Probabilistic Solutions to the Problem of Rational Consumer Choice with Random Income

Probabilistic solutions are used when the amount of decision-makers is large. Each of them chooses the optimal solution independently of the others by solving his optimization problem. In this case, the optimal solution constructed by a randomly selected person (e.g. a consumer of goods) can be considered as a random vector. In particular, probabilistic solutions arise naturally in the rational consumer choice problem if income is assumed to be random. The problem of the utility function maximization at a time when the income of a randomly selected consumer is described as a random variable is considered as the stochastic optimization problem. The properties and distribution of the probabilistic solution of the consumer choice problem for various types of the utility function and income distribution are studied.

Keywords

stochastic optimization; probabilistic solution; the problem of consumer choice; random income; utility function.

References

1. Timofeeva G., Martynenko A. Analysis of Transport Network Development via Probabilistic Modelling. Stability and Oscillations of Nonlinear Control Systems, vol. 14, pp. 1-2. DOI: 10.1109/STAB.2018.8408407
2. Timofeeva G. Investigation of Mathematical Model of Passenger Preferences. Application of Mathematics in Engineering and Economics, 2019, vol. 2172, article ID: 080001, 7 p. DOI: 10.1063/1.5133559
3. Popova O.A. Optimization Problems with Random Data. Journal of Siberian Federal University. Mathematics and Physics, 2013, vol. 6, no. 4, pp. 506-515.
4. Timofeeva G.A. Probabilistic Solutions of Conditional Optimization Problems. Proceedings of the Steklov Institute of Mathematics, 2020, vol. 26, no. 1, pp. 198-211. (in Russian)
5. Matheron G. Random Sets and Integral Geometry. New York, Wiley, 1975.
6. Aliprantis D., Border K. Infinite Dimensional Analysis: A Hitchhikers Guide. Berlin, Springer, 2007.
7. Timofeeva G.A., Ie O.N. [Properties of Probabilistic Solutions of Conditional Optimization Problem with Random Parameters]. Stability and Oscillations of Nonlinear Control Systems, 2020, vol. 22, pp. 413-416. (in Russian)
8. Polyak B. Introduction to Optimization. New York, Optimization Software, 1987.
9. Varian H.R. Intermediate. Microeconomics. A Modern Approach. New York, University of California at Berkeley, 2009.
10. McDonald J. Some Generalized Functions for the Size Distribution of Income. Econometrica, 1984, vol. 52, no. 3, pp. 647-663.
11. Chotikapanich D., Valenzuela M.R., Rao D.S. Global and Regional Inequality in the Distribution of Income: Estimation with Limited. Incomplete Data. Empirical Economics, 1997, vol. 20, pp. 533-546.
12. Arnold B. Pareto Distributions: Pareto and Related Heavy-tailed Distributions. Mimeographed manuscript, Riverside, University of California at Riverside, 1980.
13. Butaeva K.O. Considering the Problem of Personal Income Distribution in Russia. The Standard of Living of the Population of the Regions of Russia, 2016, no. 2 (200), pp. 130-136. (in Russian)
14. Singh N., Rao S. The Potluck Problem with Consumers Choice Behavior. Automation Science and Engineering, 2009, pp. 328-333. DOI: 10.1109/COASE.2009.5234114