2.1. Households

The world is assumed to consist of two countries, domestic $(D)$ and foreign $(F).$ We assume the size of the domestic economy to be $n$ relative to the world economy, which is modeled as a foreign economy.Footnote ²⁰ A continuum of domestic households exist over $[ 0,n]$ , while foreign households from $(n,1],$ where $n\in ( 0,1) .$ An agent in each country is both a consumer and a producer, producing a single differentiated good and consuming all the goods produced in both countries. Also, the population size in each country is set equal to the range of goods produced in that country, such that domestic firms produce goods on $[0,n]$ , and foreign firms produce goods on $(n,1]$ . The preferences for both domestic and foreign households is assumed to be recursive following Epstein and Zin (Reference Epstein and Zin1989) framework. The Epstein–Zin preferences are assumed similar to Benigno et al. (Reference Benigno, Benigno and Nisticò2012) and Basu and Bundick (Reference Basu and Bundick2017) as these preferences are risk-sensitive and are able to generate variations in the variables comparable to the empirical counterpart.Footnote ²¹ Following this a representative household in domestic country is captured by the following recursive utility function,Footnote ²²

(1) \begin{align} V_{D,t}=\left ( U(C_{t},H_{D,t})^{1-\eta }+\beta \left ( E_{t}\left (V_{D,t+1}\right ) ^{1-\gamma }\right ) ^{\frac{\left ( 1-\eta \right ) }{\left ( 1-\gamma \right ) }}\right ) ^{\frac{1}{\left ( 1-\eta \right ) }}. \end{align}

Here $C_{t}$ denotes the aggregate consumption index, $H_{D,t}$ denotes hours worked by the representative domestic household, $\eta$ is a measure of the inverse of the intertemporal elasticity of substitution, $\gamma$ is the measure of risk aversion, and $\beta \in (0,1)$ is the discount factor. The utility flow $U(.)$ is represented by a Cobb–Douglas function of aggregate consumption index, $C_{t}$ and leisure $\left (1-H_{D,t}\right ),$

\begin{align*} U(C_{t},H_{D,t})=\left ( C_{t}\right ) ^{\nu }\left ( 1-H_{D,t}\right ) ^{1-\nu }, \end{align*}

where $\nu$ is the weight on consumption over leisure in the utility flow function. The aggregate consumption index, $C_{t}$ , is defined as:

(2) \begin{align} C_{t}=\left [ (\mu _{D})^{1/\xi _{D}}\left ( C_{D,t}\right ) ^{\frac{\xi _{D}-1}{\xi _{D}}}+\left ( 1-\mu _{D}\right ) ^{1/\xi _{D}}\left ( C_{F,t}\right ) ^{\frac{\xi _{D}-1}{\xi _{D}}}\right ] ^{\frac{\xi _{D}}{\xi _{D}-1}} \end{align}

where $C_{D,t}$ and $C_{F,t}$ denotes the consumption index of domestic goods and foreign goods of domestic households, respectively. $\xi _{D}\gt 0$ is the elasticity of substitution between domestic goods and foreign goods for domestic households and $\mu _{D}\in (0,1)$ is the weight given to domestic goods in the aggregate consumption basket, $C_{t}.$ Footnote ²³ Analogous to equation (1), the utility function for a representative household in a foreign country is given by:

(3) \begin{align} V_{F,t}=\left ( \Gamma _{F,t}U(C_{t}^{\ast },H_{F,t})^{1-\eta }+\beta \left ( E_{t}\left ( V_{F,t+1}\right ) ^{1-\gamma }\right ) ^{\frac{\left ( 1-\eta \right ) }{\left ( 1-\gamma \right ) }}\right ) ^{\frac{1}{\left ( 1-\eta \right ) }} \end{align}

where $C_{t}^{\ast }$ denotes the aggregate consumption index, $H_{F,t}$ denotes hours worked and $\Gamma _{F,t}$ is the preference/ demand shock process. The utility flow function is given by:

\begin{align*} U(C_{t}^{\ast },H_{F,t})=\left ( C_{t}^{\ast }\right ) ^{\nu }\left ( 1-H_{F,t}\right ) ^{1-\nu } \end{align*}

The aggregate consumption bundle $C_{t}^{\ast }$ is given by:

(4) \begin{align} C_{t}^{\ast }=\left [ (\mu _{F})^{1/\xi _{F}}\left ( C_{D,t}^{\ast }\right ) ^{\frac{\xi _{F}-1}{\xi _{F}}}+\left ( 1-\mu _{F}\right ) ^{1/\xi _{F}}\left ( C_{F,t}^{\ast }\right ) ^{\frac{\xi _{F}-1}{\xi _{F}}}\right ] ^{\frac{\xi _{F}}{\xi _{F}-1}} \end{align}

where $\mu _{F}\in (0,1)$ is weight given to domestic goods in the aggregate consumption basket, $C_{t}^{\ast }$ . Following Benigno et al. (Reference Benigno, Benigno and Nisticò2012), the weights mentioned in the aggregate consumption bundles equations (2) and (4) are related to country sizes through:

(5) \begin{align} 1-\mu _{D}=\left ( 1-n\right ) \chi ; \, \mu _{F}=n\chi . \end{align}

Here, $\chi \in (0,1)$ is the (common) degree of openness between the domestic and foreign country. When $\chi =0,$ there is no trade of either goods or assets happening across the two countries and it represents an autarky case. $\chi =1$ , represents a case of complete free trade of both goods and assets between the two countries. Consumption bundles, $C_{D,t}$ , $C_{F,t},C_{D,t}^{\ast }$ , and $C_{F,t}^{\ast }$ are Dixit–Stiglitz aggregates of differentiated goods produced in two countries and are defined as:

(6) \begin{align} C_{D,t} =\left [ \left ( \frac{1}{n}\right ) ^{\frac{1}{\sigma }}\int _{0}^{n}\left ( C_{D,t}(i) \right ) ^{\frac{\sigma -1}{\sigma }}di\right ] ^{\frac{\sigma }{\sigma -1}}\text{ ; }C_{F,t}=\left [ \left ( \frac{1}{1-n}\right ) ^{\frac{1}{\sigma }}\int _{n}^{1}\left ( C_{F,t}(i) \right ) ^{\frac{\sigma -1}{\sigma }}di\right ] ^{\frac{\sigma }{\sigma -1}} \end{align}

(7) \begin{align} C_{D,t}^{\ast } &=\left [ \left ( \frac{1}{n}\right ) ^{\frac{1}{\sigma }}\int _{0}^{n}\left ( C_{D,t}^{\ast }(i) \right ) ^{\frac{\sigma -1}{\sigma }}di\right ] ^{\frac{\sigma }{\sigma -1}}\text{ ; }C_{F,t}^{\ast }=\left [ \left ( \frac{1}{1-n}\right ) ^{\frac{1}{\sigma }}\int _{n}^{1}\left ( C_{F,t}^{\ast }(i) \right ) ^{\frac{\sigma -1}{\sigma }}di\right ] ^{\frac{\sigma }{\sigma -1}}. \end{align}

Here, $\sigma$ is the elasticity of substitution between the varieties, where a variety is indexed by $i\in [ 0,1] .$ The demand for each variety of a differentiated domestic and foreign good by each country’s household is given as follows:

(8) \begin{align} C_{D,t}(i) =\left ( \frac{1}{n}\right ) \left ( \frac{P_{D,t}(i) }{P_{D,t}}\right ) ^{-\sigma }C_{D,t}\text{ ; }C_{F,t}(i) =\left ( \frac{1}{1-n}\right ) \left ( \frac{P_{F,t}(i) }{P_{F,t}}\right ) ^{-\sigma }C_{F,t} \end{align}

