Consider the simpler case with only two charities and total utility U(X,Y). For simplicity assume the second order derivatives are constant, and that the probability that
where R is some finite interval symmetric about 0. We can actually take R to be the whole real line, but the math becomes hairier. Now, each of these integrals is 0, because the uniform distribution is symmetric about each axis. The symmetry is all that is needed actually, not uniformity, so you could weaken the assumptions.
Consider the simpler case with only two charities and total utility U(X,Y). For simplicity assume the second order derivatives are constant, and that the probability that
frac{partial2U}{partialx2}=z_0,frac{partial2U}{partialxpartialy}=z_1,frac{partial2U}{partialy2}=z_2
is given by
=\phi(\mathbf{z}).)Then the second order contribution to
)is given by the integral over all possible second derivatives
%5E2z_0%20+%20\Delta%20X\Delta%20Yz_1%20+%20\frac{1}{2}(\Delta%20Y)%5E2z_2\right)\phi(\mathbf{z})%20d\mathbf{z},)which equals
%5E2\int_{R%5E3}z_0%20\phi(\mathbf{z})%20d\mathbf{z}%20+%0A\Delta%20X\Delta%20Y%20\int_{R%5E3}z_1%20\phi(\mathbf{z})%20d\mathbf{z}%20+%0A\frac{1}{2}(\Delta%20Y)%5E2\int_{R%5E3}z_2%20\phi(\mathbf{z})%20d\mathbf{z})where R is some finite interval symmetric about 0. We can actually take R to be the whole real line, but the math becomes hairier. Now, each of these integrals is 0, because the uniform distribution is symmetric about each axis. The symmetry is all that is needed actually, not uniformity, so you could weaken the assumptions.