Optimal Fiscal and Monetary Policy, Debt Crisis and Management*

Cristiano Cantore; Paul L Levine; Giovanni Melina; Joseph G Pearlman

doi:10.5089/9781475590180.001.A001

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Optimal Fiscal and Monetary Policy, Debt Crisis and Management^*

Author: Mr. Cristiano Cantore¹, Mr. Paul L Levine, Mr. Giovanni Melina, and Joseph G Pearlman

¹ 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

Contributor Notes

Authors’ E-Mail Addresses: c.cantore@surrey.ac.uk, p.levine@surrey.ac.uk, gmelina@imf.org, joseph.pearlman.1@city.ac.uk

Publication Date:: 30 Mar 2017

eISBN:: 9781475590180

Language:: English

Keywords:: WP; monetary policy; depreciation rate; debt stabilisation; steady-state debt; debt reduction policy; long-term debt

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The initial government debt-to-GDP ratio and the government’s commitment play a pivotal role in determining the welfare-optimal speed of fiscal consolidation in the management of a debt crisis. Under commitment, for low or moderate initial government debt-to-GPD ratios, the optimal consolidation is very slow. A faster pace is optimal when the economy starts from a high level of public debt implying high sovereign risk premia, unless these are suppressed via a bailout by official creditors. Under discretion, the cost of not being able to commit is reflected into a quick consolidation of government debt. Simple monetary-fiscal rules with passive fiscal policy, designed for an environment with “normal shocks”, perform reasonably well in mimicking the Ramsey-optimal response to one-off government debt shocks. When the government can issue also long-term bonds–under commitment–the optimal debt consolidation pace is slower than in the case of short-term bonds only, and entails an increase in the ratio between long and short-term bonds.

Abstract

1 Introduction

The global financial crisis has left many advanced economies with a legacy of high government debt. Despite substantial heterogeneity as regards initial conditions, macroeconomic management and post-crisis outcomes, in most key economies, governments currently face the challenge of fiscal consolidation (see Figure 1). Whilst in policy and academic circles there is agreement on the necessity of reducing the stock of government debt, an interesting and often controversial question is: how fast should debt be consolidated?

In this paper we analyse such a macroeconomic policy dilemma from a welfare-optimising viewpoint, by seeking answers to the following four question: (i) What is the welfare-optimal speed of debt consolidations? (ii) How does the picture change if the government cannot commit? (iii) What is the welfare-optimal form of simple monetary and fiscal rules to be used to achieve both one-off fiscal consolidation and to conduct stabilization policy in the face of exogenous uncertainty? (iv) Does the welfare-optimal speed of debt consolidation change if the government can optimally alter the maturity composition of its debt obligations?

We investigate these issues through the lens of a Dynamic Stochastic General Equilibrium (DSGE) model, paying particular attention to the subtle interactions between fiscal and monetary policy. The core of the model is a fairly standard New-Keynesian (NK) model featuring frictions as in Smets and Wouters (2007) with price and nominal wage rigidity. This basic set-up is augmented with a detailed fiscal sector, which is instrumental for our analysis of monetary and fiscal interactions. First, the government finances its expenditures by raising a mix of lump-sum and distortionary taxes and by issuing government bonds. Second, holding government debt is subject to sovereign default risk. Third, government expenditures are utility-enhancing and we allow for a versatile private-public consumption aggregator encompassing substitutability or complementarity. Although most of the analysis is conducted in a framework in which the government only issues short-term bonds, we provide an extension allowing the government to also issue long-term bonds. We use US data to calibrate parameter values and shocks in the model, in order to match key stylized facts and minimize a weighted loss function of key volatilities and correlations.

A number of possible interest rate and fiscal policies are compared: first, the welfare-optimal (Ramsey) policy; second, a time-consistent policy; third, optimised simple Taylor type rules (of which price-level or superinertial rules are special cases). For the simple rules, both passive and active fiscal policy stances – in the sense of Leeper (1991) – are examined. We study policy rules responding both to continuous future stochastic shocks (policy in “normal times”) and to a one-off shock to government debt (“debt crisis management”). This results in what we believe to be the first comprehensive assessment of the optimal timing and optimal combination of instruments – including the maturity composition of government debt – to achieve a fiscal consolidation, taking the size of the initial public debt overhang and government’s commitment into account. Results are as follows.

As regards our first and second questions, the initial government debt-to-GDP ratio and the government’s commitment play a pivotal role in determining the optimal speed of consolidation in the management of a debt crisis. In fact, under discretion, the cost of not being able to commit is reflected into a quick consolidation of government debt. If the government can indeed commit, a greater margin for manoeuvre is possible, and the optimal pace of consolidation is determined by the initial level of government debt. For low or moderate initial government debt-to-GDP ratios, the optimal consolidation is very slow. A faster pace is instead optimal when the economy starts from a high level of public debt (requiring high financing costs in the form of sovereign risk premia). However, if the economy is in a “bailout” regime, in which official creditors grants concessional loans, de facto permanently suppressing sovereign risk premia, then again it is optimal to enact a slow debt consolidation.

With reference to our third question, welfare calculations indicate that the ability of the simple rules to closely mimic the Ramsey optimal policy is still a feature of optimal policy with a variety of fiscal instruments. This occurs, however, only with “passive” fiscal policy. In addition, simple monetary-fiscal rules with passive fiscal policy, designed for an environment with “normal shocks”, perform reasonably well in mimicking the Ramsey-optimal response to one-off government debt shocks.

As far as our fourth question is concerned, when the government has the possibility of issuing long-term bonds, it is optimal to increase the ratio of long-to-short bonds, in the face of a debt shock. Without government’s commitment (time-consistent policy), the pace of consolidation is invariably fast. Under commitment, it is still the initial government debt-to-GDP ratio that is the main driver of the optimal consolidation speed. However, if the government also issues long-term bonds, the optimal debt consolidation pace is slower than it is in the case of short-term bonds only.

The implications of these results agree with the findings of a number of recent studies. Batini et al. (2012) show, in the context of regime-switching vector-autoregressions, that smooth and gradual consolidations are preferred to frontloaded consolidations, especially for economies in a recession. Erceg and Linde (2013) obtain similar findings in a DSGE model of a currency union. Denes et al. (2013) highlight limitations of austerity measures, while Bi et al. (2013) show that in the current economic environment, consolidation efforts are more likely to be contractionary rather than expansionary.

In seminal papers on monetary-fiscal interactions, Schmitt-Grohe and Uribe (2004a) and Schmitt-Grohe and Uribe (2007) exploit their own contribution to the computation of the solution of non-linear DSGE models in Schmitt-Grohe and Uribe (2004b) to relax the simplifying assumptions of the earlier workhorse New Keynesian model, such as the absence of capital and linearisation around a zero-inflation steady state. They examine optimal commitment and optimised simple rules as do Leith et al. (2012) who extend the model to incorporate deep habits. Another seminal contribution by Benigno and Woodford (2004) studied optimal adjustment in fiscal variables consistent with leaving debt as well as taxes at a high level in the long run when the shock(s) to the economy has adverse public finance implications. These papers study policy in what we later call “normal times” and show that government debt optimally follows a near random walk. Adam (2011) and Michal (2011) adopt a similar model but focus on what we term “crisis management”, which considers monetary and fiscal policy following a large build-up of government debt. Then, as in our results, optimal debt reduction should proceed at a slightly faster rate.¹

Siu (2004) provides a fully non-linear solution in a relatively more compact model than the one considered here but that presents various ingredients common to our analysis. The smaller size of the model in his analysis permits analysis of the implications of occasionally hitting the Zero Lower Bound (ZLB) and the comparison of small and large shocks hitting the economy. Here we employ a richer model economy at the expense of being able to consider non-linearities only under a perfect foresight solution.

There are a number of more recent works that address issues of sovereign risk and/or the benefits of commitment as in our paper. Using a standard NK model with sovereign risk, Corsetti et al. (2013) carry out a comparison of different fiscal consolidation scenarios. Our analysis differs in that we consider optimal or optimised simple rules whereas they study ad hoc policies. Kirsanova and Wren-Lewis (2012), Leith and Wren-Lewis (2013) and Ostry et al. (2015) employ a simple core NK model without capital and nominal wage rigidity and with a much simpler fiscal dimension that omits government sovereign risk and private-public consumption substitutability/complementarity. The first of these papers examines different ad hoc degrees of fiscal feedback alongside optimal monetary policy and, as in our paper, allows fiscal policy to become “active” and monetary policy “passive”, with the price level jumping in order to satisfy the government budget constraint. As in our paper, Leith and Wren-Lewis (2013) compare commitment and discretion, thus drawing conclusions regarding the importance of the former.

Our richer model, especially in terms of fiscal policy, allows the initial public debt overhang to play a significant role. In addition, optimal policy computations consistently look at a comprehensive set of issues that the literature has so far studied in isolation. First, we systematically compare optimal commitment, time-consistent, and optimized passive fiscal (active monetary) and active fiscal (passive monetary) rules, drawing conclusions about the costs of simplicity. Second, our rules have the desirable property of avoiding a frequent violation of the zero lower bound (ZLB) as in Levine et al. (2008a). Finally we extend the analysis on the fiscal side to allow for long-term debt and show that the maturity composition of debt also plays a role in the optimal speed of debt consolidation.

The remainder of the paper is organised as follows. Section 2 sets out the model. Section 3 outlines the calibration-estimation. Section 4 carries out the policy experiments. Section 5 provides an extension with long-term debt. Section 6 concludes. More technical details are appended to the paper.

2 The Model

The backbone of the DSGE model is a fairly standard New-Keynesian (NK) model with Rotemberg (1982) prices and nominal wages, featuring frictions as in Smets and Wouters (2007).² The real frictions in the model are habit formation, convex investment adjustment costs, and variable capital utilisation, while the nominal frictions are sticky prices and wages. This basic set-up is augmented with a more detailed fiscal sector, instrumental for our analysis. First, the government finances its expenditures by raising a mix of lump-sum and distortionary taxes and by issuing government bonds. Second, holding government debt is subject to sovereign default risk. Third, government expenditures are utility-enhancing. Although most of the analysis is conducted in a framework in which the government only issues short-term bonds, Section 5 contains an extension allowing the government to issue also long-term bonds. Moreover, while this section outlines the optimisation problems of all agents in the economy, Appendix A reports the full set of equilibrium conditions.

2.1 Households

A continuum of identical households j ∈ [0,1] has preferences over differentiated consumption varieties i ∈ [0,1] and derive utility from $(X_{t})^{j} = X ((X_{t}^{c})^{j}, G_{t})$ , i.e. a composite of differentiated habit-adjusted private, $(X_{t}^{c})^{j}$ , and public consumption goods, G_t, respectively, similar to that in Bouakez and Rebei (2007), Pappa (2009) and Cantore et al. (2012), which allows for $(X_{t})^{j} = (X_{t}^{c})^{j}$ as a special case and allows both for complementarity and substitutability between the two types of goods, as explained in Section 2.6.

Habits are internal (rather than external) in order to avoid counterfactual welfare externalities typical of external habits.³ As a result, the private component of the habit-adjusted consumption composite is given by

\begin{matrix} (X_{t}^{c})^{j} = C_{t}^{j} - θ C_{t - 1}^{j} & (1) \end{matrix}

where $C_{t}^{j}$ is the level of consumption and θ ∈ (0,1) is the degree of internal habit formation.

Each household j is a monopolistic provider of a differentiated labour service and supplies labour $H_{t}^{j}$ to satisfy demand,

\begin{matrix} H_{t}^{j} = {(\frac{w_{t}^{j}}{w_{t}})}^{- e_{t}^{w} η} H_{t}, & (2) \end{matrix}

where $w_{t}^{j}$ is the real wage charged by household j, w_t is the average real wage in the economy, η is the intratemporal elasticity of substitution across labour services, $e_{t}^{W}$ is a wage mark-up shock, and H_t is average demand of labour services by firms. Similarly to Zubairy (2014), the households’ budget constraint also includes a Rotemberg quadratic cost of adjusting the nominal wage, $W_{t}^{j}$ , which is zero at the steady state, and that this is proportional to the average real value of labour services as in Furlanetto (2011),

\begin{matrix} \frac{ξ^{W}}{2} {(\frac{W_{t}^{j}}{W_{t - 1}^{j}} - Π)}^{2} w_{t} H_{t} = \frac{ξ^{W}}{2} {(\frac{W_{t}^{j}}{W_{t - 1}^{j}} Π_{t} - Π)}^{2} w_{t} H_{t}, & (3) \end{matrix}

where ξ^W is the wage adjustment cost parameter and Π is the steady state value of inflation.

Households hold capital, evolving according to

\begin{matrix} K_{t + 1}^{j} = (1 - δ) K_{t}^{j} + e_{t}^{I} I_{t}^{j} [1 - S (\frac{I_{t}^{j}}{I_{t - 1}^{j}})], & (4) \end{matrix}

where $K_{t}^{j}$ is the beginning-of-period capital stock, δ is the capital depreciation rate, $I_{t}^{j}$ is investment, S(·) represents an investment adjustment cost satisfying S(1) = S′(1) = 0 and S″(1) = 0, and $e_{t}^{I}$ is an investment-specific shock. Households can also control the utilisation rate of capital. In particular using capital at rate $u_{t}^{j}$ entails a cost of $a (u_{t}^{j}) K_{t}^{j}$ units of composite good, satisfying a (u) = 0, where u is the steady-state utilisation rate.

