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