Optimal Advertising for the Nerlove-Arrow Model Under a Budget Constraint

This paper considers an optimal control problem for the dynamics of the Nerlove-Arrow advertising model, the optimal control being the rate of advertising expenditure required to maximize the present value of net profit streams (or, sales) over a finite horizon subject to a budget constraint. The maximum available advertising budget is given in present-value terms. It is shown that a singular level of advertising exists and that it is smaller than the singular or optimal stationary level obtained by Nerlove and Arrow. Along with the forms of the optimal control in all possible cases, two simple computational algorithms are also provided to obtain the optimal path including the end-game situation as a result of the finite horizon. Closed-form solutions are worked out for simple examples illustrating the use of the algorithm. An analogy with a car with a given amount of fuel (budget) concludes the paper.