Credible Forward Guidance∗
Quentin Batista†
MIT

Taisuke Nakata‡
University of Tokyo

Takeki Sunakawa§
Hitotsubashi University

First Draft: November 2018
This Draft: April 2023

Abstract
How can the central bank credibly implement a “lower-for-longer” strategy? To answer thisquestion, we analyze a series of optimal sustainable policy problems—indexed by the durationof reputational loss—in a sticky-price model with an effective lower bound (ELB) constraint
on nominal interest rates. We find that, even when the central bank lacks commitment, thecentral bank can still credibly keep the policy rate at the ELB for an extended period—
though not as extended as under the optimal commitment policy—and meaningfully mitigate
the adverse effects of the ELB constraint on economic activity.

JEL: E32, E52, E61, E62, E63
Keywords: Credibility, Effective Lower Bound, Forward Guidance, Sustainable Plan, TimeConsistency.

∗

We thank Roberto Billi, Timothy Hills, Narayana Kocherlakota, Kiminori Matsuyama, Sebastian Schmidt,
Francesco Zanetti, and seminar participants at the 2019 Workshop of the Australasian Macroeconomics Society (WAMS), the 2020 World Congress, the 5th End-of-Year Conference held at the Canon Institute of
Global Studies, the Forward Guidance Workshop 2021 Oxford and MMF, VEAMS, Asian Development Bank
Institute, Bank of Japan, Board of Governors of the Federal Reserve System, Central Bank of Peru, Hitotsubashi University, Hosei University, Kobe University, National Graduate Institute for Policy Studies, National
University of Singapore, and University of Tokyo for comments and suggestions. We also thank Katherine
Arnold and Philip Coyle for their editorial assistance. Taisuke Nakata is supported by JSPS Grant-in-Aid for
Scientific Research (KAKENHI), Project Numbers 19K24682 and 22H04927. Takeki Sunakawa is supported
by JSPS Grant-in-Aid for Scientific Research (KAKENHI), Project Numbers 18K12743 and 21K01480.
†
MIT Sloan School of Management, 100 Main Street Cambridge, MA 02142;
Email:
batista.quent@gmail.com‡
Faculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033; Email:taisuke.nakata@e.u-tokyo.ac.jp§
Faculty of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, 186-8601; Email:t.sunakawa@r.hit-u.ac.jp1

1

Introduction
In order to anchor longer-term inflation expectations at this level, the Committee
seeks to achieve inflation that averages 2 percent over time, and therefore judges
that, following periods when inflation has been running persistently below 2 percent,appropriate monetary policy will likely aim to achieve inflation moderately above
2 percent for some time.
Board of Governors of the Federal Reserve System 1
To maintain the symmetry of its inflation target, the Governing Council recognises
the importance of taking into account the implications of the effective lower bound.
In particular, when the economy is close to the lower bound, this requires especiallyforceful or persistent monetary policy measures to avoid negative deviations fromthe inflation target becoming entrenched. This may also imply a transitory period
in which inflation is moderately above target.
European Central Bank 2
Developing effective strategies to manage the adverse consequences of the effective lower

bound (ELB) constraint on nominal interest rates is an important task for economists andcentral bankers. In forward-looking models with an ELB, the commitment to keeping thepolicy rate at the ELB for an extended period—and temporarily overshooting inflation andoutput targets—is known to be effective in stimulating economic activity during a deep recession, as the anticipation of an overheated economy leads forward-looking households and firms
to increase consumption and set higher prices.3 We refer to this type of policy as overheating
commitment or lower-for-longer policy in this paper.
Recently, as shown in the quotes above, some central banks such as the Federal Reserve
Board and the European Central Bank have become more willing to allow temporary overshooting of inflation above the target. (The Bank of Japan has also adopted this type ofpolicy since November 2016.) While these policies cannot be strictly interpreted as a form ofcommitment, they share common elements with overheating commitment or lower-for-longer
policy.
One key argument against overheating commitment policy is its potential time-inconsistency.
Ex ante, it is desirable to promise to overheat the economy in the future, as the expectationsof future overheating stimulate inflation and output when the economy faces headwinds and1“2020 Statement on Longer-Run Goals and Monetary Policy Strategy” https://www.federalreserve.
gov/monetarypolicy/review-of-monetary-policy-strategy-tools-and-communications-statement-onlonger-run-goals-monetary-policy-strategy.htm.
2“The ECB’s monetary policy strategy statement” https://www.ecb.europa.eu/home/search/review/
html/ecb.strategyreview_monpol_strategy_statement.en.html.
3See, for example, Reifschneider and Williams (2000); Eggertsson and Woodford (2003); Jung, Teranishi,
and Watanabe (2005); Adam and Billi (2006).

2the ELB is a binding constraint. However, once the headwinds dissipate, the central bankwill have an incentive to renege on the promise of overheating the economy by raising thepolicy rate, because the overheating is ex post undesirable. Then, a natural question would
be how much overheating is needed to stimulate the economy while keeping monetary policy
credible.
In this paper, we study credible overheating commitment policies in a sticky-price modelwith the ELB with an eye towards understanding the best allocations the central bank cancredibly achieve when the optimal commitment policy is not credible. Specifically, we formulate and solve a series of optimal sustainable policy problems in which the central bank chooses
state-contingent allocations to maximize welfare subject to not only private-sector equilibrium
conditions, but also an incentive compatibility constraint—known as the sustainability constraint. The sustainability constraint requires that the continuation value associated withthe chosen state-contingent allocation has to be at least as large as the continuation value
associated with deviating from that allocation—and falling into a discretionary regime for N
periods—at any time and after any history of shocks. Under certain conditions discussed in
Nakata (2018), the sustainability constraint does not bind and the optimal sustainable policy
coincides with the optimal commitment policy.4 Our main interest is to characterize optimal
sustainable policies when the sustainability constraint occasionally binds.
Our main result is that, even when the optimal commitment policy is not credible, thecentral bank can still credibly keep the policy rate at the ELB for an extended period in
the aftermath of a crisis—though not as extended as under optimal commitment policy. As
in optimal commitment policy, such lower-for-longer policy generates a temporary post-crisis
overheating of the economy and mitigates the declines in output and inflation in a crisis
through expectations. Under reasonable assumptions regarding how long the central banksuffers from a loss of reputation after reneging on the promise of lower-for-longer, the welfare
cost of the ELB constraint is substantially lower under an optimal sustainable policy thanunder optimal discretionary policy.
One key feature of optimal sustainable policies is that they are less history dependent
than optimal commitment policy. As discussed in detail by Eggertsson and Woodford (2003),
a key feature of optimal commitment policy is history dependence. In particular, under
optimal commitment policy, the additional period at which to keep the policy rate at the
ELB in the aftermath of a crisis—as well as the magnitudes of the output and inflation
overshoots—increases as the realized crisis shock duration increases. When the reputational
4
Specifically, in Nakata (2018), if the central bank were to renege on the promise of overheating the
economy in the aftermath of a crisis, it would lose reputation and private-sector agents would not believe
similar promises in future crises. If private-sector agents do not believe the central bank’s promise to overheat
the economy, future ELB episodes will be associated with large declines in inflation and output. Thus, concern
for maintaining reputation gives the central bank an incentive to fulfill the promise of keeping the lower-forlonger promise. According to Nakata (2018), this incentive to maintain reputation dominates the short-run
incentive to eliminate the overheating of the economy—and as a result, the optimal commitment policy iscredible—if that the policy rate is expected to fall into the ELB in the future with sufficient frequency andthe loss of reputation lasts for a sufficiently long duration.

3

force is strong, the policy rate path under optimal sustainable policies exhibit qualitatively
similar history dependence, though the degree of history dependence is weaker than under
optimal commitment policy. When the reputational force is sufficiently weak, the policy ratepaths under optimal sustainable policies do not feature any history dependence. That is, theadditional ELB duration does not depend on the realized crisis shock duration.
We also analyze similarities between optimal sustainable policies and average inflation
targeting policies. Average inflation targeting can be seen as a way of implementing overheating commitment policies (see Budianto, Nakata, and Schmidt, 2023). When the averaging
window is one period, average inflation targeting coincides with the standard inflation targeting, which acts similarly to the optimal discretionary policy. When the averaging window
is infinity, average inflation targeting coincides with the price-level targeting, which acts similarly to the optimal commitment policy. With the averaging window in between one andinfinity, average inflation targeting can capture a variety of outcomes with differing degrees
of commitment, just as optimal sustainable policies can capture a variety of outcomes withthe punishment duration. Therefore, our optimal sustainable policies can be interpreted asoffering a microfoundation of average inflation targeting policies.
Our optimal sustainable policies are of interest to central banks for two reasons. First,
by construction, the optimal sustainable policies are time-consistent; thus, it is immune tothe criticism that the promised overshoot of inflation and output associated with any lowerfor-longer strategies may not be credible. Second, when the duration of reputational loss
is sufficiently short, the policy rate path under optimal sustainable policies are not history
dependent. Thus, it overcomes the criticism that, because the policy rate path associatedwith a lower-for-longer strategy is complex, it is difficult for central banks to clearly explainthese strategies to the public.
Our paper builds on the literature on optimal monetary policy in the New Keynesianmodel with the ELB. This literature has demonstrated the effectiveness of lower-for-longer
policies in stimulating the economy at the ELB, assuming that the central bank is equippedwith an explicit commitment technology (Eggertsson and Woodford (2003); Jung, Teranishi,
and Watanabe (2005); Adam and Billi (2006); and Nakov (2008)). Our paper contributes
to this body of work by characterizing optimal sustainable policies in a model with the ELB
and showing that the central bank can credibly engage in lower-for-longer policies even in theabsence of an explicit commitment technology.
Within the literature on optimal policy and the ELB, some authors have explored ways toimplement lower-for-longer policies at the ELB in a time-consistent way. Eggertsson (2006)
and Burgert and Schmidt (2014) show that in models with non-Ricardian fiscal policy, a
discretionary government can provide incentives to a future government to keep the policy
rate at the ELB for longer by adopting expansionary fiscal policy and raising the nominal levelof government debt. Jeanne and Svensson (2007), Berriel and Mendes (2015), and Bhattarai,
Eggertsson, and Gafarov (2013) show that central banks’ balance sheet policies can act as4

