# One discrete model of optimal advertising strategy

A discrete model of optimal advertising for a monopolist-seller of a new goods
is proposed. In the model, the dynamics given by a nonlinear difference equation. The nonlinearity depends on a parameter $σ$, $0 < σ < 1$, i.e. a continuous family of the models are considered. If $σ = 1/2$, a discrete version of the well-known model of S.P. Sethi is obtained. The seller’s objective is to maximize its profit up to the finite horizont T by the optimal advertising expenditure. This problem is a discrete multi-step optimal control problem, where advertising expenditure is a control variable. The Bellman method of dynaming programming is used to solve the problem. Explicit recurrence relations for the optimal control and the market share up to the step $t, t = 1, . . . , T$, are obtained under the assumption that the difference equation of the model has a solution. Sufficient conditions on the parameters of the model guaranteeing existence of a solution are found. The proposed algorothm is implemented as the procedure OptimalAdvertising in the package Maple. Numerical experiments with the procedure were carried out.

Key words: advertising, competition, optimal control, discrete model, dynamical programming