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I am reading Melitz (2003), and I cannot understand why the CES price index can be rewritten as an integral including $M$ and $\mu(\phi)$. I find the lecture notes by Dave Donaldson (https://dave-donaldson.com/wp-content/uploads/Lecture-3-Firm-heterogeneity-theory-I.pdf, in slide 7), and it says that:

By definition, the CES price index is given by

\begin{align*} P=\left[\int_{\omega\in \Omega} p(\omega)^{1-\sigma}d\omega\right]^{1/(1-\sigma)} \end{align*}

where $\Omega$ is the mass of available goods

Since all firms with productivity $\phi$ charge the same price $p(\phi)$, we can rearrange CES price index as:

$$P=\left[\int_{0}^{+\infty} p(\phi)^{1-\sigma}M\mu(\phi)d\phi\right]^{1/(1-\sigma)}$$ where $M$ is the mass of firms in equilibrium and $\mu(\phi)$ is the pdf of firm-productivity levels in equilibrium.

I cannot understand the math behind this transformation, like why there are $M$ and $\mu(\phi)$ in the new integral. Could anyone explain this to me? Thanks!

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Intuitively it's quite understandable, right?

Though, if we just think the Calculus, I think it should go through as follows:

Consider a joint density $\Lambda(\omega, \phi)$ with $\omega\in\Omega$ and $\phi\in[0,\infty)$; and let $f(\omega) = \int_0^{\infty} \Lambda(\omega, \phi) d\phi$ be a marginal density of $\omega$. Then we could have \begin{align} \int_{\omega\in\Omega} p(\omega)^{1-\sigma} d\omega &= \int_{\omega\in\Omega} p(\omega)^{1-\sigma} \dfrac{1}{f(\omega)} f(\omega) d\omega \\[7pt] &= \int_{\omega\in\Omega} p(\omega)^{1-\sigma} \dfrac{1}{f(\omega)} \int_0^{\infty} \Lambda(\omega, \phi) d\phi d\omega \\[7pt] &= \int_0^{\infty} \int_{\omega\in\Omega} p(\omega)^{1-\sigma} \dfrac{1}{f(\omega)} \Lambda(\omega, \phi) d\omega d\phi \\[7pt] &= \int_0^{\infty} p(\phi)^{1-\sigma} M \int_{\omega\in\Omega} \Lambda(\omega, \phi) d\omega d\phi \\[7pt] &= \int_0^{\infty} p(\phi)^{1-\sigma} M \mu(\phi) d\phi. \end{align} There is nothing special, just substitutions and the Fubini's. The fourth equality, however, is using $f(\omega) = 1/M$ in the (stationary) equilibrium. ($\because$ As $M$ is the mass of firms, we have $M$ measure of goods) And from $p(\omega)$ to $p(\phi)$ is innocuous.

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