That's the question asked in this working paper by Rachel Griffith, Martin O’Connell and Kate Smith. The abstract:
When we feel the pinch, we can substitute away from more expensive to less expensive foods. But, we can also increase the effort we expend in finding better prices. In short, our time is a valuable commodity that we treat like other economic goods.
That fact was also emphasized in a paper by Broda, Leibtag, and Weinstein in the Journal of Economic Perspectives in 2009. They used some creative methods to ask the question: do the poor pay more or less than the rich? They write:
Note that these authors aren't comparing apples to oranges. They compare the prices paid by rich and poor households for exactly the same goods.
When asked how consumers respond to higher prices or lower incomes, so often we economists refer to indifference curves and budget constrains or do a fair amount of hand-waving. Yet, as these studies show, reality is more complex. We can use our time to find better prices, or we can alter out consumption bundle to provide the same nutritional quality in a less expensive fashion.