NETWORK ECONOMICS: A VARIATIONAL INEQUALITY APPROACH

Transcription

1 NETWORK ECONOMICS: A VARIATIONAL INEQUALITY APPROACH

2 Advances in Computational Economics VOLUME 1 Series Editors Hans Amman, University of Amsterdam, Amsterdam, The Netherlands Anna Nagurney, University of Massachusetts at Amherst, Amherst, USA Editorial Board Anantha K. Duraiappah, European University Institute John Geweke, University of Minnesota Manfred Gilli, University of Geneva Kenneth L. Judd, Stanford University David Kendrick, University of Texas at Austin Daniel McFadden, University of California at Berkeley Ellen McGrattan, Duke University Reinhard Neck, University of Vienna Adrian R. Pagan, Australian National University John Rust, University of Wisconsin Berc Rustem, University of London Hal R. Varian, University of Michigan

3 NETWORK ECONOMICS: A V ARIATIONAL INEQUALITY APPROACH by Anna N agurney University of Massachusetts, USA... " SPRINGER SCIENCE+BUSINESS MEDIA, LLC

4 Library of Congress Cataloging-in-Publication Data Nagurney, Anna. Network eeonomics : a variational inequality approaeh / Anna Nagurney. p. em. -- (Advances in computational economics ; v.1) lncludes bibliographieal references and index. ISBN ISBN (ebook) DOI / Economics, Mathematieal. 2. Equilibrium (Economics) 3. Variational inequalities (Mathematics). I. TitIe. II. Series. HB135.N '.01'51--de CIP Copyright 1993 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 1993 AII rights reserved. No part ofthis publieation may be reproduced, stored in a retrieval system or transmitted in any form or by any means, meehanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, LLC. Printed an acid-free paper.

5 To my network of family, friends, colleagues, and students

6 Contents Preface Acknowledgments Glossary of Notation xv XIX xxi I Theory and Fundamentals 1 1 Variational Inequality Theory 1.1 The Variational Inequality Problem. 1.2 Basic Existence and Uniqueness Results 1.3 Partitionable Variational Inequalities 1.4 Stability and Sensitivity Analysis 1.5 Sources and Notes 2 Algorithms 2.1 The General Iterative Scheme The Projection Method The Relaxation Method 2.2 The Modified Projection Method 2.3 Decomposition Algorithms Equilibration Algorithms Demand Market Equilibration Algorithm General Equilibration Algorithms. 2.5 Sources and Notes II Partial Equilibrium - Perfect Competition 75 vii

7 viii CONTENTS 3 Spatial Price Equilibrium Static Spatial Price Equilibrium Models Quantity Formulation Price Formulation Stability and Sensitivity Analysis Policy Interventions and Disequilibrium Quantity Formulation Price Formulation Algorithms and Computation Intertemporal Spatial Market Equilibrium 3.6 Sources and Notes Traffic Network Equilibrium Traffic Network Equilibrium Models The Elastic Demand Model with Disutility Functions The Elastic Demand Model with Demand Functions The Fixed Demand Model Stability and Sensitivity Analysis Toll Policies Solution of the Link-Toll Collection Policy Solution of the Path-Toll Collection Policy Computation of Traffic Network Equilibria Sources and Notes Migration Equilibrium Costless Migration Migration with Migration Costs Migration with Class Transformations Computation of Migration Equilibria Numerical Results Sources and Notes III Partial Equilibrium - Imperfect Competition Oligopolistic Market Equilibrium Classical Oligopoly Problems Computation of Classical Oligopoly Problems A Spatial Oligopoly Model

