The Price and Demand Index in Homothetic Kimball Utility
The problem is the following:
$$
\begin{align*}
\max_{Q, q_\omega} Q \text{ s.t. } &amp;\int Y\left(q_\omega/Q\right) d\omega = 1,\\
&amp;\int p_\omega q_\omega d\omega = y
\end{align*}
$$
The first order conditions give:
$$
1 - \gamma \int Y'\left(\frac{q_\omega}{Q}\right) \frac{q_{\omega}}{Q^2} d \omega = 0,\\
 \gamma Y'\left(\frac{q_\omega}{Q}\right)\frac{1}{Q} - \lambda p_\omega = 0
$$
Then:
$$
Y' \left(\frac{q_\omega}{Q}\right) = \lambda \frac{p_\omega}{\gamma/Q}
$$
So we have that $\delta = \gamma/Q$.

What I don't understand is that the papers using this preference often set price index $P = \delta/\lambda$ and denote $\delta$ as a demand index. Whereas in CES case, the price index is $1/\lambda$.

I think the answer is already given by Baquee and Farhi themselves.
If we define $P = \gamma/(Q \lambda) = \delta/\lambda$ then we can write:
$$
\frac{q_\omega}{Q} = Y'^{-1}\left(\frac{p_\omega}{P}\right)
$$
Using $Q$ as a quantity index, we have on the left hand side the &quot;normalized&quot; for good $\omega$ and on the right we have a function of the &quot;normalized&quot; price. Under this definition, you can therefore interpret $Y'^{-1}$ as a demand function.
On the other hand, if you would use $P = 1/\lambda$  you obtain from inverting $Y'$ in the first order condition that:
$$
\frac{q_\omega}{Q} = Y'^{-1}\left(\frac{Q}{\gamma} \frac{p_\omega}{P}\right).
$$
The right hand side now also involves the quantity index $Q$, so this does not take the &quot;form&quot; of a demand function.
As a conclusion, I think the main idea of defining $P$ in this way is to obtain a nice expression and interpretation for the normalized demands as a function of some normalized prices.