1 Costs Curves Chapter 8

2 Chapter Eight Overview 1.Introduction 2.Long Run Cost Functions Shifts Long run average and marginal cost functions Economies of scale Deadweight loss – "A Perfectly Competitive Market Without Intervention Maximizes Total Surplus" 3.Short Run Cost Functions 4.The Relationship Between Long Run and Short Run Cost Functions 1.Introduction 2.Long Run Cost Functions Shifts Long run average and marginal cost functions Economies of scale Deadweight loss – "A Perfectly Competitive Market Without Intervention Maximizes Total Surplus" 3.Short Run Cost Functions 4.The Relationship Between Long Run and Short Run Cost Functions Chapter Eight

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

4 Chapter Eight Long Run Cost Functions As Quantity of output increases from 1 million to 2 million, with input prices(w, r) constant, cost minimizing input combination moves from TC 1 to TC 2 which gives the TC(Q) curve.

5 Chapter Eight What is the long run total cost function for production function Q = 50L 1/2 K 1/2 ? L*(Q,w,r) = (Q/50)(r/w) 1/2 K*(Q,w,r) = (Q/50)(w/r) 1/2 TC(Q,w,r) = w[(Q/50)(r/w) 1/2 ]+r[(Q/50)(w/r) 1/2 ] = (Q/50)(wr) 1/2 + (Q/50)(wr) 1/2 = (Q/25)(wr) 1/2 What is the graph of the total cost curve when w = 25 and r = 100? TC(Q) = 2Q Long Run Cost Functions

6 Q (units per year) TC ($ per year) TC(Q) = 2Q $4M. Chapter Eight A Total Cost Curve

7 1 M. $2M. Chapter Eight TC ($ per year) Q (units per year) TC(Q) = 2Q A Total Cost Curve

8 1 M.2 M. $2M. $4M. Chapter Eight A Total Cost Curve TC ($ per year) Q (units per year) TC(Q) = 2Q

9 Chapter Eight Long Run Total Cost Curve Definition: The long run total cost curve shows minimized total cost as output varies, holding input prices constant. Graphically, what does the total cost curve look like if Q varies and w and r are fixed? Definition: The long run total cost curve shows minimized total cost as output varies, holding input prices constant. Graphically, what does the total cost curve look like if Q varies and w and r are fixed?

10 Chapter Eight Long Run Total Cost Curve

11 Chapter Eight Long Run Total Cost Curve

12 Chapter Eight Long Run Total Cost Curve

13 Q (units per year) L (labor services per year) K TC ($/yr) 0 0 L0L0 L1L1 K0K0 K1K1 Q0Q0 Q1Q1 TC = TC 1 TC = TC 0 Chapter Eight Long Run Total Cost Curve

14 Q (units per year) L (labor services per year) K TC ($/yr) 0 0 LR Total Cost Curve Q0Q0 TC 0 =wL 0 +rK 0 L0L0 L1L1 K0K0 K1K1 Q0Q0 Q1Q1 TC = TC 1 TC = TC 0 Chapter Eight Long Run Total Cost Curve

15 Q (units per year) L (labor services per year) K TC ($/yr) 0 0 LR Total Cost Curve Q0Q0 Q1Q1 TC 0 =wL 0 +rK 0 L0L0 L1L1 K0K0 K1K1 Q0Q0 Q1Q1 TC = TC 1 TC = TC 0 TC 1 =wL 1 +rK 1 Chapter Eight Long Run Total Cost Curve

16 Chapter Eight Long Run Total Cost Curve Graphically, how does the total cost curve shift if wages rise but the price of capital remains fixed?

