Interaction of fiscal and monetary policy in a monetary union under the zero lower bound constraint

Abstract

This paper analyses the interaction of monetary and fiscal policies in a dynamic stochastic general equilibrium model of a monetary union where conventional monetary policy is restricted by the binding zero lower bound constraint. We use the OccBin toolkit to analyze the performance of optimally set monetary and fiscal policies and compare outcomes with simple rules for government spending or the interest rate. Government spending are set independently from each other. We make several findings. First, fiscal policy plays an active role in stabilizing the economy under the binding zero lower bound constraint. Second, properly designed fiscal policy can even shorten the duration of the binding constraint. Third, coordination of policies is not always preferable.

Appendices

Derivation of the flex-price equilibrium

1.1 Real exchange rate

Under flexible prices and taking into account the terms of trade $T_t=\frac{P_{F,t}}{P_{H,t}}$, the real exchange rate is given by

$$\begin{aligned} Q_t^{1-\phi }=\left( \frac{P^*_t}{P_t}\right) ^{1-\phi }=\frac{(1-\gamma )P_{H,t}^{1-\phi }+\gamma P_{F,t}^{1-\phi }}{\gamma P_{H,t}^{1-\phi }+(1-\gamma )P_{F,t}^{1-\phi } } = \frac{(1-\gamma )+\gamma T_{t}^{1-\phi }}{\gamma +(1-\gamma )T_{t}^{1-\phi }} \end{aligned}$$

(42)

Log-linearizing this relationship yields

$$\begin{aligned} \widetilde{Q}_t=(2\gamma -1)\widetilde{T}_t \end{aligned}$$

1.2 Complete markets condition

An approximation of the complete market condition (7) is given by

$$\begin{aligned} \widetilde{Q}_t= & {} \left( \varepsilon _t^F-\varepsilon _t^H\right) +\sigma ({\widetilde{c}}_t^H-{\widetilde{c}}^F_t) \end{aligned}$$

Using the above characterization of the real exchange rate this can be rewritten to yield equation (12) in the main text.

1.3 The aggregate supply equation

With flexible prices all producers set the same price with an markup over marginal costs $W_t/P_t$. However, we assume that the mark-up is neutralized by a lump-sum tax yielding the following equilibrium condition:

$$\begin{aligned} \frac{P_{H,t}}{P_t}=\frac{W_t}{P_t}=\frac{N_t^\eta }{C_t^{-\sigma }}=\frac{y(h)_t^{H,\eta }}{\zeta _t^H C_t^{-\sigma }} \end{aligned}$$

Expressing

$$\begin{aligned} \frac{P_{H,t}}{P_t}= \left( \gamma +(1-\gamma )T_t^{1-\phi }\right) ^{\frac{1}{1-\phi }} \end{aligned}$$

a log-linear approximation of the aggregate supply side is given by:

$$\begin{aligned} \sigma {\widetilde{c}}_t^H+\eta {\widetilde{y}}_t^H+(1-\gamma )\widetilde{T}_t-\varepsilon ^H_t =0 \end{aligned}$$

The foreign counterpart is given by

$$\begin{aligned} \sigma {\widetilde{c}}_t^F+\eta {\widetilde{y}}_t^F-(1-\gamma )\widetilde{T}_t-\varepsilon _t^F =0 \end{aligned}$$

1.4 Aggregate demand

The aggregate demand equation for home goods

$$\begin{aligned} Y_t^H = \left( \frac{P_{H,t}}{P_{t}}\right) ^{-\phi }\left( \gamma C^H_t+(1-\gamma )Q_t^\phi C_t^F\right) +G^H_t \end{aligned}$$

can be approximated by

$$\begin{aligned} {\widetilde{y}}_t^H= & {} c_y \left[ \phi (1-\gamma )\widetilde{T}_t+\gamma {\widetilde{c}}^H_t +(1-\gamma ){\widetilde{c}}^F_t+\phi (1-\gamma )\widetilde{Q}_t \right] +(1-c_y)\widetilde{g}^H_t \\= & {} c_y \left[ 2\phi \gamma (1-\gamma )\widetilde{T}_t+\gamma {\widetilde{c}}^H_t +(1-\gamma ){\widetilde{c}}^F_t+\right] +(1-c_y)\widetilde{g}^H_t \end{aligned}$$

where in the last line we have used equation (42) to substitute out the real exchange rate.

The foreign counterpart is given by

$$\begin{aligned} {\widetilde{y}}_t^F= & {} c_y \left[ -2\phi \gamma (1-\gamma )\widetilde{T}_t+\gamma {\widetilde{c}}^F_t +(1-\gamma ){\widetilde{c}}^H_t+ \right] +(1-c_y)\widetilde{g}^F_t \end{aligned}$$

1.5 Efficient government spending

As in Beetsma and Jensen (2005) we assume that in the flex-price equilibrium fiscal authorities coordinate choosing government spending to maximize the union-wide utility function which is a weighted average of the utility function of the representative household. Beetsma and Jensen (2005) show in their appendix how to derive a condition determining the efficient levels of government spending. The same method applies here. First, they take the first derivative of the utility function with respect to government spending taking into account that consumption levels and output which is given by aggregate demand and thus terms of trade depend on government spending. Second, they make use of the fact that the aggregate demand and aggregate supply relationships hold in equilibrium which simplifies the first-order conditions derived above to yield $V_G(G_t^H)=V_y(Y_t^H).$

Thus, an approximation leads to $-\sigma \widetilde{g}_t^H=\eta {\widetilde{y}}_t^H$ which is Eq. (13) in the main text.

The foreign counterpart is given by $-\sigma \widetilde{g}_t^F=\eta {\widetilde{y}}_t^F$.

