Leibnitz integral rule and its application to the business firm's longevity problem

The paper takes the Leibnitz Integral Rule (LIR) under variable integration limits and demonstrates how it can be applied to a business firm's dynamic problem of determining its optimum level of investment activity, when the longevity (life span) of the investment is itself a variable determined by the level of investment. Usually, the longevity of an investment is taken as given and fixed. Here, the level of investment activity has a dual effect, determining not only the longevity of the investment, but also the firm's net profit flow (which over the life span of the investment determines the firm's stock of wealth). The chosen application is the investment needed to create and maintain the salesperson-buyer business relationship involving the sale of a given product. The demonstration shows the assumptions needed to fit the LIR to the problem in order to make the fit tractable. The need for empirical research on the form of the longevityinvestment function is also discussed.