Grosset, Luca and Viscolani, Bruno (2007) Dynamic advertising with constant exogenous interference. [Technical Report] (Inedito)
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We propose a model of a firm which advertises a product in a homogeneous market where an a constant exogenous interference is present.
Using the framework of the Nerlove and Arrow's advertising-goodwill-demand model, we assume that the interference acts additively on the goodwill production term as a negative term.
Hence we admit that the firm goodwill may become negative
and associate a zero demand with negative goodwill values.
For simplicity we consider a piecewise linear demand function
and we are led to formulate a nonsmooth optimal control problem with infinite horizon.
We obtain that an optimal advertising policy exists and takes one of two forms: either a positive and constant advertising effort, or a decreasing effort, starting at a positive level and eventually reaching the zero value at a finite exit time.
In both casesa we have explicit representations of the optimal control which are obtained through the study of two auxiliary smooth optimal control problems.
It is interesting to observe that the fundamental choice between staying in the market and going out of business at some time
depends both on the interference size and on the initial state (goodwill).
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