17 L K 0 TC 0 /r Chapter Eight A Change in Input Prices

18 L0 -w 0 /r TC 0 /r TC 1 /r -w 1 /r Chapter Eight K A Change in Input Prices

19 L 0 A B -w 0 /r TC 0 /r -w 1 /r Chapter Eight TC 1 /r K A Change in Input Prices

20 L Q0Q0 0 A -w 0 /r TC 0 /r -w 1 /r Chapter Eight B TC 1 /r K A Change in Input Prices

21 Q (units/yr) TC ($/yr) TC(Q) post Chapter Eight A Shift in the Total Cost Curve

22 Q (units/yr) TC(Q) ante TC(Q) post Chapter Eight TC ($/yr) A Shift in the Total Cost Curve

23 Q (units/yr) TC(Q) ante TC(Q) post TC 0 Chapter Eight TC ($/yr) A Shift in the Total Cost Curve

24 Q (units/yr) TC(Q) ante TC(Q) post Q0Q0 TC 1 TC 0 Chapter Eight TC ($/yr) A Shift in the Total Cost Curve

25 Chapter Eight How does the total cost curve shift if all input prices rise (the same amount)? Input Price Changes

26 Chapter Eight All Input Price Changes Price of input increases proportionately by 10%. Cost minimization input stays same, slope of isoquant is unchanged. TC curve shifts up by the same 10 percent

27 Chapter Eight Long Run Average Cost Function Definition: The long run average cost function is the long run total cost function divided by output, Q. That is, the LRAC function tells us the firm’s cost per unit of output… AC(Q,w,r) = TC(Q,w,r)/Q Definition: The long run average cost function is the long run total cost function divided by output, Q. That is, the LRAC function tells us the firm’s cost per unit of output… AC(Q,w,r) = TC(Q,w,r)/Q

28 Chapter Eight Long Run Marginal Cost Function MC(Q,w,r) = {TC(Q+  Q,w,r) – TC(Q,w,r)}/  Q =  TC(Q,w,r)/  Q where: w and r are constant MC(Q,w,r) = {TC(Q+  Q,w,r) – TC(Q,w,r)}/  Q =  TC(Q,w,r)/  Q where: w and r are constant Definition: The long run marginal cost function measures the rate of change of total cost as output varies, holding constant input prices.

29 Chapter Eight Long Run Marginal Cost Function Recall that, for the production function Q = 50L 1/2 K 1/2, the total cost function was TC(Q,w,r) = (Q/25)(wr) 1/2. If w = 25, and r = 100, TC(Q) = 2Q.

30 Chapter Eight a. What are the long run average and marginal cost functions for this production function? AC(Q,w,r) = (wr) 1/2 /25 MC(Q,w,r) = (wr) 1/2 /25 b. What are the long run average and marginal cost curves when w = 25 and r = 100? AC(Q) = 2Q/Q = 2. MC(Q) =  (2Q)/  Q = 2. Long Run Marginal Cost Function

31 0 AC, MC ($ per unit) Q (units/yr) AC(Q) = MC(Q) = 2 $2 Chapter Eight Average & Marginal Cost Curves

32 0 AC(Q) = MC(Q) = 2 $2 1M Chapter Eight AC, MC ($ per unit) Q (units/yr) Average & Marginal Cost Curves

33 0 AC(Q) = MC(Q) = 2 $2 1M 2M Chapter Eight AC, MC ($ per unit) Q (units/yr) Average & Marginal Cost Curves

34 Chapter Eight Suppose that w and r are fixed: When marginal cost is less than average cost, average cost is decreasing in quantity. That is, if MC(Q) < AC(Q), AC(Q) decreases in Q. Average & Marginal Cost Curves

35 Chapter Eight Average & Marginal Cost Curves When marginal cost is greater than average cost, average cost is increasing in quantity. That is, if MC(Q) > AC(Q), AC(Q) increases in Q. When marginal cost equals average cost, average cost does not change with quantity. That is, if MC(Q) = AC(Q), AC(Q) is flat with respect to Q.

36 Chapter Eight Average & Marginal Cost Curves

37 Chapter Eight Economies & Diseconomies of Scale Definition: If average cost decreases as output rises, all else equal, the cost function exhibits economies of scale. Similarly, if the average cost increases as output rises, all else equal, the cost function exhibits diseconomies of scale. Definition: The smallest quantity at which the long run average cost curve attains its minimum point is called the minimum efficient scale.