a commitment device that allows the central bank to credibly implement lower-for-longer
policies. Billi (2017) and Nakata and Schmidt (2019b) analyze policy delegation in models
with the ELB, showing that lower-for-longer policies can be implemented in a time-consistent
way if the discretionary central bank’s standard dual-mandate objective function is replaced
by a nominal-income stabilization objective or augmented with an interest-rate smoothing
objective, respectively. Unlike these papers that either introduce a new policy instrument or
modify the central bank’s objective function, we use reputation to achieve lower-for-longer
policies in a time-consistent way.
Our paper is closely related to Nakata (2018) and Walsh (2018). Nakata (2018) has shownthat optimal commitment policy in the New Keynesian model with the ELB can be made timeconsistent by a particular trigger strategy capturing the reputational concern of the central
bank. Our paper is different from Nakata (2018) because we study the best allocations thecentral bank can credibly achieve when the optimal commitment policy is not credible, whereas
Nakata (2018) characterizes the conditions under which the optimal commitment policy iscredible. Walsh (2018) examines credibility of simple policy rules with forward guidance—
those that keep the policy rate at the ELB for a fixed number of periods after crises—andreaches a conclusion similar to that of Nakata (2018).5 Our paper is different from Walsh(2018) because we characterize the optimal allocation the central bank can credibly achieve
subject to a sustainability constraint, whereas Walsh (2018) studies credibility of simple policy
rules that may or may not be optimal.6 It turns out that there is an interesting relationshipbetween our optimal sustainable policies and the forward guidance policy of Walsh (2018),
which will be discussed in detail in Section 4.3.7
This paper is also closely related to the work of Dong and Young (2019) which uses therecursive method of Abreu, Pearce, and Stacchetti (1990), Chang (1998), and Phelan andStacchetti (2001) to characterize the entire set of sustainable plans in a fully nonlinear New
Keynesian model with the ELB. We discuss this paper in detail in Section 4.2.
Finally, our paper is related to a set of papers that characterize optimal allocations inmacroeconomic models with a sustainability constraint. Kehoe and Perri (2002) characterize
the optimal allocation in an international business cycle model in which a deviation from thepromised plan would push the economy to autarky. Fujiwara, Kam, and Sunakawa (2019b)
study the optimal sustainable policy in a two-country model in which the deviation from thepromised cooperative plan would push the countries into a non-cooperative regime. The most5

See also Sukeda (2018), which extends the analysis of Walsh (2018) to a model with a discounted Euler
equation and a discounted Phillips curve.
6
Nakata (2018) and Walsh (2018) in turn build on earlier work of reputation in macroeconomics, including
Barro and Gordon (1983), Rogoff (1987), Chari and Kehoe (1990), Chang (1998), and Phelan and Stacchetti(2001), among many others. Recent contributions include Kurozumi (2008) and Loisel (2008).
7
See also Barthélemy and Mengus (2018) who examine sustainability of optimal commitment policy ina model with the ELB constraint in which the central bank’s objective function—either a benevolent orconservative kind—is unknown to private-sector agents and there is an inflationary bias. In their model, thebenevolent central bank can make the optimal commitment policy sustainable by raising inflation prior to aliquidity trap and signaling its type to private-sector agents.

5

closely related to our paper is Sunakawa (2015) who characterizes the optimal sustainable
policy in a New Keynesian model with cost-push shocks but without the ELB constraint.
Our paper applies the same analytical framework and methodology used in these papers tomodels with the ELB.
The rest of the paper is organized as follows. Section 2 describes the model and the central
bank’s optimization problems. Section 3 presents the results. Section 4 provides additional
discussions and results. Section 5 concludes.

2

Model

2.1

Private sector

Our main model is a semi-loglinear New Keynesian model with a static Phillips curve.
The private-sector equilibrium conditions of this model are given by:

yt (st ) = Et yt+1 (st+1 ) − σ(it (st ) − Et πt+1 (st+1 ) − r∗ ) + st(1)

πt (st ) = κyt (st ) + βEt πt+1 (st+1 )

(2)
(3)

t

it (s ) ≥ iELB

where yt is output, πt is inflation, and it is the policy rate. Equations (1) and (2) are theEuler equation and the Phillips curve, respectively. Inequality (3) imposes the ELB constraint,
denoted by iELB , on the policy rate. σ is the intertemporal elasticity of substitution, r∗ > 0 is
the natural rate of interest at the deterministic steady state, and κ is the slope of the Phillips
curve.8
The exogenous shock in the Euler equation, st , is the natural rate shock and st denotes
a history of shocks up to time t. That is, st := {sk }tk=1 . Because there is uncertainty,
allocations are state-contingent and depend on st . We refer to the state-contingent sequence ofconsumption, inflation, and the nominal interest rate, {yt (st ), πt (st ), it (st )}∞t=1 , as an outcome.
Given a process for st , an outcome is said to be competitive if, for all t ≥ 1 and st ∈ St , (i)yt (st ) ∈ R, πt (st ) ∈ R, it (st ) ∈ R, where R denotes a set of real numbers, and (ii) equations(1)-(3) are satisfied.
We assume that st follows a two-state Markov process with st = r∗ > 0 in the “high” or“normal” state and st = rc < 0 in the “low” or “crisis” state. The probability of moving fromthe high/normal state to the low/crisis state is denoted by pH and will be referred to as thecrisis frequency, whereas the probability of moving from the low/crisis state to the low/crisis
8
In the working paper version, we used the model with a static Phillips curve as well as models withdiscounted Euler equation and discounted Phillips curve. See also Bilbiie (2019) who uses a model with astatic Phillips curve. See Nakata, Ogaki, Schmidt, and Yoo (2019) and Levin and Sinha (2020a,b) for asystematic analysis of optimal commitment policies with varying degrees of discounting in the Euler equation
and Phillips curve.

6

state is denoted by pL and will be referred to as the crisis persistence. Following Nakata(2018), we allow pH to be non-zero, which opens up the possibility for a reputational concern
to make lower-for-longer policies credible.
The central bank’s value at period t is given by
Vt (st ) := Et∞Xβ j u πt+j (st+j ), yt+j (st+j )

(4)

j=0

where the per-period objective function is given by the following function.
u(π, y) := −i1h 2π + λy 22(5)This quadratic objective function can be obtained as the second-order approximation to thehousehold’s welfare.9 For any outcome, there is an associated state-contingent sequence ofvalues, {Vt (st )}∞t=1 , which will be referred to as the value sequence.

2.2

Central bank

We will consider three classes of competitive outcomes that differ in how the central banksets its interest rate policy: the discretionary outcome, the commitment outcome, and thesustainable outcomes. Appendix B explains the optimization problem the central bank faces
in detail.
2.2.1

Discretionary outcome

At each time t, the discretionary central bank’s optimization problem is to choose {yt ,πt , it } to maximize the value today, taking as given the value function (Wt+1 (·)) and policy
functions for inflation and output (πt+1 (·) and yt+1 (·)) in the next period. That is,
Wt (st ) = max

πt ,yt ,itu(yt , πt ) + βEt Wt+1 (st+1 ),(6)

subject to equations (1), (2), and (3).
Let {Wdo (·), πdo (·), ydo (·), ido (·)} be the set of time-invariant value and policy functions that solve the Bellman equation above and in which the ELB binds only in the crisis
state.10 They are functions of today’s shock realization, st . The discretionary outcome isdefined as, and denoted by, the state-contingent sequence of output, inflation, and the poltt
icy rate, {ydo,t (st ), πdo,t (st ), ido,t (st )}∞t=1 such that ydo,t (s ) = ydo (st ), πdo,t (s ) = πdo (st ),and ido,t (st ) = ido (st ) and the discretionary value sequence is defined as, and denoted by,
9See, for example, Woodford (2003) and Galí (2015).
There also exists a time-invariant solution to this discretionary government’s problem in which the ELBbinds in both states. See Armenter (2017), Nakata (2018), and Nakata and Schmidt (2019a) for extensive
analyses of such deflationary Markov-perfect equilibrium.
10

7

t
{Vdo,t (st )}∞t=1 such that Vdo,t (s ) = Wdo (st ). We will also refer to the discretionary outcome

as the outcome under the optimal discretionary policy (ODP).
2.2.2

Commitment outcome

At the beginning of time one, the central bank with commitment technology chooses astate-contingent allocation, {yt (st ), πt (st ), it (st )}∞t=1 , to maximize the time-one value. Thatis,Vc,1 (s1 ) =max{yt (st ),πt (st ),it (st )}∞
t=1

E1

∞
X

β t−1 u(yt (st ), πt (st )),

(7)

t=1

subject to equations (1), (2), and (3) for all t ≥ 1 and after all histories of shocks st . The
commitment outcome, or the Ramsey outcome, is defined as the solution to this optimization
problem. In other words, the commitment outcome is a competitive outcome with the highest
time-one value. We denote the commitment outcome by {yco,t (st ), πco,t (st ), ico,t (st )}∞t=1 . The
value sequence associated with the commitment outcome is denoted by {Vco,t (st )}∞t=1 andwill be referred to as the commitment value sequence. We will also refer to the commitment
outcome as the outcome under the optimal commitment policy (OCP).
2.2.3

Sustainable outcomes

At the beginning of time one, the central bank chooses a state-contingent allocation,{yt (st ), πt (st ), it (st )}∞t=1 , to maximize the time-one value:
Vso,1 (s1 ) =max{yt (st ),πt (st ),it (st )}∞
t=1

E1

∞
X

β t−1 u(yt , πt ),

(8)

t=1

subject to equations (1), (2), and (3), and the following sustainability constraint,
Et

∞
X

N
β k u(yt+k (st+k ), πt+k (st+k )) ≥ Wdo(st ),(9)k=0

for all t ≥ 1 and after all histories of shocks, st . The left-hand side of the sustainability
constraint is the continuation value of implementing a chosen state-contingent allocation at
N (s ), is the continuation value if the central banktime t after st . The right-hand side, Wdotdeviates from the chosen state-contingent allocation, with N indicating how many periods
it takes for the central bank to restore its lost reputation (“punishment” duration). During
the periods of reputational loss, the central bank cannot engage in state-contingent policies.
That is, the central bank has to act under discretion.
N (s ) is recursively defined as follows. For N = 0,
Wdo
t
0
Wdo
(s) := Vso,1 (s),0πdo(s) := πso,1 (s),

8

0
ydo
(s) := yso,1 (s).