(9) \begin{align} C_{D,t}^{\ast }(i) =\left ( \frac{1}{n}\right ) \left ( \frac{P_{D,t}^{\ast }(i) }{P_{D,t}^{\ast }}\right ) ^{-\sigma }C_{D,t}^{\ast }\text{ ; }C_{F,t}^{\ast }(i) =\left ( \frac{1}{1-n}\right ) \left ( \frac{P_{F,t}^{\ast }(i) }{P_{F,t}^{\ast }}\right ) ^{-\sigma }C_{F,t}^{\ast } \end{align}

where $P_{D,t}(i)$ and $P_{D,t}^{\ast }(i)$ are prices of a variety $i$ of a good produced in the domestic country in domestic and foreign currency, respectively. Similarly, $P_{F,t}(i),$ and $P_{F,t}^{\ast }(i)$ are prices of a variety $i$ of a good produced in the foreign country in domestic and foreign currency, respectively. $P_{D,t},$ $P_{F,t},$ $P_{D,t}^{\ast }$ , and $P_{F,t}^{\ast }$ are the price aggregates of the aggregate consumption baskets, $C_{D,t},$ $C_{F,t},$ $C_{D,t}^{\ast }$ , and $C_{F,t}^{\ast },$ , respectively, and are defined as follows:

(10) \begin{align} P_{D,t} =\left [ \left ( \frac{1}{n}\right ) \int _{0}^{n}P_{D,t}(i) ^{1-\sigma }di\right ] ^{\frac{1}{1-\sigma }}\text{ ; }P_{F,t}=\left [ \left ( \frac{1}{1-n}\right ) \int _{n}^{1}P_{F,t}(i) ^{1-\sigma }di\right ] ^{\frac{1}{1-\sigma }} \end{align}

(11) \begin{align} P_{D,t}^{\ast } =\left [ \left ( \frac{1}{n}\right ) \int _{0}^{n}P_{D,t}^{\ast }(i) ^{1-\sigma }di\right ] ^{\frac{1}{1-\sigma }}\text{ ; }P_{F,t}^{\ast }=\left [ \left ( \frac{1}{1-n}\right ) \int _{n}^{1}P_{F,t}^{\ast }(i) ^{1-\sigma }di\right ] ^{\frac{1}{1-\sigma }} \end{align}

The law of one price is assumed to hold across all individual goods, such that, $P_{D,t}(i) =X_{t}P_{D,t}^{\ast }(i),$ and $P_{F,t}(i) =X_{t}P_{F,t}^{\ast }(i),$ where $X_{t}$ is the nominal exchange rate (price of foreign currency in terms of domestic currency). Using this relation with the price aggregates in equations (10) and (11) we also get, $P_{D,t}=X_{t}P_{D,t}^{\ast }$ and $P_{F,t}=X_{t}P_{F,t}^{\ast }.$ Demand functions for the consumption aggregates, $C_{D,t},$ $C_{F,t},$ $C_{D,t}^{\ast }$ and $C_{F,t}^{\ast }$ are as follows:

(12) \begin{align} C_{D,t}=\mu _{D}\left ( \frac{P_{D,t}}{P_{t}}\right ) ^{-\xi _{D}}C_{t}\text{ };\text{ }C_{F,t}=\left ( 1-\mu _{D}\right ) \left ( \frac{P_{F,t}^{\ast }}{P_{t}^{\ast }}\right ) ^{-\xi _{D}}C_{t}, \end{align}

(13) \begin{align} C_{D,t}^{\ast }=\mu _{F}\left ( \frac{T_{D,t}}{Q_{t}}\right ) ^{-\xi _{F}}C_{t}^{\ast }\text{ };\text{ }C_{F,t}^{\ast }=\left ( 1-\mu _{F}\right ) \left ( \frac{P_{F,t}^{\ast }}{P_{t}^{\ast }}\right ) ^{-\xi _{F}}C_{t}^{\ast } \end{align}

where $P_{t}$ and $P_{t}^{\ast }$ are the aggregate consumer price indices (CPI) in the domestic and foreign country, in domestic and foreign currency, respectively, and are defined as:

(14) \begin{align} P_{t} =\left [ \mu _{D}\left ( P_{D,t}\right ) ^{1-\xi _{D}}+\left ( 1-\mu _{D}\right ) \left ( P_{F,t}\right ) ^{1-\xi _{D}}\right ] ^{\frac{1}{1-\xi _{D}}} \end{align}

(15) \begin{align} P_{t}^{\ast } =\left [ \mu _{F}\left ( P_{D,t}^{\ast }\right ) ^{1-\xi _{F}}+\left ( 1-\mu _{F}\right ) \left ( P_{F,t}^{\ast }\right ) ^{1-\xi _{F}}\right ] ^{\frac{1}{1-\xi _{F}}} \end{align}

It can be seen that due to a heterogeneous preference structure across the two countries, purchasing power parity (PPP) does not hold at the aggregate price levels, such that $P_{t}\neq X_{t}P_{t}^{\ast }.$ PPP holds only when $\mu _{D}=\mu _{F}$ and $\xi _{D}=\xi _{F}.$ Benigno et al. (Reference Benigno, Benigno and Nisticò2012) assume $\mu _{D}\neq \mu _{F},$ such that PPP does not hold in their model too. Any deviations from PPP are measured through the real exchange rate, which is defined as the ratio of consumer price indices in the two countries in terms of domestic prices and is given by:

(16) \begin{align} Q_{t}=\frac{X_{t}P_{t}^{\ast }}{P_{t}}. \end{align}

Rewriting equation (16) gives us the following relationship between CPI in the domestic and foreign country:

(17) \begin{align} \pi _{t}^{\ast }=\pi _{t}\frac{Q_{t}}{Q_{t-1}\pi _{X,t}}. \end{align}

Here, CPI in the foreign country and domestic country are defined as $\pi _{t}^{\ast }=\frac{P_{t}^{\ast }}{P_{t-1}^{\ast }}$ and $\pi _{t}=\frac{P_{t}}{P_{t-1}}$ , respectively. Also, the change in the nominal exchange rate is defined as $\pi _{X,t}=\frac{X_{t}}{X_{t-1}}.$ The terms of trade is defined as a ratio of foreign prices to domestic prices, where both price indices are denominated in domestic currency and is given by:

(18) \begin{align} T_{t}=\frac{T_{F,t}}{T_{D,t}} \end{align}

where we define relative price ratios, $T_{D,t}=\frac{P_{D,t}}{P_{t}}$ and $T_{F,t}=\frac{P_{F,t}}{P_{t}}$ . Using these definitions of relative price ratios with equation (14), we get the following relation,

(19) \begin{align} T_{F,t}=\left [ \frac{1-\mu _{D}\left ( T_{D,t}\right ) ^{1-\xi _{D}}}{1-\mu _{D}}\right ] ^{\frac{1}{1-\xi _{D}}}. \end{align}

Similarly, equation (15) can be re-written in terms of gross foreign inflation $\left (\pi _{F,t}^{\ast }\right ),$ foreign CPI $\left (\pi _{t}^{\ast }\right ),$ and the terms of trade as:

