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Chapter 10: Production and Cost Estimation
McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.
10-2
Empirical Production Function
• An empirical production function is the mathematical form of the production function to be estimated
10-3
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
10-4
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
10-5
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
10-6
• The average & marginal products of labor are, respectively:
3 2Q AL BL
2AP Q L AL BL 23 2MP Q L AL BL
Properties of a Short-Run Cubic Production Function
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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 2Q AL BL
3 2m a
B BL L
A A and
10-8
MP & AP Curves for the Short-Run Cubic Production Function (Figure 10.1)
Q = AL3 + BL2
10-9
• 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
10-10
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
10-11
• 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
10-12
• 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
10-13
• 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
10-14
• 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
2 3TVC aQ bQ cQ
2mQ b c
10-16
• 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
10-17
• 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
10-19
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
10-20
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
10-21
Problem
Mercantile Metalworks, Inc. manufactures wire carts for grocery stores. The production manager at Mercantile wishes to estimate an empirical production function for the assembly of carts using time-series data for the last 22 days of assembly. L is the daily number of assembly workers employed and Q is the number of carts assembled (completely) for that day. Mercantile pays its assembly workers $160 per day in wages and benefits.
10-22
1. Use Excel to estimate the following short run cubic production function:
Do the parameter estimates have the appropriate algebraic signs? Are they statistically significant at the 1 percent level?
2. What are the estimated total, average, and marginal product functions from your regression results?
3. At what level of labor usage does average product reach its maximum value? In a day, how many carts are assembled when average product is at its maximum? What is average variable cost when average product is maximized?
23 BLALQ
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