Finite Horizons and the Monetary/Fiscal Policy Mix · 2020. 9. 1. · by an active monetary and a passive ﬁscal policy. When it comes to the ’70s, the literature does not seem

Finite Horizons and the Monetary/FiscalPolicy Mix∗

Kostas MavromatisDe Nederlandsche Bank and University of Amsterdam

Fiscal policy in the United States has been documentedto have been the leading authority in the ’70s (active fis-cal policy), while having switched to fiscal discipline follow-ing Volcker’s appointment (passive fiscal policy) onward. Mostpapers in the literature focus on taxes as the main instrumentto stabilize debt when fiscal policy is passive. I extend the exist-ing analysis by also allowing federal spending to adjust. I con-struct a New Keynesian Markov-switching DSGE model witha Blanchard-Yaari structure and estimate it for the UnitedStates. I find that the U.S. economy has switched regimes overtime. In line with the existing literature, I show that the econ-omy spent the ’70s in a regime where fiscal policy was activewhile monetary policy was passive. Interestingly, I show thatthe economy spent a short period in the early ’80s where bothpolicies were passive, before switching to a persistent regimewhere monetary policy has been fighting inflation aggressivelyand fiscal policy ensured that both tax revenues and federalspending adjusted to stabilize debt.

JEL Codes: E31, E58, E62.

1. Introduction

Regime switches in the monetary/fiscal policy mix in dynamic sto-chastic general equilibrium (DSGE) models have attracted much

∗The views expressed do not represent the position of De Nederlandsche Bankor the Eurosystem. I am very grateful for the comments and suggestions providedby two anonymous referees and Boragan Aruoba. I am also grateful to ChristopheKamps, Roel Beetsma, Jacopo Cimadomo, Massimo Giuliodori, Damjan Phaj-far, and Harald Uhlig. I am also grateful to seminar participants at the EuropeanCentral Bank, the University of Amsterdam, and Tilburg University as well asparticipants at several conferences. Author e-mail: [email protected].

327

328 International Journal of Central Banking September 2020

attention over the last years. The interactions between monetaryand fiscal policy are crucial for the determination of inflation andoutput dynamics as well as debt and, more importantly, of expec-tations. Whether or not monetary and fiscal policy coordinate orwhether they switch in a joint manner is a question that many papersin the literature have addressed and are still trying to address. How-ever, what is also important is not only whether the policy mix issuch that it allows for higher inflation and lower debt, or the otherway around, but also whether the fiscal authority changes the com-position of its instruments as a means to consolidate debt regardlessof the monetary policy stance. In this paper, I address those issues.I estimate a DSGE model for the U.S. economy allowing for jointswitching in monetary and fiscal policy. In deviation from existingpapers, I allow fiscal policy to switch not only between a passive andan active regime but also between using tax revenues or spending asa means to stabilize debt.

The current literature has restricted its attention to two regimes,namely an active monetary/passive fiscal (AM/PF) regime and apassive monetary/active fiscal (PM/AF) regime.1 Figure 1 (panel A)plots the debt-to-GDP ratio along with inflation and the real interestrate. Clearly, the debt-to-GDP ratio was lower during the ’70s. Thereis mixed evidence in the literature regarding that period. Accordingto the findings of Bianchi (2012), Bianchi and Ilut (2017), Davigand Leeper (2011), and Traum and Yang (2011), this period cor-responds to the passive monetary (PM) and an active fiscal (AF)policy regime and is associated with persistently high inflation anda low real interest rate.2 As shown in Leeper (1991), lack of fiscaldiscipline in a rational expectations model implies dynamics thatdepend on the joint behavior of monetary and fiscal authorities.More importantly, the effects of policy interventions differ substan-tially from those when fiscal discipline is in place. On the contrary,

1Bianchi and Ilut (2017) estimate a Markov-switching DSGE for the U.S.economy and find that apart from the AM/PF and PM/AF regimes, the U.S.economy went through a short period (in the early years of Ronald Reagan’spresidency) during which both monetary and fiscal policy were active. FollowingLeeper (1991), the model has no solution under this policy mix. For this reason,in this paper I restrict my attention to policy mixes that yield a unique boundedsolution.

2The definition of monetary and fiscal regimes is borrowed from Leeper (1991).

Vol. 16 No. 4 Finite Horizons and the Monetary/Fiscal Policy Mix 329

Figure 1. Inflation, Real Interest Rate, and FiscalVariables in the United States

Notes: Panel A: Annualized quarterly inflation, real interest rate, and debt-to-GDP ratio over the sample 1969:Q1–2008:Q4. Panel B: Interest payments as ashare of total federal expenditure. Panel C: Primary deficit as a share of privatelyheld gross federal debt.

Bhattarai, Lee, and Park (2012a) find that both policies have beenpassive during that period, while Traum and Yang (2011) find thatmonetary policy has been active and fiscal policy has been passiveduring the ’70s in the United States. The literature seems to convergeonly on the events following the appointment of Volcker onward.That is, the policy mix has since the early ’80s been characterizedby an active monetary and a passive fiscal policy. When it comes tothe ’70s, the literature does not seem to be converging to a specificpolicy mix.

A natural measure of fiscal stance is the primary surplus as afraction of outstanding debt. The reluctance of the fiscal authori-ties, from the mid-’70s until the late ’70s, to make the necessary


fiscal adjustments in order to stabilize debt can be seen in panel Cof figure 1. In particular, the first period of sustained primary deficitsbegins in the mid-’70s (Ford tax cut and tax rebate). As panel Cshows, the high deficit persisted for a couple of years. On the otherhand, from the early ’80s onward (Volcker’s appointment), inflationstarted declining steadily while the real interest rate started to risesubstantially. At the same time, the debt-to-GDP increased gradu-ally until the early ’90s, which can be partly attributed to the risinginterest payments and the subsequent recession. In fact, as panel Bshows and as also argued by Sims (2011), the interest expense was asmall fraction of the budget until the late ’70s but started to shootup from the early ’80s onward and stayed there for several years.The steady decline in deficits in the middle of the Reagan Adminis-tration was not enough to offset the sharp rise in interest expenses.According to the existing literature, the period from the ’80s onwardcorresponds to an AM/PF regime.

Historical evidence regarding U.S. fiscal policy shows that it hasfluctuated not only between a passive and an active regime but alsobetween alternative instruments (tax revenues or federal expendi-ture) in its effort to stabilize debt. During the Reagan presidency,the Tax Equity and Fiscal Responsibility Act of 1982 was approvedamid concerns regarding the growing primary deficits. The ongo-ing recession of the early ’80s caused a short-term fall in tax rev-enues. The Act was approved with the ultimate purpose of closingthe growing budget gaps. At the same time, the cuts in spendingconcerned cuts in projected spending. On the other hand, duringthe Clinton Administration sustained cuts in federal expenditurewere implemented in an effort to bring the budget deficit into a sur-plus and to reduce federal debt. Panel C depicts the sustained andgrowing primary surpluses over that period, while panel A depictsthe substantial decline in debt-to-GDP ratio. During those years,federal spending fell to almost 18.2 percent of GDP from 22.1 per-cent in 1992 (Balanced Budget Act of 1997).3 On the other hand,taxes were raised in 1993 but were cut in 1997 (Taxpayer ReliefAct of 1997). The current literature on Markov-switching DSGE

3According to the Balanced Budget Act of 1997, the targeted cut in spendingwould amount to $160 billion between 1998 and 2002.


(MS-DSGE) models remains relatively silent on the effects ofswitches between alternative fiscal instruments.4

I address the events described above by developing an MS-DSGEmodel where monetary policy switches between an active and a pas-sive regime. Not only does fiscal policy switch between being activeand passive, but also, while passive, it is allowed to change thecomposition of its fiscal instruments. That is, the model allows thereactions of tax revenues and federal spending to debt fluctuationsto vary so long as fiscal policy stays passive. With this specifica-tion (i.e., allowing for possible switching between fiscal instruments)I deviate from the current literature.5 I deviate from the currentliterature on estimated MS-DSGE models for the United States fur-ther by introducing a Blanchard-Yaari structure similar to Richter(2015).6 Agents are no longer infinitely lived. With this structurefiscal shocks have real effects in the economy, implying inflation-ary pressures following tax cuts or increases in federal expenditureregardless of the current regime and agents’ beliefs.

A number of authors attribute the lower inflation and outputvolatility in the post-Volcker era to lower shock volatility instead ofbetter policy (see Cogley and Sargent 2005; Primiceri 2005; Simsand Zha 2006; Stock and Watson 2002). For this reason I also allowthe variances of shocks to vary over time according to a Markov-switching process and independently of switches in monetary andfiscal policy.

I perform a Bayesian estimation of the model using quarterlydata. In line with estimated closed- and open-economy MS-DSGE

4A number of papers employ Bayesian estimation to estimate fiscal policyrules and to explore the economic effects of fiscal policy (Forni, Monteforte, andSessa 2009; Kamps 2007; Lopez-Salido and Rabanal 2007; Marco, Werner, andJan 2006; Straub and Coenen 2005). However, they do not account for regimeswitches of the nature considered in this paper.

5The importance of the fiscal instrument on the dynamics of economic vari-ables following fiscal shocks has been studied in simple real business cycle modelsin the literature. Leeper, Plante, and Traum (2010) and Leeper and Yang (2008)show that the dynamics and the impulse responses of key variables following bothfiscal policy and nonpolicy shocks depend on what fiscal instrument finances debt.However, those papers do not include the interactions between monetary and fis-cal policies. Moreover, they do not analyze the effects of switches across differentinstruments on expectations.

6Richter (2015) does not focus on switching between different fiscal instru-ments during an AM/PF regime.


models, I find that the monetary policy of the Federal Reserve hasindeed varied over time and is characterized by a weaker responseof the federal funds rate to inflation fluctuations during the ’70smainly (passive monetary policy), and was associated with an activefiscal policy in the spirit of Leeper (1991) and in line with Bianchi(2012, 2013) and Bianchi and Ilut (2017). I find that the probabil-ity of a switch to a regime where the reaction of monetary policyto inflation is strong and where fiscal policy is concerned with debtstabilization (passive fiscal policy) increases substantially from themid-’80s onward and stays persistently high until recently. Interest-ingly, I find that the economy spent a short period in a regime whereboth monetary and fiscal policy were passive before switching to thepersistent regime which characterized the evolution of the economyfrom the mid-’80s till today. Importantly, I find that during thisshort-lived regime, the fiscal authority used federal spending heavilyas a means to stabilize debt. As regards the policy mix from themid-’80s onward, I find that, while the Federal Reserve has beencommitted to fighting inflation, the fiscal authority has been usingboth tax revenues and federal spending in order to stabilize debt.This finding contributes further to the existing literature on esti-mated MS-DSGE models which restrict federal spending not to reactto debt fluctuations.

The paper is organized as follows. Section 2 presents some ofthe existing literature on the monetary fiscal policy mix as well ason existing models using a perpetual youth structure. In section 3,a closed-economy model with a Blanchard-Yaari structure is devel-oped. Section 4 includes the solution algorithm, the estimation tech-nique, and the data set. Section 5 summarizes the estimation results,while section 6 concludes.

