An Optimal Strategy in the Modified Nerlove-Arrow Advertising
Speaker
Abstract
| We present results of joint work with Prof. A.Rozhdestvenskii (Moscow State University). The classical Nerlove-Arrow model (1962) describes the dependency of goodwill on the current advertising efforts (money spent to advertise goods). It is based on the following beliefs on investments in advertisement by a firm. These affect its present and future benefit from sales, and build some sort of advertising capital, called goodwill. In the model, the total profit of a firm is measured by the integral (over time) of the utility function of the goodwill; the investments are connected to the goodwill by a special differential equation. The problem is to find the optimal investment policy, which maximizes the income of the firm. The utility function, the initial capital and the inflation factor are given. The classical solution of this problem leads in many cases to a so-called “impulse control”: we have to invest money with an infinite speed. To avoid this phenomenon we modify the model by introducing one natural constraint: the current advertising efforts do not exceed some fixed upper bound. We show that an optimal strategy does exist for any initial data. The solution is more complicated than that for the classical model, it involves many possible cases. Nevertheless, in each case it can be found explicitly by the Pontryagin maximum principle. |
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| Jan Brinkhuis |
