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We show how a discrete instrument can be used to identify the marginal treatment effects under a functional structure that allows for treatment heterogeneity among individuals with the same observed characteristics and self-selection based on the unobserved gain from treatment. Guided by this identification result, we perform a marginal treatment effect analysis of the interaction between the quantity and quality of children. Our estimates reveal that the family size effects vary in magnitude and even sign and that families act as if they possess some knowledge of the idiosyncratic effects in the fertility decision.
Brinch et al. (Fri,) studied this question.