Rym Aloui
email: aloui@gate.cnrs.fr.

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

Spending Multipliers with Distortionary Taxes: Does the Level of Public Debt Matter?

Keywords: Zero Lower Bound, Fiscal Policy, Distortionary Taxes, Public Debt. JEL Classification: E62, E32 Zero Lower Bound, Fiscal Policy, Distortionary Taxes, Public Debt. JEL Classification: E62, E32

We investigate the link between the size of government indebtedness and the effectiveness of government spending shocks in normal times and at the Zero Lower Bound (ZLB). We develop a New Keynesian model with capital, distortionary taxes and public debt in which the ZLB constraint on the nominal interest rate may be binding. In normal times, high steady-state levels of government debt to GDP lead to reduced output multipliers. After a negative capital quality shock that pushes the economy at the ZLB however, high steadystate debt levels produce larger output multipliers. Our results rely on the fact that fiscal policy becomes self-financing at the ZLB, and that distortionary taxes rise (respectively fall) after a spending shock at the steady state (resp. ZLB). Our results have non-trivial consequences on the design of optimized spending policies in the event of large economic downturns.

Introduction

The massive rise in government debt levels and sovereign spreads that followed the 2008 Great Recession and the 2011 recession in countries of peripheral Europe raises the question of whether the level of public debt affects the effectiveness of government spending shocks. In other words, do high levels of sovereign debt undermine the ability of governments to make use of government spending to stabilize the economy? Conventional wisdom suggests that countries with high levels of public debt have less room for fiscal stimulation than countries with low levels of public debt in the event of an economic crisis, and would therefore advocate for low debt levels on average. We investigate this question in a standard New-Keynesian model with capital accumulation where fiscal solvency is achieved through distortionary taxes on either labor or capital income. In the

Introduction

The massive rise in government debt levels and sovereign spreads that followed the 2008 Great Recession and the 2011 recession in countries of peripheral Europe raises the question of whether the level of public debt affects the effectiveness of government spending shocks. In other words, do high levels of sovereign debt undermine the ability of governments to make use of government spending to stabilize the economy? Conventional wisdom suggests that countries with high levels of public debt have less room for fiscal stimulation than countries with low levels of public debt in the event of an economic crisis, and would therefore advocate for low debt levels on average. We investigate this question in a standard New-Keynesian model with capital accumulation where fiscal solvency is achieved through distortionary taxes on either labor or capital income. In the data, countries with higher levels of debt also feature higher levels of tax rates (see Appendix A).

Because debt is high, they potentially face larger costs of debt rollover, and because taxes are high, they face larger efficiency costs of distortionary taxation. In this paper, we show that the initial level of debt lowers public spending multipliers in normal times but raises public spending multipliers at the Zero Lower Bound (ZLB). In normal times, a spending shock financed by a combination of public debt and a distortionary tax rule leads to a larger increase in public debt and in future tax rates. This imposes more distortions on production factors and reduces the positive effects of a spending shock on output. At the ZLB, in line with [START_REF] Erceg | Is There a Fiscal Free Lunch in a Liquidity Trap?[END_REF], we find that a spending shock at the ZLB is self-financing. This leads public debt and future tax rates to fall, lowers the amount of distortions imposed on production factors, and further raises output. These movements are larger for economies featuring a high initial debt-to-GDP ratio due to their higher tax levels, and because the costs of taxation are convex. The shock leading the ZLB to be binding is here a negative capital quality shock, as in [START_REF] Gertler | A Model of Unconventional Monetary Policy[END_REF], but our results are robust to considering a discount factor shock. One important difference between capital quality shocks and discount factor shocks is that the latter produce a rise in private investment while the former produce a fall in private investment. We thus consider the former to be more consistent with the Great Recession narrative, and with the factors that most likely led nominal interest rates to hit the ZLB.

Our results are qualitatively robust to keeping the steady-state level of tax rates constant and using a steady-state lump-sum tax to finance the changing steady-state level of debt. They are only slightly attenuated quantitatively speaking. Indeed, what crucially matters for our results to hold is the size of the change in tax rates that is needed to ensure fiscal solvency after spending shocks. The latter is larger when the steady-state level of debt is higher. The change in tax rates is positive in the case of a spending shock hitting the economy around the steady state because debt rises, which in turn reduces the size of the public spending multiplier, and negative in the case of a spending shock hitting at the ZLB because public debt falls, which in turn raises the value of the spending multiplier. We also show that our results hold and are quantitatively magnified when introducing hand-to-mouth households.

Quantitatively speaking, our model produces public spending multipliers that line-up quite well with the literature (see [START_REF] Ramey | Ten Years after the Financial Crisis: What Have We Learned from the Renaissance in Fiscal Research?[END_REF] for a recent overview). During normal times, short-run multipliers (at a 2 years horizon) roughly range from 0.2 to 0.3 for an empirically plausible calibration of the model. When the spending shock hits conditional on a negative capital quality shock that pushes the economy at the ZLB for a few quarters, short-run multipliers range from 1.4 to 1.45 for the same calibration. In any case, spending multipliers at the ZLB are larger than spending multipliers at the steady-state, as in [START_REF] Christiano | When Is the Government Spending Multiplier Large?[END_REF] and the subsequent literature. The impact of the initial level of debt is relatively small for short-run multipliers at the steady state, but can be very large for short-run multipliers at the ZLB. These results are quantitatively but not qualitatively sensitive to a small subset of parameters such as the Frisch elasticity on labor supply, the degree of complementarity between public and private goods in the utility function of households, or the responsiveness of the tax rules. Finally, we

show that the initial level of debt crucially affects the optimized response of government spending to a large crisis that pushes the economy to the ZLB. The size and persistence of the rise in public expenditure varies depending on the initial level of debt, which is only a consequence of the fact that the effects on output of changes in spending are stronger when the initial level of debt is higher.

Our paper relates to the literature on spending multipliers that questions how the economic environment may affect the latter. Empirically, one of the first papers to raise the question was [START_REF] Perotti | Fiscal Policy in Good Times and Bad[END_REF]. More recently, the subject has been revived by [START_REF] Auerbach | Fiscal Multipliers in Recession and Expansion[END_REF], investigating whether the business cycle position matters for the value of multipliers. Two papers, respectively by [START_REF] Corsetti | What Determines Government Spending Multipliers?[END_REF] and by [START_REF] Ilzetzki | How Big (Small?) are Fiscal Multipliers?[END_REF] question more precisely the impact of debt or fiscal stress on spending output multipliers.

Their results converge and conclude that fiscally stressed or highly indebted economies tend to be characterized by lower spending multipliers. Our results about the impact of the initial level of debt on the size of fiscal multipliers are in accordance with those results, as we find that a high level of debt lowers the spending multiplier. Further, according to [START_REF] Corsetti | What Determines Government Spending Multipliers?[END_REF], when an economy experiences a financial crisis, spending multipliers are much larger than in normal times. If one admits that financial crises are more likely to lead to ZLB episodes, this result is also consistent with more theoretical contributions like [START_REF] Christiano | When Is the Government Spending Multiplier Large?[END_REF] and the subsequent literature. Our results are also consistent with this empirical result, as we find that spending multipliers are larger during a financial crisis, triggered by a negative capital quality shock that pushes the economy at the ZLB. A recent paper by [START_REF] Boitani | Public Expenditure Multipliers in Recessions. Evidence from the Eurozone[END_REF] tests the joint conditional impact of recessions and the level of debt on the size of spending multipliers. [START_REF] Boitani | Public Expenditure Multipliers in Recessions. Evidence from the Eurozone[END_REF] find evidence of self-financing public spending shocks during recessions in Euro Area countries, and find that multipliers are larger in high deficit countries than in low deficit countries, providing empirical evidence in favor of our results.

Our paper also belongs to a model-based literature that investigates the effects of the ZLB on the size of fiscal multipliers, summarized and referenced in [START_REF] Eggertsson | What Fiscal Policy is Effective at Zero Interest Rates?[END_REF]. In particular, [START_REF] Erceg | Is There a Fiscal Free Lunch in a Liquidity Trap?[END_REF] find that fiscal policy becomes self-financing at the ZLB, a result that is also present in our paper and key to our main result. To our knowledge however, there are only very few papers questioning the effect of the initial debt level on the size of public spending multipliers. [START_REF] Corsetti | Sovereign Risk, Fiscal Policy, and Macroeconomic Stability[END_REF] do investigate this question but their analysis does not consider capital accumulation, and essentially focuses on the case of lump-sum taxes, while our main focus is on distortionary taxes. Along this dimension, our framework is closer to [START_REF] Nakata | Optimal Government Spending at the Zero Lower Bound: A Non-Ricardian Analysis[END_REF], although we consider a richer model with capital accumulation, and our main focus is not on Ramsey equilibria but on the size of government spending multipliers and optimized policies. Our paper also echoes the recent contribution of [START_REF] Bilbiie | Is Government Spending at the Zero Lower Bound Desirable?[END_REF], who derive optimal spending policies at the ZLB with an additional focus on spending multipliers. However, they consider lump-sum taxes and do not investigate the impact of the initial level of debt on spending multipliers and optimized policies.

The paper is organized as follows. Section 2 presents the model and details our calibration.

