Fixed on Flexible: Rethinking Exchange Rate Regimes after the Great Recession

Abstract

The zero lower bound problem during the Great Recession has exposed the limits of monetary autonomy, prompting a re-evaluation of the relative benefits of currency pegs and monetary unions (see, e.g., Cook and Devereux in Journal of International Economics 101:52–69, 2016). We revisit this issue from the perspective of a small open economy. While a peg can be beneficial when the recession originates domestically, we show that a float dominates in the face of deflationary demand shocks abroad. When the rest of the world is in a liquidity trap, the domestic currency depreciates in nominal and real terms even in the absence of domestic monetary stimulus (if domestic rates are also at the zero lower bound)—enhancing the country’s competitiveness and insulating to some extent the domestic economy from foreign deflationary pressure.

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Figure 1

Note: Monetary Policy Under the Taylor Rule in Home (Dashed Line) Versus Constant-Interest-Rate Period of Ten Quarters (Solid Line) Versus Fixed Exchange Rate (Dash-Dotted Line). Horizontal Axes Measure Time in Quarters. Vertical Axes Measure Deviations from the Pre-shock Path, In Percent of Steady-State Output (In Case of Quantities) or Percent (In Case of Prices).

Figure 2

Note: Unconstrained Monetary Policy in Home (Dashed Line) Versus Constant-Interest-Rate Period of Ten Quarters (Solid Line) Versus Exchange Rate Peg (Dash-Dotted Line). Horizontal Axes Measure Time in Quarters. Vertical Axes Measure Deviations from the Pre-shock Path, In Percent of Steady-State Output (In Case of Quantities) or Percent (In Case of Prices).

Notes

  1. 1.

    In the New Keynesian specification, the small open economy takes the global equilibrium as given but maintains some monopoly power on its terms of trade—see, for instance, Galí and Monacelli (2005) and De Paoli (2009) who, in turn, build on the new open economy macroeconomics literature (Obstfeld and Rogoff, 1996).

  2. 2.

    The Great Recession has also triggered new research on the desirability of capital controls and, more in general, macroprudential policies. While we abstract from these issues, it is clear that additional policy instruments may help addressing the ZLB problem or, more in general, the limits of monetary policy, see Benigno and others (2016) and the literature on policy dilemma after Rey (2016).

  3. 3.

    See also Groll and Monacelli (2016). They stress that in the absence of commitment by the central bank, an exchange rate peg may dominate a flexible exchange rate regime quite independently of the ZLB. They focus on cost push shocks.

  4. 4.

    A potentially important factor in this choice is nominal rigidities in the export market. In our paper, we consider the case of “producer currency pricing” or PCP: product prices are sticky in the currency of the exporters. In the literature, the case of “local currency pricing” or LCP (prices are sticky in the currency of the destination market) gave rise to a debate on the desirability of fixed exchange rate (see Devereux and Engel, 2003; Duarte and Obstfeld, 2008, among others). It is now clear that the optimal policy under LCP does not generally support fixed rates (with the exception of a few notable cases) and can easily imply a variance of the nominal exchange rate higher than in the case of PCP (Corsetti and others, 2010). An additional, important case is that of “Dollar pricing” or DP, where exports prices are sticky in a vehicle currency (Burstein and Gopinath, 2014; Casas and others, 2016). In our context, depending on the strength of nominal rigidities, LCP or DP could reduce the benefit from floating rates at the ZLB, without, however, necessarily changing the result. We leave an exploration of these cases to future research.

  5. 5.

    In previous work of ours, we have extensively analyzed the consequences of sovereign risk when monetary policy is constrained by the ZLB as well as in currency unions. For a discussion, see also the early version of this paper, Corsetti and others (2013b).

  6. 6.

    Amador and others (2016), instead, consider how to overcome the ZLB problem with exchange rate and international reserve policies in a world with segmented financial markets. Their analysis is motivated by the recent Swiss experience of large capital inflows.

  7. 7.

    This explicit consideration of Foreign sets apart our exercise from the typical treatment of a small open economy. Apart from this, Home is identical to the small open economy of Galí and Monacelli (2005), except that we allow for government consumption, and for tractability we restrict preferences to log utility and assume that the trade-price elasticity is unity throughout (Cobb–Douglas case).

  8. 8.

    We assume complete markets to ensure analytical tractability. By means of numerical simulations, we verify that the main results of our analysis are unchanged if we allow for international trade in non-contingent bonds only. Note that our setup implies unitary elasticities for intertemporal substitution as well as for trade. Differences between complete and incomplete markets may become larger for elasticities sufficiently below unity, and/or in the presence of non-stationary shocks or “news shocks” (Corsetti and others, 2008).

  9. 9.

    This specification of home bias follows Sutherland (2005) and De Paoli (2009). With \(\upsilon =1\), there is no home bias: if the relative price of foreign and domestic goods is unity, Home’s consumption basket contains a share n of Home-produced goods and a share of \(1-n\) of imported goods. A lower value of \(\upsilon\) implies that the fraction of domestically produced goods in final goods exceeds the share of domestic production in the world economy. If \(\upsilon = 0\), there is full home bias and no trade across countries.

