<p><em><strong>Demonstration</strong></em></p>
<ol>
<li><em><strong>Static lagrangian</strong></em></li>
</ol>
<p><span class="math-container">$$\max_{C_{j,t}C_{ij,t}}C_{j,t} $$</span>
<span class="math-container">$$s.t. \sum_{i}\frac{1}{|\Omega_i|}\int_{\omega\in\Omega} \gamma_{ij}\Upsilon\Big(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}}\Big)d\omega=1$$</span>
<span class="math-container">$$\sum_i \int_{\Omega_i} P_{ij,t}(\omega)C_{ij,t}(\omega)d\omega=y$$</span> with <span class="math-container">$y = P_{j,t}C_{j,t}$</span></p>
<p>Lagrangian is: <span class="math-container">$$\mathcal{L} = C_{j,t}+\lambda_1\Bigg(1-\sum_{i}\frac{1}{|\Omega_i|}\int_{\omega\in\Omega} \gamma_{ij}\Upsilon\Big(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}}\Big)d\omega\Bigg)+\lambda_2\Bigg(y-\sum_i \int_{\Omega_i} P_{ij,t}(\omega)C_{ij,t}(\omega)d\omega\Bigg)$$</span></p>
<p><span class="math-container">$$\frac{\partial \mathcal{L}}{\partial C_{j,t}}=0 \Rightarrow \lambda_1\sum_{i}\int_{\omega\in\Omega}\Upsilon^{'}\Big(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}}\Big)\frac{C_{ij,t}(\omega)}{C_{j,t}^2}=1$$</span></p>
<p><span class="math-container">$$\frac{\partial\mathcal{L}}{\partial C_{ij,t}}=0 \Rightarrow \lambda_1\Upsilon^{'}\Big(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}}\Big)\frac{1}{C_{j,t}} = \lambda_2 P_{ij,t}(\omega)$$</span></p>
<p>from <span class="math-container">$E_2$</span> we have :</p>
<p><span class="math-container">$$\Upsilon^{'}\Big(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}}\Big) = \frac{\lambda_2 P_{ij,t}(\omega)}{\frac{\lambda_1}{C_{j,t}}}$$</span></p>
<p>from <span class="math-container">$E_1$</span> we can write <span class="math-container">$$\lambda_1=\frac{1}{\sum_{i}\int_{\omega\in\Omega}\Upsilon^{'}\Big(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}}\Big)\frac{C_{ij,t}(\omega)}{C_{j,t}^2}}$$</span></p>
<p>By substituting the value of <span class="math-container">$\lambda_1$</span> into <span class="math-container">$E_2$</span>, we obtain:
<span class="math-container">$$\Upsilon^{'}\Big(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}}\Big)=\lambda_2 P_{ij,t}(\omega)\sum_{i}\int_{\omega\in\Omega}\Upsilon^{'}\Big(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}}\Big)\frac{C_{ij,t}(\omega)}{C_{j,t}}$$</span></p>
<p>In the context of constant elasticity of substitution, we can write <span class="math-container">$P_{j,t}=\frac{1}{\lambda_2}$</span>. as a result, <span class="math-container">$\lambda_2 P_{ij,t}=\frac{P_{ij,t}(\omega)}{P_{j,t}}$</span> because <span class="math-container">$\lambda_2=\frac{1}{P_{j,t}}$</span>.</p>
<p>All of this allows us to obtain the following relationship: <span class="math-container">$$\Upsilon^{'}\Big(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}}\Big)=\sum_{i}\int_{\omega\in\Omega}\Upsilon^{'}\Big(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}}\Big)\frac{C_{ij,t}(\omega)}{C_{j,t}}\frac{P_{ij,t}(\omega)}{P_{j,t}}$$</span></p>
