To the Problem of Constructing Analytical Indexes of Market Demand: A Variative Approach

Abstract

<jats:p><jats:italic>The article develops methods for constructing economic (analytical) indexes in the framework of the holistic theory of market demand, built in recent years. By this, the economic indexes presented in the world literature within the framework of the theory of individual demand and, accordingly, related to households, acquire practical value.</jats:italic></jats:p><jats:p><jats:italic>The introduction provides a brief overview of the main problems of modern indexology and the implementation of an economic approach dating back to the classical work of 1924 by the Soviet statistician A.A. Konüs. The properties of the most well-known «formula» indexes of Laspeyres, Paasche, and Fischer with respect to the fulfillment of the Fisher test criteria are described. These indexes play an important role in the methods proposed by the authors for constructing analytical indexes, which are determined through the function of consumer expenditures. The latter is determined by a utility function that rationalizes trade statistics. The rationalizing utility function is constructed ambiguously, and the corresponding task should be specified. Methods for its solution are proposed, developed within a non-parametric demand analysis of Afriat-Varian. The core of this analysis is the system of linear Afriat’s inequalities that determine the values of the utility function and marginal utility corresponding to statistical demand. This system can be inconsistent and unstable with respect to variations of non-exact demand statistics. In the case of compatibility, inequalities have many solutions, and the choice of different solutions of inequalities gives different values of analytical indexes. The authors suggest three types of tasks for the stable solution of Afriat’s inequalities, which define indexes with characteristics of optimism (low price indexes and high quantity indexes), pessimism (vice versa) and objectivity.</jats:italic></jats:p><jats:p><jats:italic>Therefore,</jats:italic><jats:italic> </jats:italic><jats:italic>the problem of increasing the objectivity of consumer demand indexes receives a theoretically justified toolbox methods for calculating analytical market demand indexes that take into account, in contrast to formula indices, consumer preferences.</jats:italic></jats:p>

Market demand: a holistic theory and its verification

“…Similar problems for the general normalized Afriat's system (20) with relaxation (24) and using general Konüs-Fisher indexes (29) were suggested in (Gorbunov et al 2020). The general case should be resolved if the homogeneity hypothesis was rejected due to an unacceptably great value r  of the minimum relaxation LP (23), or when a more adequate investigation of the market, free from the "Santa Claus" hypothesis, is desired.…”

“…The method for regularization and solution of Afriat's systems presented further is the developing the relaxation-penalty method for degenerate inequalities and extremal problems (Gorbunov: 2001b(Gorbunov: , 2004(Gorbunov: , 2015. In recent papers (Gorbunov and Lvov 2019;Gorbunov et al, 2020), this penalty function type method has been decomposed into a two-step one. The first step is the regularization of Afriat's inequalities, and the second is the statement of an extremal problem for choosing a solution with the desired properties.…”

“…, pe has a unique solution ( , ) x p e , which is the vertex of the polytope ( , ) X p e , and this vertex is determined by the system of equations The solution to this system is the functions 30 Two such techniques are introduced in (Gorbunov et al 2020).…”

“…These inequalities may be unstable, and methods for solving them are based on special regularization methods (Gorbunov: 1991(Gorbunov: , 1999(Gorbunov: , 2001a(Gorbunov: , 2001b. Recent papers (Gorbunov and Lvov, 2019;Gorbunov et al, 2020) present and approve variative verification methods for choosing solutions to regularized Afriat's inequalities with different desired properties. These properties are expressed through economic (Konüs / analytical) indexes.…”