the advantage of flexible targeting rules

ANDREA FERRERO

The Advantage of Flexible Targeting Rules

This paper investigates the consequences of debt stabilization for inflationtargeting. If the fiscal authority holds constant the real value of debt atmaturity under strict inflation targeting, the equilibrium dynamics are in-determinate for a wide range of parameters and steady-state fiscal stances.“Flexible” targeting rules that include a concern for stabilization of theoutput gap can restore determinacy of the equilibrium. Flexible inflationtargeting appears to be more robust than flexible debt targeting to alterna-tive parameterizations. The fiscal authority can prevent indeterminacy understrict targeting rules by committing to hold constant debt net of interest ratespending.

JEL codes: E52, E58, E62, E63Keywords: targeting rules, determinacy, flexibility.

THE GREAT RECESSION OF 2008 ignited a vigorous debate onthe appropriate conduct of public policy. In response to the financial crisis, centralbanks around the world lowered the nominal interest rate to its effective zero bound,while governments substantially increased the size of their budget deficits. As thesituation normalizes, monetary policy should revert back to its natural course, withrenewed emphasis on stabilizing inflation. At the same time, fiscal sustainability willrequire a discussion of the exit strategy from the temporary stimulus.

In this context, the design and the interaction of fiscal and monetary policiesplay a critical role for macroeconomic outcomes. Until recently, both academics andpractitioners have primarily focused on specifications of policy in terms of instrumentsetting. The most notable example of this approach is perhaps the so-called Taylorrule (Taylor 1993). According to the Taylor rule, the central bank moves its instrument

I thank to the editor (Ken West) and two anonymous referees for several criticisms and suggestions. I amgrateful to Pierpaolo Benigno for encouraging me to look into many of the issues discussed in this paper,and to Stefano Eusepi, Marc Giannoni, and Eric Leeper for useful conversations. I have also benefitedfrom comments by seminar participants at IMT Lucca, Midwest Macroeconomics Meetings (Indiana),European Meetings of the Econometric Society (Barcelona), and Computing in Economics and Finance(London). The usual disclaimers apply. The views expressed herein do not necessarily reflect the positionof the Federal Reserve Bank of New York or the Federal Reserve System.

ANDREA FERRERO is an Economist, Federal Reserve Bank of New York, Macroeconomics and MonetaryStudies Function—Research and Statistics Group (E-mail: [email protected]).

Received July 15, 2009; and accepted in revised form November 30, 2011.

Journal of Money, Credit and Banking, Vol. 44, No. 5 (August 2012)C© 2012 The Ohio State University

864 : MONEY, CREDIT AND BANKING

(typically a short-term nominal interest rate) in response to deviations of inflationand an indicator of real activity from their respective objectives.

Targeting rules represent an alternative to instrument-based specifications of policy.The idea behind targeting rules is that the policy authority aims at achieving a specificvalue for a certain variable over a given time horizon. In the case of the Bankof England, for example, “The inflation target of 2% is expressed in terms of anannual rate of inflation based on the Consumer Prices Index (CPI).” Like the Bankof England, several other central banks and governments have recently started to settheir monetary policy following a targeting rule. Following the pioneering experienceof New Zealand in 1990, more than 20 countries, both in the industrialized anddeveloping world, have adopted an explicit inflation targeting approach to monetarypolicymaking (Svensson 2008).1

Although perhaps less widespread, fiscal targeting rules represent a concrete deviceto impose accountability on governments’ discretionary decisions. The debt ceilingrule in the U.S. suddenly became popular in the summer of 2011. Other examplesinclude U.S. states, Canadian provinces, and EMU partners. All these regions areconstrained by some form of either budget or debt requirements.2 Today more thanever, balanced budget requirements and debt stabilization are on the forefront ofpolicymakers’ current agendas (Bernanke 2009).

Despite their recent surge in popularity, the properties of different targeting rules—and the consequences of their interaction—are still largely unknown. The objectiveof this paper is to study the equilibrium of an economy in which both fiscal andmonetary authorities set their policies according to targeting rules.3 Understandingthe basic functioning of targeting rules is an important first step toward explicitlyfixing quantitative criteria for monetary and fiscal authorities during the exit from theextraordinary measures that have accompanied the financial crisis.

The first result in the paper is that in a model with nominal rigidities, debt anddistortionary taxation, strict targeting rules (inflation and debt equal to their long-run values in every period) generate equilibrium indeterminacy for a wide range ofparameters and steady-state fiscal stances.4

This finding constitutes a counterexample to the relation between active/passivepolicy regimes and equilibrium determinacy (Leeper 1991). Strict inflation target-ing represents an active monetary policy regime in which the nominal interest rateresponds more than one-to-one to movements in inflation. Conversely, strict debt

1. Accountability and simplicity of communication are two important benefits of inflation targeting.See Bernanke et al. (1999) for an extensive discussion.

