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.
Highlights
In a ground-breaking analysis by Slutsky [1], the foundation for rigorous analysis of optimal consumption decision was laid
A dual problem to the utility maximization problem is the minimization of the budget subject to maintaining the utility level achieved before
One can derive a theorem concerning the relationships between the price effect of the demand of a commodity and the pure substation effect and the wealth effect in a random horizon stochastic dynamic framework as follows
Summary
In a ground-breaking analysis by Slutsky [1], the foundation for rigorous analysis of optimal consumption decision was laid. The effect of a price change on the demand of goods can be decomposed into tractable terms from the primal and dual problems yielding significant economic implications This prominent contribution in consumer theory, known as the Slutsky equation, was christened by John Hicks as the “Fundamental Equation of Value Theory”. Yeung [8] extends Slutsky’s work to a dynamic framework in which the consumer has a T-period life span with future incomes being uncertain. Another milestone in consumer theory is the Roy’s identity [9] which provides an often invoked mathematical result in consumer theory.
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