OPTIMISING ANTI-POVERTY TRANSFERS WITH QUANTILE ... - Ivie
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10<br />
Z b<br />
a<br />
k(y)dF (y),<br />
where a is the lower bound of integration, b is the upper bound, k is a derivable<br />
kernel function and F is the cdf of the living standards y.<br />
The optimisation<br />
program is<br />
max<br />
t(y)<br />
Z b<br />
−<br />
a<br />
k(y + t(y))dF (y)<br />
subject to:<br />
Z b<br />
a<br />
t(y)dF (y) =B and t(y) ≥ 0, for all y,<br />
Assume that the transfer function is continuously differentiable. The corresponding<br />
Euler necessary condition can be calculated. A convexity condition involving<br />
on the kernel function and function t(.) can make the Euler conditions<br />
sufficient.<br />
Typically, kernel function k is differentiable in the classical calculus of variations,<br />
which implies that it must be continuous. In the case of a poverty index, the<br />
kernel is the product of a function P (y, z) and the dummy for the poor 1 [y+t(y)