An analysis of "fixed-up-to" (FUT) pricing using a stochastic model of consumer behavior

"Fixed-Up-To" (FUT) tariffs, which are commonly used for pricing telecommunication services, are made up of three parts: a monthly subscription fee (or fixed fee), a usage-limit (or allowance), and an over-limit rate that is used for pricing consumptions in excess of the usage-limit. Typically, the consumption of a service is metered and billed separately for each billing cycle. Consequently, a consumer decides on using the service at an instant based on the inventory of allowance still available to him at that moment, the time left in his ongoing billing cycle, and the over-limit rate. Drawing from the finite-horizon inventory theory, we propose a stochastic model to capture this decision-making process. We provide a closed-form expression for the utility that the consumer gets from an FUT pricing plan. We then examine a monopolist's second-degree price discrimination problem in which consumers are offered a menu of FUT plans to choose from. We model two types of consumer heterogeneity: that in the rate at which opportunities to use the service arrive, and that in the average willingness-to-pay per opportunity. We characterize the optimal FUT price menu and also numerically compare FUT pricing with two-part tariffs and other alternatives. We prove that, in contrast with well-known findings in the literature on nonlinear pricing, there exist situations in the case of FUT pricing in which it is always optimal to serve both "high" and "low" type consumers regardless of their relative proportions. We also make a methodological contribution -- which is combining the finite-horizon inventory theory with the economic theory on second-degree price discrimination