In this case with N = 0, the punishment duration is zero and the central bank is not allowed
to deviate from the sustainable outcome. For any N > 0,NWdo(s) = max
π,y,i

N −1 0u(π, y) + βE[Wdo(s )|s]where the maximization is subject to the private-sector equilibrium conditions, taking as given
N −1
N −1
N −1
the value and policy functions for the next period (that is, Wdo(·), πdo(·), and ydo(·)).

Note that the sustainability constraint has to be respected each period and for each historyof shocks, just as the Euler equation, the Phillips curve, and the ELB constraint have to berespected each period and for each history of shock. The sustainable outcome with N -period
reputational loss is defined as the solution to this infinite-horizon optimization problem. We
will also refer to the sustainable outcome with N -period reputational loss as the outcome
under the optimal sustainable policy (OSP) with N -period reputational loss.
N (s ), is determined jointly with the sustainable outNote that the punishment value, Wdotcome, except when N = ∞. When N = ∞, the punishment lasts forever and its value is givenby the discretionary value, Wdo (s), which is independent of the sustainable outcome. For any
finite N , the central bank eventually restores its reputation and the economy returns to theallocations consistent with the sustainable outcome. Thus, the punishment value and thesustainable outcome are not independent of each other. All else equal, an increase (decrease)in the value associated with the sustainable outcome implies an increase (decrease) in thepunishment value.
As described in detail in Appendix A, once the sustainable outcome is computed from theoptimization problem above, we can construct a plan—a pair of central bank and privatesector strategies—that induces the sustainable outcome and that has a trigger-type structure.
In particular, we can construct a revert-to-discretion plan in which (i) the economy follows
the sustainable outcome as long as the central bank has never deviated from the policy rate
path consistent with the sustainable outcome in the past, and (ii) the economy follows thediscretionary outcome, or a temporary deviation to a discretionary regime, otherwise. By
construction, such a revert-to-discretion plan is credible, meaning that neither the central
bank nor private-sector agents have incentives to deviate from the instructions given by thestrategies. The central bank does not have an incentive to deviate from the policy rate
path consistent with the sustainable outcome because the sustainability constraint is imposedon the central bank’s optimization problem, ensuring that the continuation value under thesustainable outcome is at least as large as the punishment continuation value. Private-sector
agents do not have incentives to deviate from the private-sector strategy because the Euler
equation and the Phillips curves are satisfied, meaning that the output and inflation are
consistent with their optimizing behaviors given the central bank strategy. Even though
the deviation does not occur in equilibrium, the specification of what would happen if thecentral bank were to deviate from the sustainable outcome does affect what happens under
the sustainable outcome.
9

If the sustainability constraint does not bind at any time t and after any histories of shocks,
the sustainable outcome coincides with the commitment outcome. Also, if the sustainability
constraint always binds—which happens, for example, when the punishment length (N ) iszero or when the crisis frequency (pH ) is zero—the sustainable outcomes coincides with thediscretionary outcome. Our main interest is those cases in which the sustainability constraint
occasionally binds.

2.3

Parameter values

Table 1 shows the baseline parameter values. The quarterly frequency of crises is set to0.5/100 (=2/400). This choice is motivated by the fact that, in the United States, there havebeen two large crises that pushed the short-term nominal interest rate to the ELB over roughly
the last 100 years (400 quarters) since the creation of the Federal Reserve System. The crisis
shock persistence is set to 2/3, which implies the expected duration of the crisis shock of
3 quarters. The intertemporal elasticity of substitution, σ, is set to 1. The natural rate ofinterest in the crisis state, rc , and the slope of the forward-looking Phillips curve, κ, are chosen
so that output declines 7 percent and inflation declines 1 percentage point (annualized) inthe crisis state under the optimal discretionary policy. This severity of the crisis is consistent
with that considered in Boneva, Braun, and Waki (2016) and Nakata (2018), and is intended
to capture the severity of the Great Recession of 2007-2009 in the United States. λ is set
to 0.0012, a value consistent with a microfounded objective function when the elsticity of
substitution across intermediate goods is set to 10.
Table 1: Baseline Parameter Values
Parameter
β
σ
κ
λ
pH
pL
r∗
rc
N

Description
Discount rate
Intertemporal elasticity of substitution
Slope of the Phillips curve
Relative weight on output volatility
Crisis shock frequency
Crisis shock persistence
Natural rate in the normal state
Natural rate in the crisis state
Punishment length

Parameter Value
1
1+0.0075 ≈ 0.9925
1
0.012
0.0012
0.5/100
2/3
3/400−0.030[16, 40, ∞]We consider three values for the duration of reputational loss (16, 40, and ∞), which arechosen to cover qualitatively distinct cases that can arise. We put these values into perspective
in Section 3.3.

2.4

Solution method

The model is highly nonlinear, featuring two inequality constraints—the ELB constraint
and the sustainability constraint—and cannot be solved analytically. Following Marcet and10Marimon (2019), Kehoe and Perri (2002) and Sunakawa (2015), we recursify the infinitehorizon optimization problem of the central bank into a saddle-point functional equation
using the Lagrange multipliers on the Euler equation and the Phillips curve as pseudo-statevariables. We then apply a projection method to find the set of time-invariant policy functions
that solve the saddle-point functional equation. Appendix C describes the details of thesolution method as well as its accuracy.

3

Results

3.1

Dynamics

Figure 1 shows the dynamics of the economy under the ODP, the OCP, and OSPs with
N = [16, 40, ∞].
In this figure, the crisis shock hits the economy at time 1 and stays there until time 8.
The crisis shock disappears at time 9 and the economy is in the normal state from that point
on.
Figure 1: Dynamics

Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The policy rate and the inflation rate are expressed in annualized percent.
The output gap is expressed in percent.

Under the ODP—shown by the solid red lines—the central bank keeps the policy rate
at the ELB as long as the crisis shock persists and raises the policy rate immediately after
the crisis shock disappears. Under the OCP—shown by the solid black lines—the central
bank keeps the policy rate at the ELB even after the crisis shock disappears, engineering theovershooting of inflation and output above their targets. Since households are forward looking,the anticipation of high inflation and high output in the aftermath of the crisis stimulates
economic activity during the crisis. The declines in inflation and output are substantiallysmaller under the OCP than under the ODP.
11

Figure 2: Value of fulfilling versus reneging on the promised allocations

Note: OSP stands for optimal sustainable policy.

The allocations under the OSP with N = ∞ are identical to those under the OCP in thiscrisis scenario. As shown in the left panel of Figure 2, the value under the OSP with N = ∞—shown by the solid black—is always above the value in case the central bank deviates from the
OSP with N = ∞—shown by the dashed red line. That is, the sustainability constraint does
not bind. In our calibration, the crisis shock is sufficiently frequent so that the cost of being
unable to use lower-for-longer policies in the future forever outweighs the benefit of eliminating
the temporary overshooting of inflation and output targets. This result is consistent with thefinding of Nakata (2018) that a very small probability of being hit by the crisis shock suffices
to make the OCP credible.
When the loss of reputation is not as long, the cost of reneging on the lower-for-longer
promise in the aftermath of the crisis shock is smaller. In other words, the continuation
value in case of deviation is higher with a smaller N . The middle panel of Figure 2 shows
the value of the OSP with N = 40 and the value of deviating from the OSP with N = 40
(solid black and dashed red lines, respectively). According to the panel, the sustainability
constraint binds right after the crisis shock disappears, limiting the magnitude of the inflationand output overshooting in the aftermath of the crisis. The smaller overshoot means that
inflation and output decline by more during the crisis under the OSP with N = 40 than under
the OCP and the OSP with N = ∞. However, the declines in inflation and output are stillmuch smaller under the OSP with N = 40 than under the ODP.
Similarly, the sustainability constraint binds right after the crisis shock disappears under
the OSP with N = 16, as can be seen in the right panel of Figure 2, limiting the magnitudesof the inflation and output overshoots. The overshoots in the aftermath of the crisis are
smaller—and as a result, the declines in inflation and output are larger—under the OSP with
N = 16 than under the OSP with N = 40. Even with N = 16, the declines in inflation andoutput are still much smaller under the OSP than under the ODP.
12

Reflecting the less severe crisis under the OSP with N = 16 and N = 40, the welfare costof the ELB—shown in Table 2—is substantially lower under these OSPs than under the ODP.
With N = 40, welfare cost of the ELB is about 20 percent of that under the ODP and is only
slightly larger than under the OCP. Even with N = 16, the welfare cost of the ELB is only
about one third of that under the ODP.
Table 2: Welfare Cost of the ELB
Optimal commitment policy
Optimal sustainable policy
with N = ∞with N = 40
with N = 16
Optimal discretionary policy

abs(E[V ])38.1 (0.18)
38.1 (0.18)
40.2 (0.19)
71.3 (0.34)
207.7 (1.00)

Note: Numbers in parentheses are the welfare cost of the ELB relative to that under the optimal discretionary
policy.

