A Dynamic Programming Model of Demand for Money with a Planned Total Expenditure

Abstract

RECENT CONTRIBUTIONS BY Eppen and Fama [6, 7], Girgis [10], Miller and Orr [13], Tsiang [21] as well as the present author [5] apply inventory theory to analyze an economic agent's demand for cash arising out of an uncertain time course of cash needs. A feature common to all of them is an implicit assumption that either the economic agent has no plan whatever regarding his total income and expenditure over the period of his planning horizon or, as in Tsiang, his plan over the income period, need not necessarily be realized. Leaving the economic agent's total income and/or expenditure over the planning horizon random in this way, keeps the density functions of net payments due independent from period to period, and no doubt, makes the analysis simpler. It may also have some practical relevance, especially when the income and expenditure plans of the economic unit over the planning horizon are not exact. From a theoretical point of view, however, it seems somewhat unsatisfactory to leave the economic agent's total income and especially his total expenditure over the horizon as random and unplanned. In this paper, we shall, therefore, develop a model of demand for money based on the idea that the economic agent has some income-expenditure plan for the period of his horizon and yet he is not entirely certain about the time course of his cash needs during this period. Patinkin [14] has presented a similar model incorporating the intra-period uncertainty of receipts and payments. Restricting the uncertainty of payments and receipts only with respect to their time course during the leaves Patinkin free to determine the individual's expenditures on goods and services as well as his income for each week by using the well-known Fisherine framework of maximizing the intertemporal utility function of the individual subject to his wealth-constraint, now modified to take into account the (minimum) cost of cash management during each week.2 In this paper, we consider an individual who plans to make a given total