2. Bohn and Inman (1996) and Auerbach (2008) discuss the U.S. experience. Millar (1997) reviewsother international examples.

3. Different from the definition in Svensson (2003), in this paper, targeting rules may also involvevariables not present in the policymakers’ loss function.

4. Previous studies of the determinacy properties of the equilibrium under targeting rules have mostlyabstracted from the interaction between monetary and fiscal policy. See, for instance, Schmitt-Grohe andUribe (1997) for indeterminacy results in the neoclassical growth model under balanced budget rulesand Giannoni and Woodford (2003) or Svensson and Woodford (2003) for determinacy conditions underinflation targeting in New Keynesian frameworks.

ANDREA FERRERO : 865

targeting represents a passive fiscal policy regime in which the tax rate ensures in-tertemporal solvency for any possible level of debt. In Leeper (1991), the combinationof an active monetary policy regime and a passive fiscal regime leads to equilibriumdeterminacy.

Benhabib and Eusepi (2005) and Linnemann (2006) show that the presence of nom-inal rigidities, debt, and distortionary taxation may break the simple active–passivedichotomy. Arbitrary expectations of future inflation can lead to an indeterminateequilibrium even with an active–passive policy mix. This paper finds that a similarresult also holds for targeting rules. In this case, however, the source of indeterminacyis different and depends on arbitrary expectations of future economic activity ratherthan inflation.

The good news is that “flexible” targeting rules that include a concern for thestabilization of a measure of real activity (specifically, the output gap) can restoredeterminacy of the equilibrium. In particular, flexible inflation targeting rules (of thetype advocated by Svensson 2003) turn out to be very robust to a wide range ofparameter values. Conversely, flexible debt targeting rules are fairly sensitive to theaverage value of the tax rate, even though these types of rules feature more desirablewelfare properties.

Finally, the paper shows that a crucial element behind the indeterminacy resultsis the influence of monetary policy on fiscal solvency via the service of debt. If thegovernment perfectly stabilizes debt net of interest rate payments in each period, theequilibrium is determinate.

Overall, the main takeaway of the paper is that the interaction of fiscal and mon-etary policy may present serious challenges and nontrivial trade-offs in terms of theappropriate design of policy rules.

The paper proceeds with a brief presentation of the model in the next section.Section 2 shows that the equilibrium is indeterminate in the case of strict inflationand debt targeting rules. Section 3 demonstrates that a commitment to flexible policyrules for either monetary or fiscal policy can restore determinacy of the equilibrium,and Section 4 investigates the robustness of the results to alternative parameterconfigurations and steady-state fiscal stances. Section 5 introduces the alternativedebt targeting rule that avoids indeterminacy even in the case of a strict formulation.The last section concludes.5

1. THE MODEL

This section presents the log-linear approximation of a model with monopolisticcompetition, nominal rigidities, distortionary taxation, and public debt. This lin-earized model encompasses different specifications of distortionary taxation that

5. An online appendix, which is available at nyfedeconomists.org/ferrero/pub.html, contains furtherdetails on the model and an analysis of the welfare properties of different targeting rules.


affects aggregate supply. The online appendix shows that, up to the first order, thesame representation approximates both an economy in which the government leviesa distortionary income tax (as in Benhabib and Eusepi 2005, Linnemann 2006) andan economy with distortionary sales taxes (as in Benigno and Woodford 2003).

The key (and only) variation upon the baseline New Keynesian framework (seeClarida, Galı, and Gertler 1999, Woodford 2003) is that the fiscal authority cannotuse lump-sum taxes to satisfy its budget constraint in every period. Hence, Ricardianequivalence does not hold and fiscal decisions become relevant for the equilibriumdetermination. Specifically, the fiscal–monetary interaction occurs along two chan-nels. On the one hand, monetary policy decisions affect fiscal solvency by influencingthe real value of debt. On the other hand, fiscal policy decisions have inflationaryconsequences by acting as endogenous cost-push shocks.

The demand side of the economy is described by a standard (gap-based) expressionthat relates current economic activity negatively to the real interest rate and positivelyto expected future activity

yt = −σ (it − Etπt+1 − r∗t ) + Et yt+1, (1)

where σ > 0 is the elasticity of intertemporal substitution multiplied by the con-sumption share of GDP, yt is the output gap (i.e., output in deviations from somemeasure of potential activity), it is the nominal interest rate, πt is inflation, and r∗

tis the real interest rate that would prevail if output was at its potential level (a linearcombination of exogenous shocks).

