The Problem of Verifying the Market Demand Theory

The aim of this paper is to acquaint applied mathematicians interested in the possibilitiesof applying methods for solving inverse problems in mathematical modeling in natural sciences andengineering to economic problems with our papers in this field. These papers refer to the problemof verifying the market demand theory, developed by the first author based on the revision of theunrealistic axiomatic neoclassical theory of individual demand within the framework of generalscientific methodology. At the same time, the artificial object of individual theory—arational and independent individual who maximizes his/her utility function—was replacedby a “statistical ensemble of consumers” of the market under study, and the postulates ofindividual theory became scientific hypotheses of the new theory. The verification of this theoryconsists in elucidating the question of rationalizing the statistical market demand by the collectiveutility function. This problem refers to the inverse problems of mathematical theories of realphenomena, which are usually ill posed and have many solutions. The solution of such problemsconsists in their regularization with involvement of additional information about the desiredsolution. Our method for verifying the market demand theory is a development of thenonparametric Afriat–Varian demand analysis with using “economic indices” of market demand asadditional information, which allows obtaining solutions with various substantive properties.