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Chapter 10Production and Cost Estimation

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Learning Objectives

Specify and explain the properties of a short‐run cubic production function

Employ regression analysis to estimate a short‐run production function

Discuss two important problems concerning the proper measurement of cost: correcting for inflation and measuring economic (opportunity) costs

Specify and estimate a short‐run cost function using a cubic specification

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Empirical Production Function

An empirical production function is the mathematical form of the production function to be estimated

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Empirical Production Function

Long-run production function~ A production function in which all inputs are

variable Short-run production function

~ A production function in which at least one input is fixed

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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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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 bK where and

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The average & marginal products of labor are, respectively:

2AP Q L AL BL 23 2MP Q L AL BL

Properties of a Short-Run Cubic Production Function

3 2Q AL BL

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

Average product of labor begins to diminish beyond La units of labor

Properties of a Short-Run Cubic Production Function

3 2m a

B BL L

A A and

3 2Q AL BL

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

Q = AL3 + BL2

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

A < 0 and B > 0

Properties of a Short-Run Cubic Production Function

3 2Q AL BL

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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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Estimate using data for which the level of usage of one or more inputs is fixed~ Usually time series data are used

Estimation of a Short-Run Production Function

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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

Estimation of a Short-Run Production Function

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Nominal cost data~ Data that have not been corrected for the

effects of inflation Must eliminate effects of inflation

~ Correct for the influence of inflation by dividing nominal cost data by an appropriate price index (or implicit price deflator)

Estimation of a Short-Run Production Function

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Average variable cost & marginal cost functions are, respectively:

Properties of a Short-Run Cubic Cost Function

2 3TVC aQ bQ cQ

2AVC a bQ cQ

22 3SMC a bQ cQ

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Average variable cost reaches its minimum value at:

Properties of a Short-Run Cubic Cost Function

2mQ b c

2 3TVC aQ bQ cQ

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

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

Properties of a Short-Run Cubic Cost Function

2 3TVC aQ bQ cQ

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Cubic specification produces S-shaped TVC curve & U-shaped AVC & SMC curves

Properties of a Short-Run Cubic Cost Function

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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

Properties of a Short-Run Cubic Cost Function

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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 labor

Diminishing marginal returns

Restrictions on parameters

3 2Q AL BL

2AP AL BL 23 2MP AL BL

3m

BL

Abegin at

A < 0 and B > 0

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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

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

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Summary The first step in estimating a production function is to

specify the empirical production function, which is the exact mathematical form of the equation to be estimated

To estimate a cubic short-run production function using linear regression, transform the cubic equation into linear form; the estimated regression line must pass through the origin

A manager typically uses time-series observations on cost, output, and input prices to estimate the short-run cost function. The effects of inflation must be eliminated

Because all 3 cost curves (TVC, AVC, and SMC) employ the same parameters, one can estimate any one of them to obtain estimates of all 3 curves