M Grob
email: grob@ualberta.ca

A Maggi

E Stutzmann

Observations of the seasonality of the Antarctic microseismic signal, and its association to sea ice variability

Seismic noise spectra at all seismic stations display two peaks in the 1-20 s period band, called primary and secondary microseisms. They are caused by the coupling of ocean waves into Rayleigh waves. At most locations, microseismic power is greater during local winter (when nearby oceans are stormier) than local summer. This tendency is reversed for stations in Antarctica, where growth of local winter sea ice seems to impede microseism generation in near coastal areas. A decade of continuous data from coastal seismic stations in Antarctica show systematic seasonality in microseismic signal levels, and demonstrate associations with both broad-scale and local sea-ice conditions. Primary microseisms are known to be generated at the coast and the modulation that we observe can be associated with sea-ice variations both in the vicinity of the station and along other Antarctic coasts. The similar modulation of short-period secondary microseisms corroborates their mostly nearcoastal origin, while the continued presence of long-period secondary microseisms suggests more distant source regions. These observations could be used to extend the monitoring of climate variability prior to the availability of satellitederived climate indicators.

Introduction

Analyses of seismic noise spectra ubiquitously show broad energetic peaks over a 3-20 s period band, primarily composed of Rayleigh waves, and commonly referred to as primary (10-20 s) and secondary (3-10 s) microseisms. These microseisms are due to ocean gravity wave interactions that cause pressure oscillations, which in turn generate seismic waves at the ocean floor [START_REF] Longuet-Higgins | A theory of the origin of microseisms[END_REF][START_REF] Gerstoft | A year of microseisms in southern California[END_REF][START_REF] Kedar | The origin of deep ocean microseisms in the North Atlantic ocean[END_REF]. Primary microseisms are generated when ocean gravity waves reach shallow water near the coast and interact with the sloping seafloor, either by breaking or shoaling [START_REF] Hasselmann | A statistical analysis of the generation of microseisms[END_REF].

These seismic waves have periods similar to the incident ocean gravity waves. Secondary microseisms are more energetic, and are generated by standing or colliding waves within the ocean wave field near the coast or in the deep ocean [e.g. [START_REF] Haubrich | Microseisms: coastal and pelagic sources[END_REF][START_REF] Friedrich | Ocean-generated microseismic noise located with the Graffenberg array[END_REF][START_REF] Chevrot | Source locations of secondary microseisms in western Europe: Evidence for both coastal and pelagic sources[END_REF][START_REF] Bromirski | Ocean wave height determined from inland seismometer data: Implications for investigating wave climate changes in the NE Pacific[END_REF]. These in turn create standing pressure fluctuations at the ocean bottom at half the period of the ocean waves. The variations in microseismic power have been linked to the presence of ocean storms [START_REF] Bromirski | The near-coastal microseism spectrum: spatial and temporal wave climate relationships[END_REF][START_REF] Barruol | Characterizing swells in the southern Pacific from seismic and infrasonic noise analyses[END_REF][START_REF] Gerstoft | A year of microseisms in southern California[END_REF][START_REF] Aster | Multidecadal climate-induced variability in microseisms[END_REF][START_REF] Aster | Global trends in extremal microseism intensity[END_REF]. Stutzmann et al. [2009] analyse seismic noise spectra at all stations of the global GEO-SCOPE network, and show that microseismic power is greater during local winter at the high latitudes of both hemispheres. This seasonality is explained by the seasonal increase of nearby oceanic storms. The only GEOSCOPE station that displays a different behaviour is Dumont d'Urville (DRV) in Antarctica, where microseismic amplitude is weaker in winter than in summer. Stutzmann et al. [2009] link this phenomenon to the D R A F T May 3, 2011, 3: presence of sea ice-floe, which prevents incoming swells from reaching the coast, thereby impeding both direct coupling at the coast (responsible for the primary microseisms) and swell reflection (responsible for the secondary microseisms). In this study, we extend the analysis to seismic stations at other coastal regions of Antarctica, in order to validate the observations made at DRV.

15pm D R A F
Sea-ice extension and concentration vary seasonally (following the cycle of ice-floe formation and dislocation) and inter-annually [START_REF] Cavalieri | Sea ice concentrations from Nimbus-7 SMMR and DMSP SSM/I passive microwave data[END_REF]. They are important factors for modelling climate variability, as they influence the energy exchange between ocean and atmosphere. Sea ice has been continuously monitored by satellite since 1979, using passive microwave radiometers. We take advantage of this extensive data-set to investigate in greater detail the correlation between microseism power and ice-floe presence, and briefly discuss the use of this seismic observation for investigating climate variations. The energy spectrum is computed using the median over all windows, and is averaged over 24 hours.

Microseism analysis

Figure 1a shows a typical microseism spectrogram over a year for CASY station. The primary microseism energy drops below -150 dB towards day 140 (end of May) and starts to rise again towards day 310 (November), though it does not reach the same level as during the beginning of the year. The secondary microseism energy also drops near the end of May and, like the primary microseism, begins to increase in November. The quiet period between May and November occurs during the austral winter, when sea-ice is most prevalent; the stronger microseismic signals at the beginning of the year occur when the coast is ice-free; the strengthening of the microseisms at year end occurs when ice concentration starts decreasing.

Decreasing microseism levels are observed at all coastal stations during the austral winter, although there are local variations in average energies and in the duration of the quiet period (Figure 1b). Average microseism energies are generally comparable between stations. Inland stations have lower energies due to their greater distance from the microseism source regions. An exception to this trend is MAW, which displays exceptionally low spectral amplitudes at all periods, possibly due to an incorrect instrument response.

The strongest microseismic signals and the shortest quiet periods are observed at the most-northern station PMSA, located on the Antarctic peninsula, which is rarely iced-in and therefore more often exposed to ocean swells. The winter-time microseismic attenuation repeats annually, with minor inter-annual variations. Figure 2 shows the annual microseismic spectra for SYO from 2001 to 2009, and clearly illustrates the first-order seasonality in both ice-cover and microseism energy. Lutzow-Holm Bay, near SYO, is ice-bound even in summer, which may explain the lower summer primary microseism energy at SYO compared to other coastal stations (e.g.

D R A F T May 3, 2011, 3:15pm D R A F T
CASY and DRV, see Figure 1b). There is some inter-annual variability in the level of microseism energy and the duration of the quiet periods at SYO, which can be related to localized inter-annual variations in ice-concentration. For example, the ∼10dB reduction in summer microseismic energy at SYO during 2005, 2007 and 2008 seems to be related to residual sea ice along the coast of Enderby land (see highlighted regions in Figure 2).

The seismic frequency of the microseismic peaks varies over time. Figure 3 contains 1-day spectra for SYO and CASY taken every 40 days throughout 2007, and shows that this variation is more pronounced for the secondary microseism, which is sensitive to specific coastal and/or wave-conditions that affect the generation of opposing waves [START_REF] Aster | Global trends in extremal microseism intensity[END_REF]. The primary microseism, generated by the direct interaction between incoming swell with the sloping seafloor, displays more consistent peak frequencies, which are systematically more stable and lower in winter than in summer (blue vs red highlighted regions in Figure 3). This shift towards longer periods during winter is likely to be related to the larger winter storms that produce longer gravity waves [Webb, 1998;Stutzmann et al., 2000]. In winter, the seafloor interaction predominantly occurs farther from the Antarctic coast as the nearby regions are ice-covered. Therefore, the primary microseism

D R A F T May 3, 2011, 3:15pm D R A F T
Rayleigh waves detected propagate longer distances to reach the stations, resulting in attenuation of a greater part of their shorter-period energy.