8 CONTENTS ix Relationship Between Spatial Oligopolies and Spatial Price Equilibrium Problems Stability and Sensitivity Analysis Computation of Oligopolistic Market Equilibria Sources and Notes IV General Equilibrium Walrasian Price Equilibrium 7.1 Network Equilibrium Equivalence. 7.2 Stability and Sensitivity Analysis. 7.3 A General Walrasian Iterative Scheme The Projection Method The Relaxation Method The Projection Method for Discontinuous Aggregate Excess Demand Functions A Numerical Example Sources and Notes Financial Equilibrium 8.1 Quadratic Utility Functions 8.2 General Utility Functions Qualitative Properties 8.3 Policy Interventions Computation of General Financial Equilibria 8.5 Numerical Results 8.6 Sources and Notes ~ V Estimation Constrained Matrix Problems 9.1 The General Constrained Matrix Problem 9.2 The Splitting Equilibration Algorithm SEA for Diagonal Problems Dual Interpretation and Theoretical Analysis SEA for General Problems Computation of Large-Scale Matrices

9 x CONTENTS Serial Experiments for General Problems Parallel Experiments for General Problems Serial Experiments for Diagonal Problems Parallel Experiments for Diagonal Problems Massively Parallel Experiments Financial Flow-of-Funds Estimation Network Model of Financial Flow of Funds SEA for Financial Flow of Funds An Application 9.5 Sources and Notes Index 323

10 List of Figures 1.1 The projection y of x on the set K Geometric interpretation of (y - x)t. (z- y) 2: 0, for y = PKX and y ::I PKX Geometric interpretation of the variational inequality and its equivalent fixed point problem Depiction of bounded set KR Single origin/destination pair problem with disjoint paths Network structure of market equilibrium problem Fixed demand reformulation of elastic demand problem Bipartite market network equilibrium model Multicommodity model on a bipartite network Network equilibrium representation of market disequilibrium Intertemporal market network Parallel network structure of the subproblems The Braess network example A link-toll policy example A traffic network equilibrium example Network equilibrium formulation of costless migration The multi class migration network with three locations The transformation network for two locations and three classes Network equilibrium representation of an oligopoly model Network structure of the spatial oligopoly problem Network formulation of the pure exchange economy Network equilibrium representation of subproblem induced by the projection method xi

11 xii LIST OF FIGURES 7.3 Network equilibrium representation of subproblem induced by the relaxation method Parallel network structure of financial subproblems Parallel network structure of matrix subproblems Financial flow-of-funds accounts Network model of financial flow of funds Parallel network structure of flow-of-funds problem 314

12 List of Tables 3.1 Computational experience on disequilibrium problems Computational results for intertemporal market problems Projection method iterates for a traffic example Numerical results for nonlinear multi class migration networks Numerical results for nonlinear multi class migration networks with class transformations Parameters for the five-firm oligopoly example Projection method iterates for the five-firm example Relaxation method iterates for the five-firm example Iterates generated by the algorithm for a spatial oligopoly Parameters for a Walrasian price equilibrium example Computational comparisons of SEA, RC, and B-K Computational experience with SEA on U.S. migration tables Parallel speedup and efficiency for SEA and RC Computational experience with SEA on diagonal datasets Parallel speedup and efficiency for SEA on diagonal problems Example 1072b (485 rows X 485 columns) Example (205 rows X 205 columns) Example USDA133 (133 rows X 133 columns). 307 xiii

13 Preface Computational economics has been at the forefront in stimulating the development of mathematical methodologies for the analysis and solution of complex, large-scale problems. The past decade, in particular, has witnessed a dramatic growth of interest in this area. Supported by the increasing availability of data and advances in computer architectures, the scale and scope of problems that can now be handled are unveiling new horizons in both theoretical modeling and policy analysis. Accompanying the activity in computational economics is a need for the unification, documentation, and presentation of fundamental methodologies for use by both researchers and practitioners. This volume aims to make a contribution in this direction. The focus of this book is on network economics. Physical networks are pervasive in today's society, be they in the form of transportation networks, telecommunication networks, energy pipelines, electric power networks, etc. Mathematical networks, on the other hand, may be used to represent not only physical networks but also interactions among economic agents. In many applications, the network representation of an economic equilibrium problem may be abstract in that the nodes of the network need not correspond to locations in space and the links of the network to trade or travel routes. The identification of the network underlying an economic problem provides an added dimension to the analysis and computation of equilibria. For example, not only can the complexity of problems more readily be grasped through a graphical depiction, but issues of structural change and policy interventions may be addressed through modifications of the network. Finally, algorithms that explicitly exploit the network structure can be applied to compute the equilibrium solutions efficiently. The network framework, therefore, provides not only a mechanism for the visual representation of economic problems and a means for viewing their xv