3.2

History Dependence

Figure 3 shows the dynamics of the policy rate, the output gap, and inflation—displayedin the top, middle, and bottom rows, respectively—under three alternative realized durationsof the crisis shock. The first, second, and third columns are for the realized crisis shock
duration of 1, 4, and 8 quarters, respectively.
Under the OCP—shown by the solid black lines—the additional ELB duration is 2, 5, and6 quarters when the realized crisis shock duration is 1, 4, and 8 quarters, respectively, as canbe seen in the bottom panels of Figure 3. The magnitude of the output (inflation) overshootis 1.9 (0.15) percentage points, 3.5 (0.39) percentage points, and 4.1 (0.49) percentage points
when the realized crisis shock duration is 1, 4, and 8 quarters, respectively, as can be seenin the middle (bottom) panels of Figure 3. Thus, both the additional ELB duration andthe sizes of the inflation and output overshoots depend on the realized crisis shock duration.
This dependence can be seen in Figure 4, which shows how the additional ELB duration andthe size of the inflation and output overshoot vary with the realized crisis shock duration.11
The OSP with N = ∞ exihibits history dependence that is identical to the OCP, as thesustainability constraint does not bind in any state and the allocations under the OCP and
the OSP with N = ∞ are identical.
The history dependence of the OCP is in sharp contrast with the lack of history dependence
in the ODP. Under the ODP—shown by the solid red lines in Figure 3 and 4—the additional11

In computing the additional ELB duration and the sizes of the output overshoots, we assume that, prior
to the crisis shock, the economy has been in the normal state for some time and the Lagrange multipliers on
the Euler equation and the Phillips curve is zero in the period right before the crisis shock materializes.

13Figure 3: History Dependence (I)—Dynamics with Alternative Realized Crisis Shock Durations—

Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The policy rate and the inflation rate are expressed in annualized percent.

ELB duration and the sizes of the inflation and output overshoots do not depend on therealized crisis shock duration.
The OSP with N = 40 is history-dependent, but less so than the OCP or the OSP
with N = ∞. Under the OSP with N = 40, as the realized crisis shock duration increases
14Figure 4: History Dependence (II)Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The output gap is expressed in percent.

from 1 quarter to 2 quarters, the additional ELB duration increases from 2 quarters to 4
quarters. Thereafter, the additional ELB duration stays at 4 quarters. The sizes of theinflation and output overshoots increase initially as the realized crisis shock duration increases,
similarly to what happens under the OCP. However, as the shock duration exceeds above 3,
the sustainability constraint starts binding and the sizes of the output and inflation overshoots
stays stable, albeit not perefectly constant. Overall, the policy rate path as well as the sizesof the inflation and output overshoots are less history dependent than under the OCP.
Under the OSP with N = 16, the additional ELB duration is 2 quarters regardless of therealized shock duration. In other words, the policy rate path is not history dependent at all.
While the sizes of the output and inflation overshoots are not perfectly constant but remain
stable as the realized shock duration increases.

3.3

On “reasonable” duration of reputational loss

When the loss of reputation lasts for a long time, the power of reputation is strong and
OSPs resemble the OCP. When the loss of reputation lasts for a short time, the power of
reputation is weak and OSPs resemble the ODP. A natural question that arises is what
reasonable values of the duration of reputational loss are.
We discuss two ways to think about this question.
One way to think about the reasonable duration of reputational loss is to hypotheticallyask how long it might take for a central bank to restore its reputation once it loses it. The
reasonable value of the duration of reputational loss based on this thought experiment may
depend on various factors: whether one believes the central bank’s reputation is individualspecific or institution-specific and the tenure duration of the central bank’s governors or chairs,
if one believes in the individual specific nature of reputation.
15

The assumption that the central bank can restore its reputation after a finite number of
periods can be motivated by the fact that the tenure of governorship at central banks is finite
as well as the possibility that reputation may be specific to the leader of the central bank,as opposed to the institution. As shown in Table 3, the average tenure of the governorshipin central banks in economies that have recently faced, or are currently facing, the ELB
ranges from about 5 years (20 quarters) in the Bank of Japan to about 10 years (40 quarters)
for the Bank of Canada. The maximum tenure duration exceeds 15 years (60 quarters) atseveral central banks (the Federal Reserve, Bank of Canada, Bank of England, and Sveriges
Riksbank), as shown in Table 4.
Table 3: Average Tenure Duration of Chairpersons in Select Central Banks
Year ofNo. of leaders

No. of leaders

Average tenure

Average tenure

foundation

since foundation

since 1946

since foundation

since 1946

Federal Reserve System

1914

16

10

7.0

8.1

European Central Bank

1998

4

4

7.0

7.0

Bank of Canada

1934

10

9

9.6

9.4

Bank of Japan

1882

32

16

4.5

5.2

Bank of England

1694

121

10

2.7

8.3

Sveriges Riksbank

1901

15

12

9.0

7.1

Swiss National Bank

1907

14

10

8.1

7.4

Central Bank

Note: In computing the average tenure duration, we exlude the current chairperson/governor/president whose
tenure is yet to be concluded.

Table 4: Maximum Tenure Duration of Chairpersons in Select Central Banks
Max duration

Max duration

since foundation

since 1946

Federal Reserve System

18 yrs and 10 months (Martin)

18 yrs and 10 months (Martin)*

European Central Bank

8 yrs (Trichet and Draghi)

8 yrs (Trichet and Draghi)

Bank of Canada

20 yrs (Towers)

14 yrs (Boey)

Bank of Japan

10 yrs (Kuroda)

10 yrs (Kuroda)

Bank of England

24 yrs (Norman)

12 yrs (Cobbold)

Sveriges Riksbank

19 yrs (Rooth)18 yrs (Asbrink)

Swiss National Bank

14 yrs (Bachmann)

11 yrs (Jordan)

Central Bank

Note: The tenure of Alan Greenspan lasted for 18 years and 6 months.

Another way to think about the reasonable duration of reputational loss is to theoreticallyrefine the concept of sustainability. In models in which the commitment and discretionary
outcomes are different, there are multiple—typically infinitely many—sustainable plans. In
this paper, we study infinitely many (countable) sustainable plans indexed by the duration16

of lost reputation. By imposing further restrictions on the set of sustainable plans and thus
refining the concept of sustainability, we can select one of these sustainable plans as being
more reasonable than others.
One refinement concept for any sequential equilibria—a sustainable plan in our setup—developed by game theorists in the context of two-player games is renegotiation proofness.
Roughly speaking, renegotiation proofness requires that, even if the deviation from an equilibrium were to occur hypothetically, two players would have no incentives to renegotiate thecontract—strategies in our setup—that they have initially agreed on. According to one definition of renegotiation-proofness proposed by Pearce (1987), a sustainable plan is renegotiationproof if the punishment value associated with that plan is higher than the punishment valueof any other sustainable plans.12
To apply this concept of renegotiation proofness to our model, the dash-dotted red line in
Figure 5 shows the punishment value associated with deviating from the OSPs with different
values for N . According to the figure, the punishment continuation value is non-monotonic.
When N is large, a reduction in the punishment duration increases the punishment value:
all else equal, it is good to stay in the discretionary regime for a shorter duration, as thevalue under the discretionary regime is lower than the value under the OSP. However, when
the punishment duration is sufficiently short and the sustainability constraint binds, a shorter
punishment duration lowers the value associated with the OSPs. This non-monotonicity arises
because a shorter punishment duration limits the sizes of the inflation and output overshootsin the aftermath of crises and lowers the value associated with OSPs. As a result, when thepunishment regime ends, the economy will return to a sustainable outcome that is not as
good as the sustainable outcome with a longer punishment duration. When N is sufficiently
small, this second effect dominates the first effect, and a shorter punishment duration lowers
the punishment value.
According to Figure 5, in our model, the sustainable outcome with the least severe punishment value is the sustainable outcome with N = 27. The dash-dotted red lines in Figure 6
show the dynamics of the economy under the OSP with N = 27, together with those under the OCP and the ODP. The post-crisis ELB duration is 3 quarters under the OSP with
N = 27, 3 quarter shorter than that under the OCP. The size of the post-crisis overshooting issmaller—and output and inflation decline by more—under the OSP with N = 27 than under
the OCP. However, the declines in inflation and output are much smaller under the OSP with
N = 27 than under the ODP. The welfare cost of the ELB constraint is about 23 percent ofthat under the ODP.

12

Farrell and Maskin (1989) proposed an alternative definition of renegotiation-proofness whereby a sustainable plan is renegotiation proof if there is no Pareto-improving move to another sustainable plan after
deviating from the on-equilibrium path at any point in time. As pointed out by Matsuyama (1997), in models with benevolent government where the government’s objective function and the private-sector’s objectivefunction coincide, this definition rules out any chance for the economy to achieve allocations better than the
ODP.

17

Figure 5: Values in case of reneging

Note: Continuation value under the sustainable outcome after a crisis that has lasted for 8 quarters.

Figure 6: Dynamics with N = 27

Note: ODP, OCP, and OSP stand for optimal discretionary policy, optimal commitment policy, and optimal
sustainable policy, respectively. The policy rate and the inflation rate are expressed in annualized percent.
The output gap is expressed in percent.

3.4

The role of pH

A key force that makes the lower-for-longer promise sustainable in our model is that thecrisis shock can hit the economy in the future. Behind the sustainable outcome, there is aplan—a pair of central-bank and private-sector strategies—specifying that, if the central bankdoes not fulfill the lower-for-longer promise in the aftermath of the current recession, privatesector will not believe the same type of promise in future recessions (“the central bank loses
18

its reputation”). If the central bank cannot effectively implement lower-for-longer policies, theeconomy will suffer large declines in inflation and output. Thus, the threat of punishment byprivate-sector agents to distrust lower-for-longer promises in future recession gives the central
bank an incentive to fulfill the promise.
Figure 7: Dynamics under alternative crisis frequencies

Note: The output gap is expressed in percent.