<p>from which <span class="math-container">$$\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}})=\Upsilon'^{-1}\Bigg(\sum_{i}\int_{\omega\in\Omega}\Upsilon^{'}\Big(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}}\Big)\frac{C_{ij,t}(\omega)}{C_{j,t}}\frac{P_{ij,t}(\omega)}{P_{j,t}}\Bigg)$$</span></p>
<p><span class="math-container">$$\Rightarrow C_{ij,t}(\omega)=\gamma_{ij}C_{j,t}\Upsilon'^{-1} \Bigg(\sum_{i}\int_{\omega\in\Omega}\Upsilon^{'}\Big(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}}\Big)\frac{C_{ij,t}(\omega)}{C_{j,t}}\frac{P_{ij,t}(\omega)}{P_{j,t}}\Bigg)$$</span></p>
<p><strong>In conclusion</strong>: <span class="math-container">$$C_{ij,t}(\omega)= \gamma_{ij}\psi\Bigg(D_{j,t}\frac{P_{ij,t}(\omega)}{P_{j,t}}\Bigg)C_{j,t}$$</span></p>
<ol start="2">
<li><em><strong>Dynamic Lagrangian</strong></em></li>
</ol>
<p><span class="math-container">$$\max_{C_{j,t}N_{j,t}B_{\\\$j,t+1}B_{j,t+1}(s')} \mathbb{E}_t \sum_{t = 0}^\infty \beta^{t}U(C_{j,t}N_{j,t})$$</span>
<span class="math-container">$s.t.$</span>
<span class="math-container">$$P_{j,t}C_{j,t}+ \mathcal{E}_{\\\$j,t}(1+i_{j,t-1}^{\\\$})B_{j,t}^{\\\$}+B_{j,t}= W_{j,t}(h)N_{j,t}(h)+\Pi_{j,t}+\mathcal{E}_{\\\$j,t}B_{j,t+1}^{\\\$}+\sum_{s^{'}\in S}Q_{j,t}(s^{'})B_{j,t+1}(s^{'})$$</span></p>
<p><span class="math-container">\begin{eqnarray}
\mathcal{L}^{dyn}=\mathbb{E}_t \sum_{t = 0}^\infty \beta^{t}\Bigg[ \frac{1}{1-\sigma_c}C^{1-\sigma_c}_{j;t} - \frac{\kappa}{1+\varphi}N^{1+\varphi}_{j;t} +\lambda_t\Big(W_{j,t}(h)N_{j,t}(h)+\Pi_{j,t}\\
+\mathcal{E}_{\\\$j,t}B_{j,t+1}^{\\\$}+\sum_{s^{'}\in S}Q_{j,t}(s^{'})B_{j,t+1}(s^{'})
-P_{j,t}C_{j,t}+ \mathcal{E}_{\\\$j,t}(1+i_{j,t-1}^{\\\$})B_{j,t}^{\\\$}+B_{j,t}\Big)\Bigg]
\nonumber
\end{eqnarray}</span></p>
<p><span class="math-container">$$\frac{\partial\mathcal{L}^{dyn}}{\partial C_{j,t}}=0 \Rightarrow C_{j,t}^{-\sigma_c}=\lambda_t P_{j,t} \Rightarrow \lambda_t = \frac{C_{j,t}^{-\sigma_c}}{P_{j,t}} \Rightarrow \lambda_{t+1} = \frac{C_{j,t+1}^{-\sigma_c}}{P_{j,t+1}}$$</span></p>
<p><span class="math-container">$$\frac{\partial\mathcal{L}^{dyn}}{\partial N_{j,t}}=0 \Rightarrow \kappa N^{\varphi}_{j;t} = \lambda_t W_{j,t}(h)$$</span></p>
<p><span class="math-container">$$\frac{\partial\mathcal{L}^{dyn}}{\partial B_{j,t+1}^{\\\$}}=0 \Rightarrow \lambda_t\mathcal{E}_{\\\$j,t}= \lambda_{t+1}\mathbb{E}_t\beta\mathcal{E}_{\\\$j,t+1}(1+i_{j,t}^\\\$)$$</span></p>
<p>By substituting <span class="math-container">$\lambda_t$</span> and <span class="math-container">$\lambda_{t+1}$</span> in <span class="math-container">$E_3$</span>, we obtain: <span class="math-container">$$\frac{C_{j,t}^{-\sigma_c}}{P_{j,t}}\mathcal{E}_{\\\$j,t}=\frac{C_{j,t+1}^{-\sigma_c}}{P_{j,t+1}}\mathbb{E}_t\beta\mathcal{E}_{\\\$j,t+1}(1+i_{j,t}^\\\$)$$</span></p>
<p><strong>In conclusion</strong>: <span class="math-container">$$C_{j,t}^{-\sigma_c}=\beta(1+i_{j,t}^\\\$)\mathbb{E}_t \Big(C_{j,t+1}^{-\sigma_c}\frac{P_{j,t}}{P_{j,t+1}}\frac{\mathcal{E}_{\\\$j,t+1}}{\mathcal{E}_{\\\$j,t}} \Big)$$</span></p>

