Hey,
I have a question regarding the derivation of the relative demand using the Kimball Aggregator. E.g. Smets and Wouters (2007, AER Appendix first page), how do I get:
\frac{Y_t(i)}{Y_t}= G'^{-1}\left[\frac{P_t(i)}{P_t}\int_{0}^{1}G'\left(\frac{Y_t(i)}{Y_t}\right)\frac{Y_t(i)}{Y_t}di\right]
As far as I see, the Lagrangian is
L = P_t Y_t - \int_{0}^{1} P_t(i) Y_t(i) + \mu_{f,t}\left[\int_{0}^{1} G\left(\frac{Y_t(i)}{Y_t}\right)di-1\right]
and the first-order conditions:
\partial L / \partial Y_t = P_t +\mu_{f,t}\int_{0}^{1} G'\left(\frac{Y_t(i)}{Y_t}\right)\frac{-Y_t(i)}{Y_t^2}di=0
\partial L / \partial Y_t(i) = -P_t(i) +\mu_{f,t} G'\left(\frac{Y_t(i)}{Y_t}\right)\frac{1}{Y_t}=0
If I rearrange and combine these two equations I get:
\frac{P_t}{P_t(i)}= \left[G'\left(\frac{Y_t(i)}{Y_t}\right)\right]^{-1}\left[\int_{0}^{1}G'\left(\frac{Y_t(i)}{Y_t}\right)\frac{Y_t(i)}{Y_t}di\right]
So, I am missing a \frac{Y_t(i)}{Y_t} on the left hand side. What am I doing wrong? Probably it’s very simple, but it is getting late and I don’t see it. I’d appreciate any help or direction…
Thanks and Cheers,
Willi