Although individuals with poor food security might be expected to have reduced food intake, and therefore a lower likelihood of being overweight, some empirical evidence has indicated that overweight status is actually more prevalent among the food insecure. As obesity is associated with excessive energy intake, and hunger reflects an inadequate food supply, such observations would appear to be paradoxical. We develop an economic model that shows that this apparently paradoxical result is consistent with rational behavior regarding food availability risk and the effectiveness of food storage options. The amount of internal storage increases as the variance of food productivity in the second period increases, which is consistent with the empirical observation of a positive relationship between food insecurity and the incidence of overweight.