Sea ice observations

In Figures 1 and2, we have reproduced average monthly ice concentration images from the National Snow and Ice Data Center (NSIDC) Sea Ice Index [START_REF] Fetterer | Sea Ice Index[END_REF].

Ice concentration, i.e. the fraction, or percentage, of ocean area covered by sea ice, is estimated by exploiting the differing passive microwave signatures of water and sea-ice:

water has a highly polarized signature, while sea ice does not. The spatial resolution of sea-ice concentration estimates is 25 km. Detailed information about the satellites used, and the processing applied to obtain the Sea Ice Index images, is available from the NSIDC (http://nsidc.org/data/g02135.html).

In order to investigate the link between microseism power and ice coverage at temporal resolutions finer than one month, we also exploit daily gridded sea-ice concentrations from another NSIDC data product [START_REF] Cavalieri | Sea ice concentrations from Nimbus-7 SMMR and DMSP SSM/I passive microwave data[END_REF] greater distances. It is unclear how this could be related to the winter time microseismic noise. PMSA, being located on the Antarctic peninsula, is ice-free for most of the year, and has particularly strong microseisms. Two episodes of low levels of both primary and shorter-period secondary microseism energy at PMSA correspond to local increases in ice-concentration (Figure 4, black symbols).

Discussion

Antarctica is a perfect natural laboratory for gaining insights into the mechanisms of microseismic signal generation: the continent is surrounded by an often-stormy ocean, whose gravity wave interaction with its coasts is modulated by the presence or absence of sea-ice. Seismometer deployments directly on the Ross Ice Shelf have measured an annual attenuation by a factor of 100 of its swell-induced motion (7-40s period) at maximum ice extent [START_REF] Cathles | Seismic observations of sea swell on the floating Ross Ice Shelf, Antarctica[END_REF]. The absence of swell in shallow water surrounding the continent during the austral winter impedes the formation of local primary microseisms; furthermore, the absence of free coasts during the same period impedes the formation of reflected swell wave-trains that contribute to the generation of near-coastal, high-frequency secondary microseisms.

During the austral summer, the broad frequency band over which we have observed Longuet-Higgins [1950] showed that the largest amplitude secondary microseisms may be due to wave-wave interference in the mid ocean, although coastal reflexion may be a more common cause of microseisms of smaller amplitude. Since then, there have been many observations of both coastal sources [e.g. [START_REF] Haubrich | Microseisms: coastal and pelagic sources[END_REF][START_REF] Friedrich | Ocean-generated microseismic noise located with the Graffenberg array[END_REF][START_REF] Chevrot | Source locations of secondary microseisms in western Europe: Evidence for both coastal and pelagic sources[END_REF][START_REF] Bromirski | Ocean wave height determined from inland seismometer data: Implications for investigating wave climate changes in the NE Pacific[END_REF][START_REF] Bromirski | The near-coastal microseism spectrum: spatial and temporal wave climate relationships[END_REF],

and offshore sources, the latter principally in the North Atlantic Ocean [Stehly et al., 2006;[START_REF] Kedar | The origin of deep ocean microseisms in the North Atlantic ocean[END_REF] and the Mediterranean sea [START_REF] Chevrot | Source locations of secondary microseisms in western Europe: Evidence for both coastal and pelagic sources[END_REF]. As Antarctic stations are close to the coast, we expect the secondary microseisms generated by local coastal sources to be amplified compared to those generated by more distant sources, since they are less attenuated due to their short propagation paths. This expectation is consistent with the observed correlation between ice cover and secondary microseism amplitude, which is particularly visible at short periods.

The persistence of the long-period secondary microseisms we have observed during the austral winter is indicative of one or more source regions at considerable distance from the Antarctic coast. A possible non-coastal source region could be the Kerguelen plateau, whose water-depth (< 2000m) is similar to that of the broad region south of Iceland identified by [START_REF] Kedar | The origin of deep ocean microseisms in the North Atlantic ocean[END_REF] as an off-shore source of secondary microseisms. Systematic identification of the source regions of the austral-winter microseisms, which is beyond the scope of this paper, would give valuable insights into the relative importance of coastal and pelagic microseism sources.

D R A F T May 3, 2011, 3:15pm D R A F T

Conclusions

To first order, we have found that microseism power is anti-correlated with near-coastal ice presence, which opens the possibility of exploiting long-period seismic data to extend estimates of average ice-duration prior to the start of satellite measurements. The use of microseismic signal analysis for climate related inferences has been suggested by [START_REF] Bromirski | Ocean wave height determined from inland seismometer data: Implications for investigating wave climate changes in the NE Pacific[END_REF]; [START_REF] Aster | Multidecadal climate-induced variability in microseisms[END_REF]Stutzmann et al. [2009] for ice-free global seismic stations, by [START_REF] Barruol | Characterizing swells in the southern Pacific from seismic and infrasonic noise analyses[END_REF] for stations in French Polynesia, and by [START_REF] Kedar | The origin of deep ocean microseisms in the North Atlantic ocean[END_REF] for stations within the Labrador sea, for which the authors found that the microseisms were particularly sensitive to the exact position of the sea-ice boundary. It may be possible to extend microseismic studies in Antarctica back to the early to mid-1960s, at least for stations with a long history of recording such as DRV, which would, if successful, add one to two decades to the current, satellite-derived trends in climate indicators.

To second order, we have observed that variations in microseism power seem to be dependent on recurring local conditions, as shown from variable summer microseism levels at SYO and residual ice on the Enderby coast. More detailed study of such dependencies, including forward modeling of the microseismic signal from known wave and ice conditions, will be a great asset to understanding the regional scale variability associated with microseismic generation.

In conclusion, we have presented systematic observations of seasonality in microseismic signals from coastal stations in Antarctica, and have identified associations with both broad-scale and local sea-ice conditions. Further studies will be required to exploit these D R A F T May 3, 2011, 3 SNAA 2004, SYO 2004, MAW 2008, CASY 2007, VNDA 2006, LONW 2009, SIPL 2009, DRV 2008, SBA 2004.

  that provides a consistent time series of sea ice concentrations at a grid cell size of 25 × 25 km, spanning the coverage of several microwave instruments. The raw satellite data are processed using the NASA Team algorithm developed by the Oceans and Ice Branch of the Laboratory of Hydrophseric Processes at NASA Goddard Space Flight Center (GSFC). More detailed information is available from NSIDC (http://nsidc.org/data/nsidc-0051.html). For each coastal seismic station, we extract daily average values of sea-ice concentration over a ∼25 km-wide beam trending approximately normal to the closest coastline, and interpolate at regularly spaced distances from the coast in order to obtain average daily ice-concentration profiles. These D R A FT May 3, 2011, 3:15pm D R A F Tprofiles are plotted alongside our microseism power plots for stations SYO, DRV, SBA and PMSA in Figure4.At all stations, the presence of sea-ice is clearly correlated with a decrease in microseismic power. This correlation is most clear at SYO, at which both primary and secondary microseism power decrease sharply when ice concentration reaches 80% at approximately 100 km (extent of the continental shelf) and 200 km from the coast respectively. The longer period primary and secondary microseisms persist for more extended time periods.Two instances of anomalously high austral winter secondary microseisms at SYO occur at the same time as localized instances of ice-breakup (black arrows in Figure4). At DRV, the shortest period secondary microseisms (periods of 1-2 seconds, white bar) are no longer detected as soon as the bay ices-up, corroborating the Stutzmann et al. [2009] interpretation that they are probably due to local break-up of the nearby Astrolabe glacier. The broad-band primary microseisms (Figure 4a, black bar) persist 1-2 months after the local bay ices-up and the sea ice extends beyond the continental shelf and do not seem to correlate with local coastal conditions or the position of the local continental shelf. They may be generated by ice-free coastal regions elsewhere in Antarctica. The longest period primary and secondary microseisms remain detectable throughout the austral winter. At SBA, detection of shorter period secondary microseisms (Figure 4a, white bars) terminates as soon as Ross Bay is iced in, suggesting a local, possibly glacier related, microseism source. The longer period secondary microseisms continue throughout the winter, like at DRV, although with a broader frequency range. Near SBA, in contrast to the other stations, winter-time ice concentration is weaker close to the coast than at D R A F T May 3, 2011, 3:15pm D R A F T GROB ET AL.: SEASONALITY OF MICROSEISMIC SIGNAL AND SEA ICE VARIABILITY X -9

