An Infinite Horizon Differential Game of Optimal CLV-Based Strategies with Non-atomic Firms

Abstract

We study the structure of optimal customer acquisition and customer retention strategies as a differential game over an infinite horizon in an industry with a large number of non-atomic firms. The optimal retention effort is constant over time and the optimal acquisition effort is proportional to the size of potential customer base. Greater customer profitability leads to higher per- capita acquisition and retention efforts, larger size of firms, and lower churn rate. A greater discount rate leads to lower per-capita acquisition and retention efforts, smaller firm size, and a greater churn rate. Tougher competition lowers the firms’ acquisition and retention expenditures and it does not affect per-capita values. Both the churn rate and the share of acquisition expenditures in the total marketing budget decrease as firms grow over time. We revisit the concepts of the customer lifetime value (CLV) and the value of the firm in the dynamic equilibrium of an industry with a large number of players and demonstrate the equivalence between maximization of the value of the firm and maximization of a firm’s individual CLV.