lecture # 12 cost curves lecturer: martin paredes

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Lecture # 12 Lecture # 12 Cost Curves Cost Curves Lecturer: Martin Paredes Lecturer: Martin Paredes

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Page 1: Lecture # 12 Cost Curves Lecturer: Martin Paredes

Lecture # 12Lecture # 12

Cost CurvesCost Curves

Lecturer: Martin ParedesLecturer: Martin Paredes

Page 2: Lecture # 12 Cost Curves Lecturer: Martin Paredes

2

1. Long Run Cost Functions Shifts Average and Marginal Cost Functions Economies of Scale Deadweight Loss

2. Long Run Cost Functions Relationship between Long Run and

Short Run Cost Functions

Page 3: Lecture # 12 Cost Curves Lecturer: Martin Paredes

3

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) = wL*(Q,w,r) + r K*(Q,w,r)

where L* and K* are the long run input demand functions

Page 4: Lecture # 12 Cost Curves Lecturer: Martin Paredes

4

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) = wL*(Q,w,r) + rK*(Q,w,r)

= Q . (wr)0.5

25

( )( )

Page 5: Lecture # 12 Cost Curves Lecturer: Martin Paredes

5

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

Page 6: Lecture # 12 Cost Curves Lecturer: Martin Paredes

6

Example: Long Run Cost Curve Recall TC(Q,w,r) = Q . (wr)0.5

25 What if r = 100 and w = 25?

TC(Q,w,r) = Q . (25100)0.5

25= 2Q

Page 7: Lecture # 12 Cost Curves Lecturer: Martin Paredes

7Q (units per year)

TC (€ per year)

TC(Q) = 2Q

Example: A Total Cost Curve

Page 8: Lecture # 12 Cost Curves Lecturer: Martin Paredes

8Q (units per year)

TC (€ per year)

TC(Q) = 2Q

1 M.

€2M.

Example: A Total Cost Curve

Page 9: Lecture # 12 Cost Curves Lecturer: Martin Paredes

9Q (units per year)

TC (€ per year)

TC(Q) = 2Q

1 M. 2 M.

€2M.

€4M.

Example: A Total Cost Curve

Page 10: Lecture # 12 Cost Curves Lecturer: Martin Paredes

10

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.

Page 11: Lecture # 12 Cost Curves Lecturer: Martin Paredes

11

L (labour services per year)

K

0

•L0

K0

Q0

TC = TC0

Example: Movement Along LRTC

Page 12: Lecture # 12 Cost Curves Lecturer: Martin Paredes

12Q (units per year)

L (labour services per year)

K

TC (€/yr)

0

0

LR Total Cost Curve

Q0

TC0=wL0+rK0

•L0

K0

Q0

TC = TC0

Example: Movement Along LRTC

Page 13: Lecture # 12 Cost Curves Lecturer: Martin Paredes

13Q (units per year)

L (labour services per year)

K

TC (€/yr)

0

0

LR Total Cost Curve

Q0

TC0=wL0+rK0

••

L0 L1

K0

K1

Q0

Q1

TC = TC1

TC = TC0

Example: Movement Along LRTC

Page 14: Lecture # 12 Cost Curves Lecturer: Martin Paredes

14Q (units per year)

L (labour services per year)

K

TC (€/yr)

0

0

LR Total Cost Curve

Q0Q1

TC0=wL0+rK0

••

L0 L1

K0

K1

Q0

Q1

TC = TC1

TC = TC0

TC1=wL1+rK1

Example: Movement Along LRTC

••

Page 15: Lecture # 12 Cost Curves Lecturer: Martin Paredes

15

Example: Shift of the long run cost curve

Suppose there is an increase in wages but the price of capital remains fixed.