This incentive for the central bank to fulfill the lower-for-longer promise is larger when
the probability of a future crisis is higher. In the extreme case in which the crisis shock
is a one-time event and the normal state is an absorbing state, as typically assumed in theliterature, the incentive for the central bank to fulfill the promise does not exist, making any
degree of lower-for-longer promises unsustainable. See Nakata (2015) and Nakata (2018) for
more detailed exposition of the mechanism as well as the discussion of how other parameters—
including pL —affect the reputational cost of reneging on the lower-for-longer promise as wellas its short-run stabilization benefit.
Figure 7 illustrates the role of pH in our model. In this numerical simulation, we consider
two alternative values of pH (0.0025 and 0.0075) and the crisis shock is set to last for 8 quarters.
According to the figure, the additional duration of being at the ELB in the aftermath of acrisis is longer—and the magnitudes of the inflation and output overshoots is larger—thehigher the crisis probability (pH ) is. This numerical result is consistent with the mechanism
described in the preceding paragraphs.

4

Additional discussion and results

4.1

Relation to the loose commitment approach of Bodenstein, Hebden,
and Nunes (2012)

Under OSPs, the central bank achieves crisis-state allocations that are “in between” that
under the ODP and that under the OCP. This feature of OSPs is reminiscent of the opti19

mal policy obtained in a loose commitment approach in which the central bank reoptimizes
with a constant probability every period regardless of the incentive to renege on the priorcommitment.13 While these two approaches differ from each other in many ways, both approaches share the same spirit that they are intended to shed light on what the central bank
may be able to achieve when no explicit commitment technology is available. Indeed, recent
work by Fujiwara, Kam, and Sunakawa (2019a) shows that, when using a model without
the ELB, the allocations under the loose commitment approach with an appropriately chosen
re-optimization probability can approximate the allocation under the OSP with N -period
punishment reasonably well for any N . While we believe their result is likely to extend tothe model with ELB, it would be useful to verify the validity of their claim in our model infuture research.

4.2

Relation to Dong and Young (2019)

Our work is closely related to Dong and Young (2019). In this subsection, we discusscommonalities and differences between our work and Dong and Young (2019).
Dong and Young (2019) and our analysis both aim to characterize a set of outcomes thatare credible. i.e. that do not suffer from the time-inconsistency problem. The difference isthat we aim to characterize a subset of sustainable outcomes that are easy to interpret andthat can be found using a methodology that is a variant of a widely used the method of Marcet
and Marimon (2019), whereas Dong and Young (2019) aim to characterize the entire set ofsustainable outcomes using a rigorous computational method that is based on Chang (1998).
Our approach is more tractable than theirs in the sense that, for any punishment duration N ,we obtain the associated sustainable outcome as the solution of the standard Ramsey problem
with one additional incentive-compatibility constraint. Our tractability comes at the cost of
not being able to explore the entire set of sustainable outcomes.
The best sustainable outcome (BSO) in Dong and Young (2019) looks quite different
from any of our sustainable outcomes. Their BSO features (i) less aggressive lowering of thepolicy rate at the ELB, (ii) the policy rate never touching the ELB constraint, and (iii) larger
inflation overshooting in the aftermath of a recession than under the Ramsey policy. These
results are intriguing because (i) the policy rate is lowered to the ELB immediately upon thearrival of the shock both under the Markov Perfect policy and under the Ramsey policy and(ii) larger inflation overshooting—considered to be the key source of time-inconsistency inthis model—suggests more severe time-inconsistency at least at a first glance. In other words,
their BSO is not “in between” the Markov perfect policy and the Ramsey policy.
The exact reason for why the BSO looks the way it does is yet to be explored in the literature. The literature on sustainable outcomes suggests that the set of sustainable outcomesis large and features a sophisticated strategy of the government in many models. In contrast,
13

See Bodenstein, Hebden, and Nunes (2012) for an analysis of optimal monetary policy under loose
commitment in the model with ELB. See Levin and Sinha (2020a,b) for a more recent treatment.

20

for any given N , our sustainable outcome is “in between” the Markov perfect policy and theRamsey policy, in a way that is reminiscent of the equilibrium in the imperfect commitment
literature. Thus, some readers may find the result in our sustainable outcomes intuitive.

4.3

Relation to the simple forward guidance policies of Walsh (2018)

While our OSPs may be theoretically interesting, they may be hard to implement inreality. In this subsection and the next, we contrast our OSPs with two alternative simpler
monetary policies that may be easier to communicate with the public: (i) policy in which thecentral bank keeps the policy rate at the ELB for a fixed additional duration after the crisis
shock is gone and (ii) an average inflation targeting policy.
We have shown that, when N is sufficiently small, the policy rate paths under OSPs are
not history dependent. That is, the policy rate path after the crisis shock disappears does notdepend on the realized duration of the crisis shock. Thus, the OSPs bear some resemblance tothe simple forward guidance policies considered by Walsh (2018). Under the simple forward
guidance policies of Walsh (2018), the central bank keeps the policy rate at the ELB for a fixed
number of periods after the crisis shock disappears, regardless of the realized duration of thecrisis shock, and lets the policy rate return to the steady-state level immediately thereafter.
The only (minor) difference is that, under the optimal sustainable policy, the policy rate does
not return to the steady state level immediately after liftoff. Instead, there is typically one
period after liftoff in which the policy rate is still below the steady-state level.
The similarity between the simple forward guidance policies and OSPs with small N s
points to one benefit of OSPs over the OCP; it may be easier for central banks to explainthese OSPs to the public than the OCP. One key criticism against the OCP is that it iscomplex. As Walsh (2018) argues, because of its complexity, it may be difficult for the central
bank in practice to steer the private-sector agents’ expectations in a way consistent with the
OCP. One dimension of complexity is history dependence. The OSPs have an advantage over
the OCP because they are less history-dependent and thus simpler.
Note that, in our model, the policy rate path under the ODP and the OSPs with small N s
are history independent but state-contingent. They are state-contingent because the policy
rate path—in particular the liftoff date—depends on the realized crisis shock duration. Thus,
these policies are different from so-called calendar-based forward guidance that specifies thelikely liftoff date, if that guidance were to be narrowly or mistakenly interpreted as a nonstate-contingent commitment to raising the policy rate from the ELB at a particular dateregardless of the evolution of the economy.14
14

Even when central banks indicate a likely date of liftoff from the ELB, they typically emphasize that theliftoff date will depend on the evolution of the economic outlook. That is, if the economy were to recover fasteror more slowly than in the baseline economic projection, the central bank will raise the policy rate from the
ELB earlier or later than the most likely liftoff date under the baseline projection. In practice, it is unlikelythat any central bank will ever engage in non-state-contingent forward guidance, though market participants
may not interpret the forward guidance specifying the likely date of liftoff as state-contingent as the central
bank intends.

21

4.4

Results from the model with average inflation targeting

In this section, we analyze similarities between optimal sustainable policies and average
inflation targeting (AIT) policies—policies that feature overheating commitment as in OSPs
and that have recently attracted a lot of attentions because of new monetary policy frameworks
at the Fed and the ECB.
We consider a policy ruleit = st + φω −1 π̂t ,(10)π̂t = ωπt + (1 − ω)π̂t−1 ,

(11)

where
φ is the degree of response to inflation, and ω ∈ [0, 1] is a weight on the current inflation rate
in an exponential moving average inflation rate π̂t . The central bank sets the policy rate byreferring to the natural rate of interest st as in Cúrdia, Ferrero, Ng, and Tambalotti (2015).
In addition, the central bank responds to an average of the past inflation rates π̂t . The rest of
the equilibrium conditions consist of the Euler equation and Phillips curve, which are given
by Equations (1) and (2). We also consider the ELB constraint (3) as before.
This policy rule (10) and (11) nests the following different strategies (See also Budianto,
Nakata, and Schmidt, 2023):• When ω = 1, the Equation (10) becomes the standard Taylor ruleit = st + φπt .
The central bank follows a standard flexible inflation targeting strategy. We assume
that the central bank knows the natural rate of interest st so that the standard inflation
targeting resembles the ODP.
• When ω = 0, the Equation (10) becomes
it = st + φ(pt − p−1 ).
The central bank aims to stabilize the price level pt ≡ πt + pt−1 .15 It is well known that

15

Note thatω −1 π̂t =∞X(1 − ω)j πt−j + (1 − ω)t ω −1 π̂−1 .
j=0Assuming π̂−1 = 0, we haveit =st + φ∞
X(1 − ω)j πt−jj=022

the OCP has such a property of price level targeting.16• When ω ∈ (0, 1), the central bank aims to stabilize an exponential moving average inflation rate π̂t as given in Equation (11). We are especially interested in these intermediatecases because they can be similar to OSPs which are also intermediate between the OCP
and the ODP.
The parameter values are set to the values in Table 1. We set φ to 5.0 so that thecentral bank responds to inflation aggressively. We have checked that our results are robust
to different values of φ.
Figure 8: Dynamics under AIT policies

Note: The policy rate and the inflation rate are expressed in annualized percent. The output gap is expressedin percent.

Figure 8 shows the dynamics for different values of ω ∈ [0, 1] when the crisis shock hits
the economy at time 1 and remains there for 8 quarters as in the main exercise in Section
2 (see Figure 1). The dynamics under the AIT policy with ω = 0.01—shown by the solid
black lines—are similar to those under the OCP and the OSP with the punishment duration
N = ∞, whereas the dynamics under the AIT policy with ω = 1.0—shown by the solid
red lines—are similar to those under the ODP. As we increase the value of ω, the paths ofinflation, the output gap, and the policy rate diverge from those under OCP and converge tothose under ODP. This pattern is similar to what happens when we decrease N of OSPs in
Figure 1.