CC BY-SA 4.0
2024-02-14T13:53:05.863Z
2024-02-14T14:52:18.753Z
2024-02-14T14:52:18.753Z
darel stevy
<p>I am looking for explanation and demonstration on the Gopinath model:</p>
<p>The per-period utility function is separable in consumption and labor and given by:
<span class="math-container">$$U(C_{j;t};N_{j;t})= \frac{1}{1-\sigma_c}C^{1-\sigma_c}_{j;t} - \frac{\kappa}{1+\varphi}N^{1+\varphi}_{j;t}$$</span>
where <span class="math-container">$\sigma_c &gt; 0$</span> is the household’s coefficient of relative risk aversion, <span class="math-container">$\varphi &gt; 0$</span> is the inverse of the Frisch elasticity of labor supply and <span class="math-container">$\kappa$</span> scales the disutility of labor.</p>
<p>The consumption aggregator <span class="math-container">$C_{j;t}$</span> is implicitly defined by a Kimball (1995) homothetic demand aggregator:
<span class="math-container">$$\sum_{i}\frac{1}{|\Omega_i|}\int_{\omega\in\Omega} \gamma_{ij}\Upsilon(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{ij}C_{j,t}})d\omega=1$$</span></p>
<p>In Eq. (2), <span class="math-container">$C_{ij;t}(\omega)$</span> represents the consumption by households in country <span class="math-container">$j$</span> of variety <span class="math-container">$\omega$</span> produced by country <span class="math-container">$i$</span> at time <span class="math-container">$t$</span>. <span class="math-container">$\gamma_{ij}$</span> is a set of preference weights that captures home consumption bias in country <span class="math-container">$j$</span>, with <span class="math-container">$\sum_{i} \gamma_{ij} = 1$</span>, while <span class="math-container">$|\Omega_i|$</span> is the measure of varieties produced in country <span class="math-container">$i$</span>. The function <span class="math-container">$\Upsilon(.)$</span> satisfies the constraints <span class="math-container">$\Upsilon(1) = 1$</span>,<span class="math-container">$\Upsilon^{'}(.) &gt; 0$</span> and <span class="math-container">$\Upsilon^{''} (.) &lt; 0$</span>. As is well-known, this demand structure gives rise to strategic complementarities in pricing and variable mark-ups. It captures the classic Dornbusch (1987) and Krugman (1987) channel of variable mark-ups and pricing-to-market as described below.</p>
<p>Households in country <span class="math-container">$j$</span> solve the following dynamic optimization problem,
<span class="math-container">$$\max_{C_{j,t}W_{j,t}B_{\\\$j,t+1}B_{j,t+1}(s')} \mathbb{E}_0\sum_{t = 0}^\infty \beta^{t}U(C_{j,t}N_{j,t})$$</span></p>
<p>where <span class="math-container">$\mathbb{E}_t$</span> denotes expectations conditional on information available at time t, subject to the perperiod budget constraint expressed in home currency, <span class="math-container">$$P_{j,t}C_{j,t}+ \varepsilon_{\\\$j,t}(1+i_{j,t-1}^{\\\$})B_{j,t}^{\\\$}+B_{j,t}= W_{j,t}(h)N_{j,t}(h)+\Pi_{j,t}+\varepsilon_{\\\$j,t}B_{j,t+1}^{\\\$}+\sum_{s^{'}\in S}Q_{j,t}(s^{'})B_{j,t+1}(s^{'})$$</span></p>