2. Literature Review

The monetary/fiscal policy mix has been analyzed in the literatureextensively. On the theoretical side, the landmark paper of Leeper(1991) analyzed the stability properties in an economy featuringmonetary and fiscal policy. Canzoneri, Cumby, and Diba (2010)provide a game-theoretic approach to the interactions between themonetary and fiscal authorities in an attempt to analyze positiveand normative issues in those. Since the seminal paper of Leeper, a


number of papers have been written in an attempt to estimate thepolicy mix in the United States. Most of these papers have madeuse of a closed-economy DSGE model. Some of those studies esti-mate fixed-coefficients models, where monetary and fiscal policy arenot allowed to change behavior over time, whereas others considerchanges in the behavior of the two authorities over time. Bhattarai,Lee, and Park (2012a) estimate a closed-economy DSGE model forthe United States using postwar data and find evidence in favor ofchanges in the monetary/fiscal policy mix over time. Their model hasfixed coefficients and they estimate it in different subsamples. Theyfind that pre-Volcker, a passive monetary and fiscal policy regimeprevailed, while post-Volcker, an active monetary and passive fiscalpolicy regime was dominating. Hence, according to the terminol-ogy by Leeper (1991), since both monetary and fiscal policies werepassive pre-Volcker, there was equilibrium indeterminacy. Along thesame lines, Bhattarai, Lee, and Park (2012b) using a similar modelalso find evidence in favor of passive monetary and fiscal policy inthe pre-Volcker era and an active monetary and passive fiscal pol-icy in the post-Volcker era. Traum and Yang (2011) also estimate aclosed-economy DSGE model for the United States for different sub-samples and find that the data in the pre-Volcker period stronglyprefer an AM/PF regime, even with a prior centered in the PM/AFregion. Contrary to Bhattarai, Lee, and Park (2012a, 2012b), Traumand Yang (2011) allow both government purchases and tax revenuesto react to government debt.7 In this paper, I deviate from Traumand Yang (2011) by introducing an overlapping-generations struc-ture into the model as well as allowing for time variation in the mon-etary and fiscal feedback rules. This allows me to estimate the modelfor the whole sample period and detect whether or not the economyhas indeed been switching across regimes over time, instead of esti-mating a time-invariant model for different subsamples as Traumand Yang (2011) do.

7A number of papers employ Bayesian estimation to estimate fiscal policyrules and to explore the economic effects of fiscal policy (Forni, Monteforte, andSessa 2009; Kamps 2007; Lopez-Salido and Rabanal 2007; Marco, Werner, andJan 2006; Straub and Coenen 2005). However, they do not account for regimeswitches of the nature considered in this paper.


At the moment there is also large evidence in favor of joint regimeswitching in monetary and fiscal policy. Specifically, there are anumber of papers which estimate time-varying models allowing forswitches in the policy mix, instead of estimating a fixed-coefficientsmodel for different subsamples. Bianchi (2012) and Bianchi and Ilut(2017) estimate an MS-DSGE for the United States where mone-tary and fiscal policy switch jointly regimes. Both papers find evi-dence in favor of a passive monetary/active fiscal policy mix in the’70s and an active monetary/passive fiscal mix from the late ’80sonward. Importantly, both papers find that fiscal policy did notswitch to become passive once monetary policy started committingto a better anchoring of inflation expectations. As a result, bothpapers find that the U.S. economy spent most of the early ’80s ina regime where both policies were active. Besides models allowingfor joint monetary and fiscal policy interactions, Favero and Mona-celli (2005) estimate fiscal policy feedback rules in the United Statesfor the period 1960–2002, and find that fiscal policy in the UnitedStates has been active between 1960 and 1980, while from 1980onward it switched to become passive. Apart from estimated DSGEmodels or single-equation estimates, there are a number of paperscalibrating regime-switching models and looking at the interactionsbetween monetary and fiscal policy. Davig and Leeper (2011) andDavig, Leeper, and Chung (2007) construct an MS-DSGE model,look at how the interactions between monetary and fiscal policyaffect agents’ decisions following monetary and fiscal shocks, andalso analyze the stability properties of the model. However, littleevidence exists in the MS-DSGE literature as regards switches ofthe fiscal authorities between using tax revenues and spending as ameans to stabilize debt. As discussed in the introduction, there havebeen periods where the U.S. government focused more on spendingcuts than on tax hikes in order to control debt. In this paper, Iaccount for this possibility by estimating a three-regime model forthe United States where the fiscal authority is allowed to also varythe composition of its fiscal policy while being passive.

The interaction between monetary and fiscal policy has also beenanalyzed through the lens of the perpetual youth model due toBlanchard (1985) and Yaari (1965). Leith and Wren-Lewis (2000)use a version of the perpetual youth model to analyze the impli-cations of the fiscal stability pact in the European Economic and


Monetary Union for monetary policy. Importantly, Leith and Wren-Lewis (2000) consider two scenarios of fiscal discipline—one in whichthe fiscal authorities use taxes to control debt and another in whichthey use government spending—and they analyze the stability prop-erties of their model. Annicchiarico, Giammarioli, and Piergallini(2012) construct a New Keynesian model with capital accumulationand finite horizons to analyze the macroeconomic implications offiscal policy. They find that fiscal expansions generate a tradeoff inoutput dynamics between short-term gains and medium-term losses.They also show that the effects of fiscal shocks crucially depend uponthe conduct of monetary policy. Along similar lines, Annicchiarico,Marini, and Piergallini (2008) also build a New Keynesian modelwhere agents have finite horizons to analyze the performance of mon-etary policy under Ricardian and non-Ricardian fiscal regimes. Inter-estingly, they show that, within the class of Ricardian fiscal rules,active monetary policies are not necessary for equilibrium determi-nacy. Richter (2015) constructs a DSGE model with a fiscal limit,also making use of the perpetual youth setup in order to analyzehow intergenerational redistributions, debt maturity, and entitle-ment reforms affect the consequences of explosive federal transfers.He finds that the finite horizon structure is crucial in generating moresevere and persistent stagflation than representative-agent models.Moreover, the interactions between debt maturity and finite horizonsmay dampen the short-run effects of explosive transfers.

Analyzing fiscal consolidations, Bi, Leeper, and Leith (2013)show that expenditure-based consolidations are more successful instabilizing debt. Although they focus on the uncertainty about thetiming and the nature of fiscal consolidations, they conclude thattax-based fiscal consolidations are not as successful as expenditure-based ones. Moreover, they compute the effects of fiscal consolida-tions in the periods following each specific consolidation to show thattax-based consolidations are expansionary. In this paper, however, Ido not consider cases of short-horizon fiscal consolidations.

The current paper contributes to the existing literature in the fol-lowing respects. First, it introduces the perpetual youth model andcombines it with Markov switching in monetary and fiscal policy andthen estimates the resulting MS-DSGE model for the United States.Second, it assumes three regimes, namely two regimes where mon-etary policy is active and fiscal policy is passive and one in which


monetary policy is passive and fiscal policy is active. Specifically,while being passive, fiscal policy can vary the extent to which taxrevenues and federal spending react to fluctuations of the debt ratio.

3. The Model

I adopt the specification of the Blanchard (1985) and Yaari (1965)model of perpetual youth in discrete time similar to Devereux (2011),Richter (2015), and Smets and Trabandt (2012). Households die withprobability 1−δ each period, and every period a newborn generationi represents a fraction 1− δ of total population, where 0 ≤ δ ≤ 1. Inother words, δ captures the probability of survival from one periodto the next. Therefore,

∑∞t=0 δt = 1

1−δ represents the average house-hold lifetime. As pointed out by Smets and Trabandt (2012), analternative and empirically more plausible interpretation of 1/1 − δis that it reflects the effective planning horizon of households. Inthis paper, I adopt the planning horizon interpretation. Householdshave no bequest motive and the usual Ricardian equivalence breaksdown.

Households derive utility from the consumption of goods andsupply labor to firms. They are assumed to have external habitsin consumption. Each household is the owner of a firm producinga differentiated good. Households receive a wage from labor andprofits from firm ownership. Firms operate in a monopolisticallycompetitive market with price stickiness as in Calvo (1983). Mon-etary policy is described by an interest rate rule whose coefficientson inflation and output gap vary over time according to a Markov-switching process. In particular, there are periods where monetarypolicy satisfies the Taylor principle—active monetary policy—andperiods where it does not—passive monetary policy. Fiscal pol-icy is conducted by the government and fluctuates between threeregimes. The first and the third regime are regimes in which the fis-cal authority commits to fiscal discipline through targeting tax rev-enues and/or spending—passive fiscal policy. Crucially, the extentto which tax revenues and spending are adjusted to stabilize debtcan vary between those two regimes. The second regime is one inwhich it does not make the necessary fiscal adjustments in order tostabilize debt—active fiscal policy.


3.1 Households

In every period a new generation/cohort is born. The size of gener-ation i at time t is (1 − δ) δt−i and total population is of measure 1.The representative household l in each generation i maximizes theexpected lifetime utility function:

Ut = Et

∞∑s=t

(βδ)s−teds

[(Ci

s − κCs−1)1−σ

1 − σ−

(Hi

s

)1+γ

1 + γ

], (1)

where β ∈ (0, 1) is the subjective discount factor; Cit , Hi

t , and Bit are

consumption, labor supply, and government bond holdings of house-holds of generation i; Ct represents the average level of consumptionin the economy. The parameter κ captures the degree of externalhabit. The preference shock, dt, is assumed to have a mean zero andfollows a stationary AR(1) process: dt = ρddt−1 + σd,ζt

εd,t εd,t ∼N (0, 1). The composite consumption good Ci

t is differentiated acrossa continuum of individual goods, such that

Cit =

[∫ 1

0cit(j)

θ−1θ dj

]θ

θ−1 , (2)

where θ denotes the intratemporal elasticity of substitution. Thehousehold in each generation i chooses ci

t(j) to minimize its totalexpenditure, which implies a demand function for each good jdescribed by

cit(j) =

(pt(j)Pt

)−θ

Cit , (3)

where Pt is the aggregate price index defined as

Pt =[∫ 1

0pt(j)1−θdj

] 11−θ

. (4)

The flow budget constraint for members of generation i is summa-rized as

PtCit +

Bit

Rt=

1δBi

t−1 + WtHit + Λi

t − T t + TRt, (5)


where Bit denotes one-period riskless government bonds paying one

unit of numeraire in period t. Rt is the gross nominal interest rate onbonds purchased in period t. Wt is the nominal wage, Λi

t are nominalprofits that generation i receives lump sum, T i

t are lump-sum taxesimposed by the government to generation i, while TRi

t are nominallump-sum transfers. As in Blanchard (1985), I assume a full annu-ities market. This implies that rates of return are grossed up to coverthe probability of death. The representative household in generationi chooses at time t the set

{Ci

t , Hit , B

it

}and the sequences of con-

tingency plans{Ci

s, His, B

is

}∞s=0 in order to maximize (1) subject to

(5). The first-order conditions at an interior solution are written as

1 = βedt+1

edtEt

[RtPt

Pt+1

(λi

t+1

λit

)](6)

(Hi

s

)γ= wtλ

it, (7)

where λit =

(Ci

t − κCt−1)−σ and wt is the real wage. The budget

constraint holds with equality at the optimum and the transversalitycondition needs to be satisfied:

limT→∞

Et

{δT−tQi

t,T BiT

}= 0, (8)

where Qit,T (l) is the stochastic discount factor of the representative

household l of generation i defined as

Qit,t+s = βs edt+s

edt

(Ci

t − κCt−1

Cit+s − κCt+s−1

)σ

. (9)

Following Castelnuovo and Nistico (2010), Del Negro, Giannoni,and Patterson (2012), Nistico (2012), and Piergallini (2006), one canshow that the consumption policy function is described by

Cit = (1 − βδ)

(1δ

Bit−1

πt+ Di

t

), (10)

where

Dit = Et

[ ∞∑s=0

δsQit,t+s

(Wt+sHt+s + Λi

t+s − Tt+s + TRt+s

Pt+s

)]

(11)


describes the present discounted value of future disposable incomeadjusted for past consumption.

3.2 Aggregation

As in Annicchiarico, Giammarioli, and Piergallini (2012), giventhe overlapping-generations structure of the model, the aggregatevalue zt of a generic economic zi

t(l) is obtained as a sum acrossgenerations:

zt =t∑

i=−∞

(∫ (1−δ)δt−i

0zit(l)dl

). (12)

Define Qt,t+s as the population weighted average of thegeneration-specific stochastic discount factors:

Qt,t+s =t∑

i=−∞(1 − δ) δt−iQi

t,t+s. (13)

Since (10) is linear in generation-specific variables, I can express itas follows:

Ct = (1 − βδ)(

1δ

Bt−1

πt+ Dt

), (14)

where

Dt = Et

[ ∞∑s=0

δsQt,t+s

(Wt+sHt+s + Λt+s − Tt+s + TRt+s

Pt+s

)].

