The inverse problem of the market demand's theory is constructing a collective utility function via a trade statistics consisting of a finite set of pairs ``prices-quantities''. The main computational problem here is the solution of the Afriat's inequalities system, which determines the values of the utility function and the Lagrange multiplier on the trade statistics data, which are ``Afriat's numbers''. This inverse problem is ill-posed one because of multiplicity of inequalities system's solutions and also because of their possible inconsistency and instability. A regularization method for this problem is proposed, based on the relaxation of the Afriat's system yielding local Hausdorf continuity of its solution set, and on the use of characteristics of analytical index numbers determined via Afriat's numbers. These characteristics formalized by choice criteria are: optimism, pessimism, objectivity. The results of constructing analytical index numbers for real trade statistics of Ulyanovsk region are presented.
The article develops methods for constructing analytical («economic») indices, and their application in the analysis of consumer demand in Russia based on official statistics on the consumption of goods and services (468 items for the period 2012–2017). The «economic» direction of indexology, which is based on the use of consumer demand theory, when constructing prices indices and measuring consumption dynamics, takes into account consumer preferences instead of the subjective biases of statisticians or authors of various index formulas. This direction has a long history, dating back to the 1924 work of the Soviet economist A.A. Konüs. The development of this concept in the works of Western researchers refers only to individual or household demand due to the absence of market demand theory in neoclassical Economics. Such a holistic theory of market demand was developed in recent years in the works of V.K. Gorbunov, and on this basis were developed methods for constructing analytical indices, using the ambiguity of restoring the utility function from a finite set of data for the variant determination of indices with characteristics, namely optimistic, pessimistic, and objective.To compare the traditional and «economic» directions of indexology, the authors constructed Laspeyres, Paasche and Fischer price and quantity indices. The indices were calculated both for the summary statistics of prices and consumption of the population, and for the main groups of consumer goods: food products, non-food products and services. General (two-stage) indices of all types were constructed for group indices and the consistency in aggregation property was tested. The obtained results are a new example of a successful verification of the holistic market demand theory.
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