38 0 Q (units/yr) AC ($/yr) Q* = MES AC(Q) Chapter Eight Minimum Efficiency Scale (MES)

39 When the production function exhibits increasing returns to scale, the long run average cost function exhibits economies of scale so that AC(Q) decreases with Q, all else equal. Chapter Eight Returns to Scale & Economies of Scale

40 Chapter Eight Returns to Scale & Economies of Scale When the production function exhibits decreasing returns to scale, the long run average cost function exhibits diseconomies of scale so that AC(Q) increases with Q, all else equal. When the production function exhibits constant returns to scale, the long run average cost function is flat: it neither increases nor decreases with output.

41 Chapter Eight If  TC,Q < 1, MC < AC, so AC must be decreasing in Q. Therefore, we have economies of scale. If  TC,Q > 1, MC > AC, so AC must be increasing in Q. Therefore, we have diseconomies of scale. If  TC,Q = 1, MC = AC, so AC is just flat with respect to Q. Definition: The percentage change in total cost per one percent change in output is the output elasticity of total cost,  TC,Q.  TC,Q = (  TC/TC)(  Q /Q) = (  TC /  Q)/(TC/ Q) = MC/AC Definition: The percentage change in total cost per one percent change in output is the output elasticity of total cost,  TC,Q.  TC,Q = (  TC/TC)(  Q /Q) = (  TC /  Q)/(TC/ Q) = MC/AC Output Elasticity of Total Cost

42 Chapter Eight Short Run & Total Variable 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. Definition: The total variable cost function is the minimized sum of expenditures on variable inputs at the short run cost minimizing input combinations. 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. Definition: The total variable cost function is the minimized sum of expenditures on variable inputs at the short run cost minimizing input combinations.

43 Chapter Eight Total Fixed Cost Function Definition: The total fixed cost function is a constant equal to the cost of the fixed input(s). STC(Q,K 0 ) = TVC(Q,K 0 ) + TFC(Q,K 0 ) Where: K 0 is the fixed input and w and r are fixed (and suppressed as arguments) Definition: The total fixed cost function is a constant equal to the cost of the fixed input(s). STC(Q,K 0 ) = TVC(Q,K 0 ) + TFC(Q,K 0 ) Where: K 0 is the fixed input and w and r are fixed (and suppressed as arguments)

44 Q (units/yr) TC ($/yr) TFC Example: Short Run Total Cost, Total Variable Cost and Total Fixed Cost Chapter Eight Key Cost Functions Interactions

45 TVC(Q, K 0 ) TFC Chapter Eight Q (units/yr) TC ($/yr) Example: Short Run Total Cost, Total Variable Cost and Total Fixed Cost Key Cost Functions Interactions

46 TVC(Q, K 0 ) TFC STC(Q, K 0 ) Chapter Eight Q (units/yr) TC ($/yr) Example: Short Run Total Cost, Total Variable Cost and Total Fixed Cost Key Cost Functions Interactions

47 TVC(Q, K 0 ) TFC rK 0 STC(Q, K 0 ) rK 0 Chapter Eight Q (units/yr) TC ($/yr) Example: Short Run Total Cost, Total Variable Cost and Total Fixed Cost Key Cost Functions Interactions

48 Chapter Eight The firm can minimize costs at least as well in the long run as in the short run because it is “less constrained”. Hence, the short run total cost curve lies everywhere above the long run total cost curve. Long and Short Run Total Cost Functions

49 Chapter Eight Long and Short Run Total Cost Functions However, when the quantity is such that the amount of the fixed inputs just equals the optimal long run quantities of the inputs, the short run total cost curve and the long run total cost curve coincide.