Page 16: Lecture # 12 Cost Curves Lecturer: Martin Paredes

16

L

K

Q0

0

Example: A Change in the Price of an Input

Page 17: Lecture # 12 Cost Curves Lecturer: Martin Paredes

17

L

K

Q0•

0

A

-w0/r

TC0/r

Example: A Change in the Price of an Input

Page 18: Lecture # 12 Cost Curves Lecturer: Martin Paredes

18

L

K

Q0•

0

A

-w0/r

TC0/r

-w1/r

Example: A Change in the Price of an Input

Page 19: Lecture # 12 Cost Curves Lecturer: Martin Paredes

19

L

K

Q0•

0

A

B

-w0/r

TC0/r

TC1/r

-w1/r

Example: A Change in the Price of an Input

TC1 > TC0

Page 20: Lecture # 12 Cost Curves Lecturer: Martin Paredes

20

Q (units/yr)

TC (€/yr)

TC(Q) ante

Example: A Shift in the Total Cost Curve

Page 21: Lecture # 12 Cost Curves Lecturer: Martin Paredes

21

Q (units/yr)

TC (€/yr)

TC(Q) ante

Q0

TC0

Example: A Shift in the Total Cost Curve

Page 22: Lecture # 12 Cost Curves Lecturer: Martin Paredes

22

Q (units/yr)

TC (€/yr)

TC(Q) ante

TC(Q) post

Q0

TC1

TC0

Example: A Shift in the Total Cost Curve

••

Page 23: Lecture # 12 Cost Curves Lecturer: Martin Paredes

23

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

Page 24: Lecture # 12 Cost Curves Lecturer: Martin Paredes

24

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

Page 25: Lecture # 12 Cost Curves Lecturer: Martin Paredes

25

Example: Average and Marginal Cost

Recall TC(Q,w,r) = Q . (wr)0.5

25

Then: AC(Q,w,r) = (wr)0.5

25MC(Q,w,r) = (wr)0.5

25

Page 26: Lecture # 12 Cost Curves Lecturer: Martin Paredes

26

Example: Average and Marginal Cost

If r = 100 and w = 25, then TC(Q) = 2QAC(Q) = 2MC(Q) = 2

Page 27: Lecture # 12 Cost Curves Lecturer: Martin Paredes

270

AC, MC (€ per unit)

Q (units/yr)

AC(Q) =MC(Q) = 2

$2

Example: Average and Marginal Cost Curves

Page 28: Lecture # 12 Cost Curves Lecturer: Martin Paredes

280

AC, MC (€ per unit)

Q (units/yr)

AC(Q) =MC(Q) = 2

$2

Example: Average and Marginal Cost Curves

1M

Page 29: Lecture # 12 Cost Curves Lecturer: Martin Paredes

290

AC, MC (€ per unit)

Q (units/yr)

AC(Q) =MC(Q) = 2

$2

Example: Average and Marginal Cost Curves

1M 2M

Page 30: Lecture # 12 Cost Curves Lecturer: Martin Paredes

30

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.

Page 31: Lecture # 12 Cost Curves Lecturer: Martin Paredes

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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.

Page 32: Lecture # 12 Cost Curves Lecturer: Martin Paredes

32Q (units/yr)

AC, MC (€/yr)

0

AC

“Typical” shape of AC

Example: Average and Marginal Cost Curves

Page 33: Lecture # 12 Cost Curves Lecturer: Martin Paredes

33Q (units/yr)

AC, MC (€/yr)

0

MC AC

“Typical” shape of MC

Example: Average and Marginal Cost Curves

Page 34: Lecture # 12 Cost Curves Lecturer: Martin Paredes

34Q (units/yr)

AC, MC (€/yr)

0

MC AC

AC at minimum when AC(Q)=MC(Q)

Example: Average and Marginal Cost Curves

Page 35: Lecture # 12 Cost Curves Lecturer: Martin Paredes

35

Definitions:1. If the average cost decreases as output

rises, all else equal, the cost function exhibits economies of scale.

