Copyright © 2008 by the McGraw-Hill Companies, Inc. All rights reserved.

McGraw-Hill/IrwinManagerial Economics, 9e

Managerial Economics ThomasMauriceninth edition

Copyright © 2008 by the McGraw-Hill Companies, Inc. All rights reserved.

McGraw-Hill/IrwinManagerial Economics, 9e

Managerial Economics ThomasMauriceninth edition

Chapter 10

Production & Cost Estimation

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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 is derived as follows:

3 2Q AL BL

3 3 2 2Q aK L bK L 3 2AL BL

3 2A aK B bKWhere and

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Properties of a Short-Run Cubic Production Function

• The average & marginal products of labor are, respectively:

3 2Q AL BL

2AP Q L AL BL

2MP Q L 3AL 2BL

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Properties of a Short-Run Cubic Production Function

• Marginal product of labor begins to diminish beyond Lm units of labor

3 2Q AL BL

aLAverage product of labor begins todiminish beyond units of labor•

m aB B

L L3A 2A

and

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MP & AP Curves for the Short-Run Cubic Production Function (Figure 10.1)

Q = AL3 + BL2

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Properties of a Short-Run Cubic Production Function

• To have necessary properties of a production function, parameters must satisfy the following restrictions:

3 2Q AL BL

A 0 B 0 and

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10-8

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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10-9

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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Properties of a Short-Run Cubic Cost Function

• Average variable cost & marginal cost functions are, respectively:

2 3TVC aQ bQ cQ

2AVC a bQ cQ

2SMC a 2bQ 3cQ

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Properties of a Short-Run Cubic Cost Function

• Average variable cost reaches its minimum value at:

2 3TVC aQ bQ cQ

mQ b 2c

0 0 0a , b , c and

• To conform to theoretical properties, parameters must satisfy the following restrictions:

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10-12

Properties of a Short-Run Cost Function• Cubic specification produces S-shaped TVC curve & U-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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Summary of Short-Run Empirical Production Functions

Short-run cubic production equations

Total product

Average product of labor

Marginal product of laborDiminishing marginal returns

Restrictions on parameters

3 2Q AL BL

2AP AL BL

23 2MP AL BL

0 0A Band

3m

BL

Abegin at

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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

2 3TVC aQ bQ cQ

2AVC a bQ cQ

22 3SMC a bQ cQ

2m

bQ

c

0 0 0a , b , c