Microeconomics Question

[philnewall]

I'm really stuck on this question does anyone know the answer? Cheers guys.

[tabako]

If you just write the lemma twice (once for p_i and once for p_j) and then take the partial of each of those with respect to p (subscript the other variable), don't you just get that d2e/(dp_j * dp_i) = dh*_i/dp_j and d2e/(dp_i * dp_j) = dh*_j/dp_i

Thats hard to read but basically I think what you get after you partial the Lemma is that the Left Hand Side equals one of the sides of Young's Theorem, and the the Right Hand Side is half of the equation you want to prove holds. Do this twice for each of i,j and you should get that the equation holds.

[ncray]

Yeah, tabako's right. Maybe to be a little bit more clear,
d^2e / (dp_i*dp_j) = d/dp_i * de/dp_j = d/dp_i * h*_j = dh*_j / dp_i
And since d^2e / (dp_i*dp_j) = d^2e / (dp_j*dp_i) = d/dp_j * de/dp_i = d/dp_j * h*_i = dh*_i / dp_j

We have that dh*_j / dp_i = dh*_i / dp_j

[philnewall]

Thanks a lot!