<p>In this expression, <span class="math-container">$P_{j;t}$</span> is the price index for the domestic consumption aggregator <span class="math-container">$C_{j;t}$</span>. <span class="math-container">$\Pi_{j;t}$</span> represents domestic profits transferred to domestic households, owners of domestic firms. On the financial side, households trade a risk-free international bond denominated in dollars that pays a nominal interest rate <span class="math-container">$i^{$}_{ j;t}$</span>. <span class="math-container">$B^{\\\$}_{j;t+1}$</span> denotes the dollar debt holdings of this bond at time <span class="math-container">$t$</span>. They also have access to a full set of domestic state contingent securities (in <span class="math-container">$j$</span> currency) that are traded domestically and in zero net supply. Denoting S the set of possible states of the world, <span class="math-container">$Q_{j;t}(s)$</span> is the period-<span class="math-container">$t$</span> price of the security that pays one unit of home currency in period <span class="math-container">$t + 1$</span> and state <span class="math-container">$s \in S$</span>, and <span class="math-container">$B_{j;t+1}(s)$</span> are the corresponding holdings.</p>
<p><strong>My problem lies with the demonstrations of the results which will follow</strong>:</p>
<p>The optimality conditions of the household’s problem yield the following demand system:
<span class="math-container">$$C_{ij,t}(\omega)= \gamma_{ij}\psi(D_{j,t}\frac{P_{ij,t}(\omega)}{P_{j,t}})C_{j,t}$$</span></p>
<p>where <span class="math-container">$\psi(.) := \Upsilon^{'-1}(.) &gt; 0$</span> so that  <span class="math-container">$\psi^{'}(.) &lt; 0$</span>, <span class="math-container">$D_{j;t} := \sum_i \int_{\Omega_i}\Upsilon^{'}(\frac{|\Omega_i|C_{ij,t}(\omega)}{\gamma_{j,t}C_{ij;t}})\frac{C_{ij;t}(\omega)}{C_{j;t}}d\omega$</span> is a demand index and <span class="math-container">$P_{ij ; t}\omega)$</span> denotes the price of variety <span class="math-container">$\omega$</span> produced in country <span class="math-container">$i$</span> and sold in country <span class="math-container">$j$</span>, in currency <span class="math-container">$j$</span>. Define the elasticity of demand <span class="math-container">$\sigma_{ij;t}(\omega) := -\frac{\partial log C_{ij;t}(\omega)}{\partial log Z_{ij;t}(\omega)}$</span>, where <span class="math-container">$Z_{ij;t}(\omega) := D_{j;t}\frac{P_{ij;t}(\omega)}{P_{j;t}}$</span>.</p>
<p>The log of the optimal flexible price mark-up is <span class="math-container">$\mu_{ij;t}(\omega):= log(\frac{\sigma_{ij;t}}{\sigma_{ij;t}-1})$</span>. It is time-varying and we
let <span class="math-container">$\Gamma_{ij;t}(\omega) := \frac{\partial \mu_{ij;t}}{\partial log Z_{ij;t}(\omega)}$</span> denote the elasticity of that markup. By definition, the price index <span class="math-container">$P_{j;t}$</span> satisfies <span class="math-container">$P_{j;t}C_{j;t} =\sum_i \int_{\Omega_i} P_{ij;t}(\omega)C_{ij;t}(\omega)d\omega$</span>.</p>
<p>Thanks in advance</p>