Section 3 analyzes the Impulse Response Functions to spending shocks hitting at the steady state or conditional on a negative capital quality shock that pushes the economy at the ZLB, depending on whether the initial debt-to-GDP ratio and taxes are low or high. Section 4 summarizes our results by presenting the value of spending multipliers at various horizons and under the different cases considered, and produces an extensive sensitivity analysis. Section 5 investigates the design of optimized spending rules, and Section 6 concludes.

Model

Our framework builds on a standard New Keynesian model with capital accumulation, sticky prices, public goods entering the utility of households and a monetary policy that is restricted by a (zero) lower bound on the nominal interest rate. Public expenditure are financed through public debt and distortionary taxes on capital and labor income.

Households

Households choose consumption, labor supply, deposits and government bonds maximizing lifetime welfare

E t ∞ s=t β s-t u (c s , g s , s ) , (1)
where u ,t ≤ 0, u c,t ≥ 0 and u g,t ≥ 0 are the first-order partial derivatives with respect to the private consumption, c t , hours worked, t and the amount of public spending, g t . Parameter

β ∈ (0, 1) denotes the subjective discount factor. Households optimize subject to the following budget constraint

b g t + p t ((1 + µ) c t + k t ) = r t-1 b g t-1 + p t (1 -τ t ) w t t + R k t k t-1 + Π t . (2)
In this equation, b g t denotes the nominal amount of government bonds that returns r t between period t and period t + 1. Further, µ is a (constant) distortionary tax on consumption, w t is the real wage, and τ t is a distortionary tax on labor income.1 Variable R kt = 1 + (1 -η t ) r k t -δ is the after-tax real gross return on capital where η t is the capital income tax that comes with a deduction for capital depreciation δ, and r k t denotes the real rental rate on the capital stock. Finally, Π t comprises monopolistic profits from firms. An additional constraint to the optimization program is the law of capital accumulation

k t -(1 -δ) ξ t k t-1 = i t 1 -ϕ i /2 x 2 t , (3)
where i t is the amount of investment in physical capital, x t = i t /i t-1 -1 is the growth rate of investment, and ϕ i > 0 controls the size of investment adjustment costs. The stock of capital is potentially affected by a quality shock ξ t , that follows an AR1 process:

ξ t = (1 -ρ ξ ) ξ + ρ ξ ξ t-1 + ε ξ,t
, where ρ ξ ∈ [0, 1) and ε ξ ∼ N 0, σ 2 ξ . First-order conditions with respect to deposits, bonds and labor supply imply

E t {β t,t+1 r t /π t+1 } = 1, (4)
u ,t + (1 -τ t ) u c,t w t / (1 + µ) = 0, (5)
where β t,t+1 = βu c,t+1 /u c,t and where π t = p t /p t-1 is the inflation rate. The first equation prices government bonds. The second equation relates the marginal disutility of working to the real wage, expressed in terms of the marginal utility of consumption. We define q t p t u c,t as the Lagrangian multiplier associated with the capital accumulation constraint, and derive the following first-order conditions with respect to the capital stock and investment:

E t β t,t+1 q t+1 (1 -δ) ξ t+1 + (1 -η t ) r k t+1 + η t δ = q t , (6)
q t 1 -ϕ i /2 x 2 t -ϕ i x t (1 + x t ) + E t β t,t+1 q t+1 ϕ i x t+1 (1 + x t+1 ) 2 = 1. (7)

Firms

A perfectly competitive representative firm produces a final consumption good y t using a continuum of intermediate goods indexed in j ∈ [0, 1], according to the following production function:

y t = 1 0 y t (j) θ-1 θ dj θ θ-1 , (8)
where y t (j) denotes the time t input of intermediate good j and θ the elasticity of substitution across intermediate goods. The firm takes the price of output p t and the input price p t (j) as

given. Profit maximization leads to the following first-order condition:

y t (j) = (p t (j) /p t ) -θ y t . (9)
Substituting ( 8) into (9) yields the following relationship between the aggregate price level and the price of intermediate goods:

p t = 1 0 p t (j) 1-θ dj 1 1-θ . ( 10
)
Intermediate good j ∈ (0, 1) is produced under monopolistic competition using capital k t-1 and labor t with the following production function:

y t (j) = (ξ t k t-1 (j)) ι t (j) 1-ι , (11)
where ι ∈ (0, 1) is the share of capital in value-added. Intermediate good producers rent the effective capital stock and hire workers in perfectly competitive markets. Profits are distributed to the households at the end of each period. Recalling that r k t and w t denote the real rental rate on capital and the real wage, respectively, the real marginal cost of intermediate good producers is:

s t = ι ι (1 -ι) 1-ι -1 r k t ι (w t ) 1-ι . ( 12
)
Profits are [p t (j) /p t -s t ] p t y t (j), where p t (j) is the price of the good produced by firm j in period t. We assume there are Calvo price-setting contracts, where 1/ (1 -γ) and γ p respectively represent the average length of contracts and an indexation parameter with respect to past inflation. The optimal pricing conditions are standard and therefore not reported.

Government, Central Bank and Aggregation

Given our assumptions regarding taxes, the budget constraint writes:

b t = r t b t-1 + p t (g t -τ t w t t -η t (r kt -δ) k t-1 -µc t ) . ( 13
)
where b t is the amount of nominal bonds issued by the government. Expressed in real terms, the budget constraint writes

b r t = r t b r t-1 /π t + g t -τ t w t t -η t (r kt -δ) k t-1 -µc t . ( 14
)
where b r t = b t /p t . The stability of the debt-to-GDP ratio in the long run is ensured by the following tax rules:

τ t = τ + d τ b r t-1 -b r (15)
η t = η + d η b r t-1 -b r , (16)
where the parameters d τ and d η respectively measure the responsiveness of the labor income tax and the capital income tax to the deviation of the ratio from its initial steady state value.

Government spending evolve following a standard AR(1) process:

log (g t /g) = ρ g log (g t-1 /g) + ε g,t ., (17)
where ρ g ∈ [0, 1) and ε g ∼ N 0, σ 2 g , and where g is the initial steady state level of government expenditure. The Central Bank sets the gross nominal interest rate r t according to

r t = max (1, rn t ) , (18)
where rn t is the desired gross nominal interest rate that evolves according to:

log(rn t /r) = d π log (π t /π) + d y log (y t / y t ) , (19)
In the above equation, π t is the inflation rate and y t is the natural level of output.

(ξ t k t-1 ) ι 1-ι t = 1 0 y t (j) dj = y t dp t , (20)
Final goods:

y t = c t + i t + g t (21) Government bonds: b t = b g t (22)
Deposits:

d t = 0 (23)
where dp t = 1 0 (p t (j) /p t ) -θ dj ≥ 1 measures the dispersion of prices.

Calibration and set-up

The model is quarterly and our calibration is intended to capture an average OECD country.

We adopt a formulation of the utility where private and public goods provide direct utility:

u (c t , g t , n t ) = log ( c t ) -ω 1+ψ t / (1 + ψ) , (24)
where

c t = κc ν-1 v t + (1 -κ) g ν-1 v t ν ν-1 , ν > 0. ( 25
)
In equation ( 25), κ denotes the weight of private consumption in the effective consumption index, and ν is the elasticity of substitution between the private and the public good. When ν = 0, public and private goods are pure complements. As ν increases, private and public goods become more and more substitutable, and pure substitutability arises when ν → ∞. This specification is in the spirit of [START_REF] Leeper | Clearing Up the Fiscal Multiplier Morass: Prior and Posterior Analysis[END_REF], but our choice of a CES specification is justified by the need to capture the diminishing marginal returns to public spending in order to achieve a given level of effective consumption, ceteris paribus. Empirical evidence favor estimates pointing to a mild complementarity (see [START_REF] Bouakez | Why Does Private Consumption Rise after a Government Spending Shock?[END_REF]) and we set ν = 0.45 in the benchmark calibration. 3 We also impose the value of κ so that the marginal utilities of private and public goods are equal in the steady state, an application of Samuelson's principle. 4

The discount factor is β = 0.99 implying an annual real interest rate of 4.1 percents. The inverse of the Frisch elasticity of labor supply is typically a controversial and important parameter.

We impose ψ = 3 to capture relatively sluggish labor markets, a value that lies in-between the calibration of [START_REF] Corsetti | Sovereign Risk, Fiscal Policy, and Macroeconomic Stability[END_REF] and that of [START_REF] Galí | Understanding the Effects of Government Spending on Consumption[END_REF]. Because this parameter is both non-consensual, and key in generating our results, a sensitivity analysis will be conducted. On the production side, the share of capital is ι = 0.33, and the steadystate depreciation rate is δ = 0.018 (7% annually). The investment adjustment cost parameter, ϕ i , is set to 1.8. The steady-state mark-up is 30%, implying θ = 4.33, the Calvo parameter on price contracts is γ = 0.75 a standard value in quarterly models of price adjustments, and the indexation parameter is γ P = 0.5. The feedback parameters of the fiscal rules are d τ = d η = 0.1, which ensures medium-run fiscal solvency, and makes fiscal policy passive in the sense of [START_REF] Leeper | Equilibria under 'Active' and 'Passive' Monetary and Fiscal Policies[END_REF]. These values are quite larger than usually estimated to built intuition.

