Arnaud Badel

K Takahashi

A Zampa

T Okada

G Nishijima

T Uto

H Takewa

S Hanai

S Ioka

J Inagaki

S Awaji

First Validation of Robust REBCO Insert Concept on

Keywords: High field Superconducting magnet, Insulated REBCO magnet, numerical modelling, Quench protection

HAL is

I. INTRODUCTION

he 25 T Cryogen-Free Superconducting Magnet was installed in 2017 as a user magnet at the High Field Lab. For Superconducting Materials, of the Institute for Material Research, Tohoku University [START_REF] Awaji | First performance test of a 25 T cryogen-free superconducting magnet[END_REF]. It consists in an 11 T BiSrCaCuO HTS insert with a 14 T LTS outsert. It was the first user magnet operating above 23 T, and is still to date the only conduction-cooled magnet operating above that limit, even though several other user or commercial magnets operate now above 25 T or are expected to do so [START_REF]32 Tesla all-superconducting magnet[END_REF][START_REF] Wikus | Commercial gigahertz-class NMR magnets[END_REF][START_REF] Yanagisawa | Review of recent developments in ultra-high field (UHF) NMR magnets in the Asia region[END_REF]. Since then research activities were conducted at HFLSM together with Toshiba to develop the technology for REBCO insert for upgrading this magnet to reach 30 T while retaining the ease of use offered by cryogen-free operation of an insulated magnet [START_REF] Takahashi | Electromagnetic Characteristics Study of Two-Ply REBCO Tapes Pancake Coils[END_REF][START_REF] Takahashi | Mechanical Properties of Four-Stacked Two Tape Bundled REBCO Pancake Coils[END_REF]. Numerous small scale pancake prototypes were tested, leading to the final design for this upgrade presented in [START_REF] Awaji | Robust REBCO Insert Coil for Upgrade of 25 T Cryogen-Free Superconducting Magnet[END_REF], in which a concept of robust REBCO insert coil and its protection against thermal runaway were introduced, with a few key features:

This work was supported by JSPS KAKENHI under Grant 18H05248. (Corresponding author: Arnaud Badel.)

A. Badel is with the High Field Laboratory for Superconducting Materials, Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan and also with University of Grenoble Alpes, G2Elab/Néel Institute, CNRS, Grenoble 38042, France. (e-mail: arnaud.badel@neel.cnrs.fr).

To ensure the conduction-cooling efficiency and improve the winding structure, the pancakes are epoxy-impregnated, using a non-stick material in between turns. This pancake winding structure is expected to homogenize significantly the hoop stress distribution.

To limit the transient losses and to achieve fast dynamics, the turns are fully insulated. In this way, the insert can be discharged in a few seconds, as fast as the outsert, limiting energy transfer from one to the other during an outsert quench [START_REF] Badel | Detection and Protection Against Quench/Local Thermal Runaway for a 30 T Cryogen-Free Magnet[END_REF].

To reduce the risk that a local defect could prevent operation at nominal current, the conductor is formed of two REBCO cowound so that current can be shared [START_REF] Abe | REBCO Coil With Robust Behavior Against Local Defects Wound Using Two-Tape Bundle[END_REF]. Both tapes have their superconducting layers facing inward, in a "face -to-back" configuration as in [START_REF] Takahashi | Mechanical Properties of Four-Stacked Two Tape Bundled REBCO Pancake Coils[END_REF][START_REF] Abe | REBCO Coil With Robust Behavior Against Local Defects Wound Using Two-Tape Bundle[END_REF].

Recently, the effort was shifted from upgrading the existing 25 T CSM to the development of a new magnet, reaching 33 T. The outsert specifications for this new magnet are presented in [START_REF] Awaji | 33T cryogen-free superconducting magnet project at HFLSM[END_REF]. It will also generate 14 T but in a slightly larger bore to accommodate a larger insert generating 19 T. The REBCO insert specifications are not finalized but it will be based on the ideas introduced for the robust REBCO insert concept as mentioned above. In this work, a first large scale validation of these ideas is presented, for a prototype made of 20 pancakes. The present paper focuses on the mechanical modelling of this prototype, and its thermal runaway detection setup.

II. WINDING CHARACTERISTICS AND TEST ENVIRONMENT

The coil is made of 20 stacked pancakes. Each of these pancake are first dry-wound, then epoxy glue is applied on the pancake surfaces to bond the thin FRP flanges (0.2 mm approx.). This technique, combined with the turn-to-turn separation by fluorine-coated Polyimide achieve the so-called edge-impregnation, where the turns are glued to the thin FRP flanges but not to each other, preventing delamination. The pancakes are first glued two by two with one thin aluminum cooling sheet in between and connected electrically at their inner diameter by a silver plate to form Double Pancakes. Then these DPs are stacked, and connected electrically at the outer diameter also with silver plates.

Fujikura tapes [START_REF] Fujita | Flux-Pinning Properties of BaHfO3-Doped EuBCO-Coated Conductors Fabricated by Hot-Wall PLD[END_REF] with thick (80 µm overall) Cu coating are used. As explained in introduction, the conductor is a bundle of two tapes in face-to-back configuration. In previous studies [START_REF] Abe | REBCO Coil With Robust Behavior Against Local Defects Wound Using Two-Tape Bundle[END_REF] this ensured sufficient current redistribution between the tapes in case of local degradation of the critical current even though the distance between layers is higher than in face-to-face configuration. The turn to turn electrical insulation is a co-wound Polyimide tape (50 µm thick in this case) with Fluorine coating, which as mentioned before serves also as non-stick layer to prevent delamination. The winding characteristics are presented in Table I.

All the pancakes were tested individually at LN2 temperature assembly for quality control. After assembly the coil was equipped with strain gauges at the outer diameter for circumferential and axial strain measurements. More details on the pancake preliminary tests, the coil and its strain instrumentation can be found in [START_REF] Takahashi | Performance test of 20-stacked two-tapes-bundled REBCO pancake coils for upgrading of 25-T cryogen-free superconducting magnet to 30 T[END_REF].

The tests were conducted at the National Institute for Material Science in Tsukuba, in a 14 T Superconducting large bore outsert. These tests were primarily aimed at mechanical performance validation, which is the reason why they were made with LHe cooling. More tests in conduction cooling are planned in the next months. The current source available onsite and suitable for powering few H-range inductances is rated at 300 A. This current is 1.22 times the nominal current expected for these pancakes in the 30 T upgrade [START_REF] Awaji | Robust REBCO Insert Coil for Upgrade of 25 T Cryogen-Free Superconducting Magnet[END_REF]. As it will be explained in the next part, under 14 T these 300 A makes it possible to reach 350 MPa, the peak stress that was expected for the 30 T upgrade. This validates the mechanical design of the whole insert, even though there are 20 pancakes instead of the full stack of 52. The combined field value is slightly in excess of 25 T, 11 T being produced by the 20 REBCO pancakes.

III. MECHANICAL MODELLING AND VALIDATION OF MECHANICAL PERFORMANCES

As discussed in the previous section, the REBCO pancakes use edge impregnation. In such case the hoop stress cannot be successfully predicted using the BJR equation, which is valid only if the turns do not interact with each other and with the rest of the structure. Nor can it be predicted using the classical Wilson formula, which is valid only for a winding pack behaving as an isotropic bulk piece. Indeed with edge impregnation, the turns are not glued together, only to the flanges. In consequence, the radial stiffness of the winding is only due to flanges, which are much thinner than the conductor width (by an order of magnitude). This can be seen as a solid winding pack but with strongly anisotropic properties, whose behavior could be described following [START_REF] Arp | Stresses in Superconducting Solenoids[END_REF]. The estimation of the equivalent elastic properties of the winding structure is however complex.

Here we prefer to use an FEM model to represent the complex winding structure, including the turns, the mandrel and the glued flanges and cooling plates. To keep the FEM solving simple, the contacts are all considered as thin elastic interfaces: A layer with very low Young's Modulus (1 MPa) represent the turn to turn interface, which is not glued due to the non-stick material, while the epoxy glue between the conductor and the FRP flanges is represented realistically as a thin layer with a Young Modulus of 10 GPa (Fig. 1). Tensile stress tests were conducted on the REBCO tapes used in this coil. The equivalent Young Modulus of the tapes at cryogenic temperatures was estimated in-house at 130 GPa.

The hypothesis of weak elastic boundary between turns is only valid if the turns tend to separate from each other under load, which is the case here. If there is significant radial pressure, as in the case of pancakes with external reinforcement and/or under smaller background fields, the model must be modified. In general and to keep the model simple, an iterative approach can be conducted. The simulation is stopped when the radial stress become negative (pressure between turns), the soft elastic boundaries are removed in the area under pressure, then the simulation is run again. The process is repeated until no soft interface is under pressure anywhere. The same modelling approach was already used successfully to match experimental results in [START_REF] Takahashi | Electromagnetic Characteristics Study of Two-Ply REBCO Tapes Pancake Coils[END_REF][START_REF] Takahashi | Mechanical Properties of Four-Stacked Two Tape Bundled REBCO Pancake Coils[END_REF] and [START_REF] Vialle | A high performance insulated REBCO pancake with conductive cooling capability[END_REF]. The tensile strength of the Polyimide layer being negligible compared to that of the REBCO tapes, it is ignored in the model. That is, each turn is considered as one element in the model, with a thickness including the Polyimide co-winding. Then a ratio of 350/300, corresponding to "total turn thickness" over "thickness of 2 REBCO tapes", is applied to the hoop stress and strain results.

Fig. 2 presents the mechanical modelling results under 14 T for 300 A, highest current that can be reached with the experimental setup. As mentioned in part II, these operation condition makes it possible to reach almost 350 MPa in the winding, approximately at the middle of the winding at the center of the stack.

The stress distribution comparison between the BJR hypothesis (in blue) and the FEM model (in red) highlight the benefit of the edge impregnation, as it homogenizes the hoop stress and reduces the peek value from 407 to 346 MPa (Fig. 3). Experimental measurement were only obtained at the outer diameter, where the estimated stress 280 MPa. This should correspond 0.215 % elongation considering the Young Modulus of the conductor. The strain measurement detailed in [START_REF] Takahashi | Performance test of 20-stacked two-tapes-bundled REBCO pancake coils for upgrading of 25-T cryogen-free superconducting magnet to 30 T[END_REF] reach 0.23 % for circumferential elongation at outer turn, which is in good agreement. Fig. 4 presents an estimation of the radial stress in the FRP flanges. This stress could not be measured but appears realistic, in line with what was measured in [START_REF] Vialle | A high performance insulated REBCO pancake with conductive cooling capability[END_REF].

The successful tests of this 20-pancake-stack coil validated the mechanical design of the 30 T upgrade as presented in [START_REF] Awaji | Robust REBCO Insert Coil for Upgrade of 25 T Cryogen-Free Superconducting Magnet[END_REF] and in general the benefits of edge impregnation, an important aspect for robust REBCO insert concepts. It is also beneficial when considering the potential stress concentration effect due to inhomogeneous distribution of current density in REBCO tapes (shielding currents). This aspect will be covered in future works.

