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1 PROGRESA
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
rich kids get
w
1
+
w
0
+
&w
0
from no school and
w
1
+
1
c
from school. So, will go to school if
w
1
+
1
c > w
1
+
w
0
+
&w
0
this is the same equation as before.
to be concave, i.e. there is declining marginal utility of money.
Then we get
u
(
w
1
c
) +
(
w
1
)
> u
(
w
1
+
w
0
) +
(
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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View Full DocumentSymmetrically, 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
)
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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 Fall '10
 WayneM.Getz

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