A Quantitative Analysis of Optimal Sustainable
Monetary Policies
Takeki Sunakawa∗

Abstract
This study examines the quantitative properties of optimal sustainable monetary
policies using a monetary model with a stabilization bias. As in Kurozumi (2008),
the optimal sustainable policy is a strategy considered in the absence of commitment
technologies; however it is implemented following an optimal quasi-sustainable policy
derived by assuming that the commitment technologies are present. This study ndsthat solving for the policy function of the optimal quasi-sustainable policy yields a
result basically identical to the Ramsey-optimal commitment policy under a set of parameters commonly used in the literature. The simulation shows two further results:
policymakers have incentive to deviate from the Ramsey-optimal commitment policy
when the lagged output gap is large and the optimal quasi-sustainable policy endogenously diminishes the steadfastness of policymakers' commitment.
Keywords : Optimal monetary policy; Time inconsistency; Sustainable plans; Timelessperspective∗

Graduate School of Public Policy, the University of Tokyo. E-mail address: sunakawa@pp.u-tokyo.ac.jp.
11 Introduction
In New Keynesian (NK) models used in recent monetary policy analysis, the Ramsey-optimalcommitment policy usually yields higher social welfare than an optimal discretionary policy,

1

in which policymakers cannot commit to future policies.

Time-inconsistency plagues the

optimal commitment policy (Kydland and Prescott, 1977; Barro and Gordon, 1983) because
policymakers are tempted to abandon their previously announced policy and exploit theprivate sector's expectation.
To overcome the time-inconsistency problem, Kurozumi (2008) proposes the optimal
sustainable monetary policy by analyzing Chari and Kehoe's (1990) sustainable plans in
an NK model.

The optimal sustainable monetary policy is based on the more realistic

assumption that policymakers cannot access commitment technologies but can use reputation
among the private sector.

Therefore, when considering conduct of monetary policy, theoptimal sustainable monetary policy is an important alternative to the Ramsey-optimalcommitment policy.

However, Kurozumi's analysis considers only qualitative properties;i.e., it merely checks whether policymakers have incentives to deviate from the optimalcommitment policy.
This study presents a quantitative analysis of optimal sustainable monetary policies. As

the optimal sustainable policy is a strategy for the best sustain2
able equilibrium in the absence of commitment technologies. However, it is implemented byfollowing the optimal quasi-sustainable policy derived from the Lagrange method of Marcet
and Marimon (1994, revised in 1998 and 2011) assuming the presence of commitment tech-stated in Kurozumi (2008),

nologies. This study applies a version of the policy function iteration method in Kehoe andPerri (2002) to solve for the policy function of the optimal quasi-sustainable policy. While
doing this, it examines the equilibrium dynamics of optimal sustainable monetary policies,
such as impulse response functions and stochastic simulations.
This study nds that the optimal commitment policy is sustainable (i.e., the optimal
quasi-sustainable policy yields a result basically identical to the optimal commitment policy)
given a particular parameter set and a price markup shock as calibrated from U.S. data andcommonly found in the literature.

3

This nding contrasts to Kurozumi's nding that the

optimal commitment policy is not sustainable for some plausible parameters.

1 We assume that the optimal commitment policy considered here is under time-0

perspective. See a
discussion later in the introduction.
2 The optimal sustainable policy exploits an explicit punishment for the deviation from the current policy.
This out-of-equilibrium punishment in the optimal sustainable policy serves as the commitment technologies
in the optimal quasi-sustainable policy.
3 See Rotemberg and Woodford (1998), Woodford (2003), Adam and Billi (2006, 2007), Nakov (2008),
Giannoni (2010), and Bodenstein, Hebden, and Nunes (2012).
2

There are two types of the optimal commitment policy in the literature. No commitment
is made in the past at an initial period t0 . If the policymaker made commitment in periodt0 > −∞, the optimal commitment policy is under time-0 perspective. In contrast, such an
initial period is set in the innite past undertimeless perspective (Woodford, 1999). Dennis(2010) and Sauer (2010a,b) show that the optimal discretionary policy can be superior tothe optimal commitment policy under the timeless perspective when the lagged output gap
is large. The lagged output gap measures policymakers' previous commitment; the larger
the gap, the greater are policymakers' incentives to deviate from that commitment. In thisstudy, the optimal quasi-sustainable policy endogenously diminishes the steadfastness of
their commitment when the lagged output gap is large. In an economy with some alternative
parameters, the optimal quasi-sustainable policy yields higher social welfare than the optimalcommitment policy in terms of the unconditional expected utility. This occurs because theoptimal quasi-sustainable policy diminishes the steadfastness of policymakers' commitment
and reduces the volatility of ination and the output gap.

