The Price and Demand Index in Homothetic Kimball Utility

Question

Suppose with Kimball preferences, utility $Q$ from consuming $\left\{q_{\omega}\right\}_{\omega \in \Omega}$ is implicitly given by $$\int_{\omega \in \Omega} Y\left(\frac{q_{\omega}}{Q}\right) d \omega=1$$. The consumer thus maximizes $Q$ subject to both budget constraint $\int_{\omega \in \Omega} p_{\omega} q_{\omega} d \omega=y$ and this implicit function.

By solving this problem, we can get $$\lambda \frac{p_{\omega}}{\delta}=Y^{\prime}\left(\frac{q_{\omega}}{Q}\right)$$ and thus the demand for each variety $$q_{\omega}=Q Y^{\prime-1}(\frac{\lambda p_{\omega}}{\delta})$$, where $\lambda$ is the Lagrange multiplier for the budget constraint and $\frac{1}{{\delta}}=\int_{\omega \in \Omega} Y^{\prime}\left(\frac{q_{\omega}}{Q}\right) \frac{q_{\omega}}{Q} d \omega$.

What I don't understand is that the papers using this preference often set price index $P \equiv \delta /\lambda$ and denote $\delta$ as a demand index. Whereas in CES case, the price index is $1/\lambda$. And I manipulate the FOCs in this Kimball case and find that $Q/\lambda=y$, thus it would make sense if set $P \equiv 1/\lambda$. So why set $P \equiv \delta /\lambda$? Moreover, why is the $\delta$ term a demand index? In particular, given the argument in this answer, it seems that $\delta$ is exactly the implicit function and thus should be 1, which makes me confused.

For some reference, in a recent wp by Baqaee & Farhi ("The Darwinian Returns to Scale") they discuss this issue: "this price index $P$ does not coincide with the ideal price index $P^{Y}$ for the representative consumer: deflating income by $P$ does not yield welfare. In fact, $d \log P=d \log \bar{\delta}+d \log P^{Y}$. Since in general, $\bar{\delta}$ is not a constant, $d \log P \neq d \log P^{Y}$. It is only in the CES two price indices coincide since then $\bar{\delta}=\sigma /(\sigma-1)$ is constant. Going forward, we refer to P as “the” price index without further qualification, despite the fact that it is not the ideal price index.".

Update: By reading Baqaee & Farhi I get the idea that this manipulation of price index offers a tidy expression of relative demand and relative price. However, I still don't fully understand what the role that $1/\delta = Q/\gamma$ plays in this revised demand function, and how as a result the Kimball utility differs from the usual CES case. Most importantly, I guess this is related to what Baqaee & Farhi argues that "Monopolistic competition models with Kimball demand are parsimonious in the sense that competition across varieties is mediated by the price index; every variety competes with the price index. The flexibility of the demand system comes from the fact that it allows different varieties to face different “intensities” of competition, as captured by the different elasticities of the demand curve at different points." This seems to be the essence of the Kimball demand, but I fail to understand how is "competition across varieties is mediated by the price index; every variety competes with the price index" and why this is not the case in the CES?

Answer 1

The problem is the following:

$$ \begin{align*} \max_{Q, q_\omega} Q \text{ s.t. } &\int Y\left(q_\omega/Q\right) d\omega = 1,\\ &\int p_\omega q_\omega d\omega = y \end{align*} $$

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 "normalized" for good $\omega$ and on the right we have a function of the "normalized" 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 "form" 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.