Optimal Consumption under Uncertainties: Random Horizon Stochastic Dynamic Roy’s Identity and Slutsky Equation

David W. K. Yeung

Center of Game Theory, St Petersburg State University, St Petersburg, Russia;SRS Consortium for Advanced Study in Cooperative Dynamic Games, Shue Yan University, Hong Kong, China.

DOI: 10.4236/am.2014.52028                

Abstract

This paper extends Slutsky’s classic work on consumer theory to a random horizon stochastic dynamic framework in which the consumer has an inter-temporal planning horizon with uncertainties in future incomes and life span. Utility maximization leading to a set of ordinary wealth-dependent demand functions is performed. A dual problem is set up to derive the wealth compensated demand functions. This represents the first time that wealth-dependent ordinary demand functions and wealth compensated demand functions are obtained under these uncertainties. The corresponding Roy’s identity relationships and a set of random horizon stochastic dynamic Slutsky equations are then derived. The extension incorporates realistic characteristics in consumer theory and advances the conventional microeconomic study on consumption to a more realistic optimal control framework.

Keywords

Optimal Consumption; Uncertain Inter-Temporal Budget; Stochastic Dynamic Programming; Slutsky Equation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	E. E. Slutsky, “Sulla Teoria Del Bilancio Del Consumatore,” Giornale Degli Economisti, Vol. 51, No. 1, 1915, pp. 1-26.
[2]	R. G. D. Allen, “Professor Slutsky’s Theory of Consumers’ Choice,” Review of Economic Studies, Vol. 3, No. 2, 1936, pp. 120129.
[3]	R. G. D. Allen, “The Work of Eugen Slutsky,” Econometrica, Vol. 18, No. 3, 1950, pp. 209-216.
[4]	J. R. Hicks and R. G. D. Allen, “A Reconsideration of the Theory of Value. Parts 1-2,” Economica, New Series, Vol. 1, No. 1 & 2, 1934, pp. 52-76 & 196-219.
[5]	H. Schultz, “Interrelations of Demand, Price, and Income,” Joumal of Political Economy, Vol. 43, No. 4, 1935, pp. 433-481.
[6]	P. C. Dooley, “Slutsky’s Equation Is Pareto’s Solution. History of Political Economy,” Vol. 15, No. 4, 1983, pp. 513-517.
[7]	T. W. Epps, “Wealth Effects and Slutsky Equations for Assets,” Econometrica, Vol. 43, No. 2, 1975, pp. 301-303.
[8]	D. W. K Yeung, “Optimal Consumption under an Uncertain Inter-Temporal Budget: Stochastic Dynamic Slutsky Equations,” Vietsnik St Petersburg University: Mathematics, Vol. 10, No. 3, 2013, pp. 121-141.
[9]	R. Roy, “La Distribution Du Revenu Entre Les Divers Biens,” Econometrica, Vol. 15, No. 3, 1947, pp. 205-225.
[10]	M. L. Puterman, “Markov Decision Processes: Discrete Stochastic Dynamic Programming,” John Wiley & Sons, New York, 1994. http://dx.doi.org/10.1002/9780470316887
[11]	D. P. Bertsekas and S. E. Shreve, “Stochastic Optimal Control: The Discrete-Time Case,” Athena Scientific, 1996.
[12]	D. W. K Yeung and L. A. Petrosyan, “Subgame Consistent Cooperative Solution of Dynamic Games with Random Horizon,” Journal of Optimization Theory and Applications, Vol. 150, No. 1, 2011, pp. 78-97.
[13]	D. W. K Yeung and L. A. Petrosyan, “Subgame Consistent Solution for Cooperative Stochastic Differential Games with Random Horizon,” International Game Theory Review, Vol. 14, No. 2, 2012, pp. 1250012.01-1250012.22.
[14]	R. Bellman, “Dynamic Programming,” Princeton University Press, Princeton, 1957.
[15]	M. T. Cheung and D. W. K. Yeung, “Microeconomic Analytics,” Prentice Hall, New York, 1995.