Households buy consumption goods, $C_{t}^{j}$ ; invest in (i) investment goods, $I_{t}^{j}$ , (ii) risk-less private bonds that are mutually traded between consumers (which are in zero net supply), $B_{t}^{j}$ , (iii) nominal short-term government bond holdings, ${(B_{t}^{S})}^{j}$ ; bear the wage adjustment cost defined by equation (3) as well as the capital utilisation cost $a (u_{t}^{j}) K_{t}^{j}$ ; pay a mixture of net lump-sum, $τ_{t}^{L}$ , and distortionary taxes $τ_{t}^{C}, τ_{t}^{W}, τ_{t}^{K}$ ; receive (i) the hourly wage, W_t, (ii) the rental rate $R_{t}^{K}$ on utilised capital $u_{t}^{j} K_{t}^{j}$ , (iii) the return on nominal private bond holdings, R_t, (iv) the return on short-term nominal government bond holdings, $R_{t}^{S}$ , discounted at the ex-ante expected haircut rate, $Δ_{t}^{g}$ , (v) firms’ profits, $\int_{0}^{1} J_{i t} d i$ , (vii) transfers, ${\tilde{Ξ}}_{t}$ , and (viii) a depreciation allowance for tax purposes, $δ Q_{t} τ_{t}^{K} K_{t}^{j}$ . Therefore, households’ budget constraint reads as

\begin{array}{l} (1 + τ_{t}^{C}) C_{t}^{j} + I_{t}^{j} + τ_{t}^{L} + \frac{ξ^{W}}{2} {(\frac{w_{t}^{j}}{w_{t - 1}^{j}} Π_{t} - Π)}^{2} w_{t} H_{t} + a (u_{t}^{j}) K_{t}^{j} + \frac{B_{t}^{j}}{P_{t}} + \frac{(B_{t}^{S}) j}{P_{t}} \\ \leq (1 - τ_{t}^{W}) \frac{W_{t}^{j}}{P_{t}} H_{t}^{j} + (1 - τ_{t}^{K}) R_{t}^{K} u_{t}^{j} K_{t}^{j} + \frac{R_{t - 1} B_{t - 1}^{j}}{P_{t}} \\ + (1 - Δ_{t}^{g}) \frac{R_{t - 1}^{S} {(B_{t - 1}^{S})}^{j}}{P_{t}} + \int_{0}^{1} J_{i t} d i + {\tilde{Ξ}}_{t} + δ Q_{t} τ_{t}^{W} u_{t} K_{t}^{j} . & (5) \end{array}

Household j’s optimisation problem is then given by

\begin{matrix} \max_{{C_{t}^{j}, B_{t}^{j}, (B_{t}^{S})^{j}, K_{t + 1}^{j}, u_{t}, I_{t}^{j}, w_{t}^{j}}} E_{t} Σ_{s = 0}^{\infty} {e_{t + s}^{B} β^{t + s} U (X_{t + s}^{j}, H_{t + s}^{j})}, & (6) \end{matrix}

subject to constraints (1), (2), (4), (5), where β ∈ (0,1) is the discount factor, $U (X_{t}^{j}, H_{t}^{j})$ is the instantaneous utility, and $e_{t}^{B}$ is a preference shock.

2.2 Firms

A continuum of monopolistically competitive firms indexed by i ∈ [0,1] rents capital services, ${\tilde{K}}_{i t} \equiv u_{t} K_{i t}$ , and hires labour, H_it to produce differentiated goods Y_it with concave production function $F (e_{t}^{A} H_{i t}, {\tilde{K}}_{i t})$ – where $e_{t}^{A}$ is a labour-augmenting technology shock – which are sold at price P_it.

Assuming a standard Dixit-Stiglitz aggregator for consumption goods,

\begin{matrix} Y_{t} = {[\int_{0}^{1} {(Y_{i t})}^{1 - \frac{1}{e_{t}^{P} ζ}} d i]}^{\frac{1}{1 - \frac{1}{e_{t}^{P} ζ}}}, & (7) \end{matrix}

where $e_{t}^{P}$ is a price mark-up shock and ζ is the intratemporal elasticity of substitution across varieties of goods, the optimal level of demand for each variety, Y_it, for a given composite, is obtained by minimising total expenditure $\int_{0}^{1} P_{i t} Y_{i t} d i$ over Y_it. This leads to

\begin{matrix} Y_{i t} = {(\frac{P_{i t}}{P_{t}})}^{- e_{t}^{P} ζ} Y_{t}, & (8) \end{matrix}

where $P_{t} = {[\int_{0}^{1} P_{i t}^{1 - e_{t}^{P} ζ} d i]}^{\frac{1}{1 - e_{t}^{P} ζ}}$ is the nominal price index.

Firms face quadratic price adjustment costs $\frac{ξ^{P}}{2} {(\frac{P_{i t}}{P_{i t} - 1} - Π)}^{2} Y_{t}$ , as in Rotemberg (1982) – where parameter ξ^P measures the degree of price stickiness – and solve the following profit maximisation problem:

\max_{{{\tilde{K}}_{i t + s}, H_{i t + s}, P_{i t + s}}} J_{i t} = E_{t} {Σ_{s = 0}^{\infty} D_{t, t + s} [\begin{matrix} \frac{P_{i t + s}}{P_{t + s}} Y_{i t + s} - w_{i t} H_{i t + s} - R_{t + s}^{K} {\tilde{K}}_{i t + s} \\ - \frac{ξ^{P}}{2} {(\frac{P_{i t + s}}{P_{i t + s - 1}} - Π)}^{2} Y_{t + s} \end{matrix}]}, (9)

subject to the firm-specific demand (8) and the firm’s resource constraint,

\begin{matrix} Y_{i t} = F (e_{t}^{A} H_{i t}, {\tilde{K}}_{i t}) - F C, & (10) \end{matrix}

where FC are fixed production costs.

2.3 Government

The government finances its expenditures, G_t, by levying taxes, T_t, and by issuing one-period bonds, $B_{t}^{S}$ . The government promises to repay one-period bonds the next period and the gross nominal interest rate applied is $R_{t}^{S}$ . However, in order to introduce a sovereign risk premium, we assume that government bond contracts are not enforceable. As in Bi and Traum (2014), each period a stochastic fiscal limit expressed in terms of government debt-to-GDP ratio and denoted by $Γ_{t}^{*}$ , is drawn from a distribution, the cumulative density function (CDF) of which is represented by a logistical function, $p_{t}^{*}$ , with parameters η₁ and η₂:

\begin{matrix} p_{t}^{*} = P (Γ_{t}^{*} \leq Γ_{t - 1}) = \frac{\exp (η_{1} + η_{2} Γ_{t - 1})}{1 + \exp (η_{1} + η_{2} Γ_{t - 1})}, & (11) \end{matrix}

where $Γ_{t} \equiv B_{t}^{S} / P_{t} Y_{t}$ . If government-debt-to-GDP exceeds the fiscal limit, i.e. $Γ_{t - 1} \geq Γ_{t}^{*}$ , then the government defaults. Hence $p_{t}^{*}$ represents the probability of default. This occurs in the form of an haircut $\bar{Δ} \in [0, 1]$ applied as a proportion to the outstanding stock of government debt. In order to be able to solve the model with perturbation methods, we follow Corsetti et al. (2013) and assume that agents consider the ex-ante expected haircut rate,

\begin{matrix} Δ_{t}^{g} = {_{{\bar{Δ}}^{g}}^{0} \begin{matrix} with probability 1 - p_{t}^{*} \\ with probability p_{t}^{*} \end{matrix}, & (12) \end{matrix}

where ${\bar{Δ}}^{g} \in (0, 1]$ is the haircut rate applied in the case of default. In other words,

\begin{matrix} Δ_{t}^{g} = p_{t}^{*} {\bar{Δ}}^{g} . & (13) \end{matrix}

Let $b_{t}^{S} \equiv B_{t}^{S} / P_{t}$ be the gross inflation rate, then the government budget constraint, in real terms, reads as

\begin{matrix} b_{t}^{S} \equiv (1 - Δ_{t}^{g}) \frac{R_{t - 1}^{S}}{Π_{t}} b_{t - 1}^{S} + G_{t} - T_{t} + {\tilde{Ξ}}_{t}, & (14) \end{matrix}

where total government revenue, T_t, is given by

\begin{matrix} T_{t} = τ_{t}^{C} C_{t} + τ_{t}^{W} w_{t} h_{t} + τ_{t}^{K} [(R_{t}^{K} - δ Q_{t}) u_{t} K_{t}] + τ_{t}^{L}, & (15) \end{matrix}

and $τ_{t}^{C}, τ_{t}^{W} τ_{t}^{K}$ are tax rates on aggregate private consumption C_t, labour income, w_th_t, and the net capital rental rate, $(R_{t}^{k} - δ Q_{t}) u_{t} K_{t}$ , respectively, and $τ_{t}^{L}$ are lump-sum taxes.

As in Corsetti et al. (2013), government transfers, ${\tilde{Ξ}}_{t} \equiv Δ_{t}^{g} \frac{R_{t - 1}^{S}}{Π_{t}} b_{t - 1}^{S}$ , are set in a way that sovereign default does not alter the actual debt level. The absence of such transfers would bear the counterintuitive effect of lower risk premia prior to default, as the lower post-default debt stock would already be taken into account.

In order to reduce the number of tax instruments to one, we impose that $τ_{t}^{C}, τ_{t}^{W} τ_{t}^{K}$ and $τ_{t}^{L}$ deviate from their respective steady state by the same proportion $τ_{t}^{C} = τ_{t} τ^{C}, τ_{t}^{W} = τ_{t} τ^{W}, τ_{t}^{K} = τ_{t} τ^{K}, τ_{t}^{L} = τ_{t} τ^{L}$ and that the proportional uniform tax change, τ_t, becomes one of our fiscal policy instruments. The other instrument is represented by government spending G_t. We allow the instruments to be adjusted according to the following Taylor-type rules:

\begin{matrix} \log (\frac{τ_{t}}{τ}) = ρ_{τ} \log (\frac{τ_{t - 1}}{τ}) + ρ_{τ b} \log (\frac{b_{t - 1}^{S}}{b^{S}}) + ρ τ y \log (\frac{Y_{t}}{Y}), & (16) \end{matrix}

\begin{matrix} \log (\frac{G_{t}}{G}) = ρ_{g} \log (\frac{G_{t - 1}}{G}) - ρ_{g b} \log (\frac{b_{t - 1}^{S}}{b^{S}}) - ρ g y \log (\frac{Y_{t}}{Y}), & (17) \end{matrix}

where ρ_τ implies persistence in the tax instrument, ρ_τb is the responsiveness of the tax instrument to the deviation of government debt from its steady state, and ρ_τy is the responsiveness to output deviations. Parameters ρ_g, ρ_gb, and ρ_gy are the analogues in the expenditure rule. Notice that these are Taylor-type rules as in Taylor (1993) that respond to deviations of output and debt from their deterministic steady state values and not from their flexi-price outcomes. Such rules have the advantage that they can be implemented using readily available macrodata series rather than from model-based theoretical constructs (see e.g. Schmitt-Grohe and Uribe, 2007). Leeper et al. (2010) also show that such a specification for fiscal rules fits the data reasonably well. In Section 4.4 we describe how the parameters of the fiscal rules can be set optimally. Optimising the path of each tax instrument individually would have been computationally challenging as we would have doubled our policy instruments from three (including monetary policy) to six.⁴

2.4 Monetary policy

Monetary policy is set according to a Taylor-type interest-rate rule,

\begin{matrix} \log (\frac{R_{t}}{R}) = ρ_{r} \log (\frac{R_{t - 1}}{R}) + ρ_{r π} \log (\frac{Π_{t}}{Π}) + ρ_{r y} \log (\frac{Y_{t}}{Y}), & (18) \end{matrix}

where ρ_r is the interest rate smoothing parameter and ρ_rπ and ρ_ry are the monetary responses to inflation and output relative to their steady states. The rationale for using readily observable variables is the same as that concerning the fiscal rules.⁵

2.5 Equilibrium

In equilibrium all markets clear. The model is completed by the output, capital and labour market equilibrium conditions,

\begin{matrix} Y_{t} = C_{t} + I_{t} + G_{t} + \frac{ξ^{P}}{2} (Π_{t} - Π)^{2} Y_{t} + \frac{ξ^{W}}{2} (Π_{t}^{W} - Π)^{2} w_{t} H_{t} + a (u_{t}) K_{t} & (19) \end{matrix}

\begin{matrix} \begin{matrix} \underset{i}{Σ} K_{i t} = \underset{j}{Σ} K_{t}^{j}, & \underset{i}{Σ} H_{i t} = \underset{j}{Σ} H_{t}^{j} \end{matrix}, & (20) \end{matrix}

and the following autoregressive processes for exogenous shocks:

\begin{matrix} \log (\frac{e_{t}^{x}}{e^{x}}) = ρ_{x} \log (\frac{e_{t}^{x}}{e^{x}}) + ε_{t}^{x} & (21) \end{matrix}

where x = {B, P, I, A,W}, ρ_x are autoregressive parameters and $ε_{t}^{x}$ are mean zero, i.i.d. random shocks with standard deviations σ^x.

2.6 Functional forms

The utility function specialises as in Jaimovich and Rebelo (2008),

\begin{matrix} U (X_{t}, H_{t}) = \frac{X_{t}^{1 - σ_{c}}}{1 - σ_{c}} {(1 - ψ H^{υ})}^{1 - σ_{c}}, & (22) \end{matrix}

where σ_c > 0 is the coefficient of relative risk aversion, ψ is a scaling parameter that determines the relative weight of leisure in utility and ϑ is a preference parameter that determines the Frisch elasticity of labour supply.

