Hysteresis and Optimal Pricing of Subscriptions with Cancellation Cost

We develop a stochastic Stackelberg model of a subscription market with cancellation costs. A representative consumer chooses when to subscribe to and cancel a service as the utility derived from the subscription evolves according to a diffusion process, while the firm selects the subscription fee and cancellation cost to maximize its expected payoff. The consumer’s problem is equivalent to the classical real-options model of entry and exit under uncertainty with adjustment costs and exhibits a two-threshold policy with an inaction band and hysteresis. Unlike the standard formulation, in which the optimal thresholds are characterized implicitly through a system of nonlinear equations, we derive an explicit parametric solution in closed form. This solution reduces the firm’s optimization problem to a two-dimensional unconstrained problem and yields a detailed characterization of the optimal pricing policy. We show that the firm’s strategy exhibits three qualitatively distinct regimes depending on the initial utility level. For small utility levels, the optimal cancellation cost is zero. In an intermediate regime, the firm’s optimal policy induces the consumer to set the entry threshold equal to the initial utility level, resulting in immediate subscription. For sufficiently large utility levels, the firm induces permanent lock-in by setting a high cancellation cost and a low subscription fee: the consumer subscribes immediately and never subsequently unsubscribes. The transition between the latter two regimes is discontinuous and results from competition between two local maxima of the firm’s payoff function. We then extend the model to a heterogeneous population of consumers. The superposition of individual two-threshold subscription strategies generates a Preisach hysteresis operator describing the aggregate dependence of the firm’s revenue on the utility dynamics. The discontinuous regime transition persists under heterogeneity, demonstrating the robustness of the underlying mechanism. The Preisach representation predicts complex history dependence and long-term effects of temporary utility shocks. For a gamma distribution of consumer preferences, the firm’s expected payoff is obtained in closed form in terms of incomplete gamma functions.