Even if you're concerned only with 'the poor,' therefore, the correct question is not "are the poor less able to pay higher prices than lower prices for staple goods?’ The answer to this question is all too obvious: yes.See also this relevant Economist article from a few weeks back. In Iraq, the price of gas is kept artificially low—almost zero, in fact. This doesn't help the poor or needy at all; instead gas gets leaked into the black market for much higher prices, and chronic shortages tend to hurt the poor the most (who are less able to afford to wait in long lines).
The relevant question instead is "are the poor less able to pay higher market prices than they are able to pay to take advantage of the other methods of rationing that necessarily replace higher prices?" The answer to this question isn’t at all obvious.