Nontradable Goods and the Real Exchange Rate

Abstract

How important are nontradable goods and distribution costs to explain real exchange rate dynamics? We answer this question by estimating a general equilibrium model with intermediate and final tradable and nontradable goods. We find that the estimated model can match characteristics of the data that are relevant in international macroeconomics, such as real exchange rate persistence and volatility, and the correlation between the real exchange rate and other variables. The distinction between tradable and nontradable goods is key to understand real exchange rate fluctuations, but the introduction of distribution costs is not. Nontradable sector technology shocks explain about one third of real exchange rate volatility. We also show that, in order to explain the low correlation between the ratio of relative consumption and the real exchange rates across countries, demand shocks are necessary.

Appendices

Appendix A: The Baseline Model

In this appendix, we present the full version of a model with tradable and nontradable final consumption goods, in the spirit of Stockman and Tesar (1995) and Dotsey and Duarte (2008). We introduce sticky prices in both sectors to be able to study inflation dynamics and their role in affecting the real exchange rate.

1.1 A.1 Households

1.1.1 A.1.1 Preferences

Representative households in the home country are assumed to maximize the following utility function:

$$ U_{t}=E_{0}\left \{ \sum_{t=0}^{\infty }\beta ^{t}\psi _{t}\left[ \log \left( C_{t}-b\bar{C}_{t-1}\right) -\frac{L_{t}^{1+\varphi }}{1+\varphi } \right] \right \} , $$

(29)

subject to the following budget constraint:

$$ \frac{B_{t}^{H}}{P_{t}R_{t}}+\frac{S_{t}B_{t}^{F}}{P_{t}R_{t}^{\ast }\Phi \left( \frac{S_{t}\bar{B}_{t}^{F}}{P_{t}Y_{t}}\right) }\leq \frac{B_{t-1}^{H} }{P_{t}}+\frac{S_{t}B_{t-1}^{F}}{P_{t}}+\frac{W_{t}}{P_{t}}L_{t}-C_{t}+\Pi _{t} $$

(30)

E ₀ denotes the conditional expectation on information available at date t = 0, β is the intertemporal discount factor, with 0 < β < 1. C _t denotes the level of consumption in period t, L _t denotes labor supply. The utility function displays external habit formation with respect to the habit stock, which is last period’s aggregate consumption of the economy $\bar{C}_{t-1}$. $b\in \lbrack 0,1]$ denotes the importance of the habit stock. φ > 0 is inverse elasticity of labor supply with respect to the real wage. ψ _t is a preference shock that follows an AR(1) process in logs

$$ \log \psi _{t}=\rho _{\psi }\log \psi _{t-1}+\varepsilon _{t}^{\psi } $$

(31)

We define the consumption index as

$$ C_{t}\equiv \left[ \gamma _{c}^{1/\varepsilon }\left( C_{t}^{T}\right) ^{ \frac{\varepsilon -1}{\varepsilon }}+(1-\gamma _{c})^{1/\varepsilon }\left( C_{t}^{N}\right) ^{\frac{\varepsilon -1}{\varepsilon }}\right] ^{\frac{ \varepsilon }{\varepsilon -1}}, $$

where ε is elasticity of substitution between the final tradable ($C_{t}^{T}$) and final nontradable ($C_{t}^{N}$) goods, and γ _c is the share of final tradable goods in the consumption basket at home.

In this context, the consumer price index that corresponds to the previous specification is given by

$$ P_{t}\equiv \left[ \gamma _{c}\left( P_{t}^{T}\right) ^{1-\varepsilon }+(1-\gamma _{c})\left( P_{t}^{N}\right) ^{1-\varepsilon }\right] ^{\frac{1}{ 1-\varepsilon }}, $$

where all prices are for goods sold in the home country, in home currency and at consumer level, for both tradable and nontradable goods.

Demands for the final tradable and nontradable goods are given by:

$$\begin{array}{rll} C_{t}^{T} &=&\gamma _{c}\left( \frac{P_{t}^{T}}{P_{t}}\right) ^{-\varepsilon }C_{t}, \\ C_{t}^{N} &=&(1-\gamma _{c})\left( \frac{P_{t}^{N}}{P_{t}}\right) ^{-\varepsilon }C_{t}. \end{array}$$

1.1.2 A.1.2 Incomplete Asset Markets

For modelling simplicity, we choose to model incomplete markets with two risk-free one-period nominal bonds denominated in domestic and foreign currency, and a cost of bond holdings is introduced to achieve stationarity. Then, the budget constraint of the domestic households in real units of home currency is given by:

$$ \frac{B_{t}^{H}}{P_{t}R_{t}}+\frac{S_{t}B_{t}^{F}}{P_{t}R_{t}^{\ast }\Phi \left( \frac{S_{t}B_{t}^{F}}{P_{t}Y_{t}}\right) }\leq \frac{B_{t-1}^{H}}{ P_{t}}+\frac{S_{t}B_{t-1}^{F}}{P_{t}}+\frac{W_{t}}{P_{t}}L_{t}-C_{t}+\Pi _{t} $$

(32)

where W _t is the nominal wage, and Π _t are real profits for the home consumer. $B_{t}^{H}$ is the holding of the risk free domestic nominal bond and $B_{t}^{F}$ is the holding of the foreign risk-free nominal bond expressed in foreign country currency. S _t is the nominal exchange rate, expressed in units of home country currency per unit of foreign country. The function $\Phi \left( .\right)$ depends on the net liability position (i.e. the negative net foreign asset position) of the home country, $\bar{B} _{t}^{F}$, in percent of GDP in the entire economy, and is taken as given by the domestic household.¹⁶ $\Phi \left( .\right)$ introduces a convex cost that allows to obtain a well-defined steady state, and captures the costs of undertaking positions in the international asset market.¹⁷

1.2 A.2 Production Sector

The production of this economy is undertaken by three sectors. First, there is a final goods sector, that uses intermediate tradable inputs from both countries and operates under perfect competition, to produce the final tradable goods. This same sector also aggregates varieties of the nontradable goods to produce a final nontradable good that is sold to households. The second sector produces intermediate tradable goods, which are used as an input for the production of final goods both in the home and in the foreign country. The third sector produces nontradable goods, that are used as inputs in the production of the final nontradable good.

