Testing the Barten Model of economies of scale in household consumption: toward resolving a paradox of Deaton and Paxson

Households enhance the welfare of constituents by sharing shelter and appliances and economizing in services such as maintenance, child care, and food preparation. Resources so conserved can be applied to more personal goods such as food, clothing, or beauty care. Economizing by households is an important factor to consider when measuring poverty and inequality. In 1895, Ernst Engel observed that the share of food consumption in total consumption can be used to measure economic welfare across households of different sizes and incomes; larger or richer households spend a smaller share of their total expenditure on food. Although more elaborate methods have been developed (see, e.g., a survey by Browning [1992]), the use of food share or the budget share of other bundles of “necessities” as a proxy for welfare remains a viable method for measuring poverty and studying equivalence scales (see, e.g., Deaton and Muellbauer 1986; Lanjouw and Ravallion 1995). Engel’s method is popular for its simplicity: it has only one demand equation to be estimated, with the food share of total expenditure as a dependent variable. A single cross‐sectional survey usually provides sufficient data for the analysis, and no information on prices is required.

Barten (1964) incorporates household size into a unitary household consumption model. Deaton and Paxson (1998) extend the model and use it to derive the implication that, when per capita expenditures (PCE) are held constant, food share should increase with household size. This prediction is inconsistent with Engel’s empirical observation and with Deaton and Paxson's empirical tests showing that larger households spend a smaller share of their budget on food. Deaton and Paxson conclude that “the estimates of economies of scale that are derived by Engel’s method have no theoretical underpinning and are identified by an assertion that makes no sense” (p. 903).

Compounding the puzzle, empirical evidence in their article contradicts a second implication of the Barten model as well. The elasticity of the budget share of food with respect to household size should be larger in value in poor countries than in rich countries, but empirical estimates for countries with different income levels show the opposite.

Any inconsistency between empirical evidence and a theoretical model naturally casts doubt on the model. In this paper, we follow the assumptions of the Barten model precisely and examine food as a share of food plus the other goods known to be more public than food (i.e., housing). When we do so, we find that the food share indeed increases with family size as predicted. Likewise, when we analyze food as a share of food plus a good known to be more private than food, we observe that the food share decreases with family size, as predicted by the Barten model. We also show that “food consumed away from home” decreases with family size, suggesting that economies of scale in preparation time for home food increase the chance that food is more public than the composite of all other goods.

Finally, we address the inconsistency between the second prediction of the Barten model and the data: the elasticity of food share with respect to household size should be larger in value for poorer households, whereas the estimates of Deaton and Paxson show the opposite. The maintained assumption in welfare comparison is the same utility function across households. Since it is unlikely that this assumption holds across different countries such as the United States, South Africa, and Russia, we estimate the model by expenditure quartiles within a country. Our estimates show that the elasticity of food share with respect to household size is indeed larger for poorer households within a country, consistent with the prediction of the Barten model. Finally, in contrast to previous studies such as Muellbauer (1977), Pollak and Wales (1981), and Barnes and Gillingham (1984), we allow more flexible forms to account for the effect of family composition in testing the Barten model.

This comment is organized as follows: Section II introduces the Barten model and describes the empirical work by Deaton and Paxson that appears to contradict the model. Section III presents our alternative empirical tests that instead support the model. Section IV concludes the paper.

• http://dx.doi.org/10.1086/378534