Lecture # 12 Cost Curves Lecturer: Martin Paredes

1. Lecture # 12 Cost Curves Lecturer: Martin Paredes

2. Outline • Long Run Cost Functions • Shifts • Average and Marginal Cost Functions • Economies of Scale • Deadweight Loss • Long Run Cost Functions • Relationship between Long Run and Short Run Cost Functions

3. Long Run Cost Function Definition: The long run total cost function relates the minimized total cost to output (Q) and the factor prices (w and r). TC(Q,w,r) = wL*(Q,w,r) + r K*(Q,w,r) where L* and K* are the long run input demand functions

4. Long Run Cost Function Example: Long Run Total Cost Function • Suppose Q = 50L0.5K0.5 • We found: L*(Q,w,r) = Q . r 0.5 50 w K*(Q,w,r) = Q . w 0.5 50 r • Then TC(Q,w,r) = wL*(Q,w,r) + rK*(Q,w,r) = Q . (wr)0.5 25 ( ) ( )

5. Long Run Cost Curve Definition: The long run total cost curve shows the minimized total cost as output (Q) varies, holding input prices (w and r) constant.

6. Example: Long Run Cost Curve • Recall TC(Q,w,r) = Q . (wr)0.5 25 • What if r = 100 and w = 25? TC(Q,w,r) = Q . (25100)0.5 25 = 2Q

7. TC (€ per year) Example: A Total Cost Curve TC(Q) = 2Q Q (units per year)

8. TC (€ per year) Example: A Total Cost Curve TC(Q) = 2Q €2M. 1 M. Q (units per year)

9. TC (€ per year) Example: A Total Cost Curve TC(Q) = 2Q €4M. €2M. 1 M. 2 M. Q (units per year)

10. Long Run Cost Curve • We will observe a movement along the long run cost curve when output (Q) varies. • We will observe a shift in the long run cost curve when any variable other than output (Q) varies.

11. K Example: Movement Along LRTC Q0 TC = TC0 • K0 0 L (labour services per year) L0

12. K Example: Movement Along LRTC Q0 TC = TC0 • K0 0 L (labour services per year) L0 TC (€/yr) LR Total Cost Curve • TC0=wL0+rK0 Q (units per year) 0 Q0

13. K Example: Movement Along LRTC Q1 Q0 • TC = TC0 K1 • K0 TC = TC1 0 L (labour services per year) L0 L1 TC (€/yr) LR Total Cost Curve • TC0=wL0+rK0 Q (units per year) 0 Q0

14. K Example: Movement Along LRTC Q1 Q0 • TC = TC0 K1 • K0 TC = TC1 0 L (labour services per year) L0 L1 TC (€/yr) LR Total Cost Curve • TC1=wL1+rK1 • TC0=wL0+rK0 Q (units per year) Q1 0 Q0

15. Long Run Cost Curve Example: Shift of the long run cost curve • Suppose there is an increase in wages but the price of capital remains fixed.

16. K Example: A Change in the Price of an Input Q0 0 L

17. K Example: A Change in the Price of an Input TC0/r A • Q0 -w0/r 0 L

18. K Example: A Change in the Price of an Input TC0/r A • Q0 -w1/r -w0/r 0 L

19. K Example: A Change in the Price of an Input TC1/r B TC1 > TC0 • TC0/r A • Q0 -w1/r -w0/r 0 L

20. TC (€/yr) Example: A Shift in the Total Cost Curve TC(Q) ante Q (units/yr)

21. TC (€/yr) Example: A Shift in the Total Cost Curve TC(Q) ante • TC0 Q0 Q (units/yr)

22. TC (€/yr) Example: A Shift in the Total Cost Curve TC(Q) post • TC(Q) ante TC1 • TC0 Q0 Q (units/yr)

23. Long Run Average Cost Function Definition: The long run average cost curve indicates the firm’s cost per unit of output. • It is simply the long run total cost function divided by output. AC(Q,w,r) = TC(Q,w,r) Q

24. Long Run Marginal Cost Function Definition: The long run marginal cost curve measures the rate of change of total cost as output varies, holding all input prices constant. MC(Q,w,r) = TC(Q,w,r) Q

25. Example: Average and Marginal Cost • Recall TC(Q,w,r) = Q . (wr)0.5 25 • Then: AC(Q,w,r) = (wr)0.5 25 MC(Q,w,r) = (wr)0.5 25

26. Example: Average and Marginal Cost • If r = 100 and w = 25, then TC(Q) = 2Q AC(Q) = 2 MC(Q) = 2

27. AC, MC (€ per unit) Example: Average and Marginal Cost Curves AC(Q) = MC(Q) = 2 $2 0 Q (units/yr)

28. AC, MC (€ per unit) Example: Average and Marginal Cost Curves AC(Q) = MC(Q) = 2 $2 0 1M Q (units/yr)

29. AC, MC (€ per unit) Example: Average and Marginal Cost Curves AC(Q) = MC(Q) = 2 $2 0 1M 2M Q (units/yr)

30. Average and Marginal Cost • When marginal cost equals average cost, average cost does not change with output. • I.e., if MC(Q) = AC(Q), then AC(Q) is flat with respect to Q. • However, oftentimes AC(Q) and MC(Q) are not “flat” lines.

