That's the question asked in this working paper by Rachel Griffith, Martin O’Connell and Kate Smith. The abstract:
“Over the Great Recession real wages stagnated and unemployment increased. Concurrently, food prices rose sharply, outstripping growth in food expenditure, and leading to a reduction in calories purchased. This has led to concern about rising food poverty. We study British households to assess how they adjusted to changes in the economic environment. We show they switched to cheaper calories; implying food consumption was smoother than expenditure. We use longitudinal data to quantify the way households lowered their per calorie spending, and show they done this in part by increasing shopping e ffort, and without lowering the nutritional quality of their groceries.”
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:
“In this paper, we circumvent the problems of previous studies by using a dataset that contains actual purchases of around 40,000 households collected by Nielsen. By focusing on the actual prices paid by households, we show that poor households systematically pay less than richer households for identical goods. The poor pay less in part because they shop in cheaper stores and in part because they pay less for the same goods even in the same store. This latter effect probably arises because poorer households are more likely than richer households to buy goods on sale, even in the same store. We also confirm that the poor shop more in convenience stores—where prices are 11 percent higher than in traditional grocery stores—but show that this effect is dominated by their higher share of expenditure in supercenters where prices are 10 percent lower than in grocery stores.”
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.