Optimizing Profits from a System of Accounts Receivable

We develop a model of the period-by-period maintenance cost and interest income accumulation resulting from charges to a system of accounts receivable, and establish conditions under which the present value of the expected profit generated by the charges reaches its greatest attainable levels. The profit measure is expressed as a function of the rate at which active accounts are paid off, which is initially treated as an independent variable, then as a function of the rate of interest charged to the active accounts. In the first case we find that profitable operation of the accounts is possible only if the ratio of the charge amount to the fixed account maintenance cost is no smaller than a specific critical value which is dependent upon the rate at which charges are lost as bad debts, the interest rate charged to active accounts, and the discount rate applied to future payments on the debts. In this case the lowest allowable payoff rate maximizes the present value of expected profits. If the payoff rate is allowed to increase with the interest rate, creating a condition of decreasing marginal efficiency of increases in the interest rate, both the income-maximizing interest rate and its associated payoff rate may occur at nonextreme values within the range of definition of these measures. The form of our model and its analysis have much in common with the Optimal Dividends Problem investigated in the Finance literature.