MATHEMATICAL OPTIMIZATION AND ECONOMIC ANALYSIS

MATHEMATICAL OPTIMIZATION AND ECONOMIC ANALYSIS

Springer Optimization and Its Applications VOLUME 36 Managing Editor Panos M. Pardalos (University of Florida) Editor Combinatorial Optimization Ding-Zhu Du (University of Texas at Dallas) Advisory Board J. Birge (University of Chicago) C.A. Floudas (Princeton University) F. Giannessi (University of Pisa) H.D. Sherali (Virginia Polytechnic and State University) T. Terlaky (McMaster University) Y. Ye (Stanford University) Aims and Scope Optimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics and other sciences. The series Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository works that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multiobjective programming, description of software packages, approximation techniques and heuristic approaches. For other titles published in this series, go to www.springer.com/series/7393

MATHEMATICAL OPTIMIZATION AND ECONOMIC ANALYSIS By MIKULÁŠ LUPTÁČIK Vienna University of Economics and Business Administration, Vienna, Austria 123

Mikuláš Luptáčik Department of Economics Vienna University of Economics and Business Administration 1090 Vienna Austria mikulas.luptacik@wu-wien.ac.at ISSN 1931-6828 ISBN 978-0-387-89551-2 e-isbn 978-0-387-89552-9 DOI 10.1007/978-0-387-89552-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009937355 Mathematics Subject Classification (2000): 91B02, 91B38, 91B66, 91B76, 90C05, 90C29, 90C30, 90C46, 90C90 c Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To my wife Anni my daughter Andrea, and my sons Martin and Peter

Contents Preface... xi Part I Single-Objective Optimization 1 Scarcity and Efficiency... 3 1.1 The Mathematical Programming Problem... 4 1.2 Mathematical Programming Models in Economics... 4 1.2.1 The Diet Problem... 4 1.2.2 The Neoclassical Theory of the Household... 6 1.2.3 The Neoclassical Theory of the Firm... 7 1.2.4 The Theory of Comparative Advantage... 9 1.2.5 The Giffen Paradox... 9 1.2.6 The Transportation Problem... 10 1.2.7 Portfolio Selection Model... 11 1.2.8 Input Output Analysis and Mathematical Programming... 12 1.2.9 Data Envelopment Analysis... 17 1.3 Classification of Mathematical Programming Problems... 18 References and Further Reading... 22 2 Kuhn Tucker Conditions... 25 2.1 The Kuhn Tucker Theorem... 25 2.2 Rationale of the Kuhn Tucker Conditions... 31 2.3 Kuhn Tucker Conditions and a Saddle Point of the Lagrange Function... 32 2.4 Kuhn Tucker Conditions for the General Mathematical Programming Problem... 33 2.5 The Kuhn Tucker Conditions and Economic Analysis... 36 2.5.1 Peak Load Pricing... 37 2.5.2 Revenue Maximization under a Profit Constraint... 39

viii Contents 2.5.3 Behavior of the Firm under Regulatory Constraint... 41 2.5.4 Environmental Regulation: The Effects of Different Restrictions... 50 References and Further Reading... 56 3 Convex Programming... 59 3.1 Basic Definitions and Properties... 60 3.2 Kuhn Tucker Conditions for a Convex Programming Problem... 68 3.3 Duality Theory... 73 3.4 Economic Interpretation of Duality in Convex Programming... 78 References and Further Reading... 84 4 Linear Programming... 87 4.1 The General Linear Programming Problem... 87 4.2 Implications of Linearity Assumption for Economic Analysis... 91 4.3 Duality in Linear Programming... 93 4.4 The More-for-Less Paradox... 101 4.5 Computational Procedure: The Simplex Method... 107 4.6 Some Applications of Linear Programming in Economics... 119 4.6.1 The Theory of Comparative Advantage... 119 4.6.2 The Giffen Paradox... 123 4.6.3 Leontief Pollution Model... 127 References and Further Reading... 132 5 Data Envelopment Analysis... 135 5.1 Productivity and Technical and Allocative Efficiency... 136 5.2 Basic DEA Models... 139 5.2.1 The Input-Oriented Model under a Constant Returns-to- Scale Assumption... 140 5.2.2 The Output-Oriented Model under a Constant Returns-to-Scale Assumption... 150 5.2.3 The Additive Model under a Constant Returns-to-Scale Assumption... 153 5.2.4 DEA Models under a Variable Returns-to-Scale Assumption.. 158 5.3 Production Technologies and Efficiency Measurement... 167 5.4 Technical versus Environmental Efficiency, or How to Measure Ecoefficiency... 177 5.4.1 Composition of Technical and Environmental Efficiency... 178 5.4.2 Comprehensive Measurement of Ecoefficiency... 182 References and Further Reading... 184 6 Geometric Programming... 187 6.1 The Principle of Geometric Programming... 188 6.2 The Theory of Geometric Programming... 189