  primary microseisms is indicative of local coastal generation by well-dispersed oceanic swells from distant storms [e.g. Bromirski and Gerstoft, 2009]. The persistence of the longperiod component of these primary microseisms during the austral winter is indicative of generation in more distant source regions (possible candidates are the coastal regions of D R A F T May 3, 2011, 3:15pm D R A F T the Antarctic peninsula, South America, South Africa, southern Australia, Tasmania or New Zealand).

Figure 1 .

 1 Figure 1.(a) Variations of microseismic power spectral density for station CASY over a

  Figure 2.Seismic noise spectrum variability at SYO over a decade. Annual seismic noise

Figure 3 .

 3 Figure 3. Microseismic spectra over a year for stations SYO and CASY. Each curve represents

Figure 4 .

 4 Figure 4. Correlation between microseismic noise variations and changes in ice concentrations.

  -12 GROB ET AL.: SEASONALITY OF MICROSEISMIC SIGNAL AND SEA ICE VARIABILITY observations fully in order to make robust inferences on the detailed mechanisms of microseismic signal generation, and possibly also on trends in climate indicators. Stehly, L., M. Campillo and N.M. Shapiro (2006), A study of the seismic noise from its long-range correlation properties, J. Geophys. Res., 111(B10), B10306.Stutzmann, E., G.Roult and L. Astiz (2000), Geoscope station noise level, Bull. Seismol.

		X -15
	Soc. Am., 90, 690-701.
	Stutzmann, E., M. Schimmel, G. Patau and A. Maggi (2009), Global climate imprint on
	seismic noise, Geochem. Geophys. Geosyst., 10, Q11004, doi:10.1029/2009GC002619.
	Tanimoto, T. (2007), Excitation of microseisms, Geophys. Res. Lett., 34, L05308,
	doi:10.1029/2006GL029046.
	Webb, S.C. (1998), Broadband seismology and noise under the ocean, Rev. Geophys., 36,
	105-142.
	:15pm	D R A F T

X

Acknowledgments. The authors would like to thank Lise Retailleau and Simon Boyat for processing part of the seismic data during their Master training. The facilities of the IRIS Data Management System, and specifically the IRIS Data Management Center, were used for access to waveform, metadata or products required in this study. The IRIS DMS is funded through the National Science Foundation and specifically the GEO Directorate through the Instrumentation and Facilities Program of the National Science Foundation under Cooperative Agreement EAR-0552316. Some activities of are supported by the National Science Foundation EarthScope Program under Cooperative Agreement EAR-0733069. We have used data from the following networks: PS (PACIFIC-21), GE (GEOFON), AU(Geoscience Australia), IU (IRIS/USGS), G (GEOSCOPE), GT (Global Telemetered Seismograph Network -USAF/USGS), YT (POLENET). We thank Masaki Kanao and NIPR (National Institute of Polar Research, Japan) for running SYO station, and providing us with access to recent data through the IFREE (Institute for Research on Earth Evolution) data center at JAMSTEC (Japan Agency for Marine-Earth Science and Technology). The GEOFON network is funded and operated by GFZ Potsdam, Germany, in co-operation with almost 50 institutions worldwide. The IU stations are the part of the Global Seismic Network (GSN) that are installed, maintained and operated by the USGS Albuquerque Seismological Laboratory. Geoscience Australia operates the Australian National Seismic Network within Australia, its Territories and across its local region. D R A F T May 3, 2011, 3:15pm D R A F T


Hafedh Bouakez
email: hafedh.bouakez@hec.ca.

Aurélien Eyquem
email: aurelien.eyquem@ens-lyon.fr.

Government Spending, Monetary Policy, and the Real Exchange Rate *

Keywords: JEL classification: F31, F41 Incomplete markets, monetary policy, public spending shocks, real exchange rate, small open economy, sticky prices

A robust prediction across a wide range of open-economy macroeconomic models is that an unanticipated increase in public spending in a given country appreciates it currency in real terms. This result, however, contradicts the findings of a number of recent empirical studies, which instead document a significant and persistent depreciation of the real exchange rate following an expansionary government spending shock. In this paper, we rationalize the findings of the empirical literature by proposing a small-open-economy model that features three key ingredients: incomplete and imperfect international financial markets, sticky prices, and a nottoo-aggressive monetary policy. The model predicts that in response to an unexpected increase in public expenditures, the effective long-term real interest rate falls, causing the real exchange rate to depreciate. We establish this result both analytically, within a special version of the model, and numerically for the more general case.

Introduction

A robust prediction across a wide range of open-economy macroeconomic models is that an unanticipated increase in public spending in a given country appreciates it currency in real terms.

This result holds both in the traditional Mundell-Fleming-Dornbusch framework and in standard dynamic general-equilibrium models that assume complete financial markets. In the former, the appreciation of the real exchange rate results from the rise in aggregate demand induced by the increase in public spending, which falls more heavily on domestically produced goods, raising their relative price with respect to foreign goods. In dynamic general-equilibrium models with complete markets, the real appreciation ensues from a risk-sharing condition that relates the real exchange rate to the ratio of marginal utilities of consumption across countries. Because the increase in public spending must ultimately be financed through taxes, it lowers households' permanent income, which in turn raises the shadow value of wealth in the domestic country and appreciates its currency in real terms. This prediction, however, appears to be at odds with the data. Several empirical studies indeed find that the real exchange rate depreciates persistently in response to an unexpected exogenous increase in government expenditures [START_REF] Corsetti | Twin Deficits: Squaring Theory, Evidence and Common Sense[END_REF], [START_REF] Kim | Twin Deficit or Twin Divergence? Fiscal Policy, Current Account, and Real Exchange Rate in the U.S[END_REF], [START_REF] Müller | Understanding the Dynamic Effects of Government Spending on Foreign Trade[END_REF], [START_REF] Monacelli | Fiscal Policy, the Real Exchange Rate and Traded Goods[END_REF], [START_REF] Enders | How Do Fiscal and Technology Shocks Affect Real Exchange Rates?: New Evidence for the United States[END_REF], Ravn, Schmitt-Grohé and Uribe [2011], and [START_REF] Bouakez | Fiscal Policy and External Adjustment: New Evidence[END_REF]). This result holds across different countries, sample periods, and identification schemes, and is thus increasingly becoming a generally accepted stylized fact. To the extent that the response of the real exchange rate to public spending shocks is likely to play a critical role in determining the size of the spending multiplier in open economies, it is important to build macroeconomic models that are able to account for the empirical evidence.

