implies and is implied by additive separability of choice of treatment equations in terms of observed and unobserved variables. These results follow from properties of binary matrices developed in this paper. We investigate conditions under which unordered monotonicity arises as a consequence of choice behavior. We characterize IV estimators of counterfactuals as solutions to discrete mixture problems.

This paper develops methods for evaluating marginal policy changes. We characterize how the effects of marginal policy changes depend on the direction of the policy change, and show that marginal policy effects are fundamentally easier to identify and to estimate than conventional treatment parameters. We develop the connection between marginal policy effects and the average effect of treatment for persons on the margin of indifference between participation in treatment and nonparticipation, and use this connection to analyze both parameters. We apply our analysis to estimate the effect of marginal changes in tuition on the return to going to college.

This paper formulates and estimates multistage production functions for child cognitive and noncognitive skills. Output is determined by parental environments and investments at different stages of childhood. We estimate the elasticity of substitution between investments in one period and stocks of skills in that period to assess the benefits of early investment in children compared to later remediation. We establish nonparametric identification of a general class of nonlinear factor models. A by-product of our approach is a framework for evaluating childhood interventions that does not rely on arbitrarily scaled test scores as outputs and recognizes the differential effects of skills in different tasks. Using the estimated technology, we determine optimal targeting of interventions to children with different parental and personal birth endowments. Substitutability decreases in later stages of the life cycle for the production of cognitive skills. It increases in later stages of the life cycle for the production of noncognitive skills. This finding has important implications for the design of policies that target the disadvantaged. For some configurations of disadvantage and outcomes, it is optimal to invest relatively more in the later stages of childhood.