In the last term S is rescaled for notational convenience to be the fraction of

In the last term s is rescaled for notational

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In the last term, S is “rescaled” (for notational convenience) to be the fraction of time spent in school by the child in time period 1. The remaining time during that period, 1 - S, is spent working. For simplicity, equations (3) and (4) assume no borrowing or saving. The only way to transfer income between periods 1 and 2 is to invest in children's education. The ability to borrow or save would reduce parents’ need to invest in children's education but would not eliminate it because it is prudent for parents to diversify their investments among several different risky alternatives. Finally, equation (5) relates the child's cognitive skills to his or her income: Y c = π A, (5) where π is the productivity of cognitive skills in the labor market. 8
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Substitution of (2) into (5), of (5) into (3) and (4), and finally of (2) - (4) into (1) gives: U = Y 1 - pS + δ Y 2 + ((1-S+ δ )k π + σ ) α f(Q)g(S) (6) A. Exogenous School Quality At first, assume that school quality is exogenous, so that S is the only choice variable. Differentiating (6) with respect to S, setting the derivative equal to zero, and totally differentiating this condition gives the impacts of changes in the model's various parameters on the optimal (utility-maximizing) value of years of schooling. Optimal years of schooling increases (and thus the cognitive skills acquired by the child increase) when the following increase: the child's learning efficiency ( α ) school quality (Q) the weight ( δ ) parents give to future consumption relative to current consumption parental tastes for schooling ( σ ) 9
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