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In this paper the differential game between two profit maximizing firms considered in  is extended to the case of general (concave) effectiveness functions of advertising. For this model, a phase portrait analysis of Nash equilibrium solutions is carried out providing qualitative insights into the structure of optimal advertising rates. Applying this analysis to the Leitmann-Schmitendorf advertising model, we obtain more detailed information on the structure of the optimal controls. Moreover, Nash-optimal solutions are derived for infinite duration of the game.