Given their importance in generating our results, a sensitivity analysis with respect to these parameters will be conducted. Parameters of the Taylor rule are d π = 1.5 and d y = 0.125 and the persistence parameters of shocks are ρ ξ = 0.7 and ρ g = 0.85. In the baseline calibration,

we impose b g / (4y) = 0.8 and adjust the value of ω to get = 0.3. Both targets are realistic when it comes to match observed values in OECD countries. Whenever the debt-output ratio is varied, ω remains unchanged. In line with OECD averages, the share of public consumption in GDP is s g = g/y = 0.2, the capital income tax rate is η = 0.2, the consumption tax rate is µ = 0.15 and the labor income tax rate is adjusted to match the debt-to-GDP ratio target, implying τ = 0.2207 when b g / (4y) = 0.8, a number well within OECD average labor income tax measures. Table 1 summarizes our parameter values.

In the next paragraphs we consider various steady-state levels of debt-to-GDP ratios b g / (4y).

When varying b g / (4y), we let the labor income tax rate adjust to maintain fiscal solvency in the steady state. This approach is backed by a quick glance at the data. Indeed, we show in Appendix A that debt-to-GDP ratios relate positively to labor income taxes across years and countries, using two separate datasets. 5 In any case our results are also robust to keeping the steady-state labor income tax rate constant and using a steady-state lump-sum tax to keep the government budget constraint in balance when varying the steady-state level of public debt (see

3
This second value is also broadly consistent with the point estimates found by [START_REF] Auray | Episodes of War and Peace in an Estimated Open Economy Model[END_REF], between 0.5 and 0.6, and with the estimates of [START_REF] Leeper | Clearing Up the Fiscal Multiplier Morass: Prior and Posterior Analysis[END_REF], according to which public and private goods are complements.

4 The value of κ that is consistent with the optimal provision of public good in the steady-state is indeed uc = ug and implies:

κ = g -1 ν / c -1 ν + g -1 ν . 5
The relation is less clear for consumption tax rates but we do not consider this instrument in our analysis.

Impulse response functions

In the following experiments, we investigate the responses of our model economy under two polar cases for the debt level, b g / (4y) = 0.6 referred to as the case of low debt, and b g / (4y) = 1.15, referred to as the case of high debt. In each case, the calibration remains the same except for the steady-state level of debt and for the labor or capital income tax rate. In particular, the labor disutility parameter ω remains unchanged, so when tax rates change the steady-state level of hours worked, capital and output change as well. We get τ = 0.2050 in the case of low debtto-GDP ratio and τ = 0.2481 when the steady-state ratio is high. These numbers are reasonable in comparison to observed labor and capital income tax rates in OECD countries (See Appendix A).

Public spending shocks

Figure 1 reports the Impulse Response Functions (IRFs hereafter) of the economy to a one percent public spending shock with a labor income tax rule. The top panel reports the dynamics when the shock hits at the steady state while the bottom panel reports the dynamics conditional on a 5% capital quality shock that depresses the economy and pushes the nominal interest rate to the ZLB. 6 The model is simulated using the fully non-linear model under perfect foresight, based on a Newton-Raphson algorithm.7 Even though we solve under perfect foresight, the shock that hits in period 1 is a surprise to the agents. Therefore, because there are only shocks at period 1 under all our simulations, the model solution captures accurately the relevant non-linearities of the model, including the ZLB constraint. The black solid line shows the response of the economy with a low steady-state debt-to-GDP ratio (60%) while the red line shows the response with a high ratio (115%).

The top panel of Figure 1 shows that a spending shock has positive effects on output, does not crowd out private consumption much (remember that public and private goods are complements), and lowers investment. The real wage and hours worked both increase since the demand for final goods rises, as shown by the rise in the inflation rate. In terms of public finance, both public debt to GDP and tax rates increase, which contributes to moderate the rise in hours worked further crowd-out private investment compared to the case of lump-sum taxes. 8 The most interesting feature of this figure lies in the differences between a low or a high steady-state debt-to-GDP ratio. With a high steady-state ratio, a public spending shock leads the debt-output ratio to rise much more than with a low initial ratio. For a given dynamics of the nominal interest rate, a country with a higher debt-output ratio faces a larger rollover cost when debt increases, which requires a larger subsequent fiscal adjustment. In addition, a country with a high level of debt also faces a higher level of taxes. Since the cost of taxation is convex, the tax rate has to increase more to produce the required increase in government revenue. Thus, the shock induces both the labor income tax rate and the capital income tax rate to increase more, imposing larger distortions on hours worked, private investment and hence on output. The dynamics of the real wage is further marginally less positive in this case, which translates into slightly smoother inflation and nominal interest rate dynamics.

Let us now focus on the case where the public spending shock hits conditional on a crisis. The latter is triggered by a 5% capital quality shock. The dynamics generated by the public spending shock conditional on a crisis is obtained by taking the difference between the joint shock and a crisis shock only. The bottom panel of Figure 1 thus reports the net effects of the public spending shock. (q) Real wage (s) Inflation rate Black: b g / (4y) = 0.6 and τ = 0.2050. Red: b g / (4y) = 1.15 and τ = 0.2481.

On impact, when the ZLB is strictly binding, an increase in government spending leads to a rise in output, marginal cost, and inflation. The increase in expected inflation lowers the real interest rate, which drives up private consumption much more than when the shock hits at the steady state. This rise in private consumption expenditure leads to a further rise in output, marginal cost, and expected inflation and a further decline in the real interest rate. In addition, at the ZLB, the price of capital rises instead of falling which also boosts private investment instead of depressing it. These results line up very well with the findings of [START_REF] Christiano | When Is the Government Spending Multiplier Large?[END_REF]. Interestingly enough, output increases more in the case of a high initial debt-to-GDP ratio, leading to a larger value of the government spending multiplier. We explain these results in the following way.

When a government spending shock hits at the ZLB, consumption is more strongly crowded-in and private investment is crowded-in instead of being crowded-out. Aggregate demand is thus boosted by much more, which results in a massive rise in labor demand: hours worked and the real wage both rise more at the ZLB. Hours increase by more than 1.5% (instead of 1% out of the ZLB) and the real wage increases by more than 4% (instead of 1.5%). These effects are clearly not new and explained extensively in several papers (see [START_REF] Christiano | When Is the Government Spending Multiplier Large?[END_REF] and the subsequent literature). In addition, as in [START_REF] Erceg | Is There a Fiscal Free Lunch in a Liquidity Trap?[END_REF], fiscal policy becomes self-financing because tax bases increase massively. Hence, the debt-to-GDP ratio falls instead of rising, triggering a fall in the labor and capital income tax rates. The latter further boost the economy by lowering the overall distortions on labor supply and capital accumulation.

Finally, because countries with a higher debt level also have a higher tax level, and because the costs of taxation are convex, these movements are larger for economies featuring a large initial debt-to-GDP ratio. Therefore, when the public spending shock hits at the ZLB and produces a fall in the debt-to-GDP ratio, the high-debt economy features a larger public spending multiplier.

We expect this result to be sensitive to the labor supply elasticity, as the strength of tax base dynamics depends on how much households are willing to increase their labor supply after a spending shock, but also to the parameters of the tax rules.

Robustness

Do our results depend on adjusting the steady-state labor income tax rate when varying the steady-state level of debt? Figure 6 in Appendix B shows that they do not. Figure 6 replicates Figure 1 keeping the steady-state labor income tax rate constant and using a steady-state lumpsum tax to keep the government budget constraint balanced in the steady state. Our results remain basically unchanged. Indeed, crucial to our results is the change in the tax rates after a spending shock, not the level of the tax rates. Changes in public debt are larger when steadystate levels of public debt are higher, which requires larger changes in tax rates to stabilize public debt, which reduces the spending multiplier when the spending shock hits at the steady state and raises the spending multiplier when the shock hits at the ZLB.

Another potential extension is the inclusion of rule-of-thumb, or hand-to-mouth households, as in [START_REF] Galí | Understanding the Effects of Government Spending on Consumption[END_REF], as this assumption matters for the transmission of fiscal policy shocks. In this case, we consider two types of households. The optimizing households behave exactly as exposed in the model section, while the constrained, or hand-tomouth households, do not have access to financial markets, and therefore do not accumulate government bonds or capital. As such, they simply consume their current-period income. While they supply labor in a similar way as optimizing households, their consumption is determined by their budget constraint:

u r ,t + (1 -τ t ) u c r ,t w t / (1 + µ) = 0 (1 + µ) c r t -(1 -τ t ) w t r t = 0
where r t and c r t respectively denote the hours worked by optimizing households and their consumption level. The first-order conditions given by equations ( 4)-( 5) now apply to o t and c o t , that respectively represent the hours worked by optimizing households and their consumption level.

Both types of households face the same labor income tax rate, and the same real wage. Finally, aggregate hours worked and aggregate consumption are then given by t = λ o t + (1 -λ) r t and c t = λc o t + (1 -λ)c r t , where λ is the constant proportion of optimizing households in the economy. We impose λ = 0.75, and recalibrate the model -essentially the labor disutility parameters to get = o = r = 0.3 in the steady state. Keeping the rest of the calibration unchanged, the model with hand-to-mouth households implies c o = 0.6390 and c r = 0.2970 in the steady state.

Further, Figures 7 and8 in Appendix B replicate both panels of Figure 1 with optimizing and hand-to-mouth households.