  10. 10.

    One reason is that the relative preference shocks move Home consumption; recall (15). If the Home preference for current consumption rises (\(\xi _t\) rises), all else equal, Home consumption rises, which pushes up the wage. Conversely, the wage falls in shocks that raise the desire to consume today in Foreign.

  11. 11.

    We focus on a simple rule here for both realism and tractability. Rules like (26) need not be optimal, but tend to approximate the optimal policy quite well.

  12. 12.

    The condition can be derived combining equations (16), (20), (22) and (25).

  13. 13.

    The terms of trade converge back to the steady state in the long run because of complete financial markets and the assumption that the shock is temporary.

  14. 14.

    The reason for invoking constant interest rates rather than modeling the ZLB explicitly is purely expositional. The assumption saves a constant term in many of the equations that follow.

  15. 15.

    It is straightforward to verify that equations (20) and (21) are satisfied at these values, as is (26).

  16. 16.

    A policy targeting the natural rate in response to the demand shock is optimal under complete markets when pursued by both countries. When monetary policy in Foreign is constrained, however, there could be more efficient rules for Home, which may deliver higher welfare. For our purpose, however, a natural rate rule provides a suitable benchmark.

  17. 17.

    The inequality stated in expression (37) follows from a comparison of the two terms in square brackets. From the definition in (33) \(1/\chi >1\). At the same time \((1+\varphi (1-\upsilon ))/(1+\varphi )<1\), establishing the claim.

  18. 18.

    Here we use that prior to when the shock hits the economy is in steady state (\(p_{-1}^* = 0\)).

  19. 19.

    Note that under our scenario Home can always implement this.

  20. 20.

    See Appendix 2 for details.

  21. 21.

    At the same time, the appreciation of the terms of trade depresses domestic consumption relative to the level of consumption in the rest of the world.

  22. 22.

    The response of Home producer prices can be derived from (21). Prices fall after the shock.

  23. 23.

    In order to compute welfare, we solve the model globally under perfect foresight and assume a constant subsidy on wages that makes the steady state efficient following Galí and Monacelli (2005), see Appendix 4 for details.

  24. 24.

    Galí and Monacelli (2005), instead of taxes implement a subsidy on labor to the firm. The efficient allocation in Home is not affected by this choice.

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Corresponding author

Correspondence to Giancarlo Corsetti.

Additional information

This paper was prepared for the 2016 conference on “Exchange Rates and External Adjustment” organized by the Swiss National Bank, International Monetary Fund, and the IMF Economic Review. We thank conference participants, the editors of the IMF Economic Review, and Martin Wolf for very useful comments. We also thank our discussants Andrea Ferrero and Giovanni Lombardo. Corsetti acknowledges support from the Centre for Macroeconomics, and the Keynes Fellowship at Cambridge University. Müller acknowledges support by the German Science Foundation (DFG) under the Priority Program 1578. Francesco D'Ascanio and Frederik Kurcz provided excellent research assistance. We also thank two anonymous referees for useful comments. The usual disclaimer applies.

Appendices

Appendix 1: Linearized Model Equations

In what follows we briefly outline the approximations of the equilibrium conditions around a deterministic steady state. For this steady state, we assume that there is zero inflation. As explained in the main text, small-case letters denote log deviations of variables from their steady-state value. ‘Hats’ measure deviations from steady state in percentage points of output. We focus on the case \(n \rightarrow 0\).

Equilibrium Conditions of Approximate Model

Households supply labor such that the following relation is satisfied:

$$\begin{aligned} \tilde{w}_{t} = \varphi h_{t} + c_t, \end{aligned}$$
(58)
$$\begin{aligned} \tilde{w}^*_{t} = \varphi h^*_{t} + c_t^*, \end{aligned}$$
(59)

where \(\tilde{w}_t\) is the (consumption) real wage (the log linearization of \(W_t/P_t\)). The optimal time path of consumption satisfies:

$$\begin{aligned} c_t -\xi _t= E_t(c_{t+1} -\xi _{t+1}) -(r_{t}-E_t \pi _{t+1}), \end{aligned}$$
(60)
$$\begin{aligned} c^*_{t}-\xi _t^*=E_t \left( c_{t+1}^*-\xi _{t+1}^*\right) -\left( r^*_{t}-E_t\pi ^{*}_{t+1}\right) . \end{aligned}$$
(61)

Under complete financial markets, we have the following risk-sharing condition:

$$\begin{aligned} (c_t - \xi _t) - \left( c_t^* - \xi _t^*\right) = (1-\upsilon ) s_t. \end{aligned}$$
(62)

Intermediate good firms’ price-setting behavior gives

$$\begin{aligned} \pi _{{\mathrm{H}}t} =\beta E_t \pi _{{\mathrm{H}}t+1}+ \kappa (\tilde{w}_{t} + \upsilon s_t) \end{aligned}$$
(63)
$$\begin{aligned} \pi _{t}^*=\beta E_t \pi _{t+1}^*+ \kappa \tilde{w}^*_{t}. \end{aligned}$$
(64)

