Stéphane Auray
email: stephane.auray@ensai.fr.

Aurélien Eyquem
email: aurelien.eyquem@univ-lyon2.fr.

On the Role of Debt Maturity in a Model with Sovereign Risk and Financial Frictions

Keywords: Sovereign Default Risk, Financial Crisis, Fiscal Policy. JEL codes: E44, E62, H63 Sovereign Default Risk, Financial Crisis, Fiscal Policy. JEL Classification: E44, E62, H63

We develop a model with financial frictions and sovereign default risk where the maturity of public debt is allowed to be larger than one period. When the debt portfolio has longer average maturities, public debt increases less in the event of a crisis, reducing the size of the subsequent fiscal consolidation through distorsionary taxes or public spending, with positive effects on welfare. In addition, we provide some results suggesting that optimized fiscal responses to a crisis depend on the average maturity of the debt portfolio.

Introduction

Most DSGE models that try to analyze the effects of the 2008 financial crisis or the more recent sovereign debt crisis in Europe assume that the single support of public debt is a one-period bond. In practice however, the average maturity of public debt is much longer. Some papers have proposed simple ways of taking into account the fact that public debt has a longer maturity, including the seminal paper of Woodford (2001), or more recently [START_REF] Arellano | Default and the Maturity Structure in Sovereign Bonds[END_REF], [START_REF] Debortoli | Optimal Time-Consistent Government Debt Maturity[END_REF] and [START_REF] Faraglia | Long Term Government Bonds[END_REF]. This note investigates the importance of the average maturity of the stock of public debt in the aftermath of a financial crisis with banking frictions and sovereign risk. It is often argued that the bank-sovereign doom loop was at the heart of the double dip and 2011 debt crisis in countries of the south of the Euro Area. The initial kick however was due to the imported U.S. subprime crisis. We build a model à la [START_REF] Gertler | A Model of Unconventional Monetary Policy[END_REF] and incorporate sovereign default risk à la [START_REF] Corsetti | Sovereign Risk and Belief-driven Fluctuations in the Euro Area[END_REF] and a distortionary tax on labor income. Finally we consider that sovereign bonds are perpetuities with a coupon decay factor ρ, as in Woodford (2001), where ρ is directly related to the average maturity of bonds. We run a crisis experiment as in [START_REF] Gertler | A Model of Unconventional Monetary Policy[END_REF] under very short, short, medium or long average maturities of the stock of sovereign debt.

We show that the resulting dynamics are substantially different. It has long been established that, for a given change in yields, the fluctuations in market price are greater when the term to maturity is longer (see [START_REF] Hopewell | Bond Price Volatility and Term to Maturity: A Generalized Respecification[END_REF]). Accordingly, in our model, longer maturities favor larger movement of bond prices that mitigate the rise in public debt and hence the size of the fiscal consolidation that follows a capital quality shock that mimics the effects of a crisis. When fiscal adjustment is achieved through the distorsionary tax rate, hours worked fall less and contribute to lower the fall in output and consumption, with positive effects on welfare. When the fiscal adjustment goes through changes in public spending, as in [START_REF] Cui | Monetary-fiscal Interactions with Endogenous Liquidity Frictions[END_REF], they need to fall less for longer average maturities -they even rise under very long maturities, which contributes positively to welfare. In this sense, longer average maturities serve as a better shock absorber in the event of crises and generate welfare gains compared to shorter maturities.

Model

Our model of financial intermediation is an extension of the [START_REF] Gertler | A Model of Unconventional Monetary Policy[END_REF] model where banks can either grant loans to capital producers or hold risky sovereign bonds.

Banks

There is a unit continuum of banks. The balance sheet of the representative bank is

q t k t = Private loans d t = domestic deposits q b t b t = Sovereign bonds n t = net worth
where q t is the price of capital, q b t the real price of the sovereign portfolio, k t the amount of private capital and b t the amount of sovereign bonds. The sovereign bond portfolio is defined as perpetuities with a coupon decay factor ρ, as in Woodford (2001). The parameter ρ is directly related to the average maturity of the sovereign bond portfolio, defined as M = 1/ (1 -βρ) where β is the discount factor. We also introduce default risk (explained in details later) and assume that a fraction χ t of the portfolio may be perceived by agents as being potentially defaulted on. Banks arbitrage both assets and equate their respective expected returns