Figures 7 and8 show that our results are preserved and even magnified by the introduction of hand-to-mouth households. Let us first look at the effects of a spending shock at the steady state.

As shown already, spending shocks raise the real wage, which allows hand-to-mouth households to raise their consumption level. As a result, aggregate consumption is less crowded out, and investment more crowded out than in our baseline model. In addition, the increase in hours worked is slightly larger with hand-to-mouth households, as a result of a larger increase of the real wage. Notice however that hand-to-mouth households, because they experience a rise in their consumption level, increase their hours worked only by a very small amount due to the wealth effect. The effects of a spending shock at the steady state are thus in the same ballpark than in our baseline model, and qualitatively similar: the output effects of a government spending shock are larger when the initial level of debt is smaller, because the response of tax rates remains qualitatively unchanged. They must rise because the spending shock raises pubic debt, and more so with a large initial level of debt. Let us now discuss the effects of spending shocks with handto-mouth households at the ZLB. In this case too, the rise in the real wage is almost transmitted one for one to the consumption of hand-to-mouth households, they experience a small rise in their level of hours worked, but the very large (6%) rise in their consumption level contributes to raise aggregate consumption and aggregate output by more than in our baseline model. The output effects of a spending shock at the ZLB are thus larger with hand-to-mouth households.

Since the real wage rises more and since aggregate hours worked rise more than in our baseline model, the tax base expands more which implies a larger fall in the tax rates, especially the labor income tax rate, both for low and high initial levels of public debt. As such, the output effects are both larger but the difference is also larger. If anything, the introduction of hand-to-mouth households thus amplifies our main result. 9

Government spending multipliers

We now investigate more systematically the extent to which the initial debt-to-GDP ratio affects the value of present-value public spending multipliers. While the impact multiplier is an important measure of the effectiveness of fiscal policy, looking at longer horizons matters insofar the dynamics of subsequent fiscal adjustments matters for the overall effectiveness of fiscal policy.

Our preferred measure will therefore be the present-value multiplier at different horizons. The latter is defined as the discounted cumulative increase in output over T periods that results from the discounted cumulative increase in public spending over T periods after a spending shock in period 1:

P V M T = T j=1 β j (y t+j -y t ) T j=1 β j (g t+j -g t )
.

(26)

Table 2 below reports the values of the present-value multipliers at 2 years (T = 8) and at 5 years (T = 20) under various configurations, when the shock hits at the steady state or conditional on a crisis. We consider both a labor income tax rule and a capital income tax rule, and investigate the sensitivity of our result to the Edgeworth complementarity parameter, to the Frisch elasticity of labor supply and to the strength of the response of tax rates in the tax rules.

Quantitatively speaking, multipliers at 2 years broadly range from 0.05 to 0.5 when the shock hits at the steady state, and from 0.6 to 1.5 at the ZLB. These intervals are wide but the value of multipliers is crucially affected by the complementarity parameter ν and by the labor supply elasticity 1/ψ. Empirically, ν and ψ are more likely to be close to the baseline calibrated values ν = 0.45 and ψ = 3, suggesting multipliers at 2 years between 0.18 and 0.29 at the steady state, and between 1.4 and 1.45 at the ZLB. These are relatively consensual values in the literature.

As already suggested by the analysis of the IRFs, multipliers at the ZLB are much larger than multipliers at the steady state, both at the horizon of 2 years and 5 years. This statement is true both at low and high levels of debt. While this result may not be new, the fact that multipliers 9 See also Table 2 for the size of multipliers at different horizon with hand-to-mouth households. are large at the ZLB are clearly the explanation for the fact that fiscal policy becomes selffinancing (see [START_REF] Erceg | Is There a Fiscal Free Lunch in a Liquidity Trap?[END_REF]), leading public debt to fall and future tax rates to drop as well. Because of their distortionary nature, this in turn impacts the dynamics of factor prices and quantities, and then the dynamics of output. Indeed, when the shock hits at the steady state, multipliers are either slightly lower (2 years) or substantially lower (5 years) in the high debt economy, because debt and taxes increase after the shock. The pattern reverses at the ZLB because debt and taxes fall after the shock. These results hold qualitatively for all the parameter values considered in Table 2.

In addition, when public and private goods are complements, multipliers are always larger whatever the configuration or the calibration. This too shall not be surprising as an increase in public spending crowds-in consumption (at the ZLB) or reduces consumption crowding-out (at the steady state) when goods are complements, while consumption is clearly crowded-out when goods are substitutes. However, as will be clear when discussing the role of key parameters such as the labor supply elasticity, whether goods are complements or substitutes is far from neutral, as this not only affects the dynamics of consumption but also that of labor supply and aggregate demand.

The implication of having a significant part (25%) of the household that are financially constrained or hand-to-mouth on multipliers depends on whether the spending shock hits at the steady state or conditionally on a shock that pushes the economy to the ZLB. At the steady state, spending multipliers are lower than without hand-to-mouth agents. Because part of aggregate consumption is more clearly crowded-in after a spending shock, private investment falls more.

While both effects have opposite implications for the size of spending multipliers, the second effect dominates and the multiplier is smaller than without hand-to-mouth agents. Further, our result according to which a larger initial debt level is associated with a lower multiplier is preserved, and the difference between multipliers at low vs. higher initial levels of debt remains broadly unchanged. At the ZLB, the presence of hand-to-mouth agents increases the size of spending multipliers: because of the dynamics of the real interest rate at the ZLB, investment is not crowded out anymore and the first effect-larger crowding in of private consumption-dominates. Again, this turns out to comfort our results according to which the spending multiplier is larger with higher initial levels of public debt at the ZLB. In addition, because the presence of hand-to-mouth agents raises the absolute value of multipliers at the ZLB, the difference between multipliers at low and high debt levels increases.

Let us now discuss the implications of some key parameters on our results, starting with the labor supply elasticity 1/ψ. First, when labor becomes more elastic, labor supply becomes more sensitive to the variations of the after-tax real wage. This is the direct effect through which a more elastic labor supply (lower ψ) raises labor supply and thus output in the short run after a spending shock. It also lowers labor supply and thus output more in the medium run after an increase in the tax rate, that lowers the after-tax real wage. Second, when labor supply becomes more elastic, the real wage rises less after a spending shock, as the labor supply curve is flatter.

It attenuates the rise in the equilibrium real wage, the rise in private consumption and the rise in aggregate demand. As such, it contributes negatively to aggregate output. This indirect effect is stronger when consumption is crowded-in after a spending shock, that is, when public and private goods are complements (ν < 1) or at the ZLB. Clearly, short-run multipliers (2 year) at the steady state are always larger when labor becomes more elastic, as the direct effect dominates.

On the contrary, short-run multipliers at the ZLB are always lower when labor supply becomes more elastic, as the indirect effect dominates since consumption is strongly crowded-in. This pattern also applies to medium run (5 years) multipliers.

We now shift the discussion to the impact of tax rule parameters d τ and d η . The baseline calibrated value is quite larger than suggested by the empirical estimates, a choice that was made to illustrate our main results graphically.10 However, Table 2 shows very clearly that our main result remains qualitatively unchanged when considering more realistic values of d τ and d η .

When the spending shock hits at the steady state, a less aggressive response of taxes raise the value of multipliers. Public debt increases after the shock, so a larger rise of tax rates implies more distortions imposed on labor or capital, and therefore lower multipliers. When the shock hits at the ZLB, less aggressive tax rules actually lower the value of multipliers. As already explained and shown in Figure 1, fiscal policy becomes self-financing at the ZLB and a spending shock implies a large rise in tax bases. Public debt falls, and if taxes respond less firmly to those changes in public debt, they fall less and subsequently produce a smaller fall in the distortions imposed on capital accumulation and labor, a smaller boost to the economy and slightly lower multipliers.

To summarize, our main result is the fact that high levels of public debt in the steady state produce lower multipliers when the public spending shock hits at the steady state but larger multipliers when the shock hits at the ZLB. The chief reason relies on the combination of two features of our economy: (i ) the fact that fiscal policy becomes self-financing at the ZLB, therefore implying a fall in public debt and then a fall in distortionary taxes, while the opposite occurs in normal times; and (ii ) the fact that the rollover cost of debt and the output costs of taxation are higher when debt and taxes are larger. When debt is initially high, taxes are initially high as well, and the responses of debt and taxes are larger after a spending shock: public debt and taxes increase more in the high debt economy when the shock hits at the steady state, and fall more when the shock hits at the ZLB. Our main result is robust to considering substitutable public and private goods, and therefore does not rely on an exotic assumption, even though we argue that complementarity is empirically more realistic. It is also insensitive to the type of tax used to ensure fiscal solvency in the long run, as long as the tax used is distortionary: all the above results hold when considering tax rules separately. Finally, it holds true under various tax rule parameters and for a range of empirically realistic values for the labor supply elasticity.