The aggregate production function is given by

$$\begin{aligned} y_t = h_t , \end{aligned}$$
(65)
$$\begin{aligned} y^*_t = h^*_t. \end{aligned}$$
(66)

The terms of trade are given by

$$\begin{aligned} s_t = e_t + p_t^* - p_{{\mathrm{H}},t}, \end{aligned}$$
(67)

with the change in the price levels defining

$$\begin{aligned} \pi _t^* = p_t - p_{t-1}, \end{aligned}$$
(68)
$$\begin{aligned} \pi _{{\mathrm{H}},t} = p_{{\mathrm{H}},t} - p_{{\mathrm{H}},t-1}. \end{aligned}$$
(69)

We also have

$$\begin{aligned} \pi _t = (1-\upsilon ) \pi _{{\mathrm{H}},t} + \upsilon \left( \Delta e_t + \pi _t^*\right) . \end{aligned}$$
(70)

Finally, there is market clearing:

$$\begin{aligned} y_{t} = \upsilon s_t + (1-\upsilon ) c_t + \upsilon c_t^* + \upsilon (1-\upsilon )s_t + \hat{g}_t. \end{aligned}$$
(71)
$$\begin{aligned} y^*_{t} = c^*_{t}. \end{aligned}$$
(72)

We omit the specification of monetary and fiscal policy which is provided in the main text.

Incomplete Financial Markets In this case, instead of the risk-sharing condition (62), equilibrium requires the following UIP condition to hold:

$$\begin{aligned} r_t - r_t^* = E_t e_{t+1}- e_t. \end{aligned}$$

Also, we need to keep track of net foreign assets. Assuming that foreign-currency bonds are in zero net supply, we have:

$$\begin{aligned} \beta \hat{d}_t = \hat{d}_{t-1} + y_t - c_t - \upsilon s_t - \hat{g}_t. \end{aligned}$$

In order to close the model under incomplete markets we assume that the interest rate in home increases very mildly in the net foreign asset position \(d_t\) (see also Schmitt-Grohé and Uribe, 2003).

Canonical Representation

Foreign operates like a standard New Keynesian closed-economy model. Combining equations (61) and (72) gives the dynamic IS equation stated in the main text, see equation (16). Combining equations (59), (64), (66) and (72) gives the Phillips curve stated in the main text, see equation (17).

Home is de facto a small open economy. In main text we rely on a number of equations to determine the equilibrium outcome, given the realization of Foreign variables and Home shocks. We now derive these equations starting from equations stated in the previous subsection. Substituting for Home consumption in the market clearing condition (71) using risk sharing (62) yields

$$\begin{aligned} y_t = s_t + c_t^*+ (1-\upsilon )\left( \xi _t - \xi _t^* \right) + \hat{g}_t. \end{aligned}$$
(73)

which is equation (25) in the main text.

Combining (60) and (62) gives

$$\begin{aligned} \left( c_t^* - \xi _t^*\right) + (1-\upsilon ) s_t = E_t \left( c_{t+1}^* - \xi _{t+1}^*\right) + (1-\upsilon ) E_t s_{t+1} - (r_t - E_t \pi _{t+1} ). \end{aligned}$$

Rearranging and substituting for \(c_t^*\) using (73) gives

$$\begin{aligned} y_t = E_t y_{t+1} -\upsilon E_t s_{t+1} - \upsilon E_t \Delta \xi _{t+1}^* - (1-\upsilon ) E_t \Delta \xi _{t+1} - (r_t - E_t \pi _{t+1} ) - E_t \Delta \hat{g}_{t+1}. \end{aligned}$$

Noting from (70) and (67) that \(\pi _t = \pi _{{\mathrm{H}},t} + \upsilon \Delta s_{t}\) and using this in the expression above, we obtain equation (20) stated in the main text:

$$\begin{aligned} y_t= E_t y_{t+1} - \left( r_t - E_t \pi _{{\mathrm{H}},t+1} + (1-\upsilon ) E_t\Delta {\xi }_{t+1} +\upsilon E_t\Delta {\xi }_{t+1}^* \right) - E_t \Delta \hat{g}_{t+1}. \end{aligned}$$
(74)

To obtain the Phillips curve for Home, we start from (63) and use (58) and (65). This gives

$$\begin{aligned} \pi _{{\mathrm{H}},t} = \beta E_t \pi _{{\mathrm{H}},t+1} + \kappa \left( \varphi y_t + c_t + \upsilon s_t \right) . \end{aligned}$$

Using, in turn, equation (62) to substitute for \(c_t\) and (73) to substitute for \(c_t^*\) gives

$$\begin{aligned} \pi _{{\mathrm{H}},t} = \beta E_t \pi _{{\mathrm{H}},t+1}+ \kappa \left[ \left( 1 +\varphi \right) y_t - \hat{g}_t -\upsilon (\xi _t^* - \xi _t) \right] , \end{aligned}$$
(75)

which is expression (21) in the main text.