50 L K TC 0 /w TC 0 /r 0 Chapter Eight Long and Short Run Total Cost Functions

51 L TC 0 /w TC 1 /w TC 1 /r TC 0 /r 0 B K0K0 Chapter Eight K Long and Short Run Total Cost Functions

52 L TC 0 /w TC 1 /w TC 2 /w TC 2 /r TC 1 /r TC 0 /r 0 A C B Q1Q1 K0K0 Chapter Eight K Long and Short Run Total Cost Functions

53 L TC 0 /w TC 1 /w TC 2 /w TC 1 /r TC 0 /r Q0Q0 Expansion Path 0 A C B Q1Q1 Q0Q0 K0K0 Chapter Eight TC 2 /r K Long and Short Run Total Cost Functions

54 0 Total Cost ($/yr) Q (units/yr) TC(Q) STC(Q,K 0 ) Q0Q0 K 0 is the LR cost-minimising quantity of K for Q 0 Q1Q1 Chapter Eight Long and Short Run Total Cost Functions

55 0 Q0Q0 Q1Q1 A TC 0 Chapter Eight Total Cost ($/yr) Q (units/yr) TC(Q) STC(Q,K 0 ) K 0 is the LR cost-minimising quantity of K for Q 0 Long and Short Run Total Cost Functions

56 0 Q0Q0 Q1Q1 A C TC 0 TC 1 Chapter Eight Total Cost ($/yr) Long and Short Run Total Cost Functions TC(Q) STC(Q,K 0 ) Q (units/yr) K 0 is the LR cost-minimising quantity of K for Q 0

57 0 Q0Q0 Q1Q1 A C B TC 0 TC 1 TC 2 Chapter Eight Total Cost ($/yr) Long and Short Run Total Cost Functions TC(Q) STC(Q,K 0 ) Q (units/yr) K 0 is the LR cost-minimising quantity of K for Q 0

58 Chapter Eight Short Run Average Cost Function Definition: The Short run average cost function is the short run total cost function divided by output, Q. That is, the SAC function tells us the firm’s short run cost per unit of output. SAC(Q,K0) = STC(Q,K0)/Q Where: w and r are held fixed

59 Chapter Eight Short Run Marginal Cost Function Definition: The short run marginal cost function measures the rate of change of short run total cost as output varies, holding constant input prices and fixed inputs. SMC(Q,K0)={STC(Q+  Q,K0)–STC(Q,K0)}/  Q =  STC(Q,K0)/  Q where: w,r, and K0 are constant

60 Chapter Eight Summary Cost Functions Note: When STC = TC, SMC = MC STC = TVC + TFC SAC = AVC + AFC Where: SAC = STC/Q AVC = TVC/Q (“average variable cost”) AFC = TFC/Q (“average fixed cost”) The SAC function is the VERTICAL sum of the AVC and AFC functions

61 Q (units per year) $ Per Unit 0 AFC Example: Short Run Average Cost, Average Variable Cost and Average Fixed Cost Chapter Eight Summary Cost Functions

62 0 AVC AFC Chapter Eight Q (units per year) $ Per Unit Summary Cost Functions Example: Short Run Average Cost, Average Variable Cost and Average Fixed Cost

63 0 SAC AVC AFC Chapter Eight Q (units per year) $ Per Unit Summary Cost Functions Example: Short Run Average Cost, Average Variable Cost and Average Fixed Cost

64 0 SMC AVC AFC Chapter Eight SAC Q (units per year) $ Per Unit Summary Cost Functions Example: Short Run Average Cost, Average Variable Cost and Average Fixed Cost

65 $ per unit 0 AC(Q) SAC(Q,K 3 ) Q 1 Q 2 Q 3 Chapter Eight Q (units per year) Long Run Average Cost Function

66 0 AC(Q) SAC(Q,K 1 ) Q 1 Q 2 Q 3 Chapter Eight $ per unit Q (units per year) Long Run Average Cost Function

67 0 AC(Q) SAC(Q,K 1 ) SAC(Q,K 2 ) Q 1 Q 2 Q 3 Chapter Eight $ per unit Q (units per year) Long Run Average Cost Function

