Utility and Production: Theory and Applications

154 Pages·1999·5.173 MB·English

by  Prof. Pablo Coto-Millán (auth.)

Utility and Production
Contributions to Economics
Christoph M. Schneider Ulrich Woitek
Research and Development Business Cycles
Management: 1997. ISBN 3-7908-0997-7
From the Soviet Union to Russia
1994. ISBN 3-7908-0757-5
Michael Carlberg
International Economic Growth
Lars 010f PerssonlUlf Wiberg 1997. ISBN 3-7908-0995-0
Microregional Fragmentation
1995. ISBN 3-7908-0855-5
Massimo Filippini
Elements or the Swiss Market
ror Electricity
Ernesto Felli/Furio C. Rosatil
1997. ISBN 3-7908-0996-9
Giovanni Tria (Eds.)
The Service Sector:
Productivity and Growth Frank HosterlHeinz Welsch!
1995. ISBN 3-7908-0875-X Christoph Bohringer
COl Abatement and Economic
Structural Change in the European
Giovanni Galizzi/ Internal Market
Luciano Venturini (Eds.) 1997. ISBN 3-7908-1020-7
Economics or Innovation:
The Case or Food Industry
1996. ISBN 3-7908-0911-X Christian M. Hafner
Nonlinear Time Series Analysis
with Applications to Foreign
Exchange Rate Volatility
David T. Johnson
Poverty, Inequality and Social 1997. ISBN 3-7908-1041-X
Welfare in Australia
1996. ISBN 3-7908-0942-X
Sardar M.N. Islam
Mathematical Economics or
Multi-Level Optimisation
Rongxing Guo
1998. ISBN 3-7908-1050-9
Border-Regional Economics
1996. ISBN 3-7908-0943-8
Sven-Morten Mentzel
Real Exchange Rate Movements
Oliver Fratzscher 1998. ISBN 3-7908-1081-9
The Political Economy or Trade
Integration
1996. ISBN 3-7908-0945-4 Lei DelsenlEelke de Jong (Eds.)
The German and Dutch Economies
1998. ISBN 3-7908-1064-9
Ulrich Landwehr
Industrial Mobility and Public Policy
Mark Weder
1996. ISBN 3-7908-0949-7
Business Cycle Models
with Indeterminacy
1998. ISBN 3-7908-1078-9
Arnold PicotlEkkehard Schlicht (Eds.)
Firms, Markets, and Contracts
1996. COTTo 2nd printing 1997 Tor Rf/ldseth (Ed.)
ISBN 3-7908-0947-0 Models ror Multispecies Management
1998. ISBN 3-7908-1001-0
Thorsten Wichmann
Agricultural Technical Progress and Michael Carlberg
the Development or a Dual Economy Intertemporal Macroeconomics
1997. ISBN 3-7908-0960-8 1998. ISBN 3-7908-1096-7
continued on page 148
Pablo Coto-Millan
Utility and Production
Theory and Applications
With 10 Figures
and 13 Tables
Physica-Verlag
A Springer-Verlag Company
Series Editors
Werner A. Muller
Martina Bihn
Author
Prof. Pablo Coto-Milhin
Department of Economics
University of Cantabria
Avda. Los Castros, sIn.
E-39005 Santander, Spain
ISBN 978-3-7908-1153-7
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Coto-Millan. Pablo: Utility and production: theory and applications; with 27 tables / Pablo Colo
Millan. - Heidelberg; New York: Physica-Verl.. 1999
(Contributions to economics)
ISBN 978-3-7908-1153-7 ISBN 978-3-662-00810-2 (eBook)
DOI 10.1007/978-3-662-00810-2
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Preface
Grateful acknowledgement is made to CICYT (Comision Intenninisterial de
Ciencia y Tecnoiogia), National Plan R+D, Projects N-TXT96-2467 and N
TXT98-1453 for its financial support.
This text presents the economic theories on Utility and Production. In addition,
such theories are used to explain the real problems of consumers and firms and
several studies carried out by the author are displayed. Some collaboration by
other professors of Economics is mentioned in the corresponding footnotes. The
responsibility for errors and omissions, however, rests entirely upon tbe author.
In chapters 1 and 4 of the book try to present the theory of Utility and
Production.
Chapter 3 presents new functional forms and two empirical applications, on
demand functions and systems.
In chapters 2 and 5, the main theorems and properties presented in chapters 1
and 4 are applied.
Chapter 6 presents new functional forms and two empirical applications, on
production and cost functions respectively.
Chapters 1. 2, 4, and 5 have been used as class notes in advanced
Microeconomics lectures from 1992 to 1997. Chapters 3 and 6 have been also
discussed in different graduate courses in the 1992-1997 period. This allowed us
to detect and correct many errors and mistakes. Of course I am sure that not all the
errors have been removed. In any case I am the only one responsible for that.