Natural Rates in Home

To determine the natural level of output we set \(\pi _{{\mathrm{H}},t} = 0\) for all t in equation (75) and obtain

$$\begin{aligned} y_t^n = \frac{1}{1 + \varphi } \hat{g}_t - \frac{\upsilon }{1 + \varphi } \left( \xi _t - \xi _t^* \right) . \end{aligned}$$
(76)

Similarly, for the natural rate of interest, plug (76) into (74), with \(\pi _{{\mathrm{H}},t}=0\) for all t, and rearrange terms:

$$\begin{aligned} r_t^n = -\frac{ \varphi }{1 + \varphi } E_t \Delta \hat{g}_{t+1}-\frac{1 + \varphi (1-\upsilon )}{1 + \varphi } E_t \Delta \xi _{t+1} -\frac{\upsilon \varphi }{1 + \varphi } E_t \Delta \xi _{t+1}^*, \end{aligned}$$

which is expression (27) given in the main text. To obtain the natural terms of trade we substitute for output in equation (25). The result is shown in equation (29) in the main text.

Appendix 2: Solving the Linearized Model

Foreign variables only deviate from their steady-state values in case the demand shock originates in Foreign. Given our assumptions regarding the shock, equations (16) and (17) simplify to

$$\begin{aligned} y_L^* = \mu y_L^* - \left( -\mu \pi _L^* +\left( \mu \xi _L^* - \xi _L^*\right) \right) , \\ \pi _L^* = \beta \mu \pi _L^* + \kappa (1+\varphi ) y_L^*. \end{aligned}$$

Straightforward substitution yields (33) and (34) in the main text, and thus \(y_L^*\chi = \xi _L^*\), using the definition of \(\chi\) in the main text.

Determinacy Note that, in order to satisfy our assumption that parameters are such that they guarantee a locally unique stable rational expectations equilibrium, we require (compare Woodford, 2011)

$$\begin{aligned} (1-\beta \mu )(1-\mu ) - \mu \kappa (1+\varphi )>0. \end{aligned}$$

The response of Home variables to shocks depends on the policy regime in place. We solve the model under each policy regime.

Home: Flexible Exchange Rate and Unconstrained Monetary Policy

In this case, the flexible price allocation and the natural rates obtain. Domestic inflation is zero. In order to solve for the terms of trade, we consider equation (25) and substitute for Home output using natural output and for Foreign output using \(y_L^* = \chi ^{-1} \xi _L^*\). We obtain the following expression

$$\begin{aligned} s_L= & {} - \frac{\varphi }{1+\varphi }\hat{g}_L - \frac{1+\varphi (1-\upsilon )}{1+\varphi } \xi _L \nonumber \\&- \frac{\upsilon \varphi }{1+\varphi } \xi _L^* - \frac{\mu \kappa (1+\varphi )}{(1-\mu )(1-\beta \mu )-\mu \kappa (1+\varphi )} \xi _L^*. \end{aligned}$$
(77)

Home: ZLB Scenario

Starting from equations (20) and (21) and imposing the Markov structure on all three shocks, we get, after a bit of rearranging

$$\begin{aligned} y_L= & {} \frac{ (1 - \mu )(1 - \beta \mu ) - \mu \kappa }{(1 - \mu )(1 - \beta \mu ) - \mu \kappa (1 + \varphi )} \hat{g}_L + \frac{(1-\upsilon )(1 - \mu )(1 - \beta \mu ) + \mu \kappa \upsilon }{(1 - \mu )(1 - \beta \mu ) - \mu \kappa (1 + \varphi )} \xi _L \nonumber \\&+ \frac{ \upsilon ((1 - \mu )(1 - \beta \mu ) - \mu \kappa )}{(1 - \mu )(1 - \beta \mu ) - \mu \kappa (1 + \varphi )} \xi _L^*. \end{aligned}$$
(78)
$$\begin{aligned} \pi _L= & {} \frac{\kappa \varphi (1-\mu )}{(1 - \mu )(1 - \beta \mu ) - \mu \kappa (1 + \varphi )} \hat{g}_L + \frac{ \kappa (1-\mu ) ( 1 + \varphi (1-\upsilon ))}{(1 - \mu )(1 - \beta \mu ) - \mu \kappa (1 + \varphi )} \xi _L \nonumber \\&+ \frac{\kappa \varphi \upsilon (1-\mu )}{(1 - \mu )(1 - \beta \mu ) - \mu \kappa (1 + \varphi )} \xi _L^*. \end{aligned}$$
(79)

To solve for the terms of trade in the ZLB scenario we use equations (33) and (78) as well as equation (25). We obtain

$$\begin{aligned} s_L= & {} \frac{\mu \kappa \varphi }{(1 - \mu )(1 - \beta \mu ) - \mu \kappa (1 + \varphi )} \hat{g}_L + \frac{\mu \kappa [1+\varphi (1-\upsilon )]}{(1 - \mu )(1 - \beta \mu ) - \mu \kappa (1 + \varphi )} \xi _L \nonumber \\&- \frac{ \mu \kappa \left[ 1 + \varphi (1-\upsilon )\right] }{(1-\mu )(1-\beta \mu ) - \mu \kappa (1+\varphi )} \xi _L^*. \end{aligned}$$
(80)

