Computing Equilibria in General Equilibrium Models via Interior-point Methods

In this paper we study new computational methods to find equilibria in generalequilibrium models. We first survey the algorithms to compute equilibria thatcan be found in the literature on computational economics and we indicate howthese algorithms can be improved from the computational point of view. We alsoprovide alternative algorithms that are able to compute the equilibria in anefficient manner even for large-scale models, based on interior-point methods.We illustrate the proposed methods with some examples taken from theliterature on general equilibrium models.

[1]  D. Gale The law of supply and demand , 1955 .

[2]  Margaret H. Wright,et al.  Interior methods for constrained optimization , 1992, Acta Numerica.

[3]  Michael C. Ferris,et al.  Feasible descent algorithms for mixed complementarity problems , 1999, Math. Program..

[4]  B. Curtis Eaves,et al.  Homotopies for computation of fixed points on unbounded regions , 1972, Math. Program..

[5]  K. Arrow,et al.  EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY , 1954 .

[6]  E. Sperner Neuer beweis für die invarianz der dimensionszahl und des gebietes , 1928 .

[7]  Sherman Robinson,et al.  Income Distribution Policy in Developing Countries: A Case Study of Korea. , 1979 .

[8]  Herbert E. Scarf,et al.  The Approximation of Fixed Points of a Continuous Mapping , 1967 .

[9]  Stephen J. Wright Primal-Dual Interior-Point Methods , 1997, Other Titles in Applied Mathematics.

[10]  J. Lüroth Über Abbildung von Mannigfaltigkeiten , 1906 .

[11]  S. Smale,et al.  Global analysis and economics IIA: Extension of a theorem of Debreu , 1974 .

[12]  H. Nikaidô ON THE CLASSICAL MULTILATERAL EXCHANGE PROBLEM , 1956 .

[13]  H. Scarf The computation of equilibrium prices , 1973 .

[14]  Jorge Nocedal,et al.  An Interior Point Algorithm for Large-Scale Nonlinear Programming , 1999, SIAM J. Optim..

[15]  C. E. Lemke,et al.  Equilibrium Points of Bimatrix Games , 1964 .

[16]  Rolf R Mantel The Welfare Adjustment Process: Its Stability Properties , 1971 .

[17]  W. Rheinboldt,et al.  Pathways to Solutions, Fixed Points, and Equilibria. , 1983 .

[18]  H. Scarf The Computation of Equilibrium Prices: An Exposition , 1977 .

[19]  V. Ginsburgh,et al.  Activity analysis and general equilibrium modelling , 1981 .

[20]  P. Samuelson,et al.  Foundations of Economic Analysis. , 1948 .

[21]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[22]  F. Fairman Introduction to dynamic systems: Theory, models and applications , 1979, Proceedings of the IEEE.

[23]  A numerical assessment of the April 1973 tax changes in the United Kingdom , 1973 .

[24]  S. Dirkse,et al.  The path solver: a nommonotone stabilization scheme for mixed complementarity problems , 1995 .

[25]  L. McKenzie,et al.  ON THE EXISTENCE OF GENERAL EQUILIBRIUM FOR A COMPETITIVE MARKET , 1959 .

[26]  ON A THEOREM OF NEGISHI , 1973 .

[27]  Gerard van der Laan,et al.  A restart algorithm for computing fixed points without an extra dimension , 1979, Math. Program..

[28]  Herbert E. Scarf,et al.  The Computation of Economic Equilibria , 1974 .

[29]  M. El-Hodiri,et al.  PRORAMMING, PARETO OPTIMUM AND THE EXISTENCE OF COMPETITIVE EQUILIBRIA , 1968 .

[30]  I. Adelman,et al.  Income Distribution Policy in Developing Countries: A Case Study of Korea , 1978, The Journal of Asian Studies.

[31]  S. Smale Global analysis and economics , 1975, Synthese.

[32]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[33]  Léon Walras Éléments d'économie politique pure , 1889 .

[34]  G. Debreu,et al.  Theory of Value , 1959 .

[35]  Gene H. Golub,et al.  Matrix computations , 1983 .

[36]  Michael A. Saunders,et al.  On projected newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method , 1986, Math. Program..

[37]  T. Kehoe Computation and multiplicity of equilibria , 1991 .

[38]  R. Kellogg,et al.  A Constructive Proof of the Brouwer Fixed-Point Theorem and Computational Results , 1976 .

[39]  A. Nagurney Network Economics: A Variational Inequality Approach , 1992 .

[40]  C. E. Lemke,et al.  Bimatrix Equilibrium Points and Mathematical Programming , 1965 .

[41]  B. Curtis Eaves,et al.  Homotopies for computation of fixed points , 1972, Math. Program..

[42]  ITERATIVE SOLUTION OF NONLINEAR EQUATIONS OF HAMMERSTEIN TYPE , 2003 .

[43]  Lars Mathiesen,et al.  Computational Experience in Solving Equilibrium Models by a Sequence of Linear Complementarity Problems , 1985, Oper. Res..

[44]  S. M. Robinson Newton's method for a class of nonsmooth functions , 1994 .

[45]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[46]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[47]  T. Negishi WELFARE ECONOMICS AND EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY , 1960 .

[48]  P. Samuelson The Stability of Equilibrium: Comparative Statics and Dynamics , 1941 .

[49]  L. Chambers Practical methods of optimization (2nd edn) , by R. Fletcher. Pp. 436. £34.95. 2000. ISBN 0 471 49463 1 (Wiley). , 2001, The Mathematical Gazette.

[50]  B. Eaves,et al.  General equilibrium models and homotopy methods , 1999 .

[51]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[52]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[53]  Paul A. Samuelson,et al.  The Stability of Equilibrium: Linear and Nonlinear Systems , 1942 .

[54]  H. Kuhn Simplicial approximation of fixed points. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[55]  K. Arrow,et al.  General Competitive Analysis , 1971 .

[56]  Richaard W. Cottle Nonlinear Programs with Positively Bounded Jacobians , 1966 .

[57]  S. Smale Convergent process of price adjust-ment and global newton methods , 1976 .

[58]  K. Judd Numerical methods in economics , 1998 .

[59]  R. Fletcher Practical Methods of Optimization , 1988 .

[60]  Victor Ginsburgh,et al.  The Structure of Applied General Equilibrium Models , 1997 .

[61]  Jorge Nocedal,et al.  Assessing the Potential of Interior Methods for Nonlinear Optimization , 2003 .