Incentive Design and Pricing under Limited Inventory

Abstract

A firm faces random demand for a service it delivers on a given future date. To boost demand, the firm hires a sales agent who exerts unobservable effort continuously over time. The firm is concerned not only with increasing current demand, but also with smoothing demand over time to avoid losing goodwill if realized demand exceeds available inventory. We study the firm’s incentive design problem using a novel continuous-time principal-agent framework, in which demand drifts over time in response to an agent’s unobserved effort as well as the price the firm charges. To induce the agent’s sales effort, the firm chooses an incentive scheme that depends on the remaining inventory and the time to the service (e.g., time to departure in the case of airlines). We characterize the firm’s optimal incentive scheme under both static and dynamic pricing policies. Using parameters calibrated from the airline industry, we numerically show that under dynamic pricing, using a static incentive scheme helps the firm reap nearly all the benefits of the corresponding dynamic incentive scheme. Using a fully static strategy, on the other hand, results in a significant loss of efficiency. We also compare two partially dynamic strategies in which the firm uses dynamic pricing or dynamic contracting but not both. We show that when inventory levels are high and the demand is inelastic, the dynamic-contracting-only strategy outperforms the dynamic-pricing-only strategy; when inventory levels are low and the demand is elastic, however, the dynamic-pricing-only strategy outperforms the dynamic-contracting-only strategy.

Keywords: Incentive design, moral hazard, pricing, limited inventory, marketing-operations interface

JEL Classification: C63, D82, E2, L11