Related literature
Drawing upon game theory, Chari and Kehoe (1990) propose the concept of sustainable plansthat become optimal in dynamic decision-making for a player who cannot commit to futureplans in the present. Their standard is subgame perfect equilibria in repeated games between
a strategic player and an innite number of small agents. Chari and Kehoe (1990) follow
Abreu (1988) in using the worst sustainable equilibrium to characterize the entire set ofsustainable equilibria. The best sustainable equilibrium is then supported by the reputationof the strategic player among small agents. The equilibrium condition is summarized bythesustainability constraint, an inequality implying that any sustainable equilibrium outcome isabove the worst.
The source of the time inconsistency in NK models is not only the well-known ination
bias but also a stabilization bias following price markup shocks and a trade-o between
stabilizing ination and stabilizing the output gap. Kurozumi (2008) proposes the notionof optimal sustainable monetary policy after analyzing Chari and Kehoe's sustainable plansin an NK model.

He examines stabilization bias with an innite length of punishment

under the optimal discretionary policy and nds that the sustainability constraint is binding
for some plausible parameters and the upper bound of the markup shock.

Loisel (2008)

considers ination bias and stabilization bias with a nite duration of punishment under the
optimal discretionary policy and concludes that both can be overcome (i.e., the sustainability
constraint is not binding) by the reputation of the policymaker when the duration is a few
years. However, Kurozumi (2008) and Loisel (2008) examine only whether the sustainability

3constraint is binding and disregard the optimal sustainable policy itself.
Woodford (1999) overcomes time-inconsistency in the Ramsey-optimal commitment policy by disregarding timei.e., by assuming policymakers committed to future policies in
the indeterminate past. The optimality condition in the initial period then can be ignored
and a time-invariant policy rule can be derived. However, when the policies are evaluatedaccording to policymaker's objectives from date t0> −∞ forth, and taking the state of

the economy at date t0 as given, the optimal discretionary policy can be superior to the
optimal commitment policy under the timeless perspective (Dennis, 2010; Sauer, 2010a,b).
Thus, policymakers concerned with social welfare from date t0 forth may have incentive toabandon previous commitments.
Computations to solve time-inconsistency problems are dicult because analysis must
handle dynamic incentive constraints, which defy Bellman's principle of optimality. Marcet
and Marimon (1994, revised in 1998 and 2011) develop a Lagrange methodthe recursive
saddle point methodto consider incentive constraints in a dynamic economy. Kehoe andPerri (2002) apply the method in a two-country model with incomplete markets. Similar tothis study, they numerically solve for the policy function using a version of policy function
iteration method closely related to methods for handling occasionally binding constraints(e.g., Christiano and Fisher, 2000) because incentive constraints are occasionally binding.
Adam and Billi (2006) studied the optimal commitment policy with an occasionally binding
zero lower bound (ZLB) on nominal interest rates utilizing Marcet and Marimon's method.
This reminder of the paper proceeds as follows. Section 2 explains the model, the policy game, and computational procedures and calibrations. Section 3 discusses quantitative
properties of optimal sustainable monetary policies, such as impulse responses and stochastic
simulations. Section 4 presents an analysis using the alternative parameter sets in Kurozumi(2008) for comparison.

Section 5 concludes.

Analytical results and computational details

appear in the Appendix.

2 Framework for the analysis of optimal monetary policies
Section 2.1 presents a monetary model featuring a stabilization bias and demonstrates the4time-inconsistency problem.

Section 2.2 reproduces some results in Kurozumi (2008). The

optimal sustainable policy is a strategy for the best sustainable equilibrium.