(15)

In appendix section A.3.1, I show that aggregation across all gen-erations at time t yields the following expression for the aggregatebudget constraint:

PtCt +Bt

Rt= Bt−1 + WtHt + Λt − T t + TRt. (16)


Following the same procedure as in Smets and Trabandt (2012), inappendix section A.3.2 I derive the aggregate representation of theEuler which receives the following form:

βedt+1

edt

Rt

Πt+1

1λt

=1 − δ

δμt+1

1

λσ−1

σt+1

Bt +(

λt

λt+1

)σ−1σ 1

λt+1. (17)

Note that for δ < 1, government debt affects consumption spend-ing. This leads to a breakdown of Ricardian equivalence even thoughtaxes are lump sum. Moreover, under log-preferences, equation (17)receives a more intuitive form:

βedt+1

edt

Rt

Πt+1

1λt

=1 − δ

δμt+1Bt +

1λt+1

. (18)

3.3 Firms

Final goods are produced by monopolistically competitive firmswhich employ only labor and use a linear production technology,

Yt(j) = AtHt(j), (19)

where At is an aggregate productivity shock at date t which isassumed to follow a log-stationary AR(1) process: at = ρaat−1 +σa,ζt

εa,t εa,t ∼ N (0, 1). Firm profits are distributed to householdsat the end of each period. Each firm is the only producer of its goodand sets its price in a staggered way as in Calvo (1983). In eachperiod, a firm faces a constant probability of being able to reopti-mize its nominal price, 1 − ω, regardless of the time elapsed sinceit last adjusted its price. Following Christiano, Eichenbaum, andEvans (2005), I assume that firms that do not reoptimize their priceset their price according to the price that has been previously set,also accounting for past inflation (partial indexation). Specifically, ifa firm j does not reoptimize, it sets the price of its good accordingto the following rule:

Pt(j) = πt−1Pt−1(j). (20)

This structure adds lagged inflation in the Phillips curve. Let pt(j)denote the price that is set by a firm that can reoptimize at date t.


Given the Calvo price-setting mechanism, the price level can besummarized as

Pt =[ω (πt−1Pt−1)

1−θ + (1 − ω)pt(j)1−θ] 1

1−θ

. (21)

If firm j reoptimizes, it chooses the price that maximizes theexpected discounted sum of its profits. Profit maximization thussolves

maxp∗

t (j)Et

∞∑s=0

(δβω)sQt+s {pt(j)Xts − Pt+smct+s} yt+s(j) (22)

subject to

yt+s (j) =(

p∗t (j)Pt

)−θ

Yt+s

and

Xts =

{πt × πt+1 × · · · × πt+s−1 for s ≥ 1

1 s = 0,(23)

where mc is the real marginal cost specified as

mct =Wt

AtPt. (24)

The first-order condition associated with the firm’s choice of pt is

Et

∞∑s=0

(δβω)sQt+s

{pt(j)Xts − θ

θ − 1Pt+smct+s

}yt+s(j). (25)

Obviously, for ω = 0, the firm sets its price equal to a markup overthe current real marginal cost, mct. By definition, for s = 0, theterm Xts is equal to 1.

3.4 Fiscal Authority

The flow budget constraint of the federal government is given by

Bt = Bt−1(Rt−1) − Tt + St + TPt,


where Bt is government debt, Tt is lump-sum taxes, and St is fed-eral expenditures given by the sum of government purchases andtransfers, St = PtGt + TRt. As in Bianchi and Ilut (2017) TPt is ashock that captures a series of features that are not explicitly mod-eled here, such as changes in the term premium.8 As Bianchi andIlut point out, this shock is necessary to avoid stochastic singular-ity when estimating the model given that I treat debt, taxes, andexpenditures as observables. Expressing the variables as a fractionof output, the flow budget constraint receives the following form:

bt = bt−1Rt−1/ (ΠtYt/Yt−1) − τt + st + tpt,

where bt = Bt/PtYt, τt = Tt/PtYt, and st = St/PtYt, while Πt isCPI inflation. I define variable χt = gt/st as the fraction of federalexpenditure devoted to government purchases, where both variablesare expressed as a share of GDP (gt = Gt/Yt and st = St/PtYt).As in Bianchi and Ilut (2017), I assume that variable χt has thefollowing law of motion9:

χt = ρχχt−1 + (1 − ρχ) iyYt + σχ,stεχ,t, εχ,t ∼ N (0, 1) ,

where Y nt is the flexible-price equilibrium output.

3.5 Market Clearing

Market clearing in the goods market requires

Yt(j) = Ct(j) + G(j)

for all j ∈ [0, 1] and all t. Defining aggregate output as Yt =(∫ 10 Yt(j)

θ−1θ dj

) θθ−1

and accounting for the fact that I have

8In Bianchi and Ilut (2017) the government issues long-term debt only. There-fore, the term TPt can also capture changes in the maturity structure of federaldebt or changes in the term premium.

9In what follows, for all the variables normalized with respect to GDP (debt,government purchases, federal expenditure, tax revenues) xt denotes a lineardeviation (xt = Xt − X) from its steady state. Instead, for all other variablesxt denotes a percentage deviation (xt = log(Xt/X)) from its steady state. Thisdistinction avoids having the percentage change of a percentage.


expressed fiscal variables as fractions of nominal GDP, the aggregateresource constraint is as follows:

Yt = Ct + gtYt.

4. Markov Switching

4.1 Solution and Estimation Technique

In this section, I describe how Markov switching is introduced intothe model and how the resulting model is estimated. I follow theapproach of Chen (2017), Chen, Kirsanova, and Leith (2017), andLiu and Mumtaz (2011). First, I log-linearize the model and thenI introduce Markov switching. In appendix section A.1, I presentthe full model log-linearized. The algorithm that is used to solvethe model and obtain a time-varying minimum state variable solu-tion is that of Farmer, Waggoner, and Zha (2011). I assume thatboth monetary and fiscal policy are subject to regime shifts. Specif-ically, as regards monetary policy, I assume that the parameters inthe Taylor rule of the Federal Reserve are subject to regime shifts.As regards fiscal policy, I assume that the fiscal authority uses taxrevenues and federal spending whenever it seeks to stabilize debtand that the parameters on each of the two fiscal feedback rules aresubject to regime shifts. Monetary and fiscal policy switch regimessimultaneously. Moreover, I allow the variances of all the shocks inthe model to be subject to regime shifts as well. I allow for indepen-dent regime switching in the volatility of the structural shocks thatthe model features. To specify the MS-DSGE model, I partition theparameter vector Φ into three blocks,

Φ ={ΦZ ; Σζ ; Φ

},

where ΦZ is the set of parameters subject to regime shifts, Σζ isthe variance of the regime-switching volatilities, and Φ denotes theremaining time-invariant parameters. The Markov-switching interestrate and fiscal feedback rules are specified as

Rt = ρR,ZtRt−1 + (1 − ρZt)(φπ,Ztπt + φy,Zt Yt

)+ σR,ζt

εR,t εR,t ∼ N (0, 1) , (26)


τt = ρτ,Ztτt−1 + (1 − ρτ,Zt

)(γb,Zt

bt−1 + γyYt

)+ στ,ζt

ετ,t ετ,t ∼ N (0, 1) , (27)

and

st = ρs,Ztst−1 + (1 − ρs,Zt

)(−δb,Zt

bt−1 − δyYt

)+ σs,ζt

εs,t εs,t ∼ N (0, 1) , (28)

where π is steady-state inflation. All shocks considered in this paperare independent of one another. I allow all shock volatilities σd,ζt

,σa,ζt , σχ,ζt , σcp,ζt , σR,ζt , στ,ζt , σs,ζt , and σtp,ζt to be time varying.The superscript Z denotes the unobserved regime associated withthe monetary and fiscal policy parameters taking on values 1, 2,or 3. Monetary and fiscal policy regime follows a Markov processwith transition probabilities pij = P [Zt = i|Zt−1 = j], where i, j =1, 2, 3. Hence, I assume three regimes for the monetary/fiscal pol-icy mix. For convenience, from now on I will refer to those regimeswhich concern switches in policy parameters only as Regime-Pol.

In the MS-DSGE literature, it has been documented that U.S.monetary policy has been passive (i.e., weak reaction to infla-tion fluctuations) during the ’70s (see Bianchi 2012, 2013; Bianchiand Ilut 2017) and fiscal policy has been active (i.e., no reac-tion of taxes or expenditure to stabilize debt). Absent regimeswitches, Leeper (1991) distinguishes four regions of the parame-ter space according to existence and uniqueness of a solution. Inthe linearized version of the model, the monetary and fiscal pol-icy rule in practice determine the existence and the uniquenessof an equilibrium. Hence, there are two policy mixes that yield adeterminate equilibrium. The first is an active monetary/passivefiscal (AM/PF) mix where monetary policy satisfies the Taylorprinciple and fiscal policy passively accommodates monetary pol-icy by guaranteeing debt stability. Hence, the inflation coefficientin the interest rate rule satisfies φπ,Zt

> 1 and the coefficient ondebt in the tax rule satisfies γb,Zt

>(R − 1

)/ (1 − ρτ,Zt

) while thaton the spending rule satisfies δb,Zt >

(R − 1

)/ (1 − ρs,Zt), where

R is the steady-state interest rate, derived in appendix sectionA.2, which is a function of the subjective discount factor, β,


and the survival probability, δ. The second is a passive mone-tary/active fiscal (PM/AF) mix, where monetary policy does notsatisfy the Taylor principle and fiscal authority is not commit-ted to stabilizing the process for debt. Formally, the policy coeffi-cients in this regime satisfy φπ,Zt

< 1, γb,Zt<

(R − 1

)/ (1 − ρτ,Zt

),and δb,Zt

<(R − 1

)/ (1 − ρτ,Zt

), respectively. Finally, no station-ary equilibrium exists when both authorities are active (AM/AF),whereas when both of them are passive (PM/PF) the economy issubject to multiple equilibriums.

I restrict the policy parameters such that one regime (Regime-Pol 2), j = 2, is characterized by an active fiscal policy and by aweaker reaction of monetary policy to inflation fluctuations com-pared with Regime-Pols 1 and 3. Specifically, as I describe in theprior specification below, I center the mean of the prior for theinflation coefficient in the Taylor rule in Regime-Pol 2 at φπ,2 = 1and also impose the restriction φπ,2 < φπ,1 and φπ,2 < φπ,3. Asregards fiscal policy, I set the reaction to debt in the tax and thefederal spending rule equal to zero, γb,2 = 0 and δb,2 = 0, similarto Bianchi and Ilut (2017).10 In Regime-Pols 1 and 3, I place norestrictions on the coefficients on debt in the two fiscal feedbackrules. In other words, I allow those two to be freely estimated. Thereason behind this approach is to allow the data to identify whether,while monetary policy has been active (i.e. φπ,1 > 1 and φπ,3 > 1),the fiscal authority has been through periods during which tax rev-enues were the main instrument to stabilize debt and through otherperiods where federal spending was the main instrument. Obviously,I interpret a higher coefficient on debt in the spending rule, δb,Zt ,compared with its counterpart in the tax rule, γb,Zt

, as an indica-tion that spending is the main instrument to stabilize debt, and viceversa. The superscript ζ = 1, 2 in variances denotes the unobservedregime associated with the volatilities and which evolves indepen-dently of S. For convenience, from now on I denote the volatilityregime which concerns switches in volatilities only as Regime-Vol.The two state variables Z and ζ follow a first-order Markov chainwith the following transition probability matrices, respectively:

10Bianchi and Ilut (2017) assume that the reaction of tax revenues to the debtratio is zero whenever fiscal policy is active.


