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A Nonconvex Control Problem for the Competitive Firm

Russell G. Thompson
R. R. Hocking
Melvin D. George

Abstract

In this study, a dynamic initial investment-borrowing model involving nonconvex investment effects and borrowing limitations is formulated as a discrete-time control problem. In the model, the firm's objective is to maximize, subject to constraints, the net worth of the firm over a finite decision-making period. Initial investment and borrowing are control parameters; and the scale of capacity use is the control variable. Investment costs, which reflect the "six-tenths" rule in particular, are nonconvex. Special considerations are thus involved in deriving the investment and borrowing rules. It is shown that the optimum must be at one of the following three points: (i) no investment and no borrowing, (ii) investment of just the endowment, and (iii) investment of the maximum amount possible. This result is especially important computationally, because the problem is convex at the points described by (ii) and (iii), and trivial at the origin. Therefore, the optimum may be computed by the use of published algorithms.