A mathematical model describing the process of promoting two competing products on the market is suggested. The key parameters in the model are the intensity of advertisement, the rates of contact between nonusers and adopters of the product, and the rates at which adopters give up using the product. Since the authors are concerned with the asymptotic behavior of solutions, under a simplifying assumption, the order of the original system of three ordinary differential equations reduces to two, and the system assumes the form \[ \begin{aligned} \frac{dA_{1}}{dt}& =\left( \gamma_{1}+\lambda_{1}A_{1}\right) \left( C-A_{1}-A_{2}\right) -\alpha_{1}A_{1},\\ \frac{dA_{2}}{dt}& =\left( \gamma_{2}+\lambda_{2}A_{2}\right) \left( C-A_{1}-A_{2}\right) -\alpha_{2}A_{2},\end{aligned} \] where \(A_{i}\) is the number of users of the first and the second product respectively, \(\gamma_{i}\) is the intensity of the advertisement for each product, and \(\lambda_{i}\) is the rate of contact between users of a product with nonusers. In the case when \(\gamma_{i},\lambda_{i}>0,\) it is proved that there exists a unique positive equilibrium for the above system that is globally stable. Analogous results are derived for the case when either \(\gamma_{i}\) vanishes. Finally, the authors study the global stability of a positive equilibrium in a modified model in which delayed arguments associated with the testing periods for new products are introduced.