Exchange Rate Policy and Firm Heterogeneity

Abstract

This paper examines the exchange rate policy in a tractable framework with heterogeneous firms, incomplete financial markets and nominal rigidities. External demand shocks generate exchange rate movements leading to uncertainty in the labor demand of exporter firms. When exporter firms are homogeneous in terms of productivity, a monetary policy response to external demand shocks stabilizes the export market and improves welfare, thus providing a rationale for managed exchange rate policies.

Appendices

Appendix 1: Solution of the Model

We derive here the closed form solution of the theoretical model presented in Table 1. Similar expressions hold for Foreign. First, note that using average prices and the expressions of price indices, we have $P_{H,t}=N_{D,t}^{-\frac{1}{\sigma -1}}\widetilde{p}_{D,t}$ and $P_{F,t}=N_{X,t}^{*-\frac{1}{\sigma -1}}\widetilde{p}_{X,t}^{*}$. Plugging these expressions in the expression of domestic profits, profits from exporting and total profits on average, we have $\widetilde{D}_{D,t}=\frac{\alpha {t}}{\sigma }\frac{\mu _{t}}{N_{D,t}}$, $\widetilde{D}_{X,t}=\frac{\alpha _{t}}{\sigma }\frac{\varepsilon _{t}\mu _{t}^{*}}{N_{X,t}}-f_{X,t}W_{t}$ and $\widetilde{D}_{t}=\widetilde{D}_{D,t}+\frac{N_{X,t}}{N_{D,t}}\widetilde{D}_{X,t}$. Given the zero cutoff profits (ZCP) condition, we have $\widetilde{D}_{X,t}=W_{t}f_{X,t}\frac{\sigma -1}{\kappa -\left( \sigma -1\right) }$. By combining these two expressions of $\widetilde{D}_{X,t}$ we have $\widetilde{D}_{X,t}=\frac{\sigma -1}{\kappa }\frac{\alpha _{t}}{\sigma }\frac{\varepsilon _{t}\mu _{t}^{*}}{N_{X,t}}$. Using the ZCP condition with the expression of $\widetilde{D}_{X,t}$ and the exchange rate implied under the balanced trade $\varepsilon _{t}=\frac{\alpha _{t}^{*}}{\alpha _{t}}\frac{\mu _{t}}{\mu _{t}^{*}}$, we have $N_{X,t}=\frac{1}{\sigma }(1-\frac{\sigma -1}{\kappa })\frac{\alpha _{t}^{*}\mu _{t}}{W_{t}f_{X,t}}$. The assumption of a Pareto distribution of firm productivity implies that $\widetilde{z}_{X,t}=\left[ \frac{\kappa }{\kappa -\left( \sigma -1\right) }\right] ^{\frac{1}{\sigma -1}}\left( \frac{N_{X,t}}{N_{D,t}}\right) ^{-\frac{1}{\kappa }}$.

We are now ready to derive the number of new entrants, $N_{D,t+1}$. Free entry implies that $\widetilde{V}_{t}=f_{E,t}W_{t}$. Combined with the expression of $\widetilde{D}_{t+1}$, the Euler equation about the share holdings, $\widetilde{V}_{t}=E_{t}\left[ Q_{t,t+1}\widetilde{D}_{t+1}\right]$, is expressed as

$$\begin{aligned} E_{t}\left[ \frac{\beta P_{t}C_{t}}{P_{t+1}C_{t+1}}\left( \widetilde{D}_{D,t+1}+\frac{N_{X,t+1}}{N_{D,t+1}}\widetilde{D}_{X,t+1}\right) \right] =f_{E,t}W_{t}. \end{aligned}$$

