Consider a firm promoting a product in a fast expanding industry by using advertising as its single promotional tool. The firm's objective is to minimize the overall cost of advertising necessary for reaching certain target sales of the product by the end of a given planning period. We adopt the Student Work Control problem (SWC) framework for modeling this marketing context, in general, and the advertising-sales response function, in particular. We compare the SWC's optimal control budgeting principle with the solutions of equally effective, alternative advertising budgeting principles, which require strong regularity conditions on the path of either the advertising outlays or sales. In contrast to the other principles, SWC's optimal sales path is highly convex to the point that the firm may deliberately accept decreasing sales at the earliest periods. However, its optimal solution requires the firm to advertise in every period and to continue to accelerate its advertising outlays. The resulting Advertising Sales Response function, too, may therefore have a convex section with declining sales, a finding contributing to an optimization-driven explanation of threshold in advertising effect on sales
Aharon Hibshoosh. "Advertising Budget And Sales Paths Under The Dynamics Of The Student Work Control Problem And Regularity Requirements" Journal of Business & Economics Research (2012): 463-483.