The second Engel law: Is it a paradox?

The second Engel law says that the Engel curve for food moves out as family size increases, thus showing a decrease in welfare. What is puzzling, though, is that this regularity does not hold for equivalent income functions expressed in per capita terms. Deaton and Paxon [1998. Economies of scale, household size, and the demand for food. Journal of Political Economy 106 (5), 897-930] show that holding per capita total household expenditure constant, per capita expenditure on food falls with the number of heads. Deaton and Paxson's empirical evidence from developed and less developed countries seems to invalidate the claim of the second Engel's law. The main objective of this paper is to understand such paradoxical relationship between food consumption and household size. Our nonparametric empirical evidence, drawn from the Colombian 1985 urban survey, shows that the food share is negatively related to total household per capita expenditure in line with Deaton and Paxson's claim, but equivalent incomes shift to the right as theory predicts. The regularity of our nonparametric results is an indication of a problem in the parametric specification of the Engel curve modified by family size. In fact, using also the surveys of Italy, Nepal, Djibouti, and Bangladesh we show that a theoretically plausible modified Engel curve is coherent with the claim of the second Engel law and explains Deaton and Paxson's paradox.