1.6 The Euler equation

The Euler equation (2) can be log-linearized as in Woodford (2003) or Galí (2015):

$$\begin{aligned} {\widetilde{c}}_t^H=E_t{\widetilde{c}}_{t+1}^H-\frac{1}{\sigma } \widetilde{i}^H_t+\frac{1}{\sigma } \left( \varepsilon _t^H-E_t\varepsilon _{t+1}^H \right) \end{aligned}$$

An analogue relation holds for the foreign country.

1.7 Solving for the efficient flex-price equilibrium

To solve for the efficient allocation we substitute the government spending rules into the aggregate demand equations and solve for ${\widetilde{y}}_t^H$ and ${\widetilde{y}}_t^F$. As a next step we take the difference of both results, multiply the whole equation by $\sigma$ and compute

$$\begin{aligned} \left[ \sigma +\eta (1-c_y)\right] {\widetilde{y}}_t^R\equiv & {} \left[ \sigma +\eta (1-c_y)\right] ({\widetilde{y}}_t^F-{\widetilde{y}}_t^H)\\= & {} c_y \left[ -4 \phi \sigma \gamma (1-\gamma )\widetilde{T}_t+(2\gamma -1) \sigma {\widetilde{c}}^R_t\right] \\= & {} c_y \left[ \left( -4 \phi \sigma \gamma (1-\gamma )-(2\gamma -1)^2\right) \widetilde{T}_t+(2\gamma -1)\varepsilon ^R_t \right] \end{aligned}$$

where in the final step we have used the efficient market condition to replace relative consumption, and where $\varepsilon _t^R=\varepsilon _t^F-\varepsilon _t^H$ denotes the relative shock. Note that this equation complements the analogues result derived in Corsetti et al. (2010). However, here government spending is taken into account.

On the other side we can take differences of the aggregate supply equation, replace differences in consumption ${\widetilde{c}}_t^R$ using the complete market condition and get:

$$\begin{aligned} \varepsilon ^R_t -(2\gamma -1) \widetilde{T}_t+\eta {\widetilde{y}}_t^R-2(1-\gamma )\widetilde{T}_t-\varepsilon _t^R =0 \quad \Longrightarrow \quad \eta {\widetilde{y}}_t^R = \widetilde{T}_t \end{aligned}$$

The last two relationships can be used to express $\widetilde{T}_t$ and ${\widetilde{y}}_t^R$ in terms of the relative demand shock:

$$\begin{aligned} {\widetilde{y}}_t^R = \frac{c_y(2\gamma -1)}{\sigma +\eta (1-c_y) +\eta c_y(4 \phi \sigma \gamma (1-\gamma )+(2\gamma -1)^2)} \varepsilon _t^R \end{aligned}$$

The complete market condition can be used to derive ${\widetilde{c}}_t^R$ in terms of the shock. Taking differences of the Euler equations, taking into account the above result for relative consumption, we finally solve for the relative efficient interest rate:

$$\begin{aligned} \widetilde{\iota }^R=\frac{-\eta c_y(1-2\gamma )^2}{\sigma +\eta (1-c_y) +\eta c_y(4 \phi \sigma \gamma (1-\gamma )+(2\gamma -1)^2)}E_t\left( \varepsilon _{t+1}^R-\varepsilon _t^R \right) \end{aligned}$$

If there is no home bias in consumption, i.e., $\gamma =1/2$, the relative efficient interest rate is zero, implying that both country-specific efficient rates are equal.

To get results for country-specific variables we have to derive the aggregate variables depending on the shocks which can be done along the same line as above. However, instead of taking differences we now aggregate the corresponding equations.

First, aggregate both demand equations:

$$\begin{aligned} {\widetilde{y}}_t^W=c_y{\widetilde{c}}_t^W+(1-c_y)\widetilde{g}_t^W \end{aligned}$$

Second, aggregate both efficient policy rule for government spending:

$$\begin{aligned} -\sigma \widetilde{g}_t^W=\eta {\widetilde{y}}_t^W \end{aligned}$$

Third, aggregate both aggregate supply eqations:

$$\begin{aligned} \sigma {\widetilde{c}}_t^W= -\eta {\widetilde{y}}_t^W+\varepsilon _t^W \end{aligned}$$

Forth, use all equations to solve for

$$\begin{aligned} {\widetilde{c}}_t^W= \frac{\sigma +\eta (1-c_y)}{\sigma \left[ \sigma +\eta \right] }\varepsilon _t^W \end{aligned}$$

Finally, substitute this into the aggregate Euler equation solved for the efficient rate of interest we get

$$\begin{aligned} \widetilde{\iota }_t=\sigma E_t {\widetilde{c}}_{t+1}^W-\sigma {\widetilde{c}}_{t}^W + \varepsilon _t^W-E_t\varepsilon _{t+1}^W = \frac{\eta c_y}{ \sigma +\eta }\left( \varepsilon _t^W - E_t\varepsilon _{t+1}^W \right) \end{aligned}$$

Second-order approximations to the sticky-price equilibrium

To compute the second-order approximations to the equilibrium equations we proceed as follows: First, we take logs of the equations, then take a first-order Taylor expansion of the equations in logs, and finally we replace the first-order terms in $(x_t-x)$ by $x(\hat{x}_t+\frac{1}{2}\hat{x}_t^2)$. The resulting equation will be the second-order approximation.

