Managerial Economics
eighth edition
Thomas
Maurice
Chapter 10
Production & Cost
Estimation
McGraw-Hill/Irwin
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Managerial Economics
Empirical Production Function
• Cubic empirical specification for a
short-run production function is
derived from a long-run cubic
production function
• Cubic form of the long-run
production function is expressed as
Q  aK L  bK L
3
2
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2
2
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Managerial Economics
Properties of a Short-Run Cubic
Production Function
Q  AL  BL
3
2
• Holding capital constant, short-run
cubic production function is
derived as follows:
Q  aK L  bK L
3
2
 AL  BL
3
3
2
2
Where A  aK 3 and B  bK 2
3
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Managerial Economics
Properties of a Short-Run Cubic
Production Function
Q  AL  BL
3
2
• The average & marginal products
of labor are, respectively:
AP  Q L  AL  BL
2
MP  Q L  3AL  2BL
2
4
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Managerial Economics
Properties of a Short-Run Cubic
Production Function
Q  AL  BL
3
2
• Marginal product of labor begins to
diminish beyond Lm units of labor
• Average product of labor begins to
diminish beyond La units of labor
B
Lm  
3A
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B
and La  
2A
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Managerial Economics
MP & AP Curves for the Short-Run
Cubic Production Function (Figure 10.1)
Q = AL3 + BL2
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Managerial Economics
Properties of a Short-Run Cubic
Production Function
Q  AL  BL
3
2
• To have necessary properties of a
production function, parameters
must satisfy the following
restrictions:
A  0 and B  0
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Managerial Economics
Estimation of a Short-Run
Production Function
• To use linear regression analysis, the
cubic equation must be transformed
into linear form
• Q = AX + BW
• Where X = L3 and W = L2
• Estimated regression line must pass
through the origin
• Specify in computer routine
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Managerial Economics
Estimation of a Short-Run Cost
Function
• Estimate using data for which the level of
usage of one or more inputs is fixed
• Usually time series data are used
• Data collection may be complicated by the
fact that accounting data do not include
firm’s opportunity costs
• Capital costs should reflect not only acquisition
cost but any foregone rental income,
depreciation, & capital gains/losses
• Must eliminate effects of inflation
• Divide by appropriate price index
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Managerial Economics
Properties of a Short-Run Cubic
Cost Function
TVC  aQ  bQ  cQ
2
3
• Average variable cost & marginal
cost functions are, respectively:
AVC  a  bQ  cQ 2
SMC  a  2bQ  3cQ
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Managerial Economics
Properties of a Short-Run Cubic
Cost Function
TVC  aQ  bQ  cQ
2
3
• Average variable cost reaches its
minimum value at:
Qm   b 2c
• To conform to theoretical
properties, parameters must satisfy
the following restrictions:
a  0 , b  0 , and c  0
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Managerial Economics
Properties of a Short-Run Cost
Function
• Cubic specification produces
S-shaped TVC curve & -shaped AVC &
SMC curves
• All three cost curves employ the same
parameters
• Only necessary to estimate one of these
functions to obtain estimates of all three
• In the short-run cubic specification,
input prices are assumed constant
• Not explicitly included in cost equation
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Managerial Economics
Summary of Short-Run Empirical
Production Functions
Short-run cubic
production equations
Total product
Average product of labor
AP  AL  BL
Marginal product of labor
MP  3 AL2  2 BL
Diminishing marginal
returns
begin at Lm  
Restrictions on
parameters
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Q  AL3  BL2
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B
3A
A  0 and B  0
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Managerial Economics
Summary of Short-Run Empirical
Cost Functions
Short-run cubic
cost equations
Total variable cost
Average variable cost
Marginal cost
Average variable cost
reaches minimum at
Restrictions on
parameters
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McGraw-Hill/Irwin
TVC  aQ  bQ 2  cQ 3
AVC  a  bQ  cQ 2
SMC  a  2bQ  3cQ 2
b
Qm  
2c
a  0, b  0, c  0