How long does it take to sell your products?

We study the pricing problem of a seller with finite inventory over a finite sales horizon. Customers arrive stochastically with valuations drawn from short-tailed, light-tailed, or heavy-tailed distributions, corresponding to the standard classes in Extreme Value Theory. Our focus is on the normalized time required to sell k out of q products in an asymptotic regime in which the horizon becomes large. Under optimal static pricing, we show that for short-tailed and light-tailed valuations, the time required to sell k products converges to zero. For heavy-tailed valuations, the limiting distribution is non-degenerate but exhibits a positive probability of unsold inventory. Under optimal dynamic pricing, the time of the kth sale converges to an expression that involves a product of Beta random variables, whose parameters are determined by the tail class of the valuation distribution. In the light-tailed case, this reduces to the kth order statistic of a uniform distribution, implying that sales are asymptotically optimally balanced. For short-tailed and heavy-tailed valuations, the corresponding distributions are shifted slightly earlier and later, respectively, within the sales horizon. In addition, we show that sales times admit a natural interpretation on a logarithmic time scale, derive comparative statics, and demonstrate that static pricing cannot be said to be uniformly faster or slower than dynamic pricing. Our results address the question of what constitutes the “right time” to sell a product and provide practitioners with insights into expected sales patterns, which may contribute to greater trust in pricing software.