1.2.1 A.2.1 Final Goods Sector

The final tradable good is consumed by domestic households. This good is produced by a continuum of firms, each producing the same variety, labelled by $Y_{t}^{T}$, using intermediate home $\left( X_{t}^{h}\right)$ and foreign $\left( X_{t}^{f}\right)$ goods with the following technology:

$$ Y_{t}^{T}=\left \{ \gamma _{x}^{1/\theta }\left( X_{t}^{h}\right) ^{\frac{ \theta -1}{\theta }}+(1-\gamma _{x})^{1/\theta }\left( X_{t}^{f}\right) ^{ \frac{\theta -1}{\theta }}\right \} ^{\frac{\theta }{\theta -1}} $$

where θ is the elasticity of substitution between home-produced and foreign-produced imported intermediate goods, and γ _x is the share of home goods in the production function. We further assume symmetric home-bias in the composite of intermediate tradable goods. The corresponding composite of home and foreign intermediate tradable goods abroad is given by

$$ Y_{t}^{T^{\ast }}=\left \{ (1-\gamma _{x})^{1/\theta }\left( X_{t}^{h^{\ast }}\right) ^{\frac{\theta -1}{\theta }}+\gamma _{x}^{1/\theta }\left( X_{t}^{f^{\ast }}\right) ^{\frac{\theta -1}{\theta }}\right \} ^{\frac{ \theta }{\theta -1}} $$

$X_{t}^{h}$ and $X_{t}^{f}$, that denote the amount of home and foreign intermediate tradable inputs to produce the final tradable good at home, are also Dixit-Stiglitz aggregates of all types of home and foreign final goods, with elasticity of substitution σ:

$$ X_{t}^{h}\equiv \left[ \int \nolimits_{0}^{1}X_{t}^{h}(h)^{\frac{\sigma -1}{ \sigma }}dh\right] ^{\frac{\sigma }{\sigma -1}} $$

and

$$ X_{t}^{f}\equiv \left[ \int \nolimits_{0}^{1}X_{t}^{f}(f)^{\frac{\sigma -1}{ \sigma }}df\right] ^{\frac{\sigma }{\sigma -1}} $$

where $X_{t}^{h}(h)$ and $X_{t}^{f}(f)$ denote individual quantities from intermediate tradable goods producers at home and foreign. The equivalent quantities for foreign final tradable goods producers are $X_{t}^{h^{\ast }}(h)$ and $X_{t}^{f^{\ast }}(f)$. Optimizing conditions by final tradable goods producers deliver the following demand functions:

$$\begin{array}{rll} X_{t}^{h}(h)&=&\gamma _{x}\left( \frac{P_{t}^{h}(h)}{P_{t}^{h}}\right) ^{-\sigma }\left( \frac{P_{t}^{h}}{P_{t}^{T}}\right) ^{-\theta }Y_{t}^{T}; \\ \text{ }X_{t}^{h^{\ast }}(h) &=&(1-\gamma _{x})\left( \frac{P_{t}^{h^{\ast }}(h) }{P_{t}^{h^{\ast }}}\right) ^{-\sigma }\left( \frac{P_{t}^{h^{\ast }}}{ P_{t}^{T^{\ast }}}\right) ^{-\theta }Y_{t}^{T^{\ast }} \end{array}$$

$$\begin{array}{rll} X_{t}^{f}(f)&=&(1-\gamma _{x})\left( \frac{P_{t}^{f}(f)}{P_{t}^{f}}\right) ^{-\sigma }\left( \frac{P_{t}^{f}}{P_{t}^{T}}\right) ^{-\theta }Y_{t}^{T}; \\ \text{ }X_{t}^{f^{\ast }}(f) & =&\gamma _{x}\left( \frac{P_{t}^{f^{\ast }}(f)}{ P_{t}^{f^{\ast }}}\right) ^{-\sigma }\left( \frac{P_{t}^{f^{\ast }}}{ P_{t}^{T^{\ast }}}\right) ^{-\theta }Y_{t}^{T^{\ast }} \end{array}$$

where

$$ P_{t}^{h}\equiv \left[ \int \nolimits_{0}^{1}P_{t}^{h}(h)^{1-\sigma }dh \right] ^{\frac{1}{1-\sigma }},\text{ }P_{t}^{f}\equiv \left[ \int\nolimits_{0}^{1}P_{t}^{f}(f)^{1-\sigma }df\right] ^{\frac{1}{1-\sigma } }. $$

and

$$ P_{t}^{T}=\left[ \gamma _{x}\left( P_{t}^{h}\right) ^{1-\theta }+(1-\gamma _{x})\left( P_{t}^{f}\right) ^{1-\theta }\right] ^{\frac{1}{1-\theta }} $$

We assume that the law of one price holds for intermediate inputs, such that $P_{t}^{h}(h)=P_{t}^{h^{\ast }}(h)S_{t}$, and $P_{t}^{f}(f)=P_{t}^{f^{\ast }}(f)S_{t}$, where S _t is the nominal exchange rate.

The production of the final nontradable good is given by:

$$ Y_{t}^{N}\equiv \left[ \int \nolimits_{0}^{1}X_{t}^{N}(n)^{\frac{\sigma -1}{ \sigma }}dn\right] ^{\frac{\sigma }{\sigma -1}} $$

where we assume the same elasticity σ > 1 than in the case of final tradable goods produced within country H. The price level for nontradables is

$$ P_{t}^{N}\equiv \left[ \int \nolimits_{0}^{1}p_{t}^{N}(n)^{1-\sigma }dn \right] ^{\frac{1}{1-\sigma }} $$

1.2.2 A.2.2 Intermediate Non-Tradable Goods Sector

The intermediate nontradable sector produces differentiated goods that are aggregated by final good producing firms, and ultimately used for final consumption by domestic households only. Each firm produces intermediate nontradable goods according to the following production function

$$ Y_{t}^{N}\left( n\right) =A_{t}Z_{t}^{N}L_{t}^{N}(n) $$

(33)

where A _t is a labor augmenting aggregate world technology shock which has a unit root with drift, as in Galí and Rabanal (2005):

$$ \log A_{t}=g+\log A_{t-1}+\varepsilon _{t}^{a} $$

(34)

This shock also affects the intermediate tradable sector production function. Hence, real variables in both countries grow at a rate g. $Z_{t}^{N}$ is the country-specific productivity shock to the nontradable sector at time t which evolves according to an AR(1) process in logs

$$ \log Z_{t}^{N}=(1-\rho ^{N})\log (\bar{Z}^{N})+\rho ^{Z,N}\log Z_{t-1}^{N}+\varepsilon _{t}^{Z,N} $$

(35)

Firms in the nontradable sector face a Calvo lottery when setting their prices. Each period, with probability 1 − α _N , firms receive a stochastic signal that allows them to reset prices optimally. We assume that there is partial indexation with a coefficient φ _N to last period’s sectorial inflation rate for those firms that do not get to reset prices. As a result, firms maximize the following profits function:

$$ Max_{P_{t}^{N}(n)}E_{t}\displaystyle\sum \limits_{k=0}^{\infty }\alpha _{N}^{k}\Lambda _{t,t+k}\left \{ \left[ \frac{P_{t}^{N}(n)\left( \frac{P_{t+k-1}^{N}}{ P_{t-1}^{N}}\right) ^{\varphi _{N}}}{P_{t+k}}-MC_{t+k}^{N}\right] Y_{t+k}^{N,d}\left( n\right) \right \} $$

(36)

subject to

$$ Y_{t+k}^{N,d}\left( n\right) =\left[ \left( \frac{P_{t}^{N}(n)}{P_{t+k}^{N}} \right) \left( \frac{P_{t+k-1}^{N}}{P_{t-1}^{N}}\right) ^{\varphi _{N}} \right] ^{-\sigma }Y_{t}^{N} $$

(37)

where $Y_{t}^{N,d}\left( n\right)$ is total individual demand for a given type of nontradable good n, and $Y_{t}^{N}$ is aggregate demand for nontradable goods, as defined above. $\Lambda _{t,t+k}=\beta ^{k}\frac{ \lambda _{t+k}}{\lambda _{t}}$ is the stochastic discount factor, where $\lambda _{t}=\frac{\psi _{t}}{C_{t}-bC_{t-1}}$ is the marginal utility of consumption. $MC_{t}^{N}$ corresponds to the real marginal cost in the nontradable sector. From cost minimization:

$$ MC_{t}^{N}=\frac{W_{t}}{P_{t}Z_{t}^{N}A_{t}} $$

1.2.3 A.2.3 Intermediate Tradable Goods Sector

The intermediate tradable sector produces differentiated goods that are sold to the final sector goods producers in the home and foreign countries. Most functional forms are similar to those presented for the nontradable sector.