As ω → 0, we haveit = st + φ(pt − p−1 ).
16

Eggertsson and Woodford (2003) shows that, even in the presence of ELB, a modified price level targetingby setting the price level to an output gap-adjusted price index can mimic the OCP. The policy rule consideredhere only intends to set the price level constant and does not completely replicate the OCP.

23

4.5

Cost-push shock

Throughout the paper, we focus on the time-inconsistency problem that arises from thedemand shock—more precisely speaking, the interaction of the ELB constraint and the demand shock—and abstract from the cost-push shock—another shock commonly studied in
the literature. The cost-push shock makes the commitment outcome time-inconsistent in theabsence of a reputational mechanism, as pointed out by Walsh (2003), among many others.
Sunakawa (2015) characterizes OSPs in a New Keynesian model with cost-push shocks but
without the ELB constraint.
Nakata (2018) examines how the introduction of a cost-push shock affects the credibility of
the OCP in the model featuring the ELB and a demand shock. He finds that the introductionof the cost-push shock increase or decreases the threshold crisis frequency above which the
OCP is credible, reflecting two opposing forces. On the one hand, the introduction of acost-push shock lowers the continuation value associated with the ODP, making the reneging
on the lower-for-longer promise more costly. On the other hand, the introduction of a costpush shock increases the magnitudes of the inflation and output overshoots, increasing thetemptation to renege on the lower-for-longer promise.
We leave the task of simultaneously examining the demand and cost-push shock in ourmodel to future research.

5

Conclusion
In this paper, we have characterized OSPs in models with the ELB constraint. We findthat, even when the OCP is not credible, the central bank can still credibly commit to
keeping the policy rate at the ELB in the aftermath of a crisis—though not as long as under
the OCP—and meaningfully mitigate the adverse consequences of the ELB constraint on
economic activity in crises.
By construction, our OSPs are time-consistent and thus overcome the criticism that thetemporary overheating of the economy associated with lower-for-longer strategies is not credible. When the loss of reputation is sufficiently short-lived, these OSPs are less history dependent than the OCP. Thus, it overcomes the criticism that the implied policy rate path is
too complex for the central bank to be able to explain to the public, making the OSPs evenmore attractive.
Although we focus on the time-consistency aspect of lower-for-longer policies in this paper,
there are other aspects of these policies that could make them less attractive in reality thanin theory. For example, the public may not understand the temporary nature of the inflation
overshooting, resulting in unanchoring of the long-run inflation expectations (Kohn (2012)
and Yellen (2018)). The overheating of the economy may be less desirable for policymakers inreality than what’s implied by our model if the overheating of the economy leads to financial
instability (Yellen (2018)). It would be useful to formally analyze how these factors affect24the effectiveness and implementability of lower-for-longer strategies. We leave such analysis
to future research.

25

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30

Technical Appendix for Online Publication
This technical appendix is organized as follows:• Appendix A defines some key concepts.
• Appendix B describes the equilibrium conditions characterizing the sustainable outcomeof the model in detail.
• Appendix C describes the numerical solution method and reports the solution accuracy.
• Appendix D collects policymakers’ speeches in which the time-inconsistency problem of
the lower-for-longer policy is discussed.

A

Definition of a plan and credibility

This section defines a plan, credibility, and the revert-to-discretion plan. The definitions
closely follow Chang (1998) and Nakata (2018).

A.1

Plan

A government strategy, denoted by σg := {σg,t }∞
t=1 , is a sequence of functions that maps
a history of the nominal interest rates up to the previous period and a history of states up
to today into today’s nominal interest rate. Formally, σg,t is given by σg,1 : S → R≥0 andt
17 Given a particular realization of {s }∞ , a sequence of
σg,t : Rt−1t t=1
≥0 × S → R≥0 for all t ≥ 2.
nominal interest rates will be determined recursively by i1 = σg,1 (s1 ) and it = σg,t (it−1 , st ) for
all t > 1 and for all st ∈ St . A government strategy is said to induce a sequence of the nominalinterest rates. A private-sector strategy, denoted by σp := {σp,t }∞
t=1 , is a sequence of functions
mapping a history of nominal interest rates up to today and a history of states up to today
into today’s consumption and inflation. Formally, σp,t is given by σp,t : Rt × St → (R, R) for
all t.
Given a government and private-sector strategy, a sequence of consumption and inflation
will be determined recursively by (yt , πt ) = σp,t (it , st ) for all t ≥ 1 and for all st ∈ St . A
private sector strategy, together with a government strategy, is said to induce a sequence ofconsumption and inflation.18 A plan is defined as a pair of government and private sector
strategies, (σg ,σp ). Notice that a plan induces an outcome—a state-contingent sequence ofconsumption, inflation, and the nominal interest rate. As discussed earlier, there is a value
sequence {wt (st )}∞t=1 , associated with any outcome.

17

The first period is a special case, as there is no previous policy action.
Note that, while the nominal interest rate today depends on the history of nominal interest rates up to
the previous period, consumption and inflation today depend on the history of nominal interest rates up to
today. The implicit within-period-timing protocol behind this setup is that the government moves before theprivate sector does.
18

31

A.2

Credibility

Let us use CEtI (s) to denote a set of state-contingent sequences of the nominal interest
rate consistent with the existence of a competitive equilibrium when st = s. Formally, foreach s ∈ S, CEtI (s) := {it (s) ∈ I∞ | ∃ (yt (s), πt (s)) s.t. (yt (s), πt (s), it (s)) ∈ CEt (s)}. σg issaid to be admissible if, after any history of policy actions, it−1 , and any history of states, st ,it (s) induced by the continuation of σg belongs to CEtI (st ).
A plan, (σg , σp ), is credible if (i) σg is admissible, (ii) after any history of policy actions, it ,and any history of states, st , the continuation of σp and σg induce a (yt (st ), πt (st ), rt (st )) ∈CEt (st ), and (iii) after any history it−1 and st , it (st ) induced by σg maximizes the government’s objective over CEtI (st ) given σp . In plain languages, a plan is said to be credible ifneither the private sector nor the government has incentive to deviate from the strategies
associated with it.
An outcome is said to be credible if there is a credible plan that induces it. When a certain
plan A is credible and the plan A induces a certain outcome α, we say that the outcome α
can be made credible, or time-consistent, by the plan A.

A.3

The revert-to-discretion plan

I now define a key object of this paper, the revert-to-discretion plan, and discuss thecondition under which this plan is credible.
The revert-to-discretion plan, (σgrtd , σprtd ), consists of (i) the following government strategy:
rtd = irtd t−1 , st ) = itjσg,1so,1 (s1 ) for any s1 ∈ S, σg,t (iso,t (s ) if ij = iso,j (s ) for all j ≤ t − 1, and
rtd
t−1
t
t
rtd (it , st ) =σg,t (i , s ) = ido,t (s ) otherwise, and (ii) the following private-sector strategy: σp,trtd (it , st ) = (y (s , i ), π (s , i )) other(yso,t (st ), πco,t (st )) if rj = ico,j (sj ) for all j ≤ t, σp,t
br t t
br t t19
wise, wherehiybr (st , rt ) = Et ydo,t+1 (st+1 ) − σ it − Et πdo,t+1 (st+1 ) − r∗ + st
(12)
πbr (st , rt ) = κybr (st , rt ) + βEt πdo,t+1 (st+1 )(13)

The government strategy instructs the government to choose the nominal interest rate consistent with the sustainable outcome, but chooses the interest rate consistent with the discretionary outcome if it has deviated from the sustainable outcome at some point in the past. The
private sector strategy instructs the household and firms to choose consumption and inflationconsistent with the sustainable outcome as long as the government has never deviated fromthe sustainable outcome. If the government has ever deviated from the nominal interest rateconsistent with the sustainable outcome, the private sector strategy instructs the household
and firms to choose output and inflation today based on the belief that the government in the
future will choose the nominal interest rate consistent with the discretionary outcome. By
construction, the revert-to-discretion plan induces the sustainable outcome, and the impliedvalue sequence is identical to the sustainable value sequence.
It is relatively straightforward to show that the revert-to-discretion plan is credible. By
construction, Vso,t (st ) ≥ Vdo,t (st ) for all t ≥ 1 and all st ∈ St , making sure that the government
does not have an incentive to deviate from the instruction given by the government strategy
after any history it−1 and st in which the optimal sustainable policy has been followed.
19

Subscript br stands for best r esponse.

32

The revert-to-discretion plan that induces the optimal sustainable outcome with a finite
period punishment is defined in a similar way (see Nakata (2018) for rigorous exposition). It
is also straightforward to show that such a plan is credible.

B

Model Details
The policymaker maximizes∞
X

V1 = −E1
β t πt2 + λyt2 ,t=1

subject to
yt = Et yt+1 − σ (it − Et πt+1 − r∗ ) + st ,(14)

πt = βEt πt+1 + κyt ,(15)

it ≥ iELB ,
∞
X

2
2
N
Vt = −Et
β j πt+j+ λyt+j≥ Wdo(st ),

(16)
(17)

j=0

for all t ≥ 0. The shock, st , follows two-state Markov
chain, st ∈ {sH , sL } where sH > sL .

1 − pH pH
Transition probability matrix is given as P =, where pH is the frequency
1 − pL pL
N (s ) is the continuation value underof the crisis and pL is the persistence of the crisis. Wdot
N -period punishment.
The analytical solution for the discretionary outcome: The optimality condition forthe discretionary outcome is given by
λyt = −κπt − ψt ,where ψt is the Lagrange multiplier on the Euler equation. As we have no endogenous statevariable and the exogenous shock takes only two values in this case, each variable also takes
two values. That is, (yt , πt , it , ψt ) = (yH , πH , iH , ψH ) in the high state and (yt , πt , it , ψt ) =(yL , πL , iL , ψL ) in the low state. We assume that the ZLB is slack in the high/normal state,ψH = 0, and the ZLB is binding in the low/crisis state, ψL = ψ > 0. Thus, the equilibrium
conditions become−σ −1 yH + σ −1 (1 − pH )yH + σ −1 pH yL − iH + (1 − pH )πH + pH πL + r∗ + sH = 0,
−πH + κyH + β(1 − pH )πH + βpH πL = 0,
−λyH − κπH = 0,
−σ −1 yL + σ −1 (1 − pL )yH + σ −1 pL yL − r∗ + (1 − pL )πH + pL πL + r∗ + sL = 0,
−πL + κyL + β(1 − pL )πH + βpL πL = 0,−λyL − κπL − ψ = 0.