Plugging the expression of $\widetilde{D}_{D,t+1}$, $\widetilde{D}_{X,t+1}$ and the expression of the equilibrium exchange rate $\varepsilon _{t}=\frac{\alpha _{t}^{*}}{\alpha _{t}}\frac{\mu _{t}}{\mu _{t}^{*}}$, the previous equation is rewritten as

$$\begin{aligned} \frac{\beta }{\sigma }\frac{\mu _{t}}{N_{D,t+1}}E_{t}\left[ \left( \alpha _{t+1}+\frac{\sigma -1}{\kappa }\alpha _{t+1}^{*}\right) \right] =f_{E,t}W_{t}, \end{aligned}$$

which gives

$$\begin{aligned} N_{D,t+1}=\frac{\beta }{\sigma }\frac{\mu _{t}}{W_{t}f_{E,t}}E_{t}\left[ \alpha _{t+1}+\frac{\sigma -1}{\kappa }\alpha _{t+1}^{*}\right] . \end{aligned}$$

Next we derive the labor demand in general equilibrium. Note that $\widetilde{D}_{X,t}=\frac{1}{\sigma }\frac{\varepsilon _{t}\widetilde{p}_{X,t}}{\tau }\widetilde{y}_{X,t}-f_{X,t}W_{t}$ and $\widetilde{D}_{D,t}=\frac{1}{\sigma }\widetilde{p}_{D,t}\widetilde{y}_{D,t}$. Once we plug the expression of prices into these profits, we have $\widetilde{y}_{D,t}=\left( \sigma -1\right) \frac{\widetilde{D}_{D,t}\widetilde{z}_{D}}{W_{t}}$ and $\widetilde{y}_{X,t}=\left( \sigma -1\right) \frac{\left( \widetilde{D}_{X,t}+f_{X,t}W_{t}\right) \widetilde{z}_{X,t}}{W_{t}}$. Replacing those expressions in the labor market clearings (8), we have

$$\begin{aligned} L_{t}=N_{D,t}\left( \sigma -1\right) \frac{\widetilde{D}_{D,t}}{W_{t}}+N_{X,t}\left( \left( \sigma -1\right) \frac{\widetilde{D}_{X,t}+f_{X,t}W_{t}}{W_{t}}+f_{X,t}\right) +N_{D,t+1}f_{E,t}. \end{aligned}$$

Using the expression of $\widetilde{D}_{D,t}$, $\widetilde{D}_{X,t}$, $N_{D,t+1}$, $N_{X,t}$ and the exchange rate found previously, the above expression becomes

$$\begin{aligned} L_{t}=\frac{\mu _{t}}{W_{t}}A_{t}. \end{aligned}$$

Finally, we obtain Eq. (9) after replacing the wage setting of Eq. (6) into the above expression.

1.1 Solution of the Model Without Firm Dynamics

In our model, the monetary intervention plays a key role in mitigating the fluctuations in labor demand determined by the preference shock. In order to highlight the role of firm dynamics in labor demand fluctuations, we derive here a version of our model without selection into exporting market as described in the “lagged entry” model of Hamano and Picard (2017).

Note that by setting $f_{X,t}=0$, all firms export despite firm heterogeneity, hence $N_{X,t}=N_{D,t}$ and $\widetilde{z}_{X,t}=\widetilde{z}_{D}$. In such a specific case, we have $\widetilde{D}_{D,t}=\frac{\alpha _{t}}{\sigma }\frac{\mu _{t}}{N_{D,t}}$, $\widetilde{D}_{X,t}=\frac{\alpha _{t}}{\sigma }\frac{\varepsilon _{t}\mu _{t}^{*}}{N_{D,t}}$. Once we replace these expressions in the Euler equation with free entry condition, we get

$$\begin{aligned} E_{t}\left[ \frac{\beta P_{t}C_{t}}{P_{t+1}C_{t+1}}\left( \widetilde{D}_{D,t+1}+\widetilde{D}_{X,t+1}\right) \right] =f_{E,t}W_{t}, \end{aligned}$$

which gives the number of future domestic firms: $N_{D,t+1}=\frac{\beta }{\sigma }\frac{\mu _{t}}{W_{t}f_{E,t}}E_{t}\left[ \alpha _{t+1}+\alpha _{t+1}^{*}\right] =\frac{\beta }{\sigma }\frac{\mu _{t}}{W_{t}f_{E,t}}$. The labor market clearing is

