Specification and Estimation of Multiple-Output Production Functions

162 Pages·1976·3.657 MB·English

by  Dr. Georg Hasenkamp (auth.)

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Lecture Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. Kunzi
Mathematical Economics
120
Georg Hasenkamp
Specification and Estimation of
Multiple-Output Production Functions
Springer-Verlag
Berlin· Heidelberg· New York 1976
Editorial Board
H. Albach' A. V. Balakrishnan' M. Beckmann (Managing Editor)
P. Dhrymes . J. Green' W. Hildenbrand . W. Krelle
H. P. KUnzi (Managing Editor) . K. Ritter' R. Sato . H. Schelbert
P. Schonfeld
Managing Editors
Prof. Dr. M. Beckmann Prof. Dr. H. P. KOnzi
Brown University Universitat ZOrich
Providence, RI 02912/USA 8090 ZOrich/Schweiz
Author
Dr. Georg Hasenkamp
Sonderiorschungsbereich 21
Econometrics and Operations Research
University of Bonn
Adenauerallee 24-42
5300 Bonn/BRD
Library of Congress Cataloging in Pnbllcation Data
Hasenkamp, Georg, 1943-
Specification and estimation of multiple-dUtput
production fUnctions.
(Lecture notes in economics and mathematical systems ;
]20) (Mathematical economics)
Revision of the author's thesis, University of
Wisconsin, Madison.
Bibliography: p.
Includes index.
1. Production functions (Economic theory) 2. United
States--Industries--Mathematical models. I. Title.
II. Series. III. Series: Matbematical economics.
HBl42 .H38 338.5 76-4837
AMS Subject Classifications (1970): 62 P20
ISBN-13: 978-3-540-07625-4 e-ISBN-13: 978-3-642-45481-3
001: 10.1007/978-3-642-45481-3
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© by Springer-Verlag Berlin' Heidelberg 1976
Softcover reprint of the hardcover 1s t edition 1976
Table of Contents
Page
CHAPTER ONE
~ntroduction
CHAPTER TWO
Multiple-Output Production Functions 4
A. Introduction 4
B. Basic Theory of Production 5
1. Notation 5
2. Long-run profit function 6
3. Short-run profit function 11
a) Cost Function 12
b) Revenue Function 13
*
4. Interpretation of A 13
C. Almost-Homogeneity and Separability 15
1. Notation 15
2. A.lmost homogenei ty 15
3. Separability 19
4. Implications 20
D. The Treatment of Raw Materials in
Production Theory - or Subaggregation
of Inputs 27
1. Weakly separable forms of the
input function 27
2. Partial Leontief input function 30
3. General comment 34
E. Subaggregation of Outputs 34
1. Weakly separable form of the output
function 34
2. Partial Leontief output function 36
F. Conclusions 39
IV
CHAPTER THREE
Functional Forms for Multiple-Output Technologies 42
A. Introduction 42
B. Input Functions 43
1. Cobb-Douglas input function 43
a) Notation 43
b) The derived (long-run) cost function 45
c) The derived (short-run) cost function 47
d) The derived profit function 48
2. Constant-Elasticity-of-Substitution
input function 49
a) Notation 49
b) The derived (long-run) cost function 50
c) The derived (short-run) cost function 53
d) The derived profit function 53
C. Output Functions 54
1. Constant-Elasticity-of-Transformation
output function 54
a) Notation 54
b) The derived (long-run) revenue function 55
c) The derived profit function 56
2. Quadratic output function 56
a) Notation 56
b) The derived revenue and profit function 57
3. Generalized output function 58
D. Separable Multiple-Output Production
Functions 58
1. Models with a CD input function 59
a) Model 1 : CET output - CD input 59
b) Model 2 : DQ output - CD input 60
c) Model 3: GQ output - CD input 60
2. Models with a CES input function 60
a) Model 4 : CET output - CES input 61
b) Model 5: DQ output - CES input 61
c) Model 6 : GQ output - CES input 61
v
E. Nonseparable Multiple-Output Production
62
Functions
1. Log-Quadratic (or translog) 63
a) Production function 63
b) Profit function 63
c) Cost function 63
2. Diewert
a) Production function 64
b) Profit function 64
c) Cost function 64
Appendix: Algebraic Derivation of Aggregate Variables 65
1. Aggregate price variable for the CD input
function 65
a) Long-run cost function 65
b) Short-run cost function 66
2. Aggregate price variable for the CES input
function and CET output function 67
a) The cost function for the CES function 67
b) The revenue function for the CET output
function 69
3. Aggregate price variable for the DQ output
function 69
CHAPTER FOUR
Stochastic Specification 71
A. Introduction 71
B. The Stochastic System of output Supply and
Input Demand 72
1. Basic framework 72
2. Interpretation of the stochastic
specification 73
C. Stochastic Profit and Cost Functions 75
1. General remarks 75
2. Stochastic profit 76
3. Stochastic cost 79
VI
D. Estimation 80
1. Notation 80
2. Properties of estimator 81
3. computational aspects 84
CHAPTER FIVE
Empirical results 86
A. Introduction 86
B. Cross-Section Data for U.S. Railroads 86
1. Characteristics of the data 86
2. Formulation of empirical work 87
3. Results of empirical work with the
(logarithmic) input demand system 89
4. Interpretation of estimation results using
the input demand system 112
5. Result of empirical work with the cost
function 114
6. Comparison with Klein's result 119
C. Time Series Data for U.S. Economy 120
1. Characteristics of the data 120
2. Formulation of empirical work 122
3. Result of empirical work with the input
demand system 123
4. Interpretation of estimation results
using the input demand system 137
5. Result of empirical work with the
cost function 139
6. Comparison with results in
Christensen et.al. 144
Preface
This monograph is intended as a contribution to applied work in
production theory by treating: a) The measurement problems involved