Each firm produces tradable intermediate goods according to the following production function

$$ Y_{t}^{h}\left( h\right) =A_{t}Z_{t}^{h}L_{t}^{h}(h) $$

(38)

where $Z_{t}^{h}$ is the country-specific productivity shock to the intermediate goods tradable sector at time t which evolves according to an AR(1) process in logs

$$ \log Z_{t}^{h}=(1-\rho ^{h})\log (\bar{Z}^{h})+\rho ^{Z,h}\log Z_{t-1}^{h}+\varepsilon _{t}^{Z,h} $$

(39)

Firms in the intermediate tradable sector face the same Calvo lottery as firms in the intermediate nontradable sector, with relevant parameters α _h and φ _h :

$$ Max_{P_{t}^{h}(h)}E_{t}\displaystyle\sum \limits_{k=0}^{\infty }\alpha _{h}^{k}\Lambda _{t,t+k}\left \{ \left[ \frac{P_{t}^{h}(h)\left( \frac{P_{t+k-1}^{h}}{ P_{t-1}^{h}}\right) ^{\varphi _{h}}}{P_{t+k}}-MC_{t+k}^{h}\right] Y_{t+k}^{h,d}\left( h\right) \right \} $$

(40)

subject to

$$\begin{array}{rll} Y_{t+k}^{h,d}\left( h\right) &=&X_{t+k}^{h}(h)+X_{t+k}^{h^{\ast }}(h) \\ &=&\left[ \left( \frac{P_{t}^{h}(h)}{P_{t+k}^{h}}\right) \left( \frac{ P_{t+k-1}^{h}}{P_{t-1}^{h}}\right) ^{\varphi _{h}}\right] ^{-\sigma }X_{t}^{h} \end{array} $$

(41)

where $Y_{t}^{h,d}\left( h\right)$ is total individual demand for a given type of tradable intermediate good h, and $X_{t}^{h}$ is aggregate demand for intermediate good h, consisting of home demand, and foreign demand:

$$ X_{t}^{h}=\left[ \gamma _{x}\left( \frac{P_{t}^{h}}{P_{t}^{T}}\right) ^{-\theta }Y_{t}^{T}+(1-\gamma _{x})\left( \frac{P_{t}^{h^{\ast }}}{ P_{t}^{T^{\ast }}}\right) ^{-\theta }Y_{t}^{T^{\ast }}\right] $$

$MC_{t}^{h}$ corresponds to the real marginal cost in the nontradable sector. From cost minimization:

$$ MC_{t}^{h}=\frac{W_{t}}{P_{t}Z_{t}^{h}A_{t}} $$

1.2.4 A.2.4 Market Clearing

We assume that the demand shock is allocated between tradable and nontradable goods in the same way that private consumption is. Hence the market clearing conditions for both types of final goods, consisting of private consumption and the demand shock in the tradadable sector, are:

$$ Y_{t}^{T}=C_{t}^{T}+G_{t}^{T} $$

$$ Y_{t}^{N}=C_{t}^{N}+G_{t}^{N} $$

where $G_{t}^{N},G_{t}^{T}$ follow AR(1) processes in logs. The bond market clearing conditions are

$$ B_{t}^{H}+B_{t}^{H^{\ast }}=0 $$

(42)

$$ B_{t}^{F}+B_{t}^{F^{\ast }}=0 $$

(43)

For the nontradable intermediate goods, the market clearing condition is:

$$ Y_{t}^{N}(n)=X_{t}^{N},\text{ for all }n\in \lbrack 0,1] $$

(44)

while for the intermediate tradable goods sector it is:

$$ Y_{t}^{h}\left( h\right) =X_{t}^{h}(h)+X_{t}^{h^{\ast }}(h),\text{ for all } h\in \lbrack 0,1] $$

(45)

For the labor market:

$$\begin{array}{lll} L_{t} &=&L_{t}^{h}+L_{t}^{N}= \\ &=&\int \nolimits_{0}^{1}L_{t}^{h}(h)dh+\int \nolimits_{0}^{1}L_{t}^{N}(n)dn \end{array} $$

(46)

1.3 A.3 Optimizing, Market Clearing Conditions, and Monetary Policy

In this subsection we present the full set of equations characterizing the symmetric equilibrium. Since all agents in each economy are equal, then the per capita and aggregate consumption levels are equal ($C_{t}=\bar{C}_{t}$), as well as the net foreign assets levels ($B_{t}^{F}=\bar{B}_{t}^{F}$).

1.3.1 A.3.1 Households

The Euler equations for home and foreign households, and the optimal condition of holdings by home household of the foreign bond are:

$$\begin{array}{lll} \lambda _{t} &=&\beta E_{t}\left \{ R_{t}\frac{P_{t}}{P_{t+1}}\lambda _{t+1}\right \} \\ \lambda _{t}^{\ast } &=&\beta E_{t}\left \{ R_{t}^{\ast }\frac{P_{t}^{\ast } }{P_{t+1}^{\ast }}\lambda _{t+1}^{\ast }\right \} \\ \lambda _{t} &=&\Phi \left( \frac{S_{t}B_{t}^{F}}{P_{t}Y_{t}}\right) \beta E_{t}\left \{ R_{t}^{\ast }\frac{Q_{t+1}}{Q_{t}}\lambda _{t+1}\right \} \end{array}$$

where λ _t is the marginal utility of consumption:

$$\begin{array}{lll} \lambda _{t} &=&U_{C}\left( C_{t}\right) =\frac{\psi _{t}}{C_{t}-bC_{t-1}} \\ \lambda _{t}^{\ast } &=&U_{C}\left( C_{t}^{\ast }\right) =\frac{\psi _{t}^{\ast }}{C_{t}^{\ast }-b^{\ast }C_{t-1}^{\ast }} \end{array}$$