E t r k t+1 = E t r b t+1 (1 -χ t+1 ) (1)
where r k t+1 is the return on capital and r b t+1 = 1 + ρq b t+1 /q b t is the return on bonds. This equation shows that sovereign risk matters in the sense that agents price sovereign bonds ex-ante as if default was a potential event and are unaware of the ex-post insurance scheme. As such, sovereign risk affects the dynamics of returns both on the sovereign market but also on the capital market through the arbitrage of banks. Let a t denote the total bank asset. The balance sheet equation is

a t = q t k t + q b t b t = d t + n t (2)
As explained above, sovereign default matters ex-ante but not ex-post. Banks have access to insurance contracts and receive T t = 1 + ρq b t+1 χ t+1 b t , covering their losses in the event of sovereign default. Accordingly, net worth evolves according to

n t+1 = r a t+1 a t -r d t d t + T t (3)
where r d t is the deposit rate and r a t the rate of return on total assets defined as

r a t = r b t (1 -χ t ) q b t b t-1 + r k t q t ξ t k t-1 /a t-1 (4)
where ξ t is a capital quality shock to be defined later on. Combining both equations gives the dynamics of the bank's net worth

n t+1 = r a t+1 -r d t q a t a t + r d t n t + T t (5)
The bank maximizes expected net worth given a fixed exit probability (1 -σ), in which event net worth is rebated to the households, and discounts future outcomes at the stochastic rate β t+1 = βu c,t+1 /u c,t . We follow [START_REF] Gertler | A Model of Unconventional Monetary Policy[END_REF], and conjecture that v t is linear and assume v t = ̟ t a t + γ t n t (6)

In addition, to prevent unlimited expansion of lending due to positive arbitrage opportunities, the representative bank may divert a fraction α of its assets. This possibility adds the following incentive constraint on bank's activities

v t = ̟ t a t + γ t n t ≥ αq a t a t (7)
which will be strictly binding in equilibrium. Let φ t = a t /n t = (n t + d t ) /n t be the leverage ratio of banks, the incentive constraint writes

v t = αφ t n t (8)
Banks optimization yields the following conditions for marginal values of arguments of the value function

̟ t = E t (1 -σ) β t+1 r a t+1 -r d t + σβ t+1 ̟ t+1 Ω a t+1 (9) γ t = E t (1 -σ) + σβ t+1 γ t+1 Ω n t+1 ( 10
)
where Ω n t = n t /n t-1 is the growth rate of net worth and Ω a t = a t /a t-1 the growth rate of intermediated assets, respectively evolving according to Ω n t = (r a t -r d t-1 )φ t-1 + r d t-1 and Ω a t = (φ t /φ t-1 ) Ω n t . Using the expression of the value function finally allows to reformulate the binding incentive constraint as

φ t = γ t α -̟ t (11)

Intermediate and capital goods producers

Intermediate goods producers use capital k t-1 in the production process. They also hire labor in quantity ℓ t , that they combine to build the intermediate good, with the following production function

y m t = (ξ t k t-1 ) ι ℓ 1-ι t ( 12
)
and sell intermediate goods at real relative price p m t . In this production function, the total stock of physical capital can be affected by an AR(1) quality shock ξ t . The optimizing condition with respect to labor is

p m t (1 -ι) y m t /ℓ t = w t (13)
where w t is the real wage and the zero-profit condition implies that intermediate goods producers pay the ex-post return on capital to the capital goods producers, i.e.

r k t+1 = p m t+1 ιy m t+1 /k t + q t+1 ξ t (1 -δ) /q t (14)
Capital goods producers buy the depreciated capital of intermediate goods producers and choose investment to accrue the total amount of available capital based on the evolution of its real price q t . 1 Their profit writes

E t ∞ s=0 β t+s q t+s i t+s 1 -ϕ i /2 (i t+s /i t+s-1 -1) 2 -i t+s (17)
and optimization yields

q t -1 = q t ϕ i x t (1 + x t ) + x 2 t /2 -E t β t+1 q t+1 ϕ i x t+1 (1 + x t+1 ) 2 (18)
where x t = i t /i t-1 -1. Given this optimizing condition for investment, the law of capital accumulation gives the dynamics of the capital stock

k t -(1 -δ) ξ t k t-1 = i t 1 -ϕ i /2 x 2 t (19)

Final goods producers

Final goods producers j differentiate the intermediate good y m t in imperfectly substitutable varieties. The aggregate bundle of the final good and the corresponding aggregate price level are

y t = 1 0 y t (j) θ-1 θ dj θ θ-1 , p t = 1 0 p t (j) 1-θ dj 1 1-θ (20)
Final goods producers take into account households demands y t (j) = (p t (j) /p t ) -θ y t when optimally setting prices subject to Calvo price contracts of average length 1/ (1 -γ) with indexation parameter γ p .2

