2014/11/04 by Dominika Bogusz, Bogusz, Dominika, Mariusz Górajski +1
Business, Management and Accounting · Economics, Econometrics and Finance · #49J20 #90B60 #Analysis of PDEs (math.AP) #Consumer Market Behavior and Pricing #FOS: Mathematics #Merger and Competition Analysis #Optimization and Control (math.OC) #Wine Industry and Tourism
paper · pdf · doi:10.48550/arxiv.1411.0880
openalex publication_date 2014/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose a new optimal model of product goodwill in a segmented market where the state variable 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 advertising efforts which influence the state equation and the boundary condition. Moreover, we introduce the mathematical representation of consumer recommendations in a segmented market. Based on the semigroup approach, we prove the existence and uniqueness of optimal controls. Using a maximum principle, we construct a numerical algorithm to find the optimal solution. Finally, we examine several simulations on the optimal goodwill model and discover two types of advertising strategies.