Table of Contents
Introduction
PART I: UTILITY AND CONSUMER DEMAND ANALYSIS 5
1 Theory of Utility and Consumer Behaviour: A Comprehensive
Review of Concepts, Properties and the Most Significant
Theorems 7
1.1 Theory of Utility 7
1.2 Preference, Choice and Indifference Concept and Utility Function
Existence 8
1.3 Properties of the Utility Function lO
1.3.1 Additivity 10
1.3.2 Homogeneity 10
1.3.3 Homotheticity 10
1.3.4 Weak ,md Strong Separability 11
1.4 Basic Theory (Primal): Marshallian (or Walrasian) Demand Functions 11
1.4.1 Properties of the Marshallian (or Walrasian) Demand Functions 12
1.5 Consumer Equilibrium (Dual): Hicksian (or Compensated) Demand
Functions 12
1.5.1 Properties of the Hicksian (or Compensated) Demand Functions 13
1.6 Indirect Utility Function 13
1.7 Expenditure Function 14
1.8 Restrictions of the Demand Systems 14
1.8.1 Engel Aggregation Condition 14
1.8.2 Coumot Aggregation Condition 15
1.8.3 Homogeneity Condition 16
1.8.4 Symmetry or Integrability Condition 16
1.8.5 Negativity Condition 16
1.9 Roy's Identity 17
1.l0 HotcHing's Theorem (or Shephard's Lemma for Consumers) 18
1.11 Relationships between the UMP and the EMP 19
1.12 The Slutsky Equation 19
1. 13 Complementary and Substitutive Relationships 21
VIII Table of Contents
Basic References 22
References and Further Reading 22
2 Main Forms of Utility Functions 25
2.1 The Cobb-Douglas Utility Function 25
2.1.1 Properties 25
2.1.2 Marshallian or Ordinary Demands (Primal) 27
2.1.3 The Indirect Util~ty Function 28
2.1.4 Hicksian or Compensated Demands (Dual) 29
2.1.5 The Expenditure Function 30
2.1.6 Elasticities, Engel Curves and Expenditure Share Functions 31
2.2 The Utility Function of the Constant Elasticity of Substitution (CES) 33
2.2.1 Marshallian Demands 33
2.2.2 The Indirect Utility Function 35
2.2.3 Hicksian Demands 36
2.2.4 The Expenditure Function 36
2.2.5 Application to the Particular CES Utility Function 36
2.2.5.1 The Indirect Utility Function 37
2.2.5.2 The Expenditure Function 37
2.2.5.3 Hicksian Demands 37
2.2.5.4 The Own, Cross and Income Elasticity of the CES
Demand System 37
2.2.5.5 Restrictions of CES Demand Systems 38
2.3 The Quasi-linear Utility Functions 40
2.3.1 Marshallian Demands 40
2.3.2 The Indirect Utility Function 41
2.3.3 The Expenditure Function 41
2.3.4 Roy's Identity 42
2.3.5 Hotelling's Theorem: Hicksian Demand Functions 42
2.3.6 Application to the Particular Quasi-linear Utility Function 43
2.3.6.1 The Marshallian Demand 43
2.3.6.2 Restrictions of the Quasi-linear Demand System 43
Recommended Reading 45
3 Study of the Econometric Applications: Demand Functions and
Systems 47
3. I Demand Functions 47
3.2 Application I for Demand Functions: Walrasian (or Marshallian)
Demand Functions for Interurban Passenger Transport 48
3.2.1 Model 49
3.2.2 Data 50
3.2.3 Walrasian (or Marshallian) Demands for Interurban Passenger
Transport: Air and Road Transport 51
3.2.3.1 Air Transport Demand 51
3.2.3.2 Road Transport Demand 53
3.2.4 Results of the Empirical Research 54
Table of Contents IX
3.3 Complete Demand Systems 55
3.3.1 Linear Expenditure System (LES) 55
3.3.2 Almost Ideal Demand System 56
3.3.3 Diewert Demand Model 57
3.3.4 Translog Demand Model 58
3.4 Application II for Demand Systems: Estimation of an Almost Ideal
Demand System (AIDS): Particular Disaggregation for the Main
Transport Services 59
3.4.1 Model: Almost Ideal Demand System 59
3.4.2 Data 61
3.4.3 Estimation of the Model 61
3.4.4 Conclusions 63
Basic References 64
References and Further Reading 64
PART II: Production and Firm Supply Analysis 67
4 Theory of Production, Cost and Behaviour of the Finn:
A Comprehensive Refonnulation 69
4.1 Theory of the Firm 70
4.2 Production Possibility Set and Existence of Production Function 72
4.3 Properties of Production Function 73
4.3.1 Efficiency 74