Home: An Exchange Rate Peg

The following gives the solution to the terms of trade under an exchange rate peg with all three shocks considered. First, rewrite the marginal cost term in the Home Phillips curve (75), using equation (73):

$$\begin{aligned}&\kappa \left( 1 +\varphi \right) y_t - \kappa \hat{g}_t - \kappa \upsilon \left( \xi _t^* - \xi _t\right) , \\&\quad = \kappa \left( 1 +\varphi \right) \left( s_t + \hat{g}_t + y_t^* + (1-\upsilon ) \left( \xi _t - \xi _t^*\right) \right) - \kappa \hat{g}_t - \kappa \upsilon \left( \xi _t^* - \xi _t\right) , \\&\quad = \kappa \left( 1+\varphi (1-\upsilon )\right) \left( \xi _t - \xi _t^*\right) + \kappa (1+\varphi ) s_t + \kappa \varphi \hat{g}_t + \kappa (1+\varphi ) y_t^*. \end{aligned}$$

Subtracting Phillips curves [Foreign (17) minus Home (75)]

$$\begin{aligned} \pi _{t}^*- \pi _{{\mathrm{H}},t} =\beta E_t (\pi _{t+1}^*-\pi _{{\mathrm{H}},t+1}) + \kappa \big ( \left[ 1+\varphi (1-\upsilon )\right] ( \xi _t^*-\xi _t) - (1+\varphi ) s_t - \varphi \hat{g}_t \big ). \end{aligned}$$

Using the first difference of (67) and setting \(\Delta e_t=0\), we arrive at

$$\begin{aligned} s_t = \psi s_{t-1} + \beta \psi E_t s_{t+1} + \kappa \psi \big ([1+\varphi (1-\upsilon )] (\xi _t^* - \xi _t) - \varphi \hat{g}_t\big ), \end{aligned}$$
(81)

where \(\psi =[1+\beta +\kappa (1+\varphi )]^{-1}\), as given in the main text. Define

$$\begin{aligned} u_t \equiv \kappa \big ( (1 + \varphi (1-\upsilon ))(\xi _t^* - \xi _t) - \varphi \hat{g}_t \big ), \end{aligned}$$
(82)

such that the equation (81) reads more compactly as

$$\begin{aligned} s_t = \psi s_{t-1} + \beta \psi E_t s_{t+1} + \psi u_t . \end{aligned}$$
(83)

Using the method of undetermined coefficients, one can show that the only stable solution to (81) is

$$\begin{aligned} s_t = \delta s_{t-1} +\underbrace{\frac{\kappa \psi \left[ 1+\varphi (1-\upsilon )\right] }{1-\beta \psi [\delta +\mu ]} }_{:=\Phi } ( \xi _L^*-\xi _L)- \underbrace{\frac{\kappa \psi }{1-\beta \psi [\delta +\mu ]} \varphi }_{:= \varGamma } \hat{g}_L, \end{aligned}$$
(84)

with \(\delta = \frac{1 - \sqrt{1 - 4 \beta \psi ^2}}{2 \beta \psi }\). Note that \(\delta \in (0,1)\).

The sign of \(\Phi\) can easily be determined. Rewriting the expression slightly

$$\begin{aligned} \Phi = \frac{\kappa (1 + \varphi (1-\upsilon ))}{1 + \beta + \kappa (1+\varphi ) - \beta (\delta + \mu )}, \end{aligned}$$

it becomes apparent that \(\Phi\) is positive but always smaller than one. To see this, note that \(1 + \beta > \beta (\delta + \mu )\) (since \(\beta ,\delta \in (0,1)\) and \(\mu \in [0,1)\)), and \(\kappa (1+\varphi (1-\upsilon ))<\kappa (1+\varphi )\) [recall \(\kappa > 0\), \(\varphi >0\) and \(\upsilon \in (0,1)\)]. Exactly the same reasoning shows that \(\varGamma \in (0,1)\), as well:

$$\begin{aligned} \varGamma = \frac{\kappa \varphi }{1 + \beta + \kappa (1+\varphi ) - \beta (\delta + \mu )}. \end{aligned}$$

To determine the output response under the peg, solving (84) backward (and recalling \(s_{-1} = 0\)), we get

$$\begin{aligned} s_t = \Phi \sum _{k = 0}^{t} \delta ^{t-k} \left( \xi _k^* - \xi _k\right) - \varGamma \sum _{k = 0}^{t} \delta ^{t-k} \hat{g}_k. \end{aligned}$$
(85)

Plugging this into (25)

$$\begin{aligned} y_t = \hat{g}_t - \varGamma \sum _{k = 0}^{t} \delta ^{t-k} \hat{g}_k + (1 - \upsilon ) \xi _t - \Phi \sum _{k = 0}^{t} \delta ^{t-k} \xi _k - (1-\upsilon ) \xi _t^* + \frac{1}{\chi } \xi _t^* + \Phi \sum _{k = 0}^{t} \delta ^{t-k} \xi _k^* \end{aligned}$$
(86)

The inflation response can be recovered via (44), having derived earlier the solution to the terms of trade and foreign inflation [see (34)].