(20) \begin{align} \pi _{t}^{\ast }=\pi _{F,t}^{\ast }\left [ \frac{\mu _{F}\left ( T_{t}\right ) ^{\xi _{F}-1}+\left ( 1-\mu _{F}\right ) }{\mu _{F}\left ( T_{t-1}\right ) ^{\xi _{F}-1}+\left ( 1-\mu _{F}\right ) }\right ] ^{\frac{1}{1-\xi _{F}}} \end{align}

where $\pi _{F,t}^{\ast }=\frac{P_{F,t}^{\ast }}{P_{F,t-1}^{\ast }}.$ For the above described preferences, the total demand for each variety $i$ of the domestic produce is given by $Y_{D,t}(i) =nC_{D,t}(i) +\left ( 1-n\right ) C_{D,t}^{\ast }(i)$ , where $nC_{D,t}(i)$ and $(1-n) C_{D,t}^{\ast }(i)$ is the aggregate demand of all households in the domestic and foreign country, respectively, for variety $i$ of the domestic produce. Using the demand functions described in (8) and (9), we get

(21) \begin{align} Y_{D,t}(i) =\left ( \frac{P_{D,t}(i) }{P_{D,t}}\right ) ^{-\sigma }Y_{D,t} \end{align}

where aggregate demand for domestic good (all varieties) is given by $Y_{D,t}=$ $C_{D,t}+\left ( \frac{1-n}{n}\right ) C_{D,t}^{\ast }.$ Further, using (12) and (13) in equation (21), we can re-write $Y_{D,t}$ in terms of aggregate consumption bundles in the two countries, as given by:

(22) \begin{align} Y_{D,t}=\left ( T_{D,t}\right ) ^{-\xi _{D}}\left [ \mu _{D}C_{t}+\left ( \frac{1-n}{n}\right ) \mu _{F}Q_{t}^{\xi _{F}}\left ( T_{D,t}\right ) ^{\xi _{D}-\xi _{F}}C_{t}^{\ast }\right ] \end{align}

Similar to the domestic country, aggregate demand for a variety $i$ of the foreign good is given by:

(23) \begin{align} Y_{F,t}(i) =\left ( \frac{P_{F,t}(i) }{P_{F,t}}\right ) ^{-\sigma }Y_{F,t} \end{align}

where aggregate demand for the foreign good (all varieties), $Y_{F,t}=\frac{n}{\left ( 1-n\right ) }C_{F,t}+C_{F,t}^{\ast }.$ Aggregate demand, $Y_{F,t},$ can be rewritten in terms of aggregate consumption bundles in the two countries as:

(24) \begin{align} Y_{F,t}=\left ( T_{F,t}\right ) ^{-\xi _{D}}\left [ \frac{n}{\left ( 1-n\right ) }\left ( 1-\mu _{D}\right ) C_{t}+\left ( 1-\mu _{F}\right ) Q_{t}^{\xi _{F}}\left ( T_{F,t}\right ) ^{\xi _{D}-\xi _{F}}C_{t}^{\ast }\right ] \end{align}

Households in the domestic and foreign country maximize (1) and (3) subject to the following flow budget constraints,

(25) \begin{align} W_{D,t}H_{D,t}+\varpi _{D,t} \geq P_{t}C_{t}-B_{D,t}+E_{t}\left \{ B_{D,t+1}M_{t,t+1}\right \} \end{align}

(26) \begin{align} W_{F,t}H_{F,t}+\varpi _{F,t} \geq P_{t}^{\ast }C_{t}^{\ast }-B_{F,t}+E_{t}\left \{ B_{F,t+1}M_{t,t+1}^{\ast }\right \}, \end{align}

respectively. Here $W_{D,t}$ and $W_{F,t}$ are nominal wages in the domestic and foreign country, respectively. The nominal wages are decided in a common labor market in each country. Also, $\varpi _{D,t}$ and $\varpi _{F,t}$ are the nominal profits which households receive from owning monopolistically competitive firms in the domestic and foreign country, respectively. Each household in each country holds equal shares in all firms and there is no trade in firm shares. The asset markets are assumed to be complete both at domestic and at international levels. Households trade in state-contingent nominal securities denominated in the domestic currency. $B_{D,t+1}$ is the state-contingent payoff at time $t+1$ of a portfolio of state-contingent nominal securities held by a household in the domestic country at the end of period $t$ . The value of this portfolio can be written as $E_{t}\left \{ B_{D,t+1}M_{t,t+1}\right \},$ where $M_{t,t+1}$ is the nominal stochastic discount factor for discounting wealth denominated in the domestic currency.

Households in the foreign country also trade in state-contingent securities denominated in the domestic currency. Let $B_{t+1}$ be the state-contingent payoff (denominated in domestic currency) in period $t+1$ of the state-contingent portfolio held by foreign households at the end of period $t.$ The payoff in the foreign currency in period $t+1$ is given by $B_{F,t+1}=\frac{B_{t+1}}{X_{t+1}}.$ Also the value of the portfolio today in foreign currency in period $t$ is given by $\frac{E_{t}\left \{ B_{t+1}M_{t,t+1}\right \} }{X_{t}}=\frac{E_{t}\left \{ B_{F,t+1}X_{t+1}M_{t,t+1}\right \} }{X_{t}}.$ The nominal stochastic discount factor for discounting wealth denominated in the foreign currency can thus be defined as:

(27) \begin{align} M_{t,t+1}^{\ast }=\frac{X_{t+1}}{X_{t}}M_{t,t+1}. \end{align}

The first order conditions for maximizing utility functions (1) and (3) for consumption $\left ( C_{t},C_{t}^{\ast }\right )$ , labor $(H_{D,t},H_{F,t})$ and asset holdings $(B_{D,t+1,}B_{F,t+1})$ subject to the flow budget constraints (25) and (26), respectively, are given by the following Euler’s equations and labor supply equations:

(28) \begin{align} \text{ }\left ( D\right ) :\,{\small E}_{t}\left \{ M_{t,t+1}\right \}{\small =}{\small \beta }\left ( \frac{E_{t}\left ( V_{t+1}\right ) ^{1-\gamma }}{\left ( V_{t+1}\right ) ^{1-\gamma }}\right ) ^{\frac{\left ( \gamma -\eta \right ) }{\left ( 1-\gamma \right ) }}\left ( \frac{U_{D,t+1}}{U_{D,t}}\right ) ^{1-\eta }\left ( \frac{C_{t}}{C_{t+1}}\right ) \left ( \frac{1}{E_{t}\left \{ \pi _{t+1}\right \} }\right ) \end{align}

(29) \begin{align} (F) \,:\,E_{t}\left \{ M_{t,t+1}^{\ast }\right \} =\beta \left ( \frac{\Gamma _{F,t+1}}{\Gamma _{F,t}}\right ) \left ( \frac{E_{t}\left ( V_{t+1}^{\ast }\right ) ^{1-\gamma }}{\left ( V_{t+1}^{\ast }\right ) ^{1-\gamma }}\right ) ^{\frac{\left ( \gamma -\eta \right ) }{\left ( 1-\gamma \right ) }}\left ( \frac{U_{F,t+1}}{U_{F,t}}\right ) ^{1-\eta }\left ( \frac{C_{t}^{\ast }}{C_{t+1}^{\ast }}\right ) \left ( \frac{1}{E_{t}\left \{ \pi _{t+1}^{\ast }\right \} }\right ) \end{align}

(30) \begin{align} \left ( D\right ) :\,w_{D,t}T_{D,t}=\frac{1-\nu }{\nu }\frac{C_{t}}{1-H_{D,t}}; \, \left ( F\right ) \,:\,w_{F,t}\frac{T_{F,t}}{Q_{t}}=\frac{1-\nu }{\nu }\frac{C_{t}^{\ast }}{1-H_{F,t}} \end{align}