It is con-ducted by following the optimal quasi-sustainable policy derived from the Lagrange method

4 The model here has a micro-founded structure. See Galí (2008) and Woodford (2003) for its details.

4

of Marcet and Marimon (1994, revised in 1998 and 2011). Sections 2.3 shows the computational procedure for solving for the policy function of the optimal quasi-sustainable policy,
as in Kehoe and Perri (2002). Section 2.4 explains the calibration.

2.1

A monetary model with stabilization bias

The economy contains an innite number of private agents and a monetary policymaker.
Private agents set prices under monopolistic competition in a staggered price setting. Up tothe rst-order approximation, their behavior is represented by the Phillips curve:

πt = βEt πt+1 + κxt + ut ,for all t(1)≥ 0, where πt is current ination, Et πt+1 is expected future ination, xt is theoutput gap, and ut is the exogenous price markup shock.

β ∈ (0, 1) is the discount factorand κ > 0 is the elasticity of ination to the output gap. The markup shock ut follows an
AR(1) process:ut = ρut−1 + εt ,(2)

for all t ≥ 0 and given u−1 = 0, where ρ ∈ (−1, 1) is persistence of the markup shock, and εt is2disturbance to the shock with a variance σε > 0. εt is assumed to be bounded per Kurozumi(2008) and Loisel (2008). The policymaker chooses a sequence of ination rates and output∞
gaps {πt , xt }t=0 to maximize social welfare. That is, the second-order approximation of therepresentative household's expected utility in Period 0 is

V0 ≡ −E0∞X

β t [πt2 + λ(xt − x∗ )2 ],

(3)

t=0
where λ > 0 is a weight to stabilize the output gap relative to ination and x∗

is the target

level of the output gap, which stems from monopolistic competition and distortion in thesteady state. Subsidies are assumed to oset the monopolistic distortion so that the steadystate is ecient (x∗= 0). Therefore, there is no ination bias, and stabilization bias is thesole source of time-inconsistency.
The literature usually considers two types of optimal policies for analyzing optimal monetary policy in NK models.

Under the optimal discretionary policy, the economy follows

a Markov equilibrium in which the state variable is the only markup shock. Policymakers
maximize (3) subject to Equation (1) for all t ≥ 0, given expected future ination Et πt+1 .

5

When x∗

= 0, the optimal discretionary policy is characterized by the optimality condition
πt = −(λ/κ)xt ,for all t ≥ 0.

(4)

In this instance, Equations (1) and (4) determine ination and the output

gap given the exogenous shock process (2).

Under the optimal commitment policy, thepolicymaker can commit to future policies and inuence private agents' expectations. The
optimal commitment policy is characterized by the optimality condition

πt = −(λ/κ)(xt − xt−1 ),(5)for all t ≥ 0 with an initial condition x−1 . Equations (1) and (5) then determine inationand the output gap given the state variable xt−1 and exogenous shock process (2).
Figure 1 graphs how ination and the output gap are determined under the optimalcommitment policy with the initial condition x−1 = 0 and under the optimal discretionary
policy. It demonstrates the time inconsistency of the optimal commitment policy.

5Inationand the output gap are determined at the intersection of the upward-sloping Phillips curve
and the downward sloping optimality condition. A positive markup shock occurs in Period
0, shifting the Phillips curve upward and inducing a rise in ination and a drop in the output
gap. Under the optimal discretionary policy, given no future ination (E0 π1 = 0), there is atrade-o between stabilizing ination and stabilizing the output gap. Both are determined at

D0 . Under the optimal commitment policy, the policymaker can commit to future deationin Period 1 (E0 π1 < 0), which shifts the Phillips curve downward and accommodates thistrade-o by osetting the markup shock. Ination and the output gap are determined at C0 ,which surpasses D0 in social welfare. However, the optimality condition shifts downward in
Period 1 because of the commitment made in Period 0. Therefore, ination and the output
gap are determined at C1 , and it is worse than at D1 . If the markup shock in Period 1 isabsent, the policymaker has incentive to abandon previous commitments and to stabilize
both ination and the output gap in Period 1.
Two types of commitment policies pertain to the lagged output gap in Period 0, x−1 . In

time-0 perspective, the initial period is period 0 and x−1 = 0 because no commitment ismade in Period 0. In the timeless perspective, the initial period is in the indeterminate past,theand the initial condition x−1 is predetermined. In the timeless perspective, even in Period
0, the policymaker may have incentive to deviate from previous commitments and by doing
so can exploit private agents' ination expectations. The optimal discretionary policy canbe superior to the optimal commitment policy under the timeless perspective (Dennis, 2010;

