A Generalization of the Nerlove-Arrow Model to Multi-Firms Advertising under Uncertainty

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Abstract
This paper considers a multi-firms advertising model under uncertainty of the Nerlove-Arrow type. It is shown (under a Markovian assumption of the advertising process) that the probability of sales for a firm in a multi-firms setting is given by a Poisson distribution with a parameter given by its market share at a given time. Using the multi-firms model developed in this paper, advertising differential games are resolved. In particular, a two-firms advertising model with concave advertising costs is used to obtain optimum advertising budgets for competitive firms.

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