Here, the gross nominal interest rate in domestic country is given by $\left ( 1+R_{t}\right ) =\frac{1}{E_{t}\left \{ M_{t,t+1}\right \} }$ and the gross nominal interest rate in foreign country is given by $\left (1+R_{t}^{\ast }\right ) =\frac{1}{E_{t}\left \{ M_{t,t+1}^{\ast }\right \} }$ . Real wages in the domestic and foreign country are defined, respectively, as $w_{D,t}=\frac{W_{D,t}}{P_{D,t}}$ and $w_{F,t}=\frac{W_{F,t}}{P_{F,t}}$ . Combining the Euler equation from equation (28) and (29) with equation (27), we get the following complete asset market condition:

(31) \begin{align} \left ( \frac{V_{t+1}^{1-\gamma }E_{t}V_{t+1}^{\ast 1-\gamma }}{V_{t+1}^{\ast 1-\gamma }E_{t}V_{t+1}^{1-\gamma }}\right ) ^{\frac{\left ( \eta -\gamma \right ) }{\left ( 1-\gamma \right ) }}\left ( \frac{U\left ( C_{t+1},H_{D,t+1}\right ) }{U\left ( C_{t+1}^{\ast },H_{F,t+1}\right ) }\right ) ^{1-\eta }\left ( \frac{C_{t+1}^{\ast }}{C_{t+1}}\right ) Q_{t+1}=\left ( \frac{U\left ( C_{t},H_{D,t}\right ) }{U\left ( C_{t}^{\ast },H_{F,t}\right ) }\right ) ^{1-\eta }\left ( \frac{C_{t}^{\ast }}{C_{t}}\right ) Q_{t} \end{align}

Re-writing the above gives us,

\begin{align*} \kappa _{t+1}=\kappa _{t}\left ( \frac {V_{t+1}^{1-\gamma }E_{t}V_{t+1}^{\ast 1-\gamma }}{V_{t+1}^{\ast 1-\gamma }E_{t}V_{t+1}^{1-\gamma }}\right ) ^{\frac {\left ( \eta -\gamma \right ) }{\left ( 1-\gamma \right ) }} \end{align*}

where $\kappa _{t}=\left ( \frac{U\left ( C_{t},H_{D,t}\right ) }{U\left ( C_{t}^{\ast },H_{F,t}\right ) }\right ) ^{1-\eta }\left ( \frac{C_{t}^{\ast }}{C_{t}}\right ) Q_{t}.$ We estimate the initial value $\kappa _{0}$ from the data as the ratio of marginal utilities of nominal income across countries in the initial period. Equation (27) when combined with definitions of nominal stochastic discount factors that is, $E_{t}\left \{ M_{t,t+1}\right \} =\frac{1}{\left ( 1+R_{t}\right ) }$ and $E_{t}\left \{ M_{t,t+1}^{\ast }\right \} =\frac{1}{\left ( 1+R_{t}^{\ast }\right ) },$ gives the following uncovered interest rate parity $(UIP)$ condition (log-linearized):

(32) \begin{align} r_{t}-r_{t}^{\ast }=E_{t}\left \{ \Delta e_{t+1}\right \} \end{align}

where $r_{t},$ $r_{t}^{\ast }$ and $E_{t}\left \{ \Delta e_{t+1}\right \}$ are logs of $\left (1+R_{t}\right ),$ $\left (1+R_{t}^{\ast }\right )$ and $E_{t}\left \{ \frac{X_{t+1}}{X_{t}}\right \},$ respectively. Following Menkhoff et al. (Reference Menkhoff, Sarno, Schmeling and Schrimpf2012), Backus et al. (Reference Backus, Gavazzoni, Telmer and Zin2010) and Benigno et al. (Reference Benigno, Benigno and Nisticò2012), we define time-varying risk premiums as deviations from the UIP condition, mentioned in equation (32). The log-linearized time-varying risk premiums, $rp_{t},$ are excess returns on holding domestic currency and written as follows:

(33) \begin{align} rp_{t}=r_{t}-r_{t}^{\ast }-E_{t}\left \{ \Delta e_{t+1}\right \} . \end{align}

2.2. Firms

The domestic country produces goods on the interval $\left [ 0,n\right ]$ and the foreign country on $(n,1].$ A firm producing variety $i$ of a good in the domestic and foreign country follows a production function linear in labor as given by:

(34) \begin{align} Y_{D,t}(i) =A_{D,t}H_{D,t}(i) \end{align}

(35) \begin{align} Y_{F,t}(i) =A_{F,t}H_{F,t}(i), \end{align}

respectively. Here, $A_{D,t}$ and $A_{F,t}$ are the productivity levels (common) following exogenous processes. We will keep the productivity level of both the firms at the steady state level ( $\overline{A}_{D}$ and $\overline{A}_{F})$ as in the present paper we are focusing on the second-moment shocks to the global demand shocks. Also, $\overline{A}_{D}=\overline{A}_{F}=1,$ at the steady state. $H_{D,t}(i)$ and $H_{F,t}(i)$ are composites of all the differentiated labor supplied by household $h$ in each country as given by:

(36) \begin{align} H_{D,t}(i) =\frac{1}{n}\int _{0}^{n}H_{D,t}^{h}(i) \text{ }dh\text{ ; }H_{F,t}(i) =\frac{1}{1-n}\int _{n}^{1}H_{F,t}^{h}(i) \text{ }dh \end{align}

where $H_{D,t}^{h}(i)$ and $H_{F,t}^{h}(i)$ are the labor supplied by household $h$ to firm $i$ in the domestic and foreign country, respectively.

2.2.1. Price setting

In the benchmark model we assume that firms in both the countries have nominal price rigidities in the form of price stickiness. We follow Calvo (Reference Calvo1983) to capture price stickiness here. In each period only $(1-\alpha _{D})$ fraction of firms in the domestic country can reset their prices independent of whether they had a chance to reset them in the last period. A firm $i$ which gets a chance to reset its prices, $\overline{P}_{D,t}(i),$ maximizes a discounted sum of current and future expected values of profit, given by

(37) \begin{align} \max _{\overline{P}_{D,t}(i)}\sum \limits _{k=0}^{\infty }\alpha _{D}^{k}M_{t,t+k}\left ( \overline{P}_{D,t}(i)Y_{D,t_{+k}}(i)-MC_{D,t+k}Y_{D,t_{+k}}(i)\right ) \end{align}

where $MC_{D,t+k}$ is the nominal marginal cost of domestic firms in period $t+k$ and is the same for all firms as the nominal wage is decided in a common labor market and all firms face a common productivity level realization. The demand function $Y_{D,t+k}(i),$ for each firm $i$ in period $t+k$ is given by:

\begin{align*} Y_{D,t+k}(i)=\left ( \frac {\overline {P}_{D,t}(i)}{P_{D,t+k}}\right ) ^{-\sigma }Y_{D,t+k} \end{align*}

The optimal price chosen by firms resetting prices is given by:

