Foundations of the theory of general equilibrium (Q2813072)

The book represents the current state of the formal theory of economic equilibrium in the setting of Walras-Arrow-Debreu which is limited substantially by exchange economy (without production). The main characteristics of this theory are the axioms of independence and rationality of agents, which are consumers. The consumers are endowed with bundles of goods and exchange the goods among themselves to maximize their utility functions, which represent their preferences, at given prices. It is well known that with these axioms the natural properties of rational demand (weak axiom of revealed preference and others) which are provided by the utility maximization principle for individual demand functions are not transferred to the market demand, which is, according to Walras, the sum of demands of independent individuals (``impossibility theorems'' of W. Gorman, 1953, and of H. Sonnenschein, R. Mantel, G. Debreu, 1970th). These results have shattered this variant of equilibrium theory as not reflecting reality, according to many researchers (H. Sonnenschein, A. Kirman, W. Hildenbrand, and others). This theory cannot describe the regular character of market equilibrium (uniqueness and stability) which is being observed often during quiet periods. However, others have ignored this lack of the Arrow-Debreu equilibrium model. They were inspired by the mathematical difficulties that arise due to various peculiarities of the equilibrium sets of this model. The author of the book is one of the latter, and he is a leading researcher of this variant of equilibrium theory.NEWLINENEWLINEThe book is written at a high mathematical level. It consists of ten chapters and a mathematical appendix. Chapter 1 contains a short introduction and gives a detailed presentation of the general exchange model (EM) to be investigated in the remaining chapters in special cases and in its general form. Chapter 2 describes a linear version of the EM corresponding to Cobb-Douglas type utility functions for all consumers. The equilibrium system of this economy is linear, and its solution exists and is unique. Chapter 3 represents the EM with two goods and two consumers, which is the Edgeworth-Bowley box. Chapter 4 is devoted to individual consumer theory in the special variant where the commodity space coincides with all Euclidean space (allowing negative measures of the goods), and a consumer's wealth (or budget) equal to the value of her/his endowment. The next five chapters, from 5 to 9, investigate the equilibrium manifold of the EM in general. It is shown that the equilibrium manifold is diffeomorphic to the Euclidean space \(\mathbb{R}^{lm}\), where \(l\) is number of goods, and \(m\) is number of consumers, and topological properties of different cases of corresponding differentiable maps are deeply investigated. The last Chapter 10 gives a short presentation of an extension of general equilibrium models to ones with production, with uncertainty, and to the overlapping-generations model. The mathematical appendix contains the used facts about smooth manifolds, degree theory, fibrations and covering of manifolds, envelope theory and the other apparatus which is beyond a basics course of mathematics.NEWLINENEWLINEThere is an alternative to the equilibrium theory of Arrow-Debreu going back to Gustav Cassel (Walras's contemporary) which is based on the consolidated representation of market demand. This alternative was elaborated by \textit{A. Wald} [Econometrica 19, 368--403 (1951; Zbl 0043.35004)] who proved the existence and uniqueness of an equilibrium in the modified Cassel's economy. This result was characterized by K. Arrow and G. Debreu, as well as their followers R. Solow and others, as `economically illegitimate', because they understood market demand only within its representation by Walras. The theory of market demand, which is evidential, verifiable, and satisfies the Wald condition (the weak axiom), is developed in (Russian) works by the reviewer on the basis of the notion of the ``statistical ensemble of consumers'' [Zh. Èkon. Teor. 2009, No. 1, 85--94 (2009; Zbl 1183.91087); Потребительский спрос. Аналитическая теория и приложения (Russian). Ul'yanovsk: Ul'yanovskiĭ Gosudarstvennyĭ Universitet (2015; Zbl 1371.91002)]. This theory makes the corresponding modification of the Cassel-Wald model of economic equilibrium (having a unique equilibrium) economically legitimate and testable on the standard trade and production statistics [Ècon. Nauka Sovrem. Ross. 2013, No. 4, 19--36 (2013; Zbl 1310.91089)].