4.3.2 Differentiability and Continuity 74
4.3.3 Strict Quasi-concavity 74
4.4 The Finn's Equilibrium: Classic Demand Profit and Direct Supply
Functions 74
4.4.1 Profit Maximisation 74
4.4.2 Properties of Input Classic Demand and Output Direct Supply
Functions 75
4.4.2.1 Decreasing 76
4.4.2.2 Existence 76
4.4.2.3 Homogeneity 76
4.4.2.4 Symmetry 76
4.4.2.5 Negativity 76
4.4.2.6 Positive Semi-definite 76
4.4.3 Profit Function 76
4.4.4 Properties of the Profit Function: Hotelling's Theorem 77
4.4.4.1 Non-decreasing 77
4.4.4.2 Homogeneity 77
4.4.4.3 Convexity 77
4.4.4.4 Continuity 77
4.4.4.5 Hotelling's Theorem 77
4.5 The Firm's Equilibrium (Primal A) 78
X Table of Contents
4.6 The Finn's Equilibrium (Primal B): Marshallian Demand and Indirect
Supply Functions 78
4.6.1 Output Maximisation (Primal B) 79
4.6.2 Properties of the Input Marshallian Demand and Indirect
Supply Functions 80
4.6.2.1 Decreasing 80
4.6.2.2 Existence 80
4.6.2.3 The Lagrange Coefficient (A) 80
4.6.2.4 Homogeneity 80
4.6.2.5 Negativity 80
4.6.2.6 Symmetry 80
4.6.2.7 Negative Semi-definite 81
4.6.2.8 Roy's Identity 81
4.7 The Finn's Equilibrium: Input Classic Demand and Output Direct
Supply Functions 81
4.7.1 Loss Minimisation 81
4.7.2 Properties of Input Classic Demand and Output Direct Supply
Functions 82
4.7.3 Loss and Input Classic Demand Functions: Hotelling's
Theorem 82
4.8 The Finn's Equilibrium (Dual A) 82
4.9 The Finn's Equilibrimn (Dual B): Input Conditioned Demand and
Cost Functions 83
4.9.1 Cost Minimisation (Dual B) 83
4.9.2 Properties of the Input Conditioned Demand 84
4.9.2.1 Non-decreasing 84
4.9.2.2 Existence 84
4.9.2.3 Homogeneity 84
4.9.2.4 The Lagrange Coefficient 84
4.9.2.5 Negativity 84
4.9.2.6 Symmetry 85
4.9.2.7 Negative Semi-definite 85
4.9.3 Properties of Cost Function: Shephard's Lemma 85
4.9.3.1 Increase 85
4.9.3.2 Homogeneity 85
4.9.3.3 Concavity 85
4.9.3.4 Continuity 85
4.9.3.5 Shephard's Lemma 85
4.10 Diagranunatic Representation of the Main Relationships 86
4.11 Joint Production 91
4.11.1 Income Maximisation 93
4.11.2 Input Minimisation 94
4.12 Short-Run 96
4.12.1 Short-Run and Single Production 97
4.12.2 Short-Run and Joint Production 99
4.13 Reflections on the Main Relationships Designed 100
4.14 The Elasticity of Substitution 10 1
Basic References 103
Table of Contents XI
References and Further Reading 104
5 Main Forms of Production and Cost Functions 105
5.1 The Cobb-Douglas Production Function 105
5.1.1 Characterisation 105
5.1.2 The Marginal Rate of Technical Substitution (MRTS) 106
5.1.3 The Elasticity of Substitution 106
5.1.4 Returns to Scale 107
5.1.5 The Profit Function and Input Demand Functions 107
5.1.6 Hotelling's Theorem 110
5.1. 7 The Cost Function an Input Conditioned Demand Functions 111
5.1.8 Shephard's Lemma 113
5.1.9 LRAC and LRMC Curves 114
5.1.1 0 Applying Duality 115
5.2 The CES Production Function 117
5.2.1 The Marginal Rate of Technical Substitution (MRTS) 117
5.2.2 Returns to Scale 118
5.2.3 The Elasticity of Substitution 118
5.2.4 The Output Supply Function and Input Demand Functions 119
5.2.5 The Cost Function and Input Conditioned Demand Functions 122
5.2.6 The LRAC and LRMC 124
5.2.7 Applying the Duality 125
Recommended Reading 128
6 Study on Econometric Applications: Production and Cost
Functions 129
6.1 Production Flllctions 130
6.2 Application III for Production Functions: Analysis of the Returns to
Scale, Elasticities of Substitution and Behaviour of Shipping
Production 131
6.2.1 The Model 131
6.2.2 Data 133
6.2.3 Empirical Results 133
6.3 Cost Function 135
6.4 Other Empirical Functions 136
6.5 Application IV for Cost Functions: Elasticities of Substitution and
Behaviour of Shipping Production 138
6.5.1 The Model 138
6.5.2 Data 141
6.5.3 Empirical Results 141
6.5.4 Summary and Conclusions 144
Basic References 145
References and Further Reading 145