2. If the average cost increases as output rises, all else equal, the cost function exhibits diseconomies of scale.

3. The smallest quantity at which the long run average cost curve attains its minimum point is called the minimum efficient scale.

Page 36: Lecture # 12 Cost Curves Lecturer: Martin Paredes

36

0 Q (units/yr)

AC (€/yr)

AC(Q)

Example: Minimum Efficient Scale

Page 37: Lecture # 12 Cost Curves Lecturer: Martin Paredes

37

0 Q (units/yr)

AC (€/yr)

Q* = MES

AC(Q)

Example: Minimum Efficient Scale

Page 38: Lecture # 12 Cost Curves Lecturer: Martin Paredes

38

0 Q (units/yr)

AC (€/yr)

Q* = MES

AC(Q)

Example: Minimum Efficient Scale

Diseconomies of scale

Page 39: Lecture # 12 Cost Curves Lecturer: Martin Paredes

39

0 Q (units/yr)

AC (€/yr)

Q* = MES

AC(Q)

Example: Minimum Efficient Scale

Diseconomies of scale

Economies of scale

Page 40: Lecture # 12 Cost Curves Lecturer: Martin Paredes

40

Example: Minimum Efficient Scale for SelectedUS Food and Beverage Industries

Industry MES (% market output)

Beet Sugar (processed) 1.87Cane Sugar (processed) 12.01Flour 0.68Breakfast Cereal 9.47Baby food 2.59

Source: Sutton, John, Sunk Costs and Market Structure. MIT Press, Cambridge, MA, 1991.

Page 41: Lecture # 12 Cost Curves Lecturer: Martin Paredes

41

There is a close relationship between the concepts of returns to scale and economies of scale.

1. When the production function exhibits constant returns to scale, the long run average cost function is flat: it neither increases nor decreases with output.

Page 42: Lecture # 12 Cost Curves Lecturer: Martin Paredes

42

2. When the production function exhibits increasing returns to scale, the long run average cost function exhibits economies of scale: AC(Q) increases with Q.

3. When . the production function exhibits decreasing returns to scale, the long run average cost function exhibits diseconomies of scale: AC(Q) decreases with Q.

Page 43: Lecture # 12 Cost Curves Lecturer: Martin Paredes

43

Example: Returns to Scale and Economies of Scale

Returns to Scale DecreasingConstan

tIncreasing

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 = wAC = wQ-

0.5

Economies of ScaleDiseconomi

esNone

Economies

Page 44: Lecture # 12 Cost Curves Lecturer: Martin Paredes

44

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

Page 45: Lecture # 12 Cost Curves Lecturer: Martin Paredes

45

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.

Page 46: Lecture # 12 Cost Curves Lecturer: Martin Paredes

46

Example: Output Elasticities for Selected Manufacturing Industries in India

Industry TC,Q

Iron and Steel 0.553 Cotton Textiles 1.211Cement 1.162Electricity and Gas 0.3823

Page 47: Lecture # 12 Cost Curves Lecturer: Martin Paredes

47

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)

Page 48: Lecture # 12 Cost Curves Lecturer: Martin Paredes

48

Definitions:1. The total variable cost function is the

minimised sum spent on variable inputs at the input combinations that minimise short run costs.

2. The total fixed cost function is the total amount spent on the fixed input(s).

Page 49: Lecture # 12 Cost Curves Lecturer: Martin Paredes

49Q (units/yr)

TC ($/yr)

TFC

Example: Short Run Total Cost,

Total Variable Cost Total Fixed Cost

Page 50: Lecture # 12 Cost Curves Lecturer: Martin Paredes

50Q (units/yr)

TC ($/yr)

TVC(Q, K0)

TFC

Example: Short Run Total Cost,

Total Variable Cost Total Fixed Cost

Page 51: Lecture # 12 Cost Curves Lecturer: Martin Paredes

51Q (units/yr)

TC ($/yr)

TVC(Q, K0)

TFC

STC(Q, K0)