1.1 Log-linearizing the aggregate demand equation

The aggregate demand function for domestic output in equilibrium is given by (4):

$$\begin{aligned} Y^H_t= & {} \left( \frac{P_{H,t}}{P_{t}}\right) ^{-\phi }\left( \gamma C^H_t+(1-\gamma )Q_t^\phi C_t^F\right) +G^H_t \\= & {} \left( \gamma +(1-\gamma )T_t^{1-\phi }\right) ^{\frac{\phi }{1-\phi }} \left[ \left( \gamma C^H_t+(1-\gamma )Q_t^\phi C_t^F\right) +G^H_t\right] \end{aligned}$$

Thus, computing the necessary steps yields

$$\begin{aligned} \hat{y}^H_t= & {} c_y \left[ 2\phi \gamma (1-\gamma )\hat{T}_t+\gamma {\hat{c}}^H_t +(1-\gamma ){\hat{c}}^F_t \right] +(1-c_y)\hat{g}^H_t \\&+\frac{1}{2} c_y\left[ \gamma (1-\gamma )\phi \hat{T}^2_t+\gamma ({\hat{c}}^H_t)^2 +(1-\gamma )({\hat{c}}^F_t)^2 + 2\phi \gamma (1-\gamma ) \hat{T}^2_t({\hat{c}}^H_t+{\hat{c}}^F_t) \right] \\&-\frac{1}{2}(\hat{y}^H_t)^2+ \frac{1}{2}(1-c_y)(\hat{g}^H_t)^2 \end{aligned}$$

Similar equation holds for the foreign country:

$$\begin{aligned} \hat{y}^F_t= & {} c_y \left[ -2\phi \gamma (1-\gamma )\hat{T}_t+\gamma {\hat{c}}^F_t +(1-\gamma ){\hat{c}}^H_t \right] +(1-c_y)\hat{g}^F_t \\&+\frac{1}{2} c_y\left[ \gamma (1-\gamma )\phi \hat{T}^2_t+\gamma ({\hat{c}}^F_t)^2 +(1-\gamma )({\hat{c}}^H_t)^2 - 2\phi \gamma (1-\gamma ) \hat{T}^2_t({\hat{c}}^H_t+{\hat{c}}^F_t) \right] \\&-\frac{1}{2}(\hat{y}^F_t)^2+ \frac{1}{2}(1-c_y)(\hat{g}^F_t)^2 \end{aligned}$$

Both equations coincide with those derived in Kirsanova et al. (2007).

Aggregation of both equation yields:

$$\begin{aligned} \hat{y}_t^W= & {} c_y{\hat{c}}_t^W+(1-c_y)\hat{g}_t^W +\frac{1}{2} c_y\left[ ({\hat{c}}^H_t)^2+({\hat{c}}^F_t)^2+ 2\phi \gamma (1-\gamma ) \hat{T}^2_t \right] \nonumber \\&-\frac{1}{2}(\hat{y}^H_t)^2 -\frac{1}{2}(\hat{y}^F_t)^2+ \frac{1}{2}(1-c_y)(\hat{g}^H_t)^2 +\frac{1}{2}(1-c_y)(\hat{g}^F_t)^2 \end{aligned}$$

(43)

1.2 Log-linearizing the price setting equation

In deriving an approximation to the optimal price setting equation we exactly follow the steps as in Woodford (2003) or Galí (2015) which results in the NKPC.

Deriving the welfare function

A second-order approximation to the aggregate utility flow (1) of all representative household can be considered as a good approximation of the welfare function if fluctuations from the steady state are small enough. The negative of such an approximation yields a loss function (up to some multiplicative constants) which can be taken as policy objectives.

Following (Woodford 2003 Chapter 6.4) or Beetsma and Jensen (2005) we get for the first term in (1) the following approximation

$$\begin{aligned}&U(C_t^H, \zeta _t^H) \approx U(C, 0)+U_cC\left( \frac{C_t^H-C}{C}\right) \\&\quad +\frac{1}{2}U_{CC}C^2\left( \frac{C_t^H-C}{C}\right) ^2 +U_{\zeta }\zeta _t^H+\frac{1}{2}U_{C\zeta }\zeta _t^2+U_{C\zeta }(C_t^H-C)\zeta _t^H + {\mathcal {O}}(||\xi ||^3) \end{aligned}$$

This can be written in terms of log-deviations $\hat{x}=x_t-x=\log (X_t)-\log X$ by doing the following mechanical substitution:

$$\begin{aligned} \frac{X_t-X}{X}=\hat{x}_t+\frac{1}{2}\hat{x}_t^2+{\mathcal {O}}(||\xi ||^3) \end{aligned}$$

which implies for the utility function after some algebra and noting that $\sigma =-U_{CC}C/U_C$:

$$\begin{aligned} U(C_t^H, \zeta _t^H)\approx & {} U_{C}C \left[ {\hat{c}}^H_t+\frac{1}{2}(1-\sigma )\left( {\hat{c}}^H_t\right) ^2 -\frac{U_{C\zeta }}{U_{C}}\ln \zeta ^H{\hat{c}}_t^H \right] +t.i.p.+ {\mathcal {O}}(||\xi ||^3)\\= & {} U_{C}C \left[ {\hat{c}}^H_t+\frac{1}{2}(1-\sigma )\left( {\hat{c}}^H_t\right) ^2 -\varepsilon _t{\hat{c}}_t^H \right] +t.i.p.+ {\mathcal {O}}(||\xi ||^3) \end{aligned}$$

where in the last line we have replaced $\varepsilon ^H_t= \frac{U_{C\zeta }}{U_{C}}\ln (\zeta ^H)$ which is the percentage variation in consumption required to keep the marginal utility of expenditure at its steady state level, given disturbances to preferences.

Deriving a second-order approximation to the second term in (1) is straightforward. Following the same procedure as above we get:

$$\begin{aligned} V(G_t^H)\approx & {} V(G)+V_GG\left( \frac{G_t^H-G}{G} \right) +\frac{1}{2} V_{GG}G^2\left( \frac{G_t^H-G}{G} \right) ^2+t.i.p.+{\mathcal {O}}(||\xi ||^3) \\= & {} V_G G \left[ \hat{g}_t^H+\frac{1}{2}(1-\sigma _g)\left( \hat{g}_t^H\right) ^2\right] +t.i.p.+{\mathcal {O}}(||\xi ||^3) \end{aligned}$$

where we have defined $\sigma _g \equiv -V_{GG}(G)G/V_G(G)$. We assume for simplicity that the coefficient $\sigma _g=\sigma$. We justify this due to two reasons. First, calculations get easier. However, the results do not change qualitatively if we calibrate the model with different values for the two parameters. Second, though Beetsma and Jensen (2005) distinguish these two parameters while setting up their model in the end they choose them to be equal.