P =

⎡⎣ P11 1 − P11 0

P21 P22 1 − P21 − P221 − P33 0 P33

⎤⎦ ,

Q =[

Q11 1 − Q111 − Q22 Q22

],

where Pji = p [Zt = i|Zt−1 = j] , where i, j = 1, 2, 3, and Qmn =p [ζt = n|ζt−1 = m] , where m, n = 1, 2. The specification of thetransition probability matrix P is similar to that in Bianchiand Ilut (2017), the only difference being that, in Regime-Pol 3,the economy is restricted to switch to Regime-Pol 1 instead ofRegime-Pol 2.11 The model can be written in a matrix form as

A(Zt)Xt+1 = B(Zt)Xt + Ψ(Zt)εt + Π(Zt)ηt, where

εt ∼ N(0, Qζ), (29)

where Xt+1 = [Ct+1, πt+1, Ct, πt, Yt, Rt, τt, st, gt, χt, αt, ut, dt, tpt] εt

is a 8×1 vector of structural shocks of mean zero and whose varianceis allowed to vary over time as specified above.12 ηt is a 2 × 1 vectorof endogenous random variables.

I solve the model using the approach of Farmer, Waggoner, andZha (2011).13 Farmer, Waggoner, and Zha (2011) show that if aunique solution exists, then this can be cast as a Markov-switchingVAR of the following form:

Xt = g1,ZtXt−1 + g2,Zt

εt. (30)

11I have also estimated the model with P13 > 0 and the results are robust. Ihave set it to zero in order to reduce the parameter space given the complexitiesof the model. As regards P32, I have set it to zero in order to account for switchesin the composition of fiscal tools only while fiscal policy stays passive.

12As in Justiniano and Preston (2010), when log-linearizing the model I addan import cost-push shock, denoted by εcp,t, to the Phillips curve. The reason ismainly to have a number of structural shocks equal to the number of observablesin the state-space representation of the model.

13I log-linearize the model around the symmetric zero-inflation steady state.The log-linearized model is summarized in appendix section A.1.


Equation (30) above can be combined with an observation equa-tion giving the following state-space model with Markov switching:

Xt = g1,ZtXt−1 + g2,Zt

εt, where εt ∼ N(0, Qζ)

Dt = HXt.(31)

I assume no measurement errors. Specifically, the data for the vari-ables and the log-linearized variables are linked by the followingequations.

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

dlGDPt

Inflationt

FedFundst

d(Debt/GDP )t

d(G/GDP )t

d(TaxRev/GDP )t

d(S/GDP )t

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

Yt − Yt−1πt

Rt

bt − bt−1gt − gt−1τt − τt−1st − st−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(32)

As described above, the Markov states Z and ζ evolve independentlywith transition probability matrices P and Q, respectively. Dt rep-resents the observed data and matrix H is the loading matrix. AsLiu and Mumtaz (2011) point out, the presence of the unobservedDSGE states Xt and the unobserved Markov states makes the stan-dard Kalman filter not operational in order to provide inference onXt and to calculate the value of the likelihood. Inference now has tobe conditioned on both current and past values of Z and ζ. Follow-ing their approach, I define a new state variable Z∗ indexing bothZt and ζt and which has a four-state transition matrix, P ∗ = PQ.As in Davig and Doh (2014) and Kim and Nelson (1999), I track Z∗

t ,Z∗

t−1, and Z∗t−2, which means that I account for 63 = 216 possible

paths for the state variables at each point in time.14 As mentionedabove, I follow a Bayesian approach to estimate the model, where Icombine the approximate likelihood function with the assumed priordistributions and use a random-walk Metropolis-Hastings algorithmwith 200,000 replications in order to approximate the posterior.

14For a more detailed description of Kim and Nelson’s (1999) algorithm, seethe technical appendix in Liu and Mumtaz (2011).


4.2 Priors

A summary of the prior distributions and the relevant bounds forthe model parameters is provided in table 1. Parameters with a dashare fixed according to the specification and thus are not estimated.I specify the prior distributions following the literature on eitherclosed- or open-economy models (see, e.g., Bianchi 2013; Bianchiand Ilut 2017; Lubik and Schorfheide 2007; Smets and Wouters2003, 2007). As regards the parameters that are not estimated, Icalibrate the steady-state government purchases to GDP, gY , andfederal spending to GDP, sY , to 0.21 and 0.25, respectively, while Iset the steady-state debt-to-GDP ratio, b, equal to 1 as in Bianchi(2012). I set the intratemporal elasticity of substitution across vari-eties, ρ, equal to 8. Finally, I set parameter ιy in the law of motionfor χt equal to 0.1

In the literature, there are not many models that estimate thesurvival probability, δ. I assume a beta prior with mean 0.95 andstandard deviation 0.1. In the interest rate rule, I assume that thesmoothing parameter ρR follows a beta distribution with a mean of0.5 and standard deviation equal to 0.2 in all regimes, and the infla-tion coefficient φπ follows a gamma distribution with a mean of 1.5and a standard deviation of 0.1 in Regime-Pols 1 and 3 and a gammadistribution with mean 1 and standard deviation 0.1 in Regime-Pol 2. The coefficient on output follows also a gamma distributionwith a mean of 0.4 and a standard deviation of 0.2 in Regime-Pols 1and 3 and a gamma distribution with mean 0.15 and standard devi-ation 0.1 in Regime-Pol 2. As regards the fiscal feedback rules, Iassume that the reaction of tax revenues to lagged debt-to-GDPratio, γb,1, and the reaction of federal spending to lagged debt-to-GDP, δb,3, both follow a gamma distribution with mean 0.07 andstandard deviation 0.025. Finally, regarding the transition probabil-ities, I follow Sims and Zha (2006) in assuming a Dirichlet prior, withthe scale matrix chosen to reflect the belief that regimes are persis-tent. The relevant parameters for the Dirichlet distribution imposedare α1 = 18 and α2 = 1.

4.3 Data

The sample consists of quarterly data spanning from 1969:Q1 to2012:Q4. The real per capita GDP was constructed using data on

Vol. 16 No. 4 Finite Horizons and the Monetary/Fiscal Policy Mix 349Tab

le1.

Pri

orD

istr

ibution

san

dPos

teri

orEst

imat

es

Model

:M

odel

:In

finit

eM

odel

:N

oSw

itch

ing

Ben

chm

ark

Model

Lifet

imes

(δ=

1)

inSpen

din

g

Log

Mar

gin

alLog

Mar

gin

alLog

Mar

gin

alLik

elih

ood:–1,2

83

Lik

elih

ood:–1,3

74

Lik

elih

ood:–1,3

11

Pri

or

Spec

ifica

tion

Par

amet

ers

Med

ian

5%

95%

Med

ian

5%

95%

Med

ian

5%

95%

Dis

trib

.M

ean

Std

.

δ0.

9760

0.95

600.

9950

——

—0.

9719

0.93

450.

9927

Bet

a0.

950.

100

φπ

,10.

9059

0.71

541.

1652

0.94

900.

7518

1.09

520.

7158

0.19

310.

9552

Gam

ma

1.50

0.10

0φ

y,1

0.47

490.

3236

0.47

490.

6211

0.51

050.

6262

0.46

480.

0808

0.48

53G

amm

a0.

400.

200

ρR

,10.

5959

0.58

120.

6301

0.32

320.

3008

0.42

900.

6099

0.03

270.

8396

Bet

a0.

500.

200

φπ

,20.

7416

0.24

720.

7416

1.00

890.

8588

1.09

740.

7754

0.48

390.

9858

Gam

ma

1.00

0.10

0φ

y,2

0.14

580.

1079

0.14

970.

2201

0.17

870.

2381

0.16

610.

0769

0.19

31G

amm

a0.

150.

100

ρR

,20.

5817

0.49

120.

6160

0.65

020.

4955

0.75

360.

4300

0.17

220.

6747

Bet

a0.

500.

200

φπ

,32.

7682

2.72

802.

9227

2.24

812.

0336

2.49

051.

2132

0.99

541.

3116

Gam

ma

1.50

0.10

0φ

y,3

0.63

900.

5689

0.63

620.

2405

0.21

160.

3018

0.18

310.

0748

0.82

21G

amm

a0.

400.

200

ρR

,30.

7725

0.71

230.

8373

0.82

730.

7962

0.83

510.

6034

0.19

650.

8745

Bet

a0.

500.

200

γb,1

0.04

150.

0103

0.07

050.

0753

0.04

030.

0794

0.06

950.

0200

0.18

69G

amm

a0.

070.

025

ρτ,1

0.53

380.

4607

0.83

990.

2412

0.14

540.

3636

0.39

540.

2748

0.46

06B

eta

0.50

0.20

0γ

b,2

——

——

——

——

——

——

ρτ,2

0.49

560.

6908

0.85

370.

3931

0.22

530.

5924

0.52

260.

3044

0.76

61B

eta

0.50

0.20

0γ

b,3

0.07

940.

0446

0.07

940.

1347

0.05

080.

1601

0.08

310.

0776

0.08

46G

amm

a0.

070.

025

ρτ,3

0.72

050.

4400

0.72

940.

4273

0.23

850.

4839

0.82

290.

4498

0.87

64B

eta

0.50

0.20

0δ b

,10.

1117

0.04

950.

1117

0.18

180.

1434

0.24

56—

——

Gam

ma

0.07

0.02

5ρ

s,1

0.18

650.

3707

0.77

750.

1059

0.00

850.

1284

0.22

890.

1027

0.80

01B

eta

0.50

0.20

0δ b

,2—

——

——

——

——

——

—ρ

s,2

0.96

650.

9696

0.98

820.

7153

0.45

520.

7836

0.96

550.

7645

0.99

19B

eta

0.50

0.20

0δ b

,30.

0677

0.02

020.

0676

0.01

330.

0083

0.01

67—

——

Gam

m0.

070.

025

ρs,3

0.40

220.

2860

0.71

730.

1872

0.09

680.

3167

0.92

030.

3257

0.99

21B

eta

0.50

0.20

0P

12

0.45

730.

4340

0.50

390.

8891

0.88

560.

8976

0.10

250.

0648

0.11

24D

iric

hlet

18.0

01.

000

P21

0.03

560.

0256

0.06

940.

0381

0.03

720.

0766

0.04

520.

0390

0.11

88D

iric

hlet

18.0

01.

000

P23

0.02

320.

0126

0.14

500.

0594

0.02

480.

0829

0.04

860.

0130

0.20

93D

iric

hlet

18.0

01.

000

P31

0.07

250.

0556

0.11

710.

1386

0.12

640.

1866

0.02

970.

0136

0.26

04D

iric

hlet

18.0

01.

000

(con

tinu

ed)


Tab

le1.

(Con

tinued

)

Model

:M

odel

:In

finit

eM

odel

:N

oSw

itch

ing

Ben

chm

ark

Model

Lifet

imes

(δ=

1)

inSpen

din

g

Log

Mar

gin

alLog

Mar

gin

alLog

Mar

gin

alLik

elih

ood:–1,2

83

Lik

elih

ood:–1,3

74

Lik

elih

ood:–1,3

11

Pri

or

Spec

ifica

tion

Par

amet

ers

Med

ian

5%

95%

Med

ian

5%

95%

Med

ian

5%

95%

Dis

trib

.M

ean

Std

.

Q11

0.66

780.

5082

0.73

700.

9859

0.83

180.

9976

0.81

840.

9874

0.99

61D

iric

hlet

18.0

01.

000

Q22

0.52

770.

5229

0.81

430.

5106

0.66

410.

7680

0.75

970.

3241

0.98

56D

iric

hlet

18.0

01.

000

σ2.

5979

2.52

012.

7578

1.88

491.

5415

1.90

111.

6563

1.37

141.

8598

Nor

mal

1.50

0.37

5ω

0.75

230.

6219

0.91

710.

9103

0.90

730.

9730

0.74

780.

6418

0.96

61B

eta

0.65

0.50

0γ

1.39

271.

2035

1.45

212.

3131

2.31

302.

4609

1.28

141.

1138

1.80

71N

orm

al2.

000.

750

κ0.

4380

0.42

410.