33

These equations can be solved for (yH , πH , iH , yL , πL , ψ). Also, having these values at hand,the value functions become22
VH = −πH− λyH+ β(pH VL + (1 − pH )VH ),VL = −πL2 − λyL2 + β(pL VL + (1 − pL )VH ),which can be solved for (VH , VL ).
Solving for the sustainable outcome: The Lagrangean is given by
L ≡ E1∞Xβ t − πt2 + λyt2 − 2φt (−πt + βEt πt+1 + κyt )t=1+2ψt (−yt + Et yt+1 − σ (it − Et πt+1 − r∗ ) + st )


∞

X

2
2
N
+ϕt −Et(st ) ,β j πt+j+ λyt+j
− Wdo

j=0

= E1

∞
X



β t −Ψt πt2 + λyt2 − 2κφt yt + 2 (φt − φt−1 ) πtt=1N(st ) ,
+2ψt (−yt − σit + st ) + 2β −1 ψt−1 (σπt + yt ) − ϕt Wdo

where Ψt = ϕ0 + ϕ1 + ... + ϕt < ∞ is the sum of the Lagrange multipliers on the sustainability
constraint. The FOCs are given by∂yt :λΨt yt = −κφt − ψt + β −1 ψt−1 ,∂πt :Ψt πt = φt − φt−1 + σβ −1 ψt−1 ,∂φt :−πt + κyt + βEt πt+1 = 0,∂ψt :−yt + Et yt+1 − σ (it − Et πt+1 − r∗ ) + st = 0.

Normalizing the first and second equations by Ψt , we have
πt = φ̃t − zt φ̃t−1 + σβ −1 zt ψ̃t−1 ,λyt = −κφ̃t − ψ̃t + β −1 zt ψ̃t−1 ,where φ̃t = φt /Ψt , ψ̃t = ψt /Ψt , and zt = Ψt−1 /Ψt ∈ (0, 1]. The Karush-Kuhn-Tucker
conditions (KKTCs) must be satisfied as wellψt (it − iELB ) = 0,ψt ≥ 0,ϕt (Vt − Wdo (st )) = 0,ϕt ≥ 0.
The initial conditions on the Lagrange multipliers are such that φ0 = ψ0 = 0 and ϕ0 = 1,
which implies φ̃0 = ψ̃0 = 0 and Ψ0 = 1.

34

C

Projection method

In this section, we describe the projection method that we use to solve the model. In particular, we employ a collocation method. Our goal is to construct a parametric approximation
to each function in the set {V, φ0 , i, z, y, π, ψ 0 ,W }. Let ξ be the vector of the state variables(φ, ψ, s) and θ ≡ θV , θφ0 , θi , θz , θy , θπ , θψ0 , θW denote the vector of parameters indexing candidate approximations.
Collocation nodes. Functions of the state vector are continuous along the φ and ψ dimensions, and therefore, we need to select collocation nodes along these two dimensions. In
the φ dimension, we construct 20 evenly-spaced collocation nodes between -9 and 1. We thenadd additional collocation nodes at φ = (−3000, −2500, −2000, −1500, −1000, −500, −100,−75, −50, −25, 10, 100, 200). In the ψ dimension, we construct 26 geometrically spaced
collocation nodes between ψmin = 0 and ψmax = 300. We have two values for the exogenous
state variable s ∈ {sH , sL }. Combined across the three dimensions, this yields a total ofn = 33 × 26 × 2 = 1716 collocation nodes per function.
Parameterization of the approximation. For a given basis function Φ, we construct ourapproximations asVθ (ξ) ≡ Wθ (ξ) + Φ (ξ)0 θV
φ0θ (ξ) ≡ Φ (ξ)0 θφ0iθ (ξ) ≡Φ (ξ)0 θizθ (ξ) ≡ Φ (ξ)0 θzyθ (ξ) ≡ Φ (ξ)0 θyπθ (ξ) ≡ Φ (ξ)0 θπψθ0 (ξ) ≡Φ (ξ)0 θψ0Wθ (ξ) ≡ 1 {s = sL } θW,1 + 1 {s = sH } θW,2
Empirically, we find that the piecewise linear basis function works best to ensure stability.
We also find it convenient to approximate expectation
 functions. To this end, we construct avector of auxiliary parameters θA = θEV 0 , θEπ0 , θEy0 as follows
θEV 0 ,j ≡ p (sH | sj ) Vθ sH , φ0θ (ξj ) , ψθ0 (ξj ) + p (sL | sj ) Vθ sL , φ0θ (ξj ) , ψθ0 (ξj )θEπ0 ,j ≡ p (sH | sj ) πθ sH , φ0θ (ξj ) , ψθ0 (ξj ) + p (sL | sj ) πθ sL , φ0θ (ξj ) , ψθ0 (ξj )θEy0 ,j ≡ p (sH | sj ) yθ sH , φ0θ (ξj ) , ψθ0 (ξj ) + p (sL | sj ) yθ sL , φ0θ (ξj ) , ψθ0 (ξj )where ξj denotes the j-th collocation node and sj denotes the associated exogenous state. The
validity of this construction depends on using the piecewise linear basis function to deal withkinks due to occasionally binding constraints. Also, note that these parameters are implicitly
functions of θ. Hence, we can naturally express approximations to the expectation functionsas35

EVθ0 (ξ) ≡ Φ (ξ)0 θEV 0
Eπθ0 (ξ) ≡ Φ (ξ)0 θEπ0
Eyθ0 (ξ) ≡ Φ (ξ)0 θEy0
Our parameterization choice imposes that Vθ (ξ) ≥ W (ξ), iθ (ξ) ≥ rELB , and ψθ0 (ξ) ≥ 0, butdoesn’t require that 0 ≤ zθ (ξ) ≤ 1. Empirically, we find that imposing constraints on thislast function via the choice of parameterization occasionally destabilizes our algorithm. We
therefore proceed by verifying that our candidate solution satisfies this constraint.
Construction of the residual function. We now describe the construction of the residualfunction. For a given function hθ ∈ {Vθ , φ0θ , iθ , zθ , yθ , πθ , ψθ0 , Wθ (ξ) , EVθ0 , Eπθ0 , Eyθ0 }, we use
the shorthand notation hj ≡ hθ (ξj ). For each j, we construct the following residuals:
σ
zj ψj
β
zj ψj
e2 (ξj ) ≡ −λ−1 yj − κφ0j − ψj0 +β
e3 (ξj ) ≡ −πj + κyj + βEπj0
e1 (ξj ) ≡ −πj + φ0j − zj φj +e4 (ξj ) ≡ −yj + Eyj0 − σ ij − iELB − r∗ − Eπj0 − sj

e5 (ξj ) ≡ −Vj − πj2 − λ−1 yj2 + βEVj0e6 (ξj ) ≡ ψj0 (ij − iELB )e7 (ξj ) ≡ (1 − zj ) (Vj − Wj )Note that we use a system of equations that is slightly different than the one described in theprevious section. In particular, the functions {φ0 , ψ 0 , V, W } and state variables are scaled by
λ−1 compared to the previous section. We use this parametrization because we find that it ismore stable.
N (s ). We compute it recursively as
Another object that we need to compute is Wdoj0(sj ) = y(sj , 0, 0),
ydo

0
Wdo
(sj ) = V (sj , 0, 0),NWdo(sj ) = max
y,π,i

2

−y − λNydo(sj ) = arg max
y,π,i

−1 2

2

π +β

−y − λNNπdo(sj ) = κydo(sj ) + β

−1 2

Ns
X

0
(sj ) = π(sj , 0, 0),
πdo
!

N −1
p(sk |sj )Wdo(sk ) ,

k=1
Ns
X

π +β

!
N −1
p(sk |sj )Wdo
(sk )

k=1
Ns
X

N −1
p(sk |sj )πdo(sk ) = 0

k=1

36

,

subject to∗− y − σ (i − r ) + sj +

Ns
X

N −1
p(sk |sj )ydo
(sk ) + σ

k=1

Ns
X

N −1
p(sk |sj )πdo(sk ) = 0,k=1i ≥ iELB .
0Letting ej = e1 (ξj ) · · · e7 (ξj ) , we then construct our residual function as




e1
..
.





R (θ, N ) = 



en
N
Wθ − Wdo
N can be represented as vectors of equal length,
where we use the fact that both Wθ and Wdo
Ns = 2.
An approximate solution to the model is a solution to the following root-finding problem.