In steady state $V_G(G)=U_C(C,0)$. Moreover, $G/C=(G/Y)\cdot (Y/C)=\frac{1-c_y}{c_y}.$ Thus, we get:

$$\begin{aligned} V(G_t^H) =U_C C \frac{1-c_y}{c_y} \left[ \hat{g}_t^H+\frac{1}{2}(1-\sigma )\left( \hat{g}_t^H\right) ^2\right] +t.i.p.+{\mathcal {O}}(||\xi ||^3) \end{aligned}$$

Deriving a second-order approximation to the third term in (1) is straightforward as in Woodford (2003) or in Galí (2015). Note that by assumption

$$\begin{aligned} V(N_t(h))= & {} \frac{(N_t(h))^{1+\eta }}{1+\eta }= \frac{y_t^d(h)^{1+\eta }}{1+\eta }\\\approx & {} Y V_y \left[ \hat{y}_t(h)+\frac{1}{2}(1+\eta ) \left( \hat{y}_t(h)\right) ^2\right] +t.i.p. + {\mathcal {O}}(||\xi ||^3) \\= & {} Y U_c \left[ \hat{y}_t(h)+\frac{1}{2}(1+\eta ) \left( \hat{y}_t(h)\right) ^2\right] +t.i.p. + {\mathcal {O}}(||\xi ||^3) \end{aligned}$$

where in the last line we have used the condition that in steady state $V_y/U_C=(1-\tau )/\mu =1$. Here $\mu =\theta /(1-\theta )$ denotes the mark up of prices over marginal costs which by assumption is off-set by an appropriate subsidy $\tau$. In short, there are no distortions in steady-state. Integrating the last term and denoting with $var_h$ the variance of $\hat{y}_t(h)$ across all differentiated goods at time t, yields

$$\begin{aligned} \int _0^{1/2} V(N_t(h))dh= & {} Y u_c \left[ \hat{y}^H_t+\frac{1}{2}(1+\eta ) \left( \hat{y}^H_t\right) ^2 + \frac{1}{2}(\theta ^{-1}+\eta ) var_h \hat{y}_t(h)\right] \\&+\,t.i.p. + {\mathcal {O}}(||\xi ||^3) \end{aligned}$$

Each supplier faces a constant-elasticity demand curve which can be written as

$$\begin{aligned} \log y_t(h)= & {} \log Y_t^H-\theta (\log p_t(h)-\log P_{H,t})\\ \Longrightarrow \qquad var_h \log y_t(h)= & {} \theta ^2 var_h \log p_t(h) \equiv \theta ^2 \Delta _t \end{aligned}$$

which helps to replace the last term in the approximated disutilty function. Following (Woodford 2003, Chapter 6.2) or Beetsma and Jensen (2005) the discounted sum of price dispersion evolves over time according to

$$\begin{aligned} \sum _{t=0}^\infty \beta ^t\Delta _t = \frac{\alpha }{(1-\alpha )(1-\alpha \beta )}\sum _{t=0}^\infty \beta ^t (\pi _{t}^H)^2+t.i.p.+{\mathcal {O}}(||\xi ||^3) \end{aligned}$$

As analogues approximation hold for the foreign country we can sum up, that a second-order approximation of welfare W is given by $W= E_0\sum _{t=0}^\infty \beta ^t w_t$, where instantaneous utility is approximated by

