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For airlines selling the same seats on a scheduled flight at different fares, the demand for a fare class is affected not only by the current availability of lower fares but also by the possibility of future availability of them. To address this type of passenger behavior, this paper studies a two-class dynamic seat allocation model, which has two distinctive features. The model assumes first that the discount fare cannot be reopened once closed and, second, that a fraction of the customers are flexible, i.e., while willing to pay the full fare, they would buy discount fare tickets if available. These assumptions not only reflect customers' behavior but also are consistent with a class of existing static models that are widely accepted by the industry.For this model, we derive structural properties of the optimal policy. We show that the optimal policy is a threshold policy: The discount fare should be closed as soon as the number of seats remaining reaches a predetermined threshold, which is a function of the time remaining before departure. We show that the threshold does not always decrease over time, and that its time-monotonicity depends on how the proportion of flexible customers changes over time. Our model explains why airlines close discount fares as the departure time approaches. We also show a close relationship between the optimal policy and the policies suggested by the existing static models (the Littlewood rule and its variants). Our numerical study shows that, for parameters plausible to real applications, the latter policies, although not optimal for our dynamic model, perform well, compared to the performance of the optimal policies.