The labor supply decisions in each country are:

$$ \lambda _{t}\frac{W_{t}}{P_{t}}=L_{t}^{\varphi } $$

$$ \lambda _{t}^{\ast }\frac{W_{t}^{\ast }}{P_{t}^{\ast }}=\left( L_{t}^{\ast }\right) ^{\varphi ^{\ast }} $$

where:

$$ L_{t}=L_{t}^{h}+L_{t}^{N} $$

and

$$ L_{t}^{\ast }=L_{t}^{h^{\ast }}+L_{t}^{N^{\ast }} $$

Household demand for final tradable and nontradable goods are given by:

$$\begin{array}{rll} C_{t}^{T} &=&\gamma _{c}\left( \frac{P_{t}^{T}}{P_{t}}\right) ^{-\varepsilon }C_{t}, \\ C_{t}^{N} &=&(1-\gamma _{c})\left( \frac{P_{t}^{N}}{P_{t}}\right) ^{-\varepsilon }C_{t}. \\ C_{t}^{T^{\ast }} &=&\gamma _{c}^{\ast }\left( \frac{P_{t}^{T^{\ast }}}{ P_{t}^{\ast }}\right) ^{-\varepsilon ^{\ast }}C_{t}^{\ast }, \\ C_{t}^{N^{\ast }} &=&(1-\gamma _{c}^{\ast })\left( \frac{P_{t}^{N^{\ast }}}{ P_{t}^{\ast }}\right) ^{-\varepsilon }C_{t}^{\ast } \end{array}$$

and the CPI’s in each country are given by:

$$\begin{array}{lll} P_{t} &\equiv &\left[ \gamma _{c}\left( P_{t}^{T}\right) ^{1-\varepsilon }+(1-\gamma _{c})\left( P_{t}^{N}\right) ^{1-\varepsilon }\right] ^{\frac{1}{ 1-\varepsilon }}, \\ P_{t}^{\ast } &\equiv &\left[ \gamma _{c}^{\ast }\left( P_{t}^{T^{\ast }}\right) ^{1-\varepsilon }+(1-\gamma _{c}^{\ast })\left( P_{t}^{N^{\ast }}\right) ^{1-\varepsilon }\right] ^{\frac{1}{1-\varepsilon }} \end{array}$$

The real exchange rate is

$$ Q_{t}=\frac{S_{t}P_{t}^{\ast }}{P_{t}} $$

1.3.2 A.3.2 Final Goods Producers

The production of final tradable goods in both countries is given by:

$$ Y_{t}^{T}=\left \{ \gamma _{x}^{1/\theta }\left( X_{t}^{h}\right) ^{\frac{ \theta -1}{\theta }}+(1-\gamma _{x})^{1/\theta }\left( X_{t}^{f}\right) ^{ \frac{\theta -1}{\theta }}\right \} ^{\frac{\theta }{\theta -1}} $$

and

$$ Y_{t}^{T^{\ast }}=\left \{ (1-\gamma _{x})^{1/\theta }\left( X_{t}^{h^{\ast }}\right) ^{\frac{\theta -1}{\theta }}+\gamma _{x}^{1/\theta }\left( X_{t}^{f^{\ast }}\right) ^{\frac{\theta -1}{\theta }}\right \} ^{\frac{ \theta }{\theta -1}} $$

Demand for intermediate tradable goods is:

$$ X_{t}^{h}=\gamma _{x}\left( \frac{P_{t}^{h}}{P_{t}^{T}}\right) ^{-\theta }Y_{t}^{T};\text{ }X_{t}^{h^{\ast }}=(1-\gamma _{x})\left( \frac{ P_{t}^{h^{\ast }}}{P_{t}^{T^{\ast }}}\right) ^{-\theta }Y_{t}^{T^{\ast }} $$

$$ X_{t}^{f}=(1-\gamma _{x})\left( \frac{P_{t}^{f}}{P_{t}^{T}}\right) ^{-\theta }Y_{t}^{T};\text{ }X_{t}^{f^{\ast }}(f)=\gamma _{x}\left( \frac{ P_{t}^{f^{\ast }}}{P_{t}^{T^{\ast }}}\right) ^{-\theta }Y_{t}^{T^{\ast }} $$

where

$$ P_{t}^{h}\equiv \left[ \int \nolimits_{0}^{1}P_{t}^{h}(h)^{1-\sigma }dh \right] ^{\frac{1}{1-\sigma }},\text{ }P_{t}^{f}\equiv \left[ \int\nolimits_{0}^{1}P_{t}^{f}(f)^{1-\sigma }df\right] ^{\frac{1}{1-\sigma } }. $$

The price of final tradable goods is:

$$ P_{t}^{T}=\left[ \gamma _{x}\left( P_{t}^{h}\right) ^{1-\theta }+(1-\gamma _{x})\left( P_{t}^{f}\right) ^{1-\theta }\right] ^{\frac{1}{1-\theta }} $$

and

$$ P_{t}^{T^{\ast }}=\left[ \gamma _{x}^{\ast }\left( P_{t}^{h^{\ast }}\right) ^{1-\theta }+(1-\gamma _{x}^{\ast })\left( P_{t}^{f^{\ast }}\right) ^{1-\theta }\right] ^{\frac{1}{1-\theta }} $$

Since we assumed that the law of one price holds for intermediate goods, it also holds in the aggregate, such that $P_{t}^{h}=P_{t}^{h^{\ast }}S_{t}$, and $P_{t}^{f}=P_{t}^{f^{\ast }}S_{t}$, where S _t is the nominal exchange rate.

1.3.3 A.3.3 Nontradable Goods Producers

The price setting equations are given by the following optimal expressions:

$$ \frac{\hat{p}_{t}^{N}}{P_{t}^{N}}=\frac{\sigma }{\left( \sigma -1\right) } E_{t}\left \{ \frac{\displaystyle\sum \limits_{k=0}^{\infty }\beta ^{k}\alpha _{N}^{k}\lambda _{t+k}\left( \displaystyle\prod \limits_{s=1}^{k}\frac{\left( \Pi _{t+s-1}^{N}\right) ^{\varphi _{N}}}{\Pi _{t+s}^{N}}\right) ^{-\sigma }MC_{t+k}^{N}Y_{t+k}^{N}}{\displaystyle\sum \limits_{k=0}^{\infty }\beta ^{k}\alpha _{N}^{k}\lambda _{t+k}\left( \displaystyle\prod \limits_{s=1}^{k}\frac{\left( \Pi _{t+s-1}^{N}\right) ^{\varphi _{N}}}{\Pi _{t+s}^{N}}\right) ^{1-\sigma } \frac{P_{t+k}^{N}}{P_{t+k}}Y_{t+k}^{N}}\right \} $$

where

$$ MC_{t}^{N}=\frac{W_{t}}{P_{t}Z_{t}^{N}A_{t}}, $$

$$ Y_{t}^{N}=C_{t}^{N}+G_{t}^{N} $$

The evolution of the price level of nontradables is

$$ P_{t}^{N}\equiv \left[ \alpha _{N}\left( P_{t-1}^{N}\left( \Pi _{t-1}^{N}\right) ^{\varphi _{N}}\right) ^{1-\sigma }+(1-\alpha _{N})\left( \hat{p}_{t}^{N}\right) ^{1-\sigma }\right] ^{\frac{1}{1-\sigma }} $$

where $\Pi _{t-1}^{N}=\frac{P_{t-1}^{N}}{P_{t-2}^{N}}$.