In order to allow for complementarity between private and public consumption we specialise the consumption composite as a constant-elasticity-of-substitution (CES) aggregate,

\begin{matrix} X_{t} = {ν_{x}^{\frac{1}{σ_{x}}} [(X_{t}^{c})^{j}]^{\frac{σ_{x} - 1}{σ_{x}}} + (1 - v_{x})^{\frac{1}{σ_{x}}} G_{t}^{\frac{σ_{x} - 1}{σ_{x}}}}^{\frac{σ_{x}}{σ_{x - 1}}}, & (23) \end{matrix}

where ν_x is the weight of private goods in the aggregate and σ_x is the elasticity of substitution between private and public consumption. Such a specification is desirable in that it encompasses (i) the case of perfect substitutability between private and public consumption (when σ_x → ∞); (ii) the Cobb-Douglas case of imperfect substitutability (when σ_x → 1); (iii) the Leontief case of perfect complementarity (when σ_x → 0); and (iv) the standard case in DSGE modelling of non-utility-enhancing public consumption (when ν_x = 1). If ν_x < 1 and σ_x < ∞, then public consumption affects the marginal utility of private consumption and the marginal disutility of labour, thus influencing consumption/saving decisions and the labour supply.⁶

The remaining functional forms are as in Smets and Wouters (2007). First, investment adjustment costs are quadratic: $S (\frac{I_{t}}{I_{t - 1}}) = \frac{γ}{2} (\frac{I_{t}}{I_{t - 1}} - 1)^{2}$ , γ > 0, where γ represents the elasticity of the marginal investment adjustment cost to changes in investment. Next, the cost of capital utilisation is $a (u_{t}) = γ_{1} (u_{t} - 1) + \frac{γ_{2}}{2} {(u_{t} - 1)}^{2}$ . Following the literature, the steady-state utilisation rate is normalised to unity, u = 1. It follows that a (u) = 0, a′ (u) = γ₁, a″ (u) = γ₂ and the elasticity of the marginal utilisation cost to changes in the utilisation rate is $\frac{a^{''} (u) u}{a^{'} (u)} = \frac{γ_{2}}{γ_{1}} \equiv σ_{u}$ , which is what we calibrate. The production function is a conventional Cobb-Douglas: $F (e_{t}^{A} H_{t}, {\tilde{K}}_{t}) = {(e_{t}^{A} H_{t})}^{α} {\tilde{K}}_{t}^{1 - α}$ , where α represents the labour share of income and $\tilde{K} \equiv u_{t} K_{t}$ is capital services. Equilibrium conditions with these functional forms (for the full model including also the features of long-term government bonds described in Section 5) are provided in Appendix A.

3 Calibration

We assign numerical values to the parameters to match a number of stylised facts and moments of key macroeconomic variables of the US economy during the Great Moderation. The time period in our model corresponds to one quarter in the data. Table 1 reports all parameter values.

Table 1.

Parameter values

Table 1.

Parameter values

Parameter		Value
Discount factor	β	0.99
Capital depreciation rate	δ	0.025
Production function parameter	α	0.67
Steady-state gross inflation rate	Π	1.0075
Steady-state government spending share of output	$\frac{G}{Y}$	0.19
Steady-state consumption tax rate	τ^C	0.05
Steady-state labor income tax rate	τ^W	0.24
Steady-state capital tax rate	τ^K	0.32
Steady-state government debt-to-GDP ratio	Γ	4×0.70
Labor preference parameter	ϑ	2.8343
Elasticity of substitution in goods market	η	6
Elasticity of substitution in labor market	ζ	21
Haircut rate	$\bar{Δ}$	0.0246
Scaling factor in default probability	ζ₁	−9.7480
Slope parameter in default probability	ζ₂	1.9921
Scaling factor in utility function	ψ	4.0728
Relative risk aversion	σ_c	1.9380
Habit formation	θ	0.7110
Investment adjustment cost elasticity	γ	4.8653
Capital variable utilisation elasticity	σ_u	4.9118
Preference for private goods	ν_x	0.6815
Elast. of subst. between private/public goods	σ_x	0.6349
Rotemberg wage stickiness	ξ_W	99.9851
Rotemberg price stickiness	ξ^P	29.9947
Interest rate smoothing	ρ_r	0.7879
Monetary policy response to inflation	ρ_rπ	0.4445
Monetary policy response to output	ρ_ry	0.0342
Persistence of tax rates	ρ_τ	0.6252
Tax response to government debt	ρ_τb	0.0777
Tax response to output	^ρτy	0.1478
Persistence of government spending	ρ_τy	0.7662
Government spending response to government debt	ρ_gb	0.1197
Government spending response to output	ρ_gy	0.1624
Persistence of technology shocks	ρ_A	0.8868
Persistence of preference shocks	ρ_B	0.4149
Persistence of investment-specific shocks	ρ_I	0.6456
Persistence of price mark-up shocks	ρ_P	0.3308
Persistence of wage mark-up shocks	ρ_W	0.5714
Standard deviation of technology shocks	σ^A	0.0101
Standard deviation of preference shocks	σ^B	0.0163
Standard deviation of investment-specific shocks	σ^I	0.0591
Standard deviation of price mark-up shocks	_σP	0.0422
Standard deviation of wage mark-up shocks	_σW	0.0264

Table 1.

Parameter values

Parameter		Value
Discount factor	β	0.99
Capital depreciation rate	δ	0.025
Production function parameter	α	0.67
Steady-state gross inflation rate	Π	1.0075
Steady-state government spending share of output	$\frac{G}{Y}$	0.19
Steady-state consumption tax rate	τ^C	0.05
Steady-state labor income tax rate	τ^W	0.24
Steady-state capital tax rate	τ^K	0.32
Steady-state government debt-to-GDP ratio	Γ	4×0.70
Labor preference parameter	ϑ	2.8343
Elasticity of substitution in goods market	η	6
Elasticity of substitution in labor market	ζ	21
Haircut rate	$\bar{Δ}$	0.0246
Scaling factor in default probability	ζ₁	−9.7480
Slope parameter in default probability	ζ₂	1.9921
Scaling factor in utility function	ψ	4.0728
Relative risk aversion	σ_c	1.9380
Habit formation	θ	0.7110
Investment adjustment cost elasticity	γ	4.8653
Capital variable utilisation elasticity	σ_u	4.9118
Preference for private goods	ν_x	0.6815
Elast. of subst. between private/public goods	σ_x	0.6349
Rotemberg wage stickiness	ξ_W	99.9851
Rotemberg price stickiness	ξ^P	29.9947
Interest rate smoothing	ρ_r	0.7879
Monetary policy response to inflation	ρ_rπ	0.4445
Monetary policy response to output	ρ_ry	0.0342
Persistence of tax rates	ρ_τ	0.6252
Tax response to government debt	ρ_τb	0.0777
Tax response to output	^ρτy	0.1478
Persistence of government spending	ρ_τy	0.7662
Government spending response to government debt	ρ_gb	0.1197
Government spending response to output	ρ_gy	0.1624
Persistence of technology shocks	ρ_A	0.8868
Persistence of preference shocks	ρ_B	0.4149
Persistence of investment-specific shocks	ρ_I	0.6456
Persistence of price mark-up shocks	ρ_P	0.3308
Persistence of wage mark-up shocks	ρ_W	0.5714
Standard deviation of technology shocks	σ^A	0.0101
Standard deviation of preference shocks	σ^B	0.0163
Standard deviation of investment-specific shocks	σ^I	0.0591
Standard deviation of price mark-up shocks	_σP	0.0422
Standard deviation of wage mark-up shocks	_σW	0.0264

Three parameter values are conventional in the DSGE literature: the subjective discount factor β = 0.99 implies an annual real rate of interest of 4 percent; the capital depreciation rate δ = 0.025 corresponds to an annual depreciation of 10 percent; while the Cobb-Douglas production function parameter α = 0.67 entails a labor share of income of 2/3.

At the steady state we set a government spending share of output of 20 percent (G/Y = 0.20) and a gross inflation rate Π = 1.0075, corresponding to a net annual rate of inflation of 3 percent. The steady-state values of the tax rates are as in Christiano et al. (2014), i.e. τ^C = 0.05, τ^W = 0.24, and τ^K = 0.32, while the baseline government debt is 70 percent of annual output (Γ = 4 × 0.70).⁷

The scaling factor in the utility function ψ = 4.0728 is set to match a labor supply of 1/3 of available time at the steady state (H = 0.33), while the intratemporal elasticities of substitution in the goods and labor market (η = 6 and ζ = 21, respectively) are set as in Zubairy (2014) to match average mark-ups of 20% and around 40% (given the labor income tax rate), respectively.

To calibrate the CDF of the fiscal limit, depicted in Figure 2, we fix two points on the function in a way consistent with empirical evidence. Given two points $(Γ_{1}, p_{1}^{*})$ and $(Γ_{2}, p_{2}^{*})$ , with Γ₂ > Γ₁, parameters η₁ and η₂ are uniquely determined by

η_{2} = \frac{1}{Γ_{1} - Γ_{2}} \log (\frac{p_{1}^{*} 1 - p_{2}^{*}}{p_{2}^{*} 1 - p_{1}^{*}}), (24)

\begin{matrix} η_{1} = \log (\frac{p_{1}^{*}}{1 - p_{1}^{*}}) - η_{2} Γ_{1} . & (25) \end{matrix}

Let us assume that when the ratio of government debt to GDP is Γ2, the probability of exceeding the fiscal limit is almost unity, i.e. $p_{2}^{*} = 0.99$ . We set the fiscal limit at Γ₂ = 4 × 1.8, broadly in line with the Greek experience. Let us fix Γ₁ = 4 × 0.7, the average public-debt-to-GDP ratio in US post-WWII experience. Before the financial crisis the U.S. sovereign risk premium has been very small – around 15 annual basis points (ABP) for sovereign credit default swaps spreads (see e.g. Austin and Miller, 2011) – hence we assume that for Γ₁ = 4 × 0.7, ABP₁ = 15. At the onset of the sovereign debt crisis, the Greek sovereign risk premium skyrocketed to an order of magnitude of around 1,000 annual basis points, hence we fix ABP₂ = 1, 000.

At this point, from equation (13), we can recover the probability of default when Γ = Γ₁,

p_{1}^{*} = \frac{1 - \frac{1}{\frac{A B P_{1}}{40000} + 1}}{\bar{Δ} g},

which is $p_{1}^{*} = 0.0152$ ⁸ and parameters η₁ and η₂ of the fiscal limit CDF can be recovered by using equations (24) and (25), i.e. η₁ = –9.7480 and η₂ = 1.9921. This parameterisation implies that the sovereign risk premium rises from 15 annual basis points when government debt is 70% of annual GDP to, e.g., 143 and 452 annual basis points when the government debt ratio increases to 100%, and 120%, respectively. This captures the fact that problems related to sovereign default premia may mount at a very fast pace as public debt accumulates.

Last, we set (i) the remaining 18 structural parameters, (ii) the standard deviations, and (iii) the persistences of the five structural shocks via moment-matching of (a) the empirical standard deviations and (b) the persistences of real output, private consumption, investment, inflation, the real wage, hours worked, government spending, government revenues, and the federal funds rate; (c) the cross correlations between each macroeconomic variable and real output; and (d) the cross-correlation between private and public consumption, and between inflation and the federal funds rate. Details on data sources and transformations are provided in Appendix C.

Given the difficulty in matching exactly all moments, we construct a quadratic loss function $L = Σ_{j = 1}^{28} ω_{j} {(x_{j}^{m} - x_{j}^{d})}^{2}$ , where $x_{j}^{m}$ is the j−th moment in the model, $x_{j}^{m}$ is its analogue in the data while ω_j are weights. Given that matching volatilities is key for optimal policy exercises, we numerically search for those parameters that minimise L, assigning double weights to deviations of volatilities and splitting the remaining weights uniformly across the remaining targets.

This calibration/moment-matching procedure is similar to a more general method of moments estimation and delivers plausible parameter values, as well as volatilities, persistences and correlations of key macroeconomic variables that reasonably match the data, as can be seen in Table D.1.⁹

4 Optimal monetary and fiscal stabilisation policy

We consider two aspects of monetary and fiscal optimal stabilisation policy. The first is stabilisation policy for normal times. Rules are designed to minimise an expected conditional welfare loss starting at some steady state. In this case the optimal policy problem is purely stochastic: optimal policy is in response to all future stochastic shocks hitting the economy. By contrast, crisis management starts with the economy (the debt-GDP ratio in particular) off the steady state (for whatever reason) so that policy is required both for the economy to return to the steady state (a deterministic problem) and for it to deal with future stochastic shocks (a stochastic problem). For both problems we adopt a linear-quadratic (LQ) set-up which, for a given set of observed policy instruments, considers a model linearised around a steady state, with a welfare function that is quadratic in deviations about the steady state.