The supply side of the economy consists of a forward-looking Phillips curve,augmented by a cost-push shock that is partly endogenous

πt = κ[yt + ψ(τt − τ ∗t )] + βEtπt+1, (2)

where β ∈ (0, 1) is the individual discount factor and κ and ψ are convolution ofthe underlying structural parameters.6 The term τt − τ ∗

t in equation (2) represents a“tax gap.” The tax rate τt enters the Phillips curve because taxes directly affect thefirms’ optimal pricing decision, either increasing their marginal costs, in the case ofan income tax, or decreasing their marginal revenues, in the case of a sales tax.7 The

6. The slope of the Phillips curve relates to the fundamental parameters according to

κ ≡ (1 − α)(1 − αβ)

α(1 + θω)(ω + σ−1),

where α ∈ (0, 1) is the probability of not being able to change the price for a firm at each point in time,θ > 1 is the elasticity of substitution among differentiated goods, and ω > 0 is the inverse Frisch elasticityof labor supply. The parameter in front of the tax gap is

ψ ≡ τ

(1 − τ )(ω + σ−1),

where τ ∈ (0, 1) is the steady-state tax rate.7. Consumption taxes would affect the marginal revenues in the firms’ optimal pricing condition by

reducing the marginal utility of consumption. See the online appendix for details.


TABLE 1

BASELINE CALIBRATION AND IMPLIED STEADY-STATE VALUES

β = 0.99 Discount factorσ−1 = 0.157 Adjusted coefficient of risk aversionω = 0.473 Inverse Frisch elasticity of labor supplyκ = 0.0236 Slope of the Phillips curveτ = 20% Steady-state tax rateb/ (4Y ) = 60% Steady-state annualized debt–GDP ratio

term τ ∗t is the exogenous part of the cost-push shock. In this model, a coordinated

fiscal and monetary policy action could, in principle, offset any effect of cost-pushshocks on inflation, while maintaining a zero output gap, by appropriately using thetax rate. The term τ ∗

t is the target level of the tax rate that achieves this objective.The government budget constraint completes the list of equilibrium conditions for

this economy

Bt = (1 − β)[by yt + bτ (τt − τ ∗t )] + βEt Bt+1, (3)

where bτ ≡ τ/(τ − G/Y ), by ≡ bτ − σ−1, G/Y is the steady-state governmentspending share of GDP, and

Bt ≡ bt−1 − πt − σ−1 yt + ft . (4)

In expression (4), bt is the real value of government debt at maturity and ft is anexogenous “fiscal stress.” This term summarizes the exogenous disturbances thatprevent contemporaneous stabilization of inflation and the output gap.8

The baseline version of the New Keynesian framework is a special case of themodel presented here in which lump-sum taxes satisfy the government solvencycondition (3) in each period. In this case, the entire tax gap would correspond to afully exogenous cost-push shock.

2. STRICT TARGETING RULES AND INDETERMINACY

This section shows that under reasonable calibrations, strict targeting rules for fiscaland monetary policy can lead to an indeterminate rational expectations equilibrium.This result, however, is not a theorem but depends on parameter values. Table 1reports the baseline calibration, which follows Benigno and Woodford (2003). Thesecond part of this section provides robustness analysis to several variations in theparameters.

8. The forward-looking representation of the government budget constraint follows from substitutingthe aggregate demand equation into the backward-looking flow government budget constraint to eliminatethe real interest rate.


The monetary authority is assumed to follow an inflation targeting rule that allowsfor deviations from full price stability if output deviates from its potential level

πt + γ yt = 0, (5)

where the coefficient γ > 0 controls the intensity of the feedback from real activityonto inflation.

The fiscal authority is assumed to follow a debt targeting rule that allows fordeviations from full debt stability if output deviates from its potential level

bt + λyt = 0, (6)

where the coefficient λ > 0 represents the sensitivity of the response of debt to realactivity.

In general, the concept of flexibility is open to a variety of interpretations. Rules (5)and (6) specifically introduce a concern for the output gap as an example of departurefrom a strict formulation of the policy rules. The emphasis here is on the comparisonbetween strict and flexible specification of targeting rules, while at the same timekeeping the model analytically tractable.

Rule (5) constitutes a simple benchmark for the analysis of monetary policy in thecontext of inflation targeting. In inflation targeting countries, central banks typicallycombine the objective of price stability with a concern for some measure of realactivity (Svensson 2008). Moreover, in the baseline New Keynesian model, optimalmonetary policy takes a form similar to (5) (Clarida, Galı, and Gertler 1999, andWoodford, 2003).9 In that model, however, complete price stability in every period(γ = 0)—while suboptimal—would ensure the existence of a determinateequilibrium.