IV. ROBUST INSULATED REBCO COIL PROTECTION: THERMAL RUNAWAY ONSET DETECTION SETUP

A. Robust REBCO coil protection concept

An important aspect of the robust REBCO insert coil concept we proposed is the protection against the risk of thermal runaway. As discussed first in [START_REF] Badel | Modeling of 'quench' or the occurrence and propagation of dissipative zones in REBCO high temperature superconducting coils[END_REF], the large thermal stability of HTS coils, even operating at low temperature, makes it unlikely that a quench occur after a limited burst of energy deposition. The protection system has therefore two major and rather independent roles. On one hand protecting against major instability events, such as a loss of cooling power or more importantly a quench of the outsert magnet. On the other hand, protecting against the consequences of local dissipation due to tape performances inhomogeneities or local conductor defects, leading to a local overstepping of the critical current. The former is actually the easiest for an insulated coil, as it can be discharged fast. It was studied in [START_REF] Badel | Detection and Protection Against Quench/Local Thermal Runaway for a 30 T Cryogen-Free Magnet[END_REF] and will not be covered here. The latter is more challenging for insulated REBCO coils as a destructive thermal runaway can occur in one or a few seconds depending on the current density in the conductor, as the current cannot bypass from one turn to the other like in Non-Insulated windings [START_REF] Hahn | HTS Pancake Coils Without Turn-to-Turn Insulation[END_REF][START_REF] Lécrevisse | A (RE)BCO Pancake Winding With Metal-as-Insulation[END_REF]. The probability of a local critical current drop and its consequences are strongly mitigated by the use of a 2 tape bundle conductor [START_REF] Abe | REBCO Coil With Robust Behavior Against Local Defects Wound Using Two-Tape Bundle[END_REF]. In [START_REF] Badel | Detection and Protection Against Quench/Local Thermal Runaway for a 30 T Cryogen-Free Magnet[END_REF] it was estimated through detailed thermos-electric modelling that even though the thermal runaway itself occurs too fast for protecting the coil using a dump resistor, the dissipation levels are measurable (in the 1 -10 mV range in the current density range we are using here) a few seconds before, time enough to initiate a preemptive discharge preventing the thermal runaway to occur. The challenge is then to discriminate this dissipative signal from the other transient signals across the coil as discussed first in [START_REF] Rozier | Modeling of Dynamic Current Distribution in REBCO Insulated Coils Using a Volume Integral Formulation for Protection Purpose[END_REF] and more in details in [START_REF] Vialle | Transient voltages and energy balance in REBCO insulated magnet: experimental and numerical studies[END_REF]. Moreover, it is important to be able to localize an eventual dissipative area so that the faulty pancake can be replaced without damage to the other pancakes. It should be noted that this local dissipation detection is in principle necessary only during qualification of the coil, and not for its later nominal operation.

B. Detection system implementation and tests for 20 pancake coil

Following [START_REF] Vialle | 12 T Insulated REBCO Magnet Used as 1 MJ SMES: Protection Strategies and Preliminary Assembly Tests[END_REF] a multi-channel detection bridge is used, with independent pickup coils for inductive voltage cancelation. Here for practicality the 20 pancakes are grouped 4 by 4 leading to 5 detection channels (Fig. 5). The 5 compensation coils are wound on top of each other with 0.2 mm diameter copper wire and impregnated with epoxy to prevent wire movements. The assembly is mounted at the bottom of the REBCO insert. This position is not ideal for coupling, which mean that the inductances of these pickup coils had to be increased significantly. A localization in the clear bore of the magnet would be more suitable but that one was occupied by coil support structure in this particular test.

Fig. 6 and7, the time evolution of each 5 compensated signals is presented during a ramp up to 300 A, in self-field and under 14 T background field respectively. During the self-field tests, the ramp rate was set at 0.4 A/s. It was lowered to 0.2 A/s for the test under background field in order to get lower voltage drift due to transient effects. Multiple plateau were also made to check for pancake-to-pancake contact degradations due to stress increase. No increase of resistance duet to such degradation were observed (see [START_REF] Takahashi | Performance test of 20-stacked two-tapes-bundled REBCO pancake coils for upgrading of 25-T cryogen-free superconducting magnet to 30 T[END_REF] for more details)

In both cases, the initial values of the compensated signals at the onset of a ramp are meaningless, they only result from the compensation bridge adjustment. As described in [START_REF] Vialle | Transient voltages and energy balance in REBCO insulated magnet: experimental and numerical studies[END_REF] the voltage drift during current ramp comes from a mixture of transient dissipation and REBCO coil inductance increase when the current penetrates the tapes. These effects are rendered more difficult to evaluate quantitatively for this coil due to the transient coupling currents between the 2 bundled tapes. These are for example responsible for the slight increase of the voltage drift around t=150 ms on all 5 signals Fig. 6. More detailed modelling and quantitative comparison will be the object of future works.

Fig. 6, the noise on the signal is very low, so that observing a sudden increase of the voltage due to a dissipating zone in the few-mV range on top of the existing voltage drift trend is straightforward. Fig. 7, large spikes are visible on the signals, corresponding to conductor movement under mechanical stress. Even though such spikes will mostly disappear after the first ramp, they can be an issue to effectively detect dissipation zone during that first ramp, which is the critical one. Indeed a protection based on threshold value in the 10 mV range above the voltage trend would be impractical.

However, when superposing to the end of the voltage trend measured experimentally the simulated dissipating voltage evolution for a very pessimistic case (single local sharp defect with 50 % Ic reduction) as modelled in [START_REF] Badel | Detection and Protection Against Quench/Local Thermal Runaway for a 30 T Cryogen-Free Magnet[END_REF], it is clear that the voltage evolution in case of zone starting to dissipate locally can be distinguished from the noise and spike dynamics. There is however a need for a smarter detection system (in the present experiment it was a human) that can analyze the compensated voltage trend and identify abnormal deviations to trigger a preemptive discharge. It should be noted that even on the last simulated dissipation points (in red on both curves Fig. 8, as derived from [START_REF] Badel | Detection and Protection Against Quench/Local Thermal Runaway for a 30 T Cryogen-Free Magnet[END_REF]) there is enough time to trigger a discharge and prevent the thermal runaway to occur, even taking into account the discharge speed constrains of a full-size insert with a background magnet.

V. CONCLUSION

The robust REBCO insert coil design proposed in earlier work was tested for the first time with a large scale prototype. The combination of several key features such as two-tape bundle conductor, edge impregnation and sensitive thermal runaway onset detection could be validated up to current densities of 244 A/mm 2 and Hoop stress up to 350 MPa with no damage, the prototype generated more than 25 T in a 14 T background field with 20 pancakes. The mechanical stress appeared close to simulated values. The sensitive detection system implemented appear able to give sufficient advance warning of dissipating zone so that thermal runaway could be prevented, though such runaway was not observed due to the large margin and lack of defects. Future works in conduction cooling at higher temperature will be aimed at reducing the margin to reach dissipation and further test the system.

Fig. 1 .

 1 Fig. 1. Winding model schematics.

Fig. 2 .

 2 Fig. 2. Estimated Hoop stress at 300 A under 14 T in the upper half of the 20 pancake stack. Hoop stress exists in the flanges and mandrel but is too low to appear on scale.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Estimated Hoop Stress across the middle pancakes at 300 A under 14 T. Rin Rout

Fig. 5 .Fig. 7 .

 57 Fig. 5. Dissipative Signal Detection Bridge with 5 channels.

Fig. 6 .

 6 Fig. 6. Compensated voltages (bridge outputs) during self-field ramp up to 300 A at 0.4 A/s. In dotted line a +6 mV offset over the trend of the SP13-16 group

TABLE 1

 1

	WINDING CHARACTERISTICS
	REBCO Tape (Fujikura)	EuBa2Cu3Oy / BaHfO3
	No. of bundled tapes	2 (face-to-back)
	Tape width	4 mm
	Inner diameter (mm)	68.0
	Outer diameter (mm)	266
	Turn per pancake	279
	No. of pancakes	20
	Max source current (A)	300
	Self-inductance (H)	2.63
	Conductor current density	244
	(A/mm2)
	Peak hoop stress @ 300 A	346 MPa
	(under 14 T back)
	Total tape length	5.8 km
	Center Field	25.25 T
	(under 14 T back)

K. Takahashi, T. Okada, A. Zampa, and S. Awaji are with the High Field Laboratory for Superconducting Materials, Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan G. Nishijima is with National Institute for Materials Science, Tsukuba 305-0003, Japan.T. Uto, T. Takewa, S. Hanai, S. Ioka, and J. Inagaki are with Toshiba Energy System and Solutions Corporation, Yokohama 230-0045, Japan.

ACKNOWLEDGMENT

The authors would like to thank S. Nimori of National Institute for Materials Science and K. Fushimi of Institute for Materials Research, Tohoku University for their technical assistance with the experiments.


Nastasia Henry

Alain Venditti

Gilles Dufrenot

Céline Gimet

Tomoo Kikuchi

Leonor Modesto

Céline Poilly

Xavier Raurich

On the (de)stabilization role of protectionism: Theory and evidence *

Keywords: Public debt, tariffs, small open economy, credit constraint, global and local indeterminacy JEL classification: C62, E32, F13, F34, F43, O41

To what extent protectionism affects growth and (de)stabilizes the economies? Since 2018, some countries have resorted to protectionist measures as the United States. Although the impacts of protectionism on growth have been widely explored without reaching a consensus, few has been said on its impacts on macroeconomic stability. The present paper attempts to gauge more precisely its implications using a Barro-type (1990) endogenous growth model with public debt and credit constraint where tariffs are a proxy of protectionism. Our main result is to show that when the debt level is high, and the share of foreign goods in total consumption is large enough, increasing tariffs may have a dramatic destabilizing effect generating some expectation coordination failure between multiple equilibria and the possible existence of large self-fulfilling fluctuations. We also exhibit some trade-off between tariffs and growth as tariffs are beneficial only to the low growth equilibrium which may only appear in the globally indeterminate case. We also propose some numerical illustrations confirming the destabilizing impact of tariffs in the case of the US economy. We finally propose an Event Study analysis to confront our results. While our effects appear short lasting, two quarters, we show that the implementation of protectionism destabilizes the US economy in the short run.

Introduction

The aim of this paper is to contribute to the understanding of the consequences of protectionism on macroeconomic stability and on economic growth. Indeed, starting in 2018, protectionist measures have been introduced by some countries, especially the United States. Altogether, the various sets of the tariffs increase represent 13% of their imported goods. At the same time, following the impact of the financial crisis of 2008 and the Covid crisis of 2020, the debt level of most OECD countries has dramatically increased, raising the issue of sustainability and tax reforms to find resources to decrease the structural deficits.

We follow [START_REF] Furceri | Are tariffs bad for growth? Yes, say five decades of data from 150 countries[END_REF], proxying protectionism with tariffs (i.e. tax on imported goods). In the US, on top of starting a trade war with China, the objective of Trump was clearly to promote consumption of domestic goods and thus to promote a relocalization of productive activities in the country. The expected effect was then to boost growth. In the literature, the impact of protectionism has not clearly been identified yet. As discussed in the literature review provided in the next section, the implications of protectionism are still uncleared. On one hand, its impacts on economic growth has been widely explored, without reaching a consensus. On the other hand, its implications on macroeconomic stability has been mainly disregarded.

The contribution of the paper is to focus on three types of questions. First, we want to give clues on the possible (de)stabilizing role of protectionism, especially when economies are indebted.

Second, we aim to clarify the impact of tariffs on economic growth. Finally, we want to the gauge the link between tariffs and public debt and check whether there is an interplay between tariffs and public debt.

We consider a small open economy where endogenous growth is driven by public spending as in [START_REF] Barro | Government spending in a simple model of endogenous growth[END_REF]. 1 We use the same basic formulation as in [START_REF] Modesto | Growth and instability in a small open economy with debt[END_REF] where public debt is financed through taxes and external borrowing. A collateral constraint à la Fahri and Tirole (2012) allows usto consider that borrowing on the international markets is limited, the loans provided by the rest of the world being proportional to the capital invested in the home country. Contrary to [START_REF] Modesto | Growth and instability in a small open economy with debt[END_REF], we do not consider that the representative agent derives utility from holding domestic bonds. However, we assume that he consumes a basket of goods, composed by domestic and foreign goods. The novelty of our approach is to add an international trade dimension through tax on imported goods.