4.1 The Ramsey problem and the LQ approximation

In the LQ approximation, the steady state about which the linearisation takes place corresponds to the steady state of the solution to the non-linear deterministic Ramsey problem, modified as described below, where the welfare to be maximised is that of the representative agent. From Levine et al. (2008a) and Benigno and Woodford (2012) we know that the quadratic approximation to welfare is then given by the second-order approximation to the Lagrangian of the Ramsey problem, together with a linear term dependent on the initial value of the forward-looking variables of the system. When the latter is omitted (as is done below), the quadratic welfare approximation corresponds to the timeless approach to the solution. The timeless approach implies that the deterministic part of the solution has been followed for a long time, so that any observed deviations from steady state are purely due to the shocks hitting the system. The advantage of this LQ approximation is that the normal and crises components of policy conveniently decompose, and one optimal policy emerges conditional on the initial point.¹⁰

To be more explicit, Appendix B shows that the problem to be solved in the LQ context boils down to

\begin{matrix} \begin{matrix} \min_{x 0, y t, w t} : - a_{1}^{T} z_{0} - a_{2}^{T} x_{0} + \frac{1}{2} Σ_{t = 0}^{\infty} (y_{t}^{T} Q y_{t} + w_{t}^{T} R w_{t}), & y_{t}^{T} = [\begin{matrix} z_{t}^{T} & x_{t}^{T} \end{matrix}] \end{matrix}, & (26) \end{matrix}

given z₀, where a₁ and a₂ are constant row vectors, subject to linear dynamic constraints, z_t represents the predetermined, x_t the non-predetermined variables, w_t a vector of policy instruments, while Q and R are conformable matrices. The first stage of the fully optimal solution to this is obtained by ignoring the forward-looking nature of x_t, and the remainder of the problem can be formally represented as

\begin{matrix} \min_{x 0} : - a_{1}^{T} z_{0} - a_{2}^{T} x_{0} + \frac{1}{2} y_{0}^{T} {Sy}_{0}, & (27) \end{matrix}

where S is the solution to a Riccati equation. Writing $S= [\begin{matrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{matrix}]$ , it is easy to show that the jump in the initial value x₀ is given by $x_{0} = S_{22}^{- 1} a_{2} - S_{22}^{- 1} S_{21} z_{0}$ . The sub-optimal timeless solution is arrived at by removing the jump $S_{22}^{- 1} a_{2}$ . In our simulations that follow, we focus on a shock to debt in the initial period, and this is subsumed within the term $- S_{22}^{- 1} S_{21} z_{0}$ , so that omitting $S_{22}^{- 1} a_{2}$ does not have an impact with regard to the effect of a debt shock.

As regards the steady state, given that our paper focuses on stabilization policy and not on the optimal tax structure, the steady state about which we approximate the model and the loss function is not the Ramsey optimum but rather a modified one that corresponds to observed fiscal variables (tax and debt).¹¹

4.2 Optimal monetary-fiscal rules for normal times

In this section we examine optimal policy using both monetary and fiscal instruments. As in Cantore et al. (2012) “optimality” can mean the welfare-optimal (Ramsey) policy, or time-consistent policy or optimised Taylor-type interest rate and fiscal rules.¹² Fiscal rules use (16) for the taxation instrument τ_t and (17) for government spending G_t.¹³ Monetary policy is conducted according to (18).

One can think of this choice of rules as assigning responsibility for stabilising inflation and debt to the monetary authority and fiscal authorities respectively.¹⁴ With both the interest rate and the fiscal instruments responding to fluctuations of output, the two authorities are sharing responsibility for output fluctuations.

The assignment issue arises in a different form in Leeper (1991), who provides the original characterisation of policy rules as being “active” or “passive”. An active monetary policy rule is one in which the monetary authority satisfies the Taylor principle in that they adjust nominal interest rates such that real interest rates rise in response to excess inflation. Conversely, a passive monetary rule is one which fails to satisfy this principle. In Leeper’s terminology a passive fiscal policy is one in which the fiscal instrument is adjusted to stabilise the government’s debt stock, while an active fiscal policy fails to do this. Our simple rules allow for both these possibilities.¹⁵

For simple rules we impose two ‘feasibility’ constraints (Schmitt-Grohe and Uribe, 2007): ρrπ ≤ 5 and ρτb, ρgb ≤ 1 to avoid the threat of excessive changes in the interest rate, tax rate and government spending.¹⁶

Moreover, while we rule out superinertial monetary policy rules in our baseline results (Table 2) by imposing ρ_r ≤ 1, this possibility is explored in Table 3.

Table 2.

Optimal policy results for alternative government debt/GDP ratios

Table 3.

Optimal policy results for B/4Y=0.7 allowing for superinertial monetary policy

The aggressive nature of the optimal, time-consistent or passive simple rules often leads to high interest rate variances resulting in a ZLB problem for the interest rate. One way of getting round this problem is by a penalty function approach that adds a cost to the welfare function of —w_r(R_t — R)². The role of w_r is to lower the variance of the interest rate, thereby narrowing its probability density function and lowering the probability of the ZLB being violated (see Levine et al., 2008a). One can think of the penalty function approach to the ZLB as a policymaker giving the central bank a mandate to implement a rule that hits the ZLB with a given low probability. Thus the central bank will add a penalty term w_r(R_t — R)² to the policymaker’s welfare objective with a weight just high enough to achieve a sufficiently low nominal interest rate variance that meets this objective. The welfare outcome of a policy however continues to use the original objective having subtracted the variance term. These ZLB considerations have an important impact on the persistence parameter ρr of the optimized simple AF rules because a higher persistence reduces the variability of an interest rate policy needed to achieve a given stabilization outcome.¹⁷

In our results we choose the value of w_r in each regime such that the probability of hitting the ZLB is 0.0025, i.e. only once every 400 quarters or 100 years. Once the optimum has been found, subject to the choice of w_r, the representative agent’s utility approximation is then corrected by adding back the term w_rvar(R_t). The adjusted welfare loss shown in the tables is obtained by subtracting the latter term.

4.2.1 Results

Table 2 shows the welfare losses for the various regimes – time consistent, optimal simple passive (PF) and active fiscal (AF) policy, coupled with active and passive monetary policy, relative to optimal timeless policy. These simulations are performed for three cases of steady state state debt/GDP ratio – 70%, 90% and 120%, implying increasing risk premia. In addition, for the case in which the steady state state debt/GDP ratio is 120%, we also run simulations, under the assumption that the country never defaults – denoted as the “No Sovereign Risk” or “Bailout” case, i.e. suppressing the risk premium.

The welfare loss is reported as a consumption equivalent permanent percentage fall in private consumption below the Ramsey-optimal policy, c_e. If two policies had consumption streams ${C_{t}^{1}}$ and ${C_{t}^{2}}$ then the consumption equivalent improvement of policy 1 is found from equating the intertemporal value from streams ${C_{t}^{1}}$ and ${C_{t}^{2} (1 + \frac{c_{e}}{100})}$ where c_e is the consumption equivalent percentage increase.¹⁸

As we can see from the table, the optimal simple passive fiscal, active monetary rules generate an average welfare loss of 0.14% in consumption-equivalent terms, whereas the consumption-equivalent welfare losses under time consistent or optimal simple active fiscal, passive monetary rules are considerably larger. Note too that in the latter case the optimal rules conform consistently to the fiscal theory of the price level (FTPL), with no movement at all in any of interest rates, government spending and taxation, and the main reaction to shocks being an initial jump in the price level.

In all cases with passive fiscal policy, the optimised simple monetary policy rule is a price level rule, and both monetary and fiscal instruments do not react to output fluctuations. In addition, a low steady-state government debt/GDP ratio (70%) implies a stronger responsiveness of taxes relative to government spending (the latter displaying a high degree of inertia and a low elasticity to government debt). With high debt (120% of GDP) government spending becomes much more reactive to debt, unless the sovereign risk premium is suppressed.

The optimality of inertial rules with ρ_τ ∈ [0,1], of which the price-level rule (ρ_r = 1, ρ_ry = 0) is a limiting case is well established in the New Keynesian literature without debt considerations. Under price-level stabilization, forward-looking agents understand that current above-target inflation will be followed by below-target inflation in subsequent periods. This in turn dampens the retail firm’s willingness to change their prices, lowering the variability of inflation and welfare losses. A price level rule lowers the variance of the interest rate variance necessary to stabilize inflation making the ZLB less likely to be reached, as explained previously. In addition, a price level rule reduces the region of indeterminacy, enlarging the policy space available for optimizing rules (see Giannoni (2014)). Our results show that this good performance of a price-level rule carries over to optimized rules with a fiscal dimension, although this introduces a countervailing welfare-reducing effect of decreasing debt-deflating inflation volatility. For our calibrated model the welfare benefits of stabilizing inflation and avoiding the ZLB still outweigh the latter, resulting in the optimality of the price-level rule.

In sum, higher public debt overhangs and risk premia do not significantly alter optimal monetary policy, which features properties in agreement with what the literature found in simpler models (notably Schmitt-Grohe and Uribe, 2007 and Levine et al., 2008a). On the contrary, optimal fiscal policy becomes much more reactive to shocks in the presence of higher risk premia. Moreover, under low risk premia most of the debt stabilisation is achieved via tax changes; under high risk premia, it is optimal to adjust also government expenditures to a greater extent.

4.3 Crisis management of debt: how fast, how deep?

We now turn to the case in which policy is used to stabilise the economy in the face of a sudden shock to government debt. While in this subsection we analyse the impact of such a shock in the linearised version of the model, in Subsection 4.4 we investigate the effects of non-linearities and larger shocks using a non-linear perfect foresight solution.

In Figure 3 we depict the effects of a shock that raises the government debt/GDP ratio by 1% under the four policy regimes and each of the four debt scenarios. For the optimised simple rules we use the coefficients computed under “normal shocks” and reported in Table 2.

A number of noteworthy results emerge from the inspection of this figure. First, for the cases of debt/GDP ratio of 70% and 120% with no sovereign risk, optimal policy yields a level of debt that barely shifts at all over time. This is the standard “random walk” result that arises for debt under optimal policy when the inverse of the discount rate matches the gross real interest rate. Second, optimal simple passive fiscal rules produce trajectories very similar to those of the optimal timeless rules. Third, under optimal simple active fiscal rules, there is a large jump in prices as we would expect to see under the FTPL, and this jump is two or three times the size of that under optimal time consistent rules. Fourth, under time-consistent policy, debt consolidation is always very fast. In other words, under discretion, the cost of not being able to commit is reflected into a quick consolidation of government debt.

As a result, if the government can commit, the optimal rules and optimal passive simple rules display slow consolidation of debt levels unless the level of steady state debt is very high, e.g. at 120% of GDP. In other words, in a case similar to the recent Greek experience, the risk premium that must be paid forces debt levels to drop quickly. In this case the fiscal adjustment is more painful as output drops to a greater and more persistent extent. This outcome, however, is reversed in the presence of a successful bailout program that suppresses risk premia. Given the closed-economy nature of the model and the absence of a banking sector holding government bonds, these simulations do not encompass the costs from being locked out of international markets or from banking crises associated to sovereign defaults. The presence of these costs would amplify the speed of fiscal consolidations associated to high levels of initial government debt as a fraction of GDP.

From the figure we see that optimal simple passive rules do not always closely mimic the path of debt, e.g. for the case of a debt/GDP ratio of 70%. A better simple rule can be designed in a straightforward way via super-inertial monetary policy i.e. allowing the value of ρ_r to be greater than 1.¹⁹ For this case, the optimal simple rules lead to a very similar path for debt as the optimal rule (not reported in the figure), and we show the results on welfare for this case in Table 3, with a consumption-equivalent welfare compared with optimal policy now reduced to 0.05%. As discussed in previous section, when we impose an upper bound of 1 to ρ_r, for our calibrated model the welfare benefits of stabilizing inflation and avoiding the ZLB outweigh the countervailing welfare reducing effect of decreasing debt-deflating inflation volatility, resulting in the optimality of the price-level rule. A super-inertial rule (ρ_r > 1) takes this effect a stage further with a commitment to only temporarily raise the interest rate in response to above-target inflation. In equilibrium this results in inflation under-shooting its target in the long-run, enhancing the incentive to firms alluded to above. Thus, simple monetary-fiscal rules with passive fiscal policy, designed for an environment with “normal shocks”, turn out to perform reasonably well in mimicking the Ramsey-optimal response to one-off government debt shocks.

4.4 The role of nonlinearities, larger shocks and an interest-rate peg

We turn to the case of how to manage sudden larger jumps in government debt. In order for the size of the shock to have a role in the model dynamics, the model’s nonlinearities should be fully at play. To do this, we use the nonlinear perfect foresight solver available in Dynare.

The assumption is that the initial increase in debt is unanticipated, but agents can perfectly foresee all subsequent dynamics, including that of government debt.

The first question we intend to answer through this exercise is: do the lessons drawn from studying the effects of a 1% shock to the government debt/GDP ratio apply to cases of larger shocks? Despite the nonlinear nature of the model, it turns out that the linear approximation of the LQ methodology provides a sufficiently good guide to policy design for larger debt shocks. In fact, in Figure 4, we plot the impulses responses from the linear and nonlinear versions of the model with the optimal simple passive fiscal, active monetary rule to a shock that brings the debt/annual GDP ratio from 90% to 105%. Although there are some quantitative differences (notably that for the nonlinear simulations output takes longer to return to its base level), qualitatively there is not too much of a difference between the impulse responses of the two cases. If anything, government debt, as a fraction of GDP, is consolidated at an even slower pace in the nonlinear model, thus reinforcing the slow fiscal consolidation lesson drawn by using a linear model.

