Selling Passes to Strategic Customers

Passes are prepaid packages of multiple units of goods or services with flexible consumption times. They may take a variety of forms such as commuter passes in transportation, capped quotas in telecommunications, or memberships in health or beauty clubs. We consider a monopolist selling products or services to strategic customers by dynamically pricing passes in conjunction with individual items. The strategic behavior is captured by a dynamic choice model that endogenizes strategic purchase, utilization, and renewal of the pass. Under the control-theoretic framework, we find that the optimal pricing policy has a turnpike property; the optimal price trajectories stay near the steady state for most of the sales horizon, and the fixed-pricing policy performs very well when the horizon is long enough. In the turnpike, we show that passes should offer a quantity discount when customers are not fully strategic. As a form of advance purchase, passes allow the firm to capitalize on strategic behavior without limiting the supply. From the revenue-recognition principle, we show that a passholder can generate a higher revenue rate than a nonpassholder customer.