Chapter 10 Chapter 10Chapter 10Chapter 10Chapter 10Chapter 10

8/19/2019 Chapter 10 Chapter 10Chapter 10Chapter 10Chapter 10Chapter 10

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Managerial EconomicsManagerial Economics

10-2

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

= +3 3 2 2Q aK L bK L


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10-3

Properties of a Short-Run Cubic

Production Function

• Holding capital constant, short-run

cubic production function isderived as follows:

= +3 2Q AL BL

= +3 3 2 2

Q aK L bK L

= +3 2 AL BL

= =3 2 A aK B bK Where and


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10-4


Production Function

• The average & marginal products

of labor are, respectively:

= +3 2Q AL BL

= = +2

AP Q L AL BL

= ∆ ∆ = +2 MP Q L 3AL 2BL


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10-5


Production Function

• Marginal product of labor begins to

diminish beyond Lm units of labor

= +3 2Q AL BL

a L

verage product of labor begins to

diminish beyond units of labor

•

= − = −m a

B B L L

3 A 2 A and


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10-6

MP & AP Cures for the Short-Run

Cubic Production Function !Figure 10"1#

% & A'( )*'2


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10-7


Production Function

• To have necessary properties of a

production function, parametersmust satisfy the following

restrictions:

= +3 2Q AL BL

< > A 0 B 0and

ll


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10-8

Estimation of a Short-Run

Production Function• To use linear regression analysis, thecubic e!uation must be transformed

into linear form

• Q = AX + BW

• Where X = L3 and W = L2

• "stimated regression line must pass

through the origin

• Specify in computer routine

M i l E iM i l E i


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10-9

Estimation of a Short-Run Cost

Function• "stimate using data for which the level ofusage of one or more inputs is fixed

• Usually time series data are used

•#ata collection may be complicated by thefact 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 inde

M i l E iM i l E i


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10-


Cost Function

• verage variable cost & marginal

cost functions are, respectively:

= + +2 3TVC aQ bQ cQ

= + + 2

AVC a bQ cQ

= + + 2

SMC a 2bQ 3cQ

M i l E iM i l E i


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10-


Cost Function

• verage variable cost reaches its

minimum value at:

= + +2 3TVC aQ bQ cQ

= −m

Q b 2c

> < >0 0 0a , b , c and

• To conform to theoretical properties,

parameters must satisfy the following

restrictions:


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M i l E iM i l E i


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10-

Summar$ of Short-Run Empirical

Production FunctionsShort-run cubic production e%uations

#otal product

$verage product of labor

%arginal product of labor

Diminishing marginalreturnsestrictions on

parameters

= +3 2

Q AL BL

= +2

AP AL BL

= +2

3 2 MP AL BL

< >0 0 A Band

= −

3m

B L A

begin at

M i l E iM i l E i


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10-

Summar$ of Short-Run Empirical

Cost Functions Short-run cubiccost e%uations

#otal variable cost

$verage variable cost

%arginal cost

$verage variable costreaches minimum at

estrictions on

parameters

= + +2 3

TVC aQ bQ cQ

= + + 2

AVC a bQ cQ

= + + 2

2 3 SMC a bQ cQ

= −2

mbQc

> < >0 0 0a , b , c

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Chapter 10 Chapter 10Chapter 10Chapter 10Chapter 10Chapter 10