(38) \begin{align} \overline{P}_{D,t}(i)=\frac{\sigma }{\sigma -1}\frac{\sum \limits _{k=0}^{\infty }\alpha _{D}^{k}M_{t,t+k}MC_{D,t+k}Y_{D,t+k}(i)}{\sum \limits _{k=0}^{\infty }\alpha _{D}^{k}M_{t,t+k}Y_{D,t+k}(i)} \end{align}

where $\frac{\sigma }{\sigma -1}$ is the constant markup charged by firms. As can be seen from equation (38), the optimal price today depends on not just current but future marginal costs, and also demand conditions in the economy. A firm $i$ , which does not reset its price is assumed to keep the prices same as last year’s prices, $P_{D,t-1}(i).$ Thus, the law of motion for the aggregate producer’s price index (PPI) in the domestic country for Calvo’s model can be written as:

(39) \begin{align} P_{D,t}=\left [ \alpha _{D}\left ( P_{D,t-1}\right ) ^{1-\sigma }+\left ( 1-\alpha _{D}\right ) \left ( \overline{P}_{D,t}\right ) ^{1-\sigma }\right ] ^{\frac{1}{1-\sigma }}. \end{align}

Using the domestic household’s optimization problem we can write the stochastic discount factor $M_{t,t+k}$ as:

(40) \begin{align} E_{t}\left \{ M_{t,t+k}\right \} =\beta ^{k}\left ( \frac{E_{t}V_{D,t+k}}{V_{D,t+k}^{1-\gamma }}^{1-\gamma }\right ) ^{\frac{\left ( \gamma -\eta \right ) }{\left ( 1-\gamma \right ) }}\left ( \frac{V_{D,t}}{V_{D,t+k}}\right ) ^{\eta }\frac{\lambda _{D,t+k}}{\lambda _{D,t}}\frac{P_{t}}{P_{t+k}} \end{align}

where $\lambda _{D,t}=\nu \frac{V_{t}^{\eta }U_{D,t}^{1-\eta }}{C_{t}}$ is the Lagrangian multiplier denoting the marginal utility of income. Combined with equation (40), the price setting equation (38) can be written recursively as:

(41) \begin{align} \overline{\pi }_{D,t}=\frac{\sigma }{\sigma -1}\pi _{D,t}\frac{X_{D,t}}{Z_{D,t}} \end{align}

where $X_{D,t}$ and $Z_{D,t}$ are defined as follows:

(42) \begin{align} X_{D,t} =\lambda _{D,t}Y_{D,t}mc_{D,t}T_{D,t}+\alpha _{D}\beta \left ( \pi _{D,t+1}\right ) ^{\sigma }\left ( \frac{E_{t}\left ( V_{t+1}\right ) }{\left ( V_{t+1}\right ) ^{1-\gamma }}^{1-\gamma }\right ) ^{\frac{\left ( \gamma -\eta \right ) }{\left ( 1-\gamma \right ) }}\left ( \frac{1}{V_{t+1}}\right ) ^{\eta }E_{t}\left \{ X_{D,t+1}\right \} \end{align}

(43) \begin{align} Z_{D,t} =\lambda _{D,t}Y_{D,t}T_{D,t+k}+\alpha _{D}\beta \left ( \pi _{D,t+1}\right ) ^{\sigma -1}\left ( \frac{E_{t}\left ( V_{t+k}\right ) }{\left ( V_{t+1}\right ) ^{1-\gamma }}^{1-\gamma }\right ) ^{\frac{\left ( \gamma -\eta \right ) }{\left ( 1-\gamma \right ) }}\left ( \frac{1}{V_{t+1}}\right ) ^{\eta }E_{t}\left \{ Z_{D,t+1}\right \} \end{align}

Here, the reset domestic price inflation is defined as $\overline{\pi }_{D,t}=\frac{\overline{P}_{D,t}}{P_{D,t-1}},$ and domestic price inflation is defined as $\pi _{D,t}=\frac{P_{D,t}}{P_{D,t-1}}.$ The real marginal cost for domestic firms in terms of domestic prices is given by $mc_{D,t}=\frac{MC_{D,t}}{P_{D,t}}.$ The law of motion for the domestic producer’s prices in equation (39) can be written in terms of inflation as follows:

(44) \begin{align} \pi _{D,t}=\left [ \alpha _{D}+\left ( 1-\alpha _{D}\right ) \left ( \overline{\pi }_{D,t}\right ) ^{1-\sigma }\right ] ^{\frac{1}{1-\sigma }}. \end{align}

Since labor is the only input into production, the real marginal cost for domestic firms, $mc_{D,t},$ in terms of domestic prices would then be

(45) \begin{align} mc_{D,t}=\frac{w_{D,t}}{A_{D,t}} \end{align}

where $w_{D,t}=\frac{w_{D,t}}{P_{D,t}}$ are real wages in the domestic country.

The price setting behavior of firms in the foreign country is similar to the price setting behavior of firms in the domestic country, as described from equation (37)–(60). In the foreign country, $(1-\alpha _{F})$ proportion of the firms reset their prices to $\overline{P}_{F,t}$ and the rest $\alpha _{F}$ proportion keep it the same as last year prices, $P_{F,t-1}^{\ast }.$ Maximizing the current and future stream of profits by firms in the foreign country yields the following equation on reset foreign inflation, similar to equation (41)

(46) \begin{align} \overline{\pi }_{F,t}=\frac{\sigma }{\sigma -1}\pi _{F,t}^{\ast }\frac{X_{F,t}}{Z_{F,t}} \end{align}

where $X_{F,t}$ and $Z_{F,t}$ are defined as follows:

(47) \begin{align} X_{F,t} =\lambda _{F,t}Y_{F,t}mc_{F,t}T_{F,t}+\alpha _{F}\beta \left ( \pi _{F,t+1}^{\ast }\right ) ^{\sigma }\left ( \frac{E_{t}\left ( V_{t+1}^{\ast }\right ) }{\left ( V_{t+1}^{\ast }\right ) ^{1-\gamma }}^{1-\gamma }\right ) ^{\frac{\left ( \gamma -\eta \right ) }{\left ( 1-\gamma \right ) }}\left ( \frac{1}{V_{t+1}^{\ast }}\right ) ^{\eta }E_{t}\left \{ X_{F,t+1}\right \} \end{align}

(48) \begin{align} Z_{F,t} =\lambda _{F,t}Y_{F,t}T_{F,t}+\alpha _{F}\beta \left ( \pi _{F,t+1}^{\ast }\right ) ^{\sigma -1}\left ( \frac{E_{t}\left ( V_{t+k}^{\ast }\right ) }{\left ( V_{t+1}^{\ast }\right ) ^{1-\gamma }}^{1-\gamma }\right ) ^{\frac{\left ( \gamma -\eta \right ) }{\left ( 1-\gamma \right ) }}\left ( \frac{1}{V_{t+1}^{\ast }}\right ) ^{\eta }E_{t}\left \{ Z_{F,t+1}\right \} \end{align}

Here, the reset foreign price inflation is defined as $\overline{\pi }_{F,t}=\frac{\overline{P}_{F,t}}{P_{F,t}^{\ast }},$ and the foreign price inflation is defined as $\pi _{F,t}^{\ast }=\frac{P_{F,t}^{\ast }}{P_{F,t-1}^{\ast }}$ . The real marginal cost for the foreign firms in terms of foreign prices is given by $mc_{F,t}=\frac{MC_{F,t}}{P_{F,t}}.$ The law of motion for the foreign producer’s inflation is given by:

(49) \begin{align} \pi _{F,t}^{\ast }=\left [ \alpha _{F}+\left ( 1-\alpha _{F}\right ) \left ( \overline{\pi }_{F,t}\right ) ^{1-\sigma }\right ] ^{\frac{1}{1-\sigma }}. \end{align}

The real marginal cost for the foreign firms, $mc_{F,t},$ in terms of foreign prices would be

(50) \begin{align} mc_{F,t}=\frac{w_{F,t}}{A_{F,t}} \end{align}

where $w_{F,t}=\frac{w_{F,t}}{P_{F,t}}$ denotes real wages in the foreign country.