$$\begin{aligned} w_t/(U_CC)= & {} \left[ {\hat{c}}^H_t+\frac{1}{2}(1-\sigma )\left( {\hat{c}}^H_t\right) ^2 - \varepsilon _t{\hat{c}}_t^H \right] \\&-\,Y^H/C\left( \hat{y}^H_t+\frac{1}{2}(1+\eta )(\hat{y}^H_t)^2 \right) -\frac{1}{2} Y^H/C \, \frac{(1+\eta \theta )\alpha \theta }{(1-\alpha )(1-\alpha \beta )} (\pi _t^H)^2 \\&+\,\left[ {\hat{c}}^F_t+\frac{1}{2}(1-\sigma )\left( {\hat{c}}^F_t\right) ^2 - \varepsilon _t{\hat{c}}_t^F \right] - Y^F/C\left( \hat{y}^F_t+\frac{1}{2}(1+\eta )(\hat{y}^F_t)^2 \right) \\&-\,\frac{1}{2} Y^F/C \, \frac{(1+\eta \theta )\alpha \theta }{(1-\alpha )(1-\alpha \beta )} (\pi _t^F)^2 \\&+\,\frac{1-c_y}{c_y} \left[ \hat{g}_t^H+\frac{1}{2}(1-\sigma )\left( \hat{g}_t^H\right) ^2\right] \\&+\,\frac{1-c_y}{c_y} \left[ \hat{g}_t^F+\frac{1}{2}(1-\sigma )\left( \hat{g}_t^F\right) ^2\right] \\= & {} \frac{1}{c_y} \left\{ \hat{y}_t^W - (1-c_y)\hat{g}_t^W -\frac{1}{2} c_y\left[ ({\hat{c}}^H_t)^2+({\hat{c}}^F_t)^2+ 2\phi \gamma (1-\gamma ) \hat{T}^2_t \right] \right. \\&\left. +\,\frac{1}{2}(\hat{y}^H_t)^2 +\frac{1}{2}(\hat{y}^F_t)^2- \frac{1}{2}(1-c_y)(\hat{g}^H_t)^2 -\frac{1}{2}(1-c_y)(\hat{g}^F_t)^2 \right\} \\&+\,\frac{1}{2}(1-\sigma )\left( {\hat{c}}^H_t\right) ^2 -\varepsilon ^H_t{\hat{c}}_t^H\\&+\,\frac{1}{2}(1-\sigma )\left( {\hat{c}}^F_t\right) ^2 - \varepsilon ^F_t{\hat{c}}_t^F + \frac{1-c_y}{c_y} \left[ \hat{g}_t^H+\frac{1}{2}(1-\sigma )\left( \hat{g}_t^H\right) ^2\right] \\&+\,\frac{1-c_y}{c_y} \left[ \hat{g}_t^F+\frac{1}{2}(1-\sigma )\left( {\hat{g}}_t^F\right) ^2\right] \\&-\,\frac{1}{c_y}\left( \hat{y}^H_t+\frac{1}{2}(1+\eta )(\hat{y}^H_t)^2 \right) - \frac{1}{c_y} \left( \hat{y}^F_t+\frac{1}{2}(1+\eta )({\hat{y}}^F_t)^2 \right) \\&-\,\frac{1}{2} \frac{1}{c_y} \, \frac{(1+\eta \theta )\alpha \theta }{(1-\alpha )(1-\alpha \beta )} (\pi _t^H)^2 -\frac{1}{2} \frac{1}{c_y} \, \frac{(1+\eta \theta )\alpha \theta }{(1-\alpha )(1-\alpha \beta )} (\pi _t^F)^2 \\&+\, t.i.p.+ {\mathcal {O}}(||\xi ||^3) \\= & {} - \frac{\sigma }{2}\left( {\hat{c}}^H_t\right) ^2- \frac{\sigma }{2}\left( {\hat{c}}^F_t\right) ^2 -\phi \gamma (1-\gamma )\hat{T}^2_t - \frac{\eta }{2c_y}\left[ (\hat{y}^H_t)^2 - (\hat{y}^F_t)^2 \right] \\&-\,\frac{\sigma }{2}\frac{(1-c_y)}{c_y}(\hat{g}^H_t)^2 -\frac{\sigma }{2}\frac{(1-c_y)}{c_y}(\hat{g}^F_t)^2 - \varepsilon ^H_t{\hat{c}}_t^H - \varepsilon ^F_t{\hat{c}}_t^F \\&-\,\frac{1}{2} \frac{1}{c_y}\, \frac{(1+\eta \theta )\alpha \theta }{(1-\alpha )(1-\alpha \beta )} (\pi _t^H)^2 \\&-\,\frac{1}{2} \frac{1}{c_y}\, \frac{(1+\eta \theta )\alpha \theta }{(1-\alpha )(1-\alpha \beta )} (\pi _t^F)^2 +t.i.p.+ {\mathcal {O}}(||\xi ||^3) \end{aligned}$$

Note that in the first step we have replaced aggregate consumption with a second-order approximation of the aggregate demand equation (43) derived in “Appendix B” solved for ${\hat{c}}_t^W$. This implies that some of the linear terms and second-order terms cancel out in the second step. Further note that in steady state both countries are symmetric which implies that $Y^H/C=Y^F/C=1/c_y$.

As a next step we write everything in gaps (this means replacing every $\hat{x}$-variable with deviations $\hat{x}-\widetilde{x}$) and then add with the same parameters and signs the terms $2\hat{x}\widetilde{x}$. Higher order terms of $\widetilde{x}^2$ are summarized in t.i.p.

$$\begin{aligned} - w_t/(U_CC)= & {} \frac{\sigma }{2} \left[ \left( {\hat{c}}^H_t-{\widetilde{c}}_t^H\right) ^2+\left( {\hat{c}}^F_t-{\widetilde{c}}_t^F\right) ^2\right] \\&+\,\frac{1}{2} \frac{\eta }{c_y} \left[ \left( \hat{y}^H_t-{\widetilde{y}}_t^H\right) ^2+\left( \hat{y}^F_t-{\widetilde{y}}_t^F\right) ^2\right] +\phi \gamma (1-\gamma )(\hat{T}_t-\widetilde{T}_t)^2\\&+\,\frac{\sigma }{2}\frac{(1-c_y)}{c_y}\left[ (\hat{g}^H_t-\widetilde{g}_t^H)^2 + (\hat{g}^F_t-\widetilde{g}_t^H)^2 \right] \\&+\,\sigma {\hat{c}}_t^H{\widetilde{c}}_t^H+\sigma {\hat{c}}_t^F{\widetilde{c}}_t^F+\frac{\eta }{c_y}\hat{y}_t^H{\widetilde{y}}_t^H +\frac{\eta }{c_y}\hat{y}_t^F{\widetilde{y}}_t^F +2\phi \gamma (1-\gamma )\hat{T}_t\widetilde{T}_t\\&+\,\frac{\sigma (1-c_y)}{c_y}\hat{g}_t^H\widetilde{g}_t^H+ \frac{\sigma (1-c_y)}{c_y}\hat{g}_t^F\widetilde{g}_t^F - \varepsilon ^H_t{\hat{c}}_t^H - \varepsilon ^F_t{\hat{c}}_t^F \\&+\,\frac{1}{2}\frac{1}{c_y} \frac{(1+\eta \theta )\alpha \theta }{(1-\alpha )(1-\alpha \beta )} \left[ (\pi _t^H)^2+(\pi _t^F)^2\right] +t.i.p.+ {\mathcal {O}}(||\xi ||^3) \end{aligned}$$