Proof of Propositions

Proof of Proposition 1

It is to show that when \(\xi _t^* = \xi _L^* < 0\), under the peg the fall in output is larger than under a float (for both the unconstrained and the ZLB case). Equation (86) gives the coefficient for the peg in period t (assuming the shock started in \(t=0\)), and equation (28) gives the coefficient for the unconstrained case. This gives the following condition

$$\begin{aligned} -(1 - \upsilon ) + \chi ^{-1} + \Phi> \frac{\upsilon }{1 + \varphi } \\ \upsilon + (\chi ^{-1} -1) + \Phi >\frac{\upsilon }{1 + \varphi }, \end{aligned}$$

which holds since \(\chi ^{-1} > 1\), \(\Phi >0\) and \(\varphi >0\). The same can be shown for the ZLB scenario [equation (78) gives the coefficient for the ZLB case]:

$$\begin{aligned}&-(1 - \upsilon ) + \chi ^{-1} + \Phi> \frac{(1-\mu )(1-\beta \mu ) - \mu \kappa }{(1-\mu )(1-\beta \mu ) - \mu \kappa (1 + \varphi )} \upsilon \\&\quad (1 - \upsilon ) \mu \kappa (1 + \varphi ) + \Phi ((1-\mu )(1-\beta \mu ) - \mu \kappa (1 + \varphi )) + \mu \kappa \upsilon > 0, \end{aligned}$$

which is true since every term on the left hand side is positive (recall, \((1-\mu )(1-\beta \mu ) - \mu \kappa (1 + \varphi )>0\) by our assumption on determinacy). \(\square\)

Proof of Proposition 2

To see that the drop in output due to \(\xi _t = \xi _L < 0\) is smaller under the peg than under the ZLB, first consider the coefficient on \(\xi _L\) for the peg on impact (equation 86): \((1-\upsilon ) - \Phi\), with \(\Phi \in (0,1)\). We can now show that output in the ZLB scenario falls by more than \(1 - \upsilon\). Consider the coefficient on \(\xi _L\) in (78):

$$\begin{aligned} (1 - \upsilon )< & {} \frac{ (1 - \mu )(1-\beta \mu ) (1-\upsilon ) + \mu \upsilon \kappa }{(1 - \mu )(1-\beta \mu ) - \mu \kappa (1+\varphi )} \\ - \mu \kappa (1+\varphi )(1-\upsilon )< & {} \mu \upsilon \kappa . \end{aligned}$$

Therefore, the fall in output at the ZLB is always larger under the home savings shock than in the peg regime. \(\square\)

Proof of Proposition 3

Considering the coefficient on \(\hat{g}_t\) for output in equation (78), we can immediately see that it is larger than one. For the flexible exchange rate case, equation (28) gives a positive coefficient which is smaller than one. Lastly, the coefficient in the case of the peg (equation 86) gives \(1-\varGamma\) on impact, which is positive but smaller than one due to \(\varGamma \in (0,1)\). \(\square\)

Appendix 3: Exchange Rate Response to Foreign Shock

The main text asserts that the two terms on the right-hand sides of (39) and (43), respectively, reflect a temporary and a permanent effect of the Foreign shock on the nominal exchange rate. This Appendix lays out this argument in more detail.

Suppose that in period k the recession in Foreign persists (so that \(\xi _k=\xi _L\)). This means that the shock has persisted for a total of \(k+1\) periods (\(t=0,1,\ldots , k\)). The uncovered interest parity condition (31) holds in all periods, so it also holds in \(t=k\), reading

$$\begin{aligned} r_{k} - r_{k}^* = E_k e_{k+1} - e_k. \end{aligned}$$

Let us introduce some notation. Call \(e_{k+1}^{\text {shock}}\) the nominal exchange rate that realizes (in period \(k+1\)) if the shock persists into the next period. The latter happens with probability \(\mu\). Note that, in this notation, \(e_{k}=e_k^{\text {shock}}\) since by assumption the shock persists in period k. Call \(e_{k+1}^{\text {no\, shock\, for\, the\, first\,time}}\) the nominal exchange rate that realizes if the shock does not persist into the next period. We wish to derive an expression for \(e_{k+1}^{\text {no\, shock\, for\, the\, first\,time}}\). By way of the Markov assumption,

$$\begin{aligned} E_k e_{k+1} = \mu \, e_{k+1}^{\text {shock}} + (1-\mu ) e_{k+1}^{\text {no\, shock\, for\, the\, first\,time}}. \end{aligned}$$

