Because food choices have important implications for human health and the environment, consumers are increasingly urged to modify their purchasing and eating habits so as to comply with a set of norms (e.g., eat at least five portions of fruits & vegetables a day; reduce meat consumption). However, the effect of compliance with those norms on the composition of the entire diet is uncertain because of potentially complex substitutions. To lift this uncertainty, we propose a model which extends the theory of the consumer under rationing to the case of multiple linear constraints. The effect on the diet of compliance with norms is derived from information on consumer preferences (price and expenditure elasticities of demand), consumption levels, and technical coefficients for each food (e.g., nutritional composition, GHG emissions). The model is then used to simulate how the French diet would respond to a five percent increase in fruit and vegetable consumption, as well as the related changes in diet quality and greenhouse gas emissions.