It remains to find an expression for

$$\begin{aligned} F\equiv & {} \sigma {\hat{c}}_t^H{\widetilde{c}}_t^H+\sigma {\hat{c}}_t^F{\widetilde{c}}_t^F+\eta \frac{1}{c_y}\hat{y}_t^H{\widetilde{y}}_t^H +\eta \frac{1}{c_y}\hat{y}_t^F{\widetilde{y}}_t^F+2\phi \gamma (1-\gamma )\hat{T}_t\widetilde{T}_t\\&+ \frac{\sigma (1-c_y)}{c_y}\hat{g}_t^H\widetilde{g}_t^H+ \frac{\sigma (1-c_y)}{c_y}\hat{g}_t^F\widetilde{g}_t^F - \varepsilon ^H_t{\hat{c}}_t^H - \varepsilon ^F_t{\hat{c}}_t^F \end{aligned}$$

We rearrange these terms using the following relationship which can be shown using (25) from the main text. For two generic variables x and y the following relationship holds:

$$\begin{aligned} x_t^H y_t^H+x_t^F y_t^F = x_t^W y_t^W+\frac{1}{2} x_t^R y_t^R \end{aligned}$$

Then the above expression gets:

$$\begin{aligned} F= & {} \sigma {\hat{c}}_t^W{\widetilde{c}}_t^W+ \frac{1}{2}\sigma {\hat{c}}_t^R{\widetilde{c}}_t^R +\frac{\eta }{c_y}\hat{y}_t^W{\widetilde{y}}_t^W +\frac{1}{2}\frac{\eta }{c_y}\hat{y}_t^R{\widetilde{y}}_t^R +2\phi \gamma (1-\gamma )\hat{T}_t\widetilde{T}_t\\&+\,\frac{(1-c_y)}{c_y}\hat{g}_t^W\widetilde{g}_t^W+ \frac{1}{2} \frac{(1-c_y)}{c_y}\hat{g}_t^R\widetilde{g}_t^R - ( \varepsilon ^H_t{\hat{c}}_t^H + \varepsilon ^F_t{\hat{c}}_t^F ) \\= & {} {\hat{c}}_t^W (\sigma {\widetilde{c}}_t^W +\eta {\widetilde{y}}_t^W) \\&+\,\frac{1}{2} {\hat{c}}_t^R (\sigma {\widetilde{c}}_t^R + \eta (2\gamma -1) {\widetilde{y}}_t^R )- \hat{T}_t \left( 2\phi \gamma (1-\gamma )\widetilde{T}_t + 4\phi \gamma (1-\gamma ){\widetilde{y}}_t^R \right) \\&+\,\frac{(1-c_y)}{c_y} \hat{g}_t^W \left( \widetilde{g}_t^W+\eta {\widetilde{y}}_t^W \right) + \frac{1}{2} \frac{(1-c_y)}{c_y}\hat{g}_t^R \left( \widetilde{g}_t^R +\eta {\widetilde{y}}_t^R \right) - ( \varepsilon ^W_t{\hat{c}}_t^W + \frac{1}{2} \varepsilon ^R_t{\hat{c}}_t^R ) \end{aligned}$$

where we have replaced $\hat{y}_t^W$ by aggregated demand for the sticky price Eq. (27) and $\hat{y}_t^R$ by the difference of the demand equations (30).

Collecting terms and expressing all flex-price variables in terms of the shocks using the equilibrium found in “Appendix A” it can be shown that the expressions are zero. For example:

$$\begin{aligned} {\hat{c}}_t^W (\sigma {\widetilde{c}}_t^W +\eta {\widetilde{y}}_t^W -\varepsilon ^W_t )= & {} {\hat{c}}_t^W \left( \sigma \frac{\sigma +\eta (1-c_y)}{\sigma (\sigma +\eta )} \varepsilon _t^W\right. \\&\left. +\frac{\sigma \eta c_y}{\sigma +\eta (1-c_y)} \frac{\sigma +\eta (1-c_y)}{\sigma (\sigma +\eta )} \varepsilon _t^W - \varepsilon _t^W\right) = 0 \end{aligned}$$

To conclude the second-order approximation of the welfare function is up to a multiplicative constant given by

$$\begin{aligned} - w_t/(U_CC)= & {} \frac{\sigma }{2} \left[ \left( {\hat{c}}^H_t-{\widetilde{c}}_t^H\right) ^2 +\left( {\hat{c}}^F_t-{\widetilde{c}}_t^F\right) ^2\right] + \frac{1}{2} \frac{\eta }{c_y} \left[ \left( \hat{y}^H_t-{\widetilde{y}}_t^H\right) ^2+\left( \hat{y}^F_t-{\widetilde{y}}_t^F\right) ^2\right] \\&+\,\phi \gamma (1-\gamma )(\hat{T}_t-\widetilde{T}_t)^2\\&+\,\frac{\sigma }{2}\frac{(1-c_y)}{c_y}\left[ \left( \hat{g}^H_t-\widetilde{g}_t^H\right) ^2 + (\hat{g}^F_t-\widetilde{g}_t^H)^2 \right] \\&+\,\frac{1}{2}\frac{1}{c_y} \frac{(1+\eta \theta )\alpha \theta }{(1-\alpha )(1-\alpha \beta )} \left[ (\pi _t^H)^2+(\pi _t^F)^2\right] +t.i.p.+ {\mathcal {O}}(||\xi ||^3) \end{aligned}$$

In a final step we express this welfare function in terms of the aggregate and relative gaps rather than in country-specific gaps. Note that for a generic variable x the following condition can be derived using (25):

$$\begin{aligned} (x^H)^2+(x^F)^2=(x^W-\frac{1}{2}x^R)^2+ (x^W+\frac{1}{2}x^R)^2=2(x^W)^2+\frac{1}{2}(x^R)^2 \end{aligned}$$