R (θ, N ) = 0
Solving the root-finding problem. For each N , we want to solve a system of n × 7 + 2 =12014 nonlinear equations in 12014 unknowns. To solve these systems, we loop over valuesof N starting at N = 160. At each step, we use a trust-region algorithm initialized at thesolution of the previous step. We also solve for the case of the optimal commitment policy
without the sustainability constraint.
Error Analysis. To check the solution accraucy, Figure 9 shows the average and 99.5
percentile of the residuals in absolute terms (log 10 units) based on a 100,000-period stochastic
simulation as a function of N .
Figure 9: Simulation Residuals
L*

5

0

0

−5

−5

−10

e1
e2
e3
e4
e5
e6
e7

−15
30

60

N

90

L1

5

120

−10

e1
e2
e3
e4
e5
e6
e7

−15

150

30

60

N

90

120

150

Note: L1 is the average and L∗ is the 99.5 percentile of the residuals in absolute terms (log
10 units) based on a 100,000-period stochastic simulation.
37

D

Time-inconsistency of the commitment policy in the wordsof policymakers

The time-inconsistency of the commitment policy at the ELB is not a mere theoretical curiosity. Policymakers in many central banks have pointed out the potential timeinconsistency of the commitment-type forward guidance policy. Some have argued that the
time-inconsistency is one key reason for why most central banks refrained from making theoverheating commitment. Below are some examples:

D.1

Bean (2013)“In particular, we signalled our intention not to countenance tightening policy until unemployment has fallen to at least 7 percent.”“This guidance is intended primarily to clarify our reaction function and thus makepolicy more effective, rather than to inject additional stimulus by pre-committing
to a time-inconsistent ‘lower for longer’ policy path in the manner of Woodford(2012). While such a time-inconsistent policy may be desirable in theory, in an
individualistic committee like ours, with a regular turnover of members, it is notpossible to implement a mechanism that would credibly bind future members in
the manner required.”

D.2

Bullard (2013)“The New Keynesian, sticky price literature has been influential in U.S. monetary policymaking. The literature has been led by Michael Woodford. This line of research argues thatpolicy accommodation can be provided even when the policy rate is near zero. The extra
accommodation comes from a promise to maintain the near zero policy rate into the future,beyond the point when ordinary policymaker behavior would call for an increase in the policy
rate. This promise must be credible to have an impact.
The “Woodford period” approach to forward guidance relies on a credible announcement made today that future monetary policy will deviate from normal.
The central bank does not actually behave differently today. One might argue
that such an announcement is unlikely to be believed. Why should future monetary policy deviate from normal once the economy is growing and inflation isrising? But if the announcement is not credible, then the private sector will notreact with more consumption and investment today. That is, any effects would
be minimal.”

D.3

Carney (2012)“Today, to achieve a better path for the economy over time, a central bank may need to commit
credibly to maintaining highly accommodative policy even after the economy and, potentially,
inflation picks up. Market participants may doubt the willingness of an inflationtargeting central bank to respect this commitment if inflation goes temporarily
38

above target. These doubts reduce the effective stimulus of the commitment anddelay the recovery.”

D.4

Clarida (2019)“The benefits of the makeup strategies rest heavily on households and firms believing in advance that the makeup will, in fact, be delivered when the time comes–for example, thata persistent inflation shortfall will be met by future inflation above 2 percent. As is wellknown from the research literature, makeup strategies, in general, are not time
consistent because when the time comes to push inflation above 2 percent, conditions at that time will not warrant doing so. Because of this time inconsistency,
any makeup strategy, to be successful, would have to be understood by the public
to represent a credible commitment. That important real-world consideration isoften neglected in the academic literature, in which central bank “commitment
devices” are simply assumed to exist and be instantly credible on decree. Thus,
one of the most challenging questions is whether the Fed could, in practice, attainthe benefits of makeup strategies that are possible in models.”

D.5Cœuré (2013)“Most notably, the central bank may try to convince markets that it would keep interest rates
low, even if this would imply inflation well above its previous objective, at least temporarily.
The promise of higher future inflation, if credible, induces private agents to substitute future
for current consumption, hence providing additional stimulus today. This type of forward
guidance is closer to the academic concept of forward guidance in the strict sense—as discussed, for example, in Woodford (2012).
The main challenge of such guidance is its inherent inconsistency over time andthus lack of credibility. When the time comes, the central bank may be tempted
to deviate from its prior commitment: once the benefits of higher inflation expectations in terms of front-loaded spending have been reaped, the central bank
may not be willing to pay the bill in terms of higher inflation afterward. If the
public foresees this temptation, expectations might remain unaffected in the first
instance and the desired inter-temporal substitution of spending might not materialise. This is a possible explanation why, in practice, central banks have refrained fromusing forward guidance in a way that implies a major change in strategy. Therefore, central
banks’ forward guidance has rather aimed at providing greater clarity on the reaction function
and the assessment of future economic conditions.”

D.6

Dudley (2013)“With respect to forward guidance, it is important to distinguish between two specific forms
that this guidance may take. In the first form the central bank provides its forecast for the
future path of the policy rate and, possibly, some sense of the degree of uncertainty aroundthis path. In the second, the central bank pre-commits to a specific future path for its policy

39

rate.
Providing a forecast for the policy rate by itself does not create any budget or reputationalrisk for the Federal Reserve, so I generally do not see the first form of forward guidance as
posing much risk to central bank independence.
The second form of forward guidance—pre-commitment to a policy rate path—could
create more risk for the central bank. In particular, consider a scenario in which
the central bank decided to increase monetary accommodation by committing tomaintain a low short-term interest rate for a long time even if this commitment
resulted in inflation overshooting the central bank’s objective in the future. I
could see how this could create a potential threat to the central bank’s independence. That is because the commitment could force the central bank in the future
to conduct monetary policy in a way that was inconsistent with the inflation portion of its mandate.
Although this second form of forward guidance could create greater risk for the central bankwith respect to its future independence, this is not a policy that has been adopted by theFederal Reserve. There are implementation challenges with this approach. In particular,
it is difficult for a monetary policy committee today to institutionally bind future monetary policy committees to follow actions that could conflict with theirobjectives in the future. Without such a credible forward commitment, such policies would likely be ineffective in affecting expectations in the manner needed toprovide additional monetary policy accommodation.”

D.7

George (2019)“Third, a price-level targeting strategy is time inconsistent unless policymakers can crediblycommit to following it. If the goal is to have inflation of 2 percent on average, a period of
below 2 percent inflation would require an equal period of inflation above 2 percent. But
once inflation has moved up to 2 percent, policymakers might be tempted torenege on their prior commitment and not allow inflation to go higher. This
would undermine the future credibility of the price-level targeting strategy. To
the extent the public understood this time inconsistency problem, price-leveltargeting would not be credible to begin with, absent a commitment device.
With regular turnover among members of the FOMC, it would be difficult for
one Committee to commit a future Committee to a particular course of action.”

D.8

Lacker (2013)“Designing such conditional guidance involves trade-offs, however. Credibility requires consistency, over time, between a central bank’s statements and its actual subsequent actions. A
central bank’s statements will have greater immediate effect on the public’s expectations themore they are seen as limiting the central bank’s future choices. Yet there are likely to be circumstances, ex post, in which the central bank feels constrained by past statements. Yielding
to the temptation to implicitly renege by reworking decision criteria or citing unforeseen economic developments may have short-term appeal, but widely perceived discrepancies between
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actual and foreshadowed behavior will inevitably erode the faith people place in future central
bank statements. So central banks face an ex ante trade-off, as well, between the short-run
value of exercising discretion and the ability to communicate effectively and credibly in the
future.”D.9

Plosser (2013)“Note, however, that the central bank’s ability to influence the public’s belief about the future
path of policy and the economy depends critically on the bank’s commitment to that policy
path and the credibility of that commitment in the eyes of the public. The public mustbelieve that even after the economy begins to strengthen, the central bank willhold rates lower than it otherwise might have found desirable to do had it notbeen at the zero bound in the past."

D.10

Powell (2019)“By the time of the crisis, there was a well-established body of model-based research suggestingthat some kind of makeup policy could be beneficial. In light of this research, one might ask
why the Fed and other major central banks chose not to pursue such a policy. The answer liesin the uncertain distance between models and reality. For makeup strategies to achieve theirstabilizing benefits, households and businesses must be quite confident that the “makeup
stimulus” is really coming. This confidence is what prompts them to raise spending andinvestment in the midst of a downturn. In models, confidence in the policy is merely anassumption. In practice, when policymakers considered these policies in the wakeof the crisis, they had major questions about whether a central bank’s promiseof good times to come would have moved the hearts, minds, and pocketbooksof the public. Part of the problem is that when the time comes to deliver the
inflationary stimulus, that policy is likely to be unpopular–what is known as thetime consistency problem in economics.”

D.11

Ueda (2013)“If Max (the Taylor rule rate, zero) describes the usual central bank’s reaction function to themacroeconomic environment, the central bank can generate easing effects by offering a newreaction function to the market with a promise of a longer period at the zero rate than theabove rule suggests. To the extent that the Taylor rule represents an optimal response of the central bank to macroeconomic environment, however, this forward
guidance strategy amounts to “irresponsible” central bank behaviour. In otherwords, the strategy is time-inconsistent. This means that when the economy nolonger requires a zero rate, it is better to raise the interest rate, reneging on thepromise made. If people foresaw this ex ante, however, the strategy would become ineffective. Thus, the central bank would be sending a confusing signal if it was using
forward guidance in this sense and insisted that it was still behaving in a “responsible” way.
Also, the central bank does not seem to get much mileage out of a vague promise, such as themaintenance of a low policy rate “for an extended period,” unless there is much confusion in
the market as to where the policy rate would go in the short term.

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The BOJ seems to have faced the time-inconsistency problem in 2000.”

D.12

Williams (2012)“Although forward policy guidance has proven to be a very useful policy tool, it’s not a perfect
substitute for the kind of monetary stimulus that comes from lower interest rates. One issue
is that, for the forward guidance policy to work as desired, the public has to believe that the
FOMC will really carry out the policy as it says it will. But, the Fed doesn’t have the ability
to tie its hands that way. This point was made by Finn Kydland and Edward Prescott in
the late 1970s. Let me explain. For forward policy guidance to have its maximum effect,
the Fed must commit to keeping the short-term policy rate lower than it otherwise would
to compensate for the fact that the short-term interest rate cannot be lowered today. But
when the time comes to carry out the commitment made in its forward guidance,
it may no longer want to do so. For instance, it might be hard to resist raisingrates earlier than promised to head off an increase in inflation. So, even whencentral bankers say they will keep rates unusually low for a set time, the public
may worry that the central bank will raise rates earlier to fight budding inflationpressures.”42