In this paper, we rationalize the findings of the empirical literature by proposing a smallopen-economy model that combines three key features: (i ) incomplete and imperfect international financial markets, (ii ) sticky prices, and (iii) a not-too-aggressive monetary policy. Under incomplete financial markets, the real exchange rate is no longer pinned down by the ratio of marginal utilities of consumption, but instead by (the negative of) the long-term real interest rate net of a risk premium. The latter is assumed to be an increasing function of the economy's foreign debt, as in [START_REF] Kollmann | Monetary Policy Rules in the Open Economy: Effects on Welfare and Business Cycles[END_REF], Schmitt-Grohé and Uribe [2003], and [START_REF] Senhadji | External Shocks and Debt Accumulation in a Small Open Economy[END_REF], and is meant to capture frictions in international financial markets stemming, for instance, from the possibility of default or from agency costs. By raising demand for both domestically produced and foreign goods, an increase in public spending raises domestic inflation and deteriorates the current account. When prices are sticky and monetary policy does not react too aggressively to (expected) inflation, the resulting increase in the long-term real interest rate is smaller than the increase in the risk premium, causing the real exchange rate to depreciate. We illustrate this mechanism within a special version of our model in which we abstract from uncertainty and where we restrict the real interest rate to remain constant at its steady-state value at all times. This is achieved by assuming that the monetary authority changes the nominal interest rate one for one with expected inflation. The constancy of the real interest rate, which is only possible under sticky prices, implies that consumption also remains constant in response to transitory shocks. We can then solve the model analytically (up to a first-order approximation) and show that an increase in government purchases leads to a depreciation of the real exchange rate and to a fall in net foreign assets, consistent with the empirical findings. Intuitively, when the real interest rate is constant, output and the real exchange rate are determined independently of the supply side of the economy. Instead, their equilibrium paths are entirely pinned down by the equilibrium condition in the goods market (IS equation) and by the balance of payments (BOP) equation. By raising aggregate demand, an expansionary public spending shock shifts the IS curve outward in the output-real exchange rate space. But under a constant real interest rate, the shock causes an even larger outward shift of the BOP curve. To restore the equilibrium, the real exchange rate must depreciate, and this depreciation is larger the larger the degree of financial market imperfections.

We then proceed to the analysis of the general case via numerical simulations in order to quantify the relationship between the aggressiveness of monetary policy and the real exchange rate response to a public spending shock. We find that the values of the policy-rule parameter that generate a real depreciation fall in the range of empirically plausible numbers. Finally, we perform a sensitivity analysis by varying the values of key parameters within the range of available estimates and by assuming an alternative specification of consumer preferences. We find our main conclusion to be extremely robust to these perturbations.

Three closely related papers to ours are those by [START_REF] Kollmann | Government Purchases and the Real Exchange Rate[END_REF], [START_REF] Corsetti | Fiscal Policy Transmission with Spending Reversals[END_REF], and [START_REF] Ravn | Explaining the Effects of Government Spending Shocks on Consumption and the Real Exchange Rate[END_REF]. [START_REF] Kollmann | Government Purchases and the Real Exchange Rate[END_REF] considers a two-country model with incomplete financial markets and flexible prices, and shows that an increase in public spending in one country can depreciate its real exchange rate, provided that labor supply is highly elastic. The values of the elasticity of labor supply needed to generate a real depreciation, however, seem to be hard to reconcile with available empirical estimates. In contrast, our explanation does not require such an extreme assumption about the elasticity of labor supply. [START_REF] Corsetti | Fiscal Policy Transmission with Spending Reversals[END_REF] develop a two-country model with complete markets, sticky prices and wages, and spending reversals. The latter feature refers to the assumption that debt-financed increases in government spending will cause subsequent spending to fall below its steady state level for some time. In turn, this lowers long-term real interest rates and depreciates the currency in real terms. However, while empirical studies based on structural vector autoregressions (SVAR) invariably find that the path of government spending in the U.S. is "self-correcting", increasing for six to twelve quarters before eventually falling below trend, results for other countries do not display such a reversal, especially for long sample periods (e.g., [START_REF] Perotti | Estimating the Effects of Fiscal Policy in OECD Countries[END_REF] and [START_REF] Bouakez | Fiscal Policy and External Adjustment: New Evidence[END_REF]).

Finally, [START_REF] Ravn | Explaining the Effects of Government Spending Shocks on Consumption and the Real Exchange Rate[END_REF] also consider a two-country model with complete financial markets, but depart from time-separable preferences by assuming that consumers form deep habits, i.e., habits at the level of individual varieties of goods. The presence of deep habits induces firms to lower their markups in markets in which aggregate demand rises. Thus, in response to an increase in government spending in the domestic economy, markups on domestically sold goods will be lower than markups abroad, making those goods relatively cheaper in the domestic economy, and implying a depreciation of the real exchange rate. While our explanation involves a completely different mechanism than those discussed above, we regard these different mechanisms as complementary, rather than competing or mutually exclusive, explanations for the "puzzling" real depreciation caused by an expansionary public spending shock.

The paper is organized as follows. Section 2 presents the model. Section 3 analyzes the effects of a government spending shock first analytically within a simplified version of the model and then numerically for the general case. It also performs an extensive sensitivity analysis with respect to key parameters of the model. Section 4 concludes.

The Model Economy

The world consists of a continuum of identical small open economies that trade in final goods and financial assets. The size of each economy is small relative to the rest of the world. Each economy is populated by a continuum of identical, infinitely lived, households, and has a continuum of monopolistically competitive firms that produce a differentiated final good. The number of households and firms is normalized to one. Firms do not price discriminate between the domestic and export market, which means that the law of one price holds for all goods. Financial markets are assumed to be incomplete as households can only trade one-period non-state-contingent domestic and international bonds. Financial markets are also assumed to be imperfect in that the interest rate faced by domestic agents on their external borrowing is augmented by a premium that is increasing in the aggregate level of foreign debt.1 This debt-elastic premium implicitly captures the idea that foreign lenders may perceive higher levels of foreign debt as an indication of a higher probability of default. 2 In what follows, we describe a representative small open economy, which we call "home country", and denote by an asterisk variables pertaining to the rest of the world.

Throughout the paper, variables without time subscripts denote steady-state values.

Households The representative household in the home country maximizes its expected lifetime utility

  ∞ X =  - Ã ln   -  1+1  1 + 1 !  (1)
subject to the budget constraint

   *  +   +     =    -1  * -1  * -1 +  -1  -1 +       +   -   (2)
and to the appropriate no-ponzi-game conditions. In expression (1), the parameters 0    1 and   0 are, respectively, the subjective discount factor and the Frisch elasticity of labor supply,   is a consumption index, and   is the quantity of labor competitively supplied to the firms. In equation ( 2),  *  and   are the household's nominal holdings of, respectively, international and domestic bonds at the end of period ;   is the nominal exchange rate, defined as the price of a unit of the foreign currency in terms of the domestic currency;  * -1 and  -1 are the nominal interest rates paid on, respectively, foreign and domestic bonds between periods  -1 and ;   is the consumption-based price index (CPI hereafter),   is the real wage rate,   = R 1 0   () denotes the household's claims on firms' (indexed by ) nominal profits, and   denotes lump-sum taxes paid to the government. The term   is a risk premium that increases with the economy's aggregate level of debt as a percentage of steady-state output. More specifically,   has the following functional form.

  = exp à -    *   !  (3)
where  *  is the aggregate level of foreign debt, which the household in the small open economy takes as given, and  denotes nominal GDP in the steady state. Denoting by   ≡    *    the real exchange rate, the household's first order conditions imply

   1  -  = 0 (4)     µ      +1  +1 ¶ -1 = 0 (5)   • 1  +1  +1 µ   -  +1    +1    *   * +1    *  ¶¸= 0 (6)
in addition to the usual transversality conditions. Equation ( 4) is a standard labor supply equation, which equates the marginal rate of substitution between consumption and leisure to the real wage.