68 0 AC(Q) SAC(Q,K 1 ) SAC(Q,K 2 ) SAC(Q,K 3 ) Q 1 Q 2 Q 3 Chapter Eight $ per unit Q (units per year) Long Run Average Cost Function

69 Chapter Eight Long Run Average Cost Function Example: Let Q = K 1/2 L 1/4 M 1/4 and let w = 16, m = 1 and r = 2. For this production function and these input prices, the long run input demand curves are: Therefore, the long run total cost curve is: TC(Q) = 16(Q/8) + 1(2Q) + 2(2Q) = 8Q The long run average cost curve is: AC(Q) = TC(Q)/Q = 8Q/Q = 8 Therefore, the long run total cost curve is: TC(Q) = 16(Q/8) + 1(2Q) + 2(2Q) = 8Q The long run average cost curve is: AC(Q) = TC(Q)/Q = 8Q/Q = 8 L*(Q) = Q/8 M*(Q) = 2Q K*(Q) = 2Q

70 Chapter Eight Recall, too, that the short run total cost curve for fixed level of capital K 0 is: STC(Q,K 0 ) = (8Q 2 )/K 0 + 2K 0 If the level of capital is fixed at K 0 what is the short run average cost curve? SAC(Q,K 0 ) = 8Q/K 0 + 2K 0 /Q Recall, too, that the short run total cost curve for fixed level of capital K 0 is: STC(Q,K 0 ) = (8Q 2 )/K 0 + 2K 0 If the level of capital is fixed at K 0 what is the short run average cost curve? SAC(Q,K 0 ) = 8Q/K 0 + 2K 0 /Q Short Run Average Cost Function

71 Q (units per year) $ per unit 0 MC(Q) Chapter Eight Cost Function Summary

72 0 AC(Q) Chapter Eight Q (units per year) $ per unit MC(Q) Cost Function Summary

73 0 AC(Q) SAC(Q,K 2 ) Q 1 Q 2 Q 3 SMC(Q,K 1 ) Chapter Eight Q (units per year) $ per unit MC(Q) Cost Function Summary

74 0 AC(Q) SAC(Q,K 1 ) SAC(Q,K 2 ) SAC(Q,K 3 ) Q 1 Q 2 Q 3 MC(Q) SMC(Q,K 1 ) Chapter Eight Q (units per year) $ per unit MC(Q) Cost Function Summary

75 0 AC(Q) SAC(Q,K 1 ) SAC(Q,K 2 ) SAC(Q,K 3 ) Q 1 Q 2 Q 3 SMC(Q,K 1 ) Chapter Eight Q (units per year) $ per unit MC(Q) Cost Function Summary MC(Q)

76 Chapter Eight Economies of Scope – a production characteristic in which the total cost of producing given quantities of two goods in the same firm is less than the total cost of producing those quantities in two single-product firms. Mathematically, TC(Q 1, Q 2 ) < TC(Q 1, 0) + TC(0, Q 2 ) Stand-alone Costs – the cost of producing a good in a single-product firm, represented by each term in the right- hand side of the above equation. Economies of Scope

77 Chapter Eight Economies of Experience – cost advantages that result from accumulated experience, or, learning-by-doing. Experience Curve – a relationship between average variable cost and cumulative production volume – used to describe economies of experience – typical relationship is AVC(N) = AN B, where N – cumulative production volume, A > 0 – constant representing AVC of first unit produced, -1 < B < 0 – experience elasticity (% change in AVC for every 1% increase in cumulative volume – slope of the experience curve tells us how much AVC goes down (as a % of initial level), when cumulative output doubles Economies of Experience

78 Chapter Eight Total Cost Function – a mathematical relationship that shows how total costs vary with factors that influence total costs, including the quantity of output and prices of inputs. Cost Driver – A factor that influences or “drives” total or average costs. Constant Elasticity Cost Function – A cost function that specifies constant elasticity of total cost with respect to output and input prices. Translog Cost Function – A cost function that postulates a quadratic relationship between the log of total cost and the logs of input prices and output. Estimating Cost Functions