5 A similar gure appears in Woodford (1999).
6

Sauer, 2010a,b), even in Period 0, because the initial condition is predetermined.

[Figure 1 is inserted here]

2.2

Optimal sustainable policy

To overcome the time-inconsistency in the optimal commitment policy, Subsection 2.2 considers a policy game between an innite number of private agents and a policymaker.

2.2.1 Sustainable equilibrium and the sustainability constraint
The policymaker acts rst, after which private agents respond to its policy.

The history

relevant to agents' behavior is given by h0 = u0 and ht = (ht−1 , xt−1 , ut ) for all t > 0.

6

The

policymaker's strategies are given by xt = σt (ht ) and contingent plans for any possible future
histories(σs )s≥t+1 .

Given current history(ht , xt ) and (σs )s≥t+1 , private agents' reaction

functions are given by πt = ft (ht , xt ) and contingent plans for any possible future histories(fs )s≥t+1 . Sustainable equilibrium of the model in the previous section is dened as follows:Denition 1. A sustainable equilibrium of the model is a pair (σ, f ) of a monetary policy
strategy and private agents' reaction to the policy strategy such that(i) given the policy strategy σ and current history (ht , xt ), the continuation of private
agents' reaction f satises

ft (ht , xt ) = βEt [ft+1 (ht+1 , σt+1 (ht+1 ))] + κxt + ut ,t≥0

fs (hs , σs (hs )) = βEs [fs+1 (hs+1 , σs+1 (hs+1 ))] + κσs (hs ) + us , s ≥ t + 1
for all possible future histories induced by σ .
(ii) given private agents' reaction f and and current history ht , the continuation of thepolicy strategy σ solves

max(σ̃s )s≥t −Et∞
X

β s−t [[fs (hs , σ̃s (hs ))]2 + λ[σ̃s (hs )]2 ]

s=t
s.t.

fs (hs , σ̃s (hs )) = βEs [fs+1 (hs+1 , σ̃s+1 (hs+1 ))] + κσ̃s (hs ) + us

for all possible future histories induced by (σ̃s )s≥t .

6 Note that π is excluded from the public history because private agents are policy takers and only thetpolicymaker may have incentive to deviate from current policy, as in Chari and Kehoe (1990) and Kurozumi(2008).

7

Underany sustainable equilibrium, the policymaker acts so that the sustainability con-straint,− Et∞
X

β s−t (πs2 + λx2s ) ≥ W d (ut ),

(6)

s=t
holds in every period t



βσε2
1+κ2 /λ2+u< 0 is the≥ 0, where W d (ut ) = − (1−βρ+κ2 /λ)2 (1−βρ2 )t1−β7worst sustainable equilibrium outcome, which is induced by the optimal discretionary policy.
Kurozumi (2008) proves the following propositions:

Proposition 1. The rational expectations equilibrium (REE) under discretionary policy is
the worst sustainable equilibrium of the model.
Proposition 1 shows that W
equilibria.

d(ut ) attains the lowest value among outcomes of sustainable

Given the sustainability constraint, the worst outcome assures that the bestsustainable equilibrium is included in the set of all sustainable equilibria.

Proposition 2. Any arbitrary pair (π, x) of contingent sequences of ination rates andoutput gaps is an outcome of a sustainable equilibrium if and only if; (i) the pair (π, x)satises (1) in every period t ≥ 0 and (ii) the inequality (6) holds in every period t ≥ 0.
∞
Proposition 2 shows that any arbitrary sequence {πt , xt }t=0 is an outcome of a sustainableequilibrium if and only if Equations (1) and (6) are satised for all t ≥ 0. These constraints
dene the entire set of sustainable equilibrium outcomes.
The optimal sustainable policy is a strategy for the best sustainable equilibrium.