Households

Households face a simple optimization problem as they choose consumption, labor supply and deposits maximizing lifetime welfare

E t ∞ s=0 β s u (c t+s , g t+s , ℓ t+s ) ( 21
)
where u ℓ,t ≤ 0, u c,t ≥ 0 and u g,t ≥ 0 are the first-order partial derivatives with respect to hours worked, consumption and public spending.3 Households optimize subject to the budget 1 More formally, they maximize

Et ∞ s=0 βt+s,t+s+1 (qt+s (kt+s -(1 -δ) kt+s-1) -it+s) (15)
subject to the law of motion of capital accumulation

kt -(1 -δ) ξtkt-1 = it 1 -ϕ i /2 (it/it-1 -1) 2 . ( 16
) constraint d t + c t = r d t-1 d t-1 + (1 -τ t ) w t ℓ t + Π t ( 22
)
where d t denote deposits to banks returning r d t between t and t + 1, c t is consumption, w t denotes the real wage, τ t is a distortionary tax on labor income, ℓ t hours worked, and Π t comprises monopolistic profits from final goods producers, plus the net worth rebated by bankrupt banks, net from the starting fund allocated to new banks. FOCs give

E t β t+1 r d t = 1 (23) u ℓ,t + (1 -τ t ) u c,t w t = 0 (24)

Government and Central Bank

We adopt the approach of sovereign default from [START_REF] Corsetti | Sovereign Risk and Belief-driven Fluctuations in the Euro Area[END_REF] but with a slight variation concerning the distribution of default probabilities. Actual ex-post default is neutral while the ex-ante probability of default is crucial for the pricing of government debt and therefore for real activity. 4 Focusing on the domestic economy, the ex-ante probability of default, p t , at a certain level of sovereign indebtedness, by t = q b t b t / (4y t ), will be given by:

p t = α p e byt-by (25)
where by denotes the steady-state level of debt to GDP. Default occurs with probability p t so that χ t = ∆ if B (p t ) = 1, and

χ t = 0 if B (p t ) = 0 (26)
where B (p t ) is a Bernoulli. Given these assumptions, the consolidated budget constraint writes 5

q b t b g t = 1 + ρ t q b t b g t-1 + g t -τ t w t ℓ t (29)
The stability of public debt is ensured either by a tax rule or by a public spending rule, as in Cui ( 2016)

τ t -τ = d b τ (by t-1 -by) (30) g t -g = -d b g (by t-1 -by) (31)
The central bank controls the nominal interest rate i n t

i n t = r d t E t (π t+1 ) ( 32
)
4 Following [START_REF] Eaton | Debt with Potential Repudiation: Theoretical and Empirical Analysis[END_REF], many have modeled default as a strategic decision of the sovereign. On the other hand, [START_REF] Bi | Sovereign Default Risk Premia, Fiscal Limits, and Fiscal Policy[END_REF] considers default as the consequence of the government's inability to raise the funds necessary to honor its debt obligations. Alternatively, Juessen, Linnemann and Schabert (2016) model default as resulting from the behavior of households and their willingness to lend. Under all approaches, the probability of sovereign default is closely and non-linearly related to the level of public debt to GDP.

5 The budget constraint of the government writes

q b t b g t = 1 + ρq b t (1 -χt) b g t-1 + gt -τtwtℓt + T b t (27)
Once again, potential losses from default are fully compensated, so that only ex-ante default risk matters. As a consequence

T b t = 1 + ρtq b t χtb g t-1 (28)
which yields the reported consolidated budget constraint.

where the central bank follows a Taylor-type policy rule6

log (i n t /i n ) = ρ r log i n t-1 /i n + d π log (π t /π) + d y log (y t / y t ) (33)
where y t is the natural level of output.7

Aggregation

At the end of the period, a fraction 1 -σ of the total number of banks become households. Dividends are paid to households only when banks exit the banking sector. The net worth of continuing banks is simply carried to the next period, so that aggregate continuing banks' net worth evolve according to n e t = σΩ a t n t-1 (34)

In addition, the household provides a starting net worth to new banks, equal to a fraction ϕ/ (1 -σ) of the total assets of exiting bankers, so that the net worth of new banks is

n n t = ϕ q t ξ t k t-1 + q b t b t-1 (35)
Overall, aggregate net worth evolves according to

n t = n e t + n n t (36)
The clearing condition on the intermediate goods market is y m t = 1 0 y t (j) dj = y t dp t where dp t = 1 0 (p t (j) /p t ) -θ dj is the dispersion of prices. On the final goods market, the clearing condition simply writes y t = c t + i t + g t and the clearing condition for government bonds is just b g t = b t .

