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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

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1 Answer 1

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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: $$C_{ij,t}(\omega)= \gamma_{ij}\psi\Bigg(D_{j,t}\frac{P_{ij,t}(\omega)}{P_{j,t}}\Bigg)C_{j,t}$$

  1. 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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