Dynamic Marketing Model: The Case of Piece-Wise Constant Pricing

We study a stylized vertical distribution channel where a representative manufacturer sells a single kind of good to a representative retailer. The control of the manufacturer is the price discounts, while the control of the retailer is pass-through. In the classical setting, the arising problem is quadratic with respect to wholesale price discount and pass-through. Thus, the optimal sale price is continuous. It seems elegant mathematically but not adequate economically. Therefore we assume that the controls are constant or piece-wise constant. This way, the optimal control problem reduces to the mathematical programming problem where the profit of the manufacturer is quadratic with respect to price discount level(s), while the profit of the retailer is quadratic with respect to pass-through level(s). We study the concavity property of the profits. This allows getting the optimal behavior strategies of the manufacturer and the retailer.