To sum up, the second-order approximation of the welfare function is up to a multiplicative constant given by

$$\begin{aligned} - w_t/(U_CC)= & {} \sigma \left( {\hat{c}}^W_t-{\widetilde{c}}_t^W\right) ^2+\frac{\sigma }{4} \left( {\hat{c}}^R_t-{\widetilde{c}}_t^R\right) ^2 + \frac{\eta }{c_y} \left( \hat{y}^W_t-{\widetilde{y}}_t^W\right) ^2+\frac{\eta }{4c_y}\left( \hat{y}^R_t-{\widetilde{y}}_t^R\right) ^2\\&+\,\phi \gamma (1-\gamma )(\hat{T}_t-\widetilde{T}_t)^2+\,\frac{\sigma (1-c_y)}{c_y}\left( \hat{g}^W_t-\widetilde{g}_t^W\right) ^2 +\frac{\sigma (1-c_y)}{4c_y} \left( \hat{g}^R_t-\widetilde{g}_t^R\right) ^2 \\&+\,\frac{1}{c_y} \frac{(1+\eta \theta )\alpha \theta }{(1-\alpha )(1-\alpha \beta )} \left( \pi _t^W\right) ^2 + \frac{1}{4c_y}\frac{(1+\eta \theta )\alpha \theta }{(1-\alpha )(1-\alpha \beta )} \left( \pi _t^R\right) ^2 \\&+\,t.i.p.+ {\mathcal {O}}(||\xi ||^3) \end{aligned}$$

Deriving optimal policies

1.1 Benchmark case

The benevolent policy maker minimizes the loss function (32) subject to the economic conditions. We can take aggregate consumption as the policy instruments as this pins down the nominal interest rate via the aggregate Euler equation (28). Replacing the aggregate output gap in the Phillips curve with the aggregate demand equation written in gaps we can combine the two constraints (26) and (27) to one single constraint. The same argument applies to the two equations (29) and (30). Moreover, we consider the efficient markets condition and the evolution of the terms of trade (23). We include the zero lower bound on the nominal interest rate by combining the the inequality constraint (33) and the aggregate Euler equation (28) as in Galí (2015) to the inequality

$$\begin{aligned} c^W_t \le E_tc_{t+1}^W + \frac{1}{\sigma }E_t \pi _{t+1}^W+ \frac{1}{\sigma }\tilde{i}_t^W \end{aligned}$$

Thus Lagrangian of the problem is given by (we have denoted the gaps $\hat{x}-\widetilde{x}=x$):

$$\begin{aligned} \min _{\pi _t^W, c_t^W, g_t^W,\pi _t^R, c_t^R, g_t^R, T_t} {\mathcal {L}}_t\equiv & {} L_t + \lambda _{t} \left[ \pi _t^W - \beta E_t \pi _{t+1}^W-k\left( \sigma c^W_t + \eta \left( c_yc_t^W+(1-c_y)g_t^W \right) \right) \right] \\&+\, \mu _{1t} \left[ \pi _t^R -\beta E_t \pi _{t+1}^R+2k(1-\gamma ) {T}_t -k \left[ \sigma c^R_t\right. \right. \\&\left. \left. +\,\eta \left( c_y\left[ -4\phi \gamma (1-\gamma )T_t+(2\gamma -1)c_t^R\right] +\,(1-c_y)g_t^R \right) \right] \right] \\&+\,\mu _{2t} \left[ T_t-T_{t-1}-\pi _t^R \right] + \mu _{3t} \left[ \sigma c_t^R-(1-2\gamma ) T_t \right] \\&+\,\xi \left[ c^W_t - E_tc_{t+1}^W - \frac{1}{\sigma }E_t \pi _{t+1}^W - \frac{1}{\sigma }\tilde{i}_t^W \right] \end{aligned}$$

1.1.1 The aggregate part

The first order condition with respect to $c_t^W$ can be simplified to

$$\begin{aligned} 0= c^W_t + \frac{\eta (1-c_y)}{\sigma +\eta c_y}g_t^W-\frac{k}{2}\lambda _t + \frac{1}{\sigma +\eta c_y}\xi _t \end{aligned}$$

(44)

whereas the first order condition with respect to $g_t^W$ is given by

$$\begin{aligned} 0= c^W_t+\frac{\eta (1-c_y)+\sigma }{\eta c_y}g_t^W-\frac{k}{2}\lambda _t \end{aligned}$$

(45)

The first order condition with respect to $\pi _t^W$ yields

$$\begin{aligned} \lambda _t=\frac{2\theta }{kc_y}\pi _t^W \end{aligned}$$

(46)

First case: The zero lower bound is not binding, i.e., $\bar{\iota }^W>0$, which implies that $\xi _t=0$. Combining the first order conditions (44) and (45) yields $g_t^W=0$. Us Using (46) to replace $\lambda _t$ in either (44) or (45), and expressing $c_t^W$ in terms of $y_t^W$ with the help of (27) yields the well-known trade-off (34) in the main text.

Second case: The zero lower bound is binding, i.e., $\bar{\iota }^W=0$, which implies that $\xi _t>0$. Solving (45) for $\lambda _t$ and plugging this into (44), we can solve for

$$\begin{aligned} \xi _t = \frac{ \sigma (\sigma +\eta )}{ \eta c_y} g_t^W \end{aligned}$$

Finally, solve (46) for $\lambda _t$, and use the result for $\xi _t$ to replace the multipliers in (44). After expressing $c_t^W$ in terms of $y_t^W$ and $g_t^W$ using (27) and some rearranging we get the optimality condition (36) in the main text.

1.1.2 The relative part

Determining the optimality conditions of the relative part gets a dynamic programming problem with the last period’s terms of trade gap as the state variable, because the terms of trade affect the variables of the next period and thus the objective function of next period.