The explanation behind this result lies precisely in the size of the shock to government debt. In Figure 5, we show the impact of shocks of different magnitudes in the nonlinear model. For the sake of comparability, (i) we simulate again the model starting with a steady state of the government debt-to-GDP ratio of 90% and assuming a shock that raises debt by 15 percentage points of GDP; (ii) we then progressively halve the size of the shock until we get to a shock of about 2 percentage points of GDP; but (iii) we rescale the impulse responses by multiplying them by the same factors. If the model were linearised, then all the lines depicted in each subplot should coincide. In contrast, as the size of the shock increases, the non-linearities at play, especially those induced from the sovereign risk premium become more evident. Given that the interest rate paid on debt is elastic to the size of the stock of debt itself, greater shocks imply a more than proportional increase in interest payments. These in turn translate into a larger debt roll-over, a more protracted fiscal consolidation (government spending more persistently below-and the tax rate more persistently above-their respective steady states) and a lengthier output contraction. The larger and more protracted increase in debt and the more protracted fall in GDP turn also into a slower consolidation of debt as a fraction of GDP.

Finally, the nonlinear model allows us to study another important feature of recent debt crises, i.e. that nominal interest rates were at the ZLB, and conventional monetary policy could not be more accommodative. We investigate this issue by simulating a demand-driven recession triggering a decline in output and inflation and a surge in government debt, and by keeping the nominal interest rate pegged for a certain number of periods so that it cannot adjust following the shock. It should be clarified that the optimized monetary policy rule is in place when the peg is not at play. This interest rate peg approach has already been adopted in a different context by Cogan et al. (2010), but it is very useful also in this case, as the scale of the model does not allow us to employ fully nonlinear techniques and apply a non-negativity constraint.²⁰

Results are reported in Figure 6. We consider a preference shock inducing an increase in the debt-to-GDP ratio of approximately 1% when the central bank is free to follow the optimized simple rule computed for the case in which the steady-state debt to GDP ratio is 90%. We then compare the responses of key macroeconomic variables in the cases in which the interest rate is constrained to remain constant for 4, 6 and 8 periods. The increase in the duration of the peg from zero to 4 and 6 periods magnifies the fall in inflation, the depth of the recession, and hence the increase and the persistence of government debt as a fraction of GDP. It is noteworthy that a longer duration of the interest rate peg–e.g. 8 periods–has opposite effects. The mechanism works through inflation expectations. In fact, in a perfect foresight context, agents know with certainty that the central bank will keep the interest rate constant also at a time when the recession has come to an end. The increase in expected inflation has an immediate effect on current inflation and translates into an increase in private demand and a fall in government debt.

While in the real world, the Global Financial Crisis and the zero lower bound constraint were much more persistent than in the rosy example presented here, the model delivers two important messages: (i) a constrained monetary policy exacerbates the negative fiscal implications of a recession; and (ii) if central banks can credibly keep interest rates at the effective lower bound for a longer time, inflation expectations help the government debt consolidation process.

5 Introducing long-term government debt

To introduce long-term government debt in a straightforward way, we follow Harrison (2012) and assume that the government issues not only one-period bonds, $B_{t}^{S}$ , but also consols, $B_{t}^{C}$ . These yield one unit of currency each period for the infinite future and the value of a consol is denoted by V_t.²¹

The representation of government debt in terms of short and long bonds arises from the government budget constraint,

\begin{matrix} B_{t}^{S} + V_{t} B_{t}^{C} = (1 - Δ_{t}^{g}) [R_{t - 1}^{S} B_{t - 1}^{S} + (1 + V_{t}) B_{t - 1}^{C}] + P_{t} (G_{t} - T_{t}) + Ξ_{t}, & (28) \end{matrix}

where $\begin{matrix} Ξ_{t} = Δ_{t}^{g} [R_{t - 1}^{S} B_{t - 1}^{S} + (1 + V_{t}) B_{t - 1}^{C}] \end{matrix}$ represents the nominal transfer made by the government in case of default. The total value of long bonds is defined as $B_{t}^{L} = V_{t} B_{t}^{C}$ so that equation (28) may be rewritten as

\begin{matrix} B_{t}^{S} + B_{t}^{L} = (1 - Δ_{t}^{g}) [R_{t - 1}^{S} B_{t - 1}^{S} + R_{t}^{L} B_{t - 1}^{L}] + P_{t} (G_{t} - T_{t}) + Ξ_{t}, & (29) \end{matrix}

where $R_{t}^{L} \equiv (1 + V_{t}) / V_{t - 1}$ is the ex post one-period return on consols.

Let $b_{t}^{L} \equiv B_{t}^{L} / P_{t}$ and ${\tilde{Ξ}}_{t} \equiv Ξ_{t} / P_{t}$ , then the government budget constraint in real terms (14) is replaced by

\begin{matrix} b_{t}^{S} + b_{t}^{L} = (1 - Δ_{t}^{g}) [\frac{R_{t - 1}^{S}}{Π_{t}} b_{t - 1}^{S} + \frac{R_{t}^{L}}{Π_{t}} b_{t - 1}^{L}] + G_{t} - T_{t} + {\tilde{Ξ}}_{t} . & (30) \end{matrix}

Harrison (2012) assumes the real stock of consols to be held fixed at b^C so that the value of long-term bonds is given by

b_{t}^{L} = b^{c} V_{t},

but here we assume that $b_{t}^{C}$ is a policy instrument, and that the simple rule implemented for it is given by

\begin{matrix} \log (\frac{b_{t}^{C}}{b^{C}}) = ρ_{b c} \log (\frac{b_{t - 1}^{C}}{b^{C}}) + ρ_{b c b} \log (\frac{b_{t}^{g}}{b^{g}}) - ρ_{b c y} \log (\frac{Y_{t}}{Y}), & (31) \end{matrix}

where ρ_bc is a smoothing parameter and ρ_bcb, ρ_bcy are the feedback parameters on total real government debt $b_{t}^{g} \equiv (B_{t}^{S} + V_{t} B_{t}^{C}) / P_{t}$ and output. When we allow for the presence of long-term debt, real short-term government debt in the fiscal rules (16) and (17) needs to be replaced by total real government debt.

In order for the presence of long-term government bonds to matter in the model, there must be impediments to arbitrage behaviour that equalises asset returns. We introduce these impediments as in Andrés et al. (2004) and Harrison (2012), i.e. by assuming that households perceive long-term bonds as less liquid and hence demand additional holdings of short-term government bonds when their holdings of long-term bonds increase. This assumption captures Tobin’s claim that relative returns of different assets are affected by their relative supplies. To operationalise this mechanism, households are assumed to have a preference for keeping the ratio of short-to-long-term bond holdings constant and that departures from the preferred portfolio composition causes a welfare cost. This assumption translates into adding a convex portfolio adjustment cost, $- \frac{ν^{B}}{2} {(δ^{B} \frac{(B_{t + s}^{S})^{j}}{(B_{t + s}^{L})^{j}} - 1)}^{2}$ , in households’ utility function, which will affect the first-order conditions on ${(B_{t}^{S})}^{j}$ and ${(B_{t}^{L})}^{j}$ . Parameter δ^B is set equal to the steady-state ratio of long-term bonds to short-term bonds, rendering the cost equal to zero at the steady state, while ν^B represents the elasticity of the long-term bond rate with respect to the portfolio mix. Following Harrison (2012), we set δ^B = 3 and ν^B = 0.1, in accordance with empirical evidence for the US.

5.1 Results

In the presence of long government bonds, we obtain qualitatively similar results to the case of short bonds only, as far as welfare rankings and optimised simple rules are concerned (see Table 4).

Table 4.

Optimal policy results for alternative government debt/GDP ratios allowing for long-term government debt

Hence, to appreciate some interesting differences, it is convenient to plot the impulse responses of key macroeconomic variables to a sudden increase in government debt, under the various debt/GDP scenarios, comparing optimal (Ramsey) and time-consistent policies across the baseline (short-term bonds only) versus the extended model (short and long-term bonds combined), as we do in Figure 7. When the government has the possibility of issuing long-term bonds, it is always optimal to increase the ratio of long-to-short bonds, in the face of a positive debt shock. Without government’s commitment (time-consistent policy), the pace of consolidation is invariably fast, as it is in the model with short-term bonds only. Under commitment, it is still the initial government debt-to-GDP ratio to be the main driver of the optimal consolidation speed. However, if the government also issues long-term bonds, the optimal debt consolidation pace is slower than it is in the case of short-term bonds only, with the difference being more visible at higher debt levels implying greater sovereign risk premia.

6 Conclusions

Our paper contributes to a large recent literature of both an empirical and DSGE-modelling nature that has studied the optimal speed of fiscal consolidation and the monetary-fiscal rules that should be used to achieve this. It adds to a remarkable consensus on the subject, in part documented in our introduction, that – with some important caveats – optimal consolidation should be slow.

Our contribution is, first, to examine this important policy question within a DSGE framework with both a full range of frictions and a rich fiscal component, allowing the initial public debt overhang and the maturity composition of debt to play a significant role. Second, our optimal policy computations consistently look at a comprehensive set of issues that the literature has so far looked in isolation. In particular, our rules have the desirable property of avoiding a frequent violation of the ZLB, and examine crucial properties of policymaking such as commitment, time-consistency, timelessness, and costs of simplicity assessed via the computation of simple passive fiscal (active monetary) and active fiscal (passive monetary) rules.

The bottom line of the analysis is that the speed of debt consolidation is fast only if the government lacks commitment–i.e. the cost of not being able to commit is reflected in a quick consolidation of government debt–or if the initial public debt overhang is very high and the government cannot access official bailout schemes curbing risk premia. In addition, the possibility for the government to commit to the re-payment of long-term bonds calls for further gradualism in the debt consolidation strategy.

Consider for instance our simulation of the optimised simple fiscal-monetary (passive fiscal, active monetary) commitment rule with short-term debt. With an initial moderately high steady-state debt/GDP ratio of 90% that jumps to 105% (for whatever reason), sovereign risk implies a debt reduction policy, but one that allows the debt-GDP ratio to rise for about 10 quarters, and then to slowly fall back to 105% after about a further 20 quarters. This 7-8 year adjustment is clearly a big departure from fiscal consolidation programmes we have seen implemented in the post-financial-crisis era.

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Appendix

A Equilibrium conditions

A.1 Utility function and marginal utilities

\begin{matrix} U (X_{t}, H_{t}) = \frac{X_{t}^{1 - σ c}}{1 - σ_{c}} {(1 - ψ H^{υ})}^{1 - σ c} - \frac{ν^{B}}{2} {(δ^{B} \frac{b_{t}^{S}}{b_{t}^{L}} - 1)}^{2} & (A .1) \end{matrix}

\begin{matrix} X_{t} = {ν_{x}^{\frac{1}{σ x}} {[(X_{t}^{c})]}^{\frac{σ_{x - 1}}{σ_{x}}} + (1 - ν_{x})^{\frac{1}{σ_{x}}} G_{t}^{\frac{σ_{x - 1}}{σ_{x}}}}^{\frac{σ_{x}}{σ_{x - 1}}} & (A .2) \end{matrix}

\begin{matrix} U_{X^{c}, t} = e_{t}^{B} ν_{x}^{\frac{1}{σ_{x}}} X_{t}^{- σ_{c}} {(1 - ψ H_{t}^{υ})}^{1 - σ_{c}} {(\frac{X_{t}}{X_{t}^{c}})}^{\frac{1}{σ_{x}}} & (A .3) \end{matrix}

\begin{matrix} U_{H, t} = - e_{t}^{B} υ ψ X_{t}^{1 - σ_{c}} {(1 - ψ H_{t}^{υ})}^{- σ_{c}} H_{t}^{υ - 1} & (A .4) \end{matrix}

A.2 Consumption/saving

\begin{matrix} X_{t}^{c} = C_{t} - θ C_{t - 1} & (A .5) \end{matrix}

\begin{matrix} U_{X_{t}^{c}} - (1 + τ_{t}^{C}) λ_{t} = θ β E_{t} [U_{X_{t + 1}^{c}}] & (A .6) \end{matrix}

\begin{matrix} 1 = E_{t} [D_{t, t + 1} \frac{R_{t}}{Π_{t + 1}}] & (A .7) \end{matrix}

\begin{matrix} D_{t, t + 1} = β \frac{λ_{t + 1}}{λ_{t}} & (A .8) \end{matrix}

\begin{matrix} 1 = E_{t} [(1 - Δ_{t}^{g}) β \frac{λ_{t + 1}}{λ_{t}} \frac{R_{t}^{S}}{Π_{t + 1}}] - \frac{ν^{B}}{λ_{t}} \frac{δ^{B}}{b_{t}^{L}} (δ^{B} \frac{b_{t}^{S}}{b_{t}^{L}} - 1) & (A.9) \end{matrix}

\begin{matrix} 1 = E_{t} [(1 - Δ_{t}^{g}) β \frac{λ_{t + 1}}{λ_{t}} \frac{R_{t + 1}^{L}}{Π_{t + 1}}] + \frac{ν^{B}}{λ_{t}} \frac{δ^{B} b_{t}^{S}}{{(b_{t}^{L})}^{2}} (δ^{B} \frac{b_{t}^{S}}{b_{t}^{L}} - 1) & (A.10) \end{matrix}