$$\begin{aligned} L_{t}=N_{D,t}\left( \frac{\widetilde{y}_{D,t}}{\widetilde{z}_{D}}+\frac{\widetilde{y}_{X,t}}{\widetilde{z}_{D}}\right) +N_{D,t+1}f_{E,t}, \end{aligned}$$

where $\widetilde{y}_{D,t}=\left( \sigma -1\right) \frac{\widetilde{D}_{D,t}\widetilde{z}_{D}}{W_{t}}$ and $\widetilde{y}_{X,t}=\left( \sigma -1\right) \frac{\widetilde{D}_{X,t}\widetilde{z}_{D}}{W_{t}}$. Together with $N_{D,t+1}$, we obtain the labor demand in Hamano and Picard (2017):

$$\begin{aligned} L_{t}=\frac{\mu _{t}}{W_{t}}\left[ \frac{\sigma -1}{\sigma }+\frac{\beta }{\sigma }\right] . \end{aligned}$$

The equilibrium wage is

$$\begin{aligned} W_{t}=\Gamma \left\{ E_{t-1}\left[ \mu _{t}^{1+\varphi }\right] \right\} ^{\frac{1}{1+\varphi }}, \end{aligned}$$

which corresponds to the wage in our model when $\kappa =\sigma -1$: when firm heterogeneity is the largest as possible, the adjustment at the extensive margin due to labor demand uncertainty is the smallest.

Appendix 2: Social Planner

In this section, we show the solution of a benevolent social planner. Due to the assumption of one period to build and the full depreciation of firms after one period of production, we express the expected utility only for two consecutive periods without loss of generality as

$$\begin{aligned}&{E}_{t-1}\left[ {\mathcal {U}}\right] \equiv {E}_{t-1}\left[ U_{t}\right] +\beta {E}_{t-1}\left[ U_{t+1}\right] \\&={E}_{t-1}\left[ \text {ln}C_{t}\right] -\frac{\eta }{1+\varphi }{E}_{t-1}\left[ L_{t}^{1+\varphi }\right] +\beta \left\{ {E}_{t-1}\left[ \text {ln}C_{t+1}\right] -\frac{\eta }{1+\varphi }{E}_{t-1}\left[ L_{t+1}^{1+\varphi }\right] \right\} . \end{aligned}$$

Using the good market clearing conditions $\widetilde{c}_{D,t}=\widetilde{y}_{D,t}, \widetilde{c}_{X,t}=\widetilde{y}_{X,t}^{*}, \widetilde{c}_{D,t}^{*}=\widetilde{y}_{D,t}^{*}, \widetilde{c}_{X,t}^{*}=\widetilde{y}_{X,t}$, we get:

$$\begin{aligned} {E}_{t-1}\left[ {\mathcal {U}}\right]= & {} {E}_{t-1}\left[ \alpha _{t}\left( 1+\frac{1}{\sigma -1}\right) \mathrm {ln}N_{D,t}\right. \\&\left. +\alpha _{t}\mathrm {ln}\widetilde{y}_{D,t}+\alpha _{t}^{*}\left( 1+\frac{1}{\sigma -1}\right) \mathrm {ln}N_{X,t}^{*}+\alpha _{t}^{*}\mathrm {ln}\widetilde{y}_{X,t}^{*}\right] \\&-\frac{\eta }{1+\varphi }{E}_{t-1}\left[ L_{t}^{1+\varphi }\right] \\&+\beta {E}_{t-1}\left[ \alpha _{t+1}\left( 1+\frac{1}{\sigma -1}\right) \mathrm {ln}N_{D,t+1}+\alpha _{t+1}\mathrm {ln}\widetilde{y}_{D,t+1}\right. \\&\left. +\alpha _{t+1}^{*}\left( 1+\frac{1}{\sigma -1}\right) \mathrm {ln}N_{X,t+1}^{*}+\alpha _{t+1}^{*}\mathrm {ln}\widetilde{y}_{X,t+1}^{*}\right] \\&-\frac{\beta \eta }{1+\varphi }{E}_{t-1}\left[ L_{t+1}^{1+\varphi }\right] . \end{aligned}$$