7425

0.79

340.

7932

0.79

800.

6801

0.53

250.

7895

Bet

a0.

500.

200

γy

0.05

300.

0253

0.07

810.

0977

0.09

610.

1394

0.12

190.

1129

0.16

84N

orm

al0.

200.

100

δ y0.

0069

0.00

300.

0726

0.02

140.

0146

0.02

560.

1426

0.06

520.

2678

Nor

mal

0.20

0.10

0ρ

α0.

7105

0.65

460.

7717

0.78

510.

6694

0.81

610.

7174

0.53

030.

8819

Bet

a0.

900.

256

ρu

0.28

350.

0389

0.49

360.

1831

0.06

170.

2366

0.42

430.

1021

0.42

93B

eta

0.90

0.25

6ρ

d0.

4774

0.47

570.

6174

0.80

950.

4748

0.85

940.

5238

0.30

840.

7649

Bet

a0.

500.

200

ρχ

0.96

880.

9626

0.97

740.

6767

0.60

800.

7432

0.93

430.

6933

0.95

80B

eta

0.90

0.25

6ρ

tp0.

2992

0.11

340.

4055

0.51

310.

3813

0.70

350.

9663

0.90

700.

9797

Bet

a0.

500.

200

100σ

α,1

3.82

372.

5119

3.99

943.

7632

3.77

594.

0017

0.86

730.

3747

1.19

69In

v.G

amm

a1.

001.

000

100σ

α,2

1.36

101.

0173

2.28

609.

6442

4.26

549.

9621

2.64

501.

3313

2.89

83In

v.G

amm

a1.

001.

000

100σ

u,1

1.16

820.

1306

1.29

980.

6367

0.29

110.

9665

1.48

820.

0057

1.56

04In

v.G

amm

a1.

001.

000

100σ

u,2

2.79

401.

4370

2.87

125.

9296

5.50

806.

4926

8.75

788.

3695

9.12

16In

v.G

amm

a1.

001.

000

100σ

d,1

3.03

982.

5150

3.18

830.

5540

0.34

091.

0791

6.01

445.

9061

6.85

30In

v.G

amm

a1.

001.

000

100σ

d,2

7.84

487.

4916

8.48

555.

6665

5.54

105.

8163

7.25

956.

8128

8.57

16In

v.G

amm

a1.

001.

000

100σ

tp,1

0.02

150.

0021

0.10

250.

0190

0.00

240.

1682

4.80

394.

1636

7.12

20In

v.G

amm

a1.

001.

000

100σ

tp,2

2.51

642.

4107

2.62

447.

4701

7.42

297.

7497

7.99

324.

9533

8.89

59In

v.G

amm

a1.

001.

000

100σ

s,1

2.45

071.

9907

3.20

540.

0412

0.00

070.

3563

2.37

802.

0733

3.64

83In

v.G

amm

a1.

001.

000

100σ

s,2

4.89

614.

2459

5.11

340.

7009

0.54

101.

0166

4.91

224.

8031

5.16

84In

v.G

amm

a1.

001.

000

100σ

τ,1

0.18

560.

1268

0.19

610.

0136

0.01

200.

0168

0.16

800.

1131

0.51

36In

v.G

amm

a1.

001.

000

100σ

τ,2

9.41

118.

6033

9.72

290.

1212

0.00

310.

2157

3.49

253.

1235

3.93

27In

v.G

amm

a1.

001.

000

100σ

χ,1

2.37

171.

8218

3.45

462.

4630

2.40

232.

8085

4.14

663.

5349

5.20

26In

v.G

amm

a1.

001.

000

100σ

χ,2

2.11

272.

1063

2.53

312.

4038

2.31

053.

1560

5.49

695.

8036

6.33

72In

v.G

amm

a1.

001.

000

100σ

R,1

2.21

141.

9804

2.34

581.

3362

1.14

931.

5843

0.70

170.

0747

1.97

98In

v.G

amm

a0.

500.

500

100σ

R,2

4.79

574.

5006

5.20

662.

4849

2.13

542.

5950

4.71

874.

3262

5.58

25In

v.G

amm

a0.

500.

500


nominal GDP (NIPA table 1.1.5, line 1) over the GDP deflator(NIPA table 1.1.4, line 1) and the civilian non-institutional pop-ulation over 16 (given by LNU00000000Q, at the Bureau of LaborStatistics). The table and line numbers refer to the NIPA (nationalincome and product accounts) tables on the Bureau of EconomicAnalysis website. The data for consumption, government purchases,federal spending, tax revenues, and debt were linearly detrended toget stationary series. Specifically, private consumption consists ofthe sum of personal consumption expenditures on nondurable goods(table 1.1.5, line 5) and services (table 1.1.5, line 6) divided by theGDP deflator and by population. I construct inflation as the firstdifference of the GDP deflator. GDP deflator data are obtained asdescribed above. I use monthly federal funds rate series from the Fed-eral Reserve Bank of St. Louis’s FRED (Federal Reserve EconomicData) website, averaged to receive quarterly series of the federalfunds rate. Government purchases consist of government consump-tion expenditures and gross investment (table 1.1.5, line 22) dividedby the GDP deflator and by population. Federal expenditure is thesum of government purchases plus net purchases of nonproducedassets (table 3.2, line 43), minus consumption of fixed capital (table3.2, line 44), minus wage accruals less disbursements (table 3.2, line33) plus net current transfer payments (table 3.2, lines 22–16), sub-sidies (table 3.2, line 32), and net capital transfers (table 3.2, lines42–38) divided by the GDP deflator and by population. Tax revenuesare given by the difference between current receipts (table 3.2, line37) and current transfer receipts (table 3.2, line 16) divided by theGDP deflator and by population. Interest payments are measuredusing total interest payments to persons and businesses and the restof the world (table 3.2, line 29). Finally, I compute government debtby taking the market value of privately held gross federal debt fromthe Federal Reserve Bank of Dallas website. The quarterly series isconstructed by summing up the monthly series.

5. Parameter Estimates

Table 1 reports the posterior parameter estimates. Since the focusof the paper is on the finite horizon of agents and on the inter-actions between monetary and fiscal policy across regimes, I will


focus on the parameters describing those eventualities. The poste-rior mean of the survival probability parameter is equal to 0.9760,implying a planning horizon of agents equal to 41 quarters (or 10years), approximately. Recall that in this paper I interpret δ as aparameter determining the effective planning horizon of households,as mentioned in the description of household decisions in section 2.

The estimation of the monetary/fiscal policy parameters revealsthe following facts. The monetary/fiscal policy mix has switchedthree regimes over time. Specifically, the estimation reveals that theU.S. economy has been through two determinate regimes and oneindeterminate regime. In line with the existing literature, a passivemonetary/active fiscal policy mix (Regime-Pol 2) is the most recur-rent during the ’70s until the early ’80s (see figure 2). From the mid-’80s onward the economy switched to a regime (Regime-Pol 3) wheremonetary policy was active and fiscal policy was passive, adjustingboth taxes and spending in reaction to debt fluctuations. I displaythe estimated posterior distributions of the reaction coefficients inthe monetary policy rule in figure 3. As regards the fiscal feedbackrules, the reaction coefficients on lagged debt-to-GDP ratios are closeto each other (γb,3 = 0.0794 and δb,3 = 0.0677).15 However, theeconomy spent a very short period during the early ’80s where bothmonetary and fiscal policy were passive, with the latter adjustingspending heavily in order to stabilize debt (Regime-Pol 1). In fact,the coefficient on debt in the spending rule is δb,1 = 0.1117, whilethat in the tax rule is γb,1 = 0.0415. During that period the medianof the posterior of the coefficient on inflation in the Taylor rule is esti-mated at φπ,1 = 0.9059. This result is in sharp contrast with existingevidence on estimated MS-DSGE models. In particular, Bianchi andIlut (2017) find that in the early ’80s the U.S. economy spent a shortperiod where both policies were active until it later switched to anAM/PF policy mix. However, Bianchi and Ilut do not account forthe possibility of federal spending being used as an additional oralternative instrument to stabilize debt. Moreover, I find that overthat period monetary policy was passive, although the posterior dis-tribution of φπ,1 lies to the right of φπ,2 (see figure 3). This could

15In fact, the mode of the posterior distribution of the inflation coefficient isabove 1 in Regime-Pol 3, while in Regime-Pol 1 and 2 it is below 1 and muchlower.


Figure 2. MS-DSGE Model, Posterior Median Estimates:Probabilities of Regimes

Notes: In the top panel, the black area displays the probability of Regime-Pol 1,the light gray area the probability of Regime-Pol 2, and the dark gray areathe probability of Regime-Pol 3. The bottom panel displays the probability ofRegime-Vol 1.

indicate that monetary policy in the United States started to adjustgradually, in the face of the persistent inflationary pressures of theprevious decade, until it switched to become active from the mid-’80sonward. The model estimation clearly identifies that the reaction ofthe Federal Reserve has changed over time with the mass of theestimated Regime-Pol 2 distribution of φπ,2 lying to the left of thedistribution of Regime-Pols 1 and 3 coefficients, φπ,1 and φπ,3.

The estimation of the model also shows that there have beenchanges in the volatilities of the shocks. Specifically, most the shockvolatilities are lower in Regime-Vol 1 than in Regime-Vol 2, withthe productivity and the government purchases share, χt, shocks tobe the only exceptions. For those two shocks, their volatilities are


Figure 3. Estimated Posterior Distribution of InflationReaction Coefficient in Both Regimes

Notes: The distribution in Regime-Pol 1 is displayed by the dashed line, the dis-tribution in Regime-Pol 2 by the solid line, and the distribution in Regime-Pol 3by the dashed-dotted line.

estimated to be higher in Regime-Vol 1. As shown in the bottompanel in figure 2, the probability of Regime-Vol 1 increases in theearly to mid-’80s and stays high until the wake of the recent financialcrisis.16

5.1 Impulse Responses

In this section, I look at the impulse responses in each regimeindividually. The model features eight shocks, namely a preference

16Bianchi and Ilut (2017) estimate an MS-DSGE model for the United Statesand also find evidence for a high-volatility regime in the early ’80s along with anincrease in the probability of this regime almost during the same period in theearly 2000s. Bianchi (2013) finds that the high-volatility regime is the dominantregime during the ’70s until the mid-’80s, with a small break in the late ’70s.


shock, technology shock, cost-push shock, tax shock, federal spend-ing shock, government purchases shock, monetary policy shock, andterm-premium shock. In order to save space, I look at the responsesto a monetary policy, a tax shock, a federal spending shock, anda preference shock. Specifically, I look at a contractionary mon-etary policy shock, a tax hike shock, a positive federal spendingshock, and a positive preference shock. The responses of inflation,output, the federal funds rate, and the debt-to-GDP are displayedin figure 4. In each case, I compute the median impulse responsefunction by solving the model using the median posterior estimateof each parameter. Everything thus is evaluated at the posteriormedian.

Let me compare the responses in the two determinate regimes,namely Regime-Pols 2 and 3. Inflation and output revert back tothe steady state faster in Regime-Pol 3 than in Regime-Pol 2 in allcases, the only exception being after a preference shock. As regardsoutput, the responses in Regime-Pols 2 and 3 following a preferenceshock look very similar. Notice the change in sign in the responsesof inflation in these two regimes following a monetary policy shock.Inflation increases in Regime-Pol 2, while it falls in Regime-Pol 3.In this case, higher inflation and a weak response of monetary pol-icy to inflation keep the real rate and hence the debt ratio lower inRegime-Pol 2 than in Regime-Pol 3. In general, the weak response ofmonetary policy to inflation in Regime-Pol 2 leads to higher inflationvolatility regardless of the shock hitting the economy. As a result,when shocks are inflationary, the real interest rate stays persistentlylower compared with Regime-Pol 3, which implies lower debt ser-vice costs. This allows the debt ratio to fluctuate at lower levels inRegime-Pol 2 than in Regime-Pol 3.