Example: Short Run Total Cost,

Total Variable Cost Total Fixed Cost

Page 52: Lecture # 12 Cost Curves Lecturer: Martin Paredes

52Q (units/yr)

TC ($/yr)

TVC(Q, K0)

TFC

rK0

STC(Q, K0)

rK0

Example: Short Run Total Cost,

Total Variable Cost Total Fixed Cost

Page 53: Lecture # 12 Cost Curves Lecturer: Martin Paredes

53

Example: Short Run Total Cost Suppose : Q = K0.5L0.25M0.25

w = €16m = €1r = €2

Recall the input demand functions:LS* (Q,K0) = Q2

4K0

MS*(Q,K0) = 4Q2

K0

Page 54: Lecture # 12 Cost Curves Lecturer: Martin Paredes

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

Page 55: Lecture # 12 Cost Curves Lecturer: Martin Paredes

55

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)

Page 56: Lecture # 12 Cost Curves Lecturer: Martin Paredes

56

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.

Page 57: Lecture # 12 Cost Curves Lecturer: Martin Paredes

57L

K

0

Q0

Example: Short Run and Long Run Total Costs

Page 58: Lecture # 12 Cost Curves Lecturer: Martin Paredes

58L

K

TC0/w

TC0/r

0

Q0

Example: Short Run and Long Run Total Costs

•A

Page 59: Lecture # 12 Cost Curves Lecturer: Martin Paredes

59L

K

TC0/w

TC0/r

0

Q0

K0

Example: Short Run and Long Run Total Costs

•A

Page 60: Lecture # 12 Cost Curves Lecturer: Martin Paredes

60L

K

TC0/w

TC0/r

0

Q1

Q0

K0

Example: Short Run and Long Run Total Costs

•A

Page 61: Lecture # 12 Cost Curves Lecturer: Martin Paredes

61L

K

TC0/w

TC0/r

0

B

Q1

Q0

K0

Example: Short Run and Long Run Total Costs

•A

Page 62: Lecture # 12 Cost Curves Lecturer: Martin Paredes

62L

K

TC0/w TC2/w

TC2/r

TC0/r

0

B

Q1

Q0

K0

Example: Short Run and Long Run Total Costs

•A

Page 63: Lecture # 12 Cost Curves Lecturer: Martin Paredes

63L

K

TC0/w TC1/w TC2/w

TC2/r

TC1/r

TC0/r ••

0

C

B

Q1

Q0

K0

Example: Short Run and Long Run Total Costs

•A

Page 64: Lecture # 12 Cost Curves Lecturer: Martin Paredes

64L

K

TC0/w TC1/w TC2/w

TC2/r

TC1/r

TC0/r •••

Expansion path

0

A

C

B

Q1

Q0

K0

Example: Short Run and Long Run Total Costs

Page 65: Lecture # 12 Cost Curves Lecturer: Martin Paredes

65

Example: Short Run and Long Run Total Costs

0Q (units/yr)

TC(Q)

Total Cost (€/yr)

Page 66: Lecture # 12 Cost Curves Lecturer: Martin Paredes

66

Example: Short Run and Long Run Total Costs

0Q (units/yr)

TC(Q)

Q0

ATC0

Total Cost (€/yr)

Page 67: Lecture # 12 Cost Curves Lecturer: Martin Paredes

67

C

Example: Short Run and Long Run Total Costs

0Q (units/yr)

TC(Q)

Q0 Q1

•A

TC0

TC1

Total Cost (€/yr)

Page 68: Lecture # 12 Cost Curves Lecturer: Martin Paredes

68

C

Example: Short Run and Long Run Total Costs

0Q (units/yr)

TC(Q)

STC(Q,K0)

Q0 Q1

•A

TC0

TC1

Total Cost (€/yr)

Page 69: Lecture # 12 Cost Curves Lecturer: Martin Paredes

69

C

Example: Short Run and Long Run Total Costs

0Q (units/yr)