As argued in the text, the planner maximizes ${E}_{t-1}\left[ {\mathcal {U}}\right] +{E}_{t-1}\left[ {\mathcal {U}}^{*}\right]$ with respect to $\widetilde{y}_{D,t},\widetilde{y}_{D,t}^{*},N_{D,t+1},N_{D,t+1}^{*},\widetilde{y}_{X,t},\widetilde{y}_{X,t}^{*},N_{X,t},N_{X,t}^{*}$ subject to two types of technological constraints, namely (5) and (8) for each country. The solution is given by Table 2. The optimal labor supply in Home is given by

$$\begin{aligned} L_{t}=\left( \frac{1}{\eta }{\mathcal {A}}_{t}\right) ^{\frac{1}{1+\varphi }}, \end{aligned}$$

(15)

where ${{\mathcal {A}}_{t}}\equiv \left( 2+\frac{1}{\sigma -1}-\frac{1}{\kappa }\right) \alpha _{t}+\beta (\frac{1}{\sigma -1}+\frac{1}{\kappa })E_{t}\left[ \alpha _{t+1}\right]$. The planner lets Home households work more when the preference attached to goods produced in the Home country is high ($\alpha _{t}>\alpha _{t}^{*}$). As a result, as shown in Table 2, the number of exporters in Home is higher than in Foreign ($N_{X,t}>N_{X,t}^{*}$) and the average domestic production in Home ($\widetilde{y}_{D,t}>\widetilde{y}_{D,t}^{*}$) is higher than in Foreign. The extent of this gap depends negatively upon the marginal disutility of labor supply $\eta L_{t}^{\varphi }$ and $\eta L_{t}^{*\varphi }$. Further, given $N_{X,t}>N_{X,t}^{*}$ and noting that $N_{D,t}$ and $N_{D,t}^{*}$ are the state of the economy, the average productivity of Home exporters is lower than the average productivity of Foreign exporters ($\widetilde{z}_{X,t}<\widetilde{z}_{X,t}^{*}$). As a consequence, the average production of Home exporters is smaller than that of Foreign exporters ($\widetilde{y}_{X,t}<\widetilde{y}_{X,t}^{*}$). When the expected preference attached to goods produced in the Home country is high ($E_{t}\left[ \alpha _{t+1}\right] >E_{t}\left[ \alpha _{t+1}^{*}\right]$), the planner lets Home households work more. The future number of firms in the Home country is then higher than in the Foreign country ($N_{D,t+1}>N_{D,t+1}^{*}$).

Appendix 3: Complete Financial Markets

Let us show that the allocation of the social planner is very close to the one in our framework once we allow for complete financial markets and flexible wages. To begin with, we characterize the equilibrium exchange rate. Under complete asset markets, the marginal utility stemming from one additional unit of nominal wealth is equal across countries. Given our preferences defined in Eq. (1), this implies

$$\begin{aligned} \varepsilon _{t}=\frac{\mu _{t}}{\mu _{t}^{*}}. \end{aligned}$$

Note that complete markets allow households to ensure against demand shocks, and as a consequence, the exchange rate is also independent from demand shocks. Table 3 reports the solution of the model under complete asset markets (with wage rigidity).

To what extent the allocation under complete markets differs from that implied by the social planner? Without wage rigidities, the equilibrium wage is $W_{t}=\Gamma \mu _{0}{\mathcal {A}}_{t}^{\frac{\varphi }{1+\varphi }}$ and $W_{t}^{*}=\Gamma \mu _{0}{\mathcal {A}}_{t}^{*\frac{\varphi }{1+\varphi }}$, where monetary stances serve just as the “nominal anchors” which determine the wage level in each country. As a result, the real variables are independent from monetary stances. In particular, the equilibrium labor supply under complete asset markets and flexible wages is

$$\begin{aligned} L_{t}=\left[ \frac{\sigma -1}{\sigma }\frac{\left( \theta -1\right) \left( 1+\xi \right) }{\eta \theta }{\mathcal {A}}_{t}\right] ^{\frac{1}{1+\varphi }}. \end{aligned}$$