We will consider two cases depending on the level of public debt: low and high public debt.

Contrary to the standard framework of a closed economy2 , we show that when public debt is low, global indeterminacy based on the existence of multiple stationary equilibria (actually two balanced growth rates) cannot be ruled out3 . Indeed, under a low debt burden, both, a low and a high balanced growth rate can be sustained. However, there is no significative impact of tariffs. 4When public debt is high instead, we provide the main conclusion of our paper. We show that there exists a unique stationary equilibrium if and only the tax on imports and/or the share of the foreign good into total utility are low enough. As soon as tariffs and the share are large enough, two balanced growth rates coexist leading to global indeterminacy, the existence of expectations coordination failures and then the possible existence of large self-fulfilling fluctuations. In this case, because the debt burden is high, the low growth equilibrium cannot be sustained anymore if there is no enough additional revenues from tariffs (for the government). But, this is no longer true when the representative household consumes a large share of the foreign good which is significantly taxed.

In such a case, the government revenues can sustain the debt reimbursement even if the growth rate remains low. We then show that under a large debt level, if the share of foreign good into consumption is large enough, increasing tariffs may have a dramatic destabilizing effect generating some expectation coordination failure between multiple equilibria and the possible existence of large self-fulfilling fluctuations.

We also provide some comparative statics exercise focusing on the impact of tariffs on the stationary equilibria. We emphasize that the high balanced growth rate is always negatively affected by tariffs while the low balanced growth rate, when it exists, is always positively affected.

This difference appears to be explained by the origin of growth. At the high steady state, growth is driven by productive spending from the government which is relatively large compared to capital.

At the low steady state, growth is driven by private capital and thus private investment instead.

Any increase of tariffs has two opposite effects: first households can dedicate less revenues to productive investment, and second, the government experiences additional resources that are used to increase public spending. Along the high growth equilibrium, the small increase of tariffs relative to the size of government spending has a limited impact on the government spending capacities that weakly increase while the tariffs strongly impacts the households' income. The first effect is then dominant and growth declines. Along the low growth equilibrium on the contrary, the first effect is dominated by the second one since the increase of tariffs generates a relatively large increase of government spending that compensates the negative impact on households income. Our results therefore suggest the existence of a trade-off between tariffs and growth. Indeed, tariffs may enhance growth for the low equilibrium while they have a negative impact on the high equilibrium.

Focusing on the local stability property, we show that the high BGP is always characterized by local indeterminacy, while the low BGP, when it exists, is always a saddle-point. In the case of a high BGP, assume that along an equilibrium path, the agents expect an increase of the growth rate.

Due to the access to international market, they may borrow, consume and invest more (preventing a crowding out effect) and they will expect a higher public spending. Then, for given tariffs, the government's revenue can increase significantly leading to an increase of public spending that generates a higher growth rate. The expectations are therefore self-fulfilling.

When it exists, the low BGP has quite different properties. The economy is now characterized by a low growth rate and thus by a low public spending to capital ratio. The credit constraint is then more tightened since the collateral needed to borrow is relatively low. The inflows of capital thus remain limited. Assume again that along an equilibrium, agents are expecting an increase of the growth rate. Being more constrained, the agents consume less and decrease their investment of productive capital. Unable to rely on tariffs in this case, since growth, consumption and tariffs' income remain low, the government does not invest enough and growth cannot increase. The expectations cannot be self-fulfilling and the equilibrium remains locally determinate.

Building on these theoretical results, we then provide some numerical illustrations and empirical evidence supporting our main conclusions. First, considering a set of six countries characterized by large levels of debt, France, Greece, Japan, Spain, United Kingdom and United States, we notice that between 2018 and 2019, the US are the only country to have significantly increased its tariffs. While all the other countries are characterized by the existence of two BGPs, we show that the US moved from one BGP in 2019 to two BGPs in 2019 after the tariffs increase. We therefore conclude that tariffs may have had a destabilizing effect on the US economy.

In a second step, we aim to find empirical evidence for this conclusion, exploiting the introduction of protectionist measures by the US in 2018. Using an Event Study approach, we document the implications of this trade shock on US volatility of growth. To precisely capture the effects of the shock on the volatility, we propose various specifications based on different definitions of macroeconomic stability. The effects appear short lasting, two quarters, suggesting thus that the implementation of protectionism destabilizes the US economy in the short run.

The rest of this paper proceeds as follows: Section 2 propose a literature review. Section 3 presents the model. In Section 4, We study the existence and possible multiplicity of BGPs, while the Section 5 is dedicated to some comparative statics and policy implications. Section 6 analyses the local stability and Section 7 provides a numerical illustration. In Section 8 we employ an empirical strategy based on an Event Study approach and the Section 9 concludes. All technical detail are relegated to the Appendix.

Related literature

Looking at the link between international trade, macroeconomic stability and economic growth, many papers in the literature focus on the role of trade openness. In particular, Nishimura et al.

(2010) consider a two-country, two-good, two-factor general equilibrium model with sector-specific externalities and show that some country's expectation-driven fluctuations can spread throughout the world once trade opens even if the other country has determinacy under autarky.5 Globalization and market integration then appear to have destabilizing effects on a country's competitive equilibrium.6 On the contrary, [START_REF] Doi | A two-country dynamic model of international trade and endogenous growth: multiple balanced growth paths and stability[END_REF] formulate a two-country endogenous growth model, which explain joint determination of long-run trade patterns and world growth rates and prove the existence and local stability of a continuum of balanced growth paths. The destabilizing effect does not hold here but the continuum of equilibria generates some indeterminacy. Some similar conclusions are found around the Compensation hypothesis [START_REF] Iversen | The dynamics of welfare state expansion: trade openness, de-industrialization and partisan politics[END_REF], [START_REF] Down | Trade openness, country size and economic volatility: the compensation hypothesis revisited[END_REF], [START_REF] Kim | Openness, external risk, and volatility: implications for the compensation hypothesis[END_REF], Ehrlich and Hern (2014)). According to this view, higher exposure to trade leads to less domestic macroeconomic stability as soon as trade partners share risks. The latter induces a higher demand for compensation through more transfers. Open countries expand security programs that inflate public expenditures, making countries even more vulnerable to shocks. In the same vain, [START_REF] Krugman | Lessons of Massachusetts for EMU[END_REF] shows that it may be explained by more specialization (i.e. geographical concentration of an industry). Following countries tend to specialize more their production, being then more subject to regional shock. On the empirical side, some evidence suggest that the link between trade openness and macroeconomic stability (i.e. growth volatility) is not straightforward where other factors play a role. Bejan (2004) emphasizes the role of the government size in an Econometric analysis based on 111 countries. Developed and developing countries exhibit different patterns. Trade openness allows to smooth volatility for developed countries whereas developing counties experience more volatility. [START_REF] Jansen | Income volatility in small and developing economies: export concentration matters[END_REF], [START_REF] Cavallo | Does openness to trade make countries more vulnerable to sudden stops, or less? Using gravity to establish causality[END_REF] shed light on other factors, as export concentration and product diversification for example.

Concerning the impact of trade on growth, again many papers in the literature considers the role of trade openness. [START_REF] Ho | Financial market globalization and growth with interdependent countries[END_REF] examines the effect of externalities on the consequences of financial market globalization in a two-country growth model augmented with domestic credit market imperfections. He finds that depending on the externalities formation financial market globalization can improve growth at the world level, or in the rich country only, and may in some cases imply that both the rich and the poor countries become locked in a stage with no meaningful growth. For the impact of tariffs on the growth rate the results are most of the time not conclusive. [START_REF] Osang | Import tariffs and growth in a small open economy[END_REF] consider a small open economy where growth is endogenously driven by human capital accumulation. They examine the effects of an unanticipated increase in one of the tariff rates under different replacement regimes: a lump-sum transfer (LST) or an investment tax credit (ITC). An increase in the tariff of the consumption good is shown not to affect growth in the LST scenario while it positively affects growth under an ITC as the accumulation of capital is accelerated. 7 Naito (2003) examines how a revenue-neutral tariff reform affects growth in an endogenous growth small open economy model with two final goods. In contrast to the previous paper, he argues that tariff-reforms have ambiguous effects on growth, which depend on the pattern of trade and the elasticities of substitution between the inputs and consumption of final goods. Closer to our framework, [START_REF] Osang | Differential tariffs, growth, and welfare in a small open economy[END_REF] also analyze the effects of consumption and investment tariffs on growth. However, conversely to the previous work, they develop an endogenous growth model in which the economy faces restricted access to the world capital market. A higher consumption tariff, by reducing the growth rate of consumption, is shown to have a negative impact on the long-run growth rate. On the empirical side, Furceri et al. (2020) use a local projection method on a data set composed by 150 countries over the period 1963-2014 and emphasize the detrimental effect of protectionism on economic growth. Following the implementation of protectionist measures, economy experiences a rise in unemployment and inequalities, together with a significant decrease in labor productivity. All in all, long run growth declines.8

Theoretical framework

Our framework builds on [START_REF] Barro | Government spending in a simple model of endogenous growth[END_REF] model where production benefits from externalities due to public spending. We consider a decentralized, continuous time intertemporal model of a small open economy composed by three agents: a large number of identical competitive firms, a constant population of identical infinitely lived households and a government. Firms and households operate in competitive markets, they are price takers. The government levies import tariffs on all imported goods, taxes on the global output of the country and issues public debt. Tariffs, tax revenues and debt are used to produce a public good affecting the aggregated production function to maintain its budget balanced at each period of time. The country imports a consumption good. We assume that the country can borrow on the international market subject to a borrowing constraint based on its domestic capital as a collateral. Since we consider small open economy, the price of imported good is taken as given. All the prices are expressed in units of the domestically produced good, the numeraire good.

Production

We consider a perfectly competitive economy where the final output y is produced using capital k and labor supplied in one unit. As in [START_REF] Barro | Government spending in a simple model of endogenous growth[END_REF], the production benefits from an externality due to public spending G, and is given by y = k s G 1-s . Public spending is thus the driver of endogenous growth.9 The rental rate of capital r(t) and the wage rate w(t) satisfy:

r(t) = sx(t) 1-s
(1)

w(t) = (1 -s)x(t) 1-s k(t), (2)
where x ≡ G/k.

Households

The size of population is normalized to one.

rd(t) ≥ -θ(1 -τ )r(t)k(t) (4)
B h (t) ≥ 0.

(

) 5
where ρ > 0 corresponds to the discount rate. We normalize the price of Home good to unity and we denote P * the price of imported good. 10 Considering the per capita net foreign asset (NFA) expressed in terms of foreign goods, we derive that d = P * d representing the NFA expressed in domestic good. τ is the tax rate on income which is assumed to be constant and such that τ ∈ [0, 1).

The international interest rate is denoted by r, expressed in domestic goods and we assume that it is constant and strictly positive. We follow the same type of formulation as [START_REF] Farhi | Bubbly liquidity[END_REF] for the borrowing constraint, where θ ∈ [0, 1] is a parameter that captures credit market imperfection. 11

In our framework, the portfolio decisions of the households are based of three assets: domestic public debt, physical capital and foreign assets. These assets are imperfect substitutes, letting the residents not indifferent between holding international asset and domestic public debt. 12 While, foreigners are indifferent between both assets since they offer the same return, r.