The production function is:

$$ Y_{t}^{N}=A_{t}Z_{t}^{N}L_{t}^{N}. $$

In the foreign country these expressions are:

$$ \frac{\hat{p}_{t}^{N^{\ast }}}{P_{t}^{N^{\ast }}}=\frac{\sigma }{\left( \sigma -1\right) }E_{t}\left \{ \frac{\displaystyle\sum \limits_{k=0}^{\infty }\beta ^{k}\alpha _{N^{\ast }}^{k}\lambda _{t+k}\left( \displaystyle\prod \limits_{s=1}^{k} \frac{\left( \Pi _{t+s-1}^{N^{\ast }}\right) ^{\varphi _{N^{\ast }}}}{\Pi _{t+s}^{N^{\ast }}}\right) ^{-\sigma }MC_{t+k}^{N^{\ast }}Y_{t+k}^{N^{\ast }} }{\displaystyle\sum \limits_{k=0}^{\infty }\beta ^{k}\alpha _{N^{\ast }}^{k}\lambda _{t+k}\left( \displaystyle\prod \limits_{s=1}^{k}\frac{\left( \Pi _{t+s-1}^{N^{\ast }}\right) ^{\varphi _{N^{\ast }}}}{\Pi _{t+s}^{N^{\ast }}}\right) ^{1-\sigma }\frac{P_{t+k}^{N^{\ast }}}{P_{t+k}}Y_{t+k}^{N^{\ast }}}\right \} $$

where

$$ MC_{t}^{N^{\ast }}=\frac{W_{t}}{P_{t}Z_{t}^{N^{\ast }}A_{t}}, $$

$$ Y_{t}^{N^{\ast }}=C_{t}^{N^{\ast }}+G_{t}^{N^{\ast }} $$

The evolution of the price level of nontradables is

$$ P_{t}^{N^{\ast }}\equiv \left[ \alpha _{N^{\ast }}\left( P_{t-1}^{N^{\ast }}\left( \Pi _{t-1}^{N^{\ast }}\right) ^{\varphi _{N^{\ast }}}\right) ^{1-\sigma }+(1-\alpha _{N^{\ast }})\left( \hat{p}_{t}^{N^{\ast }}\right) ^{1-\sigma }\right] ^{\frac{1}{1-\sigma }} $$

where $\Pi _{t-1}^{N^{\ast }}=\frac{P_{t-1}^{N^{\ast }}}{P_{t-2}^{N^{\ast }}}$.

The production function is:

$$ Y_{t}^{N^{\ast }}=A_{t}Z_{t}^{N^{\ast }}L_{t}^{N^{\ast }}. $$

1.3.4 A.3.4 Intermediate Traded Goods Producers

The price setting equations are given by the following optimal expressions:

$$ \frac{\hat{p}_{t}^{h}}{P_{t}^{h}}=\frac{\sigma }{\left( \sigma -1\right) } E_{t}\left \{ \frac{\displaystyle\sum \limits_{k=0}^{\infty }\beta ^{k}\alpha _{h}^{k}\lambda _{t+k}\left( \displaystyle\prod \limits_{s=1}^{k}\frac{\left( \Pi _{t+s-1}^{h}\right) ^{\varphi _{h}}}{\Pi _{t+s}^{h}}\right) ^{-\sigma }MC_{t+k}^{h}Y_{t+k}^{h}}{\displaystyle\sum \limits_{k=0}^{\infty }\beta ^{k}\alpha _{h}^{k}\lambda _{t+k}\left( \displaystyle\prod \limits_{s=1}^{k}\frac{\left( \Pi _{t+s-1}^{h}\right) ^{\varphi _{h}}}{\Pi _{t+s}^{h}}\right) ^{1-\sigma } \frac{P_{t+k}^{h}}{P_{t+k}}Y_{t+k}^{h}}\right \} $$

where

$$ MC_{t}^{h}=\frac{W_{t}}{P_{t}Z_{t}^{h}A_{t}}, $$

$$ Y_{t}^{h}=X_{t}^{h}+X_{t}^{h^{\ast }}. $$

The evolution of the price level of final tradables is

$$ P_{t}^{h}\equiv \left[ \alpha _{h}\left( P_{t-1}^{h}\left( \Pi _{t-1}^{h}\right) ^{\varphi _{h}}\right) ^{1-\sigma }+(1-\alpha _{h})\left( \hat{p}_{t}^{h}\right) ^{1-\sigma }\right] ^{\frac{1}{1-\sigma }} $$

where $\Pi _{t-1}^{h}=\frac{P_{t-1}^{h}}{P_{t-2}^{h}}$.

The production function is:

$$ Y_{t}^{h}=A_{t}Z_{t}^{h}L_{t}^{h} $$

In the foreign country, this expressions are:

$$ \frac{\hat{p}_{t}^{f^{\ast }}}{P_{t}^{f^{\ast }}}=\frac{\sigma }{\left( \sigma -1\right) }E_{t}\left \{ \frac{\displaystyle\sum \limits_{k=0}^{\infty }\beta ^{k}\alpha _{f^{\ast }}^{k}\lambda _{t+k}\left( \displaystyle\prod \limits_{s=1}^{k} \frac{\left( \Pi _{t+s-1}^{f^{\ast }}\right) ^{\varphi _{f^{\ast }}}}{\Pi _{t+s}^{f^{\ast }}}\right) ^{-\sigma }MC_{t+k}^{f^{\ast }}Y_{t+k}^{f^{\ast }} }{\displaystyle\sum \limits_{k=0}^{\infty }\beta ^{k}\alpha _{f^{\ast }}^{k}\lambda _{t+k}\left( \displaystyle\prod \limits_{s=1}^{k}\frac{\left( \Pi _{t+s-1}^{f^{\ast }}\right) ^{\varphi _{f^{\ast }}}}{\Pi _{t+s}^{f^{\ast }}}\right) ^{1-\sigma }\frac{P_{t+k}^{f^{\ast }}}{P_{t+k}}Y_{t+k}^{f^{\ast }}}\right \} $$

where

$$ MC_{t}^{f^{\ast }}=\frac{W_{t}}{P_{t}Z_{t}^{f^{\ast }}A_{t}}, $$

$$ Y_{t}^{f^{\ast }}=X_{t}^{f}+X_{t}^{f^{\ast }}. $$

The evolution of the price level of final tradables is

$$ P_{t}^{f^{\ast }}\equiv \left[ \alpha _{f^{\ast }}\left( P_{t-1}^{f^{\ast }}\left( \Pi _{t-1}^{f^{\ast }}\right) ^{\varphi _{f^{\ast }}}\right) ^{1-\sigma }+(1-\alpha _{f^{\ast }})\left( \hat{p}_{t}^{f^{\ast }}\right) ^{1-\sigma }\right] ^{\frac{1}{1-\sigma }} $$

where $\Pi _{t-1}^{f^{\ast }}=\frac{P_{t-1}^{f^{\ast }}}{P_{t-2}^{f^{\ast }}}$.