Contents ix 6.3 Models of Geometric Programming in Economics... 195 6.3.1 The Economic Lot Size Problem... 196 6.3.2 The Minimization of Cost... 197 6.3.3 The Economic Interpretation of Dual Variables as Elasticity Coefficients... 200 6.3.4 An Open Input Output Model with Continuous Substitution between Primary Factors... 201 6.4 Transformation of Some Optimization Problems into Standard Geometric Programming Models... 206 References and Further Reading... 209 Part II Multiobjective Optimization 7 Fundamentals of Multiobjective Optimization... 213 7.1 Examples of Multiobjective Programming Models in Economics... 214 7.1.1 Welfare Economics... 214 7.1.2 Quantitative Economic Policy... 218 7.1.3 Optimal Monetary Policy... 219 7.1.4 Optimal Behavior of a Monopolist Facing a Bicriteria Objective Function... 221 7.1.5 Leontief Pollution Model with Multiple Objectives... 222 7.1.6 A Nonlinear Model of Environmental Control... 224 7.2 Kuhn Tucker Conditions for the Multiobjective Programming Problem... 225 7.3 Duality for Multiobjective Optimization Problems... 232 7.3.1 Duality for Multiobjective Optimization Problems in Parametric Form... 232 7.3.2 Duality Theory for Convex Lexicographic Programming... 233 7.4 Behavior of the Firm Facing a Bicriteria Objective Function under Regulatory Constraint... 235 References and Further Reading... 237 8 Multiobjective Linear Programming... 243 8.1 Linear Vector Optimization Problems... 243 8.2 Duality in Multiple-Objective Linear Programming... 248 8.3 Interactive Procedures and the Zionts Wallenius Method... 257 8.4 The Leontief Pollution Model with Multiple Objectives... 262 References and Further Reading... 267 9 Multiobjective Geometric Programming... 271 9.1 Vector Minimization Problems in Geometric Programming... 271 9.2 Duality for Multiobjective Geometric Programming in Parametric Form... 275

x Contents 9.3 A Nonlinear Model of Environmental Control... 280 9.4 Optimal Behavior of a Monopolist Facing a Bicriteria Objective Function... 283 References and Further Reading... 287 Index... 289

Preface The problem of allocating scarce resources among competing ends is central to economic analysis. Resources are not sufficiently available to produce all of the goods and services to satisfy human wants and therefore choices must be made concerning how resources will be used. Particularly in its neoclassical phase, since about 1870 economic analysis tends to presuppose that the economic agents are optimizing. Production units, or firms, maximize profit and households maximize their utility or well-being. In general, there exist a variety of objectives, besides profit maximization, high sales revenue or market share, environmental goals or the different goals followed by economic policy. The scarcity of resources acts as a bottleneck in the furthering of the objectives and represents the opportunity set from which the choices can be made. The problem of optimal allocation of scarce resources can thus be summarized as the optimization of some objective(s) subject to constraints. Constrained optimization, referred to as a mathematical programming model, is useful in economic analysis for providing deeper insights into the behavior of economic agents as well as for preparing of decision support systems for businessmen and policymakers. This book is intended to offer the reader a systematic exposition of both singleand multiobjective optimization models with the focus on their use for economic analysis. The emphasis is given to the exposition of mathematical optimization as an instrument for qualitative analysis and to a wide range of applications in economics, including efficiency analysis, industrial economics (with focus on regulatory economics), international economics, input output economics, quantitative economic policy and environmental economics. Part I of the book is devoted to single-objective optimization and starts with the notion of scarcity and efficiency and with the formulation of different economic problems leading to optimization models (Chapter 1). Kuhn Tucker conditions as the necessary optimality conditions for the general mathematical programming problem are explored and their application as an instrument of qualitative economic analysis is presented in Chapter 2. Chapter 3 deals with convex programming and with the economic implications of the convexity property. For an economist, the problem of optimal allocation of scarce resources is immediately related to the pricing problem, referred to in the language of

