# Gopinath et al (2019)

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 $$P_{j;t}C_{j;t} =\sum_i \int_{\Omega_i} P_{ij;t}(\omega)C_{ij;t}(\omega)d\omega$$.

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)$$