4progresa - 1 PROGRESA So far we focused on supply-side...

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1 PROGRESA So far, we°ve focused on supply-side innovations Who should be e/ected most by these? Recall our model ±kids go to school if ° ( w 1 ° w 0 ) > c + w 0 What if some people are poor and some people are rich? So, parents have income w 1 or w 0 ; i.e. we have rich parents who went to school and poor parents who did not. Let°s suppose that kids get to share this income when they are young. So, then, rich kids get w 1 + w 0 + °w 0 from no school and w 1 + °w 1 ° c from school. So, will go to school if w 1 + °w 1 ° c > w 1 + w 0 + °w 0 this is the same equation as before. Nothing in that equation. But, suppose we care about people°s utility over income. Utility seems likely to be concave, i.e. there is declining marginal utility of money. Then we get u ( w 1 ° c ) + °u ( w 1 ) > u ( w 1 + w 0 ) + °u ( w 0 ) and you attend if ° ( u ( w 1 ) ° u ( w 0 )) > u ( w 1 + w 0 ) ° u ( w 1 ° c ) (1) 1
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Symmetrically, for poor people you attend if ° ( u ( w 1 ) ° u ( w 0 )) > u (2 w 0 ) ° u ( w 0 ° c ) (2) The Left hand sides are the same, more rich people will attend if the rich right hand side is smaller. u ( w 1 + w 0 ) ° u ( w 1 ° c ) < u (2 w 0 ) ° u ( w 0 ° c ) So, this is true if so a < b means that u ( a + y ) ° u ( a ° x ) > u ( b + y ) ° u ( b ° x ) if utility is concave (since the second derivative of a concave function is negative). since w 0 < w 1 ; we know that more rich people will attend school, no matter what c is. Suppose in particular u ( w ) = ln w: Who is impacted more by changes in costs? Remember, people will go to school more if the right hand side of equation 1 or 2 becomes smaller. The left hand side is una/ected. So changes in costs will e/ect groups more if the right hand side changes more, i.e. if the derivative of the right hand side is bigger. For rich people, the right hand side is now ln( w 1 + w 0 ) ° ln ( w 1 ° c ) and dRHS dc = 1 w 1 ° c For poor people, the right hand side is ln (2 w 0 ) ° ln ( w 0 ° c ) dRHS dc = 1 w 0 ° c > 1 w 1 ° c (for w 0 ; w 1 > c ) So, as you decrease costs poor people enroll more than rich. But, bene²ts may still be largest among rich (since more of them are going to school), and cost di/erences may not be large enough to substantially 2
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increase enrollment among the poor (especially if w 0 is not a lot larger than c ).
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