Comparing the above solution with (15), we can state that the equilibrium allocation under complete financial markets and flexible wages is identical to the one implied by the social planner once monopolistic distortions both in goods and labor markets are removed. Indeed, by setting $\frac{\sigma -1}{\sigma }\frac{\left( \theta -1\right) \left( 1+\xi \right) }{\theta }=1$, the labor supply is equal to the one in the planner problem. In order to compare the allocation of the competitive equilibrium with the allocation of the social planner who does not have prices, we express the allocation with the above labor supply. Note that without wage rigidities, Eq. (6) implies that the equilibrium wage is $W_{t}=\Gamma ^{1+\varphi }\mu _{t}L_{t}^{\varphi }$ and $W_{t}^{*}=\Gamma ^{1+\varphi }\mu _{t}^{*}L_{t}^{*\varphi }$. By plugging the expressions for the model solution under complete asset markets of Table 3, we can show the Corollary of Proposition 1.

Finally, we can write the terms of trade under complete markets and flexible wages as

$$\begin{aligned} TOT_{t}=\left( \frac{L_{t}^{*}}{L_{t}}\right) ^{\varphi }\frac{\widetilde{z}_{X,t}}{\widetilde{z}_{X,t}^{*}}. \end{aligned}$$

Following a positive demand shock for Home produced goods, the Home terms of trade appreciate because $L_{t}/L_{t}^{*}$ increases and $\widetilde{z}_{X,t}/\widetilde{z}_{X,t}^{*}$ decreases. The extent of the appreciation is higher for a lower elasticity of labor supply, $1/\varphi$. This expression is considered as the desired terms of trade by the social planner. Shutting down monopolistic power and firm heterogeneity, i.e., without variation in the cutoff level of productivity, the expression of the desired terms of trade by the social planner collapses into the one in Devereux (2004).

Appendix 4: Incomplete Financial Markets and Flexible Wages

The allocation with flexible wages under incomplete financial markets is obtained by removing the expectation operator in the solution of the benchmark economy presented in Table 1: wages are flexible and are not set one period in advance. The equilibrium wage is then $W_{t}=\Gamma A_{t}^{\frac{\varphi }{1+\varphi }}\mu _{t}$ and $W_{t}^{*}=\Gamma A_{t}^{\frac{\varphi }{1+\varphi }}\mu _{t}$ and the monetary stance is just a nominal anchor. Accordingly, the nominal exchange rate $\varepsilon _{t}$ has no impact on the real allocation. Plugging the equilibrium flexible wage in the solution of Table 1, we prove Proposition 3.

In the peculiar case of infinite elasticity of labor supply, that is when $\varphi =0$, the allocation with flexible wages (and incomplete financial markets) is exactly the same as under the flexible exchange rate policy. When labor supply is infinitely elastic, the flexible exchange rate can therefore compensate for the wage rigidity. However, this does not imply that a flexible exchange rate is the dominant one. This allocation is indeed far from the first best allocation. Following a positive demand shift for Home goods, the relative number of Home exporters decreases, whereas it would increase increases in the planner solution. The adjustments at the extensive and intensive margins are inefficient even when wages are flexible in our setting with incomplete financial markets. This also implies that the fluctuations in the terms of trade under a flexible exchange rate do not reproduce the complete markets allocation. The comparison between the flexible exchange rate and the flexible wage therefore highlights the role of incomplete financial markets for the choice of the exchange rate policy.

Appendix 5: Exchange Rate Policy

1.1 Polar Exchange Rate Policies

In competitive equilibrium, ${E}_{t-1}\left[ L_{t+1}^{1+\varphi }\right]$ is constant, thus the expected utility of Home representative household for any consecutive time period is given by Eq. (11). Using the solution of Table 1 and reporting time invariant variables as a constant, we get