Expression (6) is the obvious counterpart of (5) for fiscal policy. The rule imposesa restriction on the path of government debt. Nothing is special about specifying thefiscal targeting rule in terms of debt rather than deficit. Substitution of (6) into thegovernment budget constraint (3) would recast the fiscal targeting rule in terms of abudget requirement. In particular, the strict version of the rule (λ = 0) correspondsto a balanced budget condition in each period. Legislations around the world oftenuse a similar constraint on fiscal authorities, as the experience of several U.S. statesand the Stability and Growth Pact in the European Monetary Union suggest.10

The first result of this paper is that if both policy authorities follow a strict formu-lation of their targeting rules, the equilibrium is indeterminate.

To see why this result holds, suppose the monetary and fiscal authorities commit tofollow rules (5) and (6) with γ = 0 and λ = 0, respectively. Replacing the targeting

9. Specifically, targeting rule (5) represents optimal monetary policy under discretion, while in thecase of commitment the indicator of real activity would be the one period change of the output gap yt .In both cases, γ is a function of the model parameters.

10. When λ = 1, the fiscal authority moves debt one to one with any deviation of output from potential.


rules in the Phillips curve (2) and the government budget constraint (3) yields11

0 = yt + ψ(τt − τ ∗t ) (7)

and

βσ−1Et yt+1 = [(1 − β)by + σ−1]yt + (1 − β)bτ (τt − τ ∗

t ) − ft + βEt ft+1.

(8)

Substituting for the tax gap from the first expression into the second gives a forward-looking first-order difference equation in the output gap

βσ−1Et yt+1 = [(1 − β)(by − ψ−1bτ ) + σ−1]yt + ε

ft , (9)

where ε ft ≡ βEt ft+1 − ft . A necessary (and in this case also sufficient) condition

for a determinate equilibrium to exist is that equation (9) delivers a unique stablesolution for yt . This condition obtains if and only if |ρy| > 1 where12

ρy ≡ (1 − β)(by − ψ−1bτ ) + σ−1

βσ−1. (10)

Under the baseline calibration, the inequality in (10) is not satisfied and no determinateequilibrium exists under strict inflation targeting and constant debt rules.

Equations (7) and (8) are useful to study the mechanics behind the indeterminacyresult. Consider an arbitrary sunspot shock that creates expectations of higher futureoutput. For a given tax rate, expression (8) requires that output increases alreadyin the current period. Determinacy would then only occur if the policy response isstrong enough so that an increase in taxes more than offsets the upward pressure oncurrent activity. But the aggregate supply relation (7) suggests the opposite: if outputis higher, the tax rate decreases. Back to expression (8), combinations of highercurrent output and a lower tax gap are consistent with expectations of higher futureoutput. Therefore, a nonfundamental sunspot shock can become self-fulfilling.

This type of indeterminacy result differs from previous contributions that havefocused on instrument rules. Linnemann (2006), for example, considers arbitraryrevisions to inflation expectations. This sunspot shock, coupled with an active interestrate rule, drives up the real interest rate, which depresses current consumption anddemand. Higher real rates and lower income increase the cost of servicing publicdebt and lower the tax base. A balanced budget then requires higher taxes that, in

11. If the nominal interest rate does not appear in the targeting rules, the aggregate demand equation (1)is block-exogenous and residually determines the nominal interest rate it given the solution for the otherendogenous variables.

12. Equation (9) is a degenerate system in one non-predetermined unknown. The well-known deter-minacy criterion in Blanchard and Kahn (1980) requires the eigenvalue of this system to lie outside theunit circle.


0 20 40 60 80 100τ

DETERMINACY

FIG. 1. Determinacy Regions as a Function of the Steady-State Tax Rate τ under πt = 0 and bt = 0.

turn, increase inflation via the adjustment in the labor market.13 A strict inflationtargeting rule, like (5) with γ = 0, clearly prevents any inflationary sunspot. Yet,indeterminacy can still occur. The reason is that the fiscal–monetary policy mix istoo rigid. The fiscal authority only cares about debt, while the monetary authorityonly cares about inflation. In these circumstances, arbitrary expectations of higheroutput and lower taxes (or vice versa) may leave debt and inflation on target withoutviolating any of the equilibrium conditions.

