Tight bounds on the relative performances of pricing optimization mechanisms in storable good markets

In the storable good monopoly problem, a monopolist sells a storable good by announcing a price in each time period. Each consumer has a unitary demand per time period with an arbitrary valuation. In each period, consumers may buy none, one, or more than one good (in which case the extra goods are stored for future consumption incurring in a linear storage cost). We compare the performance of two important monopoly pricing optimization mechanisms: price optimization using pre-announced prices and price optimization without commitments (contingent mechanism). In pre-announced pricing the prices in each time period are stated in advance; in a price contingent mechanism each price is stated at the start of the time period, and these prices are dependent upon past purchases. We prove that monopolist can earn up to $O(logT+logN)$ times more profit by using a pre-announced pricing mechanism rather than a price contingent mechanism. Here $T$ denotes the number of time periods and $N$ denotes the number of consumers. This bound is tight; examples exist where the monopolist would earn a factor $\Omega (logT+logN)$ more by using a pre-announced pricing mechanism.