Services such as FedEx charge up‐front fees but reimburse customers for delays. However, lead‐time pricing studies ignore such delay refunds. This paper contributes to filling this gap. It studies revenue‐maximizing tariffs that depend on realized lead times for a provider serving multiple time‐sensitive customer types. We relax two key assumptions of the standard model in the lead‐time pricing literature. First, customers may be risk averse (RA) with respect to payoff uncertainty, where payoff equals valuation, minus delay cost, minus payment. Second, tariffs may be arbitrary functions of realized lead times. The standard model assumes risk‐neutral (RN) customers and restricts attention to flat rates. We report three main findings: (1) With RN customers, flat‐rate pricing maximizes revenues but leaves customers exposed to payoff variability. (2) With RA customers, flat‐rate pricing is suboptimal. If types are distinguishable, the optimal lead‐time‐dependent tariffs fully insure delay cost risk and yield the same revenue as under optimal flat rates for RN customers. With indistinguishable RA types, the differentiated first‐best tariffs may be incentive‐compatible even for uniform service, yielding higher revenues than with RN customers. (3) Under price and capacity optimization, lead‐time‐dependent pricing yields higher profits with less capacity compared to flat‐rate pricing.