Figure 1 plots the determinacy regions as a function of the steady-state tax rateτ , holding fixed all other parameters. Interestingly, indeterminacy occurs for therange of steady-state tax rates in which the majority of tax rates (tax revenues asa fraction of GDP) in industrialized economies actually fall. Values of the steady-state tax rate below 20% are of little interest because, holding constant the ratiobetween government spending and GDP, the implied steady-state government debtwould be negative. At the other extreme, values above 40%, while possibly morerelevant in practice, imply an economy lying on the “slippery” slope of the Laffercurve (Trabandt and Uhlig 2011). In particular, the value of the steady-state tax rate

13. Higher taxes reduce labor supply, hence increasing the real wage. Firms therefore face highermarginal costs that drive up inflation.


that maximizes government revenues is

τ Laffer = 1 − 1

1 + (ω + σ−1)= 39.89%,

while the upper threshold for existence of a determinate equilibrium is

τ h = 1 − 1

1 + (ω + σ−1)= 38.65%. (11)

The difference between the two tax rates lies in the correction for the steady-stateconsumption share that accounts for the differential percentage point.

The indeterminacy result under strict targeting rules is very much robust to alter-native calibrations. The right-hand side of (11) is increasing in the sum of the inverseFrisch elasticity of labor supply ω and the adjusted coefficient of relative risk aver-sion. This sum is equal to 0.63 under the baseline calibration. The values for σ−1 andω in Benigno and Woodford (2003) are clearly at the lower end of the spectrum ofwhat macroeconomists generally use. For instance, Galı, Gertler, and Lopez-Salido(2007) choose a benchmark of 5 for the coefficient of relative risk aversion and 1 forthe inverse Frisch elasticity but also consider values of 1 for the former and 2 and 5for the latter. All combinations would imply a value of τ h of at least 67% to ensure adeterminate equilibrium, hence widening the indeterminacy region.

The bottom line is that a strict formulation of inflation and debt targeting rules canendanger the stability of the economy by bringing about indeterminate dynamics. Thispossibility is very much robust to alternative parameterizations. In fact, values of theparameters commonly used in the literature would actually widen the indeterminacyregion, as depicted in Figure 1.

In spite of this indeterminacy result, targeting rules remain appealing for a varietyof other reasons, such as the simplicity of communication and accountability of thepolicymakers’ performances. The point of this paper is not to dismiss the usefulnessof targeting rules based on the indeterminacy result of this section but rather toemphasize the concept of flexibility in the design of the appropriate targeting rules.The remaining sections undertake this task.

3. ESCAPING INDETERMINACY VIA FLEXIBLE TARGETING RULES

This section shows that the idea in Benhabib and Eusepi (2005) and Linnemann(2006) of departing from a strict formulation of policy to help resolve indeterminacyissues also applies to the case of targeting rules. A commitment by either the monetaryor fiscal authority to a flexible targeting rule delivers a determinate equilibrium for awide range of parameter configurations.


3.1 Flexible Inflation Targeting

Suppose that the monetary authority commits to a flexible inflation targeting rule(πt + γ yt = 0), while the fiscal authority continues to follow a strict debt targetingrule (bt = 0). In this case, after replacing the targeting rules into the Phillips curveand the government budget constraint, the system that describes the dynamics of theeconomy becomes

βγEt yt+1 = (κ + γ )yt + κψ(τt − τ ∗t ) (12)

and

β(σ−1 − γ )Et yt+1 = [(1 − β)by + σ−1 − γ ]yt + (1 − β)bτ (τt − τ ∗t ) + ε

ft .

(13)

Solving (12) for the tax gap and plugging the result into (13) yield a modified versionof equation (9) that reads as

β(σ−1 − ωγ )Et yt+1 = (βσ−1ρy − ωγ )yt + εf

t , (14)

where ωγ ≡ γ [1 + (1 − β)(κψ)−1bτ ]. Obviously, if γ = 0, expression (14) coin-cides with (9). In this case, a unique stable solution for yt obtains if and only if|�y| > 1 where

�y ≡ βσ−1ρy − ωγ

β(σ−1 − ωγ ).

Under the baseline calibration, the equilibrium is determinate for any value of γlarger than 0.0093. A very small amount of flexibility in the inflation targeting ruleis sufficient to rule out the sunspot equilibria discussed in the previous section.

The monetary authority is now willing to depart from perfect price stability topartially stabilize output. In particular, the monetary response counteracts the effectsof arbitrary expectations of higher output in the future on current activity. If currentoutput drops, taxes need to increase to keep debt on target. As a consequence, thesunspot is not consistent with both the aggregate supply condition and the governmentbudget constraint.

The minimum value for the flexibility parameter in the inflation targeting rule(5) that guarantees determinacy of the equilibrium (γmin) is not very sensitive tovariations of the steady-state tax rate τ in the interval [18.7%, 38.6%], for whichindeterminacy occurs under strict targeting rules. While γmin is increasing in τ , theamount of flexibility in the inflation targeting rule necessary to ensure determinacyat the peak of the Laffer curve is still rather small: γmin|τ=38.6% = 0.0875. Figure 2tabulates the determinacy region as a function of τ in the relevant interval.