<dsge>
Gopinath et al (2019)
darel stevy
# Gopinath et al (2019)

(Asked by: username_0 on 2023-10-10)

I am looking for explanation and demonstration on the Gopinath model:

The per-period utility function is separable in consumption and labor and given by:
$$U(C\_{j;t};N\_{j;t})= \frac{1}{1-\sigma\_c}C^{1-\sigma\_c}\_{j;t} - \frac{\kappa}{1+\varphi}N^{1+\varphi}\_{j;t}$$
where $\sigma\_c > 0$ is the household’s coefficient of relative risk aversion, $\varphi > 0$ is the inverse of the Frisch elasticity of labor supply and $\kappa$ scales the disutility of labor.

The consumption aggregator $C\_{j;t}$ is implicitly defined by a Kimball (1995) homothetic demand aggregator:
$$\sum\_{i}\frac{1}{|\Omega\_i|}\int\_{\omega\in\Omega} \gamma\_{ij}\Upsilon(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}})d\omega=1$$

In Eq. (2), $C\_{ij;t}(\omega)$ represents the consumption by households in country $j$ of variety $\omega$ produced by country $i$ at time $t$. $\gamma\_{ij}$ is a set of preference weights that captures home consumption bias in country $j$, with $\sum\_{i} \gamma\_{ij} = 1$, while $|\Omega\_i|$ is the measure of varieties produced in country $i$. The function $\Upsilon(.)$ satisfies the constraints $\Upsilon(1) = 1$,$\Upsilon^{'}(.) > 0$ and $\Upsilon^{''} (.) < 0$. As is well-known, this demand structure gives rise to strategic complementarities in pricing and variable mark-ups. It captures the classic Dornbusch (1987) and Krugman (1987) channel of variable mark-ups and pricing-to-market as described below.

Households in country $j$ solve the following dynamic optimization problem,
$$\max\_{C\_{j,t}W\_{j,t}B\_{\\\$j,t+1}B\_{j,t+1}(s')} \mathbb{E}\_0\sum\_{t = 0}^\infty \beta^{t}U(C\_{j,t}N\_{j,t})$$

where $\mathbb{E}\_t$ denotes expectations conditional on information available at time t, subject to the perperiod budget constraint expressed in home currency, $$P\_{j,t}C\_{j,t}+ \varepsilon\_{\\\$j,t}(1+i\_{j,t-1}^{\\\$})B\_{j,t}^{\\\$}+B\_{j,t}= W\_{j,t}(h)N\_{j,t}(h)+\Pi\_{j,t}+\varepsilon\_{\\\$j,t}B\_{j,t+1}^{\\\$}+\sum\_{s^{'}\in S}Q\_{j,t}(s^{'})B\_{j,t+1}(s^{'})$$

In this expression, $P\_{j;t}$ is the price index for the domestic consumption aggregator $C\_{j;t}$. $\Pi\_{j;t}$ represents domestic profits transferred to domestic households, owners of domestic firms. On the financial side, households trade a risk-free international bond denominated in dollars that pays a nominal interest rate $i^{$}\_{ j;t}$. $B^{\\\$}\_{j;t+1}$ denotes the dollar debt holdings of this bond at time $t$. They also have access to a full set of domestic state contingent securities (in $j$ currency) that are traded domestically and in zero net supply. Denoting S the set of possible states of the world, $Q\_{j;t}(s)$ is the period-$t$ price of the security that pays one unit of home currency in period $t + 1$ and state $s \in S$, and $B\_{j;t+1}(s)$ are the corresponding holdings.

**My problem lies with the demonstrations of the results which will follow**:

The optimality conditions of the household’s problem yield the following demand system:
$$C\_{ij,t}(\omega)= \gamma\_{ij}\psi(D\_{j,t}\frac{P\_{ij,t}(\omega)}{P\_{j,t}})C\_{j,t}$$

where $\psi(.) := \Upsilon^{'-1}(.) > 0$ so that $\psi^{'}(.) < 0$, $D\_{j;t} := \sum\_i \int\_{\Omega\_i}\Upsilon^{'}(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{j,t}C\_{ij;t}})\frac{C\_{ij;t}(\omega)}{C\_{j;t}}d\omega$ is a demand index and $P\_{ij ; t}\omega)$ denotes the price of variety $\omega$ produced in country $i$ and sold in country $j$, in currency $j$. Define the elasticity of demand $\sigma\_{ij;t}(\omega) := -\frac{\partial log C\_{ij;t}(\omega)}{\partial log Z\_{ij;t}(\omega)}$, where $Z\_{ij;t}(\omega) := D\_{j;t}\frac{P\_{ij;t}(\omega)}{P\_{j;t}}$.

The log of the optimal flexible price mark-up is $\mu\_{ij;t}(\omega):= log(\frac{\sigma\_{ij;t}}{\sigma\_{ij;t}-1})$. It is time-varying and we
let $\Gamma\_{ij;t}(\omega) := \frac{\partial \mu\_{ij;t}}{\partial log Z\_{ij;t}(\omega)}$ denote the elasticity of that markup. By definition, the price index $P\_{j;t}$ satisfies $P\_{j;t}C\_{j;t} =\sum\_i \int\_{\Omega\_i} P\_{ij;t}(\omega)C\_{ij;t}(\omega)d\omega$.