Thus, the value function of the relative part is given by

$$\begin{aligned} V_t(T_{t-1})=\min _{T_t, c_t^R, g_t^R,\pi _t^R} \left( L_t + \beta V_{t+1} (T_t) \right) \end{aligned}$$

Setting up the Lagrangian as above the first order conditions of the relative part are given

$$\begin{aligned} \text {w.r.t.}\, \pi _t^R:\qquad 0= & {} \theta \pi _t^R/(2c_yk)+\mu _{1t}-\mu _{2t} \\ \text {w.r.t.}\, c_t^R:\qquad 0= & {} \sigma c_t^R/2+ \eta y_t^R(2\gamma -1)/2- \mu _{1t}k(\sigma +\eta c_y(2\gamma -1))+\mu _{3t}\sigma \\ \text {w.r.t.}\, g_t^R:\qquad 0= & {} \sigma g_t^R+\eta y_t^R- 2c_yk\eta \mu _{1t} \\ \text {w.r.t.}\, T_t:\qquad 0= & {} 2 \phi \gamma (1-\gamma )(T_t-\eta y_t^R)+\mu _{2t} +2k(1-\gamma )(1+\eta c_y 2\phi \gamma )\mu _{1t}- \mu _{3t}(1-2\gamma )\\&+\,\beta \frac{\partial V_{t+1}}{\partial T_t} \end{aligned}$$

Use the envelope theorem to determine $\frac{\partial V_{t+1}}{\partial T_t} = -\mu _{2,t+1}$.

1.2 Independent monetary policy

The central bank’s reaction function is derived by minimizing the loss function (37) subject to the aggregate Phillips curve (26) taking into account that the output gap is determined by (27). Setting up the Lagrangian, taking the derivatives with respect to $c_t^W$, and $\pi _t^W$ and taken as given expectations and government spending the first-order conditions can be rearranged to yield the optimality condition (38).

1.3 Fiscal policy

We minimize the loss function (39) subject to the constraints analogues to the relative part of the benchmark scenario (Fig. 11).

The first order conditions for the home countries are given by (here, $\mu ^H_{1t}$ is the multiplier attached to (17), with output replaced by (18), and $\mu ^H_{2t}$ denotes the multiplier attached to the terms of trade dynamics, and $\mu ^H_{3t}$ is the one attached to the risk-sharing condition:)

$$\begin{aligned} \text {w.r.t.}\, \pi _t^H:\qquad 0= & {} \frac{\theta }{c_y k} \pi _t^H+\mu ^H_{1t}+\mu ^H_{2t} \\ \text {w.r.t.}\, c_t^H:\qquad 0= & {} \sigma c_t^H+ \eta \gamma y_t^H - \mu ^H_{1t}k(\sigma +\eta \gamma c_y)-\sigma \mu _{3t}^H \\ \text {w.r.t.}\, g_t^H:\qquad 0= & {} \sigma g_t^H + \eta y_t^H- c_y k\eta \mu ^H_{1t} \\ \text {w.r.t.}\, T_t:\qquad 0= & {} 2 \phi \gamma (1-\gamma )\eta y_t^H+ \phi \gamma (1-\gamma )T_t+\mu ^H_{2t}-\beta \mu ^H_{2,t+1}- k(1-\gamma )(1+\eta c_y 2\phi \gamma )\mu ^H_{1t}\\&-\,\mu ^H_{3t} (1-2\gamma ) \end{aligned}$$

The first order conditions for the foreign country are similar except different signs when referring to the terms of trade:

$$\begin{aligned} \text {w.r.t.}\, \pi _t^F:\qquad 0= & {} \frac{\theta }{c_y k} \pi _t^F+\mu ^F_{1t}-\mu ^H_{2t} \\ \text {w.r.t.}\, c_t^F:\qquad 0= & {} \sigma c_t^F+ \eta \gamma y_t^F - \mu ^F_{1t}k(\sigma +\eta \gamma c_y)+\sigma \mu _{3t}^F \\ \text {w.r.t.}\, g_t^F:\qquad 0= & {} \sigma g_t^F + \eta y_t^F- c_y k\eta \mu ^F_{1t} \\ \text {w.r.t.}\, T_t:\qquad 0= & {} -2 \phi \gamma (1-\gamma )\eta y_t^F + \phi \gamma (1-\gamma )T_t +\mu ^F_{2t}-\beta \mu ^F_{2,t+1}+ k(1-\gamma )(1+\eta c_y 2\phi \gamma )\mu ^F_{1t}\\&-\,\mu ^F_{3t} (1-2\gamma ) \end{aligned}$$

Impulse response functions

Fig. 1

Impulse response functions of aggregated variables to a symmetric demand shock in the benchmark scenario

Fig. 2

Impulse response functions of relative variables to an asymmetric demand shock in the benchmark scenario

Fig. 3

Impulse response functions of the home variables to an asymmetric demand shock in scenario 4

Fig. 4

Impulse response functions of the nominal interest rate across the different scenarios (symmetric shock)

Fig. 5

Impulse response functions of the government spending gap across the different scenarios (symmetric shock)

Fig. 6

Impulse response functions of home consumption spending gap for different $\delta ^H$ in Scenario 4 ($\delta ^F=0$)

Fig. 7

Impulse response functions of home output gap for different $\delta ^H$ in Scenario 4 ($\delta ^F=0$)

Fig. 8

Impulse response functions of the terms of trade for different $\delta ^H$ in Scenario 4 ($\delta ^F=0$)

Fig. 9

Impulse response functions of the aggregated output gap in scenario 4 for different $\delta$ (symmetric shock)

Fig. 10

Impulse response functions of aggregate inflation in scenario 4 for different $\delta$ (symmetric shock)

Fig. 11

Impulse response functions of the nominal interest rate to a symmetric demand shock when both policy makers follow a rule for different values of $\delta$