Besides this portfolio choice, households consume to gain utility. They consume a single consumption good that is a composite of the domestic and the foreign consumption goods, denoted c h (t) and c f (t) respectively. As [START_REF] Osang | Import tariffs and growth in a small open economy[END_REF], the consumption good c(t) is expressed as follow:

c(t) = c f (t) α c h (t) 1-α . (6)
with α ∈ [0, 1], the share of imported foreign goods in total consumption. The consumption bundle is a combination of foreign and domestic goods, where the two goods are imperfect substitutes. In our model, we introduce tariffs τ c that are imposed on the foreign composite. We express the total consumption spending of the household as:

P c (t)c(t) = c h (t) + (1 + τ c )P * c f (t), (7)
Maximizing (6) subject to (7), leads to

c h (t) = (1 -α)P c (t)c(t) and c f (t) = αP c (t)c(t) (1 + τ c )P * . (8)
which implies a constant price P c such that

P c (t) = 1 α α 1 1 -α 1-α (1 + τ c ) α P * α . ( 9
)
Our model is build on a unique dynamical equation corresponding to the budget constraint of the representative household. Therefore, it is more convenient to solve the model to use the standard method of calculus of variations based on the consideration of the Euler equation.

Let us then introduce the following Lagrangian:

L = e -ρt ln((1-τ )(r(t)k(t)+w(t))+rd(t)+rB h (t)-k(t)-ḋ(t)-Ḃh (t)) Pc(t) +λ(t)[rd(t) + θ(1 -τ )r(t)k(t)] + µ(t)B h (t).
e -ρt (1 -τ )r(t) P c c(t) + λ(t)(1 -τ )θr(t) = e -ρt P c c(t) ρ + ċ(t) c(t) (10) e -ρt P c c(t) + λ(t) r = e -ρt P c c(t) ρ + ċ(t) c(t) (11) e -ρt r P c c(t) + µ(t) = e -ρt P c c(t) ρ + ċ(t) c(t) (12)
Any solution needs also to satisfy the transversality conditions:

lim t→+∞ e -ρt ω(t)/c(t) = 0, with ω(t) = {k(t), d(t), B h (t)} (13)
Using ( 10) and ( 11), the Lagrange multiplier associated to the borrowing constraint is given by:

λ(t) = e -ρt c(t) (1-τ )r(t)-r r-θ(1-τ )r(t) , (14)
and we easily get a condition that ensures λ(t) > 0:

(

1 -τ )r(t) > r > θ(1 -τ )r(t). (15)
Glancing out the second Lagrange multiplier µ(t), we easily point out that it is strictly positive.

Using indeed Kuhn Tucker conditions, λ(t), ( 11) and ( 12), we find that λ(t)r = µ(t), but from ( 15)

and the return of international asset, we obtain that µ is strictly positive. Hence, we prove from Kuhn Tucker conditions that domestic agents do not hold any domestic bonds (i.e. B h (t) = 0), and thus all public debt is held only by foreigners, i.e. B(t) = B f (t). This result is explained by the fact that, since the borrowing constraint is binding, (1 -τ )r k (t) > r, the domestic public asset is strictly dominated by capital.

We focus on configurations where the credit constraint expressed in ( 4) is binding (i.e. condition ( 15) is satisfied). Substituting ( 14) in ( 10), we obtain the consumption growth rate:

ċ(t) c(t) = (1-τ )r(t)(1-θ) 1-θ(1-τ )r k (t)/r -ρ. (16)
In ( 16), we can remark that the assets are not perfect substitutes in this case. The growth rate of consumption is not constant, whereas if the assets would have been perfect substitutes, the consumption growth rate will be constant and equal to r -ρ.

As mentioned previously, every borrowing on the international market is subject to a collateral.

Using ( 16), we express the expected return of capital as:

(1 -τ )r(t)(1 -θ) 1 -θ(1 -τ )r(t)/r . ( 17
)
The marginal benefit of investing one unit of capital is equal to (1-τ )r(t)(1-θ) while the expected cost of investing one unit of capital is given by 1 -θ(1 -τ )r(t)/r. One key feature of our model is the imperfect substituability between two assets. It is noteworthy that capital used in production has different return than the international asset.

Government

The government levies tax on production and on imported goods and issues debt to finance public spending G. The government budget constraint is then given by:

Ḃ(t) = G(t) + rB(t) -τ (w(t) + r(t)k(t)) -τ c P * c f (t), (18)
where B(t) corresponds to the newly-issued government bonds, G(t) the amount of public investments in time t and rB(t) the debt repayment.

Since the last financial crisis, debt stability became a major concern for governments. To reduce the debt-to-GDP ratio, some reforms have been led as the Stability and Growth Pact (SGP) for example. 13 We replicate these types of reforms assuming that the government adjusts its debtto-GDP ratio according to a specific rule. As in [START_REF] Minea | Dette publique, croissance et bien-être : une perspective de long terme[END_REF], we denote b ≡ B/y as the ratio of debt over GDP, where the public debt must converge to a certain level b * , using the following rule:

ḃ(t) b(t) = -ϕ 1 -b b(t) , with ϕ. > 0 (19)
In ( 19), two policy parameters allow the government to design its policy: b and ϕ. Firstly, b

represents the target to be reached such that any difference with this threshold requires a debt adjustment. The adjustment of public debt is calibrated with the second policy parameter: ϕ.

Intertemporal equilibrium

We describe now the intertemporal equilibrium. Let us denote v = c/y the consumption as a proportion of GDP.

Lemma 1 The intertemporal equilibrium is determined by the following three dynamical equations:

ẋ(t) x(t) = r-θ(1-τ )sx(t) 1-s r x(t) s -τ b(t) +r- ατc (1-α)(1+τc ) v(t) b(t) +ϕ(1-b * b(t) ) 1-s - x(t) 1-s [(1-τ)(1-θs)-v(t) 1-α ] 1-s , v(t) v(t) = r(1-τ )sx(t) 1-s (1-θ) r-θ(1-τ )sx(t) 1-s -ρ -x(t) s -τ b(t) -r + ατc (1-α)(1+τc) v(t) b(t) -ϕ 1 -b * b(t) , ≡ V (b(t), x(t), v(t)) ḃ(t) b(t) = -ϕ 1 -b * b(t) , with ϕ > 0. (20)
Proof. See appendix 10.1

We have a three-dimensional dynamic system which involves three variables: b(t), x(t) and v(t).

Among these three variables, one is pre-determined, b(t) and the two remaining, x(t) and v(t), are forward. Any intertemporal equilibrium path also needs to satisfy the transversality conditions (described in Section 3.2).

4 The balanced growth path (BGP): uniqueness versus multiplicity

In this section, we analyse the conditions for the existence of a unique or multiple BGPs. We show that the results are driven by three elements. In the first place, the share of foreign goods in total consumption and tariffs are central in the existence of Balanced Growth path. Depending on their values, they promote multiplicity. We then shed light on the importance of the interest rate that have a strong impact on the characterization of the BGP.

A Balanced Growth path is a steady state of the dynamical system (20), i.e. is a stationary solution (b,v,x) solving ḃ(t) = v(t) = ẋ(t) = 0. Along the BGP, the following equality is satisfied:

It follows that at the equilibrium, the growth rate should be propertional to the total deficit

Γ(x) = H(x), (21)
where:

Γ(x) ≡ (1 -τ )sx 1-s (1 -θ) 1 -θ(1 -τ )sx 1-s /r -ρ, (22)
H(x) ≡ x s -τ b * + r - ατ c (1 -τ )(1 -s)x 1-s + ρ r-θ(1-τ )sx 1-s r (1 + τ c )b * x 1-s . ( 23
)
Using ( 1), (2) and the binding (4) we necessarily have x < x < x with:

x ≡ r (1 -τ )sθ 1 1-s and x ≡ r (1 -τ )s 1 1-s . ( 24
)
We assume

Assumption 1 r > ρ and θ > s/(2 -s).

The first part of the assumption ensures that the growth rate ( 22) is positive whatever the value of θ, while the second part allows to simplify the analysis, ensuring that Γ(x) is a convex function.14

Replacing x by the upper bound x and x, yields respectively to:

r ≡ τ 1-s s (1 -τ )s and r ≡ τ 1-s s (1 -τ )θs = rθ. ( 25
)
We will consider in the rest of the paper that r ∈ (r, r).

Beside the analysis of existence and uniqueness of the steady state, we need also to determine whether the steady state is characterized by a primary surplus or rather by a primary deficit as this property will be partially related to the number of equilibria. A primary surplus (deficit) is obtained if and only if τ y+τ c P *c f -G > (<)0. It is therefore immediate to derive that a stationary solution x features a primary surplus if

x s -τ -ατ c v/[(1 -α)(1 + τ c )] < 0, and a primary deficit if x s -τ -ατ c v/[(1 -α)(1 + τ c )] > 0.
4.1 The case of large debt: a fundamental role of tariffs

In the following Proposition, we first consider the case of a high enough debt-output ratio. We

show that there is a unique BGP, as illustrated in Figure 1, if the share α of imported foreign goods in total consumption is low enough, or if the tariff τ c on imported good is low enough. On the contrary, for large values of both α and τ c , two BGPs may occur. Moreover, depending on the value of the interest rate r, the steady states can be characterized by a primary deficit or surplus:

Proposition 4.1 Under Assumption 1, let r > ρ and r ∈ (r, r). Consider the critical values:

b ≡ τ - r (1-τ )s s 1-s ρ < b = b + (1 -τ )(1 -s) ρ (26)
and assume b * ∈ ( b, b). Then, there exist 1 > θ > θ > 0 such that when θ ∈ (θ, θ), the following cases hold:

1. There is a unique stationary solution x ∈ (x, x) of ( 22) in the following cases:

(a) for any τ c ∈ (0, 1) if α ≤ ᾱ(θ) with:

ᾱ(θ) ≡ rρ[ b-b] (1-τ )[(1-s)r+sρ(1-θ)] (b) if α > ᾱ(θ) and τ c ≤ τc (θ) with: τc (θ) ≡ ρr[ b-b] (1-τ )[(1-s)r+sρ(1-θ)][α-ᾱ(θ)]
Moreover, there exists r 0 ∈ (r, r) such that the steady state is characterized by a primary surplus (deficit) for r < (>)r 0 .

2. There exists ϵ > 0, such that there are two stationary solutions x 1 , x 2 ∈ (x, x) of ( 22), with

x 1 < x 2 , if α > ᾱ(θ) and τ c ∈ ( τc (θ), τc (θ) + ϵ). Moreover, x 1 is always characterized by a primary surplus, while x 2 is characterized by a primary surplus when r ∈ (r, r 0 (ϵ)) and a primary deficit when r ∈ (r 0 (ϵ), r) with r 0 (ϵ) close to r 0 .

Proof. See appendix 10.2. shown by Minea and Villieu (2013) that multiplicity of BGPs is ruled out under a log linear utility function in consumption. Indeed, if agents expect an increase of public expenditures, this will induce a higher future income. To finance this increase in public spending, a larger debt emission is required, which crowds out private investment having a negative impact on future income. Then, expectations may not be self-fulfilling and uniqueness is obtained.

In contrast, in our framework, multiplicity is driven first by the coexistence of two key mechanisms, (i) the inflow of international assets and (ii) the existence of an investment multiplier due to the credit constraint with collateral, and second by the existence of tariffs. As a small open economy can import international funds, such a crowding out effect is no longer relevant. Therefore, a higher public spending may now be compatible with an increase of productive investment.

The resulting effect on growth is magnified by the collateral role of capital which generates an investment multiplier. In this case, an expected increase of public spending can be self-fulfilling because of higher future income and growth, which sustain a long run equilibrium with larger public spending. However, in [START_REF] Modesto | Growth and instability in a small open economy with debt[END_REF], it is shown that this mechanism strongly depends on the size of public debt. Indeed, in the case of high debt, it is shown that no matter what is the value of the preference parameter for domestic debt, uniqueness of the BGP holds.