A.3 Investment

\begin{matrix} K_{t + 1} = (1 - δ) K_{t} + e_{t}^{I} I_{t} [1 - S (\frac{I_{t}}{I_{t} - 1})] & (A.11) \end{matrix}

\begin{matrix} Q_{t} = E_{t} {D_{t, t + 1} [(1 - τ_{t + 1}^{K}) u_{t + 1} R_{t + 1}^{K} + δ Q_{t + 1} τ_{t + 1}^{K} u_{t + 1} - a (u_{t + 1}) + (1 - δ) Q_{t + 1}]} & (A.12) \end{matrix}

\begin{matrix} e_{t}^{I} Q_{t} = (1 - S (\frac{I_{t}}{I_{t - 1}}) - S' (\frac{I_{t}}{I_{t - 1}}) \frac{I_{t}}{I_{t - 1}}) + E_{t} (e_{t + 1}^{I} D_{t, t + 1} + Q_{t + 1} S' (\frac{I_{t + 1}}{I_{t}}) {(\frac{I_{t + 1}}{I_{t}})}^{2}) = 1 & (A.13) \end{matrix}

\begin{matrix} a' (u_{t}) = (1 - τ_{t}^{K}) R_{t}^{K} + δ Q_{t} τ_{t}^{K} & (A.14) \end{matrix}

\begin{matrix} S (\frac{I_{t}}{I_{t - 1}}) = \frac{γ}{2} {(\frac{I_{t}}{I_{t - 1}} - 1)}^{2} & (A.15) \end{matrix}

\begin{matrix} S' (\frac{I_{t}}{I_{t - 1}}) = γ (\frac{I_{t}}{I_{t - 1}} - 1) & (A.16) \end{matrix}

\begin{matrix} a (u_{t}) = γ_{1} (u_{t} - 1) + \frac{γ_{2}}{2} (u_{t} - 1)^{2} & (A.17) \end{matrix}

\begin{matrix} a' (u_{t}) = γ_{1} + γ_{2} (u_{t} - 1) & (A.18) \end{matrix}

A.4 Wage setting

\begin{matrix} (1 - τ_{t}^{W}) (e_{t}^{W} η - 1) w_{t} - e_{t}^{W} η \frac{w_{t}}{{\tilde{μ}}_{t}} + ξ^{W} (Π_{t}^{W} - Π) w_{t} Π_{t}^{W} = E_{t} [D_{t, t + 1} ξ^{W} (Π_{t + 1}^{W} - Π) w_{t + 1} Π_{t + 1}^{W} \frac{H_{t + 1}}{H_{t}}] & (A.19) \end{matrix}

\begin{matrix} {\tilde{μ}}_{t} = w_{t} / M R S_{t} & (A.20) \end{matrix}

\begin{matrix} M R S_{t} = - U_{H, t} / λ_{t} & (A.21) \end{matrix}

\begin{matrix} Π_{t}^{W} = \frac{w_{t}}{w_{t - 1}} Π_{t} & (A.22) \end{matrix}

A.5 Production

\begin{matrix} F (e_{t}^{A} H_{t}, {\tilde{K}}_{t}) = {(e_{t}^{A} H_{t})}^{α} {(u_{t} K_{t})}^{1 - α} & (A.23) \end{matrix}

\begin{matrix} F_{H, t} = α \frac{F (e_{t}^{A} H_{t}, {\tilde{K}}_{t})}{H_{t}} & (A.24) \end{matrix}

\begin{matrix} F_{K, t} = (1 - α) \frac{F (e_{t}^{A} H_{t}, {\tilde{K}}_{t})}{u_{t} K_{t}} & (A.25) \end{matrix}

\begin{matrix} Y_{t} = F (e_{t}^{A} H_{t}, {\tilde{K}}_{t}) - F C & (A.26) \end{matrix}

\begin{matrix} R_{t}^{K} = M C_{t} F_{K, t} & (A.27) \end{matrix}

\begin{matrix} w_{t} = M C_{t} F_{H, t} & (A.28) \end{matrix}

\begin{matrix} (1 - e_{t}^{P} ζ) + e_{t}^{P} ζ M C_{t} - ξ^{P} (Π_{t} - Π) Π_{t} + ξ^{P} E_{t} [D_{t, t + 1} (Π_{t + 1} - Π) Π_{t + 1}] Y_{t + 1} / Y_{t} = 0 & (A.29) \end{matrix}

A.6 Government

\begin{matrix} p_{t}^{*} = P (Γ_{t}^{*} \leq Γ_{t - 1}) = \frac{\exp (η_{1} + η_{2} Γ_{t - 1})}{1 + \exp (η_{1} + η_{2} Γ_{t - 1})} & (A.30) \end{matrix}

\begin{matrix} Δ_{t} = p_{t}^{*} Δ & (A.31) \end{matrix}

\begin{matrix} Γ_{t} = b_{t}^{g} / Y_{t} & (A.32) \end{matrix}

\begin{matrix} b_{t}^{g} = b_{t}^{S} + b_{t}^{L} & (A.33) \end{matrix}

\begin{matrix} \log (\frac{b_{t}^{C}}{b^{C}}) = ρ_{b c} log (\frac{b_{t - 1}^{C}}{b^{C}}) + ρ_{b c B} log (\frac{b_{t}^{g}}{b^{g}}) - ρ_{b c Y} log (\frac{Y_{t}}{Y}) & (A.34) \end{matrix}

\begin{matrix} R_{t}^{L} = \frac{1 + V_{t}}{V_{t - 1}} & (A.35) \end{matrix}

\begin{matrix} b_{t}^{S} + b_{t}^{L} = (1 - Δ_{t}^{g}) [\frac{R_{t - 1}^{S}}{Π_{t}} b_{t - 1}^{S} + \frac{R_{t}^{L}}{Π_{t}} b_{t - 1}^{L}] + G_{t} - T_{t} + {\tilde{Ξ}}_{t} & (A.36) \end{matrix}

\begin{matrix} {\tilde{Ξ}}_{t} = Δ_{t}^{g} [\frac{R_{t - 1}^{S}}{Π_{t}} b_{t - 1}^{S} + \frac{R_{t}^{L}}{Π_{t}} b_{t - 1}^{L}] & (A.37) \end{matrix}

\begin{matrix} T_{t} = τ_{t}^{C} C_{t} + τ_{t}^{W} w_{t} h_{t} + τ_{t}^{K} [(R_{t}^{K} - δ Q_{t}) u_{t} K_{t}] + τ_{t}^{L} & (A.38) \end{matrix}

\begin{matrix} τ_{t}^{C} = τ_{t} τ^{C} & (A.39) \end{matrix}

\begin{matrix} τ_{t}^{W} = τ_{t} τ^{W} & (A.40) \end{matrix}

\begin{matrix} τ_{t}^{K} = τ_{t} τ^{K} & (A.41) \end{matrix}

\begin{matrix} τ_{t}^{L} = τ_{t} τ^{L} & (A.42) \end{matrix}

\begin{matrix} \log (\frac{τ_{t}}{τ}) = ρ_{τ} \log (\frac{τ_{t - 1}}{τ}) + ρ τ b log (\frac{b_{t - 1}^{g}}{b^{g}}) + ρ_{τ y} log (\frac{Y_{t}}{Y}) & (A.43) \end{matrix}

\begin{matrix} \log (\frac{G_{t}}{G}) = ρ_{g} \log (\frac{G_{t - 1}}{G}) + ρ_{g b} log (\frac{b_{t - 1}^{g}}{b^{g}}) - ρ_{g y} log (\frac{Y_{t}}{Y}) & (A.44) \end{matrix}

A.7 Monetary policy

\begin{matrix} \log (\frac{R_{t}}{R}) = ρ_{τ} \log (\frac{R_{t - 1}}{R}) + ρ τ π log (\frac{Π_{t}}{Π}) + ρ_{τ y} log (\frac{Y_{t}}{Y}) & (A.45) \end{matrix}

A.8 Resource constraint

\begin{matrix} Y_{t} = C_{t} + I_{t} + G_{t} + \frac{ξ^{P}}{2} {(Π_{t} - \bar{Π})}^{2} Y_{t} + \frac{ξ^{W}}{2} (Π_{t}^{W} - \bar{Π})^{2} w_{t} H_{t} + a (u_{t}) K_{t} & (A.46) \end{matrix}

A.9 Autoregressive processes

\begin{matrix} \log (\frac{e_{t}^{B}}{e^{B}}) = ρ_{x} \log (\frac{e_{t}^{B}}{e^{B}}) + ε_{t}^{B} & (A.47) \end{matrix}

\begin{matrix} \log (\frac{e_{t}^{P}}{e^{P}}) = ρ_{P} \log (\frac{e_{t}^{P}}{e^{P}}) + ε_{t}^{P} & (A.48) \end{matrix}

\begin{matrix} \log (\frac{e_{t}^{I}}{e^{I}}) = ρ_{I} \log (\frac{e_{t}^{I}}{e^{I}}) + ε_{t}^{I} & (A.49) \end{matrix}

\begin{matrix} \log (\frac{e_{t}^{A}}{e^{A}}) = ρ_{A} \log (\frac{e_{t}^{A}}{e^{A}}) + ε_{t}^{A} & (A.50) \end{matrix}

\begin{matrix} \log (\frac{e_{t}^{W}}{e^{W}}) = ρ_{A} \log (\frac{e_{t}^{W}}{e^{W}}) + ε_{t}^{W} & (A.51) \end{matrix}

B The Ramsey problem and the LQ approximation

The problem is to maximise $E_{0} Σ_{t = 0}^{\infty} β^{t} u (y_{t}, w_{t})$ such that

\begin{matrix} E_{t} f (y_{t}, y_{t + 1}, y_{t - 1}, w_{t}, ɛ_{t}) = 0 & (B.1) \end{matrix}

where $y_{t}^{T} \equiv [z_{t}^{T} x_{t}^{T}]$ , z_t is a vector of predetermined variables, x_t that of non-predetermined, ‘jump’ variables, w_t is a vector of instruments and ε_t is a vector of exogenous shocks. For convenience, assume that there are no higher order leads or lags greater than y_t+1 and y_t-1.²²

Now write the Lagrangian for the problem as

\begin{matrix} L = Σ_{t = 0}^{\infty} β^{t} [u (y_{t}, w_{t}) + λ_{t + 1}^{T} f (y_{t}, y_{t + 1}, y_{t - 1}, w_{t}, ɛ_{t})] & (B.2) \end{matrix}

From Levine et al. (2008b) and Benigno and Woodford (2012) we know that, for a purely backward-looking system, an approximate solution for this problem is obtained by solving for the deterministic steady state of the optimum, and then solving the stabilisation problem obtained by maximising the second order approximation to the Lagrangian, subject to the linearised constraints about this steady state.

First-order conditions are given by

\begin{matrix} \frac{\partial L}{\partial w_{t}} = u_{2} + λ_{t + 1}^{T} f_{4} (y_{t}, y_{t + 1}, y_{t - 1}, w_{t}, ɛ_{t}), & (B.3) \end{matrix}

\begin{matrix} \frac{\partial L}{\partial y_{t}} = & u_{1} + λ_{t + 1}^{T} f_{1} (y_{t}, y_{t + 1}, y_{t - 1}, w_{t}, ɛ_{t}) + \frac{1}{β} λ_{t}^{T} f_{2} (y_{t - 1}, y_{t}, y_{t - 2}, w_{t - 1}, ɛ_{t - 1}) \\ + β λ_{t + 2}^{T} f_{3} (y_{t + 1}, y_{t + 2}, y_{t}, w_{t + 1}, ɛ_{t + 1}) & (B.4) \end{matrix}

where the subscripts in {u_i, f_j} refer to the partial derivatives of the ith, jth variable in u, f.

Now partition λ_t = [λ_1,t λ_2,t] so that λ_1,t, the co-state vector associated with the backward-looking component of (B.1) is of dimension (n — m) x 1 and λ_2,t, the co-state vector associated with the forward-looking component is of dimension m × 1.²³ An important optimality condition is:

\begin{matrix} λ_{2, 0} = 0; (ex ante optimal) & (B.5) \end{matrix}

\begin{matrix} λ_{2, 0} = {\bar{λ}}_{2}; (' timeless' solution) & (B.6) \end{matrix}

where ${\bar{λ}}_{2}$ is the deterministic steady state of λ_2,t. To complete our solution we require 2n boundary conditions. Then together with (B.5) or (B.6), Z₀ given are n of these. The ‘transversality condition’ lim_t→∞ λ_t = λ gives us the remaining n.

To arrive at the expression (²⁷) in the main text and the contribution of initial values to the problem with rational expectations, write the second-order approximation to the Lagrangian (for the deterministic case) as a deviation about its steady state value $\bar{L}$ :

\begin{matrix} L - \bar{L} = & Σ β^{t} [{\bar{u}}_{1} {\tilde{y}}_{t} + {\bar{u}}_{2} {\tilde{w}}_{t} + {\bar{λ}}^{T} ({\bar{f}}_{1} {\tilde{y}}_{t} + {\bar{f}}_{2} {\tilde{y}}_{t + 1} + {\bar{f}}_{3} {\tilde{y}}_{t - 1} + {\bar{f}}_{4} {\tilde{w}}_{t}) \\ + {\tilde{λ}}_{t + 1}^{T} ({\bar{f}}_{1} {\tilde{y}}_{t} + {\bar{f}}_{2} {\tilde{y}}_{t + 1} + {\bar{f}}_{3} {\tilde{y}}_{t - 1} + {\bar{f}}_{4} {\tilde{w}}_{t}) + \frac{1}{2} \underset{i, j}{Σ} (\bar{u} + {\bar{λ}}^{T} f)_{i j} {\tilde{x}}_{i t} {\tilde{x}}_{j t}], & (B.7) \end{matrix}

where $\bar{λ}$ is the steady state of the Lagrange multiplier and ${\tilde{y}}_{t}, {\tilde{w}}_{t}$ represents deviations about $\bar{y}, \bar{w}$ , where the latter are the steady states of the optimal problem, subscripts on f, u refer to partial derivatives and ${\tilde{x}}_{i t}$ refers to the various elements of ${\tilde{y}}_{t}, {\tilde{y}}_{t + 1}, {\tilde{y}}_{t - 1}, {\tilde{w}}_{t}$ .