Calibration

We calibrate the model to the Euro Area but follow Gertler and Karadi (2011) in many respects. The time unit is a quarter. The functional form of preferences is

u (c t , g t , n t ) = (1 -κ) log (c t ) + κ log (g t ) -ωℓ 1+ψ t /1 + ψ (37)
The discount factor is β = 0.99. The inverse of the Frisch elasticity on labor supply is ψ = 3.8

The preference parameter κ is set so as to equate the marginal utility of private consumption and public spending u c = u g . On the production side, the share of capital is ι = 0.33, the depreciation rate is δ = 0.018 (7% annually). We impose a 1pp spread (in annual terms) over the risk-less rate for r k which pins down the capital to output ratio. Investment adjustment costs are ϕ i = 2, the Calvo parameters are γ = 0.75 and γ p = 0.25, and the steady-state mark-up is 30%, implying θ = 4.33. In the banking sector, we impose a steady-state leverage ratio of φ = 4. The survival probability of bankers is σ = 0.975. On the monetary and fiscal policy side, we set the steady-state probability of default at α p = 0.5% and the value of ∆ = 0.55, implying a roughly 2pp sovereign spread (in annual terms) with respect to the risk-less deposit rate for the calibrated debt-output ratio. We assume standard Taylor rule parameters, i.e. ρ r = 0.8, d π = 1.5 and d y = 0.125. The stability of debt to GDP is ensured either through taxes or through public spending. In the former case, Equation ( 30) is in place and we set d b τ = 0.33 while at d b g = 0. In the latter case, Equation ( 31) is in place with d b g = 0.1 while d b τ = 0. Based on the average maturity of sovereign debt in Euro Area countries in 2007 (7 years), the average duration of the portfolio is M =1/ (1 -βρ) = 28 implying that the steady-state value of the decay is ρ = 0.9740. Alternative values are considered in the paper: M = 4 (1 year maturity), M = 8 (2 years maturity) and M = 100 (25 years maturity). Using OECD data for 2007, we build a measure of hours worked and set ℓ = 0.2571. Proceeding similarly, the share of public expenditure in GDP and the level of public debt to GDP are respectively s g = 0.2025 and b g / (4y) = 0.6959.9 The steady-state labor income tax rate is then adjusted to match the debt-to-GDP ratio target: τ = 0.4762. This calibration implies that our model has a unique determinate equilibrium in which, in the terminology of Leeper (1991), monetary policy is active and fiscal policy is passive.10 Table 1 summarizes our parameter values. The model is solved non-linearly under perfect foresight using a Newton-type algorithm. 11 We consider four alternative steady-state maturities for public debt: 1, 2, 7 and 25 years respectively. 11 The algorithm is a built-in routine of Dynare. It is an application of the Newton-Raphson algorithm that takes into consideration the special structure of the Jacobian matrix in dynamic models with forward-looking variables. The details of the algorithm are explained in Juillard (1996).

Figure 1 plots the Impulse Response Functions (IRFs hereafter) of the economy after a 5% capital quality risk shock with persistence 0.66. As capital quality falls the price of capital q t falls as well, deteriorating banks balance sheet and contributing to lower investment. The stock of capital shrinks, bringing output and consumption down. This dynamics tends to raise the debt to GDP ratio, which contributes to raise sovereign risk and further raises the stock of debt. Sovereign spreads rise one for one with private spreads and the price of bonds falls. The government has to change the tax rate to stabilize debt in the medium run, which results in an additional variation of hours worked, consumption, investment and output.

The major difference with the various maturities lies in the size of the response of bond prices: the longer the average maturity, the larger the fall in bond prices induced by the shock. Since the shock implies a surge in sovereign rates -partly through the financial accelerator and partly through sovereign risk -bond prices fall. The sensitive of prices to the rise in the yields is much longer for longer maturities. Hence, for a given rise of sovereign rates, longer maturities contribute to lower the rise in public debt, and so the rise in distorsionary taxes. The stabilizing movement of sovereign bonds price is so large for very long maturities that it reverses the initial response of the value of the stock of debt, leading to a temporary fall in distortionary taxes, an increase in hours worked, consumption and output. This "positive" outcome comes at the cost of larger sovereign and private spreads, and hence at the cost of a larger fall in investment. Summarizing, longer maturities reduce the fall in output, consumption and hours worked, and magnify the fall in investment.