Given the regime-switching environment, inflation expectationsstay anchored in Regime-Pol 2 even though monetary policyresponds weakly to inflation fluctuations. Specifically, the specifi-cation of the transition matrix implies that while the economy liesin Regime-Pol 2, agents are aware of the fact that the economy mayswitch to Regime-Pol 3 with probability P23 = 1−P21 −P22. There-fore, this possibility allows for some anchoring of inflation expec-tations. However, the anchoring of inflation expectations is weakerthan in the case where agents are infinitely lived, δ = 1. This isbecause agents discount the future more under a Blanchard-Yaari


Figure 4. Impulse Response Functions

Notes: This figure displays the responses of inflation, output, real interest rate,and debt-to-GDP to a monetary, tax, federal expenditure, and preference shock.The solid lines display the median of the impulse responses in Regime-Pol 1,where both monetary and fiscal policy is passive; the dashed-dotted lines displaythe median in Regime-Pol 2, where monetary policy is passive and fiscal policy isactive; and the dashed lines display the median in Regime-Pol 3, where monetarypolicy is active and fiscal policy is passive.

structure, which implies that they place less weight on a futurestrong monetary response to inflation fluctuations when formingtheir expectations.17 Additionally, the anchoring of inflation expec-tations is further weakened, compared with the δ = 1 case, by the

17In fact, Del Negro, Giannoni, and Patterson (2012) embed a Blanchard-Yaaristructure in the medium-scale model of Smets and Wouters (2007) and show thatannouncements of policy changes in the future generate smaller effects on currentaggregate variables compared with a model with infinitely lived agents.


possibility of a switch to the indeterminate Regime-Pol 1. This wouldnot happen if Regime-Pol 2 stayed forever.18

Let me now look at the responses following fiscal shocks. Clearly,after a tax increase or a federal spending increase, inflation is bet-ter controlled in Regime-Pol 3. The strong reaction of the centralbank to inflation fluctuations in this regime keeps inflation expec-tations anchored. The finite lifetimes of households allow the effectof the monetary policy stance on inflation expectations to dominatethe negative effect stemming from the probability of a switch toRegime-Pol 1 where monetary policy is passive. The same argumentholds in Regime-Pol 2. The weak reaction of monetary policy toinflation dominates the effect of a possible switch to Regime-Pol 3in the future. In Regime-Pol 1 the responses of inflation and outputalso seem to be mainly affected by the current monetary policy andfiscal policy stance rather than by the probability of a future switchto Regime-Pol 2. In fact, even though passive, the monetary reactionto inflation is stronger in Regime-Pol 1 than Regime-Pol 2. All in all,it seems that the finite-lifetimes structure of the model makes thecurrent monetary/fiscal policy stance more important in households’decisions than the probability of a future switch to another regime.These effects become clearer in the next section, where I estimatethe same model but with infinite lifetimes, δ = 1.

5.2 Alternative Specifications

5.2.1 The Model with Infinite Lifetimes: δ = 1

In this section I reestimate the model under the assumption thatagents are infinitely lived. Specifically, I fix the survival probability

18Bianchi and Ilut (2017) perform beliefs counterfactuals to show the anchoringof inflation expectations when agents anticipate a switch to an active mone-tary/passive fiscal policy mix in the future. In their approach, though, agents donot face a probability of death, as such an active monetary/passive fiscal policymix can have zero effects on inflation because agents anticipate a switch to aRicardian regime in the future, as long as the probability of the latter is highenough. In the framework of the current paper, though, this can never happen.This is because I have assumed that agents face a probability of death whichleads to a debt-to-GDP ratio having real effects on output even though taxes arelump sum.


Figure 5. MS-DSGE Model with Infinite Lifetimes(δ = 1), Posterior Median Estimates: Probabilities of

Regimes

Note: The black area displays the probability of Regime-Pol 1, the light grayarea the probability of Regime-Pol 2, and the dark gray area the probability ofRegime-Pol 3.

to 1, δ = 1, and reestimate the model following the same proce-dure as described in section 4. The rest of the assumptions of themodel stay the same as well as the priors specification as summarizedoriginally at table 1.

The estimation results reveal some crucial differences betweenthe benchmark model with finite lifetimes and the one with infi-nite lifetimes. First, the estimation of the model favors two regimes.Specifically, as shown in figure 5, the economy switches betweentwo regimes, namely a regime where monetary policy is passiveand fiscal policy is active (Regime-Pol 2) during the ’70s and aregime where monetary policy is active and fiscal is passive (Regime-Pol 3) from the ’80s onward. Interestingly, the economy switches


earlier to Regime-Pol 3 when lifetimes are infinite compared withthe benchmark case (see figure 2). In the benchmark model withfinite lifetimes, the economy switched to Regime-Pol 3 around 1984after having spent a short time in Regime-Pol 1. The parameters ofRegime-Pol 1 show both policies to be passive in that regime. How-ever, the estimated smoothed state probabilities indicate that theeconomy has spent nearly no time in this regime, as opposed to thebenchmark model.19

The second important difference between the two models is thatin the model with infinite lifetimes tax revenues are substantiallymore sensitive to debt fluctuations (γb,3 = 0.1347) than federalexpenditure (δb,3 = 0.013) in Regime-Pol 3. In the benchmarkmodel, instead the sensitivity of the two fiscal tools in that regimewas similar (γb,3 = 0.079 and δb,3 = 0.067). As far as monetary pol-icy is concerned, the coefficients on inflation across regimes do notseem to differ much in the two models.

Third, some key parameters are also different in the two models.The Phillips curve is flatter in the model with infinite lifetimes, witha posterior mode of the Calvo parameter equal to ω = 0.9102, con-trary to ω = 0.6565 in the benchmark model. Additionally, output ismore persistent than in the benchmark case, with a posterior medianof the degree of habits κ = 0.7934, compared with κ = 0.4380.Finally, households appear to be less risk averse when lifetimes areinfinite, with a posterior median for the degree of relative risk aver-sion of σ = 1.8548 relative to σ = 2.5979 in the benchmark case.

Fourth, the responses of the variables differ substantially in somecases. However, those differences should be treated with caution,as not only the lifetimes differ but also the probabilities of theregimes, the policy parameters, and the rest of the deep parame-ters. In figure 6, I compare the responses of the variables from thetwo models following a monetary policy shock (in both cases a one-standard-deviation shock for valid comparisons). The responses ofinflation and output when lifetimes are infinite are either dampened

19The model estimation reveals also two volatility regimes, namely a high-volatility regime and a low-volatility regime. The former dominates from theearly ’80s onward, while the latter dominates in the ’70s. This is in line with thebenchmark model. I do not display in figure 5 the estimated filtered probabilitiesof the volatility regimes in order to save space. The results are available thoughupon request.


Figure 6. Impulse Response Functions

Notes: This figure displays the impulse responses of inflation, output, real rate,and the debt-to-GDP to a monetary policy shock. The solid lines display theresponses from the benchmark model and the dashed lines display the responsesfrom the model with infinite lifetimes.

or revert back to the steady state faster than in the benchmarkmodel, in all three regimes. This is mainly because inflation expec-tations are better anchored when agents have an infinite lifetime.In this case, agents discount the future less and are aware that theeconomy may switch to the most recurrent Regime-Pol 3 where mon-etary policy reacts aggressively to inflation fluctuations. As such,they expect lower inflation on average. Instead, when agents havefinite lifetimes, they put more weight on the current contractionarymonetary policy and less weight on the higher likelihood to stayin a regime where the central bank commits to keep inflation low(i.e., Regime-Pol 3). The fact that inflation expectations are betteranchored when lifetimes are infinite can be observed more clearly


when looking at the responses in Regime-Pol 2. The response ofinflation in the benchmark model is substantially more amplifiedthan in the estimated model with δ = 1. In the benchmark model,inflation stays persistently higher because agents discount the futuremore than when δ = 1. As such, they put less weight on the possi-bility of a future switch to the most recurrent Regime-Pol 3 wheremonetary policy reacts aggressively to inflation. Agents with infinitehorizons instead place a higher weight on that event to happen inthe future, which keeps their expectations well anchored.

5.2.2 No Switching in Federal Expenditure

In this section, I estimate another version of the model with the fol-lowing features. I allow households to face a probability of death intheir maximization problem, as in the benchmark model, but nowI assume that the fiscal authority uses only tax revenues to stabi-lize debt. In the process for federal spending, I set the reaction tothe debt ratio, δb, equal to zero in all regimes. However, I allowthe autoregressive coefficient, ρs,Zt , to vary across regimes. The newprocess is now specified as follows:

st = ρs,Zt st−1 − (1 − ρs,t) δyYt + σs,ζtεs,t εs,t ∼ N (0, 1) . (33)

The model estimation reveals that the U.S. economy has spentthe ’70s in Regime-Pol 2 where monetary policy was passive andfiscal policy was active, in line with the literature and the previ-ous estimates. The economy now seems to switch to Regime-Pol 1from the mid-to-late ’70s until the mid-’80s. The probability of thatregime increases above that of Regime-Pol 3 during that period.This could well mean that the persistently high inflation of the late’70s till the very first quarters of 1980 could still be attributed to apassive monetary policy, although fiscal policy had already startedto stabilize debt. In fact the latter started to slightly increase in themid-’70s until it started to increase abruptly in the early to mid-’80s owing to higher real rates. Contrary to the benchmark model,the economy is likely to have shifted to Regime-Pol 1 after Regime-Pol 2 earlier and is likely to have stayed there longer than whatthe benchmark model suggests. However, from the mid-’80s onwardthe probability of Regime-Pol 3 increases continuously above that ofRegime-Pol 1. This indicates that the likelihood that the economy


Figure 7. MS-DSGE Model with No Switching Spending,Posterior Median Estimates: Probabilities of Regimes

Note: The black area displays the probability of Regime-Pol 1, the light grayarea the probability of Regime-Pol 2, and the dark gray area the probability ofRegime-Pol 3.

switched to a regime where monetary policy was active and fiscalpolicy was passive increased from the mid-’80s onward. I display thesmoothed probabilities in figure 7 below.

As far as the rest of the parameters are concerned, the posteriormedian of the survival probability δ = 0.9717 is close to the bench-mark estimation. Monetary policy is reacting to inflation in a similarway as in the benchmark model in Regime-Pol 2 with a posteriormedian of φpi,2 = 0.7754. In Regime-Pol 3, although active, thecoefficient on inflation is lower than in the benchmark case, with aposterior median of φpi,2 = 1.2132. As regards Regime-Pol 1, mone-tary policy is more passive in the model where federal spending doesnot react to debt than it is in the other two models.


5.2.3 Comparisons across Models and Key Findings

The estimation of the three models leads to three important observa-tions. First, the data seem to support the shorter-horizon approach.Both the benchmark model and the model without switching inspending have a higher likelihood than the model with infinite plan-ning horizons. Both models point toward similar values of the sur-vival probability δ (9 to 10 years). As mentioned in section 3, I adoptthe planning horizon interpretation of 1/1 − δ, which means that inboth models households have a planning horizon of 9 to 10 years.This can be considered as a plausible number of years to plan ahead.

Second, of the two models with finite lifetimes, the one withswitching in federal spending to debt fluctuations fits the data bet-ter. Hence, it seems that accounting for regime switches in spendingengineered by a varying response to debt improves the model fit.Third, the two models with finite lifetimes find that the economyhas also spent a period in an indeterminate regime where both mon-etary and fiscal policy have been passive. When federal spendingalso reacts to debt in a time-varying fashion (benchmark model),the economy seems to have spent a very short period in the early’80s, when both policies are passive. When federal spending does notreact to debt, the model also indicates a switch to a regime whereboth policies have been passive in the early ’80s. However, in thatvery model the economy seems to have stayed longer in that regimebefore switching to a determinate regime where monetary policy wasactive and fiscal policy was passive.

Third, all models agree that the economy spent most of the ’70sin a regime where monetary policy was passive and fiscal policy wasactive. All models also find that the economy switched to a regimewhere monetary policy has been active while fiscal policy has beenpassive at least from the early ’80s onward.