$$\begin{aligned} {E}_{t-1}\left[ {\mathcal {U}}\right]= & {} {E}_{t-1}\left[ \alpha _{t}\mathrm {ln}\mu _{t}\right] -{E}_{t-1}\left[ \alpha _{t}\mathrm {ln}W_{t}\right] \\&+\left( \frac{1}{\sigma -1}+1-\frac{1}{\kappa }\right) \left\{ {E}_{t-1}\left[ \alpha _{t}^{*}\mathrm {ln}\mu _{t}^{*}\right] -{E}_{t-1}\left[ \alpha _{t}^{*}\mathrm {ln}W_{t}^{*}\right] \right\} \\&+\frac{\beta }{\sigma -1}\left\{ {E}_{t-1}\left[ \alpha _{t+1}\mathrm {ln}\mu _{t}\right] -{E}_{t-1}\left[ \alpha _{t+1}\mathrm {ln}W_{t}\right] \right\} \\&+\frac{\beta }{\kappa }\left\{ {E}_{t-1}\left[ \alpha _{t+1}^{*}\mathrm {ln}\mu _{t}^{*}\right] -{E}_{t-1}\left[ \alpha _{t+1}^{*}\mathrm {ln}W_{t}^{*}\right] \right\} +\mathrm {cst}. \end{aligned}$$

Recall that $\alpha _{t}=\frac{1}{2}\upsilon _{t}$ and $\alpha _{t}^{*}=\frac{1}{2}\upsilon _{t}^{*}$, and we assume zero serial correlation across shocks. Finally, plugging the expression of wages in equilibrium, we get

$$\begin{aligned} {E}_{t-1}\left[ {\mathcal {U}}\right]= & {} \frac{1}{2}\left\{ {E}_{t-1}\left[ \upsilon _{t}\mathrm {ln}\mu _{t}\right] -\frac{1}{1+\varphi }\mathrm {ln}E_{t-1}\left[ \left( A_{t}\mu _{t}\right) ^{1+\varphi }\right] \right\} \\&+\frac{1}{2}\left( \frac{1}{\sigma -1}+1-\frac{1}{\kappa }\right) \left\{ {E}_{t-1}\left[ \upsilon _{t}^{*}\mathrm {ln}\mu _{t}^{*}\right] -\frac{1}{1+\varphi }\ln E_{t-1}\left[ \left( A_{t}^{*}\mu _{t}^{*}\right) ^{1+\varphi }\right] \right\} \\&+\frac{1}{2}\frac{\beta }{\sigma -1}\left\{ {E}_{t-1}\left[ \mathrm {ln}\mu _{t}\right] -\frac{1}{1+\varphi }\ln E_{t-1}\left[ \left( A_{t}\mu _{t}\right) ^{1+\varphi }\right] \right\} \\&+\frac{1}{2}\frac{\beta }{\kappa }\left\{ {E}_{t-1}\left[ \mathrm {ln}\mu _{t}^{*}\right] -\frac{1}{1+\varphi }\ln E_{t-1}\left[ \left( A_{t}^{*}\mu _{t}^{*}\right) ^{1+\varphi }\right] \right\} +\mathrm {cst}. \end{aligned}$$

We then replace the equilibrium variables under the two polar exchange rate policies to evaluate their impact on welfare.

1.2 Welfare and Firm Heterogeneity

We provide here the proof of Proposition 5. Once we take a Taylor expansion up to the second order of $E_{t-1}\left[ A_{t}^{1+\varphi }\right]$ and $E_{t-1}\left[ \left( A_{t}\upsilon _{t}\right) ^{1+\varphi }\right]$, and we evaluate these functions at $\upsilon _{t}=1$, we get

$$\begin{aligned}&E_{t-1}\left[ A_{t}^{1+\varphi }\right] =A^{1+\varphi }+\frac{1}{2}\left( 1+\varphi \right) f_{FL}(\kappa )Var(\upsilon _{t})+\cdots \\&E_{t-1}\left[ \left( A_{t}\upsilon _{t}\right) ^{1+\varphi }\right] =A^{1+\varphi }+\frac{1}{2}\left( 1+\varphi \right) f_{FX}(\kappa )Var(\upsilon _{t})+\cdots \end{aligned}$$

