A Continuous-Time Model of Subscription and Repeat Purchase

Problem definition: We present a continuous-time model of consumers’ joint decisions of membership subscription and repeat purchase. A consumer pays a flow membership fee; her valuation follows a Brownian motion, purchase needs arrive via a Poisson process, and she chooses whether to buy and when to unsubscribe. The model is analytically tractable thanks to the connection between consumer lifetime and hitting time of Brownian motions.

Methodology/results: Consumers’ unsubscription decision solves an optimal-stopping problem with a closed-form valuation cutoff; purchase occurs when valuation exceeds the transaction price. These thresholds generate three fluid consumer modes—buyer, tracker, and churner. We find that consumers who are about to terminate subscription tend to purchase less frequently, and who are currently subscribing are more likely to continue subscription. This endogenous customer loyalty is captured by a one-parameter retention formula $S(t)=2\Phi \text{}\left(\sqrt{T^{*}/t}\right)-1$, which can accommodate rapid early attrition and an asymptotic slow decay.

Managerial implications: For empirical estimation, we cast latent-valuation filtering as a heat-diffusion problem and develop a kernel-iteration likelihood approximation with vanishing discretization errors. Three empirical studies that feature two distinct firm-customer relationships demonstrate that our model performs on par with or better than existing models. Based on the model estimates, firms can infer consumers’ latent engagement and churn risk from purchase histories and devise the optimal package of membership fee and transaction price.