Identification of semiparametric model coefficients, with an application to collective households

Abstract

We prove identification of coefficients for a set of semiparametric specifications that are related to multiple index models. Potential applications of these results include models of observed heterogeneity in production functions and in consumer demand systems.We then generalize these results to identify a class of collective household consumption models. We extend the existing literature by proving point identification, rather than the weaker generic identification, of all the features of the collective household model, including price effects. We estimate the model using Japanese consumption data, and find substantial variation in resource shares and indifference scales across households of different sizes.

The more the poorer? Resource sharing and scale economies in large families

“…Figure A1 in the Appendix shows the distribution of household size in selected countries. 2 Parallel to our work, Lewbel and Lin (2020) and Donni and El Badaoui (2020) provide identification results that, at least partly, rely on preference stability between singles and individuals in couples or price variation. Alternative methods achieve set-identification of resource shares using axiomatic revealed preference methods (Cherchye et al, 2011(Cherchye et al, , 2015(Cherchye et al, , 2016(Cherchye et al, , 2017.…”

Inefficient Collective Households: Cooperation and Consumption

“…Given sufficient data, the household's demand equations ( 7) could be estimated as described by BCL or by Lewbel and Lin (2021). BCL show identification of the model on the basis of observing both household demand functions for all goods and individual demand functions for all goods.…”

“…Thus, without further assumptions, BCL does not identify the resource shares of children (who do not live alone as singles). However, Lewbel and Lin (2021) provide examples of additional assumptions that allow the BCL to be identified and estimated for children as well as adults. Primary among these additional assumptions is the presence of private, assignable goods, which are also used by Dunbar, Lewbel and Pendakur (2013) in their cross section, Engel curve identification arguments.…”

“…BCL and Lewbel and Lin (2021) give alternative conditions under which the demand functions, resource share functions, and A matrix in BCL are nonparametrically either point identified or generically identified (see also Chiappori and Ekeland 2009). A similar argument to theirs shows identification of our general model above.…”

“…In particular, suppose we impose the minimal regularity needed to guarantee that equation ( 9) is maximized by only one value of f , which depends on v. For example, if f is discrete then we only need to rule out knife edge cases where multiple values of f yield the exact same value for the function Ψ regardless of the values of the other covariates. Then, conditional on any given value of v and hence of f , our general model reduces to BCL, and so the demand functions, resource share functions, and A matrix in our model can also be identified, using the arguments in BCL and Lewbel and Lin (2021).…”