To

analyze it, Kurozumi (2008) derives the optimal quasi-sustainable policy, which induces thebest sustainable equilibrium outcome in thepresence of commitment technologies.

The∞
policymaker chooses {πt , xt }t=0 to maximize (3) subject to Equations (1) and (6) for t ≥ 0.
Kurozumi argues that the optimal sustainable policy now becomes a policy strategy which
species to continue the optimal quasi-sustainable policy as long as it has been adopted in thepast; otherwise, the strategy species to switch to the optimal discretionary policy forever.
This implies that the optimal sustainable policy is conducted by following the optimal quasisustainable policy and yields the best sustainable equilibrium outcome in theabsence ofcommitment technologies (Kurozumi, 2008, p. 1278).
If the policymaker abandons its previous policy, the trigger strategy species switching
to the optimal discretionary policy, which is o the equilibrium path, indenitely.

Onemight wonder if other equilibria would be chosen if the trigger strategy involved milder
punishmentfor example, the nite duration in Loisel (2008).

Since no theory currently

7 See Appendix A.1 for the derivation of W d (u ). There are no temporary gains from deviating, as in atmodel with inationary bias (Ireland, 1997), because the policymaker acts rst.
8

suggests how other equilibria are chosen, it is assumed that the policymaker and private
agents switch to the optimal discretionary policy indenitely, rather than to a less punishingpolicy.

2.2.2 Characterizing the optimal sustainable policy
To characterize the optimal sustainable policy, this study uses the Lagrange method ofMarcet and Marimon to derive the optimal quasi-sustainable policy.

The Lagrangian in

Period 0 is set up as

L0 ≡ E0∞X
β t −(πt2 + λx2t ) + 2φt (πt − βEt πt+1 − κxt − ut )t=0"
−ϕt Et∞X#)β s−t (πs2 + λx2s ) + W d (ut )

,

(7)

s=t
where φt and ϕt are Lagrange multipliers on each constraint (1) and (6) for t ≥ 0. Abel's summation formula is used with the law of iterated expectations E0

P∞

Pt
t

P∞

t
t=0 β ϕt [Et

P∞

s=t β

s−t

(πs2 + λx2s )] =22i=0 ϕi (πt +λxt ) to eliminate the innite sum in the sustainability constraints,
t=0 β
P∞ t
P∞ t
and E0t=0 β (φt − φt−1 )πt to eliminate the forward term of
t=0 β φt (πt − βEt πt+1 ) = E0

E0

the ination rate:

L0 = E0∞X
β t −Ψt (πt2 + λx2t ) + 2(φt − φt−1 )πt − 2φt (κxt + ut ) − ϕt W d (ut ) ,t=0
where Ψt = Ψt−1 + ϕt with an initial condition Ψ−1 = 1 and φ−1 = 0.

8

From the rst orderconditions of πt and xt , the optimality condition of the optimal quasi-sustainable policy is

λπt = −
κ




Ψt−1
xt −
xt−1 ,
Ψt(8)

for all t ≥ 0 with an initial condition x−1 . This is dierent from the optimality conditionsof the optimal commitment policy (5) and the optimal discretionary policy (4). In the optimal quasi-sustainable policy (8), the lagged output gap xt−1 also appears in the optimality
conditions, and the ratio of the sum of Lagrange multipliers on the sustainability constraint

Ψt−1 /Ψt = Ψt−1 /(Ψt−1 + ϕt ) ∈ (0, 1] appears as a coecient on the lagged output gap.
If the sustainability constraint is slack, the optimal quasi-sustainable policy appears identical to the optimal commitment policy (see Equation (5) and Equation (8) with Ψt−1 /Ψt =1). If the sustainability constraint is binding, ϕt > 0, then 0 < Ψt−1 /Ψt < 1 and the optimal
8 The Lagrangean corresponds to the saddle point problem in a recursive form considered in Marcet andMarimon (2011).