Equation ( 5) is the Euler equation, relating the expected growth rate of consumption to the real interest rate. Equation ( 6) is a no-arbitrage condition that equates the expected rates of return on foreign and domestic bonds. A log-linear approximation of this equation yields a modified uncovered interest rate parity condition that includes a risk premium.

The consumption index,    is an aggregate of consumption of goods produced in the home country (  ) and goods produced in the rest of the world (  )

  = ³ (1 -) 1  (  ) -1  +  1  (  ) -1  ´ -1  (7)
where the parameter  ∈ [0 1] measures the share of imported goods in total consumption, thus reflecting the degree of openness of the small open economy,3 and the parameter  is the elasticity of substitution between domestic and foreign goods. The associated CPI is

  = ³ (1 -) (  ) 1- +  (  ) 1- ´1 1-  (8)
where   and   are the price sub-indexes associated with   and    respectively. The optimal allocation of consumption between home and foreign goods gives rise to the following demand functions:

  = (1 -) µ     ¶ -      =  µ     ¶ -    (9)
The consumption sub-indexes   and   are bundles of differentiated varieties produced domestically and abroad, and are given by, respectively

  = µZ 1 0   () -1   ¶  -1 ,   = µZ 1 0   () -1   ¶  -1  (10)
where   () (respectively   ()) is consumption of a typical variety  produced in the home country (respectively in the rest of the world) and   1 is the elasticity of substitution between varieties originating in the same country. The price sub-indexes associated with   and   are given by, respectively

  = µZ 1 0   () 1-  ¶ 1 1- ,   = µZ 1 0   () 1-  ¶ 1 1-  ( 11
)
where   () (respectively   ()) is the domestic-currency price of a typical variety  produced in the home country (respectively in the rest of the world). Demand functions for domestically produced and foreign varieties are given by

  () = µ   ()   ¶ -      () = µ   ()   ¶ -    (12)
Note that because the law of one price holds for all goods, we have

  () =    *  () ∀ and thus   =    *  .
Firms Each monopolistically competitive firm is the supplier of a single variety , produced with the following linear technology:

  () =   ()  (13)
where   () is the quantity of labor hired by firm  Since firms do not segment markets by country, the export price of a typical firm ,  *  () will be equal to   ()   which also implies that  *  =     . Prices are set à la [START_REF] Calvo | Staggered Prices in a Utility-maximizing Framework[END_REF]. That is, each period, a fraction (1 -) of firms are randomly selected to set new prices while the remaining fraction  of firms keep their prices unchanged. Let   denote the price set by a typical firm at period . 4 Assuming that the government consumption indexes are analogous to those of private consumption, the total demand facing this firm in period  ( ≥ ) is

  = µ     ¶ - ¡   +  *  +   +  *  ¢ = µ     ¶ - µ     ¶ - ((1 -) (  +   ) +    ( *  +  *  ))  (14)
The optimal price,    is chosen to maximize

 =   ∞ X = () -   ¡   -    ¢    (15)
where

  =      
The first-order condition for this problem yields

  =   -1 P ∞ = () -   (      ) P ∞ = () -   (  (    ))  (16)
Aggregating across all firms, the price index for domestically produced goods is given by

  = ³ (1 -)   1- +  1- -1 ´1 1-  (17)
Government Since Ricardian equivalence holds in this model, we abstract from public debt and assume that the government finances its expenditures through lump-sum taxes. That is

    =    ( 18
)
Public spending is exogenous and follows the autoregressive process

ln µ    ¶ =   ln µ  -1  ¶ +     (19)
where 0 ≤   ≤ 1 and    is a serially uncorrelated disturbance with zero mean.

Monetary authority

The monetary authority in the small open economy sets the nominal interest rate according to the following rule:

ln µ    ¶ =   ln µ  -1  ¶ + (1 -  )    ³  +1  ´ (20)
where   =    -1 is the CPI inflation rate between periods  -1 and , 0 ≤    1 captures the degree to which the monetary authority smooths out changes in the nominal interest rate, and   ≥ 1 measures the aggressiveness with which it responds to expected inflation. The assumption that the monetary authority responds to expected, rather than current inflation, is only made for tractability as it allows us to obtain analytical results in Section 3.1, but it is not essential to the general point made in this paper.

Equilibrium, steady-state and log-linearized model

Equilibrium Given that we consider a small open economy, we assume that domestically produced goods receive a negligible weight in the world consumption bundle. This implies that

 *  =  *  
For the purpose of deriving the market clearing conditions, let

  ≡ ³ R 1 0   () -1   ´ -1 denote
aggregate output. The goods market clearing condition is then given by

  = µ     ¶ - ((1 -) (  +   ) +    ( *  +  *  ))  (21)
Furthermore, aggregating the production functions of all domestic firms (equation 13) yields

∆    = Z 1 0   ()  = Z 1 0   ()  ≡    (22)
where

∆  ≡ R 1 0 ³   ()  
´-  measures the dispersion of domestic producer prices. 5 Labor-market clearing implies that

  =    (23)
Since households in the domestic economy are identical, the domestic bonds market is in zero net supply, i.e.,

    ≡   = 0 and  *   *  =  *   *  ≡  *  .
Imposing these conditions and consolidating the households' and government budget constraints yield the following balance of payments (BOP) equation:

   *  =    * -1    * -1 +       -  -   (24)
where we have substituted the firms' current dividends, used the fact that

    = R 1 0   ()   ()
, and imposed the labor-market clearing condition.

Conditionally on both monetary policy and the exogenous variables, a symmetric equilibrium for this economy is a collection of 9 sequences ( Steady state We define a symmetric flexible-price initial steady state in which exogenous variables take identical values in the small open economy and in the rest of the world, and net foreign assets are zero, i.e.,  * = 0 6 Moreover, we assume without loss of generality that all nominal prices are equal to unity in steady state. This ensures that  = 1. Denoting by  the steady-state ratio of government spending to output, the steady-state allocations are

 ,   ,   ,  *  ,   ,      ,   ,
 =  = µ  -1  (1 -)  ¶  1+   = (1 -)  (25)
Log-linearization In order to solve the model, we log-linearize its equilibrium conditions around the steady state. Denoting by a circumflex the log-deviation of a variable from its steady-state 5 The variable   ≡  1 0 () = ∆ is the appropriate measure of GDP in this model. However, as shown by [START_REF] Galí | Monetary Policy and Exchange Rate Volatility in a Small Open Economy[END_REF], ∆  is of second order, which implies that GDP and aggregate output are equivalent up to a first order approximation. 6 This symmetric steady state is convenient to derive the analytical solution discussed in the next section, but our results still hold when we log-linearize the model around a steady state with non-zero net foreign assets. value, 7 the log-linearized version of the model is given by (see the Appendix for the derivation)

b   =   b  +1 -(  -  b  +1 ) (26) b   =   b  +1 + ( b  *  -  b  * +1 ) -(  -  b  +1 ) - b  *   (27) b  *  =  -1 b  * -1 + b   - µ (1 -) b   + b   +  1 - b   ¶  (28) b   =   b  +1 + (1 -) (1 -)  µ b   +  1 - b   ¶  (29) b   = b   +  -1 b    (30) b   = b   +  1 - (b   -b  -1 )  (31) b   = b    (32) b   = (1 -) ((1 -) b   + b   ) +  µ 2 - 1 - ¶ b   +  ((1 -) b  *  + b  *  )  (33) b   =   b  -1 + (1 -  )     b  +1  (34) b   =   b  -1 +     (35)

Public Spending Shocks and the Real Exchange Rate

Assume that all foreign variables remain at their steady-state values (b

 *  = b  *  = b  *  = b  *  = 0)
, an assumption that we shall maintain in the rest of the paper. Combining the balance of payments (equation 28) and the goods market clearing condition (equation 33) yields:

b   = -b  *  +  -1 b  * -1 - (1 -) b   +  1 - ( (2 -) -1) b    (36)
This equation states that for given paths of private consumption and net foreign assets, an increase in government spending implies a depreciation of the real exchange rate, provided that  (2 -) 

1. How plausible is this parametric restriction? Available estimates of the elasticity of substitution between domestic and foreign goods indicate that  is slightly above 1, although values as large as 15 or 2 are frequently used in calibrated open-economy models. Estimates of the degree of trade openness suggest that plausible values for  are in the range of (01 04). Clearly, even the most conservative values of  and  imply that the condition  (2 -)  1 is met. Therefore, the rest of the analysis will assume that this condition holds.