Thanks in advance

**Tags:** dsge

---

## 1 answers

### Answer by username_0 on 2024-02-14. Score: 1

***Demonstration***

1. ***Static lagrangian***

$$\max\_{C\_{j,t}C\_{ij,t}}C\_{j,t} $$
$$s.t. \sum\_{i}\frac{1}{|\Omega\_i|}\int\_{\omega\in\Omega} \gamma\_{ij}\Upsilon\Big(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}}\Big)d\omega=1$$
$$\sum\_i \int\_{\Omega\_i} P\_{ij,t}(\omega)C\_{ij,t}(\omega)d\omega=y$$ with $y = P\_{j,t}C\_{j,t}$

Lagrangian is: $$\mathcal{L} = C\_{j,t}+\lambda\_1\Bigg(1-\sum\_{i}\frac{1}{|\Omega\_i|}\int\_{\omega\in\Omega} \gamma\_{ij}\Upsilon\Big(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}}\Big)d\omega\Bigg)+\lambda\_2\Bigg(y-\sum\_i \int\_{\Omega\_i} P\_{ij,t}(\omega)C\_{ij,t}(\omega)d\omega\Bigg)$$

$$\frac{\partial \mathcal{L}}{\partial C\_{j,t}}=0 \Rightarrow \lambda\_1\sum\_{i}\int\_{\omega\in\Omega}\Upsilon^{'}\Big(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}}\Big)\frac{C\_{ij,t}(\omega)}{C\_{j,t}^2}=1$$

$$\frac{\partial\mathcal{L}}{\partial C\_{ij,t}}=0 \Rightarrow \lambda\_1\Upsilon^{'}\Big(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}}\Big)\frac{1}{C\_{j,t}} = \lambda\_2 P\_{ij,t}(\omega)$$

from $E\_2$ we have :

$$\Upsilon^{'}\Big(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}}\Big) = \frac{\lambda\_2 P\_{ij,t}(\omega)}{\frac{\lambda\_1}{C\_{j,t}}}$$

from $E\_1$ we can write $$\lambda\_1=\frac{1}{\sum\_{i}\int\_{\omega\in\Omega}\Upsilon^{'}\Big(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}}\Big)\frac{C\_{ij,t}(\omega)}{C\_{j,t}^2}}$$

By substituting the value of $\lambda\_1$ into $E\_2$, we obtain:
$$\Upsilon^{'}\Big(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}}\Big)=\lambda\_2 P\_{ij,t}(\omega)\sum\_{i}\int\_{\omega\in\Omega}\Upsilon^{'}\Big(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}}\Big)\frac{C\_{ij,t}(\omega)}{C\_{j,t}}$$

In the context of constant elasticity of substitution, we can write $P\_{j,t}=\frac{1}{\lambda\_2}$. as a result, $\lambda\_2 P\_{ij,t}=\frac{P\_{ij,t}(\omega)}{P\_{j,t}}$ because $\lambda\_2=\frac{1}{P\_{j,t}}$.

All of this allows us to obtain the following relationship: $$\Upsilon^{'}\Big(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}}\Big)=\sum\_{i}\int\_{\omega\in\Omega}\Upsilon^{'}\Big(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}}\Big)\frac{C\_{ij,t}(\omega)}{C\_{j,t}}\frac{P\_{ij,t}(\omega)}{P\_{j,t}}$$

from which $$\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}})=\Upsilon'^{-1}\Bigg(\sum\_{i}\int\_{\omega\in\Omega}\Upsilon^{'}\Big(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}}\Big)\frac{C\_{ij,t}(\omega)}{C\_{j,t}}\frac{P\_{ij,t}(\omega)}{P\_{j,t}}\Bigg)$$

$$\Rightarrow C\_{ij,t}(\omega)=\gamma\_{ij}C\_{j,t}\Upsilon'^{-1} \Bigg(\sum\_{i}\int\_{\omega\in\Omega}\Upsilon^{'}\Big(\frac{|\Omega\_i|C\_{ij,t}(\omega)}{\gamma\_{ij}C\_{j,t}}\Big)\frac{C\_{ij,t}(\omega)}{C\_{j,t}}\frac{P\_{ij,t}(\omega)}{P\_{j,t}}\Bigg)$$