The production function is:

$$ Y_{t}^{f^{\ast }}=A_{t}Z_{t}^{f^{\ast }}L_{t}^{f^{\ast }} $$

1.3.5 A.3.5 Monetary Policy

Monetary policy in both countries is conducted with a Taylor rule that targets CPI inflation and output growth deviation from steady-state values:

$$ R_{t}=\bar{R}^{(1-\rho _{r})}R_{t-1}^{\rho _{r}}\left( \frac{P_{t}/P_{t-1}}{ \Pi }\right) ^{(1-\rho _{r})\gamma _{\pi }}\left( \frac{Y_{t}/Y_{t-1}}{1+g} \right) ^{(1-\rho _{r})\gamma _{y}}\exp (\varepsilon _{t}^{r}) $$

$$ R_{t}^{\ast }=\bar{R}^{\ast (1-\rho _{r}^{\ast })}\left( R_{t-1}^{\ast }\right) ^{\rho _{r}^{\ast }}\left( \frac{P_{t}^{\ast }/P_{t-1}^{\ast }}{\Pi ^{\ast }}\right) ^{(1-\rho _{r}^{\ast })\gamma _{\pi }^{\ast }}\left( \frac{ Y_{t}^{\ast }/Y_{t-1}^{\ast }}{1+g}\right) ^{(1-\rho _{r}^{\ast })\gamma _{y}^{\ast }}\exp (\varepsilon _{t}^{r^{\ast }}) $$

1.3.6 A.3.6 Demand Shocks

$$ G_{t}^{T}=(\bar{G}^{T})^{(1-\rho _{G^{T}})}(G_{t-1}^{T})^{\rho _{G^{T}}}\exp (\varepsilon _{t}^{G^{T}}) $$

$$ G_{t}^{N}=(\bar{G}^{N})^{(1-\rho _{G^{N}})}(G_{t-1}^{N})^{\rho _{G^{N}}}\exp (\varepsilon _{t}^{G^{N}}) $$

$$ G_{t}^{T^{\ast }}=(\bar{G}^{T^{\ast }})^{(1-\rho _{G^{T^{\ast }}})}(G_{t-1}^{T^{\ast }})^{\rho _{G^{T^{\ast }}}}\exp (\varepsilon _{t}^{G^{T^{\ast }}}) $$

$$ G_{t}^{N^{\ast }}=(\bar{G}^{N^{\ast }})^{(1-\rho _{G^{N^{\ast }}})}(G_{t-1}^{N^{\ast }})^{\rho _{G^{N^{\ast }}}}\exp (\varepsilon _{t}^{G^{N^{\ast }}}) $$

1.3.7 A.3.7 Trade Balance and Net Foreign Asset Dynamics

We present the evolution of the trade balance and net foreign assets of the home country, since the definition those in the foreign country will mirror those in the home country. Holdings of foreign bonds depend on the trade balance (NX _t ) as follows

$$ \frac{S_{t}B_{t}^{F}}{P_{t}R_{t}^{\ast }\Phi \left( \frac{S_{t}B_{t}^{F}}{ P_{t}Y_{t}}\right) }=\frac{S_{t}B_{t-1}^{F}}{P_{t}}+NX_{t} $$

Since international trade only occurs at the intermediate goods level, net exports equal exports minus imports of intermediate goods:

$$ NX_{t}=\frac{P_{t}^{h}X_{t}^{h^{\ast }}-P_{t}^{f}X_{t}^{f}}{P_{t}} $$

Finally, we define nominal GDP to be equal to aggregate nominal private and public consumption, hence $P_{t}Y_{t}=P_{t}^{T}(C_{t}^{T}+G_{t}^{T})+P_{t}^{N}(C_{t}^{N}+G_{t}^{N})$.

Appendix B: Log-Linear Version of the Model

Euler equations

$$\begin{array}{lll} b\Delta c_{t}&=&-\left( 1+g-b\right) \left( r_{t}-E_{t}\triangle p_{t+1}\right) +\left( 1+g\right) E_{t}\Delta c_{t+1}\\ &&+\left( 1+g-b\right) \left( 1-\rho _{\psi }\right) \widehat{\psi }_{t} \end{array} $$

(47)

$$\begin{array}{lll} b^{\ast }\Delta c_{t}^{\ast }&=&-\left( 1+g-b^{\ast }\right) \left( r_{t}^{\ast }-E_{t}\triangle p_{t+1}^{\ast }\right) +\left( 1+g\right) E_{t}\Delta c_{t+1}^{\ast }\\ &&+\left( 1+g-b^{\ast }\right) \left( 1-\rho _{\psi }^{\ast }\right) \widehat{\psi }_{t}^{\ast } \end{array} $$

(48)

Risk sharing

$$\begin{array}{lll} {\kern-4pt} E_{t}\left( q_{t+1}-q_{t}\right) &=&\left[ \frac{\left( 1+g\right) E_{t}\Delta c_{t+1}-b\Delta c_{t}}{\left( 1+g-b\right) }\right] {\kern-2pt} -{\kern-2pt} \left[ \frac{\left( 1+g\right) E_{t}\Delta c_{t+1}^{\ast }-b^{\ast }\Delta c_{t}^{\ast }}{\left( 1+g-b^{\ast }\right) }\right] \\ &&+\left( 1-\rho _{\psi }\right) \widehat{\psi }_{t}-\left( 1-\rho _{\psi }^{\ast }\right) \widehat{\psi }_{t}^{\ast }+\chi b_{t} \end{array} $$

(49)

where $\chi \equiv -\Phi ^{\prime }\left( 0\right) Y,$ $b_{t}=\left( \frac{ S_{t}B_{t}^{F}}{P_{t}}\right) Y^{-1}.$

The labor supply schedules are given by:

$$\tilde{\omega}_{t} =\varphi l_{t}+\left[ \frac{1+g}{1+g-b}\right] \widetilde{c}_{t}-\frac{b}{\left( 1+g-b\right) }\widetilde{c}_{t-1}+\frac{b}{ \left( 1+g-b\right) }\varepsilon _{t}^{a} $$

(50)

$$\tilde{\omega}_{t}^{\ast } =\varphi l_{t}^{\ast }+\left[ \frac{1+g}{ 1+g-b^{\ast }}\right] \widetilde{c}_{t}^{\ast }-\frac{b^{\ast }}{\left( 1+g-b^{\ast }\right) }\widetilde{c}_{t-1}^{\ast }+\frac{b^{\ast }}{\left( 1+g-b^{\ast }\right) }\varepsilon _{t}^{a} $$

(51)

Technology

$$ \widetilde{y}_{t}^{h}=l_{t}^{h}+z_{t}^{h}-\varepsilon _{t}^{a} $$

(52)

$$ \widetilde{y}_{t}^{h^{\ast }}=l_{t}^{h^{\ast }}+z_{t}^{h^{\ast }}-\varepsilon _{t}^{a} $$

(53)

$$ \widetilde{y}_{t}^{N}=l_{t}^{N}+z_{t}^{N}-\varepsilon _{t}^{a} $$

(54)

$$ \widetilde{y}_{t}^{N^{\ast }}=l_{t}^{N^{\ast }}+z_{t}^{N^{\ast }}-\varepsilon _{t}^{a} $$

(55)

Consumer price inflation

$$\triangle p_{t} =\gamma _{c}\triangle p_{t}^{T}+\left( 1-\gamma _{c}\right) \triangle p_{t}^{N} \\[3pt] $$

(56)

$$\triangle p_{t} =\gamma _{c^{\ast }}\triangle p_{t}^{T^{\ast }}+\left( 1-\gamma _{c^{\ast }}\right) \triangle p_{t}^{N^{\ast }} $$

(57)