9

quasi-sustainable policy is intermediate between the optimal commitment policy and the optimal discretionary policy (see Equations (5) and (4)). Under the optimal quasi-sustainable
policy, the policymaker endogenously reduces the steadfastness of its commitment via thelagged output gap, if the current period sustainability constraint is binding.

2.3

9Solving for the optimal quasi-sustainable policy

In the optimal quasi-sustainable policy, the equilibrium conditions in period t are summarized
as

Ψt−1
κ
xt−1 ,
x t = − πt +λΨtΨt = Ψt−1 + ϕt ,πt = βEt πt+1 + κxt + ut ,∞X
β s−t (πs2 + λx2s ) ≥ W d (ut ),−Et
s=t
for t ≥ 0, given initial conditions x−1 and Ψ−1= 1. This is the four-equation system of∞∞four endogenous variables {xt , πt , Ψt , ϕt }t=0 given a sequence of exogenous shock {ut }t=0 .
Therefore, the system can be solved in principle, but not via the usual methods because
there is an innite sum of future variables in the sustainability constraints, which occasionally bind given the exogenous stochastic process. Instead, a version of the policy function
iteration method is used, as in Kehoe and Perri (2002).
the current policy is dened as Vt ≡ −Et

P∞

s=t β

s−t

10

To this end, the value of continuing(πs2 + λx2s ) (Note that the policymakermaximizes social welfare in Period 0, V0 ). The system is written recursively as

Vt = −(πt2 + λx2t ) + βEt Vt+1 ,κ
xt = − πt + zt xt−1 ,λπt = κxt + ut + βEt πt+1 ,Vt ≥ W d (ut ),where

zt ≡ Ψt−1 /Ψt ∈ (0, 1] is the steadfastness of commitment, which equals 1 if thesustainability constraint is slack, and less than 1 if it is binding.

ut and xt−1 are the state

9 One may ask whether the binding constraint imposes a cost on the policymaker that might increase

its incentive to abandon the optimal quasi-sustainable policy. The question is moot, as the presence ofcommitment technologies is assumed.
10 In Kehoe and Perri (2002), the worst sustainable equilibrium value is a function of the endogenous statevariable. In this study, it is a function only of the exogenous state variable.

10variables. Vt , πt , xt and zt are the jump variables.

11

Let s = (u, x−1 ) ∈ U × X where U and

X are closed sets. A state-space representation is then obtained:
V (s) = −([π(s)]2 + λ[x(s)]2 ) + βXp(u0 |u)V (u0 , x(s)),u0κx(s) = − π(s) + z(s)x−1 ,λXπ(s) = κx(s) + u + βp(u0 |u)π(u0 , x(s)),
u0dV (s) ≥ W (u),where π(s), x(s), and z(s) are the policy functions and V (s) is the value function.

p(u0 |u)represents a transition probability matrix, which approximates the AR(1) process in Equation (2).

This system has a recursive structure with regard to V (s) and π(s), making it

possible to apply a policy function iteration method to the system. The occasionally binding constraint V (s) ≥ W
grid point s∈ U × X.

d(u) must be addressed; hence, two cases are considered at each

The constraint is slack or binding.

Appendix A.2 provides thecomputational details.
This approach characterizes the behavior of the optimal quasi-sustainable policy directly
and solves the equilibrium conditions for the policy functions instead of solving the sustainability constraint for the lower bound of the discount factor β , as explained in Section 4.
Then the approach calculates the probabilities of binding constraints (and other statistics),
unlike Kurozumi. These results are quantitative; Kurozumi's results are qualitative, as hechecked only whether the optimal commitment policy is sustainable.

2.4

Calibration

Parameter values (β, κ, λ, ρ, σε ) are calibrated, and the ranges of state spaces U and X are set.
Table 1 summarizes the parameter values. For the slope of the Phillips curve and the weight2
to the output gap, κ = 0.024 and λ = 0.048/4 = 0.003 are chosen. These were originallyestimated by Rotemberg and Woodford (1998) and Woodford (2003) and are common in theliterature (Adam and Billi, 2006; 2007; Nakov, 2008; Giannoni, 2010; Bodenstein, Hebden,
and Nunes (2012)).