The problem approximates to maximisation of the discounted sum of the second-order terms in (B.7) subject to the linearised constraints of f = 0. This is because the linear deviations cancel out with one another because of the first order conditions.

Note however that there are two terms in the first line of (B.7) that remain uncancelled:

\begin{matrix} - \frac{1}{β} {\bar{λ}}^{T} f_{2} {\tilde{y}}_{0} + {\bar{λ}}^{T} f_{3} {\tilde{y}}_{- 1} . & (B.8) \end{matrix}

This can be seen by expanding the first line of (B.7):

\begin{matrix} {(L - \bar{L})}_{1 s t o r d e r} & = Σ β^{t} [u_{1} {\tilde{y}}_{t} + u_{2} {\tilde{w}}_{t} + {\bar{λ}}^{T} (f_{1} {\tilde{y}}_{t} + f_{2} {\tilde{y}}_{t + 1} + f_{3} {\tilde{y}}_{t - 1} + f_{4} {\tilde{w}}_{t}) \\ = u_{1} {\tilde{y}}_{0} + u_{2} {\tilde{w}}_{0} + {\bar{λ}}^{T} (f_{1} {\tilde{y}}_{0} + f_{2} {\tilde{y}}_{1} + f_{3} {\tilde{y}}_{- 1} + f_{4} {\tilde{w}}_{0}) \\ + β (u_{1} {\tilde{y}}_{1} + u_{2} {\tilde{w}}_{1} + {\bar{λ}}^{T} (f_{1} {\tilde{y}}_{1} + f_{2} {\tilde{y}}_{2} + f_{3} {\tilde{y}}_{0} + f_{4} {\tilde{w}}_{1})) \\ + β^{2} (u_{1} {\tilde{y}}_{2} + u_{2} {\tilde{w}}_{2} + {\bar{λ}}^{T} (f_{1} {\tilde{y}}_{2} + f_{2} {\tilde{y}}_{3} + f_{3} {\tilde{y}}_{0} + f_{4} {\tilde{w}}_{1})) + …), & (B.9) \end{matrix}

and recalling from (B.4) for the ${\tilde{y}}_{0}$ term that

\begin{matrix} u_{1} + {\bar{λ}}^{T} (f_{1} + \frac{1}{β} f_{2} + β f_{3}) = 0 & (B.10) \end{matrix}

Clearly there is no maximisation with respect to $\tilde{y} - 1$ , but one is free to set the forward-looking variables in ${\tilde{y}}_{0}$ . Thus one must account for the forward-looking terms in $- \frac{1}{β} {\bar{λ}}^{T} f_{2} {\tilde{y}}_{0}$ when optimising policy.

C Data

In this section we describe the data sources and how we constructed the observables to be used in the moment-matching procedure. In Table C.1 we present the original dataset and the data sources. From these sources data we constructed the 9 observables²⁴ (Table C.2) and we considered the subsample 1984:Q1-2008:Q2 for the matching moment procedure. All real variables were filtered using an HP(1600) filter.

Table C.1.

Data sources

Table C.2.

Data transformations – observables

D Moments

Table D.1.

Moments of key macroeconomic variables

Table D.1.

Moments of key macroeconomic variables

Moment	Data	Model
Standard deviations (in %)
Real output	1.09	1.89
Private consumption	0.88	1.40
Private investment	4.06	6.94
Inflation	0.23	0.91
Real wage	1.10	1.82
Hours worked	2.20	1.96
Government spending	1.07	1.84
Government revenue	3.18	3.93
Interest rate	0.61	0.84
Autocorrelations
Real output	0.8516	0.8824
Private consumption	0.8387	0.9742
Private investment	0.9148	0.9551
Inflation	0.5960	0.3791
Real wage	0.8197	0.8975
Hours worked	0.9113	0.6480
Government spending	0.7105	0.9811
Government revenue	0.8649	0.7606
Interest rate	0.9560	0.8731
Cross-correlations with output
Private consumption	0.8723	0.6414
Private investment	0.9201	0.8779
Inflation	0.1795	0.1260
Real wage	−0.0539	0.7476
Hours worked	0.3714	0.4113
Government spending	−0.1550	−0.1008
Government revenue	0.7603	0.6920
Interest rate	0.3460	0.0129
Cross-correlations
Inflation/Interest rate	0.4111	0.6540
Private/public consumption	−0.0334	−0.0938

Table D.1.

Moments of key macroeconomic variables

Moment	Data	Model
Standard deviations (in %)
Real output	1.09	1.89
Private consumption	0.88	1.40
Private investment	4.06	6.94
Inflation	0.23	0.91
Real wage	1.10	1.82
Hours worked	2.20	1.96
Government spending	1.07	1.84
Government revenue	3.18	3.93
Interest rate	0.61	0.84
Autocorrelations
Real output	0.8516	0.8824
Private consumption	0.8387	0.9742
Private investment	0.9148	0.9551
Inflation	0.5960	0.3791
Real wage	0.8197	0.8975
Hours worked	0.9113	0.6480
Government spending	0.7105	0.9811
Government revenue	0.8649	0.7606
Interest rate	0.9560	0.8731
Cross-correlations with output
Private consumption	0.8723	0.6414
Private investment	0.9201	0.8779
Inflation	0.1795	0.1260
Real wage	−0.0539	0.7476
Hours worked	0.3714	0.4113
Government spending	−0.1550	−0.1008
Government revenue	0.7603	0.6920
Interest rate	0.3460	0.0129
Cross-correlations
Inflation/Interest rate	0.4111	0.6540
Private/public consumption	−0.0334	−0.0938

Cantore: University of Surrey; Levine: University of Surrey; Melina: IMF and City University of London; Pearlman: City University of London. This version of the paper is forthcoming in Macroeconomics Dynamics. Previous versions, circulated under the title “Optimal Fiscal and Monetary Rules in Normal and Abnormal Times”, were presented at seminars at the BI Norwegian Business School, the Universities of Loughborough, Manchester, Oxford and Reading; at a European Monetary Forum Conference held at the Bank of Greece, “Financial Crisis – Models and Policy Responses”, Athens, 30-31 March 2012; at the conference “Monetary and Fiscal Policy Rules with Labour Market and Financial Frictions”, University of Surrey, 14-15 September 2012; the 6th International Conference on “Computational and Financial Econometrics” (CFE 2012), Oviedo, Spain, 1-3 December 2012; and the Barcelona Summer Forum “Fiscal Sustainability in the XXI Century”, Barcelona, Spain 9-10 June 2016. Comments by participants at these events are gratefully acknowledged, especially those of a discussant, Katsuyuki Shibayama, at the Surrey Conference, Dora Benedek, Sheherya Malik, and two anonymous referees. All remaining errors are ours. We also acknowledge financial support from ESRC project RES-062-23-2451.

This result emerges in these papers from a second-order perturbation stochastic solution that captures budget risk considerations. In our model these emerge from the sovereign risk premium.

The only difference is the use of Rotemberg rather than Calvo contracts. It is well-known that for a low steady state inflation rate the differences in the dynamic properties are small up to first order. Moreover the Calvo approach cannot aggregate prices with a time-varying elasticity of demand as in (²) and (⁸).

External habit formation, i.e. on the average level of consumption rather than on own’s consumption, creates an externality whereby households supply too many hours of work in the steady state. As a result, policies curbing employment can be welfare enhancing.

Examples of papers in the literature that optimize two or more tax instruments in the context of simpler NK models are Correia et al. (2008) and Correia et al. (2013), who use tax policy to neutralize price stickiness and to stimulate the economy at the zero lower bound, respectively.

In the context of a NK model, Cantore et al. (2012) compare simple interest-rate rules embedding the model-based definition of the output gap to rules employing deviations of output from the steady state. They find that when the two types of rule are designed to be optimal, they result in almost identical real and inflation outcomes.

In our matching-moments procedure below, it turns out that σ_x < 1 indicating empirical support for complementarity between private and public consumption (on this see also the Bayesian estimates of Cantore et al. (2014)).

Later we examine the effects of high debt levels.

To see this, note that from equation (¹³), the haircut rate, ${\bar{Δ}}^{g}$ , consistent with ABP₂ and $p_{2}^{*}$ is ${\bar{Δ}}^{g} = \frac{1 - \frac{1}{\frac{A B P_{2}}{40000} + 1}}{p_{2}^{*}} .$ In the absence of long-term government bonds, equations (A.7) and (A.9) imply the following steady-state sovereign risk premium: $\frac{R^{S}}{R} = \frac{1}{(1 - Δ^{g})} = 1 + \frac{A B P}{40000},$ using which Δ^g can be written as a function of a chosen premium expressed in annual basis points, $Δ^{g} = 1 - \frac{1}{1 + \frac{A B P}{40000}}$ . Finally, from equation (¹³), ${\bar{Δ}}^{g} = Δ^{g} / p^{*}$

We prefer a moment-matching approach to a Bayesian estimation approach as the latter produces counter-factual volatilities for macroeconomic variables such as consumption and hours that are crucial for the policy analysis.

Of the three policy regimes compared in the paper, optimal timeless commitment, optimized simple rules and optimal time-consistent policy (discretion), the first two can be computed without an LQ approximation of the non-linear set-up using perturbation methods for the stochastic solution, although the zero lower bound constraint still requires the penalty function approach of our paper. A Markov-perfect time-consistent solution requires global methods which are, as yet, not feasible for a medium-sized NK model with many state variables. For a small RBC model however see Dennis and Kirsanova (2015).

To achieve such a steady state in the LQ approximation procedure, we make an underlying assumption that fiscal policymakers assign a small cost to quadratic deviations of the fiscal variables about some target values corresponding to historically observed values, so that the steady state solution to the first order conditions are forced to these values. As a result of the small cost of these quadratic deviations, we can ignore their contribution to the LQ approximation to welfare. In addition, we pin down the government debt-to-GDP ratio by appropriately changing (non-distortionary) lump sum taxes. The advantage of this approach is that the steady states of the key macroeconomic variables are the same across (i) the three policy regimes, (ii) for the case of short bonds only versus that of long and short bonds; and (iii) different steady-state government debt/income ratios.

The LQ solutions for these three policy regimes are now standard – see, for example, Levine et al. (2008a) for details. Regarding discretionary policy, recent important contributions by Blake and Kirsanova (2012) and Dennis and Kirsanova (2013) raise the possibility of multiple discretionary equilibria. These are of two types: “point-in-time”, which give multiple responses of the private sector to a given policy rule and those arising from more than one discretionary policy. The iterative algorithm we use rules out the former. The latter can in principle be found by experimenting with different initialisations; however for the model and loss function employed in this paper we have not been able to find more than one equilibrium.

A previous version of this paper compared the use of one or the other, but we found this of little of importance with regard to the main thrust of the paper.

For a recent discussion of the assignment issue see Kirsanova et al. (2009).

Cochrane (2011) proposes passive fiscal rules to avoid the arbitrary assumption of a non-explosive path for the price level needed in the standard Blanchard-Kahn rational expectations solution. But Sims (2013) points out that introducing a very small feedback from inflation to the tax-rate, together with ZLB constraint on the nominal interest rate and an upper bound on government asset accumulation, are sufficient to rule out such explosive paths.

In fact ρτb, ρgb ≤ 0.25 is the minimal feedback for either instrument separately to stabilise the government debt-income ratio when there is no risk premium – effectively the case for a debt to income ratio of 70%. However, for the higher debt to income ratios of this paper, interest payments on debt are higher, so this minimal feedback must be higher. For consistency throughout, we have therefore imposed an upper bound of 1, but robustness exercises have shown that changing this bound has very little effect on the performance of optimal passive rules for debt to income ratios of 70%.

See Woodford (2003), Proposition 2.9, Chapter 2 in the context of a simple analytical NK model.

To a very good approximation for our LQ set-up, we can compute c_e about the common steady state of our policies as follows. Expanding utility U(C_t,H_t) as a Taylor series about this steady state, the welfare change $Δ U = U_{C} Δ C = U_{C} C \frac{Δ C}{C}$ Losses ΔU relative to the optimal commitment policy reported in the Table have been scaled by 1 — β making the intertemporal and single-period welfare the same in the vicinity of the steady state. The percentage change in consumption is $c_{e} = \frac{Δ C}{C} \times 10^{2}$ . Thus $c_{e} = \frac{Δ C \times 10^{2}}{U_{C} C}$ . For the steady state of this model, U_CC = 2.60. It follow that a welfare loss difference of ΔU = 0.01 gives a consumption equivalent percentage difference of c_e ≈ 0.385%.

For example, see Woodford (2003), Chapter 2.

Bringing the nominal interest rate from its steady state value to zero would require a very large shock and the Dynare’s algorithm solving the nonlinear model does not always converge in the presence of very large shocks, especially in the presence of a non-negativity constraint within a large model. An alternative procedure could be using the linearised version of the model, hitting the economy with a very large preference shock that would push the interest rate to the negative territory, and introducing exogenous interest rate shocks to bring it to zero.

Modelling long-term bonds as consols is a useful alternative to assuming that the long-term bond is a zero-coupon fixed-maturity bond and that there is no secondary market for long-term bonds. In fact the use of consols allows assuming that they can indeed be traded each period and that the optimal long-bond holdings depend on the one-period return on consols.

If there are then just add to the vector y_t another variable that includes one of the lagged variables.

In practice, assigning equations or variables as forward or backward looking is a non-trivial issue, since some variables that appear as forward looking may be functions of other forward looking variables. For full details of how to handle this for the LQ approximation see Levine and Pearlman (2011).