Using this in the interest parity condition, we have

$$\begin{aligned} \begin{array}{rcl} r_{k} - r_{k}^* &{} = &{} \mu \,e_{k+1}^{\text {shock}} + (1-\mu ) e_{k+1}^{\text {no\, shock\, for\, the\, first\,time}} - e_{k} \\ &{} = &{} \mu \, e_{k+1}^{\text {shock}} + (1-\mu ) e_{k+1}^{\text {no\, shock\, for\, the\, first\,time}} - e_{k}^{\text {shock}} \\ &{} = &{} \mu \left( e_{k+1}^{\text {shock}} - e_{k}^{\text {shock}}\right) + (1-\mu ) \left( e_{k+1}^{\text {no\, shock\, for\, the\, first\,time}} - e_{k}^{\text {shock}}\right) . \end{array} \end{aligned}$$

Since the shock is active in period k, the lower bound will bind in Foreign. This means \(r_k^*=0\). Therefore,

$$\begin{aligned} r_{k} = \mu \left( e_{k+1}^{\text {shock}} - e_{k}^{\text {shock}}\right) + (1-\mu ) \left( e_{k+1}^{\text {no\, shock\, for\, the\, first\,time}} - e_{k}^{\text {shock}}\right) . \end{aligned}$$
(87)

Unconstrained Monetary Policy in Home

Consider first the case when Home monetary policy is unconstrained. For the terms \(e_{k+1}^{\text {shock}}\) and \(e_k^{\text {shock}}\) formula (39) (evolution of the exchange rate while the shock lasts) applies. With this, we have that

$$\begin{aligned} e_{k+1}^{\text {shock}} - e_{k}^{\text {shock}} = - \pi _L^*. \end{aligned}$$

Using this in (87), we have

$$\begin{aligned} r_{k} = - \mu \pi _L^* + (1-\mu ) \left( e_{k+1}^{\text {no\, shock\, for\, the\, first\,time}} - e_{k}^{\text {shock}}\right) . \end{aligned}$$

While the shock binds, \(r_k=r_L^n\). Therefore, the above implies

$$\begin{aligned} r_{L}^n + \mu \pi _L^* = (1-\mu ) \left( e_{k+1}^{\text {no\, shock\, for\, the\, first\,time}} - e_{k}^{\text {shock}}\right) , \end{aligned}$$

or, equivalently,

$$\begin{aligned} e_{k+1}^{\text {no\, shock\, for\, the\, first\,time}} = e_{k}^{\text {shock}} + \frac{r_{L}^n + \mu \pi _L^*}{1-\mu }. \end{aligned}$$

In words, in the period in which the shock ceases to exist, the nominal exchange rate appreciates relative to the previous period (\(r_{L}^n + \mu \pi _L^*<0\)). Using (39) to substitute for \(e_{k}^{\text {shock}}\), we have that the nominal exchange rate in period \(k+1\) (if the shock ceased to exist that period) is given by

$$\begin{aligned} e_{k+1}^{\text {no\, shock\, for\, the\, first\,time}} = - (k+1) \cdot \pi _L^*. \end{aligned}$$

In words, once the shock ceases to exist, the exchange rate permanently reflects the depreciation that insulates Home from the Foreign deflationary crawl. That is, the second term in (39) is permanent. The first term in (39), instead, reflects a temporary depreciation.

ZLB in Home

For the ZLB scenario in Home (and still with the Foreign shock), one can follow steps analogous to those above. Start from (87). Observe that under the ZLB in Home also \(r_k=0\). Otherwise apply the same steps used above (but using (43) to substitute for \(e_{k}^{\text {shock}}\) and \(e_{k+1}^{\text {shock}}\)). Then, in \(k+1\) (if this is the first period in which the shock ceases to exist), the nominal exchange rate takes on the value

$$\begin{aligned} e_{k+1}^{\text {no\, shock\, for\, the\, first\,time}} = (k+1)\cdot \left( \pi _{{\mathrm{H}},L} - \pi _L^* \right) , \end{aligned}$$

highlighting that the first term in (43) reflects a temporary depreciation, while the second term is permanent.

Appendix 4: Efficient Steady State

Our welfare comparisons build on a calibration with an efficient steady state. This Appendix first derives the allocation that the social planner of a small open economy would choose. It, then, specifies a labor income tax that would decentralize this allocation. The derivations here follow closely Galí and Monacelli (2005), using the notation for our model.

Planner’s Allocation

Consider the problem of a planner of the small open Home economy (\(n\rightarrow 0\)). Consider the flex-price steady-state allocation without government spending. International risk sharing implies

$$\begin{aligned} C = C^* S^{1-\upsilon }. \end{aligned}$$

Home output is given by

$$\begin{aligned} Y = \left( \frac{P_{{\mathrm{H}}}}{P}\right) ^{-1}\left[ (1-\upsilon )C + \upsilon S^{1-\upsilon } C^*\right] . \end{aligned}$$

The Home price level is given by

$$\begin{aligned} P = P_{{\mathrm{H}}}^{1-\upsilon }P_{{\mathrm{F}}}^\upsilon . \end{aligned}$$

