We consider a seller who wishes to sell K goods by time T. Potential buyers enter IID
over time and are patient. At any point in time, profit is maximized by awarding the good
to the agent with the highest valuation exceeding a cutoffs. These cutoffs are characterized
by a one-period-look-ahead rule and are deterministic, depending only on the number of
units left and the time remaining. The cutoffs decrease over time and in the inventory size,
with the hazard rate of sales increasing as the deadline approaches. In the continuous time
limit, the optimal allocation can be implemented by posted-prices with an auction at time
T. Unlike the cutoffs, the prices depend on the history of past sales.