xii Preface mathematical programming as the dual problem. Therefore, the basic duality theory is presented with the focus on its economic interpretation. In Chapter 4, the theory of linear programming as the simplest and most widely spread class of convex programming is developed. The chapter concentrates on the implications of a linearity assumption for economic analysis and the applications of linear programming in economics. Data envelopment analysis (DEA) as one of the most important recent applications of linear programming in economics is treated in Chapter 5. DEA represents a widely used approach for measuring efficiency and productivity even when dealing with multiple inputs and outputs without the need to assign prespecified weights to either. Chapter 6 completes the first part of the book with geometric programming as a special class of nonlinear programming focusing on various applications in economics and management science. In Part II of the book, multiobjective optimization is presented as an instrument of economic analysis providing a deeper insight into the trade-off choices that have to be made with respect to the objectives. Chapter 7 deals with the extension of the Kuhn Tucker conditions and with the duality theory for multiobjective optimization. Examples from different fields of economics and the analysis of the behavior of a firm facing a bicriteria objective under regulatory constraint demonstrate the possibilities for the application of multiobjective optimization in economic analysis. As in single-objective optimization, the most developed part of multiobjective optimization is multiobjective linear programming, treated in Chapter 8. The extension of geometric programming from the first part of the book to problems with multiple objectives is the subject of Chapter 9. A list of references is added to each chapter separately with the aim of providing references for more detailed study and further reading related to particular topics. Because of the increasing complexity of recent economic problems, the use of mathematical techniques including optimization plays a very important part in economics education and in applied economic research. The book is intended for university economists, graduate and postgraduate students and for quantitative oriented economists in applied research who want to expand the array of mathematical techniques at their disposal. Students of mathematics and operations research interested in economic applications of mathematical programming can also benefit from using this book. As a prerequisite to follow the text, the basics of calculus and linear algebra are needed. Definitions, theorems, and propositions are stated rigorously, but due to the mathematical prerequisite and to emphasize an economic interpretation, most of the proofs are omitted and referred to in the literature. Following not only the principle of the division of labor by the classical economist Adam Smith, optimization under uncertainty referred to as stochastic programming and questions of choice in dynamic economic models (there are some excellent monographs in this field) are not discussed in the book. These problems and models require essentially higher mathematical background and I aimed to provide a not-toovoluminous text. A number of students and colleagues have contributed to this book directly or indirectly. The book is an outgrowth of many courses in optimization and mathematical

Preface xiii economics that I have taught at Vienna University of Economics and Business Administration, Vienna University of Technology and Comenius University Bratislava, Slovakia. Inspiring questions that students have raised in my courses have often helped me both to clarify and to deepen my own perception of particular topics. I am indebted to numerous authors and researchers who contributed to the study of mathematical optimization and economics. Relevant literature sources are listed at the end of each chapter. I owe much to Gustav Feichtinger for his unending encouragement and support during the long and fruitful time I shared with him at the Vienna University of Technology. I wish to express my gratitude to my first teachers at the University of Economics in Bratislava, Juraj Fecanin, Milan Hamala, Jozef Sojka, and Ladislav Unčovský, who introduced me to optimization and mathematical economics. I thank Bernhard Böhm and František Turnovec for permission to include part of the research outcome published in our joint papers into the book. I am grateful to Clemens Hödl, Wolfgang Katzenberger, Carl-Louis Sandblom, Susanne Warning, Wendy Williams, and Michael Weber who read the manuscript (or its parts) and suggested many improvements. The book went through several drafts, and I am deeply indebted to Viera Zajačiková for her patience and excellent typing of the manuscript. I thank Daniel Ševčovič and Robert Zvonár for preparing the figures. I want to thank the publisher s two anonymous referees for their very helpful comments. Any errors or omissions in the book are the responsibility of the author only and I will be grateful if they are pointed out to me. Finally, I would like to thank the publisher for constructive cooperation and patience, understanding and encouragement during the years it took to complete the book. Last but not least, I wish to express my thanks to my family my wife Anni and children Martin, Andrea, and Peter for their encouragement and understanding during the time-consuming task of preparing this book. Mikuláš Luptáčik Vienna, January 2009