<p>Let us show that:
<span class="math-container">$$
\int c_j {\rm d}j = \left(\int q_j^{1 - \eta} {\rm d}j \right)^{\frac{1}{1 - \eta}}.
$$</span>
This is true if:
<span class="math-container">$$
\begin{align*}
&amp;\left(\int c_j {\rm d}j\right)^{1 - \eta} = \int q_j^{1 - \eta} {\rm d}j,\\
\iff &amp;\left(\int c_j {\rm d}j \right)^{1 - \eta} = \int\left[\left(\int c_{ij} {\rm d}i \right)^{1 - \eta} \right] {\rm d}j,\\
\iff &amp; \left(\int c_j {\rm d}j\right)^{1 - \eta} = \int\left[\int c_i \left(\frac{p_j}{P}\right)^{-1/\eta} {\rm d}i\right]^{1 - \eta} {\rm d}j,\\
\iff &amp; \left(\int c_j {\rm d}_j\right)^{1 - \eta} = \left(\int \left(\frac{p_j}{P}\right)^{-\frac{1 - \eta}{\eta}} {\rm d}j \right) \left(\int c_i {\rm d}i\right)^{1 - \eta},\\
\iff &amp;\left(\int c_j {\rm d}_j\right)^{1 - \eta} =\underbrace{\frac{\int p_j^{1 - \frac{1}{\eta}} {\rm d}j}{P^{1 - \frac{1}{\eta}}}}_{=1} \left(\int c_i {\rm d}_i\right)^{1 - \eta},\\
\iff &amp; \left(\int c_j {\rm d}j\right)^{1 - \eta} = \left(\int c_i {\rm d}_i\right)^{1 - \eta}. 
\end{align*}
$$</span></p>

CC BY-SA 4.0
2024-04-02T09:17:23.860Z
2024-04-02T09:17:23.860Z
tdm
<p>I am reading the <a href="https://economics.mit.edu/sites/default/files/inline-files/incomplete%20information%20in%20macro.pdf" rel="nofollow noreferrer">following paper</a> by Angeletos and Lian and got stuck on equation (3). The problem is the standard optimal consumption bundle choice. Namely, aggregate consumption of &quot;farmer&quot; <span class="math-container">$i$</span> is</p>
<p><span class="math-container">$$
c_i = \left(\int c_{ij}^{1-\eta} \mathrm{d}j\right)^{\frac{1}{1 - \eta}},
$$</span></p>
<p>and the budget constraint of the farmer is <span class="math-container">$\int p_j c_{ij}\mathrm{d}j = p_i q_i$</span>, where <span class="math-container">$q_i$</span> is production of farmer <span class="math-container">$i$</span>. Maximization of <span class="math-container">$c_i$</span> subject to a fixed aggregate expenditure yields the standard result</p>
<p><span class="math-container">$$
\int p_j c_{ij}\mathrm{d}j = Pc_i 
$$</span>
and</p>
<p><span class="math-container">$$
\frac{c_{ij}}{c_i} = \left(\frac{p_j}{P}\right)^{-\frac{1}{\eta}}
$$</span>
for all <span class="math-container">$j$</span>, where <span class="math-container">$P = (\int p_j^{1 - 1/\eta}\mathrm{d}j)^{1/(1- 1/\eta)}$</span> is the ideal price index. Now the authors impose the market clearing condition <span class="math-container">$\int c_{ji} \mathrm{d}j = q_i$</span> for all <span class="math-container">$i$</span> and conclude from this that</p>
<p><span class="math-container">$$
p_i = P\left(\frac{q_i}{Q}\right)^{-\eta}
$$</span>
for all <span class="math-container">$i$</span>, where <span class="math-container">$Q = (\int q_j^{1 - \eta} \mathrm{d}j)^{1/(1-\eta)}$</span> is supposed to be aggregate output. I cannot derive this result under the above assumptions.</p>
<p>What I tried was starting from</p>
<p><span class="math-container">$$
c_{ji} = \left(\frac{p_i}{P}\right)^{-\frac{1}{\eta}}c_j
$$</span></p>
<p>(note this is just the optimal demand with switched indices). Integrating over <span class="math-container">$j$</span>, and solving for <span class="math-container">$p_i$</span>, I get</p>
<p><span class="math-container">$$
p_i = P\left(\frac{q_i}{\int c_j \mathrm{d}j}\right)^{-\eta}.
$$</span>
This is almost what I want, but I would need that <span class="math-container">$Q = \int c_j \mathrm{d}j$</span>, which although maybe intuitive, does not seem to hold (compare by plugging in the definitions of <span class="math-container">$c_j$</span> and <span class="math-container">$q_j$</span>). What am I missing? Thanks in advance!</p>