The terms of trade equation (18) can be written as $T_{t}=\frac{X_{t}P_{F,t}^{\ast }}{P_{D,t}}.$ Rewriting this gives us the following relation between the terms of trade, the nominal exchange rate change and producer price inflation between the two countries:

(51) \begin{align} T_{t}=T_{t-1}\pi _{X,t}\frac{\pi _{F,t}^{\ast }}{\pi _{D,t}}. \end{align}

Under a flexible price equilibrium, $\alpha _{D}=$ $\alpha _{F}=0$ , such that all firms reset their prices in each period. This would imply, $P_{D,t}=\overline{P}_{D,t}$ , $P_{F,t}^{\ast }=\overline{P}_{F,t}$ and $Disp_{D,t}=Disp_{F,t}=1.$ The reset price in each period would simply be a markup over marginal cost in both the countries that is, $P_{D,t}=\frac{\sigma }{\sigma -1}MC_{D,t}$ and $P_{F,t}=\frac{\sigma }{\sigma -1}MC_{F,t}.$

2.3. Equilibrium

2.3.1. Aggregate goods market equilibrium in a SOE

In this section we will describe the equilibrium for the benchmark case of the SOE. To characterize the SOE we follow Benigno and Paoli (Reference Benigno and De Paoli2010) and limit $n\rightarrow 0,$ such that $1-\mu _{D}$ $\rightarrow \chi$ and $\mu _{F}\rightarrow 0$ from equation (5). It can be seen that the share of domestic goods in the consumption basket of domestic households, $\mu _{D},$ now depends only upon the degree of openness (inversely), while the share of domestic goods in the consumption basket of foreign households, $\mu _{F},$ is negligible.Footnote ²⁴ The real exchange rate in equation (16) is now given by:

(52) \begin{align} Q_{t}=\frac{X_{t}P_{F,t}^{\ast }}{P_{t}}=\frac{P_{F,t}}{P_{t}}=T_{F,t} \end{align}

(since $P_{t}^{\ast }=P_{F,t}^{\ast }$ under the limit $n\rightarrow 0$ in CPI equation (15)). The demand function equations (12) and (13), aggregate demand equations (22) and (24), relative price and inflation relations in equations (19) and (20) reduce to the following, respectively:

(53) \begin{align} C_{D,t} =\left ( 1-\chi \right ) \left ( T_{D,t}\right ) ^{-\xi _{D}}C_{t}\ \ ; \, C_{F,t}=\chi \left ( T_{F,t}\right ) ^{-\xi _{D}}C_{t} \end{align}

(54) \begin{align} C_{D,t}^{\ast } =0\text{ }\ ;\text{ }C_{F,t}^{\ast }=\left ( \frac{T_{F,t}}{Q_{t}}\right ) ^{-\xi _{F}}C_{t}^{\ast } \end{align}

(55) \begin{align} Y_{D,t} =\left ( T_{D,t}\right ) ^{-\xi _{D}}\left [ \left ( 1-\chi \right ) C_{t}+\chi Q_{t}^{\xi _{F}}\left ( T_{D,t}\right ) ^{\xi _{D}-\xi _{F}}C_{t}^{\ast }\right ] \end{align}

(56) \begin{align} Y_{F,t} =C_{t}^{\ast } \end{align}

(57) \begin{align} T_{F,t} =\left [ \frac{1-\left ( 1-\chi \right ) \left ( T_{D,t}\right ) ^{1-\xi _{D}}}{\chi }\right ] ^{\frac{1}{1-\xi _{D}}} \end{align}

(58) \begin{align} \pi _{t}^{\ast } =\pi _{F,t}^{\ast }, \end{align}

2.3.2. Aggregate labor market equilibrium

Equilibrium in the labor market would require aggregate labor supply to be equal to aggregate labor demand. For the domestic country, labor is aggregated as follows:

\begin{align*} H_{D,t}=\frac {1}{n}\int \limits _{0}^{n}H_{D,t}(i) di \end{align*}

Using labor demand of a firm $i,$ $H_{D,t}(i),$ from equation (34), and demand for the firms’s output, $Y_{D,t}(i),$ from equation (21), we re-write equilibrium in labor market as:

(59) \begin{align} H_{D,t} =\frac{Y_{D,t}}{A_{D,t}}Disp_{D,t} \end{align}

(60) \begin{align} \text{where }Disp_{D,t} =\left ( \pi _{D,t}\right ) ^{\sigma }\left [ \alpha _{D}Disp_{D,t-1}+\left ( 1-\alpha _{D}\right ) \left ( \overline{\pi }_{D,t}\right ) ^{-\sigma }\right ] \end{align}

is the recursive form of price dispersion term, $Disp_{D,t}=\frac{1}{n}\int \limits _{0}^{n}\left ( \frac{P_{D,t}(i) }{P_{D,t}}\right ) ^{-\sigma }di$ . Analogously, equilibrium in the foreign labor market implies,

(61) \begin{align} H_{F,t} =\frac{Y_{F,t}}{A_{F,t}}Disp_{F,t} \end{align}

(62) \begin{align} \text{where }Disp_{F,t} =\left ( \pi _{F,t}^{\ast }\right ) ^{\sigma }\left [ \alpha _{F}Disp_{F,t-1}+\left ( 1-\alpha _{F}\right ) \left ( \overline{\pi }_{F,t}\right ) ^{-\sigma }\right ] \end{align}

is the recursive form of the price dispersion term, $Disp_{F,t}=\frac{1}{1-n}\int \limits _{n}^{1}\left ( \frac{P_{F,t}^{\ast }(i) }{P_{F,t}^{\ast }}\right ) ^{-\sigma }di.$ For a given wages and prices, labor supply equations (30) along with labor demand equations (59) and (61) determines the labor market equilibrium.

2.3.3. Trade balance and welfare

The trade balance is captured through net exports (net trade of goods) in domestic and foreign country. The value of net exports for the domestic country in terms of domestic consumer prices, $NX_{D,t},$ is defined as the value of total imports (in domestic consumer prices) subtracted from the value of total exports (in domestic consumer prices) and is given by:

(63) \begin{align} NX_{D,t}=T_{D,t}C_{D,t}^{\ast }-T_{F,t}C_{F,t} \end{align}

Similarly, the value of net exports for the foreign country in terms of foreign consumer prices (foreign currency), $NX_{F,t},$ is defined as the value of total imports (in foreign consumer prices) subtracted from the value of total exports (in foreign consumer prices) and is given by:

(64) \begin{align} NX_{F,t}=\frac{T_{F,t}}{Q_{t}}C_{F,t}-\frac{T_{D,t}}{Q_{t}}C_{D,t}^{\ast } \end{align}

A positive and a negative net exports are referred to as trade surplus and trade deficit, respectively. The utility-based welfare criterion defines welfare as an expected lifetime utility of a representative household.Footnote ²⁵ The welfare function in the domestic/ foreign country would thus be a following lifetime utility of a representative household, described in equation (1)/(3):

\begin{align*} Welfare_{i,t}=E_{t}\sum _{t=0}^{\infty }\beta ^{t}V_{i,t} \end{align*}

where $i=D$ or $F.$ We define welfare losses in the domestic country and foreign country as $-Welfare_{D,t}$ and $-Welfare_{F,t}$ , respectively.

