An Economic Model of Membership Subscription and Repeat Purchase Based on Continuous Information Tracking

Abstract

We present an economic model of consumers' membership subscription and repeat purchase behaviors based on their information tracking decisions in continuous time. The model features constant volatility in consumers' belief updating that signifies evolving product designs, as well as a constant flow cost that is flexible to capture both time cost of information tracking and potentially membership subscription fee. The model is analytically tractable thanks to the connection between consumer lifetime and hitting time of Brownian motions.

Three modes of consumers arise endogenously: buyers, trackers and defectors, depending on their product valuations. We find that consumers who are about to drop out tend to purchase less frequently, and who are currently alive tend to stay alive. This \textit{endogenous customer loyalty} is captured by a \textit{one-parameter retention rate formula} that features an initial steep drop and an asymptotic slow decay. For empirical estimation, we leverage a unique feature of our model by showing that it can be transformed into a \textit{heat diffusion equation}, which we solve numerically by both finite-difference and finite-element methods and benchmark them against standard random walk simulations.

Three comprehensive studies that feature distinct firm-customer relationships demonstrate that our model is empirically flexible. Based on the model estimates, a marketing manager can devise the optimal package of transaction price and membership fee, and infer which segment a consumer belongs to based on her past purchase history. Interestingly, consumers may benefit from a higher information tracking cost due to a lower price set by the firm.

Keywords: information tracking, repeat buying, membership subscription, Brownian motion, optimal stopping, heat diffusion equation

JEL Classification: M31