Optimal Equity and Financing Model of Krouse and Lee: Corrections and Extensions

Abstract

Krouse and Lee (1973) have formulated an optimal financing problem of a firm in the dynamic setting of optimal control theory. Specifically, the problem is to find a financing mix of retained earnings and external equity over time in a way that maximizes the present value of the entire future dividends stream accruing to the firm's initial stockholders subject to a given maximum allowable growth rate for the firm.

In their solution of the problem using the maximum principle, however, Krouse and Lee have made some errors. As a result, they have ended up solving a finite horizon problem instead of the infinite horizon problem they claim to have solved. Furthermore, the resulting finite horizon problem has, thus, been left without a bequest function. And it is a common knowledge that finite horizon problems without a bequest function are unrealistic. The purpose of this paper is, therefore, threefold:

i) To point out the mistakes committed by Krouse and Lee and how these affected their solution. ii) To present the correct solution for the infinite horizon problem. Also noted are the delicate mathematical points which are required i n passing from finite horizon solutions to infinite horizon solutions. iii) Finally, to extend the finite horizon problem to include a bequest (or salvage value) function. Furthermore, we obtain a complete solution for the problem when the bequest function is linear. That is, we show how the solution changes when the importance of the bequest function changes relative to the present-value of dividends captured.

Keywords: Optimal Financing, optimal control, Miller-Modigliani theory, finacing mix, the maximum principle, retained earnings, equtiy financing, financial engineering

JEL Classification: C61, M2, G3