### ON A DISCRETE MODEL OF OPTIMAL ADVERTISING

#### Abstract

The continuous models are considered in the most works on optimal advertising. Articles on the discrete-time models are more rare because in this case it is difficult to obtain an explicit solution. In this paper a new discrete model of optimal advertising for a monopolist-seller of a new goods is proposed. In the model, the dynamics is given by a nonlinear difference equation. The non-linearity depends on a parameter $\sigma$, $0<\sigma<1$, i.e. a continuous family of the models is considered. The discrete versions of the Vidale -- Wolfe model and the Sethi model are particular cases of this model. The seller's problem is to maximize its profit up to the finite horizon $T$ by the optimal advertising expenditure. This problem is a discrete multistep optimal control problem, where an advertising expenditure is a control variable. For our model the optimal control problem can be solved explicitly. The Bellman method of dynaming programming is used to study the problem. Explicit recurrence relations for the optimal control and the market share up to the step $t$, $t=1,\ldots, T$, are obtained under the assumption that the difference equation of the model has a solution. Sufficient conditions on the parameters of the model, which ensure the existence of a solution, are found. The proposed algorithm is implemented as the procedure OptimalAdvertising in the package Maple. Numerical experiments with the procedure were carried out.

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