Cost Curves

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

TC0wL0rK0
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

TC0wL0rK0
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

TC1wL1rK1

TC0wL0rK0
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 gt 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
Q (units/yr)
Q0
22
TC (/yr)
Example A Shift in the Total Cost Curve
TC(Q) post
TC(Q) ante

TC1

TC0
Q (units/yr)
Q0
23
Long Run Average Cost Function

• Definition The long run average cost curve
  indicates the firms 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) lt 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) gt 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
Industry MES ( 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

1. When the production function exhibits increasing
   returns to scale, the long run average cost
   function exhibits economies of scale AC(Q)
   increases with Q.
2. 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
Returns to Scale Decreasing Constant Increasing
Production Function Q L0.5 Q L Q L2
Labour Demand L Q2 L Q L Q0.5
Total Cost Function TC wQ2 TC wQ TC wQ0.5
Average Cost Function AC wQ AC w AC wQ-0.5
Economies of Scale Diseconomies None Economies
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 gt 1, then MC gt AC
• AC must be increasing in Q.
• The cost function exhibits economies of scale.
• If ?TC,Q lt 1, then MC gt 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 Electri
city 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)
51
TC (/yr)
Example Short Run Total Cost, Total
Variable Cost Total Fixed Cost
STC(Q, K0)
TVC(Q, K0)
TFC
Q (units/yr)
52
TC (/yr)
Example Short Run Total Cost, Total
Variable Cost Total Fixed Cost
STC(Q, K0)
TVC(Q, K0)
rK0
TFC
rK0
Q (units/yr)
53

• Example Short Run Total Cost
• Suppose Q K0.5L0.25M0.25
• w 16
• m 1
• r 2
• Recall the input demand functions
• LS (Q,K0) Q2
• 4K0
• MS(Q,K0) 4Q2
• K0

54

• Example (cont.)
• Short run total cost
• STC(Q,K0) wLS mMS rK0
• 8Q2 2K0
• K0
• Total fixed cost
• TFC(K0) 2K0
• Total variable cost
• TVC(Q,K0) 8Q2
• K0

55
Relationship Between Long Run and Short Run Total
Cost Functions

• Compared to the short-run, in the long-run the
  firm is less constrained.
• As a result, at any output level, long-run total
  costs should be less than or equal to short-run
  total costs
• TC(Q) ? STC(Q,K0)

56
Relationship Between Long Run and Short Run Total
Cost Functions

• In other words, any short run total cost curve
  should lie above the long run total cost curve.
• The short run total cost curve and the long run
  total cost curve are equal only for some output
  Q, where the amount of the fixed input is also
  the optimal amount of that input used in the
  long-run.

57
Example Short Run and Long Run Total Costs
K
Q0
L
0
58
Example Short Run and Long Run Total Costs
K
TC0/r
Q0
A

L
0
TC0/w
59
Example Short Run and Long Run Total Costs
K
TC0/r
Q0
A

K0
L
0
TC0/w
60
Example Short Run and Long Run Total Costs
K
Q1
TC0/r
Q0
A

K0
L
0
TC0/w
61
Example Short Run and Long Run Total Costs
K
Q1
TC0/r
Q0
A


B
K0
L
0
TC0/w
62
Example Short Run and Long Run Total Costs
K
TC2/r
Q1
TC0/r
Q0
A


B
K0
L
0
TC0/w TC2/w
63
Example Short Run and Long Run Total Costs
K
TC2/r
Q1
TC1/r
C

TC0/r
Q0
A


B
K0
L
0
TC0/w TC1/w TC2/w
64
Example Short Run and Long Run Total Costs
K
TC2/r
Expansion path
Q1
TC1/r
C

TC0/r
Q0
A


B
K0
L
0
TC0/w TC1/w TC2/w
65
Total Cost (/yr)
Example Short Run and Long Run Total Costs
TC(Q)
0
Q (units/yr)
66
Total Cost (/yr)
Example Short Run and Long Run Total Costs
TC(Q)
A