Optimized public spending rules in a crisis

We now consider endogenous variations of public spending. Our goal is to assess the extent to which higher levels of debt imply different optimized responses of public spending to a crisis shock. On the one hand, public spending might be more effective in stimulating output and hence escape a crisis. On the other hand, too much public spending could become costly if fiscal policy becomes non-self-financing and thus provokes a rise in future distortionary taxes. Thus, what is the welfare-maximizing path of public spending? Do higher initial levels of debt imply larger or smaller changes in public spending? Or should this more effective tool be used less intensively? To anwser these questions, we abstract from spending shocks per se and consider that the government follows a simple spending rule. The rule makes government spending react directly to the initial capital quality shock with some persistence:

g t -g = ρ * g (g t-1 -g) -d * g ε ξ,t (27)
The coefficients ρ * g , d * g are chosen to maximize the Hicksian consumption equivalent that solves:

E t ∞ s=t β s-t u (c s , g s , s ) - ∞ s=t β s-t u (c (1 + /100)) , g, ) = 0 (28)
Variable is expected to be negative after a large economic downturn. It measures the steadystate percentage of consumption loss associated with experiencing macroeconomic fluctuations, induced in our case by a negative capital quality shock. As seen in Appendix B, the shock implies an important drop in private consumption, inducing large current-period utility losses. ν. As such, raising public spending raises the welfare of households.11

Second, the size of the government intervention is affected by the initial debt-to-GDP ratios, and by the Edgeworth complementarity parameter ν. We shown in the previous sections that government spending policies are more effective (in the sense of a larger output multiplier) at the ZLB when the initial debt-to-GDP ratio is high. Figure 2 confirms this, as the optimized response of public spending is slightly smaller in this case, but produces comparable or larger positive effects on output, consumption and investment. This holds true whether public and private goods are complements or substitutes. In the latter case, public spending rise less, around 2 pp of steady-state GDP, while the increase is quite larger, around 3 pp of GDP, when public and private goods are substitutes.

Table 3: Optimized policy rules. Note: c denotes the welfare loss from the crisis with constant public spending, in percents of permanent consumption. c+g denotes the welfare loss from the crisis with an active optimized spending rule. ∆ is the welfare gain from active spending policies: ∆ = c+g -c . ρg and dg are the optimized coefficients of the spending rule.

Columns "2 years" and "5 years" respectively denote the present-value multipliers of output at the horizon of 8 quarters and 20 quarters.

In addition, Table 3 confirms the results given by the IRFs: high initial levels of debt and complementarity between public and private goods are associated with less aggressive spending policies, given that the latter are more effective at stabilizing output. Indeed, output multipliers are larger at the horizon of 4 or 20 quarters when debt levels are initially high or when public and private goods are complements in utility.12 Closest to the above analysis are [START_REF] Bilbiie | Is Government Spending at the Zero Lower Bound Desirable?[END_REF] and [START_REF] Nakata | Optimal Government Spending at the Zero Lower Bound: A Non-Ricardian Analysis[END_REF], who derive Ramsey policies subject to ZLB episodes. Some of our results are in perfect accordance with theirs. In particular, the fact that government interventions imply an increase in public spending. Further, the size of government interventions are broadly comparable, as well as the size of welfare gains from active policies at the ZLB. Our results differ however along crucial dimensions. First, [START_REF] Bilbiie | Is Government Spending at the Zero Lower Bound Desirable?[END_REF] assume lump-sum taxation. As such, the initial level of debt is irrelevant to the analysis. Second, Nakata (2017) finds that higher initial levels of debt should imply more aggressive spending policies while we find the opposite.

In our view, this is due to the fact that the labor income tax follows a systematic feedback rule while this variable is an optimized policy instrument in [START_REF] Nakata | Optimal Government Spending at the Zero Lower Bound: A Non-Ricardian Analysis[END_REF]. In any case the design of optimal policies is not the main focus of our paper. We simply derive optimized rules to contrast the potential impact of the initial level of debt on the design of government interventions at the ZLB, and show that it potentially has non-trivial consequences.

Conclusion

This paper investigates the relation between the initial level of debt and the effectiveness of public spending shocks in a New-Keynesian model with capital accumulation, distortionary taxes and a lower bound constraint on the nominal interest rate. We find that countries with high debt are more fragile in the event of a crisis, as they experience larger economic downturns. In line with the literature, we also find that the spending multipliers are lower with a high initial debt-to-GDP ratio during normal times, and larger than during normal times when the ZLB is binding.

The novel result of the paper is that economies with a high initial debt-to-GDP ratio feature larger spending multipliers at the ZLB. The result is driven by the fact that fiscal policy becomes self-financing at the ZLB, allowing the debt-to-GDP ratio to fall after a spending shock, then producing a fall in distortionary taxes. Because the efficiency costs of taxation are convex, the subsequent fall in tax rates are larger when debt is initially higher, leading to larger positive effects on output. This raises the spending multipliers. Finally, this result implies that optimized spending policies at the ZLB are affected by the initial level of debt: a lower increase in public spending is required with high initial levels of public debt, as fiscal policy is more effective at stimulating output and provides larger reductions in the welfare losses from the crisis that pushed the economy at the ZLB.

A Debt-to-GDP and tax rates in the data

We take a quick look at the relation between debt-to-GDP ratios and tax rates using various data sources. First, we take tax data from the dataset compiled by [START_REF] Mcdaniel | Average Tax Rates on Consumption, Investment, Labor and Capital in the OECD 1950-2003[END_REF]. Following the method of [START_REF] Mendoza | Effective Tax Rates in Macroeconomics: Cross-country Estimates of Tax Rates on Factor Incomes and Consumption[END_REF], she computes the consumption, labor income and capital income tax rates for 15 advanced countries from 1950 to 2014. 13 We relate each of these tax rates with debt-to-GDP ratios taken from the updated [START_REF] Reinhart | From Financial Crash to Debt Crisis[END_REF] dataset for the same countries and the same time span in the following scatter plots. Notes: Tax rates are taken from the updated dataset of [START_REF] Mcdaniel | Average Tax Rates on Consumption, Investment, Labor and Capital in the OECD 1950-2003[END_REF]. Debt-to-GDP ratios are taken from the updated dataset of [START_REF] Reinhart | From Financial Crash to Debt Crisis[END_REF].

The above graphs are not a thorough empirical analysis but clearly suggest that higher levels of debt-to-debt are associated with larger tax rates on average over time. They are consistent with the idea of long-run fiscal sustainability. In addition, they suggest that labor income tax rates are more significantly and positively related to debt-to-GDP ratios. To confirm this first-pass analysis, we also plot various measures of labor income tax rates using OECD data. We take the OECD measures of total labor wedges at for different levels of wages (67%, 100%, 133% and 167% of the average real wage) from the OECD Tax Database for 34 OECD countries between 2000 and 2016, and relate those labor income tax measures to the debt-to-GDP levels reported in the Central Government Debt OECD dataset for the same countries over the same period.14

Again, we pool all data together to get an idea of the average relation between debt-to-GDP and labor income tax rates. The second set of graphs confirms the positive relation between Note: Labor income tax rates are taken from the OECD tax database. They are computed using personal income taxes and social security contribution rates on gross labor income. Debt-to-GDP ratios are taken OECD Central Government Debt dataset.

debt-to-GDP ratios and labor income tax rates. Overall, this simple descriptive analysis shows that our assumption that consists in adjusting either the labor or the capital tax rate in the steady state when we vary the steady-state debt-to-GDP ratio is broadly consistent with the data. (q) Real wage (s) Inflation rate

B Additional figures

Figure 1 :

 1 Figure 1: IRFs to a 1% public spending shock. At steady state.

Figure 2

 2 Figure2reports the optimized dynamic response of government spending after a 5 percent negative capital quality shock, as well as the net effects of this spending policy on macroeconomic aggregates. The latter are computed as the difference between the responses with an optimized policy and the responses with constant public spending.

Figure 2 :

 2 Figure2: Optimized response of government spending after a 5% negative capital quality shock and net effect on macroeconomic aggregates. c t and g t complements (ν = 0.45)

Figure 4 :

 4 Figure 4: Labor income tax rates and debt-to-GDP for 34 OECD countries, 2000-2016 (a) 67% of the average wage

Figure 5 :

 5 Figure 5: IRFs to a 5% negative capital quality shock.

  g / (4y) = 0.6 and τ = 0.2050. Red: b g / (4y) = 1.15 and τ = 0.2481.

Figure 6 :

 6 Figure 6: IRFs to a 1% public spending shock with a constant steady-state labor tax. At steady state.

  Black: b g / (4y) = 0.6. Red: b g / (4y) = 1.15. Varying steady-state levels of public debt are financed using a steadystate lump-sum tax or transfer instead of adjusting the steady-state labor income tax rate. The latter remains fixed at τ = 0.2207.

Figure 7 :

 7 Figure 7: IRFs to a 1% public spending shock with hand-to-mouth households at the steady state.(a) Output

  g / (4y) = 0.6 and τ = 0.2050. Red: b g / (4y) = 1.15 and τ = 0.2481.

Figure 8 :

 8 Figure 8: IRFs to a 1% public spending shock with hand-to-mouth households at the ZLB.

  g / (4y) = 0.6 and τ = 0.2050. Red: b g / (4y) = 1.15 and τ = 0.2481.