where $A\equiv \frac{\sigma -1}{2\sigma }\left[ 1+\beta \left( \frac{1}{\sigma -1}+\frac{1}{\kappa }\right) \right]$, $\Phi =\frac{\sigma -1}{2\sigma }\left( \frac{1}{\sigma -1}-\frac{1}{\kappa }\right)$, $f_{FX}(\kappa )\equiv \left[ \varphi \left( A-\Phi \right) ^{2}A^{\varphi -1}-2\Phi A^{\varphi }\right]$ and $\quad f_{FL}(\kappa )\equiv \varphi \Phi ^{2}A^{\varphi -1}$. We then obtain

$$\begin{aligned}&\frac{2\sigma \kappa ^{2}}{\sigma -1}\left[ \frac{\partial f_{FL}(\kappa )}{\partial \kappa }-\frac{\partial f_{FX}(\kappa )}{\partial \kappa }\right] =2\beta \varphi \left( A-\Phi \right) \Lambda ^{\varphi -1}+2\left( A-\Phi \right) ^{\varphi }A^{\varphi -1}+2A^{\varphi }\\&\quad +\,2\left( 1-\beta \right) \varphi \Phi A^{\varphi -1}+\beta \left( \varphi -1\right) \varphi A^{\varphi -1}\left( A-2\Phi \right) >0, \end{aligned}$$

since $A-\Phi =\frac{\sigma -1}{2\sigma }\left[ \frac{\sigma }{\sigma -1}+\frac{1}{\kappa }+\beta \left( \frac{1}{\sigma -1}+\frac{1}{\kappa }\right) \right] >0$ and $A-2\Phi =\frac{\sigma -1}{2\sigma }\left[ 1+\frac{2}{\kappa }+\beta \left( \frac{1}{\sigma -1}+\frac{1}{\kappa }\right) \right] >0$. It follows that $\frac{\partial \Delta \ln W_{t}}{\partial \kappa }<0$.

1.3 Optimal Non-cooperative Monetary Policy

The Home monetary authority maximizes (11) with respect to $\mu _{t}$ and takes $\mu _{t}^{*}$ as given:

$$\begin{aligned} \max _{\mu _{t}}{E}_{t-1}\left[ {\mathcal {U}}\right] . \end{aligned}$$

The first-order condition with respect to $\mu _{t}$ is

$$\begin{aligned}&\frac{1}{2}\left\{ \frac{\upsilon _{t}}{\mu _{t}}-\frac{1}{E_{t-1}\left[ \left( A_{t}\mu _{t}\right) ^{1+\varphi }\right] }\frac{\left( A_{t}\mu _{t}\right) ^{1+\varphi }}{\mu _{t}}\right\} \\&\quad +\,\frac{1}{2}\frac{\beta }{\sigma -1}\left\{ \frac{1}{\mu _{t}}-\frac{1}{E_{t-1}\left[ \left( A_{t}\mu _{t}\right) ^{1+\varphi }\right] }\frac{\left( A_{t}\mu _{t}\right) ^{1+\varphi }}{\mu _{t}}\right\} =0. \end{aligned}$$

A similar condition holds for the monetary authority in Foreign. Replacing $A_{t}$, $A_{t}^{*}$ and the optimal policies $\mu _{t}$ and $\mu _{t}^{*}$ in the expression of the exchange rate, we get

$$\begin{aligned} \varepsilon _{t}=\frac{\upsilon _{t}^{*}}{\upsilon _{t}}\frac{\mu _{t}}{\mu _{t}^{*}}=\frac{\upsilon _{t}^{*}}{\upsilon _{t}}\left[ \frac{1+\frac{1}{\sigma -1}-\frac{1}{\kappa }+\beta \left( \frac{1}{\sigma -1}+\frac{1}{\kappa }\right) -\left( \frac{1}{\sigma -1}-\frac{1}{\kappa }\right) \upsilon _{t}^{*}}{1+\frac{1}{\sigma -1}-\frac{1}{\kappa }+\beta \left( \frac{1}{\sigma -1}+\frac{1}{\kappa }\right) -\left( \frac{1}{\sigma -1}-\frac{1}{\kappa }\right) \upsilon _{t}}\right] \left[ \frac{\upsilon _{t}+\frac{\beta }{\sigma -1}}{\upsilon _{t}^{*}+\frac{\beta }{\sigma -1}}\right] ^{\frac{1}{1+\varphi }}. \end{aligned}$$