Public spending rule

Figure 2 plots the IRFs to the very same shock but when the government stabilizes the stock of public debt using public spending instead of the labor income tax rate.

The story shares a lot with the previous case: output, consumption, hours worked and investment fall, private and sovereign spread rise, and the debt-output ratio increases. In addition, as in the previous case, the adjustment of sovereign bond prices is much larger for longer maturities than for short maturities. Again, this stabilizing effect is large enough to overturn the response of the value of debt in the first quarters after the shock from positive to negative for very long maturities. However, the fiscal adjustment is different in that public spending is the instrument to stabilize debt, and the labor income tax remains constant. Hence, the additional adverse effects from higher taxes are absent in this case: output and consumption fall less, hours worked fall more. The dynamics of consumption is affected by public spending through the crowding-out effect by which any fiscal consolidation using public spending will make private consumption rise, which contributes positively to the dynamics of output. As in the previous case, the positive spillovers from long average maturities of the sovereign debt portfolio favor debt sustainability and lower the size of the required fiscal adjustment.

Welfare

What are the implications of these results for the welfare losses generated by the crisis? Table below reports the welfare losses under different average maturities, under alternative fiscal rules First, Table 2 shows that, when the government follows a tax rule, longer maturities unambiguously help stabilizing debt, taxes and thus consumption, which generates positive effects on welfare. Welfare losses are clearly decreasing functions of maturities, at all horizons, and the reduction in welfare losses can be quite substantial. Second, Table 2 also shows that, when the government follows a spending rule, longer maturities generate smaller welfare losses at short horizon against a negligible larger lifetime welfare loss. Finally, our results point to the much smaller welfare losses from the shock under public spending adjustments than under distorsionary tax adjustments, at all horizons. Even though the analysis of optimal policies is clearly beyond the scope of this note, our results suggest that adjusting public spending rather than distortionary taxes is the way to go for governments experiencing large negative shocks, at least for short to moderately long maturities.

To confirm this intuition, we can analyze optimized rules by picking the parameters of the fiscal rules d b τ and d b g -allowing both rules to be in play simultaneously -that minimize the lifetime welfare losses generated by a negative shock on capital quality. The optimized parameters and corresponding welfare losses are reported in Table 3 below. 13 Table 3 confirms that, for short to moderately long maturities, the best instrument to stabilize debt remains public spending. 14 The response of the tax rule is muted for all maturities but the longest, and the optimized response of public spending should be substantially stronger than the value imposed in the baseline calibration. The welfare gains of optimized rules are observed in the last line of Table 3 by looking at the lifetime welfare losses and comparing them to the corresponding losses with simple rules, in particular with the spending rule (last line of Table 12 Welfare losses are computed as the Hicksian consumption equivalent that makes households indifferent between experiencing the crisis and remaining at the initial steady state. The calculation is made for different dates T after the shock. When T is small, the calculation captures the short-run gains or losses, when T is larger, is gets closer to the lifetime welfare gains or losses. The "true" welfare effect corresponds to T = ∞ but considering shorter horizons also sheds light on the short-run effects of the shock under alternative policies and for different average maturities.

13 Figure 4 in Appendix B reports the IRFs under optimized rules.

14 Viegas and Ribeiro (2016) report similar results. However, our point is far from general and remains contextualized to this model and the associated assumptions, in particular regarding the form of preferences and fiscal rules. 2). They can be as large as 0.44% of permanent consumption, especially for short maturities.

For longer maturities, the potential lifetime welfare gains are smaller while still non-negligible, around 0.24% of permanent consumption. At shorter horizons, optimized rules tend to increase the welfare losses for short average maturities, while they yield smaller losses (even gains) under longer maturities.

Overall, our results point to the importance of taking into account the structure of the sovereign debt portfolio, its average maturity and its effects on the banking system when analyzing the transmission of shocks and the effects of monetary and fiscal policies. These features happen to make a large difference in terms of the dynamics of public debt, sovereign risk and aggregate macroeconomic dynamics. Notes: M/F means monetary or fiscal policy and A/P means active or passive policy in the sense of Leeper (1991).