As Proposition 4.1 makes clear, the international trade dimension is central in our framework. Indeed, when public debt is high enough, the share of foreign goods and tariffs drive the existence of two BGPs. We note that when domestic households do not consume an important share of goods, or when tariffs are low enough, a unique BGP exists. This configuration is in a sense similar to the framework of [START_REF] Modesto | Growth and instability in a small open economy with debt[END_REF].

To understand why uniqueness occurs under a large debt-output ratio as soon as the tariffs income is limited,15 we rewrite the government budget constraint (20

) as Γ(x)b * = G/y -τ + rb * - ατc (1-α)(1+τc) v.
When τ c and/or α is low, for a too low x, growth is not sufficient to allow the repayment of a high level of debt. Hence, a large debt-output ratio is not compatible with the government budget constraint and cannot be sustained, so that a too low steady state cannot exist. We then recover the result of [START_REF] Modesto | Growth and instability in a small open economy with debt[END_REF].

On the other hand, when tariffs and the share of foreign goods into total consumption are high enough, a second BGP exists, which is characterized by a lower growth. Henceforth, when two BGPs coexist, the economy may be located at either the low (x 1 ) or high growth (x 2 ) steady state. In such a configuration, households consume an important share of goods that are highly taxed. The government therefore earns some extra revenues that allow to sustain a low equilibrium characterized by a low growth rate and a large debt at the same time. In such a configuration there is a potential of expectations coordination problem. Indeed, the crowding out effect on private investment generated by the large debt can be more than compensated by government expenditures allowed by tariffs income. It follows that even if agents expect a low growth, the related equilibrium can be self-fulfilled as they expect the government will be able to sustain the large debt burden from the tariffs income. We then conclude that when the debt level is high, and the share of foreign goods in total consumption is large enough, increasing tariffs may have a dramatic destabilizing effect generating some expectation coordination failure between multiple equilibria and the possible existence of large self-fulfilling fluctuations.

We now discuss the properties of the steady state in terms of primary deficit/surplus. In the case of a unique steady state we find a primary deficit/surplus when r is high/low. A sufficiently high interest rate r > r 0 pushes down growth since ∂Γ(x)/∂r < 0. To sustain the reimbursement of the high level of public debt, a sufficient high level of growth is however needed, which means that public spending should be high enough. This explains that there is a primary deficit. Of course, when r < r 0 , we have exactly the opposite situation.

When two BGPs exist, we argue here that the high steady state x 2 has the same properties as the one in the case of uniqueness. On the other hand, x 1 is always characterized by a primary surplus. This steady state is characterized by a lower growth but at the same time by low public spending. Since we are in a case of large tariffs, the total income of the government is large enough to guarantee that the low steady state is characterized by a primary surplus no matter what is the value of the interest rate.

The case of low debt: tariffs do not really matter

Let us focus now on the case where public debt is low enough, i.e. b * < b. Unlike the previous case, we show that two BGPs always exist.

Proposition 4.2 Under Assumption 1, let r ≡ τ 1-s s (1 -τ )s θ and b ≡ r (1-τ )s θ s 1-s -τ r 1-θ θ-θ -ρ
For given (b * , θ, r), there exist ρ 0 > 0, θ ∈ (θ, 1) and r ∈ [r, r] such that if ρ < ρ 0 , r ∈ ( r, r) and b * < min{ b, b}, there are two stationary solutions x 1 , x 2 ∈ (x, x) of ( 22), with x 1 < x 2 , in the following cases:

1. for any τ c ∈ [0, 1], if α ≤ α(θ) with:

α(θ) ≡ [ρ0-ρ]( b-b * ) (1-τ ) 1-s+ρs θ-θ r 2. if α > α(θ) and τ c ≤ τc ( θ), with: τc ( θ) ≡ [ρ0-ρ]( b-b * ) (1-τ ) 1-s+ρs θ-θ r (α-α(θ))
Moreover, x 2 is always characterized by a primary deficit, and there exists r 0 ∈ [r, r] such that x 1 is characterized by a primary deficit when r ∈ [r, r 0 ] and a primary surplus when r ∈ [r 0 , r].

Proof. See appendix 10.3

Interestingly, when public debt is low enough, two BGPs exist. To understand this result consider again the government budget constraint (20) rewritten as

Γ(x)b * = G/y -τ + rb * - ατc (1-α)(1+τc) v.
If the debt-output ratio is low enough, the government budget constraint is sustainable even with a low growth rate, which explains the existence of the low steady state (x 1 ). In contrast, at a high steady state (x 2 ), the growth rate is high enough to sustain the government budget whatever the level of debt. Given a sufficiently low level of public debt, the multiplicity of BGPs is essentially explained by the coexistence of the same two key mechanisms: (i) the inflow of international assets and (ii) the existence of an investment multiplier due to the credit constraint with collateral. Compared to the case with a high debt, the role of tariffs income is here not crucial as it just appears as a complement of resources allowing to support the two steady states. 16 We clearly observe indeed that even if α = 0, the two steady states still exist as initially shown by [START_REF] Modesto | Growth and instability in a small open economy with debt[END_REF]. However, as we will see in the next section, tariffs have a crucial role on the value of the long-run growth rate.

5 Comparative statics: a trade-off between tariffs and growth

We now provide comparative statics. We focus on the behavior of the equilibrium when tariffs are modified. Consider the two equations characterizing the intertemporal equilibrium, namely Γ(x)

and H(x). As we have already noticed before, Γ(x) corresponds to the growth rate of the economy.

Let us now turn to the economic interpretation of H(x). Multiplying H(x) by b * , gives:

H(x)b * = x s -τ - ατ c (1 -α)(1 + τ c ) v + rb * . (27)
In ( 27), we easily recognize the primary balance and the debt burden, rb * . We finally argue that (27) expresses the total public expenditures net of taxes proceedings. It follows therefore that at the equilibrium, the growth rate should be proportional to the total deficit of the government.

Notice that only H(x) depends on tariffs and its impact appears to be negative. We formulate the following proposition where we consider the case of two BGPs since in the case of uniqueness the same result as the high steady state x 2 is obtained:

Proposition 5.1 Under Assumption 1, Propositions 4.1 and 4.2, for a given τ c ∈ [0, 1] :

∂x 1 ∂τ c > 0 and ∂x 2 ∂τ c < 0
Proof. See Appendix (10.4)

In Figure 2, we depict an increase of tariffs.

The increase of tariffs allows to decrease the deficit of the government. However it has different impacts on growth depending on the amount of public spending relative to capital. On one hand at

x x ′ 1 x 1 x 2 x ′ 2 x x Γ(x) H(x) H ′ (x)
Figure 2: The impact of an increase of τ c

Note: In this figure we display the effects of increasing tariffs. Dotted curve represents H(x) following an increase of tariffs, while the solid H(x) curve represents H(x) before the increase of tariffs. We easily remark that steady states move in opposite directions following tariffs' increase.

the low steady state x 1 with a relatively low public spending over capital, any increase of tariffs is pro-growth. On the other hand at the high steady state x 2 , with a relatively large public spending over capital, this increase is detrimental for growth. We argue that this difference comes from the origin of growth. At x 2 , growth is driven by productive spending from the government which is relatively large compared to capital. On the contrary, at x 1 , growth is driven by private capital and thus private investment. Following an increase of tariffs two effects counteract. Firstly, households can dedicate less revenues to productive investment and this has a negative impact on growth.

At the same time, government experiences additional resources that are used to increase public spending, boosting growth. The source of growth now matters. At x 2 , growth comes mainly from productive spending. But the small increase of tariffs relative to the size of government spending has a limited impact on the government spending capacities that weakly increase while the tariffs strongly impacts the households' income. Therefore, the first effect dominates the second one and growth declines. At x 1 on the contrary, the first effect is dominated by the second one. Indeed, even though growth is driven by private investment, the increase of tariffs generates a relatively large increase of government spending that compensates the negative impact on households income. Our results therefore suggest the existence of a trade-off between tariffs and growth. Indeed, tariffs may enhance growth for the low equilibrium while they have a negative impact on the high equilibrium.

Local stability analysis

We now investigate the local stability properties of the BGPs. Let us consider the three-dimensional dynamic system as given by equations ( 20) in Lemma 1 and that can be written as follows

ḃ(t) = -ϕ(b(t) -b * ) v(t) = V (b(t), x(t))v(t) ẋ(t) = X(b(t), v(t), x(t))x(t). (28)
For the local stability analysis, we linearize this three-dimensional dynamic system given around the steady state. From the eigenvalues of the Jacobian matrix, we determine the properties of the steady states. It is worth noting that -ϕ is a negative eigenvalue. 17 iii) When ρ < ρ 0 , r ∈ ( r, r) and b * < min{ b, b}, if α ≤ α(θ), or α > α(θ) and τ c ≤ τc ( θ), then the high steady state x 2 is locally indeterminate and the low one x 1 is saddle-point stable.

Proof. See Appendix (10.6)

When the balance growth path is unique, the equilibrium is locally indeterminate. The main result of this paper is then to show that when debt is large, the fact that domestic households consume an important share of foreign goods under large enough tariffs is the key ingredient to explain global indeterminacy, i.e. the existence of expectations coordination failures that may lead to the occurrence of large sunspot fluctuations.

Let us present the mechanisms at stake in the two cases: uniqueness and multiplicity. We first discuss the existence of self-fulfilling expectations when BGP is unique. We consider an equilibrium along which agents suddenly formulate expectations about a possible higher growth rate. Since assets are imperfect substitutes and due to the access to international market, agents may borrow, consume and invest more (preventing a crowding out effect). The investment multiplier allows to achieve a higher growth rate. Moreover, since the consumption of the foreign good increases, tariffs income are quite large, improving the government's revenue dedicated to increase its spending.

All these mechanisms highlight the possibility for self-fulfilling expectations leading to multiple transitional paths and expectation-driven fluctuations when the BGP is unique.

When two BGPs coexist, a low steady state x 1 coexists with a high one, x 2 . Nevertheless,

x 2 keeps the same local indeterminacy property as in the uniqueness case as all the mechanisms previously mentioned remain exactly the same. We explain the mechanisms are stake for the lowest steady state. The economy is now characterized by a low growth rate and thus by a low government spending to capital ratio. The credit constraint is then more tightened since the collateral needed to borrow is relatively lower. Assume again that along an equilibrium, agents are expecting an increase of the growth rate. Being able to get less inflows of capital, they cannot increase significantly their consumption and their investment in productive capital. At the same time, the government becomes more constrained, being unable to rely on sufficient revenues, since growth, consumption and tariffs' income remain low. As a result, the government does not invest more and growth cannot increase. Therefore, the expectations cannot be self-fulfilling.

However, due to expectations coordination failures already mentioned earlier, the existence of global indeterminacy with two steady states leads to possible large fluctuations around them.

Numerical illustration

This section is devoted to a numerical illustration of Properties presented in Section 4. The data used have been collected on OECD, IMF and the World Bank websites for the year 2019 (and 2018 for the tariffs rate) and for six countries: France, Greece, Japan, Spain, United Kingdom and United States. 18 We select 2019 data to model the impacts of US protectionism measures as we can notice that contrary to all the other countries, the US have strongly increased their tariffs between 2018 and 2019. This year then appears as the most suitable one to explore the impacts of tariffs increases. For the calibration step, we follow [START_REF] Modesto | Growth and instability in a small open economy with debt[END_REF], where θ = 0.06 and ρ = 0.01. For each separate country, we use the specific data of We conclude that with the 2019 data all the countries are characterized by the existence of two equilibria. We find the same result for all the countries except the US using the tariffs of 2019.