2.4. Monetary policy rules

2.4.1. Simple taylor rule: Baseline policy rule

In the baseline case we assume that the central banks in both the domestic and the foreign country set a monetary policy rule on the nominal interest rates using a simple Taylor rule (Taylor (Reference Taylor1993)). Here the central bank attempts to stabilize both inflation and output. In this case, we assume that the measure of inflation targeted by a central bank targets is the CPI in their respective countries. The rules are given by:

(65) \begin{align} \text{TR-CPI} \,:\,\left ( 1+R_{t}\right ) =\overline{R}\left ( \frac{\pi _{t}}{\overline{\pi }}\right ) ^{\phi _{\pi }}\left ( \frac{Y_{D,t}}{Y_{D,t-1}}\right ) ^{\phi _{y}} \end{align}

(66) \begin{align} \text{TR-CPI*} \,:\,\left ( 1+R_{t}^{\ast }\right ) =\overline{R}^{\ast }\left ( \frac{\pi _{t}^{\ast }}{\overline{\pi }^{\ast }}\right ) ^{\phi _{\pi }^{\ast }}\left ( \frac{Y_{F,t}}{Y_{F,t-1}}\right ) ^{\phi _{y}^{\ast }} \end{align}

for the domestic and foreign country, respectively. Here, $\overline{R}=\frac{1}{\beta }$ and $\overline{R}^{\ast }=\frac{1}{\beta }$ are the steady state values of nominal interest rate, $R_{t},$ and $R_{t}^{\ast },$ respectively. We get these steady state values from Euler equations (28) and (29). Here, $\overline{\pi }$ and $\overline{\pi }^{\ast }$ are the steady state values of CPI, in the domestic and foreign country, respectively. The parameters $(\phi _{\pi },\text{ }\phi _{y})$ and $(\phi _{\pi }^{\ast },\text{ }\phi _{y}^{\ast })$ capture the responsiveness of the interest rates to the deviation of inflation from its steady state level and deviation of output from its flexible price level counterpart in the respective countries. We will estimate the $\phi _{\pi },$ $\phi _{y}$ to match the IRFs of model with that of the data for the baseline case in a SOE. Thus, TR-CPI with usual calibration will also be part of the alternate monetary policy rules.

2.4.2. Alternate monetary policy rules

For comparative analysis, we only vary the monetary policy rule in the domestic economy/ SOE. The monetary policy rule for the foreign economy is assumed to be a simple Taylor rule as described in equation (66) for all the alternative monetary policy cases we consider for the domestic economy. The Taylor rule we consider in the baseline case, as described in equation (65) is a CPI-based rule. The first alternate rule we consider is a Taylor rule with PPI as given by:

(67) \begin{align} \text{TR-PPI: }\left ( 1+R_{t}\right ) =\overline{R}\left ( \frac{\pi _{D,t}}{\overline{\pi }_{D}}\right ) ^{\phi _{\pi }}\left ( \frac{Y_{D,t}}{Y_{D,t-1}}\right ) ^{\phi _{y}} \end{align}

Here, $\pi _{D,t}$ is producer price inflation in the domestic country and $\overline{\pi }_{D}$ is it’s steady state value. This is an interesting case because it has been shown in Gali and Monacelli (Reference Galí and Monacelli2005) that under a floating exchange regime it is optimal for the central bank of a SOE to target producer price inflation. Later, Engel (Reference Engel2011) showed that under local currency pricing, exchange rate flexibility does not matter and the optimal policy for a central bank is to completely stabilize CPI.Footnote ²⁶

As argued in Calvo and Reinhart (Reference Calvo and Reinhart2002) and Reinhart (Reference Reinhart2000), EMEs use their foreign exchange reserves and monetary policy with interest rates as an instrument to stabilize exchange rate movements in a floating exchange rate regime. There also exists empirical evidence showing that central banks in emerging markets consider exchange rate movements while setting their monetary policy (Cuevas and Topak (Reference Cuevas and Topak2008); Aizenman et al. (Reference Aizenman and Noy2011)). Given this, the next set of rules we consider are Taylor rules (both CPI and PPI) with nominal exchange rates, as given by

(68) \begin{align} \text{TR-CPI-ER}\,:\, \left ( 1+R_{t}\right ) =\overline{R}\left ( \frac{\pi _{t}}{\overline{\pi }}\right ) ^{\phi _{\pi }}\left ( \frac{Y_{D,t}}{Y_{D,t-1}}\right ) ^{\phi _{y}}\left ( \frac{X_{t}}{X_{t-1}}\right ) ^{\phi _{X}} \end{align}

(69) \begin{align} \text{TR-PPI-ER} \,:\,\left ( 1+R_{t}\right ) =\overline{R}\left ( \frac{\pi _{D,t}}{\overline{\pi }_{D}}\right ) ^{\phi _{\pi }}\left ( \frac{Y_{D,t}}{Y_{D,t-1}}\right ) ^{\phi _{y}}\left ( \frac{X_{t}}{X_{t-1}}\right ) ^{\phi _{X}} \end{align}

Here, $\frac{X_{t}}{X_{t-1}}$ denotes a change in the nominal exchange rate and the policy rate responds positively to a positive change in the nominal exchange rate. This is because a depreciation of currency would imply an increase in expected future inflation (due to a rise in import prices) and an increase in output (because of a higher demand for exports and import substitution). A rise (fall) in the interest rate is thus required to stabilize the economy from the effects of the depreciation (appreciation).

From the empirical evidence shown in Section 1.1, it is evident that the movement of the exchange rates (both nominal as well as real) is high and significant in emerging markets in presence of global uncertainty shocks. We also observed that the nominal interest rates increase as a response to an increase in global uncertainty and thus can reinforce the adverse effects of uncertainty shock. At the same time the interest rates do not seem to stabilize exchange rates. Aizenman et al. (Reference Aizenman and Noy2011) showed that when monetary policy is geared to stabilize inflation, output and exchange rates, exchange rates are not much stabilized as a part of mixed strategy in an IT (inflation-targeting) regime. Given the inability of IRRs to absorb the effect of the shock under consideration, we examine, an alternative instrument for conducting monetary policy, namely, exchange rates. This puts a rule on exchange rates directly and does not let them float freely. These set of rules are called ERR where a central bank manages exchange rates to target inflation and output. The Monetary Authority of Singapore (MAS) has been following this rule since 1981 (McCallum (2006)). We consider a simple ERR as described in Heiperzt et al. (Reference Heiperzt, Mihov and Santacreu2017):

(70) \begin{align} \text{ERR: }\frac{X_{t}}{X_{t-1}}=\left ( \frac{Y_{D,t}}{Y_{D,t-1}}\right ) ^{-\phi _{y}^{e}}\left ( \frac{\pi _{t}}{\overline{\pi }}\right ) ^{-\phi _{\pi }^{e}} \end{align}

Here, $\phi _{y}^{e}$ and $\phi _{\pi }^{e}$ are the response parameters of nominal exchange to the change in output and inflation. Note that the exchange rate responds negatively to an increase in inflation and output to stabilize the economy. This is because increase in inflation and output can be stabilized when nominal exchange rates fall (an appreciation). An appreciation reduces inflation (by reducing the price of imports) and also reduces output (by reducing the foreign demand for domestic goods and reducing the domestic good’s demand by domestic households). We also consider an extreme case of a fixed exchange rule (PEG) where the central bank completely stabilizes the nominal exchange rate, as given by

(71) \begin{align} \text{PEG: }\frac{X_{t}}{X_{t-1}}=1 \end{align}

When, $\phi _{y}^{e}\rightarrow 0$ and $\phi _{\pi }^{e}\rightarrow 0,$ the ERR (70) approaches a PEG rule in (71). As values of $\phi _{y}^{e}$ and $\phi _{\pi }^{e}$ increase, the exchange rate adjusts more to stabilize the economy. Note that interest rates are endogenously determined in the economy under ERR and PEG rule.