14 xvi similarities and differences but, in addition, a novel theoretical approach. In this book connections are established among a variety of network economic equilibrium problems. The theory of variational inequalities is the methodology of choice in this book. Among the goals are: to educate, to illuminate, and to unify what is a powerful methodology for the study of economic equilibria. This book begins with the foundations for both qualitative analysis and for computational procedures by overviewing variational inequality theory. Variational inequalities were originally introduced for the study of partial differential equations, with applications derived principally from mechanics and defined on infinite-dimensional spaces. This book, in contrast, focuses on economics and on finite-dimensional variational inequality problems. After the foundations are in place, the book proceeds to apply the variational inequality methodology to a spectrum of economic equilibrium problems. The book consists of five principal parts, beginning with the theory and fundamentals, and closing with estimation. Each of the nine chapters follows with a sources and notes section and with its set of references. Each application chapter utilizes variational inequality theory to formulate the equilibrium conditions and to establish qualitative properties of the equilibrium pattern, such as existence and uniqueness. Examples of problems are included for illustrative purposes. Numerical algorithms are proposed for the models, and computational results are also presented. The fundamentals of the theory are presented in Part 1. An introduction to variational inequality theory is given in the first chapter. In this chapter the variational inequality problem is related to such classical mathematical programming problems as fixed point problems, optimization problems, and nonlinear equations. Methods of establishing existence of solutions and qualitative properties of uniqueness and stability are also presented. The second chapter presents general variational inequality algorithms for the computation of equilibria, along with conditions for convergence. Since variational inequality problems are typically solved as series of simpler variational inequality problems, usually, optimization problems, equilibration algorithms that exploit network structure are also presented. These two chapters provide the foundations for the chapters that follow, are self-contained, and also serve as reference material. Part II deals with perfectly competitive partial equilibrium models, beginning with a variety of spatial price equilibrium models, moving through traffic network equilibrium models, and ending with human migration mod-

15 xvii els. The former two problems are fundamental to network economics in that they have provided much of the stimulus and motivation for the methodological advances. Indeed, both spatial price and traffic network problems were historically reformulated as optimization problems, under a certain restrictive symmetry assumption, and studied independently. Such an assumption - as we shall demonstrate - is no longer needed. Variational inequality theory now permits us to incorporate asymmetric interactions among the underlying economic functions to handle, for example, multiple commodities, multiple modes of transportation, and different classes of migrants. In addition, transportation networks can be very complex topologically, and, thus, their study and understanding is crucial. In this part, sensitivity analysis results, which were motivated by the observance of paradoxical behavior in transportation networks, are also presented. It is also established how certain policy interventions can be incorporated directly into the network equilibrium framework. Connections among all the models in this section are made using both variational inequality theory and the underlying network structure of the problems. Algorithms that are proposed for these problems include those that can be implemented on parallel architectures. Part III addresses imperfectly competitive equilibrium models, in particular, aspatial and spatial oligopolistic market equilibrium problems. Game theoretic concepts are first introduced in this part to highlight an additional equilibrium concept that is amenable to study using variational inequality theory. The network structures of both aspatial and spatial oligopoly problems are then derived and exploited for algorithmic purposes. Several numerical examples are also presented. Finally, the connection between spatial oligopolies and perfectly competitive spatial price equilibrium problems is established. Part IV considers general equilibrium models, beginning with Walrasian price equilibrium problems, in the form of the pure exchange economic equilibrium problem, and ending with general financial equilibrium problems. In this part, the networks are again abstract in that the nodes do not correspond to locations in space and the links to trade or travel routes. The algorithms presented fully exploit the underlying network structure of these problems. Numerical examples are presented for illustrative purposes. Part V deals with the estimation problems preceding model development, in particular, the formulation and solution of constrained matrix problems, for the use in the creation of base-line general equilibrium datasets. All of these problems are nonlinear network optimization problems. In particular,