TC0
0
Q0
Q (units/yr)
67
Total Cost (/yr)
Example Short Run and Long Run Total Costs
TC(Q)

TC1
C
A

TC0
0
Q0
Q1
Q (units/yr)
68
Total Cost (/yr)
Example Short Run and Long Run Total Costs
STC(Q,K0)
TC(Q)

TC1
C
A

TC0
0
Q0
Q1
Q (units/yr)
69
Total Cost (/yr)
Example Short Run and Long Run Total Costs
STC(Q,K0)
TC(Q)

B
TC2

TC1
C
A

TC0
0
Q0
Q1
Q (units/yr)
70
Total Cost (/yr)
Example Short Run and Long Run Total Costs
STC(Q,K0)
TC(Q)

B
TC2

TC1
C
A

TC0
K0 is the LR cost-minimising quantity of K for Q0
0
Q0
Q1
Q (units/yr)
71
Short Run Average Cost Function

• Definition The short run average cost function
  indicates the short run firms cost per unit of
  output.
• It is simply the short run total cost function
  divided by output, holding the input prices (w
  and r) constant.
• SAC(Q,K0) STC(Q,K0)
• Q

72
Short Run Marginal Cost Function
Definition The short run marginal cost curve
measures the rate of change of short run total
cost as output varies, holding all input prices
and fixed inputs constant. SMC(Q,K0)
?STC(Q,K0) ?Q
73

• Notes
• The short run average cost can be decomposed into
  average variable cost and average fixed cost.
• SAC AVC AFC
• where
• AVC TVC/Q
• AFC TFC/Q
• When STC TC, then also SMC MC

74
Per Unit
Example Short Run Average Cost,
Average Variable Cost Average Fixed Cost
AFC
0
Q (units per year)
75
Per Unit
Example Short Run Average Cost,
Average Variable Cost Average Fixed Cost
AVC
AFC
0
Q (units per year)
76
Per Unit
Example Short Run Average Cost,
Average Variable Cost Average Fixed Cost
SAC
AVC
AFC
0
Q (units per year)
77
Per Unit
Example Short Run Average Cost,
Average Variable Cost Average Fixed Cost
SMC
SAC
AVC
AFC
0
Q (units per year)
78
Relationship Between Long Run and Short Run
Average Cost Functions

• Just as with total costs curves, any short run
  average cost curve should lie above the long run
  average cost curve.
• In fact, the long run average cost curve forms a
  boundary or envelope around the set of short-run
  average cost curves.

79
per unit
The Long Run Average Cost Curve as an Envelope
Curve
SAC(Q,K1)
0
Q (units per year)
80
per unit
The Long Run Average Cost Curve as an Envelope
Curve
SAC(Q,K1)
SAC(Q,K2)
0
Q (units per year)
81
per unit
The Long Run Average Cost Curve as an Envelope
Curve
SAC(Q,K3)
SAC(Q,K1)
SAC(Q,K2)
0
Q (units per year)
82
per unit
The Long Run Average Cost Curve as an Envelope
Curve
SAC(Q,K3)
SAC(Q,K4)
SAC(Q,K1)
SAC(Q,K2)
0
Q (units per year)
83
per unit
The Long Run Average Cost Curve as an Envelope
Curve
SAC(Q,K3)
SAC(Q,K4)
SAC(Q,K1)
AC(Q)
SAC(Q,K2)




0
Q1 Q2 Q3 Q4
Q (units per year)
84
Summary

1. Long run total cost curves plot the minimized
   total cost of the firm as output varies.
2. Movements along the long run total cost curve
   occur as output changes.
3. Shifts in the long run total cost curve occur as
   input prices change.

85
Summary

1. Average costs tell us the firms cost per unit of
   output.
2. Marginal costs tell us the rate of change in
   total cost as output varies.
3. Relatively high marginal costs pull up average
   costs, relatively low marginal costs pull average
   costs down.