2.5. Exogenous shock processes

We followed Basu and Bundick (Reference Basu and Bundick2017) and Fernández-Villaverde et al., (Reference Fernández-Villaverde, Guerrón-Quintana, Rubio-Ramírez and Uribe2011) to describe the shock processes with uncertainty shocks. For the global uncertainty shocks we assume a shock to the second-moment of a foreign country’s preference/ demand process as mentioned above. The demand shock process in equation (3) take the following form,

(72) \begin{align} \Gamma _{F,t}=\left ( 1-\delta _{F}\right ) \overline{\Gamma }_{F}+\delta _{F}\Gamma _{F,t-1}+v_{F,t-1}\varepsilon _{F,t} \end{align}

where $\varepsilon _{F,t}$ is shock to the first-moment of demand shock. The standard deviation $v_{F,t-1}$ in the foreign demand are not constant and are described by the following AR $(1)$ processes,

(73) \begin{align} v_{F,t}=\left ( 1-\delta _{\sigma _{F}}\right ) \overline{v}_{F}+\delta _{\sigma _{F}}v_{F,t-1}+\varpi _{F}\vartheta _{F,t} \end{align}

Here, $\vartheta _{F,t}$ is shock to the second-moment or an uncertainty shock to the underlying demand. In other words, uncertainty shocks here refer to the shocks to standard deviation of the underlying process. It is assumed that the stochastic shocks, $\varepsilon _{F,t}$ and $\vartheta _{F,t},$ are independent and normally distributed random variables.

2.6. Solution method

We are interested in looking at the effects of shocks to the second moments (or uncertainty shocks) of the demand/ preference levels of the foreign country on a SOE (domestic country). To capture the complete effect of the second-moment shocks on the endogenous variables of the model, we need to take the third order approximation of the model equations as explained in Fernández-Villaverde et al., (Reference Fernández-Villaverde, Guerrón-Quintana, Rubio-Ramírez and Uribe2011) and later also applied in Basu and Bundick (Reference Basu and Bundick2017). Following this, we do a third order Taylor series approximation of the model using the Dynare software package in MATLAB to find a solution to our benchmark model.Footnote ²⁷ All the approximations are done around the stochastic steady state.

2.7. Calibration

In this section we attempt to calibrate the SOE to a prototypical EME and the foreign country, which comprises the world, to an AE. We estimate the degree of openness parameter, $\chi,$ to be 0.6, as the average trade share to GDP of EMEs. To get this we use World Bank’s country level trade data for year 2015 (World Bank (Reference Bank2018)). The value of $\kappa$ , which is the initial parameter in the asset market condition is estimated to be 3.8. We calculate this using the OECD database on national accounts (OECD (2019)). Details on the calculation of $\chi$ and $\kappa$ is provided in the Data Appendix A.1. The value of intertemporal elasticity of substitution varies from 0.2 to 0.5 following Fernández-Villaverde et al. (Reference Fernández-Villaverde, Guerrón-Quintana, Rubio-Ramírez and Uribe2011) and Benigno et al. (Reference Benigno, Benigno and Nisticò2012), respectively. The elasticity of intertemporal of substitution here is set to 0.33 such that the inverse of the intertemporal elasticity of substitution parameter, $\eta,$ for the domestic and the foreign country, is 3. For the value of the elasticity of substitution between domestic and foreign goods, $\xi _{D}$ and $\xi _{F},$ we use 1.5 as calculated in Benigno et al. (Reference Benigno, Benigno and Nisticò2012). The values are kept same for both the foreign as well domestic country due to lack of any data evidence on EMEs for the same. The goods are thus assumed to be relatively substitutable in the benchmark calibration for both the countries. The discount factor, $\beta$ is assumed to be the same in both the countries and is set to 0.994 following Basu and Bundick (Reference Basu and Bundick2017). We use standard value for the weight on the consumption over leisure that is, 1/3 as also calibrated in Benigno et al. (Reference Benigno, Benigno and Nisticò2012). The parameter for the elasticity of substitution between varieties, $\theta$ , is set to 6 following Benigno et al. (Reference Benigno, Benigno and Nisticò2012) such that the steady state markup for a firm is 20%. In the baseline calibration we fix the value of stickiness parameter for the foreign country, $\alpha _{F},$ to be 0.66 following Sbordone (Reference Sbordone2002) and Gali et al. (Reference Gali and Lopez-Salido2001). These papers provide empirical evidence for stickiness parameter for the US and Europe, respectively. For the domestic country/SOE the parameter for stickiness, $\alpha _{D}$ is set slightly higher to 0.75 such that domestic firms revise prices in 4 quarters.Footnote ²⁸ We also compare our baseline sticky price calibration results to a completely flexible price calibration, where $\alpha _{D}=0$ and $\alpha _{F}=0$ . The preference shock parameters for the foreign country at first-order and second-order are partially calibrated from Basu and Bundick (Reference Basu and Bundick2017) $,$ such that we set persistence shock parameter $\delta _{F}=0.94$ and the steady state values for the demand shock, $\overline{\Gamma }_{F}=1.$ The rest of the shocks parameters namely, the persistence to the uncertainty shock parameter, $\delta _{\sigma _{F}}=0.62,$ the steady state values its standard deviation of demand, $\overline{v}_{F}=0.085$ and the scaling parameter for the uncertainty shock $\varpi =0.45$ are set to match the impulse response function of the nominal interest rates with the empirical counterpart as shown in Figure 4. The response of the rest of the economy is put for an assessment. This is how we take the IRFs generated by the model to the IRFs generated using data in the empirical section.

Figure 4.

Interest rate response before and after IRF matching for Taylor parameter $ \phi _{y}.$

For the baseline calibration of the Taylor rule as described in equations (65) and (66), for both the countries, we set the weight on inflation to be $\phi _{\pi }=\phi _{\pi }^{\ast }=1.5$ . These are the standard values used in the literature (Taylor (Reference Taylor1993)). While the weight on output growth for the foreign country, $\phi _{y}^{\ast }$ is set to its standard value $0.5$ , for the domestic country we estimate the weight to match the impulse response in data and find $\phi _{y}=0.1$ . In other words, the response of the EMEs to the output change is muted in presence of uncertainty shock. We also consider models with alternate monetary policies. The parameter for Taylor rules with an exchange rate where weight on the exchange rate change, $\phi _{X}$ , is set to 0.05 uses estimates from Cuevas and Topak (Reference Cuevas and Topak2008). The ERR parameters, $\phi _{\pi }^{e},$ that is, weight on the inflation gap, and $\phi _{y}^{e},$ that is, weight on the output gap are set to 0.30 and 0.02, respectively, similar estimates from Parrado (Reference Parrado2004) and Heiperzt et al. (Reference Heiperzt, Mihov and Santacreu2017). We also show the impulse response functions for ERRs with varying value of inflation sensitivity parameter, $\phi _{\pi }^{e},$ to 0.60 and 0.16, respectively. The parameters are summarized in Table 1.

Table 1.

Summary of parameter values