<macroeconomics><mathematical-economics>
Demand aggregation with CES
Wittgenstein's Poker
# Demand aggregation with CES

(Asked by: username_0 on 2024-03-30)

I am reading the [following paper](https://economics.mit.edu/sites/default/files/inline-files/incomplete%20information%20in%20macro.pdf) by Angeletos and Lian and got stuck on equation (3). The problem is the standard optimal consumption bundle choice. Namely, aggregate consumption of "farmer" $i$ is

$$
c\_i = \left(\int c\_{ij}^{1-\eta} \mathrm{d}j\right)^{\frac{1}{1 - \eta}},
$$

and the budget constraint of the farmer is $\int p\_j c\_{ij}\mathrm{d}j = p\_i q\_i$, where $q\_i$ is production of farmer $i$. Maximization of $c\_i$ subject to a fixed aggregate expenditure yields the standard result

$$
\int p\_j c\_{ij}\mathrm{d}j = Pc\_i
$$
and

$$
\frac{c\_{ij}}{c\_i} = \left(\frac{p\_j}{P}\right)^{-\frac{1}{\eta}}
$$
for all $j$, where $P = (\int p\_j^{1 - 1/\eta}\mathrm{d}j)^{1/(1- 1/\eta)}$ is the ideal price index. Now the authors impose the market clearing condition $\int c\_{ji} \mathrm{d}j = q\_i$ for all $i$ and conclude from this that

$$
p\_i = P\left(\frac{q\_i}{Q}\right)^{-\eta}
$$
for all $i$, where $Q = (\int q\_j^{1 - \eta} \mathrm{d}j)^{1/(1-\eta)}$ is supposed to be aggregate output. I cannot derive this result under the above assumptions.

What I tried was starting from

$$
c\_{ji} = \left(\frac{p\_i}{P}\right)^{-\frac{1}{\eta}}c\_j
$$

(note this is just the optimal demand with switched indices). Integrating over $j$, and solving for $p\_i$, I get

$$
p\_i = P\left(\frac{q\_i}{\int c\_j \mathrm{d}j}\right)^{-\eta}.
$$
This is almost what I want, but I would need that $Q = \int c\_j \mathrm{d}j$, which although maybe intuitive, does not seem to hold (compare by plugging in the definitions of $c\_j$ and $q\_j$). What am I missing? Thanks in advance!

**Tags:** macroeconomics, mathematical-economics


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## 1 answers

### Answer by username_1 on 2024-04-02. Score: 2 | Accepted answer

Let us show that:
$$
\int c\_j {\rm d}j = \left(\int q\_j^{1 - \eta} {\rm d}j \right)^{\frac{1}{1 - \eta}}.
$$
This is true if:
$$
\begin{align\*}
&\left(\int c\_j {\rm d}j\right)^{1 - \eta} = \int q\_j^{1 - \eta} {\rm d}j,\\
\iff &\left(\int c\_j {\rm d}j \right)^{1 - \eta} = \int\left[\left(\int c\_{ij} {\rm d}i \right)^{1 - \eta} \right] {\rm d}j,\\
\iff & \left(\int c\_j {\rm d}j\right)^{1 - \eta} = \int\left[\int c\_i \left(\frac{p\_j}{P}\right)^{-1/\eta} {\rm d}i\right]^{1 - \eta} {\rm d}j,\\
\iff & \left(\int c\_j {\rm d}\_j\right)^{1 - \eta} = \left(\int \left(\frac{p\_j}{P}\right)^{-\frac{1 - \eta}{\eta}} {\rm d}j \right) \left(\int c\_i {\rm d}i\right)^{1 - \eta},\\
\iff &\left(\int c\_j {\rm d}\_j\right)^{1 - \eta} =\underbrace{\frac{\int p\_j^{1 - \frac{1}{\eta}} {\rm d}j}{P^{1 - \frac{1}{\eta}}}}\_{=1} \left(\int c\_i {\rm d}\_i\right)^{1 - \eta},\\
\iff & \left(\int c\_j {\rm d}j\right)^{1 - \eta} = \left(\int c\_i {\rm d}\_i\right)^{1 - \eta}.
\end{align\*}
$$


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