6. Concluding Remarks

Monetary policy and fiscal policy in the United States have beenwidely documented to have switched over time. There have beenperiods during which fiscal policy has been the leading authority,while monetary policy has been accommodating (pre-Volcker era).On the other hand, there have been periods over which monetary


policy is leading (active) and fiscal policy commits to fiscal disci-pline (passive). Most studies have focused on tax revenues being theonly or the main fiscal instrument. However, evidence shows that,while passive, fiscal policy has at times switched toward using federalexpenditure instead of taxes in order to stabilize debt. One exampleis the continuous spending cuts during the Clinton Administration.

I have addressed the above facts using a Markov-switching DSGEmodel with a Blanchard-Yaari structure. The model features threeregimes. I estimate the model for the United States and find thatthe U.S. economy experienced regime shifts in the monetary/fiscalpolicy mix. In particular, in line with the existing literature, I havefound that monetary policy was passive during the ’70s, while fis-cal policy was active. The estimation of the model showed that thisregime prevailed until the early ’80s where the economy switchedto an indeterminate regime where both policies were passive. Fromthe mid-’80s onward, I found that the economy switched to a regimeassociated with an active monetary and a passive fiscal policy.

I have contributed to the current literature by allowing federalspending to also react to debt fluctuations. I provided empirical evi-dence motivating this approach. The estimation revealed that theU.S. government has used both tax revenues and spending almostequally, from the mid-’80s onward, in order to stabilize debt. Inter-estingly, though, I found that the U.S. economy spent a very shortperiod during the early ’80s in an indeterminate regime, where theU.S. government used spending heavily as a means to consolidate.Finally, by estimating alternative specifications I have shown, first,that the model with finite lifetimes fits the data better than a modelwith infinite lifetimes and, second, that allowing for federal spendingto also react to debt fluctuations improves the performance of themodel.

Appendix

A.1 The Linearized Model

The model is linearized with respect to taxes, government expendi-ture, and debt, whereas it is log-linearized with respect to all theother variables. I obtain a system of equations:


• Aggregate Euler equation:

Ct = �1Ct−1 + �2EtCt+1 − �3(Rt − Etπt+1 + (ρd − 1)dt)

+ �4

(bt + μt+1 + bYt

), (A.1)

where

�1 = κ

(1 +

1ς

· σ − 1σ

((1 − κ)C

)1−σ)

(A.2)

�2 =σ − 1

σ+

1ν

(1 − κ)2C (A.3)

�3 =1 − κ

ς· 1σ

(A.4)

�4 =(1 − κ)(1 − δ)

ςνδμ· Y

σ(A.5)

with

ς = 1 +1ν

· σ − 1σ

((1 − κ)C

)1−σ +κ

ν(1 − κ)2C + κ

σ − 1σ(A.6)

ν =(1 − δ)

δμbY + (1 − κ) C. (A.7)

The process for μt is specified as follows:

μt =μ − 1

μ

×[μt+1 +

σ − 11 − κ

((1 + κ)Ct − Ct+1 − κCt−1

)+ Δdt+1

],

(A.8)

where μ is equal to

μ =1

1 − δβ. (A.9)


• Phillips curve:

πt =1

1 + βδEtπt−1 +

βδ

1 + βδEtπt+1 +

(1 − ω)(1 − ωβδ)ω (1 + βδ)

×(γYt + σCt − σκCt−1 − (γ + σ) αt

)+ ut. (A.10)

• Government budget constraint:

bt = Rbt−1 + bR(Rt−1 − πt − Yt + Yt−1 − at) − τt + st + tpt.(A.11)

• Market clearing:

Yt = Ct +1

1 − gYgt. (A.12)

• Monetary policy rule:

Rt = ρR,ZtRt−1 + (1 − ρR,Zt)(φy,Zt Yt + φπ,Zt

πt

)+ σR,ζt

εR,t.

(A.13)

• Ratio between government purchases and federal expenditure:

χt = ρχχt−1 + (1 − ρχ) iyYt + σχ,ζtεχ,t. (A.14)

• Tax rule:

τt = ρτ,Zt τt−1 + (1 − ρτ,Zt)(γb,Zt bt−1 + γyYt

)+ στ,ζtετ,t.

(A.15)

• Federal spending rule:

st = ρs,Zt st−1 + (1 − ρs,Zt)(−δb,Zt bt−1 − δyYt

)+ σs,ζtεs,t.

(A.16)

• Technology:

αt = ρααt−1 + σa,ζtεα,t. (A.17)


• Cost-push shock:

ut = ρuut−1 + σu,ζtεu,t. (A.18)

• Preference shock:

dt = ρddt−1 + σd,ζtεd,t. (A.19)

• Risk premium shock:

tpt = ρdtpt−1 + σtp,ζtεtp,t. (A.20)

• Definition of χt:

χt =1

gY − 1gt − 1

sst. (A.21)

A.2 Steady State

The zero-inflation steady state of the model is summarized as fol-lows. From the representative household’s labor supply decision, Ihave for each country that

Hγ = wλ, (A.22)

while from the firms’ production function in each country, I havethat

Y = AH. (A.23)

Using the demand for each good and the market clearing condi-tion, the steady-state level of aggregate output is specified as

Y =1

1 − gYC. (A.24)

Using the aggregate Euler equation, the steady-state gross real inter-est rate is equal to

R =1β

+(1 − δ)δβμ

bY λ1/σ. (A.25)

Finally, from the government budget constraint, the steady-statelevel of debt-to-GDP ratio is specified as follows:

b =s − τ

1 − R. (A.26)


A.3 Deriving Aggregate Budget Constraint and Euler Equation

A.3.1 Deriving Aggregate Budget Constraint

The budget constraints of the different generations living in theeconomy at a given time t are given as follows:

(1 − δ)(

PtCit +

Bit

Rt

)= (1 − δ)

(WtH

it + Λi

t − T t + TRt

)

(1 − δ) δ

(PtC

it +

Bit

Rt

)= (1 − δ) δ

(WtH

it + Λi

t − T t + TRt

)+ (1 − δ) δ

1δBi

t−1

(1 − δ) δ2(

PtCit +

Bit

Rt

)= (1 − δ) δ2 (

WtHit + Λi

t − T t + TRt

)+ (1 − δ) δ2 1

δBi

t−1

(1 − δ) δ3(

PtCit +

Bit

Rt

)= (1 − δ) δ3 (

WtHit + Λi

t − T t + TRt

)+ (1 − δ) δ3 1

δBi

t−1

. . .

Let me denote the relation between generation-specific variable xit

and aggregate variable xt as follows:

xt =t∑

i=−∞(1 − δ) δt−ixi

t

xt−1 =t∑

i=−∞(1 − δ) δt−1−ixi

t−1,

so that the aggregate budget constraint reads as follows:

PtCt +Bt

Rt= Bt−1 + WtHt + Λt − T t + TRt. (A.27)


A.3.2 Deriving Aggregate Euler Equation

The flow budget constraint for members of generation i is summa-rized as

PtCit +

Bit

Rt=

1δBi

t−1 + WtHit + Λi

t − T t + TRt.

Define variable Δit as

Δit = WtH

it + Λi

t − T t + TRt. (A.28)

Writing the budget constraint above for periods t + 1, t + 2, . . . , Ireceive [

Pt+1Cit+1 +

Bit+1

Rt+1− Δi

t+1

]δ = Bi

t[Pt+2C

it+2 +

Bit+2

Rt+2− Δi

t+2

]δ = Bi

t+1.

Substituting for Bit in the generation i’s budget constraint in period

t, I receive

PtCit +

δ

RtPt+1C

it+1 +

δ

RtRt+1Bi

t+1 − δ

RtRt+1Δi

t+1 − Δit =

1δBi

t−1.

Substituting for Bit+1 in the equation above, I receive

PtCit +

δ

RtPt+1C

it+1 +

δ

RtRt+1

[Pt+2C

it+2 +

Bit+2

Rt+2− Δi

t+2

]

× δ − δ

RtRt+1Δi

t+1 − Δit =

1δBi

t−1

or

PtCit +

δ

RtPt+1C

it+1 +

δ2

RtRt+1Pt+2C

it+2 +

δ2

RtRt+1Rt+2Bi

t+2

− δ2

RtRt+1Δi

t+2 − δ

RtRt+1Δi

t+1 − Δit =

1δBi

t−1.


By iterating forward, I receive∞∑

s=0

δs∏s−1s=0 Rt+s

Pt+sCit+s −

∞∑s=0

δs∏s−1s=0 Rt+s

× Δit+s + lim

s→∞

δs∏s−1s=0 Rt+s

Bit+s

Rt+s=

1δBi

t−1.

By imposing the transversality condition,

lims→∞

δs∏s−1s=0 Rt+s

Bit+s

Rt+s= 0,

I receive∞∑

s=0

δs∏s−1s=0 Rt+s

Pt+sCit+s −

∞∑s=0

δs∏s−1s=0 Rt+s

Δit+s =

1δBi

t−1. (A.29)

Now I need to work with the first term on the left-hand side ofthe equation above. Using the first-order condition (6) and iteratingforward, I receive the following expression:

δs∏s−1s=0 Rt+s

Pt+s

λit+s

=Pt

λit

(1

(δβ)sedt

edt+s

)−1

. (A.30)

Using the definition λit =

(Ci

t − κCt−1)−σ, equation (A.29) can be

rewritten as follows:∞∑

s=0

δs∏s−1s=0 Rt+s

Pt+s

λit+s

(λi

t+s

)σ−1σ + κ

∞∑s=0

δs∏s−1s=0 Rt+s

Pt+sCt+s−1

−∞∑

s=0

δs∏s−1s=0 Rt+s

Δit+s =

1δBi

t−1. (A.31)

Substituting (A.30) into (A.31), I receive the following:

Pt

λit

∞∑s=0

(δβ)s

(edt+s

edt

) (λi

t+s

)σ−1σ + κ

∞∑s=0

δs∏s−1s=0 Rt+s

Pt+sCt+s−1

−∞∑

s=0

δs∏s−1s=0 Rt+s

Δit+s =

1δBi

t−1. (A.32)


By rearranging and aggregating, I receive

Pt

λt=

edt∑∞s=0 (δβ)s

edt+sλσ−1

σt+s(

Bt−1 − κ

∞∑s=0

δs∏s−1s=0 Rt+s

Pt+sCt+s−1 +∞∑

s=0

δs∏s−1s=0 Rt+s

Δt+s

).

(A.33)

Adding and subtracting δRt

edt

∑∞s=0(δβ)sedt+sλ

σ−1σ

t+s

Bt, I receive

Pt

λt=


edt+sλσ−1

σt+s

(Bt−1 − κPtCt−1 + Δt)

− δ

Rt


edt+sλσ−1

σt+s

Bt +δ

Rt


edt+sλσ−1

σt+s

×(

Bt − κ

∞∑s=0

δs∏s−1s=0 Rt+s+1

Pt+sCt+s

+∞∑

s=0


Δt+s+1

).

(A.34)

Note that

Pt+1

λt+1=

edt+1∑∞s=0 (δβ)s

edt+s+1λσ−1

σt+s+1

×(

Bt − κ

∞∑s=0


Pt+sCt+s

+∞∑

s=0


Δt+s+1

). (A.35)


Substituting (A.35) in (A.36), I receive

Pt

λt=


edt+sλσ−1

σt+s

×

⎡⎣Bt−1 − κPtCt−1 + Δt − δ

RtBt

+δ

Rt

∑∞s=0 (δβ)s

edt+s+1λσ−1

σt+s+1

edt+1

Pt+1

λt+1

⎤⎦ . (A.36)

Using the aggregate budget constraint (A.27) to substitute out forBt−1, I get

Pt

λt=


edt+sλσ−1

σt+s

×

⎡⎣PtCt − κPtCt−1 +

1 − δ

RtBt

+δ

Rt

∑∞s=0 (δβ)s

edt+s+1λσ−1

σt+s+1

edt+1

Pt+1

λt+1

⎤⎦ . (A.37)

Note that Ct − κCt−1 = λ− 1

σt . Substituting this expression into

(A.38), I receive

Pt

λt=


edt+sλσ−1

σt+s

×

⎡⎣Ptλ

− 1σ

t +1 − δ

RtBt +

δ

Rt

∑∞s=0 (δβ)s

edt+s+1λσ−1

σt+s+1

edt+1

Pt+1

λt+1

⎤⎦ .