2

  Parametersd π and d y are the elasticities of rn t to inflation and the output gap, respectively. The central bank sets r t equal to rn t if and only if its policy rule implies a non-negative level for the nominal interest rate. Otherwise, the ZLB binds and r t equals one. A competitive equilibrium in our model is defined as a situation where (i) households and firms optimize for a given path of policy instruments, and (ii) prices clear markets according to:

	Intermediate goods:

Table 1 :

 1 Baseline parameter values

	Discount factor	β = 0.99
	Edgeworth preference parameter	ν = 0.45
	Inverse of the Frisch elasticity of labor supply	ψ = 3
	Steady-state depreciation rate of capital	δ = 0.018
	Production function, capital parameter	ι = 0.33
	Investment adjustment cost parameter	ϕ i = 1.8
	Steady-state mark-up	θ/ (θ -1) = 1.3
	Calvo parameter	γ = 0.75
	Indexation parameter	γ p = 0.5
	Fraction of time spent working	= 0.3
	Labor disutility parameter	ω = adjusted
	Government debt to annual GDP	b g / (4y) = 0.8
	Government spending to GDP	s g = 0.2
	Labor income tax rate	τ = 0.2207
	Capital income tax rate	η = 0.2
	Consumption tax rate	µ = 0.15
	Tax rule parameters	d τ = d η = 0.1
	Taylor rule, response to inflation	d π = 1.5
	Taylor rule, response to output gap	d y = 0.125
	Persistence of public spending shock	ρ g = 0.85
	Persistence of capital quality shock	ρ ξ = 0.7
	Figure 6 in Appendix B).

Table 2 :

 2 Multipliers at different horizons.

				c t and g t complements (ν = 0.45)
		Low debt (b g / (4y) = 0.6)		High debt (b g / (4y) = 1.15)
		Steady state	ZLB		Steady state	ZLB
		2 years 5 years	2 years 5 years	2 years 5 years	2 years 5 years
	ψ = 3	0.207 -0.263	1.403	1.124	0.177 -0.359	1.492	1.321
	ψ = 2	0.308 -0.196	1.363	1.079	0.277 -0.298	1.480	1.326
	ψ = 1	0.494 -0.061	1.191	0.848	0.468 -0.159	1.292	1.059
	d τ = d η = 0.05	0.255 -0.131	1.426	1.164	0.239 -0.188	1.470	1.286
	d τ = d η = 0.04	0.265 -0.099	1.424	1.162	0.253 -0.146	1.464	1.269
	d τ = d η = 0.03	0.275 -0.064	1.420	1.156	0.266 -0.100	1.453	1.244
	d τ = d η = 0.02	0.286 -0.027	1.413	1.146	0.280 -0.052	1.437	1.210
	d τ = 0.1, d η =	0.232 -0.194	1.389	1.119	0.210 -0.271	1.452	1.269
	d τ = 0.05, d η =	0.273 -0.073	1.424	1.164	0.263 -0.114	1.460	1.257
	d τ = 0.02, d η =	0.295	0.006	1.416	1.148	0.291 -0.011	1.437	1.196
				c t and g t substitutes (ν = 1)
		Low debt (b g / (4y) = 0.6)		High debt (b g / (4y) = 1.15)
		Steady state	ZLB		Steady state	ZLB
		2 years 5 years	2 years 5 years	2 years 5 years	2 years 5 years
	ψ = 3	0.057 -0.301	0.765	0.501	0.039 -0.359	0.864	0.672
	ψ = 2	0.092 -0.318	0.705	0.407	0.073 -0.381	0.826	0.615
	ψ = 1	0.158 -0.351	0.584	0.195	0.141 -0.412	0.709	0.414
	d τ = 0.05	0.106 -0.171	0.782	0.561	0.096 -0.205	0.836	0.662
	d τ = 0.04	0.116 -0.138	0.785	0.573	0.109 -0.167	0.829	0.657
	d τ = 0.03	0.127 -0.104	0.787	0.585	0.121 -0.126	0.820	0.650
	d τ = 0.02	0.138 -0.066	0.789	0.597	0.134 -0.081	0.811	0.642
	d τ = 0.1, d η =	0.078 -0.238	0.756	0.516	0.064 -0.285	0.842	0.657
	d τ = 0.05, d η =	0.121 -0.117	0.777	0.574	0.114 -0.143	0.824	0.655
	d τ = 0.02, d η =	0.146 -0.036	0.786	0.605	0.144 -0.046	0.807	0.639
			c t and g t complements (ν = 0.45) + hand-to-mouth households *
		Low debt (b g / (4y) = 0.6)		High debt (b g / (4y) = 1.15)
		Steady state	ZLB		Steady state	ZLB
		2 years 5 years	2 years 5 years	2 years 5 years	2 years 5 years
	ψ = 3	0.184 -0.291	1.582	1.281	0.144 -0.406	1.715	1.509
	ψ = 2	0.292 -0.211	1.535	1.237	0.251 -0.332	1.722	1.555
	ψ = 1	0.483 -0.066	1.419	1.107	0.447 -0.184	1.671	1.541
	d τ = 0.05	0.226 -0.180	1.552	1.265	0.203 -0.252	1.620	1.399
	d τ = 0.04	0.236 -0.151	1.545	1.259	0.217 -0.212	1.598	1.370
	d τ = 0.03	0.246 -0.118	1.537	1.252	0.231 -0.167	1.576	1.338
	d τ = 0.02	0.257 -0.082	1.528	1.244	0.245 -0.117	1.554	1.302
	d τ = 0.1, d η =	0.206 -0.231	1.552	1.259	0.175 -0.326	1.658	1.443
	d τ = 0.05, d η =	0.243 -0.126	1.531	1.248	0.226 -0.180	1.586	1.350
	d τ = 0.02, d η =	0.266 -0.050	1.517	1.235	0.258 -0.075	1.539	1.277

The consumption tax rate is introduced in the model to allow for a more realistic baseline calibration. In particular, we want both capital and labor income tax rates to be on the increasing part of the Laffer Curve to allow these instruments to be effective in stabilizing the level of debt. However, µ will not be considered as an active policy instrument.

As in[START_REF] Gertler | A Model of Unconventional Monetary Policy[END_REF], variations in the mark-up serve as a proxy for variations in the output gap.

Appendix B provides the IRFs to this shock only. In line with[START_REF] Gertler | A Model of Unconventional Monetary Policy[END_REF], a capital quality shock induces a fall in output, consumption, investment, real wages, hours worked, the price of equity, and inflation, pushing the economy to the ZLB for 4 quarters. It also raises public debt and tax rates. While[START_REF] Gertler | A Model of Unconventional Monetary Policy[END_REF] set a smaller shock value, notice that our model does not feature banking frictions and therefore requires a larger innovation to produce a similar negative output dynamics -as well as a binding ZLB.

The algorithm is a built-in routine of Dynare 4.5.1[START_REF] Adjemian | Dynare: Reference Manual, Version 4[END_REF]). 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[START_REF] Juillard | Dynare : A Program for the Resolution and Simulation of Dynamic Models with Forward Variables through the Use of a Relaxation Algorithm[END_REF]. The algorithm does not rely on linearization.

8 IRFs under lump-sum taxes do not show any difference whether debt is high or low, under any of the situations we consider. These IRFs are available upon request.

* : The share of hand-to-mouth households is 1 -λ = 0.25.

[START_REF] Corsetti | What Determines Government Spending Multipliers?[END_REF] show that this large value is necessary to guarantee the determinacy of the equilibrium.

Our results, although derived in a different environment, are qualitatively and quantitatively consistent with those reported by[START_REF] Bilbiie | Is Government Spending at the Zero Lower Bound Desirable?[END_REF] and[START_REF] Nakata | Optimal Government Spending at the Zero Lower Bound: A Non-Ricardian Analysis[END_REF].

Notice that the values of output multipliers differ quite substantially from those reported in the first sections of the paper. These differences are attributed to the fact that the persistence and size of increases in public spending both depend on the rule, and are no longer determined exogenously. Given this, one should not seek to compare directly the values reported in Table3with those reported in Table2.

The following countries are considered: Australia, Austria, Belgium, Finland, France, Germany, Italy, Netherlands, Spain, Sweden, Switzerland, United Kingdom, United States.

The following countries are considered: Australia, Austria, Belgium, Canada, Chile, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Israel, Italy, Japan, Korea, Luxembourg, Mexico, Netherlands, New Zealand, Norway, Poland, Portugal, Slovak Republic, Slovenia, Spain, Sweden, Switzerland, Turkey, United Kingdom, United States.

comments. Aurélien Eyquem gratefully acknowledges financial support of the ANR program entitled ANR-15-CE33-0001-02 FIRE.


Guillaume Hoffmann

Vincent Tognetti
email: vincent.tognetti@univ-rouen.fr

Laurent Joubert
email: laurent.joubert@univ-rouen.fr

On the influence of dynamical effects on reactivity descriptors

Keywords: Michael acceptors, Electrophilicity, Conceptual DFT, Reactivity indices, Molecular Dynamics, Mayr's scale

In this paper, we investigate the role of structural dynamical effects on reactivity properties of selected Michael acceptors. To this aim, quantum molecular dynamics simulations were performed. Conceptual density functional theory descriptors were then evaluated on selected snapshots, providing statistical indicators suitable to assess the discrepancies between static and dynamical approaches. The implications of these results for building a predictive model correlated to Mayr's electrophilicity index are then discussed, paving the way towards a more realistic account of experimental results.

