelope of the short-run average total cost curves ... 69. $/output unitACs(y;x2) ACs(y;x2) ACs(y;x2) The long-run av. total cost AC(y) curve is the lower envelope of the short-run av. total cost curves. y 70. Short-Run & Long-Run Marginal Cost Curves Q: Is the long-run marginal cost curve the lower envelope of the firms short-run marginal cost curves? 71. Short-Run & Long-Run Marginal Cost Curves Q: Is the long-run marginal cost curve the lower envelope of the firms short-run marginal cost curves? A: No. 72. Short-Run & Long-Run Marginal Cost Curves The firms three short-run average total cost curves are ... 73. $/output unitACs(y;x2) ACs(y;x2) ACs(y;x2)y 74. $/output unitMCs(y;x2)MCs(y;x2)ACs(y;x2) ACs(y;x2) ACs(y;x2)MCs(y;x2)y 75. $/output unitMCs(y;x2)MCs(y;x2)ACs(y;x2) ACs(y;x2) ACs(y;x2)MCs(y;x2)AC(y)y 76. $/output unitMCs(y;x2)MCs(y;x2)ACs(y;x2) ACs(y;x2) ACs(y;x2)MCs(y;x2)AC(y)y 77. $/output unitMCs(y;x2)MCs(y;x2)ACs(y;x2) ACs(y;x2) ACs(y;x2)MCs(y;x2) MC(y), the long-run marginal cost curve. y 78. Short-Run & Long-Run Marginal Cost Curves For any output level y > 0, the longrun marginal cost of production is the marginal cost of production for the short-run chosen by the firm. 79. $/output unitMCs(y;x2)MCs(y;x2)ACs(y;x2) ACs(y;x2) ACs(y;x2)MCs(y;x2) MC(y), the long-run marginal cost curve. y 80. Short-Run & Long-Run Marginal Cost Curves For any output level y > 0, the longrun marginal cost is the marginal cost for the short-run chosen by the firm. This is always true, no matter how many and which short-run circumstances exist for the firm. 81. Short-Run & Long-Run Marginal Cost Curves For any output level y > 0, the longrun marginal cost is the marginal cost for the short-run chosen by the firm. So for the continuous case, where x 2 can be fixed at any value of zero or more, the relationship between the long-run marginal cost and all of the short-run marginal costs is ... 82. Short-Run & Long-Run Marginal Cost Curves $/output unitSRACs AC(y)y 83. Short-Run & Long-Run Marginal Cost Curves $/output unitSRMCs AC(y)y 84. Short-Run & Long-Run Marginal Cost Curves $/output unitSRMCsMC(y) AC(y)y For each y > 0, the long-run MC equals the MC for the short-run chosen by the firm.