It is worth noting in

Empirical strategy

Our numerical exercise suggests that because of the strong increase of their tariffs rate between 2018 and 2019, the US may have been subject to a greater volatility of GDP in 2019. We need however to check whether such a conclusions could be observed in the data. We then employing an

Event study approach ton confirm our claim. Event study methodology appears to be suitable to understand if and how the increase of tariffs by the US have generated macroeconomic (in)stability.

Data and methodology

In this section, we describe the source of data and the construction of the growth volatility (i.e. macroeconomic stability).

Data

In this section, we focus on the United States. The data used in this exercise are quarterly data from 2000Q1 to 2020Q4, collected on IMF, OECD and World Bank websites. We collected: the growth rate, tariff rate, fiscal deficit, public debt, interest rates, inflation rate and exchange rate volatility (nominal). In this paper, we document the impacts of the increase of tariffs in 2019 by the US on macroeconomic stability. We assume that volatility of growth can be a proxy for macroeconomic stability [START_REF] Ramey | Cross-country evidence on the link between volatility and growth[END_REF][START_REF] Sangnier | Does trust favor macroeconomic stability?[END_REF]. We first estimate growth volatility using GARCH (Generalized Auto Regressive Conditional Heterosckedasticity) 19 .

Modeling macroeconomic stability with GARCH

We start from a simple GDP growth equation:

Y t = βX t + ε t (29) ε t ∼ N (0, σ 2 t )
We can model the volatility of growth (conditional variance of ε t ) as:

σ 2 t = ω + q i=1 α i ε 2 t-1 + p i=1 β j σ 2 t-j ( 30
)
where

ε 2 t = ω + α(L)ε 2 t + V t .
In (30), α represents how persistent the time series is, ε 2 t-1 corresponds to the contribution of disturbances of the previous quarter to the actual volatility, β represents how persistent the 19 GARCH model are often used to model volatility in financial markets. This type of models allows to deal with heteroskedasticity that could alter the conclusions drawn since other models tend to create observations clusters 22 conditional variance is, L a lag operator and finally σ 2 t-j corresponds to the contribution of past volatility. Moreover α i and β j are constant and ≥ 0. If,

q i=1 α i + p j=1 β j ≤ 1.
We have a positive and stable conditional variance of ε t , we also affirm that ε t asymptotically stationary to the second order. We can define V t , the variance as :

V t ≡ ε 2 t -σ 2 t ,
we impose in (29) that disturbances are distributed as a Gaussian law. In fact, the disturbances do not follow necessarily a Normal distribution, particularly when time series has fat tail, Student-t error distribution appears more suitable. For this reason, we choose to simulate GARCH model with four types of error distributions: Normal, skew-Normal, Student-t and skew-Student. To choose the appropriate distributions we implement tests 20 and use the Akaike information criterion.

After investigations 21 , we choose the Skewed-Student distribution for the error terms.

The Event Study approach

Now, we start the Event study to understand the impacts of the increase of tariffs on US growth volatility. But, let us briefly the US context before the presentation of the Event Study.

The US context -In January 2018, the United States implemented five sets of measures representing 12.6% of the total goods imported by the US. The main goods targeted by those measures are semi conductors, steel, aluminium, etc,. from China, European Union, South Korea and Mexico. The tax on semi conductors from China for example have been increased by 30% for example.

In tables 3 22 and 4 23 are reported the tariffs rate on primary products and on manufactured products from 2014 to 2020. We exploit this trade shock using an Event study to analyze whether it generated more volatility. Let us now describe the central specification.

The baseline specification -As mentioned in Section 8.1.1, we follow the BMS approach where fiscal and monetary factors are added. Our specification can be written as: where R t is the growth volatility for the time period t relative to the event; K t the expected (predicted) growth volatility and e t the component of growth volatility that can be considered as abnormal, unexpected, X t is the vector of control variables composed by the inflation rate, public debt, fiscal deficit, interest rates, the nominal exchange rate volatility. In (31), we easily obtain a measure of the change of growth volatility associated to the event, i.e. the abnormal growth volatility, e t from it. We then define the event window, we set it at nine quarters, including quarters before and four quarters after24 the tariffs increase.

R t = α t K t + β t X t + e t , (31

Results

This section is devoted to the presentation of the main results, firstly from the central specification (BMS approach) and then from the various robustness checks.

Main results

Figure 3 plots the results from our experiment25 .

Figure 3: Event study on US tariffs increase Note: The x-axis represents the time horizon, quarterly. On the y-axis are displayed the growth volatility. The dotted lines represent the confidence intervals. It is worth noting that the increase of tariffs starting 2018 significantly generated more volatility for at least two quarters. After three quarters, the effects do not seem significant anymore.

Following the increase of tariffs in January 2018, the US economy experiences more volatility of growth. Nonetheless, we remark that the effects are short-lasting, two quarters. These results could be associated to different behaviors, from the production sectors and or households like a reallocation of production factors or some modifications of the consumption bundle.

Robustness Checks

We perform various robustness checks to assess the stability and the validity of our results. To do so, we first add control variables and then we change the time window.

Control variables-In our central specification, we control for some factors according to the BMS approach. In this subsection, we add new control variables in the specification from others as the Narrow Monetary Stability approach or an approach closely related to the five Maastricht convergence criteria. The Narrow Monetary Stability (NMS) focuses solely on the value and stability of inflation (as discussed by [START_REF] Fischer | Macroeconomic stability and growth[END_REF]. According to this view, macroeconomic stability is dependent on maintaining inflation at a stable and desirable level. In the latter, macroeconomic stability is promoted only if the criteria are satisfied26 (see among others [START_REF] Obstfeld | The great depression as a watershed: international capital mobility over the long run[END_REF])

We obtain similar results: the increase of tariffs generates more growth volatility for two quarters, whatever the approach used.

Time window-As second robustness check, we also vary the time window to assess the robustness of our results. We first increase to 11 and 13 quarters the time window. Then, we decrease it to 7 quarters. We do not observe significant differences compared to the main specification with 9 quarters.

Conclusion

In this paper, we have considered a small open economy with endogenous growth driven by public spending where agents consume some foreign good which is subject to tariffs taxation and are subject to some borrowing constraint. The government finances its expenditures through taxes on income, imported goods and by the issue of debt. Tariffs are here considered as proxy of protectionism.

We have first proved that when debt is sufficiently high, the trade dimension is central. Indeed, when agents consume a sufficiently large share of the foreign good and tariffs are high enough, two BGPs coexist while uniqueness holds without foreign good consumption. The high BGP is always negatively affected by some tariffs increase and is always locally indeterminate. On the contrary, the low BGP is always positively affected by some increase of tariffs and is always locally determinate. We then shown that tariffs may have a dramatic destabilizing effect generating some expectation coordination failure between multiple equilibria and the possible existence of large self-fulfilling fluctuations. We also exhibit some trade-off between tariffs and growth as tariffs are beneficial only to the low growth equilibrium which may only appear in the globally indeterminate case.

Building on these theoretical results, we provide some numerical illustration and empirical evidence supporting our main conclusions. Exploiting that the US are the only country to have significantly increased its tariffs in 2019, we show numerically that its economy is characterized by more volatility after this change. Moreover, using an Event Study approach, we find empirical evidence for the existence of more growth volatility for the US economy from 2019, although these effects are short-lasting.

10 Appendice

Proof of Lemma 1

Using the equilibrium prices (1) and ( 2), the constraint on public debt (19) and G/y = x s , we derive from equation ( 18)

x(t) = Ḃ(t) k(t) + τ x(t) 1-s + τcP * c f (t) k(t) -rB(t) k(t) (32)
Let us introduce the variable v ≡ c/y. From (8) we get

P * c f (t) k(t) = α (1+τc) Pcc(t) k(t) = α (1+τc)(1-α) v(t)x(t) 1-s
Substituting this expression into (32) and using the fact that ḃ/b = Ḃ/B -ẏ/y give the growth rate of production:

ẏ(t) y(t) = x(t) s -τ b(t) + r - ατc (1+τc)(1-α) v(t) b(t) + ϕ 1 -b * b(t) (33)
and the growth rate of consumption ( 16) can be rewritten as:

Γ(x(t)) ≡ ċ(t) c(t) = ċh (t) c h (t) = (1-τ )sx(t) 1-s (1-θ) 1-θ(1-τ )sx(t) 1-s /r -ρ. (34)
From ( 8) we also get

Pcc(t) k(t) = 1 1-α c h (t) k(t) = 1 1-α v(t)x(t) 1-s .
Moreover, a binding credit constraint (4) means that d = -θ(1 -τ )sy/r and also implies that ḋ/d = ẏ/y. Using these results and (3), we then derive the growth rate of capital: 33) and (34), we easily get:

k(t) k(t) = x(t) 1-s (1 -τ )(1 -θs) -1 1-α v(t) + θ(1-τ )s r x(t) 1-s ẏ(t) y(t) = x(t) 1-s (1 -τ )(1 -θs) -1 1-α v(t) + θ(1-τ )s r x(t) 1-s x(t) s -τ b(t) + r - ατc (1+τc)(1-α) v(t) b(t) + ϕ 1 -b * b(t) (35) Note that x = G/k = (y/k) 1 1-s . Using (
ẋ(t) x(t) = r-θ(1-τ )sx(t) 1-s r x(t) s -τ b(t) +r- ατc (1+τc )(1-α) v(t) b(t) +ϕ(1-b * b(t) ) 1-s - x(t) 1-s [(1-τ)(1-θs)-v(t) 1-α ] 1-s ≡ X(b(t), v(t), x(t)). (36)
Recalling that v ≡ c/y, we finally get:

v(t) v(t) = r(1-τ )sx(t) 1-s (1-θ) r-θ(1-τ )sx(t) 1-s -ρ -x(t) s -τ b(t) -r + ατc (1+τc)(1-α) v(t) b(t) -ϕ 1 -b * b(t) ≡ V (b(t), v(t), x(t)).
(37) 10.2 Proof of Proposition 4.1

From ( 22)-( 23) we derive:

Γ ′ (x) = r 2 (1 -τ )s(1 -s)(1 -θ)x -s [r -θ(1 -τ )sx 1-s ] 2 (38) H ′ (x) = s x s-1 b * + ρατ c (1 -s) (1 + τ c )b * x 2-s (39)
We have Γ ′ (x) > 0 and H ′ (x) > 0. We also easily see that H ′′ (x) < 0, while Γ ′′ (x) has the same sign than θ(1 -τ )(2 -s)x 1-s -r. Since x > x, Γ ′′ (x) > 0 is ensured by Assumption 1. In addition, we have:

Γ(x) = +∞ and Γ(x) = r -ρ > 0 H(x) = ( r (1-τ )sθ ) s 1-s -τ b * + r -ατc(1-τ ) (1+τc)b * r [(1 -s)r + ρs(1 -θ)] H(x) = ( r (1-τ )s ) s 1-s -τ b * + r -ατc(1-τ )(1-s) (1+τc)b *
Considering that Γ(x) is convex and H(x) is concave with Γ(x) > H(x), there is a unique solution x ∈ (x, x) if Γ(x) < H(x). When α = 0 or τ c = 0, Γ(x) < H(x) if and only if b * > b, with b as given by ( 26). Consider now α > 0 and τ c > 0.

1. (a) We easily derive that Γ(x) < H(x) for any τ c ∈ (0, 1) if α < ᾱ(θ) with

ᾱ(θ) ≡ rρ(b * -b) (1-τ )[(1-s)r+ρs(1-θ)]
Moreover, straightforward computations show that for any θ ∈ (0, 1), ᾱ 2. Assume now that α > ᾱ(θ) and τ c > τc (θ). By continuity, there exists ϵ > 0 such that if τ c ∈ (τ c (θ), τc (θ) + ϵ), there exist two stationary solutions x 1 , x 2 ∈ (x, x) of ( 21) with x 1 < x 2 .

