  We consider the problem of maximizing the expected revenue from selling $k$
homogeneous goods to $n$ unit-demand buyers who arrive sequentially with
independent and identically distributed valuations. In this setting the optimal
posted prices are dynamic in the sense that they depend on the remaining
numbers of goods and buyers. We investigate how much revenue is lost when a
single static price is used instead for all buyers and goods, and prove upper
bounds on the ratio between the maximum revenue from dynamic prices and that
from static prices. These bounds are tight for all values of $k$ and $n$ and
vary depending on a regularity property of the underlying distribution. For
general distributions we obtain a ratio of $2-k/n$, for regular distributions a
ratio that increases in $n$ and is bounded from above by $1/(1-k^k/(e^{k}k!))$,
which is roughly $1/(1-1/(\sqrt{2\pi k}))$. The lower bounds hold for the
revenue gap between dynamic and static prices. The upper bounds are obtained
via an ex-ante relaxation of the revenue maximization problem, as a consequence
the tight bounds of $2-k/n$ in the general case and of $1/(1-1/(\sqrt{2\pi
k}))$ in the regular case apply also to the potentially larger revenue gap
between the optimal incentive-compatible mechanism and the optimal static
price. Our results imply that for regular distributions the benefit of dynamic
prices vanishes while for non-regular distributions dynamic prices may achieve
up to twice the revenue of static prices.