16 xviii a unified approach for the solution of a spectrum of quadratic constrained matrix problems is presented, and a complete theoretical analysis provided. The algorithm is then implemented on both coarse-grained and massively parallel architectures, and applied to a variety of problems, including the estimation of input/output and social accounting matrices. Finally, the algorithm is extended to handle the estimation of financial flow-of-funds accounts. Each of the chapters is meant to be self-contained for an individual's own interest and use. Each application has been selected so as to highlight certain aspects of the methodology while, at the same time, being of interest to both generalists and specialists, and researchers and practitioners alike. It is hoped that this book will help to attract students into computational economics in the future.

17 xix Acknowledgments This book is a synthesis and culmination of research conducted by my colleagues, collaborators, students, and me, both jointly and independently, on variational inequalities and networks over the last decade. Their enthusiasm for the subject and their friendship made this journey an enriching one. The book's conception took place while I held a National Science Foundation Visiting Professorship for Women at the Center for Transportation Studies at the Massachusetts Institute of Technology, from , under NSF Grant No. RII Further work was made possible while I was a Visiting Scholar at the Sloan School at MIT, from , under the sponsorship of a University Faculty Fellowship from the University of Massachusetts, Amherst. The book evolved while I taught a seminar on the topic at the School of Management at the University of Massachusetts. It was completed while I was on sabbatical leave at the Lefschetz Center for Dynamical Systems at Brown University and supported by a National Science Faculty Award for Women, NSF Grant No. DMS Additional research support during this time was provided by the Economic Research Service of the United States Department of Agriculture. I am very grateful for the support and hospitality of my home and host institutions during the writing of this book. I especially appreciate the intellectual freedom that has characterized my tenure at the University of Massachusetts at Amherst. The resources of the Cornell National Supercomputer Facility at Cornell University and the Northeast Parallel Architectures Center (NPAC) at Syracuse University were utilized in the computational research. I am grateful to Ding Zhang and June Dong for their careful reading of the manuscript and many helpful suggestions. I would also like to thank Vidya Shankar Thumsi for the preparation of the illustrations. I am grateful to Marie Stratta and to Kluwer Academic Publishers for assistance throughout the manuscript preparation process. I also acknowledge the fundamental contributions of the late Stella Dafermos. Finally, my sincerest appreciation goes to my husband, Professor Ladimer S. Nagurney, for his enthusiastic and consistent support throughout my academic career, and for his willingness to carry the luggage in our world-wide travels.

18 XXI Glossary of Notation This is a glossary of symbols used in this book. Other symbols are defined in the book, as needed. A vector is assumed to be a column vector, unless noted otherwise. E c u,n,x V 3 R R n R+ o /lx/l = (Li=l xnt xt Iyl [a,b]j(a,b) VI VF U argminxek I( x) sup/(x) inf I(x) AT A-I I an element of subset of union, intersection, Cartesian product for all there exists the real line Euclidean n-dimensional space Euclidean n-dimensional space on the nonnegative orthant such that; also I assignment operator is equivalent to maps to tends to inner product composition length of x E Rn with components (Xl, X2,, x n) transpose of a vector x absolute value of y a closed intervalj an open interval in R gradient of I: R n 1-+ R the n X n Jacobian of a mapping F: Rn 1-+ Rn partial derivative of I with respect to Xj also If the set of x E [( attaining the minimum of I( x) the supremum of the function I the infimum of the function I transpose of the matrix A the inverse of the matrix A the identity matrix infinity perpendicular complement to subspace of H the difference of sets Sl and S2