With respect to other parameters, the threshold for the flexibility parameter γmin

that ensures determinacy is increasing in the adjusted coefficient of risk aversion σ−1

but decreasing in the inverse Frisch elasticity of labor supply ω. Hence, alternative


18.7 22.7 26.7 30.7 34.7 38.7

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

τ

γ

DETERMINACY

FIG. 2. Determinacy Regions as a Function of the Steady-State Tax Rate τ under πt + γ yt = 0 (γ > 0) and bt = 0.

combinations of those two parameters in line with the literature, such as those dis-cussed in the previous section, typically preserve the result that a small amount offlexibility to monetary policy brings about a determinate equilibrium. For instance,if both σ−1 and ω are equal to 1, the threshold for determinacy is γmin = 0.0239. Ifσ−1 = 5 and ω = 2, the threshold moves up slightly to γmin = 0.0384.

3.2 Flexible Debt Targeting

Suppose now that the fiscal authority commits to a flexible debt targeting rule(bt + λyt = 0), while the monetary authority follows a strict inflation targeting rule(π t = 0). In this case, replacing the targeting rules into the government budgetconstraint yields

βσ−1Et yt+1 = [(1 − β)by + σ−1 − βλ]yt + λyt−1 + (1 − β)bτ (τt − τ ∗

t ) + εf

t .

(15)

Solving the Phillips curve, which coincides with (7), for the tax gap and replacingthe result in the equation above gives

βσ−1 Et yt+1 = β(σ−1ρy − λ)yt + λyt−1 + εf

t . (16)


Also, in this case, if λ = 0, expressions (16) and (9) coincide. The determinacyproperties of the model depend on the roots of the characteristic equation

P (μ) ≡ μ2 −(ρy − λ

σ−1

)μ− λ

βσ−1= 0.

The equilibrium is determinate if and only if the absolute value of the roots ofP (μ) lie on opposite sides of the unit circle.14 Under the benchmark calibration, theequilibrium is determinate for values of λ ∈ [0.093, 12.667]. A fairly small amountof flexibility in the fiscal response to departures of real activity from its target issufficient to guarantee determinacy of the rational expectations equilibrium. Thefeedback from the output gap, however, should not be excessive to avoid explosivedynamics.15

In this case, it is the fiscal authority that is willing to let debt vary in exchangefor better stabilization of real activity. The fiscal response now counterbalances theeffects of arbitrary expectations of higher output in the future on current activity.Expectations of higher output in the future correspond to lower debt levels. Thefiscal authority increases taxes to reduce debt, generating inflationary pressures. Themonetary authority can keep inflation on target by contracting aggregate demand,thus generating a negative effect on current output. As a consequence, the sunspot isnot consistent with the equilibrium relations of this economy.

Values of the steady-state tax rate higher than 20% shrink the determinacy intervalfor the fiscal policy parameter λ. The thresholds are much more sensitive in thiscase than for flexible inflation targeting rules. If τ = 30%, a determinate equilibriumrequires λ ∈ [0.151, 1.137] (see Figure 3).

A higher coefficient of risk aversion moves the interval for λ that ensures determi-nacy to the right with substantial skewness in favor of the high values. When σ−1 = 1or 5, the interval is, respectively, [0.80, 40.76] or [4.1, 174.1]. A higher inverse Frischelasticity of labor supply widens the determinacy interval. For ω = 1, determinacyoccurs for λ ∈ [0.01, 30.23].

4. DISCUSSION

Flexibility in either monetary or fiscal policy helps avoid unpleasant indeterminacyresults that arise under a strict formulation of targeting rules. Broadly speaking, thisfinding is robust to several alternative parameter configurations. The average value ofthe tax rate, however, appears to be critical for the design of flexible fiscal targetingrules. From this perspective, flexible inflation targeting is less sensitive to uncertaintyabout parameter values.

14. Equation (16) corresponds to a system of two equations in two unknowns, one of which ispredetermined. The online appendix suggests a simple way to check whether the absolute value of theroots of P(μ) splits across the unit circle without the need of calculating their actual values.

15. The determinacy region further widens if the flexible debt targeting rule (6) allows for additionallags of the output gap.


18.7 23.7 28.7 33.7 38.7

15

13.5

12

10.5

9

7.5

6

4.5

3

1.5

0

τ

λ

DETERMINACY

FIG. 3. Determinacy Regions as a Function of the Steady-State Tax Rate τ under πt = 0 and bt + λyt = 0 (λ > 0).

