Faculty of Economics and Sociology of the University of Lodz
Abstract
We propose a new dynamic model of product goodwill where a product is sold in many
market segments, and where the segments are indicated by the usage experience of consumers. The
dynamics of product goodwill is described by a partial di erential equation of the Lotka{Sharpe{
McKendrick type. The main novelty of this model is that the product goodwill in a segment of
new consumers depends not only on advertising e ort, but also on consumer recommendations,
for which we introduce a mathematical representation. We consider an optimal goodwill model
where in each market segment the control variable is the company's advertising e orts in order to
maximize its pro ts. Using the maximum principle, we numerically nd the optimal advertising
strategies and corresponding optimal goodwill paths. The sensitivity of these solutions is analysed.
We identify two types of optimal advertising campaign: `strengthening' and `supportive'. They
may assume di erent shapes and levels depending on the market segment. These experiments
highlight the need for both researchers and managers to consider a segmented advertising polic