TC(Q)

STC(Q,K0)

Q0 Q1

••

A

B

TC0

TC1

TC2

Total Cost (€/yr)

Page 70: Lecture # 12 Cost Curves Lecturer: Martin Paredes

70

C

Example: Short Run and Long Run Total Costs

0Q (units/yr)

TC(Q)

STC(Q,K0)

Q0

K0 is the LR cost-minimisingquantity of K for Q0

Q1

••

A

B

TC0

TC1

TC2

Total Cost (€/yr)

Page 71: Lecture # 12 Cost Curves Lecturer: Martin Paredes

71

Definition: The short run average cost function indicates the short run firm’s 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

Page 72: Lecture # 12 Cost Curves Lecturer: Martin Paredes

72

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

Page 73: Lecture # 12 Cost Curves Lecturer: Martin Paredes

73

Notes: The short run average cost can be

decomposed into average variable cost and average fixed cost.

SAC = AVC + AFCwhere:

AVC = TVC/QAFC = TFC/Q

When STC = TC, then also SMC = MC

Page 74: Lecture # 12 Cost Curves Lecturer: Martin Paredes

74Q (units per year)

€ Per Unit

0

AFC

Example: Short Run Average Cost,

Average Variable Cost Average Fixed Cost

Page 75: Lecture # 12 Cost Curves Lecturer: Martin Paredes

75Q (units per year)

€ Per Unit

0

AVC

AFC

Example: Short Run Average Cost,

Average Variable Cost Average Fixed Cost

Page 76: Lecture # 12 Cost Curves Lecturer: Martin Paredes

76Q (units per year)

€ Per Unit

0

SACAVC

AFC

Example: Short Run Average Cost,

Average Variable Cost Average Fixed Cost

Page 77: Lecture # 12 Cost Curves Lecturer: Martin Paredes

77Q (units per year)

€ Per Unit

0

SMC SACAVC

AFC

Example: Short Run Average Cost,

Average Variable Cost Average Fixed Cost

Page 78: Lecture # 12 Cost Curves Lecturer: Martin Paredes

78

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.

Page 79: Lecture # 12 Cost Curves Lecturer: Martin Paredes

79

Q (units per year)

€ per unit

0

SAC(Q,K1)

The Long Run Average Cost Curve as an Envelope Curve

Page 80: Lecture # 12 Cost Curves Lecturer: Martin Paredes

80

Q (units per year)

€ per unit

0

SAC(Q,K1)

SAC(Q,K2)

The Long Run Average Cost Curve as an Envelope Curve

Page 81: Lecture # 12 Cost Curves Lecturer: Martin Paredes

81

Q (units per year)

€ per unit

0

SAC(Q,K1)

SAC(Q,K2)

The Long Run Average Cost Curve as an Envelope Curve

SAC(Q,K3)

Page 82: Lecture # 12 Cost Curves Lecturer: Martin Paredes

82

Q (units per year)

€ per unit

0

SAC(Q,K1)

SAC(Q,K2)

The Long Run Average Cost Curve as an Envelope Curve

SAC(Q,K4)SAC(Q,K3)

Page 83: Lecture # 12 Cost Curves Lecturer: Martin Paredes

83

Q (units per year)

€ per unit

0

• ••

SAC(Q,K1)

SAC(Q,K2)

Q1 Q2 Q3 Q4

The Long Run Average Cost Curve as an Envelope Curve

AC(Q)

SAC(Q,K4)

SAC(Q,K3)

Page 84: Lecture # 12 Cost Curves Lecturer: Martin Paredes

84

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.

Page 85: Lecture # 12 Cost Curves Lecturer: Martin Paredes

85

4. Average costs tell us the firm’s cost per unit of output.

5. Marginal costs tell us the rate of change in total cost as output varies.

6. Relatively high marginal costs pull up average costs, relatively low marginal costs pull average costs down.