whenever several outputs are jointly produced, and b) The (possible)
econometric framework for estimating production function parameters
whenever (relative) prices vary sufficiently over the observations.
Thus, the "pure" theory of production, and the theory of non-linear
regression is not treated in sufficient detail: for these topics
we refer to the available literature~ for a comprehensive list of
references we refer to Uebe (1975). Some years ago, papers by
L.J. Lau and W.E. Diewert initiated my interest in duality within
production theory, and then I started to appreciate R.W. Shephard's
fundamental contribution to this topic.
This monograph is a (minor) revision of a Ph.D. dissertation
submitted to the Department of Economics, The University of Wisconsin,
Madison, in August 1973. The Graduate School and the Social Systems
Research Institute provided funds for the reported computer work,
Lawrence R. Klein and Laurits R. Christensen provided data, and as a
Graduate Student I received financial support in form of an
University Fellowship~ all this is very much appreciated. With some
nostalgia I think back to the enjoyable period of learning during
the preparation of my diFosertation~ I appreciate the spirit within
the department, and the guidance of my committee Arthur S. Goldberger,
Dennis J. Aigner, and Laurits R. Christensen. Many thanks are also
due to Frau Monika Nicolaisen, Econometrics Unit, Institut fUr
~konometrie und Operations Research, Universitat Bonn, who quite
cheerfully (-- in spite of the Tables) typed the final version of
this monograph.
Bonn, November 1975 Georg Hasenkamp
CHAPTER ONE
INTRODUCTION
Economic research on production usually specifies that a
'single "output" is determined by a production function from a set
of inputs. The measure for "output" usually takes on one of the
different forms: physical unit of a single output, monetary value
of a single output, total monetary value of several distinct
physical types of output, or the monetary value of the outputs minus
the value of raw materials. The last approach of measuring "output"
is usually referred to as "value added".
The use of total_monetary value as the single output measure
for the production function has been criticized by Mundlak (1963,
1964). Whenever the production process involves several jOintly
produced physical types of outputs, Mundlak argues, a production
function is needed which incorporates the distinct outputs in
physical units. Besides handling the measurement problem, the use
of such a mUltiple-output production function with physical
components has the added advantage of providing us with both an
output supply system and an input demand system.
Klein (1947, 1953), in his study of U.s. railroads, pioneered
the use of such multiple-output production functions. He proposed
a Cobb-Douglas form for both the outputs and the inputs, despite
the fact that the Cobb-Douglas output form did not provide correct
convexity conditions. The transcendental multiple-output production
function introduced by Mundlak (1963, 1964) has the correct
convexity conditions, but this function is not homogeneous and does
not provide a parametric elasticity of transformation among the
outputs. Both of these possible shortcomings can be avoided with the
constant-elasticity-of-transformation (CET) multiple-output
function proposed by Powell and Gruen (1968). Mundlak and Razin
(1969, 1971) generalized this CET output function in an important
aspect, to include also sub-aggregate measures for outputs.
2
In addition, Eads, Nerlove, and Raduchel (1969), Diewert
(1973, 1974 a), Hall (1973), and Christensen, Jorgenson, and Lau
(1973) discuss functions which are homogeneous, but with no
parametric representation of the elasticities of transformation or
substitution. Eads et al. (1969) used their multiple-output
production in a cost-minimization model of U. S. airlines. Christensen
et al. (1973) perform an empirical study with u.s. macro data, by
using their proposed "translog" production function. For
theoretical work on multiple-output production functions, reference
should be made to Shephard (1970), McFadden (1970), Jorgenson and
Lau (1974), and in particular to Lau (1972), who discusses the
implications of production function properties for derived profit
functions.
In this monograph we will first discuss the elementary theory
of multiple-output production under the maintained assumption of
cost-minimizing, revenue-maximizing, or profit-~aximizing behavior
of an individual producer. In developing this theory, we shall pay
special attention to the case in which the outputs can be
aggregated by a function, and the inputs can also be aggregated by
a function. Using this assumption of separability of inputs and
outputs in the mUltiple-output production function, we shall
utilize properties of the derived ("dual") cost, revenue, and
profit functions to develop a framework for our empirical work.
Next, we consider the Griliches and Ringstad (1971, Chapter 5) and
Arrow (1972) discussion of the treatment of raw materials, and
extend their analysis to the multiple-output case. The problem
raised by Mundlak (1963) concerning the monetary value of multiple
outputs is also discussed.
We shall pay special attention to those functional forms of
multiple-output production which facilitate the algebraic
derivation of dual costs, revenue, or profit functions, and output
supply and input demand systems. Our proposed ,functions are
convenient for the estimation of production function parameters;
e.g., returns to scale, elasticity of substitution, and elasticity