Chapter 9: Production and Cost in the Long Run

McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

9-2

Production Isoquants

• In the long run, all inputs are variable & isoquants are used to study production decisions• An isoquant is a curve showing all possible

input combinations capable of producing a given level of output

• Isoquants are downward sloping; if greater amounts of labor are used, less capital is required to produce a given output

9-3

A Typical Isoquant Map (Figure 9.1)

9-4

Production Function

8-4

9-5

Typical Isoquants

9-5

9-6

Marginal Rate of Technical Substitution

• The MRTS is the slope of an isoquant & measures the rate at which the two inputs can be substituted for one another while maintaining a constant level of output

KMRTS

L

The minus sign is added to make MRTS a positive number since ∆K / ∆L, the slope of the isoquant, is negative

9-7

• The MRTS can also be expressed as the ratio of two marginal products:

Marginal Rate of Technical Substitution

L

K

MPMRTS

MP

L

K

MPKMRTS

L MP

As labor is substituted for capital, MPL declines & MPK rises causing MRTS to diminish

9-8

6–8

Marginal Rate of Technical Substitution

MRTSL

K

MP

MP

KMPLMP

KMPLMPYL

KMRTS

k

L

KL

KL

)()(0

)()(

Slope of production isoquant

Slope of production isoquant reflects relative marginal product of labor and capital

9-9

Isocost Curves

• Show various combinations of inputs that may be purchased for given level of expenditure (C) at given input prices (w, r)

C wK L

r r

• Slope of an isocost curve is the negative of the input price ratio (-w/r)• K-intercept is C/r

• Represents amount of capital that may be purchased if zero labor is purchased

9-10

Isocost curve

.

LK

Lr

w

r

CK

KLC

rKwLC

C

w

r

10

5

10

100

105

100$

5$

10$

9-11

Isocost Curves (Figures 9.2 & 9.3)

9-12

Optimal Combination of Inputs

• Two slopes are equal in equilibrium• Implies marginal product per dollar spent on last

unit of each input is the same

• Minimize total cost of producing Q by choosing the input combination on the isoquant for which Q is just tangent to an isocost curve

or L L K

K

MP MP MPw

MP r w r

9-13

Optimal Input Combination to Minimize Cost for Given Output (Figure 9.4)

9-14

Optimal Allocation of Inputs

2$,3

1$,2

Suppose

rMP

wMP

K

L

•Is the firm using the right combination of inputs?•If not, how should the firm reallocate its expenditure?•Use the last dollar rule

9-15

Optimal Allocation of Inputs

labor to relativecapital of useDecrease

capital totivelabor rela of useIncrease

capitalon spent dollar last than the

moreoutput by increaseslabor on spent dollar Last

5.12$

3

21$

2

r

MPw

MP

K

L

9-16

Output Maximization for Given Cost (Figure 9.5)

9-17

Optimization & Cost

• Expansion path gives the efficient (least-cost) input combinations for every level of output• Derived for a specific set of input prices• Along expansion path, input-price ratio is

constant & equal to the marginal rate of technical substitution

9-18

Expansion Path (Figure 9.6)

9-19

Long-Run Costs

• Long-run total cost (LTC) for a given level of output is given by:

LTC = wL* + rK*

Where w & r are prices of labor & capital, respectively, & (L*, K*) is the input combination on the expansion path that minimizes the total cost of producing that output

9-20

Long-Run Costs

• Long-run average cost (LAC) measures the cost per unit of output when production can be adjusted so that the optimal amount of each input is employed• LAC is U-shaped

• Falling LAC indicates economies of scale

• Rising LAC indicates diseconomies of scale

LTCLAC

Q

9-21

Long-Run Costs

• Long-run marginal cost (LMC) measures the rate of change in long-run total cost as output changes along expansion path• LMC is U-shaped

• LMC lies below LAC when LAC is falling

• LMC lies above LAC when LAC is rising

• LMC = LAC at the minimum value of LAC

LTCLMC

Q

9-22

Derivation of a Long-Run Cost Schedule (Table 9.1)

Least-cost combination of

Output Labor (units)

Capital (units)

Total cost

(w = $5, r = $10)

LAC LMC

100

500

600

200

300

400

700

LMC

10

4052

1220

30

60

7

2230

8

10

15

42

$120

420

560

140

200

300

720

$1.20

0.840.93

0.700.67

0.75

1.03

$1.20

1.201.40

0.200.60

1.00

1.60

9-23

Long-Run Total, Average, & Marginal Cost (Figure 9.8)

9-24

Long-Run Average & Marginal Cost Curves (Figure 9.9)

9-25

Economies of Scale

• Larger-scale firms are able to take greater advantage of opportunities for specialization & division of labor

• Scale economies also arise when quasi-fixed costs are spread over more units of output causing LAC to fall

• Variety of technological factors can also contribute to falling LAC

9-26

9-26

Returns to Scale

• If all inputs are increased by a factor of c & output goes up by a factor of z then, in general, a producer experiences:• Increasing returns to scale if z > c; output goes up

proportionately more than the increase in input usage

• Decreasing returns to scale if z < c; output goes up proportionately less than the increase in input usage

• Constant returns to scale if z = c; output goes up by the same proportion as the increase in input usage

f(cL, cK) = zQ

9-27

Returns to Scale

9-27

9-28

Economies & Diseconomies of Scale (Figure 9.10)

9-29

Constant Long-Run Costs

• Absence of economies and diseconomies of scale• Firm experiences constant costs in the long

run• LAC curve is flat & equal to LMC at all output

levels

9-30

Constant Long-Run Costs (Figure 9.11)

9-31

Minimum Efficient Scale (MES)

• The minimum efficient scale of operation (MES) is the lowest level of output needed to reach the minimum value of long-run average cost

9-32

Minimum Efficient Scale (MES) (Figure 9.12)

9-33

MES with Various Shapes of LAC (Figure 9.13)

9-34

Economies of Scope• Exist for a multi-product firm when the joint cost of

producing two or more goods is less than the sum of the separate costs for specialized, single-product firms to produce the two goods:

LTC(X, Y) < LTC(X,0) + LTC(0,Y)

• Firms already producing good X can add production of good Y at a lower cost than a single-product firm can produce Y:

LTC(X, Y) – LTC(X,0) < LTC(0,Y)

• Arise when firms produce joint products or employ common inputs in production

9-35

Purchasing Economies of Scale

• Purchasing economies of scale arise when large-scale purchasing of raw materials enables large buyers to obtain lower input prices through quantity discounts

9-36

Purchasing Economies of Scale (Figure 9.14)

9-37

Learning or Experience Economies

• “Learning by doing” or “Learning through experience”

• As total cumulative output increases, learning or experience economies cause long-run average cost to fall at every output level

9-38

Learning or Experience Economies (Figure 9.15)

9-39

Relations Between Short-Run & Long-Run Costs

• LMC intersects LAC when the latter is at its minimum point

• At each output where a particular ATC is tangent to LAC, the relevant SMC = LMC

• For all ATC curves, point of tangency with LAC is at an output less (greater) than the output of minimum ATC if the tangency is at an output less (greater) than that associated with minimum LAC

9-40

Long-Run Average Cost as the Planning Horizon (Figure 9.16)

9-41

Restructuring Short-Run Costs

• Because managers have greatest flexibility to choose inputs in the long run, costs are lower in the long run than in the short run for all output levels except that for which the fixed input is at its optimal level• Short-run costs can be reduced by adjusting fixed

inputs to their optimal long-run levels when the opportunity arises

9-42

Restructuring Short-Run Costs (Figure 9.14)