Estimation of collective household models with Engel curves

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Why Are Older Women Missing in India? The Age Profile of Bargaining Power and Poverty (2020, background): “In particular, provided that β f = βm = βc = β , the slopes (24)Menon et al. (2012) show that for Italian households resource shares do not exhibit much dependence on household expenditure, therefore supporting identification of resource shares based on this particular assumption.”
Children's Resources in Collective Households: Identification, Estimation and an Application to Child Poverty in Malawi (2013, methodology): “This assumption implies joint restrictions on the preferences of household members and on the household’s bargaining or social welfare function Ũs (see, proposition 2 of Browning, Chiappori, and Lewbel 2008).”
Sharing of Resources Within the Family and the Economics of Household Decision Making (2013, methodology): “…strategy is to use data for both single person and couple households, by assuming that preferences do not differ in important respects between single people and those in couples with otherwise similar characteristics (Browning, Chiappori, & Lewbel, 2006; Couprie, 2007; Lewbel & Pendakur, 2008).”