(A.38)


Multiplying and dividing the first term of the right-hand side abovewith λ

σ−1σ , I receive

Pt

λt=


edt+sλσ−1

σt+s

×

⎡⎣Pt

λtλ

σ−1σ

t +1 − δ

RtBt +

δ

Rt

∑∞s=0 (δβ)s

edt+s+1λσ−1

σt+s+1

edt+1

Pt+1

λt+1

⎤⎦ .

(A.39)

Rearranging the equation above, I get⎡⎣1 − edtλ

σ−1σ

t∑∞s=0 (δβ)s

edt+sλσ−1

σt+s

⎤⎦ Pt

λt=


edt+sλσ−1

σt+s⎡

⎣1 − δ

RtBt +

δ

Rt

∑∞s=0 (δβ)s

edt+s+1λσ−1

σt+s+1

edt+1

Pt+1

λt+1

⎤⎦ . (A.40)

Multiplying and dividing the right-hand side of (A.40) by λσ−1

σt , I

receive⎡⎣1 − edtλ

σ−1σ

t∑∞s=0 (δβ)s

edt+sλσ−1

σt+s

⎤⎦ Pt

λt=

edtλσ−1

σt∑∞

s=0 (δβ)sedt+sλ

σ−1σ

t+s⎡⎣ 1 − δ

Rtλσ−1

σt

Bt +δ

Rt

∑∞s=0 (δβ)s

edt+s+1λσ−1

σt+s+1

edt+1λσ−1

σt

Pt+1

λt+1

⎤⎦ . (A.41)

Multiplying and dividing the last term of the right-hand side in(A.41) by λ

σ−1σ

t+1 , I receive

(μt − 1)δ

Pt

λtRt =

1 − δ

δλσ−1

σt

Bt +Pt+1

λt+1μt+1, (A.42)

where

μt =∞∑

s=0

(δβ)s edt+sλσ−1

σt+s

edtλσ−1

σt


μt = 1 +∞∑

s=1


σt+s

edtλσ−1

σt

μt = 1 +edt+1δβ

edt

∞∑s=0


σt+s

edtλσ−1

σt

μt = 1 +edt+1λ

σ−1σ

t+1

edtλσ−1

σt

δβμt+1. (A.43)

Finally, multiplying the left-hand side and the right-hand side of(A.44) by λ

σ−1σ

t and using (A.43) to substitute out for μt, I receiveexpression (17):

βedt+1

edt

Rt

Πt+1

1λt

=1 − δ

δμt+1

1

λσ−1

σt+1

Bt +(

λt

λt+1

)σ−1σ 1

λt+1. (A.44)

References

Annicchiarico, B., N. Giammarioli, and A. Piergallini. 2012. “Bud-getary Policies in a DSGE Model with Finite Horizons.” Researchin Economics 66 (2): 111–30.

Annicchiarico, B., G. Marini, and A. Piergallini. 2008. “MonetaryPolicy and Fiscal Rules.” B.E. Journal of Macroeconomics 8(1, February): 1–42.

Bhattarai, S., J. W. Lee, and W. Y. Park. 2012a. “Monetary-FiscalPolicy Interactions and Indeterminacy in Postwar US Data.”American Economic Review 102 (3): 173–78.

———. 2012b. “Policy Regimes, Policy Shifts, and U.S. BusinessCycles.” Technical Report.

Bi, H., E. M. Leeper, and C. Leith. 2013. “Uncertain Fiscal Consol-idations.” Economic Journal 123 (566): F31–F63.

Bianchi, F. 2012. “Evolving Monetary/Fiscal Policy Mix in theUnited States.” American Economic Review 102 (3): 167–72.

———. 2013. “Regime Switches, Agents’ Beliefs, and Post-WorldWar II U.S. Macroeconomic Dynamics.” Review of EconomicStudies 80 (2): 463–90.


Bianchi, F., and C. Ilut. 2017. “Monetary/Fiscal Policy Mix andAgents’ Beliefs.” Review of Economic Dynamics 26 (October):113–39.

Blanchard, O. J. 1985. “Debt, Deficits, and Finite Horizons.” Jour-nal of Political Economy 93 (2): 223–47.

Calvo, G. A. 1983. “Staggered Prices in a Utility-Maximizing Frame-work.” Journal of Monetary Economics 12 (3): 383–98.

Canzoneri, M., R. Cumby, and B. Diba. 2010. “The InteractionBetween Monetary and Fiscal Policy.” In Handbook of Mone-tary Economics, Vol. 3, ed. B. M. Friedman and M. Woodford,935–99 (chapter 17). Elsevier.

Castelnuovo, E., and S. Nistico. 2010. “Stock Market Conditionsand Monetary Policy in a DSGE Model for the U.S.” Journal ofEconomic Dynamics and Control 34 (9): 1700–31.

Chen, H. 2017. “The Effects of the Near-Zero Interest Rate Policyin a Regime-Switching Dynamic Stochastic General EquilibriumModel.” Journal of Monetary Economics 90 (October): 176–92.

Chen, X., T. Kirsanova, and C. Leith. 2017. “How Optimal is USMonetary Policy?” Journal of Monetary Economics 92 (Decem-ber): 96–111.

Christiano, L. J., M. Eichenbaum, and C. L. Evans. 2005. “Nomi-nal Rigidities and the Dynamic Effects of a Shock to MonetaryPolicy.” Journal of Political Economy 113 (1): 1–45.

Cogley, T., and T. J. Sargent. 2005. “Drift and Volatilities: Mone-tary Policies and Outcomes in the Post WWII US.” Review ofEconomic Dynamics 8 (2): 262–302.

Davig, T., and T. Doh. 2014. “Monetary Policy Regime Shifts andInflation Persistence.” Review of Economics and Statistics 96 (5):862–75.

Davig, T., and E. M. Leeper. 2011. “Monetary–Fiscal Policy Inter-actions and Fiscal Stimulus.” European Economic Review 55 (2):211–27.

Davig, T., E. M. Leeper, and H. Chung. 2007. “Monetary and FiscalPolicy Switching.” Journal of Money, Credit and Banking 39 (4):809–42.

Del Negro, M., M. Giannoni, and C. Patterson. 2012. “The ForwardGuidance Puzzle.” Staff Report No. 574, Federal Reserve Bankof New York.


Devereux, M. B. 2011. “Fiscal Deficits, Debt, and Monetary Policyin a Liquidity Trap.” In Monetary Policy under Financial Turbu-lence, Vol. 16 of Central Banking, Analysis, and Economic Poli-cies Book Series, eds. L. F. Cespedes, R. Chang, and D. Saravia,369–410 (chapter 10). Central Bank of Chile.

Farmer, R. E., D. F. Waggoner, and T. Zha. 2011. “Minimal StateVariable Solutions to Markov-Switching Rational ExpectationsModels.” Journal of Economic Dynamics and Control 35 (12):2150–66.

Favero, C., and T. Monacelli. 2005. “Fiscal Policy Rules and Regime(In)Stability: Evidence from the U.S.” IGIER Working Paper No.282.

Forni, L., L. Monteforte, and L. Sessa. 2009. “The General Equi-librium Effects of Fiscal Policy: Estimates for the Euro Area.”Journal of Public Economics 93 (3–4): 559–85.

Justiniano, A., and B. Preston. 2010. “Monetary Policy and Uncer-tainty in an Empirical Small Open-Economy Model.” Journal ofApplied Econometrics 25 (1): 93–128.

Kamps, C. 2007. “Dynamic Scoring in an Estimated DSGE Modelof the US Economy.” European Central Bank.

Kim, C.-J., and C. Nelson. 1999. State-Space Models with RegimeSwitching: Classical and Gibbs-Sampling Approaches with Appli-cations. 1st ed., Vol. 1. The MIT Press.

Leeper, E. M. 1991. “Equilibria under ‘Active’ and ‘Passive’ Mon-etary and Fiscal Policies.” Journal of Monetary Economics 27(1): 129–47.

Leeper, E. M., M. Plante, and N. Traum. 2010. “Dynamics of FiscalFinancing in the United States.” Journal of Econometrics 156(2): 304–21.

Leeper, E. M., and S.-C. S. Yang. 2008. “Dynamic Scoring: Alterna-tive Financing Schemes.” Journal of Public Economics 92 (1–2):159–82.

Leith, C., and S. Wren-Lewis. 2000. “Interactions between Monetaryand Fiscal Policy Rules.” Economic Journal 110 (462): 93–108.

Liu, P., and H. Mumtaz. 2011. “Evolving Macroeconomic Dynam-ics in a Small Open Economy: An Estimated Markov SwitchingDSGE Model for the UK.” Journal of Money, Credit and Banking43 (7): 1443–74.


Lopez-Salido, J. D., and P. Rabanal. 2007. “Government Spend-ing and Consumption Hours Preferences.” Manuscript, FederalReserve Board.

Lubik, T. A., and F. Schorfheide. 2007. “Do Central Banks Respondto Exchange Rate Movements? A Structural Investigation.”Journal of Monetary Economics 54 (4): 1069–87.

Marco, R., R. Werner, and V. Jan. 2006. “Fiscal Policy in an Esti-mated Open-Economy Model for the EURO Area.” Computingin Economics and Finance 2006, No. 43, Society for Computa-tional Economics.

Nistico, S. 2012. “Monetary Policy and Stock-Price Dynamics in aDSGE Framework.” Journal of Macroeconomics 34 (1): 126–46.

Piergallini, A. 2006. “Real Balance Effects and Monetary Policy.”Economic Inquiry 44 (3): 497–511.

Primiceri, G. E. 2005. “Time Varying Structural Vector Autoregres-sions and Monetary Policy.” Review of Economic Studies 72 (3):821–52.

Richter, A. W. 2015. “Finite Lifetimes, Long-term Debt and theFiscal Limit.” Journal of Economic Dynamics and Control 51(February): 180–203.

Sims, C. A. 2011. “Stepping on a Rake: The Role of Fiscal Policy inthe Inflation of the 1970s.” European Economic Review 55 (1):48–56.

Sims, C. A., and T. Zha. 2006. “Were There Regime Switches in U.S.Monetary Policy?” American Economic Review 96 (1): 54–81.

Smets, F., and M. Trabandt. 2012. “Government Debt, the ZeroLower Bound and Monetary Policy.” Preliminary Paper.

Smets, F., and R. Wouters. 2003. “An Estimated Dynamic Stochas-tic General Equilibrium Model of the Euro Area.” Journal of theEuropean Economic Association 1 (5): 1123–75.

———. 2007. “Shocks and Frictions in US Business Cycles: ABayesian DSGE Approach.” American Economic Review 97 (3):586–606.

Stock, J. H., and M. W. Watson. 2002. “Has the Business CycleChanged and Why?” NBER Working Paper No. 9127.

Straub, R., and G. Coenen. 2005. “Non-Ricardian Households andFiscal Policy in an Estimated DSGE Model of the Euro Area.”Computing in Economics and Finance 2005, No. 102, Society forComputational Economics.


Traum, N., and S.-C. S. Yang. 2011. “Monetary and Fiscal PolicyInteractions in the Post-War U.S.” European Economic Review55 (1): 140–64.

Yaari, M. 1965. “Uncertain Lifetime, Life Insurance, and the Theoryof the Consumer.” Review of Economic Studies 32 (2): 137–50.

Documents

Finite Horizons and the Monetary/Fiscal Policy Mix · 2020. 9. 1. · by an active monetary and a passive ﬁscal policy. When it comes to the ’70s, the literature does not seem