Introduction

Conceptual density functional theory (CDFT) [1][2][3][4] and the respective chemical reactivity indices that it generates were shown successful in the prediction and characterization of empirical and theoretical concepts, such as electrophilicity [5][6][7][8][9], nucleophilicity [10][11][12], gas-phase basicities [13], Woodward-Hoffmann rule's [14], σ-holes [15,16], trans-effect [17,18], and covering all research fields ranging from organic to inorganic chemistries. In particular, they provide an efficient prediction of the feasibility of a chemical reaction based only on the reactants physicochemical properties. In the case of electrophilicity and nucleophilicity, these theoretical descriptors can be confronted to the corresponding experimental parameters defining the popular Mayr's scales [19][20][21].

Noteworthy, these scales are widely used by experts in organic synthesis for a quick assessment of the outcome of a reaction between known compounds. Indeed, a reliable rule of thumb is that the reaction of a nucleophile and an electrophile will be quantitative if the sum of their Mayr's parameters is higher than -5. Currently, there are 1456 available experimental parameters for reagents such as diverse as carbocations, aldehydes, ketones, imines, alkenes, ylides, aliphatic amines…, thus covering most of the usual arsenal used in organic synthesis.

It currently remains a theoretical challenge to predict these experimental parameters (based on kinetic measurements) for any type of molecule. Certainly, such a predicting tool would be highly valuable for organic chemists dealing with compounds for which experimental parameters have not been yet determined, and even for biologists, for example in the assessment of toxicity of molecules [22]. Besides this predictive power, such a model could also convey chemical rationalization, which could help improving the current reagents by a well-targeted in silico strategy.

In a preceding study [23], we developed a model able of categorizing the electrophilicity reactivity of α-β-unsaturated compounds, which epitomize Michael acceptors, a family of chemical reagents instrumental in the formation of carbon-carbon bonds and ubiquituous in organic synthesis. This model, as almost all of related works (see also reference 24), was based on a static approach of chemical reactivity, and failed at providing a quantitative model with respect to experimental data, a fact also underlined by Mayr in recent studies devoted to Michael acceptors [25,26] and later extended to other reagents like ketones [27].

One may thus wonder whether this failure might be ascribable to the neglect of dynamical features. Indeed, due to thermal agitation, the reactants inside a beaker are generally not frozen in their most stable conformation. This structural breathing may in some cases lead to explore geometries that can significantly differ from the optimized ones, and which would accordingly exhibit significantly different reactivity behaviors.

From the best of our knowledge, the calculation of chemical reactivity descriptors extracted from dynamical trajectories is extremely scarce, probably because static approaches often revealed sufficient to correctly predict experimental results, as exemplified by the benzhylidrium carbocations family for which many high linear correlations were reported and that has served as a workhorse for electrophicility studies for a long time.

As a matter of fact, the predominant focus on rigid or hindered species of such type could account for the prevalence of the static theoretical approach that is, furthermore, much less timeconsuming from a computational perspective. Nevertheless, we want to point out that the dynamical behavior of chemical reactivity indices was seminally investigated by Liu [28] in gas phase for few hydrated clustered cations or anions (like Ca 2+ (H2O)15 or Cl -(H2O)30), but with the main focus not being organic chemical reactivity. Another important study is that presented as a poster by Oller and Vöhringer-Martinez during the QUITEL 2018 conference held in Santiago (Chile), which reports a methodology based on molecular dynamics to define Boltzmann weighted reactivity descriptors [START_REF] Oller | Boltzmann weighted local reactivity descriptors of α,βunsaturated systems by molecular dynamics simulations[END_REF].

In the current work, molecular dynamics simulations in solution on selected Michael acceptors were performed at a semi-empirical quantum chemical level of theory in order to follow the evolution over time of some of the most relevant reactivity descriptors considered in our previous paper [23], and to assess to what extent they differ from their static properties and may afford hints on how to improve reactivity prediction by means of conceptual DFT.

Computational Details

All density functional theory calculations were performed using the Gaussian 09 package [START_REF] Frisch | Gaussian 09 Revision D.01[END_REF].

Consistently with our previous study [23], molecular geometries were first optimized using the meta-hybrid M06-2X exchange-correlation functional [START_REF] Zhao | [END_REF] in conjunction with the aug-cc-pVTZ basis set, with recommended 'ultrafine' integration grid. Solvent effects (dimethylsulfoxide, DMSO) were taken into account by a polarizable continuum model (in the last implementation of the integral equation formalism (IEFPCM) [32]) for all computations (static and dynamical approaches). These optimized structures were then used to start unbiased molecular dynamics simulations at the quantum chemical semi-empirical PM6 level of theory [33] (within IEFPCM), which allows long simulations at a low computational cost, in the NVT canonical ensemble. The time step for propagation was set to 1 fs. The PM6 wavefunction was converged at any step (Born-Oppenheimer-type dynamics). The total simulation time was equal to 100 ps, ensuring representative sampling. Temperature (T=298.15 K) was controlled by velocity scaling.

Structures were then extracted every 1 ps. The 20 first ones were considered to be part of the equilibration process, while the 80 following ones were considered for data production. Indeed, as shown in Graph S1 in the supplementary information file, temperature was found to be properly converged after 20 ps for every simulation. Subsequent single point calculations at the M06-2X/aug-cc-pVTZ level of theory were carried out to evaluate the chosen conceptual DFT reactivity descriptors.

These descriptors can be classified into two main classes: molecular (also known as "global") and atomic ones. Those belonging to the first category are here: three avatars of the electronic chemical potential (µ, µ + , µ -), the molecular hardness η, softness S, and hyperhardness γ [34], the electrophilicity index ω [35,36], and the electroaccepting power ω + . Atomic descriptors considered here are: condensed electrophilic and nucleophilic Fukui functions f + , f -, and the dual descriptor f (2) [37,38], supplemented by grand-canonical extensions (µ + f + , µ -f -, ωf + , Sf (2) , ωf (2) , s (2) , s + =S f + , s -=S f -). We refer the interested reader to our previous paper for the corresponding mathematical definitions [23].

Besides, two main computational approaches can be implemented to evaluate these descriptors: either finite difference linearization (FDL) or the Koopmans-like frozen molecular orbital (FMO) scheme, which are not equivalent [39]. For each descriptor, both were here considered. Atomic condensation was performed within the framework of Bader's atoms-in-molecules theory (QTAIM) [START_REF] Bader | Atoms in Molecules: A Quantum Theory[END_REF][START_REF] Popelier | Atoms in Molecules An Introduction[END_REF] using Todd Keith's AIMAll software [START_REF] Keith | [END_REF].

It is important to notice that two carbon atoms denoted Cα and Cβ (see Figure 1) may be involved in the carbon-carbon bond formation process during a Michael addition, so that condensation of the two atomic corresponding basins was performed. In ref. 23, we also defined the total electrophilicity of the Cα=Cβ double bond, f ( ), by simply summing the electrophilicity of each atom, f ( ) and f ( ).

In total, for each molecule, 9 global descriptors and 22 condensed ones were evaluated at both the FDL and FMO theory levels at each selected point along the dynamical trajectory.

Atomic units (a.u.) are used throughout the paper unless explicitly otherwise stated. Numerical values corresponding to the statistical analysis of all descriptors are gathered in the supplementary information file (Tables S1-S80). Experimental values were extracted from Mayr's online database [43].

Results and discussion

In order to investigate dynamical effects on chemical reactivity indices, we selected 11 molecules (represented in Figure 1) out of our previous dataset [23] that consisted of 35 Michael acceptors.

Let us recall that we divided it into three subgroups: "good" Michael acceptors, "medium", and "low" ones. In order to be as representative as possible, we thus selected in the current paper molecules stemming from each of these three representative families. Note that two molecules of these sets are symmetric (compounds 1 and 8) at their energy minimum, a property that will not be kept during the molecular dynamics at a given time. However, the time-averaged properties for the two carbons that are equivalent in the optimized structures were found, as expected, almost equal at the end of the simulation.

In summary, this new working subset covers a range of experimental electrophilicity values from (max), and the standard deviation is 0.009 a.u., so that the coefficient of variation is equal to 16%. This last value is not at all negligible, showing that even for a rigid molecule as 11, dynamical effects can be of importance. Interestingly, the average and median values suggest that the molecule is actually less reactive than expected from the static approach.

One may wonder if this is a general result for the FDL electrophilicity index. It is actually the case for compound 10 (static value: 0.091, average: 0.075) and 3 (static: 0.122, average: 0.115).

Note that these two molecules show different behaviors: important discrepancies between the two approaches for 10, small ones for 3. Conversely, the trend was found opposite for 2 (static value: 0.079, average: 0.094) and 1 (static: 0.120, average: 0.124). It is also instructive to compare 10 and 2: the electrophilicity index is higher from the static point view for 10, while it is higher in 2 from the average side.

It is also instructive to compare values for FMO electrophilicity index values. At the static level, the values are almost identical for compounds 2 and 10 (0.060 and 0.062, respectively), while the average values considerably differ: 0.069 (molecule 2) with respect to 0.052 (10). This is in line with the fact that Mayr's electrophilicity is significantly higher for 2 than for 10 (-12.1 vs. -23.0).

All these preliminary results prove that considering a dynamical approach can invert reactivity orders, and possibly significantly alter the reactivity hierarchy.

As a second example, we now have a look at an atomic descriptor, and more specifically at the dual descriptor condensed on carbon Cβ for compound 10, namely (E)-4-phenylbut-3-en-2-one.