Steady state characterization. We characterize the steady state of the economy. Considering ( 33) and (34) and solving for v yields:

v = 1-α x 1-s (1 -τ )(1 -s)x 1-s + ρ r-θ(1-τ )sx 1-s r (40)
We can express the primary deficit as:

DP (x) = x s -τ - ατ c (1 + τ c ) (1 -τ )(1 -s)x 1-s + ρ r -θ(1 -τ )sx 1-s r (41)
We easily get DP ′ (x) > 0. Consider the bounds x and x as given by (24). We get:

DP (x) = r (1 -τ )s s 1-s -τ - ατ c (1 -τ ) (1 + τ c )r [r(1 -s) + ρs(1 -θ)] < 0 (42)
since r < r. Moreover, since r > r, we have

DP (x) = r (1 -τ )sθ s 1-s -τ - ατ c (1 -τ )(1 -s) (1 + τ c ) ⪌ 0 (43)
if and only if

α ⪋ 1 + τ c τ c (1 -τ )(1 -s) r (1 -τ )sθ s 1-s -τ ≡ α(θ) (44)
We can easily derive that under b * ∈ ( b, b) we have ᾱ(θ) > α(θ) and that if θ > θ with

θ = max s 2 -s ; τ 1 -s + τ s 1-s s (45) then ᾱ(θ) < 1 if τ c > r (1-τ )sθ s 1-s -τ 1 -s + τ s - r (1-τ )sθ s 1-s ≡ τc (θ) (46)
Moreover, there exists θ ∈ (θ, 1) such that when θ ∈ (θ, θ) we have τc (θ) > τc (θ). Obviously we have τc (θ) > τc (θ) + ϵ 1. We immediately derive that in the case of a unique BGP, i.e. if α < ᾱ(θ), or α > ᾱ(θ) and τ c < τc (θ), then DP (x) > 0 and there exists x 0 ∈ (x, x) such that DP (x 0 ) = 0. It follows therefore that H(x 0 ) = r and thus the unique BGP x * is such that x * > x 0 if and only if Γ(x 0 ) < H(x 0 ) = r, i.e.

F (r) ≡ r 2 + r ρ -s(1 -τ )x 1-s 0 -θρs(1 -τ )x 1-s 0 > 0
We conclude that F (r) ⪌ 0 if and only if r 0 with

r 0 = s(1 -τ )x 1-s 0 -ρ + ρ -s(1 -τ )x 1-s 0 2 + 4θρs(1 -τ )x 1-s 0 2 ∈ (r, r) (47)
Notice that if r = r 0 , the BGP is by definition equal to x 0 . We have then proved that the unique BGP is characterized by a primary surplus (deficit) when r < (>)r 0 . 2. In the case of two BGPs such that x 1 < x 2 , i.e. when α > ᾱ(θ) and τ c ∈ (τ c (θ), τc (θ) + ϵ), notice first that the equality Γ(x) = H(x) can be equivalently written [Γ(x) -r]b * = DP (x). Since b * > b, we easily derive that when r = r 0 we have x 2 = x 0 . We conclude by continuity using the same argument as in 1. above that x 1 is always characterized by a primary surplus, while there exists r 0 (ϵ) close to r 0 such that x 2 is characterized by a primary surplus (deficit) when r < (>)r 0 .

Proof of Proposition 4.2

If b * < b we get Γ(x) > H(x). As Γ(x) > H(x), it follows that there are two steady states if there exists a x ∈ (x, x) such that Γ( x) < H( x). Let us define

x ≡ r (1-τ )s θ 1 1-s
where θ ∈ (θ, 1) ensures that x can take any value between x and x. Γ( x) < H( x) is equivalent to:

r (1-τ )s θ s 1-s -τ -b * r 1-θ θ-θ -ρ > α(1-τ )τc 1-s+ ρs( θ-θ) r 1+τc (48)
Assume from now on that b * < b and ρ < ρ 0 with b

≡ r (1-τ )s θ s 1-s -τ ρ0-ρ and ρ 0 ≡ r 1-θ θ-θ so that the right-hand side (RHS) is positive. Note that b > 0 for r > r with r = τ 1-s s (1 -τ )s θ ∈ (r, r) (49)
Assume that for a given θ ∈ (0, 1), τ c < τc (θ) and consider the value x 0 ∈ (x, x) such that DP (x 0 ) = 0. Let us then choose θ such that x < x 0 . It follows therefore that r < r 0 . Then, it is easy to check that inequality ( 48) is equivalent to

(ρ 0 -ρ)( b -b * ) > τ c α(1 -τ ) 1 -s + ρs( θ-θ) r -(ρ 0 -ρ)( b -b * ) (50)
This inequality holds for any τ c ≥ 0 if and only if α ≤ α with

α ≡ (ρ0-ρ)( b-b * ) (1-τ ) 1-s+ ρs( θ-θ) r
On the contrary, when α > α, inequality (50) holds if and only if τ c < τ c with

τ c ≡ (ρ0-ρ)( b-b * ) (1-τ ) 1-s+ ρs( θ-θ) r (α-α)
Assuming r ∈ ( r, r) and b < min{ b, b}, we then conclude that if α ≤ α, or α > α and τ c < τ c , there are two stationary solutions x 1 , x 2 ∈ (x, x), with x 1 < x 2 . Let us study now whether the steady states are characterized by a primary deficit or a primary surplus. Recall that the equality Γ(x) = H(x) can be equivalently written [Γ(x) -r]b * = DP (x). Since now b * < b, we easily derive that when r = r 0 as given by (47) we have x 1 = x 0 . We conclude by continuity using the same argument as in 1. above that x 2 is always characterized by a primary deficit, while x 1 is characterized by a primary surplus (deficit) when r > (<)r 0 .

Proof of Proposition 5.1

Since H(x) = Γ(x) at a steady state and tariffs appear only in H(x), we have:

dx dτ c = ∂H/∂τ c Γ ′ (x) -H ′ (x) (51) = - α (1 -τ )(1 -s)x 1-s + ρ r-θ(1-τ )sx 1-s r (1 + τ c ) 2 b * x 1-s [Γ ′ (x) -H ′ (x)] (52)
In then case of a unique BGP x * , we know that H ′ (x) < Γ ′ (x) and thus dx * /dτ c < 0. In the case of two steady states, the impact of the tariffs depends on the difference

H ′ (x) -Γ ′ (x). At x 1 , since H ′ (x 1 ) > Γ ′ (x 1 ), we get dx 1 /dτ c > 0, while at x 2 , since H ′ (x 2 ) < Γ ′ (x 2 ), we get dx 2 /dτ c < 0.
10.5 Proof of Proposition 5.2

Since H(x) = Γ(x) at a steady state and b * only enters H(x), we have: :

dx db * = ∂H/∂b * Γ ′ (x) -H ′ (x) (53) = - DP (x) b * 2 [Γ ′ (x) -H ′ (x)] (54)
with DP (x) the primary deficit as given by ( 41). We already know that x 1 is such that Γ

′ (x 1 ) - H ′ (x 1 ) < 0 while x 2 is such that Γ ′ (x 2 )-H ′ (x 2 ) > 0.
The case of a unique steady state corresponds to the same properties as x 2 . We then easily deduce that the sign of the primary deficit will drive the effects of increasing public debt.

In the case of a unique steady state x * , using Proposition 4.1, since Γ ′ (x * ) -H ′ (x * ) > 0, we easily derive that dx * /db * < 0 when r > r 0 while dx * /db * > 0 when r < r 0 .

In the case of two steady states we have to distinguish the configurations b * > b and b * < b.

-If b * > b we know that the low steady state is such that Γ ′ (x 1 ) -H ′ (x 1 ) < 0 and DP (x 1 ) < 0. We then get dx 1 /db * < 0. For the high steady state x 2 , since Γ ′ (x 2 ) -H ′ (x 2 ) > 0, we conclude from Proposition 4.1 that dx 2 /db * < 0 when r > r 0 while dx 2 /db * > 0 when r < r 0 .

-If b * < b we know that the high steady state is such that Γ ′ (x 2 ) -H ′ (x 2 ) > 0 and DP (x 2 ) > 0 which imply dx 2 /db * < 0. For the low steady state x 1 , since Γ ′ (x 1 ) -H ′ (x 1 ) < 0, we conclude from Proposition 4.2 that dx 1 /db * > 0 when r < r 0 while dx 1 /db * < 0 when r > r 0 .

Proof of Proposition 6

Linearizing the dynamical system (20) around a steady state (b * , v * , x * ) gives the following Jacobian matrix

J =       -ϕ 0 0 V 1 (b * , v * , x * )v * V 2 (b * , v * , x * )v * V 3 (b * , v * , x * )v * X 1 (b * , v * , x * )x * X 2 (b * , v * , x * )x * X 3 (b * , v * , x * )x *      
We easily derive

D = -ϕ [V 2 (b * , v * , x * )X 3 (b * , v * , x * ) -V 3 (b * , v * , x * )X 2 (b * , v * , x * )] x * v * T = -ϕ + V 2 (b * , v * , x * )v * + X 3 (b * , v * , x * )x * S = -ϕ [V 2 (b * , v * , x * )v * + X 3 (b * , v * , x * )x * ] + [V 2 (b * , v * , x * )X 3 (b * , v * , x * ) -V 3 (b * , v * , x * )X 2 (b * , v * , x * )] x * v *
It follows that the eigenvalues of J are solution of the following polynomial

P(λ) = λ 3 -T λ 2 + Sλ -D = (λ + ϕ) λ 2 -λ(T + ϕ) -D ϕ
We then get three eigenvalues (λ 1 , λ 2 , λ 3 ) such that We then get after simplifications In the cases where there exists a unique steady state, it must be such that H ′ (x * ) -Γ ′ (x * ) < 0. It follows that λ 2 > 0 and λ 3 < 0. Since λ 1 = -ϕ < 0, we conclude that two eigenvalues are negative and one is positive implying that the steady state is locally indeterminate. Let us finally consider the cases where two steady states x 1 < x 2 exist and are necessarily such that Γ ′ (x 1 ) -H ′ (x 1 ) < 0 and Γ ′ (x 2 ) -H ′ (x 2 ) > 0. It follows that the highest steady state x 2 satisfies -D/ϕ < 0, and we conclude again that two eigenvalues are negative and one is positive implying local indeterminacy.

λ 1 = -ϕ λ 2 + λ 3 = T + ϕ = V 2 (b * , v * , x * )v * + X 3 (b * , v * , x * )x * λ 2 λ 3 = -D ϕ = [V 2 (b * , v * , x * )X 3 (b * , v * , x * ) -V 3 (b * , v * , x * )X 2 (b * , v * , x * )] x * v * Note first that X(b, v, x) = r-θ(1-τ )sx 1-s r [Γ(x)-V (b,v,x)]-x 1-s [(1-τ)(1-θs)-v 1-α ]
Let us consider now the lowest steady state x 1 which is such that -D/ϕ > 0 implying λ 2 λ 3 > 0. We need then to study the sign of λ 2 + λ 3 and thus to compute We then derive

T + ϕ = V 2 (b * , v * , x * )v * + X 3 (b * , v * , x * )x * = ρ + ατcv *
(1-α)(1+τc)b * + r-θs(1-τ )x * 1-s r(1-s)

H ′ (x * )x * -Γ ′ (x * )x * -ρατc(1-s) (1+τc)b * x * 1-s
Using the expression of v * as given by ( 40), we finally get

T + ϕ = ρ + r-θs(1-τ )x * 1-s r(1-s) [H ′ (x * ) -Γ ′ (x * )] x * + ατc(1-τ )(1-s) (1+τc)b *
Since the lowest steady state x 1 si such that Γ ′ (x 1 ) -H ′ (x 1 ) < 0, we derive T + ϕ > 0 and we conclude that one eigenvalue is negative and two eigenvalues have a positive real part. It follows that x 1 is saddle-point stable. coefficients associated to the tariffs. In parentheses, the standard deviations. The two approaches that we consider for the notion of stability are used there. In Column (1) the control variables selected correspond to the Broad Monetary Stability approach. In Column (2), these are related to the Narrow Monetary Stability approach. The results suggest that the increase of tariffs have generated more growth volatility. We do not find any statistically difference between both approaches.

