production costs eco61 udayan roy fall 2008. bundles of labor and capital that cost the firm $100
TRANSCRIPT
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Production Costs
ECO61Udayan Roy
Fall 2008
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Bundles of Labor and Capital That Cost the Firm $100
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Isocost LinesK
, U
nits
of
capi
tal p
erye
ar
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
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A Family of Isocost LinesK
, U
nits
of
capi
tal p
erye
ar
a
e
$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….
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A Family of Isocost LinesK
, U
nits
of
capi
tal p
erye
ar
a
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….
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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.
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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.
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Cost MinimizationK
, Uni
ts o
f cap
ital p
er h
our
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?
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Cost MinimizationK
, Uni
ts o
f cap
ital p
er h
our
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-
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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
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Cost MinimizationK
, Uni
ts o
f cap
ital p
er h
our
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
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.
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Cost MinimizationK
, Uni
ts o
f cap
ital p
er h
our
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.
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Change in Input PriceK
, U
nits
of
cap
ital p
er
ho
ur
x
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
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How Long-Run Cost Varies with Output
• expansion path - the cost-minimizing combination of labor and capital for each output level
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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
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Expansion Path and Long-Run Cost Curve (cont’d)
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Long-Run Cost Curves
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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.
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Returns to Scale and Long-Run Costs
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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
8-20
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Figure 8.10: Output Expansion Path and Total Cost Curve
8-21
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Figure 8.28: Returns to Scale and Economies of Scale
8-22