A Determinacy analysis

The unstable (resp. determinate / indeterminate) region corresponds to parameter values for which the number of explosive roots exceeds (resp. is equal to / is less than) the number of forward variables. Determinacy regions are not affected by the average maturity so these maps are valid for any steady-state value of the average maturity of the sovereign debt portfolio.

B IRFs after a negative capital quality shock with optimized fiscal rules

Figure 1 :

 1 Figure 1: IRFs to a capital quality shock with different average maturities of the sovereign bonds portfolio -Labor income tax rule

Figure 2 :

 2 Figure 2: IRFs to a capital quality shock with different average maturities of the sovereign bonds portfolio -Public spending rule

Figure 3 :

 3 Figure 3: Determinacy analysis in the (d π , d b τ ) space and in the (d π , d b g ) space

Figure 4 :

 4 Figure 4: IRFs to a capital quality shock with different average maturities of the sovereign bonds portfolio -Optimized fiscal rule

Table 1 :

 1 Parameter values.We investigate the transmission of a capital quality shock, that mimics quite accurately the effects of the 2008 financial crisis with different steady-state average sovereign debt maturities affect.

	Discount factor

Table 2 :

 2 Welfare losses from a negative capital quality shock, in percents.

		M = 4 M = 8 M = 28 M = 100
		Tax rule -d b τ , d b g = (0.33, 0)
	T = 4	1.7713 1.6784	1.3179	0.8130
	T = 8	2.9328 2.8213	2.3955	1.8123
	T = 32 5.2111 5.1640	4.9671	4.6809
	T = ∞ 3.8055 3.7931	3.7298	3.6177
		Spending rule -d b τ , d b g = (0, 0.1)
	T = 4	0.5998 0.5011	0.1522	-0.2419
	T = 8	1.6913 1.5603	1.1085	0.6022
	T = 32 4.5287 4.4839	4.3460	4.2049
	T = ∞ 3.3180 3.3275	3.3543	3.3774

Table 3 :

 3 Welfare losses from a negative capital quality shock under optimized rules, in percents.

		M = 4 M = 8 M = 28 M = 100
	d b τ d b g	0.0000 0.0000 0.5177 0.5295	0.0000 0.4711	0.1799 0.2459
	T = 4 1.0164 0.6813 -0.2631 -0.5186
	T = 8 3.2540 2.8093	1.3617	0.5207
	T = 32 5.2879 5.2409	5.1235	4.8581
	T = ∞ 2.8853 2.9002	2.9675	3.1306

  Juillard, Michel, "Dynare : A Program for the Resolution and Simulation of Dynamic Models with Forward Variables through the Use of a Relaxation Algorithm," CEPREMAP Working Papers (Couverture Orange) 9602, CEPREMAP 1996. Viegas, Miguel and Ana Paula Ribeiro, "Assessing Welfare Impacts of Some Debtconsolidation Episodes in the European Union," Macroeconomic Dynamics, 2016, 20 (5), 1146-1173. Woodford, Michael, "Fiscal Requirements of Price Stability," Journal of Money Credit and Banking, 2001, 33, 669-728.

	Leeper, Eric M., "Equilibria under 'Active' and 'Passive' Monetary and Fiscal Policies," Jour-nal of Monetary Economics, February 1991, 27 (1), 129-147.

The optimal pricing conditions are standard and therefore not reported.

We assume that households value public spending in the utility function since public spending will be used as a potential policy instrument to stabilize the economy.

None of the scenarios investigated in this paper implies that the nominal interest rate hits the Zero Lower Bound after the shock. While such a case would be an interesting extension (obtained for instance by assuming a zero persistence in the Taylor rule), we do not consider this possibility.

Variations in the mark-up will serve as a proxy for variations in the output gap.

This value is much larger than the value considered by Gertler and Karadi (2011) -they use 0.276 -but their calibration relates to the U.S. where the labor market is much more responsive than in the Euro Area.

Notice that considering alternative average maturities does not affect the steady state, but only affects the price-quantity nexus of public debt. Longer maturities are associated with larger bond price and less quantities while shorter maturities are associated with low bond prices and larger quantities. The total steady-state value of the portfolio q b b g however remains the same under all calibrations.

As shown in Appendix A, alternative monetary-fiscal regimes producing stable and unique equilibria existin particular one in which monetary policy is passive and fiscal policy is active -but their analysis is beyond the scope of the present paper.

The authors gratefully acknowledge financial support of the Chair ACPR/Risk Foundation: "Regulation and