Let us recall that the dual descriptor allows for predicting if an atom is mainly electrophilic (positive condensed value) or nucleophilic (negative condensed value). The time evolution of the dual descriptor

( )
is represented in the right part of Figure 2. It appears that the static, median and average values are all positive, but that the two first quantities differ a lot, since these three values were found equal to 0.012, 0.057, 0.055, respectively.

Another salient feature is that there are several sign changes along the trajectory, the most important one occurring around t = 55 ps. This means that the carbon atom then changes its reactivity from electrophilic to nucleophilic. One then would have expected the average value to be lower than the static one due to these negative contributions. This is actually not the case because they are largely compensated by important positive values (see for instance the positive peak before t = 80 ps). As a consequence, this compound exhibits a very high coefficient of variation value since it amounts to 84%.

All of these statistical quantities were evaluated for all molecules. By doing so, the average, minimal, maximal, and median values become new reactivity descriptors that can be used to predict Mayr's experimental electrophilicity, denoted EMayr, for the subset of 11 Michael acceptors. In fact, for a given descriptor, the dynamical values are generally not highly correlated to the static ones, as shown in Tables S81-S88 in the supplementary information file that gather selected correlation matrices. For instance, the coefficient of determination between the average and the static

( )
values was found equal to 0.66. This implies that these dynamical descriptors are true new descriptors.

A simple way for comparing their prediction ability with static ones is to compare their coefficient of determination (for monolinear regressions) towards Mayr's experimental electrophilicity. To this aim, Figure 3 shows the values from dynamical simulations with respect to the ones for all tested descriptors.

This analysis showed that for some descriptors, dynamics effects do not significantly modify the correlation to the experimental value. This is for instance the case for μ , and .

Indeed, for the electronic chemical potential, we found = 0.74 and = 0.72.

Likewise, all descriptors with very low R 2 values (lower than 0.2) in the static case also exhibit low R 2 dynamical values.

On the contrary, we obtained important differences for some other reactivity descriptors. This is notably the case for , μ and , the corresponding points being circled in Figure 3. For the electrophilicity index, we got = 0.71 and = 0.42. These results indicate that some descriptors that were not highly correlated to experimental electrophilicity in the static approach can be now considered as promising tools for the global prediction. More generally, it was found that values are generally higher than static ones, corresponding to points located in the blue area in Figure 3. Similar conclusions hold for the other statistical descriptors, in particular the time-averaged values.

More precisely, the highest R 2 values obtained using all static and dynamical values in monolinear regression procedure was found with the median value of , for which R 2 is equal to 0.77. If only static descriptors were retained, this highest R 2 value dropped down to 0.55.

Henceforth, there is a little improvement, but it can be considered not sufficient to achieve quantitative correlation. In fact, as discussed in our previous paper [23], at least two descriptors should be used for correlation when the dataset is composed of very different molecules in order to grasp the various possible reactivity behaviors (charge or orbital control, role of the two carbons of the double bond in the chemical process…).

We then performed bilinear regression analysis. In order to only deal with consistent combinations, only FDL (resp. FMO) descriptors were paired with another FDL (resp. FMO) one. For the very same reasons, we only coupled statistical descriptors of the same kind: for instance, an average value with another average value, excluding mixed combinations (an average value with a maximal one, for example). Ten descriptor subsets were thus subjected to bilinear regressions, defined by FMO static, FDL static, FMO average, FDL average, FMO median, FDL median, FMO maximal, FDL maximal, FMO minimal, and finally FMO maximal values. Table 1 collects the obtained models with the highest R 2 values.

When using only static descriptors, the best models showed R 2 of about 0.8. Noteworthy, such values indicate a prediction capacity that is similar to the one we reported for the full dataset of 35 molecules of our previous paper [23] (this likeness also confirms that the current subset can be considered as statistically representative of the whole one). The model with the highest R 2 value was the following one:

EMayr = -54.39 + 344.92 μ , + 821.34

, . (1)
It is worth noticing that it involves only global descriptors, which is an advantage since it is independent of any atomic partition and since its evaluation is more straightforward than condensed values. Nevertheless, the R 2 value remains rather low. One can then wonder if it can be improved using statistical global descriptors. This is actually the case, involving other descriptors:

EMayr = -42.99 -169.60 μ , " # $ -55.35

, " # $ , (2)
with R 2 = 0.86. Substantial predictivity enhancement can thus be achieved only incorporating condensed descriptors from the dynamical approach. The most promising models were actually obtained using the medians (R 2 = 0.93) according to:

EMayr = -
and with the average values (R 2 = 0.97) following:

EMayr = -14.58 -44.84 % , " # $ ( ) + 1996.99 , " # $ ( ).

Note that here refers to the median of the whole (& ) (& ) value distribution along the trajectory (at times ti). It is not equal to the average value of ω multiplied by the average of values.

Our last model (represented in Figure 4) is thus particularly appealing since its mean absolute deviation with respect to Mayr's experimental values is equal to 0.6 relative units, to be compared to 1.6 that was obtained with equation 1 from the static approach.. A striking feature of equations 3 and 4 is that they involve properties related to carbon α and not to carbon β, while the bond between nucleophile and electrophile occurs on carbon β. This is an important point we already discussed in our previous paper [23].

with R 2 = 0.84, featuring an obvious worsening of the model.

Obviously, bilinear models are relevant if they combine descriptors that are not highly correlated.

We thus computed the intercorrelation matrix gathering the R 2 values between each pair of descriptors entering equations 1-5. It corresponds to Table S88 in the supplementary information file. It immediately appears that such descriptors cannot be considered correlated, so that those involved in the bilinear models are really complementary.

Lastly, in order to roughly assess the reliability of such models, we considered a molecule that was not included in the fitting set, namely fumaronitrile. Its Mayr's electrophilicity is equal to -15.71, filling the missing gap between compounds 4 (EMayr = -16.76) and 3 (EMayr = -13.85). The static values for % ( ) and were found to be equal to 0.527 and 0.013, respectively, to be compared to the corresponding time-averaged ones: 0.564 and 0.012. The use of equation 4

then predicted an electrophilicity equal to -15.31. The difference between this last value and the experimental one is equal to 0.4, and it is thus lower than the mean absolute deviation calculated on the 11 molecules used to build the model. This new result supports once more the promising predictive power of dynamical reactivity descriptors.

- 11 .

 11 31 to -23.54, which is that of the known experimental one for Michael acceptors, with values well distributed along it. To be more comprehensive, let us recall that the nucleophilic addition on a Michael acceptor that features conjugated multiple bonds (for instance C=C and C=O as in 11, or C=C and C≡N as in 7) affords the formation of a bond between the nucleophile (often a carbanion) and carbone Cβ of the acceptor (see Figure1for atom labeling).As a first example for the analysis of variation of descriptors along a trajectory, we now scrutinize the electrophilicity index (calculated using FDL) for N,N-dimethylprop-2-enamide (molecule 11). Its values during the simulation are displayed in the left part of Figure2. The static value was found equal to 0.061 a.u. It can be compared to the average value (calculated on the last 80 ps): 0.057 a.u., and to the median: 0.054 a.u. The obtained values spanned between 0.046 a.u. (min) and 0.077 a.u.

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Time (in ps) evolution of the electrophilicity index for molecule 11 (left part, atomic units) and of Cβ condensed dual descriptor for molecule 10 (right part, atomic units), obtained by single-point calculations at the M06-2X/aug-cc-pVTZ level of theory on snapshots from PM6 NVT molecular dynamics simulation.

Figure 3 .

 3 Figure 3. Coefficients of determination ( ) for monolinear regressions using median values of each reactivity descriptor with respect to coefficients of determination ( ) using static values of the same descriptors.

Figure 4 .

 4 Figure 4. Best bilinear model for the prediction of Mayr's electrophilicity scale (relative units) according to equation 4. The straight line represents perfect agreement with experimental data.

  Actually, if we restrained the bilinear regression to Cβ descriptors, the best model corresponds to:

	EMayr = -32.37 -14,59	( )	,	'	(	( ) + 1053.59	,	'	( ( ),

Acknowledgements

The authors would like to gratefully acknowledge the LABEX SynOrg and the Normandy Region for funding and support, and the CRIANN high-performance computing facility. Part of this project was supported by the RIN AGAC project.

Finally, it should also be noticed that, unfortunately, such models are far from being easily interpreted since it involves a competition between % , " # $ ( ) and

, " # $ that both enter equation 4 with opposite signs. There are in fact cases where the sum of these two contributions is positive and other ones for which it reveals negative. We believe that such facts could not simply be accounted for without resorting to a detailed mechanistic study of these reactions, which is outside the scope of this paper and that we will report in due course.

Conclusions

In this letter, we have shown that dynamical effects can trigger dramatic changes of reactivity descriptor values with respect to those obtained from a static approach. We also gave evidence that average and median statistical data gathered along a trajectory allows for a much better prediction of Mayr's electrophilicity parameters. It also paves the way toward the definition of temperature-dependent reactivity indices (by choosing the appropriate thermostat temperature for the molecular dynamics simulations), and therefore toward more realistic explanation of experimental data. Besides, such a protocol is general and could be applied to other families than Michael acceptors.

Supplementary information

Tables with statistical analysis of all descriptors and graphs showing temperature convergence. , ω f ( ) 0.84 μ , f ( ) 0.88