The design of monetary and fiscal policy rules, however, also presents an interestingnormative question: Which of the two types of flexible targeting rules displays moredesirable welfare properties? The short answer is that in this model, flexibility infiscal policy is highly desirable.16

A flexible debt targeting rule with a very high value of λ, conditional on respectingintertemporal solvency, induces much smaller welfare losses than a flexible infla-tion targeting rule.17 The intuition is that in this economy, optimal policy requirespermanent variations of debt to smooth out the distortionary effects of tax changes(a result reminiscent of Barro 1979). A targeting rule that allows for very persistent(permanent in the limit) changes to government debt approximates that outcome.These considerations raise an interesting trade-off between robustness and welfare inthe design of targeting rules for policymaking that deserves further scrutiny.

5. DETERMINACY WITH BALANCED BUDGET

Flexible targeting rules provide a framework for monetary and fiscal policy designthat guarantees transparency and, at the same time, ensures a well-behaved (i.e.,

16. The online appendix presents the full details of the welfare analysis.17. An alternative way to think about this result is that strict inflation targeting is close to optimal while

balanced budget requirements imply high welfare costs.


determinate) macroeconomic outcome. In practice, however, exogenous politicalconsiderations may impair fiscal flexibility. If the political legislation prevents thefiscal authority from varying government debt, one possible solution would be togrant flexibility to the monetary authority. This policy configuration would have theadvantage of being robust to parameter uncertainty, as discussed in the previoussection. Yet, similar constraints could contemporaneously affect monetary policy.18

This section presents a strict debt targeting rule that yields a determinate equilibriumeven under strict inflation targeting.19

The debt variable bt in the model corresponds to the log-deviation from steady-stateof the real value of debt at maturity

bt ≡ log

(bt

b

),

where bt ≡ (1 + it ) Bt/Pt .The debt rule (6) with λ = 0 requires the fiscal authority to adjust real debt so that

its value at maturity remains constant in every period and state of the world. Thisrule, in conjunction with strict inflation targeting, is potentially destabilizing becausemultiple combinations of output and taxes constitute an equilibrium in response to asunspot shock in future real activity.

An alternative specification of fiscal targeting rules would instead aim at stabilizingonly the real stock of debt excluding interest rate payments on outstanding liabilities.The flexible version of this fiscal rule is

bt − it + φyt = 0, (17)

while the strict version obtains for φ = 0.20

The reference of rule (17) to the nominal interest rate implies that the system ofequations characterizing the equilibrium also includes the aggregate demand relation(1). Replacing the targeting rules into the private sector equilibrium conditions andthe government budget constraint yields

Et yt+1 = yt + σ (it − r∗t ), (18)

0 = yt + ψ(τt − τ ∗t ), (19)

18. In an extreme interpretation, the European Monetary Union is an example of this tight policyformulation. The ECB mandate clearly gives a priority to inflation stabilization (narrow inflation targeting).The Stability and Growth Pact requires member countries’ debt to eventually be below a certain threshold(narrow debt targeting).

19. Another option to obtain determinacy with strict inflation and debt targeting would be to resort to adistortionary consumption tax. The results in this case are fairly sensitive to the calibration. See the onlineappendix for details.

20. Benhabib and Eusepi (2005) and Schmitt-Grohe and Uribe (2007) also study balanced budget rulesof this type in models in which the monetary authority follows a Taylor rule. See Ferrero (2009) for anapplication to the case of a currency union.


and

βσ−1Et yt+1 = t[(1 − β)by + σ−1]yt + (1 − β)bτ (τt − τ ∗

t ) + εf

t + βit − bt−1.

(20)

The strict version of rule (17), together with a strict inflation targeting rule, precludesaltogether the possibility of sunspot equilibria driven by arbitrary expectations aboutfuture output. Substituting expression (18) into (20) eliminates the expectation offuture output from the government budget constraint. After also plugging expression(19) into the government budget constraint, output admits a simple closed-formsolution

yt = − 1

βσ−1(1 − ρy)

(bt−1 − ε

yt

), (21)

where εyt ≡ ε

ft + βr∗

t . The determinacy properties of the model then depend uponthe dynamics of the equation for the evolution of debt. After combining the solutionfor the output gap (21) with the Euler equation (1) and the fiscal rule (17), the solutionfor debt is

bt = ρbbt−1 + εbt , (22)

where εbt ≡ ρb[(Et u

ft+1 − u f

t ) + β(Etr∗t+1 − ρyr∗

t )] and

ρb ≡ 1

1 + β(1 − ρy).