Tradable inflation

$$ \triangle p_{t}^{T}=\gamma _{x}\triangle p_{t}^{h}+\left( 1-\gamma _{x}\right) (\triangle p_{t}^{f^{\ast }}+\Delta s_{t}) $$

(58)

$$ \triangle p_{t}^{T^{\ast }}=\gamma _{x^{\ast }}(\triangle p_{t}^{h}-\Delta s_{t})+\left( 1-\gamma _{x^{\ast }}\right) \triangle p_{t}^{f^{\ast }} $$

(59)

Price setting in the nontradable sector

$$ \triangle p_{t}^{N}-\varphi _{N}\triangle p_{t-1}^{N}=\beta E_{t}\left( \triangle p_{t+1}^{N}-\varphi _{N}\triangle p_{t}^{N}\right) +\kappa _{N}\left( \widetilde{w}_{t}-z_{t}^{N}-t_{t}^{N}\right) $$

(60)

$$ \triangle p_{t}^{N^{\ast }}-\varphi _{N^{\ast }}\triangle p_{t-1}^{N^{\ast }}=\beta E_{t}\left( \triangle p_{t+1}^{N^{\ast }}-\varphi _{N}^{\ast }\triangle p_{t}^{N^{\ast }}\right) +\kappa _{N^{\ast }}\left( \widetilde{w} _{t}^{\ast }-z_{t}^{N^{\ast }}-t_{t}^{N^{\ast }}\right) $$

(61)

where $\kappa _{N}=\left( 1-\alpha _{N}\right) \left( 1-\beta \alpha _{N}\right) /\alpha _{N},$ $\kappa _{N^{\ast }}=\left( 1-\alpha _{N^{\ast }}\right) \left( 1-\beta \alpha _{N^{\ast }}\right) /\alpha _{N^{\ast }},$ $t^{N}=p_{t}^{N}-p_{t},$ and $t^{N^{\ast }}=p_{t}^{N^{\ast }}-p_{t}$.

Price setting in the intermediate tradable good sector

$$ \triangle p_{t}^{h}-\varphi _{h}\triangle p_{t-1}^{h}=\beta E_{t}\left( \triangle p_{t+1}^{h}-\varphi _{h}\triangle p_{t}^{h}\right) +\kappa _{h}\left( \widetilde{w}_{t}-z_{t}^{h}-t_{t}^{h}-t_{t}^{T}\right) $$

(62)

$$\triangle p_{t}^{f^{\ast }}-\varphi _{f^{\ast }}\triangle p_{t-1}^{f^{\ast }}=\beta E_{t}\left( \triangle p_{t+1}^{f^{\ast }}-\varphi _{f}^{\ast }\triangle p_{t}^{f^{\ast }}\right) +\kappa _{f^{\ast }}\left( \widetilde{w} _{t}^{\ast }-z_{t}^{f^{\ast }}-t_{t}^{f^{\ast }}-t_{t}^{T^{\ast }}\right)\\ $$

(63)

where $\kappa _{h}=\left( 1-\alpha _{h}\right) \left( 1-\beta \alpha _{h}\right) /\alpha _{h},$ $\kappa _{f^{\ast }}=\left( 1-\alpha _{f^{\ast }}\right) \left( 1-\beta \alpha _{f^{\ast }}\right) /\alpha _{f^{\ast }},t_{t}^{h}=p_{t}^{h}-p_{t}^{T},t_{t}^{f^{\ast }}=p_{t}^{f^{\ast }}-p_{t}^{T^{\ast }},$ $t^{T}=p_{t}^{T}-p_{t},$ and $t^{T^{\ast }}=p_{t}^{T^{\ast }}-p_{t}$.

Final consumption demand

$$ \widetilde{c}_{t}^{T}=-\varepsilon t_{t}^{T}+\widetilde{c}_{t} $$

(64)

$$ \widetilde{c}_{t}^{T^{\ast }}=-\varepsilon ^{\ast }t_{t}^{T^{\ast }}+ \widetilde{c}_{t}^{\ast } $$

(65)

$$ \widetilde{c}_{t}^{N}=-\varepsilon t_{t}^{N}+\widetilde{c}_{t} $$

(66)

$$ \widetilde{c}_{t}^{N^{\ast }}=-\varepsilon ^{\ast }t_{t}^{N^{\ast }}+ \widetilde{c}_{t}^{\ast } $$

(67)

Intermediate tradable and nontradable demand

$$ \widetilde{x}_{t}^{h}=-\theta t_{t}^{h}+\widetilde{y}_{t}^{T} $$

(68)

$$ \widetilde{x}_{t}^{h^{\ast }}=-\theta t_{t}^{h^{\ast }}+\widetilde{y} _{t}^{T^{\ast }} $$

(69)

$$ \widetilde{x}_{t}^{f}=-\theta t_{t}^{f}+\widetilde{y}_{t}^{T} $$

(70)

$$ \widetilde{x}_{t}^{f^{\ast }}=-\theta t_{t}^{f^{\ast }}+\widetilde{y} _{t}^{T^{\ast }} $$

(71)

Relative Price Index

Let’s define $t_{t}=\frac{p_{t}^{f}}{p_{t}^{h}}$, since the law of one price holds $t_{t}=-t_{t}^{\ast }=\frac{p_{t}^{h^{\ast }}}{p_{t}^{f^{\ast }}},$ then we can write the following relative prices as a function of t _t .

$$t_{t}^{h} =-\left( 1-\gamma _{x}\right) t_{t} \\ $$

(72)

$$t_{t}^{h^{\ast }} =-\gamma _{x}t_{t} \\ $$

(73)

$$t_{t}^{f} =\gamma _{x}t_{t} \\ $$

(74)

$$t_{t}^{f^{\ast }} =\left( 1-\gamma _{x}\right) t_{t} $$

(75)

Relative prices

$$ t_{t}=t_{t-1}+\triangle s_{t}+\triangle p_{t}^{f^{\ast }}-\triangle p_{t}^{h} $$

(76)

$$ t_{t}^{T}=t_{t-1}^{T}+\triangle p_{t}^{T}-\triangle p_{t} $$

(77)

$$ t_{t}^{N}=-\frac{\gamma _{c}}{1-\gamma _{c}}t_{t}^{T} $$

(78)

$$ t_{t}^{T^{\ast }}=t_{t-1}^{T^{\ast }}+\triangle p_{t}^{T^{\ast }}-\triangle p_{t}^{\ast } $$

(79)

$$ t_{t}^{N^{\ast }}=-\frac{\gamma _{c}^{\ast }}{1-\gamma _{c}^{\ast }} t_{t}^{T^{\ast }} $$

(80)

$$ q_{t}=q_{t-1}+\triangle s_{t}+\triangle p_{t}^{\ast }-\triangle p_{t} $$

(81)

Taylor rules

$$ r_{t}=\rho _{r}r_{t-1}+\left( 1-\rho _{r}\right) \gamma _{\pi }\triangle p_{t}+\gamma _{y}\triangle y_{t}+\varepsilon _{t}^{r} $$

