A Note on Modeling Simple Dynamic Cash Balance Problems

Sethi and Thompson (1970) illustrated the applications of the maximum principle to solve simple dynamic cash balance problems. There we introduced the idea of penalty function to solve the cash balance problem with bounded state variables arising out of disallowing overdrafts and short selling. This resulted in the adjoint equations containing terms in the state variables. We then stated the need for solving a two-point boundary value problem. In this note, we show that for the example under consideration we can get the optimal solution without solving a two-point boundary value problem. That we can do so depends on two observations: (1) the adjoint equations are independent of the terms in state variables if they are within their bounds, and (2) the Hamiltonian can be easily maximized in a way that keeps the state variables at nonnegative levels