**MSC2010** 91B42, 91B82, 90C30

### Inverse problem of the market demand theory and analytical indices of demand

#### V. K. Gorbunov^{1}, A. G. Lvov^{2}

Annotation | 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. |
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Keywords | inverse problem of the market demand's theory, analytical indices, Afriat's inequalities, regularization methods, relaxation of inequalities |

^{1}**Vladimir K. Gorbunov**, Professor of Digital Economics Department, Ulyanovsk State University (42 Lev Tolstoy St., Ulyanovsk 432017, Russia), D.Sc. (Physics and Mathematics), ORCID: http://orcid.org/0000-0001-5276-0501, vkgorbunov@mail.ru

^{2}**Alexander G. Lvov**, Associate Professor of Digital Economics Department, Ulyanovsk State University (42 Lev Tolstoy St., Ulyanovsk 432017, Russia), PhD (Economics), ORCID: http://orcid.org/0000-0002-6726-8234, aglvov@mail.ru

**Citation**: V. K. Gorbunov, A. G. Lvov, "[Inverse problem of the market demand theory and analytical indices of demand]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,21:1 (2019) 89–110 (In Russian)

**DOI** 10.15507/2079-6900.21.201901.89-110