The coefficient ρb governs the dynamics of debt and, hence, the determinacy prop-erties of the model under strict inflation targeting rule and the strict debt targetingrule in (17) with φ = 0. In particular, the equilibrium is determinate if and only if|ρb| < 1. This condition is satisfied under the benchmark calibration since |ρy| < 1.Moreover, the determinacy regions as a function of the steady-state tax rate are ex-actly the opposite (τ ∈ [18.7%, 38.6%]) with respect to the case of Section 2 (seeFigure 4).21

The intuition for why the equilibrium is determinate under the strict formulationof the debt targeting rule (17) even with strict inflation targeting is that, in this case,the modified fiscal rule is very similar to a special case of a flexible debt targetingrule analyzed in Section 3.2. Substituting the aggregate demand equation (1) understrict inflation targeting into the fiscal targeting rule yields

bt + σ−1 (yt − Et yt+1) − r∗t = 0.

21. The robustness analysis to different values of the coefficient of risk aversion and the inverse of theFrisch elasticity of labor supply coincides with the results of Section 2.


0 20 40 60 80 100τ

DETERMINACY

FIG. 4. Determinacy Regions as a Function of the Steady-State Tax Rate τ under πt = 0 and bt − it = 0.

Moreover, as mentioned above, the expectation term in the equation above exactlyoffsets any sunspot shock in the government budget constraint that affects futureoutput. As in Leeper (1991), a combination of active monetary policy (strict inflationtargeting) and passive fiscal policy (strict debt targeting) guarantees determinacy ofthe rational expectation equilibrium.

Flexibility in the modified debt targeting rule (17) displays properties comparableto the case discussed in Section 3.2. If φ > 0, the solution for debt is again a first-orderautoregressive process

bt = �bbt−1 + εbt , (23)

where now εbt ≡ �b{[Et u

ft+1 − (1 + φ/σ−1)u f

t ] + β(Etr∗t+1 − ρyr∗

t )}(1 + φ/σ−1)and

�b ≡ 1 + φ/σ−1

1 + β[(1 − ρy) + φ/σ−1].

Clearly, if φ = 0, (22) and (23) coincide since �b = ρb and εbt = εb

t . Determinacyrequires that |�b| < 1. Since under the baseline calibration |ρy | < 1, the equilibrium isdeterminate unless φ is too high (again the threshold is 12.667). In this circumstance,


18.7 22.7 26.7 30.7 34.7 38.7

13

11.75

10.5

9.25

8

6.75

5.5

4.25

3

1.75

0.5

τ

φ

DETERMINACY

FIG. 5. Determinacy Regions as a Function of the Steady-State Tax Rate τ under πt = 0 and bt − it + φyt = 0 (φ > 0).

debt would have an explosive dynamic and no rational expectations equilibriumwould exist.

The robustness analysis to variations in the coefficient of risk aversion and the in-verse Frisch elasticity of labor supply essentially coincides with the previous formu-lation of flexible debt targeting rules as far as the upper threshold for φ is concerned.The key difference is that changes in parameter values do not influence the lowerthreshold. Balancing the budget via a strict version of rule (17) is always a viablepossibility.

Figure 5 portraits the determinacy region as a function of τ for φ > 0. Flexibledebt targeting remains feasible for very much the same range of steady-state tax ratesassociated with determinacy under strict debt targeting.22

6. CONCLUSIONS

A flexible formulation of targeting rules for monetary and fiscal policy preventsindeterminate equilibria for a wide range of parameter configurations and steady-state tax rates. Conversely, a combination of strict debt and inflation targeting gives

22. Also, in this case, the welfare gains of granting flexibility to the fiscal authority rather than to themonetary authority remain substantial.


rise to indeterminacy because the policy mix is too rigid. Arbitrary expectations ofhigher future output and lower taxes (or vice versa) may leave debt and inflation ontarget without violating the private sector equilibrium conditions and the governmentbudget constraint.

Flexibility in either inflation or debt targeting avoids this unpleasant outcomeby counteracting the effects of arbitrary expectations of higher output in the futureon current activity. Which form of flexibility is more appropriate depends on theobjective of policy design: flexible inflation targeting is more robust to parameteruncertainty, but flexible debt targeting displays more desirable welfare properties.

Determinacy with a balanced budget rule under strict inflation targeting may occurif the fiscal authority stabilizes debt net of interest rate spending. This alternativeformulation of policy corresponds to a special flexible debt targeting rule in whicharbitrary expectations of future output do not directly affect fiscal solvency. Deter-minacy then obtains, provided that debt dynamics are not explosive.

The broad message of the paper is that considering the design of fiscal and mon-etary policy rules in isolation might lead to undesirable equilibrium outcomes. Theinteraction between the two forms of policy is a crucial dimension to evaluate incrafting a graceful exit from the extraordinary measures taken in response to thefinancial crisis of 2007–2008.

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the advantage of flexible targeting rules

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