Note that the resulting series of hours (as in Table C.2) is then demeaned before it is used for the estimation/calibration.

B/4Y=0.7
Rule	[ρ_r, ρ_rπ, ρ_ry]	[ρ_τ, ρ_τπ, ρ_τy]	[ρ_g, ρ_gb, ρ_gy]	w_r	Adjusted Loss	cons eq %
Optimal	n/a	n/a	n/a	371	−0.0038
TCT	n/a	n/a	n/a	135	0.0083	0.45
Simple (PF)	1, 0.232, 0	0.051, 0.25, 0	0.789, 0.018, 0	385	0.0000	0.14
Simple (AF)	0,0,0	0,0,0	0,0,0	135	0.0069	0.40
B/4Y=0.9
Rule	[ρ_r, ρ_rπ, ρ_ry]	[ρ_τ, ρ_τπ, ρ_τy]	[ρ_g, ρ_gb, ρ_gy]	w_r	Adjusted Loss	cons eq %
Optimal	n/a	n/a	n/a	296	−0.0046
TCT	n/a	n/a	n/a	102	0.0088	0.49
Simple (PF)	1, 0.236, 0	0.504, 0.25, 0	0.733, 0.046, 0	381	0.0000	0.17
Simple (AF)	0,0,0	0,0,0	0,0,0	79	0.0066	0.41
B/4Y=1.2
Rule	[ρ_r, ρ_rπ, ρ_ry]	[ρ_τ, ρ_τπ, ρ_τy]	[ρ_g, ρ_gb, ρ_gy]	w_r	Adjusted Loss	cons eq %
Optimal	n/a	n/a	n/a	221	−0.0021
TCT	n/a	n/a	n/a	63	0.0032	0.20
Simple (PF)	1,0.261,0	0.579,1,0	0.161, 0.749,0	291	0.0015	0.13
Simple (AF)	0,0,0	0,0,0	0,0,0	60	0.0054	0.28
B/4Y=1.2	Bailout
Rule	[ρ_r, ρ_rπ, ρ_ry]	[ρ_τ, ρ_τπ, ρ_τy]	[ρ_g, ρ_gb, ρ_gy]	w_r	Adjusted Loss	cons eq %
Optimal	n/a	n/a	n/a	310	−0.0047
TCT	n/a	n/a	n/a	244	0.011	0.58
Simple (PF)	1,0.232,0	0,0.25,0	0,0,0	382	0.0002	0.18
Simple (AF)	0,0,0	0,0,0	0,0,0	60	0.0054	0.38

B/4Y=0.7
Rule	[ρ_r, ρ_rπ, ρ_ry]	[ρ_τ, ρ_τπ, ρ_τy]	[ρ_g, ρ_gb, ρ_gy]	w_r	Adjusted Loss	cons eq %
Optimal	n/a	n/a	n/a	371	−0.0038
TCT	n/a	n/a	n/a	135	0.0083	0.45
Simple (PF)	1, 0.232, 0	0.051, 0.25, 0	0.789, 0.018, 0	385	0.0000	0.14
Simple (AF)	0,0,0	0,0,0	0,0,0	135	0.0069	0.40
B/4Y=0.9
Rule	[ρ_r, ρ_rπ, ρ_ry]	[ρ_τ, ρ_τπ, ρ_τy]	[ρ_g, ρ_gb, ρ_gy]	w_r	Adjusted Loss	cons eq %
Optimal	n/a	n/a	n/a	296	−0.0046
TCT	n/a	n/a	n/a	102	0.0088	0.49
Simple (PF)	1, 0.236, 0	0.504, 0.25, 0	0.733, 0.046, 0	381	0.0000	0.17
Simple (AF)	0,0,0	0,0,0	0,0,0	79	0.0066	0.41
B/4Y=1.2
Rule	[ρ_r, ρ_rπ, ρ_ry]	[ρ_τ, ρ_τπ, ρ_τy]	[ρ_g, ρ_gb, ρ_gy]	w_r	Adjusted Loss	cons eq %
Optimal	n/a	n/a	n/a	221	−0.0021
TCT	n/a	n/a	n/a	63	0.0032	0.20
Simple (PF)	1,0.261,0	0.579,1,0	0.161, 0.749,0	291	0.0015	0.13
Simple (AF)	0,0,0	0,0,0	0,0,0	60	0.0054	0.28
B/4Y=1.2	Bailout
Rule	[ρ_r, ρ_rπ, ρ_ry]	[ρ_τ, ρ_τπ, ρ_τy]	[ρ_g, ρ_gb, ρ_gy]	w_r	Adjusted Loss	cons eq %
Optimal	n/a	n/a	n/a	310	−0.0047
TCT	n/a	n/a	n/a	244	0.011	0.58
Simple (PF)	1,0.232,0	0,0.25,0	0,0,0	382	0.0002	0.18
Simple (AF)	0,0,0	0,0,0	0,0,0	60	0.0054	0.38

B/4Y=0.7
Rule	[ρ_r, ρ_rπ, ρ_ry]	[ρ_τ, ρ_τπ, ρ_τy]	[ρ_g, ρ_gb, ρ_gy]	w_r	Adjusted Loss	cons eq %
Optimal	n/a	n/a	n/a	371	−0.0038
Simple (PF)	1, 0.232, 0	0.051, 0.25, 0	0.789, 0.018, 0	385	0.0000	0.14
Superinertial	1.59,0.215,0.034	0,0.15,0	0.39,0.18,0	370	0.0025	0.05

B/4Y=0.7
Rule	[ρ_r, ρ_rπ, ρ_ry]	[ρ_τ, ρ_τπ, ρ_τy]	[ρ_g, ρ_gb, ρ_gy]	w_r	Adjusted Loss	cons eq %
Optimal	n/a	n/a	n/a	371	−0.0038
Simple (PF)	1, 0.232, 0	0.051, 0.25, 0	0.789, 0.018, 0	385	0.0000	0.14
Superinertial	1.59,0.215,0.034	0,0.15,0	0.39,0.18,0	370	0.0025	0.05

B/4Y=0.7
Rule	[ρ_r, ρ_rπ, ρ_ry]	[ρ_τ, ρ_τb, ρ_τy]	[ρ_g, ρ_gb, ρ_gy]	[ρ_bc, ρ_bcb, ρ_bcy]	w_r	Adj Loss	c eq %
Optimal	n/a	n/a	n/a	n/a	331	−0.0051
TCT	n/a	n/a	n/a	n/a	87	0.0059	0.40
Simple (PF)	1.46,0.37,0.03	0,0.16,0	0.87,0,0.10	0.11,0.76,0.82	233	−0.0010	0.15
Simple (AF)	0,0,0	0,0,0	0,0,0	0,0	n/a	0.0061	0.41
B/4Y=0.9
Optimal	n/a	n/a	n/a	n/a	329	−0.0050
TCT	n/a	n/a	n/a	n/a	86	0.0051	0.37
Simple (PF)	1.35,0.36,0.02	0,0.50,0	0.86,0.01,0.12	0.01 0.86,0.98	233	−0.0008	0.15
Simple (AF)	0,0,0	0,0,0	0,0,0	0,0	n/a	0.0063	0.41
B/4Y=1.2
Optimal	n/a	n/a	n/a	n/a	323	−0.0046
TCT	n/a	n/a	n/a	n/a	88	0.0038	0.31
Simple (PF)	1.23,0.35,0.02	0.47,1,0	0.71,0.13,0.41	0.49,0.34,1	232	0.0006	0.19
Simple (AF)	0,0,0	0,0,0	0,0,0	0,0,0	n/a	0.0061	0.39
B/4Y=1.2	no sov risk
Optimal	n/a	n/a	n/a	n/a	650	−0.0082
TCT	n/a	n/a	n/a	n/a	88	0.0054	0.50
Simple (PF)	1.42,0.37,0.02	0, 0.21,0	0.88,0,0.10	0.882,1	240	−0.0010	0.27
Simple (AF)	0,0,0	0,0,0	0,0,0	0,0,0	n/a	0.0061	0.53

Label	Description	Source	Frequency
GDP	Nominal GDP	BEA NIPA Table 1.1.5	Q
PCE	Personal Consumption expenditure (total)	BEA NIPA Table 1.1.5	Q
PFI	Private Fixed Investment	BEA NIPA Table 5.3.5	Q
GCE	Government consumption expenditure and gross investment	BEA NIPA Table 1.1.5	Q
GR	Government revenues	BEA NIPA Table 3.1	Q
RGDP	Real GDP (base year 2005)	BEA NIPA Table 1.1.6	Q
CNP16OV	Civilian non-institutional population, over 16	BLS	Q
CE16OV	Civilian Employment sixteen years and over	BLS	Q
LBMNU	Non-farm business hours worked	BLS	Q
LBCPU	Hourly non-farm business compensation	BLS	Q
FFR	Federal Funds Rate	St. Louis FRED	Q

Label	Description	Source	Frequency
GDP	Nominal GDP	BEA NIPA Table 1.1.5	Q
PCE	Personal Consumption expenditure (total)	BEA NIPA Table 1.1.5	Q
PFI	Private Fixed Investment	BEA NIPA Table 5.3.5	Q
GCE	Government consumption expenditure and gross investment	BEA NIPA Table 1.1.5	Q
GR	Government revenues	BEA NIPA Table 3.1	Q
RGDP	Real GDP (base year 2005)	BEA NIPA Table 1.1.6	Q
CNP16OV	Civilian non-institutional population, over 16	BLS	Q
CE16OV	Civilian Employment sixteen years and over	BLS	Q
LBMNU	Non-farm business hours worked	BLS	Q
LBCPU	Hourly non-farm business compensation	BLS	Q
FFR	Federal Funds Rate	St. Louis FRED	Q

Variable	Description	Construction
GDP_Deflator	GDP deflator	$\frac{G D P}{R G D P} 100$
index	Population index	$\frac{C N P 16 O V}{C N P 16 O V_{2005 : 2}}$
CE16OV_index	Employment index	$\frac{C N P 16 O V}{C N P 16 O V_{2005 : 2}} 100$
Observables	Description	Construction
R_GDP	Real per capita gross domestic product	$L N (\frac{\frac{G D P}{i n d e x}}{G D P_{D e f l a t o r}}) 100$
GOV_SP	Real per capita government spending	$L N (\frac{\frac{G D P}{i n d e x}}{G D P_{D e f l a t o r}}) 100$
GOV_RV	Real per capita government revenues	$L N (\frac{\frac{G D P}{i n d e x}}{G D P_{D e f l a t o r}}) 100$
HOURS	Per capita hours worked	$L N (\frac{\frac{L B M N U * C E 16 O V_{i n d e x}}{100}}{i n d e x}) 100$
WAGE	Real wage	$L N (\frac{L B C P U}{G D P_{D e f l a t o r}}) 100$
FFF	Quarterly Federal Funds rate	$\frac{F F R}{4}$
Π	Inflation	∆GDP_Deflator 100
CON	Real per capita consumption	$L N (\frac{\frac{G D P}{i n d e x}}{G D P_{D e f l a t o r}}) 100$
INV	Real per capita investment	$L N (\frac{\frac{G D P}{i n d e x}}{G D P_{D e f l a t o r}}) 100$

Variable	Description	Construction
GDP_Deflator	GDP deflator	$\frac{G D P}{R G D P} 100$
index	Population index	$\frac{C N P 16 O V}{C N P 16 O V_{2005 : 2}}$
CE16OV_index	Employment index	$\frac{C N P 16 O V}{C N P 16 O V_{2005 : 2}} 100$
Observables	Description	Construction
R_GDP	Real per capita gross domestic product	$L N (\frac{\frac{G D P}{i n d e x}}{G D P_{D e f l a t o r}}) 100$
GOV_SP	Real per capita government spending	$L N (\frac{\frac{G D P}{i n d e x}}{G D P_{D e f l a t o r}}) 100$
GOV_RV	Real per capita government revenues	$L N (\frac{\frac{G D P}{i n d e x}}{G D P_{D e f l a t o r}}) 100$
HOURS	Per capita hours worked	$L N (\frac{\frac{L B M N U * C E 16 O V_{i n d e x}}{100}}{i n d e x}) 100$
WAGE	Real wage	$L N (\frac{L B C P U}{G D P_{D e f l a t o r}}) 100$
FFF	Quarterly Federal Funds rate	$\frac{F F R}{4}$
Π	Inflation	∆GDP_Deflator 100
CON	Real per capita consumption	$L N (\frac{\frac{G D P}{i n d e x}}{G D P_{D e f l a t o r}}) 100$
INV	Real per capita investment	$L N (\frac{\frac{G D P}{i n d e x}}{G D P_{D e f l a t o r}}) 100$

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Abstract

1 Introduction

2 The Model

2.1 Households

2.2 Firms

2.3 Government

2.4 Monetary policy

2.5 Equilibrium

2.6 Functional forms

3 Calibration

4 Optimal monetary and fiscal stabilisation policy

4.1 The Ramsey problem and the LQ approximation

4.2 Optimal monetary-fiscal rules for normal times

4.2.1 Results

4.3 Crisis management of debt: how fast, how deep?

4.4 The role of nonlinearities, larger shocks and an interest-rate peg

5 Introducing long-term government debt

5.1 Results

6 Conclusions

References

Appendix

A Equilibrium conditions

A.1 Utility function and marginal utilities

A.2 Consumption/saving

A.3 Investment

A.4 Wage setting

A.5 Production

A.6 Government

A.7 Monetary policy

A.8 Resource constraint

A.9 Autoregressive processes

B The Ramsey problem and the LQ approximation

C Data

D Moments

Same Series

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