**In conclusion**: $$C\_{ij,t}(\omega)= \gamma\_{ij}\psi\Bigg(D\_{j,t}\frac{P\_{ij,t}(\omega)}{P\_{j,t}}\Bigg)C\_{j,t}$$

2. ***Dynamic Lagrangian***

$$\max\_{C\_{j,t}N\_{j,t}B\_{\\\$j,t+1}B\_{j,t+1}(s')} \mathbb{E}\_t \sum\_{t = 0}^\infty \beta^{t}U(C\_{j,t}N\_{j,t})$$
$s.t.$
$$P\_{j,t}C\_{j,t}+ \mathcal{E}\_{\\\$j,t}(1+i\_{j,t-1}^{\\\$})B\_{j,t}^{\\\$}+B\_{j,t}= W\_{j,t}(h)N\_{j,t}(h)+\Pi\_{j,t}+\mathcal{E}\_{\\\$j,t}B\_{j,t+1}^{\\\$}+\sum\_{s^{'}\in S}Q\_{j,t}(s^{'})B\_{j,t+1}(s^{'})$$

\begin{eqnarray}
\mathcal{L}^{dyn}=\mathbb{E}\_t \sum\_{t = 0}^\infty \beta^{t}\Bigg[ \frac{1}{1-\sigma\_c}C^{1-\sigma\_c}\_{j;t} - \frac{\kappa}{1+\varphi}N^{1+\varphi}\_{j;t} +\lambda\_t\Big(W\_{j,t}(h)N\_{j,t}(h)+\Pi\_{j,t}\\
+\mathcal{E}\_{\\\$j,t}B\_{j,t+1}^{\\\$}+\sum\_{s^{'}\in S}Q\_{j,t}(s^{'})B\_{j,t+1}(s^{'})
-P\_{j,t}C\_{j,t}+ \mathcal{E}\_{\\\$j,t}(1+i\_{j,t-1}^{\\\$})B\_{j,t}^{\\\$}+B\_{j,t}\Big)\Bigg]
\nonumber
\end{eqnarray}

$$\frac{\partial\mathcal{L}^{dyn}}{\partial C\_{j,t}}=0 \Rightarrow C\_{j,t}^{-\sigma\_c}=\lambda\_t P\_{j,t} \Rightarrow \lambda\_t = \frac{C\_{j,t}^{-\sigma\_c}}{P\_{j,t}} \Rightarrow \lambda\_{t+1} = \frac{C\_{j,t+1}^{-\sigma\_c}}{P\_{j,t+1}}$$

$$\frac{\partial\mathcal{L}^{dyn}}{\partial N\_{j,t}}=0 \Rightarrow \kappa N^{\varphi}\_{j;t} = \lambda\_t W\_{j,t}(h)$$

$$\frac{\partial\mathcal{L}^{dyn}}{\partial B\_{j,t+1}^{\\\$}}=0 \Rightarrow \lambda\_t\mathcal{E}\_{\\\$j,t}= \lambda\_{t+1}\mathbb{E}\_t\beta\mathcal{E}\_{\\\$j,t+1}(1+i\_{j,t}^\\\$)$$

By substituting $\lambda\_t$ and $\lambda\_{t+1}$ in $E\_3$, we obtain: $$\frac{C\_{j,t}^{-\sigma\_c}}{P\_{j,t}}\mathcal{E}\_{\\\$j,t}=\frac{C\_{j,t+1}^{-\sigma\_c}}{P\_{j,t+1}}\mathbb{E}\_t\beta\mathcal{E}\_{\\\$j,t+1}(1+i\_{j,t}^\\\$)$$

**In conclusion**: $$C\_{j,t}^{-\sigma\_c}=\beta(1+i\_{j,t}^\\\$)\mathbb{E}\_t \Big(C\_{j,t+1}^{-\sigma\_c}\frac{P\_{j,t}}{P\_{j,t+1}}\frac{\mathcal{E}\_{\\\$j,t+1}}{\mathcal{E}\_{\\\$j,t}} \Big)$$


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