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I am reading the following paper 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!

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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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