Our Agenda
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The Theory of Production:
Production and Cost in the Long-run
Phongthorn Wrasai
Production and Cost in the Long-run
Isocost
Isoproduct or Isoquant
Least Cost Combination
Expansion Path
The Meaning of Returns to Scale
Long-run Costs of Production: LTC, LAC, LMC
Relationship between Expansion Path and LTC
Relationship between Long-run and Short-run Costs
Economies and Diseconomies of Scale
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Production and Cost in the LR
Analysis
• In the long run, the firm can adjust all inputs so
that its cost of production is as low as possible.
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• If the manager of the firm would like to produce
a given level of output at the lowest possible
cost and is free to choose any input combination
she pleases, the main question is:
Which one should she choose?
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Now we are in the Long run consideration.
Suppose, we have two variable inputs: K, L.
Price of capital input (K) = r baht/unit.
Price of labor input (L) = w baht/unit.
Total cost of producing any particular output = C
baht.
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Tools for our Analysis
Isocost
Amount of Capital Input (K)
C = w*L + r*K
• Isocost: shows all possible combinations of labor
and capital that can be purchased for a given
total cost.
K = C/r – (w/r)L
C/r
• Isoquant: shows
Amount of Labor Input (L)
C/w
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Total Production Functions
TPK(l = l1)
TPL(k = k1)
TPK(l = l0)
TPL(k = k0)
K
L
l1
Change in Cost
k1
Change in Price of an Input
Source: Baye (2006), Managerial Economics, McGraw-Hill.
l0
k0
0
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Q
The Production Mountain
C
Output
contours
C
B
B
A
A
K
A’ B’
C’
C’
B’ A’
L
0
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Source: Frank (2006), Microeconomics and Behavior, McGraw-Hill.
K
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Isoquant Map derived from the
Production Mountain
Increasing Output
Q3
L
K
Q2
Q1
0
0
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L
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An Isoquant
Marginal Rate of Technical Substitution
Enlarged picture
K
MRTS =
−ΔK
ΔL
A
2
A
K
Rate at which one input can be exchanged for another input without
changing the total level of output.
MRTS
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or
L
=
=
Q = 10 Units
L
0
=
−ΔK
ΔL
Q constant
- dK
dL
Q constant
- MPL
MPK
= Slope of Isoquant
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Least Cost Combination
From A to C
Δ Q = M PK ⋅ Δ K .
From C to B
K
MPL w
= .
MPK r
Δ Q = M PL i Δ L .
A
Cross-multiplying,
( M PL i Δ L ) + ( M PK i Δ K ) = 0.
ΔK
B
C
ΔL
Q1
L
− M PL
−ΔK
=
= M RTS
ΔL
M PK
0
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MPL MPK
.
=
w
r
When costs are at minimum, the extra output we get from the last
baht spent on an input must be the same for all inputs.
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Expansion Path
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Returns to Scale
Constant Returns to Scale (CRS)
Q: How much does output change if a firm increases all its
inputs proportionately?
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Returns to scale: Change in output resulting from
equiproportional change in all inputs
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3 Types:
¾ Constant Returns to Scale (CRS)
¾ Increasing Returns to Scale (IRS)
¾ Decreasing Returns to Scale (DRS)
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• If output doubles when all inputs doubles,the
production function is said to exhibit constant
returns to scale (CRS).
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Increasing Returns to Scale (IRS)
Decreasing Returns to Scale (DRS)
• If output more than doubles when all inputs are
doubled, the production function is said to
exhibit increasing returns to scale (IRS).
• If output less than doubles when all inputs are
doubled, the production function exhibits
decreasing returns to scale (DRS).
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Returns to Scale: Mathematical
Approach
Supposing that we increase K and L by λ.
• Cobb - Douglas Production Function:
Q 2 = A( λ K )α ( λ L )β
Q = F ( K, L).
Q
1
= A K
α
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= A λ ( α + β ) K α Lβ
Lβ .
= λ
( α +β )
= λ
( α +β )
A K α Lβ
Q1 .
where A, α, and β are all positive constants.
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α +β
Varying Scale Economies
and Returns to Scale
α +β =1
Constant Return to Scale.
α +β >1
Increasing Return to Scale.
α +β <1
Decreasing Return to Scale.
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Long-run Costs of Production:
LTC, LAC, LMC
Source: Perloff (2007), Figure 6.5, page 170.
© 2007 Pearson Addison-Wesley. All rights reserved.
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Relationship between
Expansion Path and LTC
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Relationship between
Long-run and Short-run Costs
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Economies and Diseconomies 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 rises.
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