The problem of verifying the market demand theory

The aim of this paper is to acquaint applied mathematicians interested in the possibilities of applying methods for solving inverse problems in mathematical modeling in natural sciences and engineering to economic problems with our papers in this field. These papers refer to the problem of verifying the market demand theory, developed by the first author based on the revision of the unrealistic axiomatic neoclassical theory of individual demand within the framework of general scientific methodology. At the same time, the artificial object of individual theory—a rational and independent individual who maximizes his/her utility function—was replaced by a “statistical ensemble of consumers” of the market under study, and the postulates of individual theory became scientific hypotheses of the new theory. The verification of this theory consists in elucidating the question of rationalizing the statistical market demand by the collective utility function. This problem refers to the inverse problems of mathematical theories of real phenomena, which are usually ill posed and have many solutions. The solution of such problems consists in their regularization with involvement of additional information about the desired solution. Our method for verifying the market demand theory is a development of the nonparametric Afriat—Varian demand analysis with using “economic indices” of market demand as additional information, which allows obtaining solutions with various substantive properties.