31. Average and Marginal Cost • When marginal cost is less than average cost, average cost is decreasing in quantity. • I.e., if MC(Q) < AC(Q), AC(Q) decreases in Q. • When marginal cost is greater than average cost, average cost is increasing in quantity. • I.e., if MC(Q) > AC(Q), AC(Q) increases in Q. • We are implicitly assuming that all input prices remain constant.

32. AC, MC (€/yr) Example: Average and Marginal Cost Curves “Typical” shape of AC AC 0 Q (units/yr)

33. AC, MC (€/yr) Example: Average and Marginal Cost Curves “Typical” shape of MC MC AC • 0 Q (units/yr)

34. AC, MC (€/yr) Example: Average and Marginal Cost Curves MC AC • AC at minimum when AC(Q)=MC(Q) 0 Q (units/yr)

35. Economies and Diseconomies of Scale Definitions: If the average cost decreases as output rises, all else equal, the cost function exhibits economies of scale. If the average cost increases as output rises, all else equal, the cost function exhibits diseconomies of scale. The smallest quantity at which the long run average cost curve attains its minimum point is called the minimum efficient scale.

36. AC (€/yr) Example: Minimum Efficient Scale AC(Q) 0 Q (units/yr)

37. AC (€/yr) Example: Minimum Efficient Scale AC(Q) 0 Q (units/yr) Q* = MES

38. AC (€/yr) Example: Minimum Efficient Scale AC(Q) Diseconomies of scale 0 Q (units/yr) Q* = MES

39. AC (€/yr) Example: Minimum Efficient Scale AC(Q) Diseconomies of scale Economies of scale 0 Q (units/yr) Q* = MES

40. Example: Minimum Efficient Scale for Selected US Food and Beverage Industries IndustryMES (% market output) Beet Sugar (processed) 1.87 Cane Sugar (processed) 12.01 Flour 0.68 Breakfast Cereal 9.47 Baby food 2.59 Source: Sutton, John, Sunk Costs and Market Structure. MIT Press, Cambridge, MA, 1991.

41. Returns to Scale and Economies of Scale • There is a close relationship between the concepts of returns to scale and economies of scale. • When the production function exhibits constant returns to scale, the long run average cost function is flat: it neither increases nor decreases with output.

42. Returns to Scale and Economies of Scale When the production function exhibits increasing returns to scale, the long run average cost function exhibits economies of scale: AC(Q) increases with Q. When . the production function exhibits decreasing returns to scale, the long run average cost function exhibits diseconomies of scale: AC(Q) decreases with Q.

43. Example: Returns to Scale and Economies of Scale

44. Output Elasticity of Total Cost Definition: The output elasticity of total cost is the percentage change in total cost per one percent change in output. • TC,Q = (% TC) = TC. Q = MC • (% Q) Q TC AC • It is a measure of the extent of economies of scale

45. Output Elasticity of Total Cost • If TC,Q > 1, then MC > AC • AC must be increasing in Q. • The cost function exhibits economies of scale. • If TC,Q < 1, then MC > AC • AC must be increasing in Q • The cost function exhibits diseconomies of scale.

46. Example: Output Elasticities for Selected Manufacturing Industries in India IndustryTC,Q Iron and Steel 0.553 Cotton Textiles 1.211 Cement 1.162 Electricity and Gas 0.3823

47. Short Run Cost Functions Definition: The short run total cost function tells us the minimized total cost of producing Q units of output, when (at least) one input is fixed at a particular level. • It has two components: variable costs and fixed costs: STC(Q,K0) = TVC(Q,K0) + TFC(Q,K0) • (where K0 is the amount of the fixed input)

48. Short Run Cost Functions Definitions: The total variable cost function is the minimised sum spent on variable inputs at the input combinations that minimise short run costs. The total fixed cost function is the total amount spent on the fixed input(s).

49. TC ($/yr) Example: Short Run Total Cost, Total Variable Cost Total Fixed Cost TFC Q (units/yr)

50. TC ($/yr) Example: Short Run Total Cost, Total Variable Cost Total Fixed Cost TVC(Q, K0) TFC Q (units/yr)