It is straightforward to note that

$$\begin{aligned} \frac{\partial \varepsilon _{t}}{\partial \upsilon _{t}}\mid _{\upsilon _{t}=1}=-1+\frac{\frac{1}{\sigma -1}-\frac{1}{\kappa }}{1+\beta \left( \frac{1}{\sigma -1}+\frac{1}{\kappa }\right) }+\frac{1}{\left( 1+\varphi \right) \left( 1+\frac{\beta }{\sigma -1}\right) }>-1. \end{aligned}$$

For a given demand shock, the fluctuations of the nominal exchange rate under the optimal non-cooperative policy are therefore lower than those under the flexible exchange rate policy, which are proportional to the demand shocks. Moreover, the fluctuations under the optimal policies are more limited, the higher is the Pareto shape $\kappa$. This leads to Proposition 6.

1.4 Optimal Cooperative Monetary Policy

Under cooperation, the objective of monetary policy in the Home country is

$$\begin{aligned} \max _{\mu _{t}}{E}_{t-1}\left[ {\mathcal {U}}+{\mathcal {U}}^{*}\right] . \end{aligned}$$

The first-order condition with respect to $\mu _{t}$ is

$$\begin{aligned}&\frac{1}{2}\left( \frac{1}{\sigma -1}+2-\frac{1}{\kappa }\right) \left\{ \frac{\upsilon _{t}}{\mu _{t}}-\frac{1}{E_{t-1}\left[ \left( A_{t}\mu _{t}\right) ^{1+\varphi }\right] }\frac{\left( A_{t}\mu _{t}\right) ^{1+\varphi }}{\mu _{t}}\right\} \\&\quad +\,\frac{\beta }{2}\left( \frac{1}{\sigma -1}+\frac{1}{\kappa }\right) \left\{ \frac{1}{\mu _{t}}-\frac{1}{E_{t-1}\left[ \left( A_{t}\mu _{t}\right) ^{1+\varphi }\right] }\frac{\left( A_{t}\mu _{t}\right) ^{1+\varphi }}{\mu _{t}}\right\} =0. \end{aligned}$$

A similar condition holds for the monetary authority in Foreign. Replacing $A_{t}$, $A_{t}^{*}$ and the optimal policies $\mu _{t}$ and $\mu _{t}^{*}$ in the expression of the exchange rate, we get

$$\begin{aligned} \varepsilon _{t}=\frac{\upsilon _{t}^{*}}{\upsilon _{t}}\frac{\mu _{t}}{\mu _{t}^{*}}=\frac{\upsilon _{t}^{*}}{\upsilon _{t}}\left[ \frac{1+\frac{1}{\sigma -1}-\frac{1}{\kappa }+\beta \left( \frac{1}{\sigma -1}+\frac{1}{\kappa }\right) -\left( \frac{1}{\sigma -1}-\frac{1}{\kappa }\right) \upsilon _{t}^{*}}{1+\frac{1}{\sigma -1}-\frac{1}{\kappa }+\beta \left( \frac{1}{\sigma -1}+\frac{1}{\kappa }\right) -\left( \frac{1}{\sigma -1}-\frac{1}{\kappa }\right) \upsilon _{t}}\right] \left[ \frac{\upsilon _{t}+\beta \left( \frac{\frac{1}{\sigma -1}+\frac{1}{\kappa }}{\frac{1}{\sigma -1}+2-\frac{1}{\kappa }}\right) }{\upsilon _{t}^{*}+\beta \left( \frac{\frac{1}{\sigma -1}+\frac{1}{\kappa }}{\frac{1}{\sigma -1}+2-\frac{1}{\kappa }}\right) }\right] ^{\frac{1}{1+\varphi }}. \end{aligned}$$

The non-cooperative and cooperative monetary policies are identical when $\kappa =\sigma -1$, that is for the largest degree of firm heterogeneity. Instead, when $\kappa$ is larger, the gains of international monetary policy cooperation are larger, as the coordinated response to demand shocks prevent abrupt adjustments in the extensive margins of trade.

Table 3 The closed form solution of the model with complete financial markets