Abstract
We propose a new optimal control model of product goodwill in a segmented market where the state variable behaviour is described by a partial differential equation of the Lotka–Sharp–McKendrick type. In order to maximize the sum of discounted profits over a finite time horizon, we control the marketing communication activities which influence the state equation and the boundary condition. Moreover, we introduce the mathematical representation of heterogeneous electronic word of mouth. Based on the semigroup approach, we prove the existence and uniqueness of optimal controls. Using a maximum principle, we describe a numerical algorithm to find the optimal solution. Finally, we examine several examples on the optimal goodwill model and discover two types of marketing strategies.
Highlights
In this paper we investigate the mathematical model of the product goodwill
We propose the new way of creating goodwill among potential customers by taking into account electronic word of mouth
In each time t we assume that the value of goodwill in the segment of new consumers G(t, 0) is influenced by the positive electronic word of mouth (eWOM) recommendation and defensive and offensive marketing efforts
Summary
In this paper we investigate the mathematical model of the product goodwill. For the first time in this type of models, we consider segmentation with respect the consumer. They interpreted goodwill as the part of the demand for products that is created by current and past advertising efforts, see Nerlove and Arrow (1962), and assumed that the stock of goodwill depreciates over time at a constant rate and depends positively on the advertising effort They described the dynamics of goodwill in a non-segmented market by an ordinary differential equation. The same state equation but with a different interpretation is proposed by Barucci and Gozzi (1999) They consider a goodwill model with market segmentation and describe a monopolistic firm selling infinitely many products with new goods continuously launched onto the market.
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