Production Costs

ECO61Udayan Roy

Fall 2008

Bundles of Labor and Capital That Cost the Firm $100

Isocost LinesK

, U

nits

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capi

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a

d

e

$100 isocost

L, Units of labor per year

$100$10

10 =

$100$5

= 20

Isocost Equation

K = r

- L wr

C

Initial Values

C = $100w = $5 r = $10

15

2.5

10

5

7.5

5

c

b

L = 5

K = 2.5

For each extra unit of capital it uses, the firm must use two fewer units of labor to hold its cost constant.

Slope = -1/2 = -w/r

A Family of Isocost LinesK

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$150 isocost$100 isocost

L, Units of labor per year

$150$10

15 =

$100$10

10 =

$100$5

= 20$150

$5= 30

Isocost Equation

K = r

- L wr

C

Initial Values

C = $150w = $5 r = $10

An increase in C….

A Family of Isocost LinesK

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capi

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e

$150 isocost$100 isocost$50 isocost

L, Units of labor per year

$150$10

15 =

$100$10

10 =

$50$10

5 =

$50$5

= 10$100

$5= 20

$150$5

= 30

Isocost Equation

K = r

- L wr

C

Initial Values

C = $50w = $5 r = $10

A decrease in C….

Costs• The firm’s total cost equation is:

C = wL + rK.– Therefore,

Lr

w

r

CK

r

wLCK

wLCrK

Note that if C is constant—as along an isocost line—then a one-unit increase in L requires K to change by –w/r units. That is, the slope of the isocost line is –w/r.

Combining Cost and Production Information.

• The firm can choose any of three equivalent approaches to minimize its cost:

– Lowest-isocost rule - pick the bundle of inputs where the lowest isocost line touches the isoquant.

– Tangency rule - pick the bundle of inputs where the isoquant is tangent to the isocost line.

– Last-dollar rule - pick the bundle of inputs where the last dollar spent on one input gives as much extra output as the last dollar spent on any other input.

Cost MinimizationK

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x

500 L, Units of labor per hour

100

q = 100 isoquant

$3,000isocost

$2,000isocost

$1,000isocost

Isocost Equation

K = r

- L wr

C

Initial Values

q = 100C = $2,000w = $24 r = $8

Isoquant Slope

MPL

MPK

= -MRTS-

Which of these three Isocost would allow the firm to produce the 100 units of output at the lowest possible cost?

Cost MinimizationK

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y

x

z

11650240L, Units of labor per hour

100

303

28

q = 100 isoquant

$3,000isocost

$2,000isocost

$1,000isocost

Isocost Equation

K = r

- L wr

C

Initial Values

q = 100C = $2,000w = $24 r = $8

Isoquant Slope

MPL

MPK

= MRTS-

Cost Minimization• At the point of tangency, the slope of the isoquant

equals the slope of the isocost. Therefore,

r

MP

w

MP

r

w

MP

MP

MP

MPMRTS

r

wMRTS

KL

K

L

K

LLK

LK

last-dollar rule: cost is minimized if inputs are chosen sothat the last dollar spent on labor adds as much extra output as the last dollar spent on capital.

Slope of isoquant Slope of isocost

Cost MinimizationK

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x

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11650240L, Units of labor per hour

100

303

28

q = 100 isoquant

$3,000isocost

$2,000isocost

$1,000isocost

Initial Values

q = 100C = $2,000w = $24 r = $8

w r

MPL MPK=

MPL = 0.6q/LMPK = 0.4q/K

= 24 8

1.2 0.4= = 0.05

Spending one more dollar on labor at x gets the firm as much extra output as spending the same amount on capital.

Cost MinimizationK

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y

x

z

11650240L, Units of labor per hour

100

303

28

q = 100 isoquant

$3,000isocost

$2,000isocost

$1,000isocost

Initial Values

q = 100C = $2,000w = $24 r = $8

w

r

MPL

MPK

MPL = 0.6q/LMPK = 0.4q/K

= 24

8

2.5

0.13=

= 0.1

if the firm shifts one dollar from capital to labor, output falls by 0.017 because there is less capital but also increases by 0.1 because there is more labor for a net gain of 0.083 more output at the same cost….

= 0.02

So …the firm should shifteven more resources from capital to labor—which increases the marginal product of capital and decreases the marginal product of labor.

Change in Input PriceK

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77500 L, Workers per hour

100

52

q = 100 isoquant

Originalisocost,$2,000

New isocost,$1,032

w r

MPL MPK=

Minimizing Cost Rule

A decrease in w….Initial Values

q = 100C = $2,000w = $24 r = $8w2 = $8C2 = $1,032

v

How Long-Run Cost Varies with Output

• expansion path - the cost-minimizing combination of labor and capital for each output level

Expansion Path

K,

Un

its o

f ca

pita

l pe

r h

ou

r

x

y

z

10075500 L, Workers per hour

150

200

100

Expansion path

$3,000isocost

$2,000isocost

$4,000isocost

q = 100 Isoquant

q = 200 Isoquant

q = 300 Isoquant

Expansion Path and Long-Run Cost Curve (cont’d)

Long-Run Cost Curves

Economies of Scale

• economies of scale - property of a cost function whereby the average cost of production falls as output expands.

• diseconomies of scale - property of a cost function whereby the average cost of production rises when output increases.

Returns to Scale and Long-Run Costs

Figure 8.7: Least-Cost Method, No-Overlap Rule Example

Q = 140

Square Feetof Space, K

1 2 3 4 5 6

500

1000

1500

2000

2500

Number of Assembly Workers, L

B

A

C = $3500

D

C = $3000

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Figure 8.10: Output Expansion Path and Total Cost Curve

8-21

Figure 8.28: Returns to Scale and Economies of Scale

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