Of course, whether the real exchange rate appreciates or depreciates in equilibrium will depend on the general-equilibrium adjustment of consumption and net foreign assets. In this paper, we

show that a real depreciation occurs as long as consumption does not fall too much in response to the 7 The only exception is   *  which denotes

 *   
increase in government spending, which implies that the long term real interest rate, This version allows us to solve analytically for the equilibrium path of the real exchange rate. We then turn to the analysis of the more general version of the model using numerical simulations.

  P ∞ =0 ( b  + - b  +1+ ),

A special case

Consider a simplified version of our model in which we abstract from uncertainty and where we set   = 0 and   = 1. In this case, the monetary authority changes the nominal interest rate one for one with expected inflation, thereby implying that the short and long-term real interest rates remain constant in every period, i.e., b

  -b  +1 = P ∞ =0 ( b  + -b  +1+ ) = 0 for all  This in turn
implies that consumption remains constant in equilibrium if the economy is affected by transitory shocks, i.e., b   = 0 for all . Note that this equilibrium exists only if prices are sticky, as the real interest rate is independent of monetary policy and must adjust in a flexible-price equilibrium.

In the absence of intertemporal substitution (stemming from the constancy of the real interest rate), it can be easily seen that the dynamics of net foreign assets and the real exchange rate can be solved for independently of the rest of the model. More specifically, these two variables are determined by the following two-equation dynamic system:

b  *  - -1 b  * -1 = b   -b    (37) b  +1 -b   =  b  *   (38)
where

 ≡  1- ( (2 -) -1)  0 A graphical
  = -  -1 -1  b  * -1 +   b    (39) b   = -  -1  b  * -1 +   b    (40)
8 Indeed, iterating equation ( 26) forward yields:

   = lim  →∞     + -  ∞  =0 (   + -  +1+ )
which shows that the reponse of consumption to a transitory shock (so that lim  →∞     + = 0) is entirely determined by the (negative of) response of the long-term real interest rate. appreciates and net foreign assets fall for some time (inducing a current-account deficit) before starting to increase. Both variables then move down the saddle path until the economy returns to its initial equilibrium.

We now provide an analytical solution to system (37)-(38). To do so, we use the method of undetermined coefficients. That is, we conjecture that the solution takes the form

b   =  1 b  * -1 +  2 b    (41)
and determine  1 and  2 as a function of the deep parameters. By straightforward algebra, it can be shown that the coefficient  1 solves the following second-order equation:

 2 1 -(1 +  - -1 ) 1 - -1 = 0 (42)
The root that leads to a stable equilibrium is given by

 1 = 1 +  - -1 - q (1 +  - -1 ) 2 + 4 -1 2  0 (43)
The parameter  2 is thus given by

 2 = -( 1 -) 1 -  -( 1 -)  0 (44)
Note that  2 is unambiguously positive, which means that the real exchange rate depreciates following a positive government spending shock. One can also easily check that the current account deteriorates following such a shock. Intuitively, given that the real interest rate is constant, the balance of payments equilibrium cannot be achieved with too large a decline in net foreign assets, as such a scenario would violate the transversality condition. The increase in government spending must therefore be accompanied by a real depreciation (see equation 36).

The effects of government spending shocks in this special case with a constant real interest rate can also be understood using an IS-BOP framework. The IS curve describes the goods-market equilibrium in the small open economy, and is characterized by the following equation:

b   = µ  +  1 - ¶ b   +  (1 -) b    (45)
Note that this curve is increasing in b   , i.e., the relative price of foreign goods. To obtain an equation for the BOP curve in the (b    b   ) space, we start by using (41) to solve for b  +1 in (38).

This yields b 

*  = 1  1 - b   -  2    1 - b    Inserting this expression into equation (28), we obtain b   = - -1 b  * -1 + µ 1  1 - +  1 - ¶ b   + µ  -  2    1 - ¶ b    (46)
The slope of this curve,

³ 1  1 - +  1-
´, is negative for sufficiently low values of , but even when it is positive, it never exceeds that of the IS curve. 9 In other words, the BOP curve is either downward sloping or is steeper than the IS curve in the (b

   b   ) space.
Equations ( 45) and ( 46) show that an increase in b   shifts both the IS and BOP curves outward. This is shown in Figure 2, which illustrates the case of a downward sloping BOP curve. The shift in the IS curve is maximal given that it is not attenuated by a decline in private consumption (recall that b   = 0). The shift in the BOP curve reflects the fact that the real exchange rate needs to depreciate, for a given level of output, to restore the balance-of-payments equilibrium. Because the BOP curve shifts more than does the IS curve (note that  - government spending leads to an increase in output and to a depreciation of the real exchange rate.

 2    1 -   (1 -)), the rise in
These effects are larger the steeper the BOP curve (i.e., the larger the value of ).

Recall, however, that the depreciation of the real exchange rate is necessary in this special case because the equilibrium cannot be restored through a change in consumption, or alternatively, a change in the long-term real interest rate. In the general case where monetary policy changes the nominal interest rate more than one for one with expected inflation, the response of the real exchange rate will depend on the magnitude of the adjustment of the long-term real interest rate.

We explore this matter in the next section.

The general case

In this section, we show that the result highlighted in the special case discussed above holds under less restrictive assumptions about monetary policy. In particular, we show that the real exchange rate continues to depreciate in response to a positive public spending shock if monetary policy adjusts the nominal interest rate more than one for one to expected inflation, without being overly aggressive. When   takes on values that are strictly larger than 1, however, the linearized model no longer admits an analytical solution and we have to proceed with numerical simulations. To do so, we need to assign values to the structural parameters.

9 Indeed, 

→∞  1  1 - +  1-  =  1-   +  1- 
Parameter values We calibrate the model at a quarterly frequency. We start by discussing the parameters whose values are relatively conventional in the literature. The discount factor,  is set to 099 so that the implied steady-state real interest rate is 41% annually. The preference parameter  is a scaling parameter and is set to 1. We choose the elasticity of substitution across domestic varieties of final goods, , to equal 6, which yields a steady-state markup of 20%, as in [START_REF] Rotemberg | An Optimization-based Econometric Framework for the Evaluation of Monetary Policy[END_REF]. The steady-state share of public spending in total output,  is set to 025. The parameter governing trade openness, , is calibrated to 03 to match the average share of imports over GDP in industrialized economies. The smoothing parameter,   , is set to 08, consistently with the estimates reported in the literature (e.g., [START_REF] Smets | Comparing Shocks and Frictions in US and Euro Area Business Cycles: A Bayesian DSGE Approach[END_REF]).