The terms of trade are defined as

$$\begin{aligned} S = P_{{\mathrm{F}}}/P_H, \end{aligned}$$

Last, the production function is

$$\begin{aligned} Y=H. \end{aligned}$$

Combining the risk-sharing condition and the demand equation gives:

$$\begin{aligned} Y = \left( \frac{P_{{\mathrm{H}}}}{P}\right) ^{-1}C \end{aligned}$$

Using this with the price aggregate and the terms of trade, we have

$$\begin{aligned} Y = S^\upsilon C. \end{aligned}$$

Substituting the real exchange rate Q from the risk-sharing condition, and using \(C^*=Y^*\), we have, after rearranging:

$$\begin{aligned} C=Y^{1-\upsilon } {Y^*}^\upsilon . \end{aligned}$$

Using the production function, the constraint that the small open economy’s planner faces is

$$\begin{aligned} C=H^{1-\upsilon } {Y^*}^\upsilon . \end{aligned}$$

The small open economy’s planner solves

$$\begin{aligned} \max _{C,H} \ln (C) - \frac{1}{1+\varphi }H^{1+\varphi }\;\text {s.t.}~C=H^{1-\upsilon } {Y^*}^\upsilon , \end{aligned}$$

the first-order condition of which gives

$$\begin{aligned} H^{\varphi } = \frac{1}{C} (1-\upsilon ) Y^* H^{-\upsilon }. \end{aligned}$$

Using that \(Y=H\) and substituting from \(C=Y^{1-\upsilon } {Y^*}^\upsilon\), we get

$$\begin{aligned} H^{1+\varphi } = \frac{1}{C} C (1-\upsilon ) = (1-\upsilon ) \end{aligned}$$

That is, the allocation that is optimal from the perspective of the small open economy’s planner is given by

$$\begin{aligned} H = (1-\upsilon )^\frac{1}{1+\varphi }, \end{aligned}$$

and

$$\begin{aligned} C = (1-\upsilon )^\frac{1-\upsilon }{1+\varphi }{Y^*}^\upsilon . \end{aligned}$$

Decentralization

This steady state can be decentralized through a distortionary tax on household’s labor income (and a lump-sum transfer). Let labor income after taxes be \(wH(1-\tau )\). Then the level of taxes that decentralizes the planner’s allocation is \(\tau = 1-\frac{(1-\upsilon )\epsilon }{\epsilon -1}\).Footnote 24

We implement the same tax in Foreign in order to focus on a symmetric steady state.

Appendix 5: Fiscal Simulations Complementing Section VI

This Appendix shows numerical simulations to illustrate the quantitative relevance of the results in Section VI. We proceed with the same parametrization of the model as in Section V. We assume that the shock to government spending follows an AR(1) process and set the persistence parameter to 0.8. Figure 3 traces the effect of an increase of government spending by one percent of GDP (the shock itself depicted in the upper-left figure). In the case of a free float, as long as monetary policy is unconstrained, the fiscal expansion has moderate effects. As the monetary authority is ensuring price stability, more government spending leads to a monetary contraction and real appreciation. The increase in government spending raises output, but only at the expense of domestic consumption. The multiplier is substantially below one. Conversely, fiscal policy is quite powerful when the domestic policy rates are temporarily constant at the ZLB. Persistently higher government spending raises expected inflation, thus lowering the long-term real rate: private consumption rises substantially. At the same time, the fall in long-term rates causes the nominal exchange rate to depreciate. Domestic consumption rises with domestic inflation. Comparing the ZLB case across Figure 1 in the main text and Figure 3 above illustrates the “benign coincidence” emphasized above: under those circumstances in which the demand shocks can become more damaging because of the ZLB, fiscal policy is a powerful substitute for monetary stabilization, if exchange rates are flexible.

Figure 3
figure3

Note: Unconstrained Monetary Policy (Dashed Line) Versus Constant-Interest-Rate Period of Ten Quarters (Solid Line) and Exchange Rate Peg (Dash-Dotted Line). Horizontal Axes Measure Time in Quarters. Vertical Axes Measure Deviations from the Pre-shock Path, In Percent of Steady-State Output (In Case of Quantities) or Percent (In Case of Prices).

Effect of Domestic Government Spending Increase

This benign coincidence breaks down, however, when the country pursues a currency peg. Figure 3 shows that fiscal policy is not particularly effective in a fixed exchange rate regime. Note that this is precisely the regime where the adverse external shock is most consequential for Home output and consumption—compare, again, Figure 1 in the main text and Figure 3 here. The mechanism governing the transmission of fiscal policy operates via price-level dynamics: in order to restore purchasing power parity in the medium and the long run, under a peg, the initial positive response of inflation to a government spending expansion will be offset over time. This is in sharp contrast to the evolution of Home prices when Home monetary policy is constrained by the ZLB but pursues flexible exchange rates. There, the Home price level keeps increasing over the entire life of the fiscal expansion.

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Corsetti, G., Kuester, K. & Müller, G.J. Fixed on Flexible: Rethinking Exchange Rate Regimes after the Great Recession. IMF Econ Rev 65, 586–632 (2017). https://doi.org/10.1057/s41308-017-0038-0

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