This paper explores, theoretically and experimentally, a fixed price mechanism by which, if aggregate demand exceeds supply, bidders are proportionally rationed. If demand is uncertain, equilibrium consists in overstating true demand to alleviate the effects of being rationed. Overstating is more intense the lower the price, with bids reaching their upper limit for sufficiently low prices. In the experiment, despite of a significant proportion of equilibrium play, subjects tend to (under)overbid the equilibrium strategy when rationing is (high) low, with only this latter effect being persistent over time. We explain the experimental evidence by a simple model in which the probability of a deviation is decreasing in the expected loss associated with it.