Next, we turn to the parameters whose values are relatively more controversial. For each of those parameters, our strategy is to choose a benchmark value that lies within the range of available estimates and then perform a sensitivity analysis. Starting with the elasticity of substitution between domestic and foreign goods,  available micro estimates tend to be much larger than macro estimates (see [START_REF] Ruhl | The International Elasticity Puzzle[END_REF] for an extensive review of the literature). We choose to follow the macroeconomic literature and set  = 15 in the benchmark calibration (as in [START_REF] Backus | Dynamics of the Trade Balance and the Terms of Trade: The J-Curve?[END_REF], for example). In the sensitivity analysis, we vary  between 1 and 2. [START_REF] Merz | Search in the Labor Market and the Real Business Cycle[END_REF] reports that existing estimates of the Frisch elasticity of labor supply, , vary between 003 and 3. In general, micro estimates converge towards low values of , whereas macro estimates tend to be large. Following [START_REF] Galí | Markups, Gaps, and the Welfare Costs of Business Fluctuations[END_REF], we consider a benchmark value of 05 and investigate the sensitivity of the results to values of  between 02 and 1. We set the Calvo probability of not changing prices, , to 075, which implies an average duration of price spells of 4 quarters. In Section 3.3, we let this probability span the entire range of possible values, i.e., (0 1). In our model, the risk-premium parameter, , measures the (negative of the) elasticity of the interest rate differential with respect to net foreign assets (see equation 27). [START_REF] Lane | Long-Term Capital Movements[END_REF] estimate the elasticity of the real interest rate differential (in %) to the ratio of foreign debt to exports using annual data. Their estimates range between 083 and 324. In our benchmark calibration, we choose to be conservative and use Lane and Milesi-Ferretti's lowest estimate. As we will show in the sensitivity analysis, higher values of  actually strengthen our results. Once converted to a quarterly basis, Lane and Milesi-Ferretti's lowest estimate implies  = 00017. 10 Finally, we calibrate the autocorrelation coefficient of the government spending 10 Denote by κ the estimate reported by [START_REF] Lane | Long-Term Capital Movements[END_REF]. In our model,  *  is the ratio of net foreign assets to quarterly GDP and  is the steady-state share of exports in GDP. Thus, the ratio of net foreign assets to annual exports is  * (4). Since interest rates are quarterly in the model, Lane and Milesi-Ferretti's estimate implies 400 ( - * ) = -κ * (4) process,   to 09, and study the sensitivity of the results when this parameter varies between 06 and 1. The benchmark values assigned to the parameters are summarized in Table 1.

Table 1: The responses of output, consumption, net foreign assets, the nominal interest rate and the CPI inflation rate are qualitatively similar across the two cases. The increase in public spending increases aggregate demand, thus putting pressure on production capacities and input prices and inducing higher domestic-price inflation. The resulting increase in CPI inflation leads the monetary authority to raise the nominal interest rate, which, under the assumption that    1 raises the long-term real interest rate,

  P ∞ =0 ( b  + -b  +1+
). This in turn causes consumption to fall. As in the special case discussed above, net foreign assets decline. Increasing   from 11 to 15 only changes the magnitude of these responses, but not their sign.

In contrast, the real exchange rate exhibits very different responses under the two scenarios, appreciating when   = 15 and depreciating when   = 11. In order to understand this result, it where  and  * are the (net) domestic and world real interest rates, respectively. The expression above yields the following mapping between the estimate κ and our parameter :

 = κ(1600)
For instance, if  = 03 and κ = 083, we obtain  = 00017, which is the value used in our benchmark calibration.

  = lim  →∞   b  + -  ∞ X =0 ( b  + -b  +1+ ) -  ∞ X =0 b  * +  (47)
If the shock is temporary, the term lim  →∞   b  + must be equal to zero, and the response of the real exchange rate will depend on the difference between the response of the long-term real interest rate and that of the long-term risk premium, -  P ∞ =0 b  * + , or, equivalently, on the response of the effective long-term real interest rate,

  P ∞ =0 ( b  + -b  +1+ +  b  * + ).
Figure 3 shows that under aggressive monetary policy (  = 15), the rise in the long-term real interest rate is larger in magnitude than the increase in the long-term risk premium. Therefore the effective long-term real interest rate increases, causing the real exchange rate to appreciate. Under less aggressive monetary policy (  = 11), the risk premium dominates, the effective rate falls initially, and the real exchange rate depreciates.

Figure 4 depicts the relationship between the initial response of the real exchange rate to a positive government spending shock and the parameter   , conditional on our benchmark calibration.

The figure shows that this response is monotonically decreasing    and that a real depreciation is obtained with empirically plausible values of   .

Sensitivity analysis

We now study the sensitivity of our results to alternative values of key parameters, namely, the elasticity of substitution between domestic and foreign goods, the Frisch elasticity of labor supply, the Calvo probability of not changing prices, the risk-premium parameter, and the autocorrelation of the government spending shock. In each case, we focus on the initial response of the real exchange when both the parameter of interest and the policy parameter,   , vary. We also consider an alternative specification of households' preferences whereby government spending enters the utility function.

Elasticity of substitution between domestic and foreign goods () As stated above, the necessary condition for the real exchange rate to depreciate in response to an expansionary government spending shock, i.e.,  (2 -) -1  0, holds for any sensible value of  But how sensitive is this depreciation across the range of plausible values of ? Figure 5 depicts the initial response of the real exchange rate for values of  ranging from 1 to 2, and values of   ranging from 1 to 15.

It shows that the magnitude of the exchange rate response is essentially insensitive to the value of  (within the range considered), regardless of the degree of aggressiveness of monetary policy. means that the labor supply curve does not shift after an increase in public spending. In that version of the model, the parameter  plays no role in the determination of the real exchange rate.

In the general case, however, changes in consumption shift the labor supply curve. An expansionary government spending shock will therefore shift the aggregate supply curve rightward through its effect on labor supply, and the resulting effect on the exchange rate will be larger the larger is .

As pointed out by [START_REF] Kollmann | Government Purchases and the Real Exchange Rate[END_REF], however, under flexible prices, generating a real depreciation requires very large values of , which lie far outside the range of available estimates.11 In contrast, as shown in Section 3.2, our model generates a real depreciation using a value of  that is in line with empirical estimates. Interestingly, Figure 6 below shows that, as  tends towards the lower bound of existing estimates, a real depreciation is still possible to obtain, but it requires less and less aggressive monetary policy.

Calvo probability of not changing prices () While the equilibrium path of the exchange rate is independent of the degree of price rigidity in the special case with constant real interest rate, this is no longer true in the more general version of the model. Figure 7 indeed shows that the magnitude of the exchange rate response to a government spending shock increases exponentially with  and that the model can generate a real depreciation even with an aggressive monetary policy if prices are sufficiently rigid. Risk premium parameter () In the special case with a constant real interest rate, higher values of  make the BOP curve steeper and lead to a larger depreciation of the real exchange rate following an expansionary public spending shock. 12 To verify whether this result also holds in the more general case, we consider values of  that range from 0.015 to 0.15. Figure 8 shows that this is indeed the case. For any given value of the policy parameter   , the real exchange rate response is increasing in . Intuitively, higher values of  imply a larger premium on the country's borrowing, for a given level of foreign debt. This translates into larger decline in the effective long term real interest rate and, therefore, a larger depreciation of the real exchange rate.

Autocorrelation of the government spending shock (  ) Figure 9 shows that the way in which the persistence of the public spending shock affects the exchange rate response depends on the policy parameter,   . For relatively small values of   , this response increases monotonically with   , 13 but for larger values of   , the relationship becomes non-monotonic, being negative for values of   that are less than 09 and positive for larger values Again, this means that one can obtain a depreciation of the real exchange rate in response to a positive public spending shock even with an aggressive monetary policy, provided that the shock is sufficiently persistent.