  λ(t) corresponds to the Lagrange multiplier associated to the borrowing constraint while µ(t) is the Lagrange multiplier of the domestic debt. The first order conditions are derived from the Euler equation ∂L ∂ω = d dt ∂L ∂ ω , with ω = {k, d, B h }:

Figure 1 :

 1 Figure 1: Uniqueness versus Multiplicity of BGP

  . The two others variables x(t) and v(t) are forward, thereby if the two remaining eigenvalues have a positive real part, then the steady state ( b, v * , x * ) is a saddle-point stable. While, if λ 2 or λ 3 has negative real part, then the steady state is locally indeterminate. Following the conditions presented in the Propositions 4.1 and 4.2, we formulate the following proposition: Proposition 6.1 Under Assumption 1, let r > ρ, r ∈ (r, r), and consider Propositions 4.1 and 4.2. Then the following results hold: i) When b * ∈ ( b, b) and θ ∈ (θ, θ), if α ≤ ᾱ(θ), or α > ᾱ(θ) and τ c ≤ τc (θ), then the unique steady state x is locally indeterminate.ii) When b * ∈ ( b, b) and θ ∈ (θ, θ), if α > ᾱ(θ) and τ c ∈ ( τc (θ), τc (θ) + ϵ), then the high steady state x 2 is locally indeterminate while the low one x 1 is saddle-point stable.

  (θ) < 1 if b * ∈ ( b, b) with b as given in (26). (b) On the contrary if α > ᾱ(θ), we easily derive that Γ(x) < H(x) for any τ c < τc (θ) with τc (θ) ≡ rρ(b * -b) (1-τ )[(1-s)r+ρs(1-θ)](α-ᾱ(θ))

  Straightforward computations giveV 2 (b * , v * , x * ) = ατc (1-α)(1+τc)b * V 3 (b * , v * , x * ) = Γ ′ (x * ) -sx * s-1 b * X 2 (b * , v * , x * ) = -r-θ(1-τ )sx 1-s r V2(b * ,v * ,x * )+ x * 1-s 1-α 1-s X 3 (b * , v * , x * ) = -θ(1-τ )sx -s r Γ(x * ) + r-θ(1-τ )sx 1-s r [Γ ′ (x * )-V3(b * ,v * ,x * )] 1-s -x * -s (1 -τ )(1 -θs) -v * 1-α with Γ(x * ) = r(1-θ)(1-τ )sx * 1-s r-θ(1-τ )sx * 1-s -ρNote that at the steady state we getx * -s (1 -τ )(1 -θs) -v *

X 3

 3 (b * , v * , x * )x * = s 1-s r-θ(1-τ )sx 1-s r x * s b * -Γ(x * )Straightforward computations then yield-D ϕ = x * 2-s v * (1-s)(1+α) [H ′ (x * ) -Γ ′ (x * )]

-

  V 2 (b * , v * , x * )v * + X 3 (b * , v * , x * )x * .Let us write X(b * , v * , x * ) as followsX(b * , v * , x * ) = Φ(x * ) 1-s with Φ(x) = B(x) A(x) -ατcv * (1-α)(1+τc)b * -x 1-s (1 -τ )(1 -θs) -v *1-α , andA(x) = x s -τ b * + r + ϕ 1 -b * b B(x) = r-θs(1-τ )x 1-s rRecall that at the steady state b * = b. We obviously getX 3 (b * , v * , x * ) = Φ ′ (x * ) 1-sFrom this we can computeΦ ′ (x * ) = B ′ (x * ) A(x * ) -ατcv * (1-α)(1+τc)b * + A ′ (x * )B(x * ) -(1 -s)x * -s (1 -τ )(1 -θs) -v * 1-αWe easily deriveA ′ (x * ) = s x * s-1 b * > 0 B ′ (x * ) = -θ(1-s)s(1-τ )x * -s state we get Φ(x * ) = 0 which implies v * x * -s 1-α = x * -s (1 -τ )(1 -θs) -B(x * ) x * A(x * ) -ατcv * (1-α)(1+τc)b * Substituting all this into the expression of Φ ′ (x * ) yields Φ ′ (x * ) = A ′ (x * )B(x * ) + A(x * ) -ατcv * (1-α)(1+τc)b * B ′ (x * ) -(1-s) x * B(x * )Considering thatB ′ (x * ) -(1-s) x * B(x * ) = -θ(1-s)s(1-τ )x * -s r Φ ′ (x * ) = A ′ (x * )B(x * ) -(1-s) x * A(x * ) -ατcv * (1-α)(1+τc)b * 36Recalling that for any steady state we have Γ(x * ) = A(x * ) andA ′ (x * ) = H ′ (x * ) -ρατc(1-s) (1+τc)b * x * 2-s Γ ′ (x * ) = (1-s)r x * [r-θ(1-τ )sx * 1-s ] [Γ(x * ) + ρ] (57) we derive Φ ′ (x * ) = r-θs(1-τ )x * 1-s r H ′ (x * ) -Γ ′ (x * ) -ρατc(1-s) (1+τc)b * x * 2-s + ρ(1-s)x * and thus X 3 (b * , v * , x * )x * = ρ + r-θs(1-τ )x * 1-s r(1-s)H ′ (x * )x * -Γ ′ (x * )x * -ρατc(1-s) (1+τc)b * x * 1-s

Table 1 :

 1 Local stability analysis

	The steady

17

Since the equation which is driving public debt is linear

Table 2 :

 2 Table 2 and we check the conditions provided in Propositions 4.1 and 4.2. Data for countries sample

		s	τ	b *	r	τc(2018) τc(2019)	α
		(%GDP) (%GDP) (%GDP)	(%)	(%)	(%)	(%GDP)
	France	30.7	44.89	97.62	0.13	3.10	3.08	32.55
	Greece	29.30	39.48	184.91	2.59	3.10	3.08	41.86
	Japan	36.45	31.41	235.45	-0.11	3	3.23	17.4
	Spain	34.16	34.68	95.54	0.66	3.08	3.10	32.02
	UK	29.30	32.72	85.24	0.94	3	2.65	31.92
	US	35.8	24.97	108.46	2.14	3.2	13.8	14.54

Note: Data for 6 developed countries in 2019, where: s corresponds to 1 -labour share, labour share has been collected on OECD website, τ the tax pressure on OECD, b * on IMF, r represents the long-term interest rate on OECD, τc is the tariff rate on World Bank and α the share of import goods in total consumption on World Bank website.

  Table 2 that only US experienced a significant change in tariffs rate between 2018 and 2019 as it has been multiplied by more than 4. Using 2018 rate shows that US are characterized by a unique equilibrium. According to our framework, the increase of tariffs by the US in 2018 triggered instability generating multiplicity of equilibrium and therefore large fluctuations associated to global indeterminacy.

)

  20 Shapiro-Wilk test for Normality of the series, Ljung-Box test for auto correlation testing and the Jarque Bera test to see if the error terms follow a Normal distribution.

21

See details Appendix 22 See Appendix 23 See Appendix

Table 3 :

 3 US tariffs rate on primary products Note. Data for US from 2014 to 2020 of tariffs rate on primary products.

	10.7 Figures and Tables
		tariffs rate primary products
	2014	2.5
	2015	2.4
	2016	2.4
	2017	6.6
	2018	6
	2019	39.5
	2020	2.7

Table 4 :

 4 US tariffs rate on manufactured products Note. Data for US from 2014 to 2020 of tariffs rate on manufactured products.

	Dependent variable	Growth volatility
		Broad Monetary Stability	Narrow Monetary Stability
	Tariffs	0.18917 *	0.17349 *
		(0.091)	(0.092)

Table 5 :

 5 Event StudyNotes: *p<0.10, **p<0.05, ***p<0.01 . Robust standard errors in parentheses. This table summarizes the

See alsoMorimoto et al. (2017).

as studied by[START_REF] Chéron | Growth and public debt: what are the relevant trade-offs?[END_REF],[START_REF] Futagami | Debt policy rule, productive government spending, and multiple growth paths[END_REF],Maebayashi et al. (2017) orMinea and Villieu (2013).

The same result is found in[START_REF] Modesto | Growth and instability in a small open economy with debt[END_REF].

Except that too large tariffs may prevent the existence of any equilibria. As they act as an externality, such a result is not surprising.

For some references on expectation-driven fluctuations, see Le[START_REF] Van | Instability and fluctuations in intertemporal equilibrium models: Presentation[END_REF].

See also[START_REF] Ghiglino | Trade, redistribution and indeterminacy[END_REF]. Le[START_REF] Le Riche | Intra-industry trade, involuntary unemployment and macroeconomic stability[END_REF] derive similar results in a one-sector model of differentiated products with productive labor externalities, considering two OLG countries, one with wage rigidity and the other with full employment.

See also[START_REF] Chaudhry | Tariffs, trade and economic growth in a model with institutional quality[END_REF] where the innovation degree of the export sector and the quality of institutions are key.

see also[START_REF] Bairoch | Free trade and European economic development in the 19th century[END_REF] and[START_REF] Eichengreen | A dynamic model of tariffs, output and employment under flexible exchange rates[END_REF] where protectionism affects negatively economic growth but only on the short run.

See also[START_REF] Boucekkine | Special functions for the study of economic dynamics: The case of the Lucas-Uzawa model[END_REF] and[START_REF] Brito | Local and global indeterminacy in two-sector models of endogenous growth[END_REF] for Lucas-type endogenous growth models based on human capital.

In SGP, the member states for which the current debt-to-GDP is above a threshold (60% fixed by the Maastricht treaty) must reduce the distance between their ratio and the threshold by an average rate of one-twentieth per year. If the debt-to-GDP ratio is beyond this threshold, government must reduce it at a steady peace level.

All our results on existence, uniqueness and multiplicity of the steady states could be obtained even under θ < (2 -s), but at the cost of cumbersome technical details.

As suggested by Figure1, uniqueness corresponds to a configuration where a unique "large" steady state x occurs while a lower one is outside of the admissible set (x, x) as it is unsustainable.

It is worth noticing however that too large tariffs may prevent the existence of any equilibria. As they act as an externality, such a result is not surprising.

The selection of the sample is motivated by their heterogeneity on: public debt level, tariffs and tax pressure. Countries like Japan, Greece and United States are characterized by high public debt; France by strong tax pressure, United States have the highest tariffs. Spain and United Kingdom dedicate one third of their consumption to the consumption of foreign goods.

In Section 8.3.2, other time windows are used

The coefficients associated to the impact of tariffs increase on growth volatility are reported in Table4available in Appendix. The coefficients are positive and statistically different from zero, suggesting that the increase of tariffs by US in 2018 has generated more volatility in the US. The coefficients and our interpretation from are in line with the Figure3.

low and stable inflation, long-run interest rates, public sector deficit to GDP, public debt to GDP and a stable currency fluctuations

The project leading to this paper has received funding from the French government under the "France 2030" investment plan managed by the French National Research Agency (reference: ANR-17-EURE-0020), and from the Excellence Initiative of Aix-Marseille University -A*MIDEX.