Providing two different households with the same standard of living, making them equallywell off, requires some denition of well-being. In the early literature on equivalencescales, a household's well-being was dened in terms of needs, such as having a nutritionally adequate diet.Engel (1895) observed that household's food expenditures are an increasing functionof income and of family size, but that richer households tend to spend a smaller share oftheir total budget on food than poorer households. He therefore proposed that this foodbudget share could be a measure of a household's welfare or standard of living. Theresulting Engel equivalence scale is dened as the ratio of incomes of two different sizedhouseholds that have the same food budget share. This is essentially the method used bythe United States Census Bureau to measure poverty. Roughly, the bureau rst denes thepoverty line for a typical household as three times the cost of a nutritionally adequate diet,then uses food shares (Engel scales) to derive comparable poverty lines for households ofdifferent sizes and compositions, and nally adjusts the results annually by the consumerprice index to account for ination (see Fisher 1997).2Similar to Engel scales, given two households that differ only in their number or agedistribution of children, Rothbarth (1943) equivalence scales are dened as the ratio ofincomes of the two households when each household purchases the same quantity of somegood that is only consumed by adults, such as alcohol, tobacco, or adult clothing.Modern equivalence scales measure well-being in terms of utility, using cost (expenditure) functions estimated from consumer demand data via revealed preference theory.Having Engel or Rothbarth scales equal valid cost function based equivalence scales requires strong restrictions regarding the dependence of demand functions on characteristicssuch as age and family size, and on the links between demand functions and utility forthese different household types.One strand of the equivalence scale literature focuses on the former issue, and so dealsprimarily with the empirical question of how best to model the dependence of household Marshallian demand functions on demographic characteristics. Examples are Sydenstricker and King (1921), Prais and Houthakker (1955), and Barten (1964) scales, in whicha different Engel type scale is constructed for every good people purchase, roughly corresponding to a different economies of scale measure for each good. Other examples areGorman's (1976) general linear technologies, Lewbel's (1985) modifying functions, andPendakur's (1999) shape-invariance.The second, closely related literature, focuses on the joint restrictions on both preferences and interpersonal comparability of utility required for measuring the relative costs ofproviding one household with the same utility level as another. Examples include Jorgenson and Slesnick (1987), Lewbel (1989), Blackorby and Donaldson (1993), and Donaldsonand Pendakur (2004, 2006)