12 One can also check that  2  =  2   (+)

+  2  1  (-)  1   (-)  0
13 In the special case of no intertemporal substitution (  = 1) this can be clearly seen from equation ( 41) and the expression of  2 . Alternative specification of the utility function The model presented above predicts that consumption is crowded out following an increase in public spending, except in the special case where the real interest rate is held constant. Several empirical papers, however, find that consumption increases following an expansionary government spending shock. In what follows, we show that allowing for an alternative specification of consumer preferences whereby public spending increases the marginal utility of private consumption (i.e., public and private goods are Edgeworth complements) can generate a crowding-in of private consumption in response to a positive government spending shock,14 without altering the mechanism that gives rise to a real depreciation.

To see this, assume that the representative consumer maximizes the following lifetime utility:

  ∞ X =  - Ã ln (  -  ) -  1+1  1 + 1 !  ( 48
)
where  is a parameter satisfying the condition 0    1-  . The fact that  is positive ensures that the marginal utility of consumption is increasing in  (i.e.,  and  are edgeworth complements).

In this case, equations ( 26) and ( 30) are replaced by, respectively

b   =   b  +1 -  1 - (b   -  b  +1 ) - 1 - (1 + ) 1 - ³  -  b  +1 ´ (49) b   = 1 - 1 - (1 + ) b   -  1 - (1 + ) b   +  -1 b    (50)
whereas equations ( 27)-( 29) and ( 31)-(35) remain unchanged. In the special case where we abstract from uncertainty and where the monetary authority changes the nominal interest rate one for one with expected inflation, the resulting constancy of the real interest rate implies that the marginal utility of consumption remains constant, which by equation ( 49) implies that

b   =  1 - b    (51)
Since  1-  0, an increase in government spending raises private consumption by an amount that increases linearly with . Equations ( 27), ( 28), ( 33) and ( 51) yield again a two-equation system that allows us to solve for net foreign assets and the real exchange rate independently of the rest of the variables

b  *  - -1 b  * -1 = b   - (1 + ) b    (52) b  +1 -b   =  b  *   (53)
The equilibrium path of the real exchange rate is again given by equation ( 41), where  2 is now given by

 2 = -(1 + )( 1 -) 1 -  -( 1 -)  0
That is, the real exchange rate depreciates following an expansionary government spending shock, and this depreciation is larger the larger is . Intuitively, as  increases (that is, as edgeworth complementarity between private and public expenditures becomes stronger), the outward shift of the BOP curve exceeds that of the IS curve by a larger amount. 15 This leads to a larger increase in output and a stronger depreciation of the real exchange rate, which are both necessary to accommodate the increase in private consumption.

Conclusion

Empirical studies conclude that the real exchange rate depreciates sharply after an expansionary public spending shock. This paper has shown that a sticky-price small-open-economy model with incomplete and imperfect financial markets can account for this empirical regularity, at least qualitatively, provided that monetary policy is not too aggressive in fighting inflation. Admittedly, however, the magnitude of the real depreciation implied by the model is small in comparison with that documented in the empirical literature. Nonetheless, the mechanism highlighted in this paper can be viewed as a step forward towards a successful quantitative account of the effects of fiscal policy shocks on the real exchange rate.

15 Note that the IS and BOP curves are now characterized by the following two equations, respectively:

   =   +  1 -     +  (1 + ) (1 -)       = - -1   * -1 +  1  1 - +  1 -    +   (1 + ) -  2    1 -   

  and  ) ∞ =0 satisfying the households' first-order conditions (equations 4-6, with equation 23 substituted in for   ), the definition of the CPI (equation 8), the pricing condition (equation 16), the definition of the price index for domestically produced goods (equation 17), the aggregate production function (equation 22), the goods market clearing condition (equation 21), and the balance of payments (equation 24).

  solution of this system is shown in Figure 1, which depicts the phase diagram in the ( b  * -1  b   ) space. The loci ∆ b  *  = 0 and ∆b  +1 = 0 are characterized by the following equations, respectively: b

Figure 1 :

 1 Figure 1: Illustrating the effects of an increase in government spending using the phase diagram.

Figure 2 :

 2 Figure 2: Illustrating the effects of a government spending shock using the IS-BOP framework.

  Steady state ratio of public spending to output  = 025 Trade openness  = 03 Nominal-interest-rate smoothing parameter   = 08 Risk-premium parameter  = 00017 Elasticity of substitution between goods  = 15 Frisch elasticity of labor supply  = 05 Calvo probability of not changing prices  = 075 Autocorrelation of public spending shocks   = 09 Impulse response functions We now discuss the impulse responses generated by the model under the benchmark calibration when the economy is hit by a 1 percent positive government spending shock. To shed light on the importance of monetary policy for the response of the real exchange rate, we study two scenarios: one in which the monetary authority fights inflation aggressively (  = 15) and one in which monetary policy is moderately aggressive (  = 11). The results are depicted in Figure 3.

Figure 3 :

 3 Figure 3: Impulse responses to a 1 percent increase in public spending under aggressive (  = 15) and moderately aggressive (  = 11) monetary policy.

Figure 4 :

 4 Figure 4: Initial response of the real exchange rate as a function of   

Figure 5 :Figure 6 :

 56 Figure 5: Initial response of the real exchange rate in the (   ) space.

Figure 7 :

 7 Figure 7: Initial response of the real exchange rate in the (   ) space.

Figure 8 :

 8 Figure 8: Initial response of the real exchange rate in the (   ) space.

Figure 9 :

 9 Figure 9: Initial response of the real exchange rate in the ¡      ¢ space.

  does not increase too much. 8 Below, we illustrate this point by focusing on a special case of the model in which the real interest rate, and hence consumption, remain constant in equilibrium.

This assumption has been used by several authors, such as[START_REF] Kollmann | Monetary Policy Rules in the Open Economy: Effects on Welfare and Business Cycles[END_REF] or Schmitt-Grohé and Uribe[2003].

This premium also induces stationarity in the log-linearized version of the model. Without it, the model's equilibrium dynamics exhibit a unit root (see[START_REF] Schmitt-Grohé | Closing Small Open Economy Models[END_REF]).

The fraction (1 -) therefore measures the degree of home bias in consumption.

The optimal price does not depend on the firm index because all the firms that have the opportunity to change their prices at a given time choose the same price.

[START_REF] Kollmann | Government Purchases and the Real Exchange Rate[END_REF] considers values of the Frisch elasticity equal to 2, 5 and infinity.

The idea that Edgeworth complementarity between private and public spending can cause private consumption to rise in response to an expansionary government spending shock has been suggested by[START_REF] Bouakez | Why Does Private Consumption Rise After a Government Spending Shock?[END_REF].

Appendix

Equations ( 4), ( 5), ( 6), ( 8), ( 16), ( 17), ( 22), ( 21), ( 24), as well as the monetary policy rule (20) and the government spending shock process (19) are log-linearized around the initial steady state defined in Section 2.2.

The log-linearized versions of equations ( 4)-( 6) are, respectively

which are, respectively, equations ( 30), ( 26), ( 27) in the main text.

Replacing   by    *  in equation ( 8) and dividing both sides by  -1 yield

where

Noting that b   =  --1 + b    and using equation (A5) to substitute for  and -1 , we obtain

which is equation (31) in the main text.

The optimal pricing condition ( 16) can be rewritten as

Expressing   and   recursively yields

where we have used the fact that   = 1. Dividing both sides of equation ( 17) by   , we obtain

Combining equations (A10), ( A11) and (A13), we obtain

which is equation ( 29) in the main text.

The log-linearized versions of the remaining equations (22, 21, 24, 20, and 19) are, respectively

which are equations ( 28), ( 32), ( 33), (34), and (35) in the main text.