<p>Intuitively it's quite understandable, right?</p>
<p>Though, if we just think the Calculus, I think it should go through as follows:</p>
<p>Consider a joint density <span class="math-container">$\Lambda(\omega, \phi)$</span> with <span class="math-container">$\omega\in\Omega$</span> and <span class="math-container">$\phi\in[0,\infty)$</span>; and let <span class="math-container">$f(\omega) = \int_0^{\infty} \Lambda(\omega, \phi) d\phi$</span> be a marginal density of <span class="math-container">$\omega$</span>.
Then we could have
<span class="math-container">\begin{align}
\int_{\omega\in\Omega} p(\omega)^{1-\sigma} d\omega 
&amp;= \int_{\omega\in\Omega} p(\omega)^{1-\sigma} \dfrac{1}{f(\omega)} f(\omega) d\omega \\[7pt]
&amp;= \int_{\omega\in\Omega} p(\omega)^{1-\sigma} \dfrac{1}{f(\omega)} \int_0^{\infty} \Lambda(\omega, \phi) d\phi d\omega \\[7pt]
&amp;= \int_0^{\infty} \int_{\omega\in\Omega} p(\omega)^{1-\sigma} \dfrac{1}{f(\omega)} \Lambda(\omega, \phi) d\omega d\phi \\[7pt]
&amp;= \int_0^{\infty} p(\phi)^{1-\sigma} M \int_{\omega\in\Omega} \Lambda(\omega, \phi) d\omega d\phi \\[7pt]
&amp;= \int_0^{\infty} p(\phi)^{1-\sigma} M \mu(\phi) d\phi.
\end{align}</span>
There is nothing special, just substitutions and the Fubini's.
The fourth equality, however, is using <span class="math-container">$f(\omega) = 1/M$</span> in the (stationary) equilibrium. (<span class="math-container">$\because$</span> As <span class="math-container">$M$</span> is the mass of firms, we have <span class="math-container">$M$</span> measure of goods)
And from <span class="math-container">$p(\omega)$</span> to <span class="math-container">$p(\phi)$</span> is innocuous.</p>

CC BY-SA 4.0
2024-05-11T05:50:49.713Z
2024-05-13T05:20:51.307Z
2024-05-13T05:20:51.307Z
JiYong Jung
<p>I am reading Melitz (2003), and I cannot understand why the CES price index can be rewritten as an integral including <span class="math-container">$M$</span> and <span class="math-container">$\mu(\phi)$</span>. I find the lecture notes by Dave Donaldson (<a href="https://dave-donaldson.com/wp-content/uploads/Lecture-3-Firm-heterogeneity-theory-I.pdf" rel="nofollow noreferrer">https://dave-donaldson.com/wp-content/uploads/Lecture-3-Firm-heterogeneity-theory-I.pdf</a>, in slide 7), and it says that:</p>
<p>By definition, the CES price index is given by</p>
<p><span class="math-container">\begin{align*}
    P=\left[\int_{\omega\in \Omega} p(\omega)^{1-\sigma}d\omega\right]^{1/(1-\sigma)}
\end{align*}</span></p>
<p>where <span class="math-container">$\Omega$</span> is the mass of available goods</p>
<p>Since all firms with productivity <span class="math-container">$\phi$</span> charge the same price <span class="math-container">$p(\phi)$</span>, we can rearrange CES price index as:</p>
<p><span class="math-container">$$P=\left[\int_{0}^{+\infty} p(\phi)^{1-\sigma}M\mu(\phi)d\phi\right]^{1/(1-\sigma)}$$</span>
where <span class="math-container">$M$</span> is the mass of firms in equilibrium and <span class="math-container">$\mu(\phi)$</span> is the pdf of firm-productivity levels in equilibrium.</p>
<p>I cannot understand the math behind this transformation, like why there are <span class="math-container">$M$</span> and <span class="math-container">$\mu(\phi)$</span> in the new integral. Could anyone explain this to me? Thanks!</p>

<international-trade><ces-function>
CES Price Index in Melitz (2003)
LEcon
# CES Price Index in Melitz (2003)

(Asked by: username_0 on 2024-05-10)

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!

**Tags:** international-trade, ces-function


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## 1 answers

### Answer by username_1 on 2024-05-11. Score: 2

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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