(82)

$$ r_{t}^{\ast }=\rho _{r}^{\ast }r_{t-1}^{\ast }+\left( 1-\rho _{r^{\ast }}\right) \gamma _{\pi }^{\ast }\triangle p_{t}^{\ast }+\gamma _{y}^{\ast }\triangle y_{t}+\varepsilon _{t}^{r^{\ast }} $$

(83)

Net foreign assets and net exports

$$ \beta \widetilde{b}_{t}-\frac{1}{1+g}\widetilde{b}_{t-1}=\widetilde{nx}_{t} $$

(84)

$$ \widetilde{nx}_{t}=\frac{X^{f}}{Y}\left( \widetilde{x}_{t}^{h^{\ast }}- \widetilde{x}_{t}^{f}-t_{t}\right) $$

(85)

where $\widetilde{b}_{t}=\frac{\bar{B}_{t}^{F}S_{t}}{P_{t}Y}$ is the debt to GDP ratio, and $\widetilde{nx}_{t}=\frac{NX_{t}}{Y}$, and where we have assumed balanced trade in the steady state. To solve for the steady-state ratios, we have that:

$$ \frac{Y^{T}}{Y}=\frac{G^{T}}{G}=\frac{C^{T}}{C}=\gamma _{c} $$

$$ \frac{Y^{N}}{Y}=\frac{G^{N}}{G}=\frac{C^{N}}{C}=1-\gamma _{c} $$

$$ \frac{X^{h}}{Y^{T}}=\gamma _{x},\text{ }\frac{X^{f}}{Y^{T}}=1-\gamma _{x} $$

Therefore tradable GDP over total GDP is

$$ \frac{X^{h}}{Y}=\frac{X^{h}}{Y^{T}}\frac{Y^{T}}{Y}=\gamma _{x}\gamma _{c}. $$

Hence

$$ \frac{X^{f}}{Y}=\frac{X^{f}}{X^{T}}\frac{X^{T}}{Y^{T}}\frac{Y^{T}}{Y} =(1-\gamma _{x})\gamma _{c}. $$

Market clearing:

$$ \tilde{y}_{t}^{T}=(1-\gamma )\tilde{c}_{t}^{T}+\gamma \tilde{g}_{t}^{T} $$

(86)

where γ equals the fraction of government spending over total output. For the nontradable good, the market clearing condition is:

$$ \tilde{y}_{t}^{N}=(1-\gamma )\tilde{c}_{t}^{N}+\gamma \tilde{g}_{t}^{N} $$

(87)

Finally, for the intermediate tradable goods sector:

$$ \widetilde{y}_{t}^{h}=\gamma _{x}\widetilde{x}_{t}^{h}+\left( 1-\gamma _{x}\right) \widetilde{x}_{t}^{h^{\ast }} $$

(88)

Total real GDP

$$ \widetilde{y}_{t}=\gamma _{c}(t_{t}^{T}+\widetilde{y}_{t}^{T})+\left( 1-\gamma _{c}\right) (t_{t}^{N}+\widetilde{y}_{t}^{N}) $$

(89)

total labor

$$ l_{t}=\frac{X^{h}}{X^{h}+Y^{N}}l_{t}^{h}+\frac{Y^{N}}{X^{h}+Y^{N}}l_{t}^{N} $$

(90)

For the foreign country:

$$ \tilde{y}_{t}^{T^{\ast }}=(1-\gamma ^{\ast })\tilde{c}_{t}^{T^{\ast }}+\gamma ^{\ast }\tilde{g}_{t}^{T^{\ast }} $$

(91)

$$ \tilde{y}_{t}^{N^{\ast }}=(1-\gamma ^{\ast })\tilde{c}_{t}^{N^{\ast }}+\gamma ^{\ast }\tilde{g}_{t}^{N^{\ast }} $$

(92)

$$ \widetilde{y}_{t}^{f^{\ast }}=\gamma _{x}^{\ast }\widetilde{x}_{t}^{f^{\ast }}+\left( 1-\gamma _{x}^{\ast }\right) \widetilde{x}_{t}^{f} $$

(93)

$$ \widetilde{y}_{t}^{\ast }=\gamma _{c}^{\ast }(t_{t}^{T^{\ast }}+\widetilde{y} _{t}^{T^{\ast }})+\left( 1-\gamma _{c}^{\ast }\right) (t_{t}^{N^{\ast }}+ \widetilde{y}_{t}^{N^{\ast }}) $$

(94)

$$ l_{t}=\frac{Y^{f^{\ast }}}{Y^{h}+Y^{N}}l_{t}^{f^{\ast }}+\frac{Y^{N^{\ast }} }{Y^{f^{\ast }}+Y^{N^{\ast }}}l_{t}^{N\ast } $$

(95)

Mapping variables in the model with observable variables.

$$\widetilde{c}_{t}-\widetilde{c}_{t-1} =\bigtriangleup c_{t}-\varepsilon _{t}^{a} \\[3pt] $$

(96)

$$\widetilde{c}_{t}^{\ast }-\widetilde{c}_{t-1}^{\ast } =\bigtriangleup c_{t}^{\ast }-\varepsilon _{t}^{a} \\[3pt] $$

(97)

$$\widetilde{y}_{t}-\widetilde{y}_{t-1} =\bigtriangleup y_{t}-\varepsilon _{t}^{a} \\[3pt] $$

(98)

$$\widetilde{y}_{t}^{\ast }-\widetilde{y}_{t-1}^{\ast } =\bigtriangleup y_{t}^{\ast }-\varepsilon _{t}^{a} $$

(99)

Shocks

$$ \psi _{t}=\rho _{\psi }\psi _{t-1}+\varepsilon _{t}^{\psi } $$

$$ \psi _{t}^{\ast }=\rho _{\psi ^{\ast }}\psi _{t-1}^{\ast }+\varepsilon _{t}^{\psi ^{\ast }} $$

$$ z_{t}^{h}=\rho ^{Z,h}z_{t-1}^{h}+\varepsilon _{t}^{Z,h} $$

$$ z_{t}^{f^{\ast }}=\rho ^{Z,f^{\ast }}z_{t-1}^{f^{\ast }}+\varepsilon _{t}^{Z,f^{\ast }} $$

$$ z_{t}^{N}=\rho ^{Z,N}z_{t-1}^{N}+\varepsilon _{t}^{Z,N} $$

$$ z_{t}^{N^{\ast }}=\rho ^{Z,N^{\ast }}z_{t-1}^{N^{\ast }}+\varepsilon _{t}^{Z,N^{\ast }} $$

$$ g_{t}^{T}=\rho ^{G,T}g_{t-1}^{T}+\varepsilon _{t}^{G,T} $$

$$ g_{t}^{N}=\rho ^{G,N}g_{t-1}^{N}+\varepsilon _{t}^{G,N} $$

$$ g_{t}^{T^{\ast }}=\rho ^{G,T^{\ast }}g_{t-1}^{T^{\ast }}+\varepsilon _{t}^{G,T^{\ast }} $$

$$ g_{t}^{N^{\ast }}=\rho ^{G,N^{\ast }}g_{t-1}^{N^{\ast }}+\varepsilon _{t}^{G,N^{\ast }} $$

and $\varepsilon _{t}^{a},\varepsilon _{t}^{r},\varepsilon _{t}^{r^{\ast }}$ are iid shocks.