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McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter 10Production and Cost Estimation
10-2
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
10-3
Empirical Production Function
An empirical production function is the mathematical form of the production function to be estimated
10-4
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
10-5
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
10-6
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
10-7
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
10-8
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
10-9
MP & AP Curves for the Short-Run Cubic Production Function (Figure 10.1)
Q = AL3 + BL2
10-10
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
10-11
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
10-12
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
10-13
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
10-14
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
10-15
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
10-16
Average variable cost reaches its minimum value at:
Properties of a Short-Run Cubic Cost Function
2mQ b c
2 3TVC aQ bQ cQ
10-17
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
10-18
Cubic specification produces S-shaped TVC curve & U-shaped AVC & SMC curves
Properties of a Short-Run Cubic Cost Function
10-19
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
10-20
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
10-21
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
10-22
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
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