Computation of Fisher-Gale equilibrium by auction
The paper suggests an algorithm for computing an equilibrium in the Fisher model of a competitive market, which is an early development of the Walras equilibrium theory with two finite sets of agents that are thought independent and rational: consumers (buyers) having fixed quantities of money, and producers of goods (sellers). Each producer supplies fixed amounts of goods, and consumers spend all their money on obtaining bundles that maximize their individual utility functions. This conceptual normative model of artificial economy (in view of impossibility to know the numbers of elementary agents and to estimate their characteristics in real markets and economies) was later generalized by \textit{K. J. Arrow} and \textit{G. Debreu} [Econometrica 22, 265--290 (1954; Zbl 0055.38007)]. The latter is the cornerstone of contemporary neoclassical equilibrium theory till now in spite of strong deficiencies of this model which was revealed at the middle of 20-th century. These deficiencies are being discussed, particularly, in the papers by \textit{A. Kirman} [``Whom or what does the representative individual represent?'', J. Econ. Perspect. 6, No. 2, 117--136 (1992; \url{doi:10.1257/jep.6.2.117}); ``The economic crisis is a crisis for economic theory'', CESifo Econ. Stud. 56, No. 4, 498--535 (2010)], by \textit{V. Polterovich} [``Crisis of economics'', Ekonomicheskaya Nauka Sovremennoy Rossii, No. 1, 46--66 (1998)], and by \textit{V. K. Gorbunov} [Econ. Nauka Sovrem. Ross. 2013, No. 4, 19--36 (2013; Zbl 1310.91089); Dokl. Math. 98, No. 2, 537--539 (2018; Zbl 1419.91448); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 482, No. 3 (2018)]. However, the Fisher market model can be considered as a model of international trade with agents-states. Such a real object can be observed and its quantitative analysis will need computational methods. The authors construct their computational method in frame of the \textit{E. Eisenberg} and \textit{D. Gale}'s method [Ann. Math. Stat. 30(1), 165--168 (1959; Zbl 0087.13805)] where the problem of finding the Fisher equilibrium was reduced to maximizing an aggregated logarithmic utility function subject to market feasibility constraints. This convex programming problem was initially elaborated in assuming linearity utility functions, later \textit{E. Eisenberg} did it for concave and homogeneous utility functions of degree one [``Aggregation of utility functions'', Manag. Sci. 7, No. 4, 337--350 (1961; \url{doi:10.1287/mnsc.7.4.337})], and in recent paper by \textit{L. Chen} et al. [``A note on equilibrium pricing as convex optimization'', Lect. Notes Comput. Sci. 4858, 7--16 (2007; \url{doi:10.1007/978-3-540-77105-0_5})] ''the case of concave and nonhomogeneous utility functions in potential or logarithmic form has been successively tackled''. In the paper under reviewing the authors develop their subgradient-type algorithm for solving the nonsmooth convex minimization problems [\textit{Yu. Nesterov} and \textit{V. Shikhman}, J. Optim. Theory Appl. 165, No. 3, 917--940 (2015; Zbl 1330.90078)] to compute a Fisher-Gale equilibrium for the general concave class of utility functions within Eisenberg-Gale's approach. The authors' novelty is introduction in the price-adjustment (tatonnement) process an auction procedure with individual utility prices which the consumers settle and update there. The main theorem 4.1 proves that ``the sequence of highest offer prices, historical averages of consumption bundles and historical averages of utility prices generated by our algorithm, converge to the set of Fisher-Gale equilibria''. In addition, the theorem presents the rate of convergence of the proposed algorithm.
scientific article; zbMATH DE number 6915968
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