12

Estimates by Adam and Billi (2006; 2007) serve as markup shock

parameters; ρ = 0.0 and σε = 0.154 based on quarterly U.S. data from 1983:1 to 2002:4.
Finally, the discount factor to match a 3.5% annual real interest rate is calibrated; it implies

11 z

t = Ψt−1 /(Ψt−1 + ϕt ) is a jump variable because it depends on the Lagrange multiplier in the current
period ϕt .
12 Note that π is the quarterly ination rate in the model, whereas Woodford estimates using an annualtination rate. Therefore, I divide the original estimated value of λ by 42 .

11that β = 1/(1 +3.5%) = 0.9913.
4

β = 0.9 is used to demonstrate the dierence between the

optimal quasi-sustainable policy and the two other. Several alternative parameters sets areused to compare quantitative results with Kurozumi's qualitative results in Section 4. The

13

upper bound of the disturbance is mσε and m = 6 is set.

Maximum and minimum values of

X are chosen to be consistent with the optimal volatility of the output gap. The number ofgrids is nu = 31 for U and nx = 15 for X . Tauchen's (1986) method is used to approximate

0
the AR(1) process (Equation (2)) with parameters (ρ, σε , m) for U and p(u |u). Use of theTauchen approximation makes ut ∈ U virtually bounded, as in Kurozumi (2008) and Loisel(2008). For comparison, the policy and value functions under the optimal commitment anddiscretionary policies are numerically obtained by policy function iteration method.
[Table 1 is inserted here]

3 Numerical results
This section shows impulse response functions of disturbances to the markup shock andstochastic simulations. Section 3.1 demonstrates dierences among the optimal quasi-sustainable
policy, the optimal time-0 commitment policy (i.e., the optimal commitment policy under thetime-0 perspective), and the optimal discretionary policy with a low discount factor β = 0.9
and a large shock. The optimal quasi-sustainable policy is intermediate between the optimal
time-0 commitment policy and the optimal discretionary policy. By contrast, in Section 3.2,
using the high discount factor β = 0.9913 common in the literature, the dierence between
the optimal quasi-sustainable policy and the optimal commitment policy ispractically zero.

This result contrasts with Kurozumi's result with the upper bound of the markup shock andlagged output gap.

3.1

Impulse responses and policy functions: the case with β = 0.9

Figure 2 computes the impulse response functions of a large disturbance ε0 (5 × 0.154% =0.77%), with a low discount factor β = 0.9 and initial condition x−1 = 0, under the time-0
perspective.

The chosen discount factor β= 0.9 is extremely low for a quarterly model.

Therefore, the policymaker gives less consideration to a long-term relationship with private
agents and the optimal commitment policy is less likely to be sustainable. Stochastic simulations with a high discount factor and reasonably-sized shocks appear in the next subsection.
Under the optimal discretionary policy, the economy responds in Period 0 only; as there isno persistence in the markup shock (ρ = 0), ination rises and the output gap drops. Under

13 This is larger than m = 3, the value commonly used in the literature (Judd, 1992).
12the optimal time-0 commitment policy, the policymaker promises deation after Period 1 toaccommodate a trade-o between stabilizing ination and the output gap in Period 0.
Under the optimal quasi-sustainable policy, the policymaker promises less deation after
Period 1 and less stabilization in Period 0 than under the optimal time-0 commitment policy.
Period 1 features an incentive for the policymaker to deviate from the past commitment,
and the optimal quasi-sustainable policy diminishes the steadfastness of the policymaker's
commitment. The sustainability constraint in Period 1 is binding (ϕt > 0), and the ratio of
the sum of Lagrange multipliers on the sustainability constraint Ψt−1 /Ψt takes a value below
1.
[Figure 2 is inserted here]
The optimal quasi-sustainable policy falls between the optimal time-0 commitment policy
and the optimal discretionary policy. To inspect the mechanism behind this result, the policyafunction of zt (ut , xt−1 ) = Ψt−1 /Ψt is examined in Figure 3. . In Period 0, the markup shockrises (u0 > 0) and the lagged output gap is zero (x−1 = 0). Therefore, the value of continuing
the current policy V (u0 , x−1 ) exceeds the value of switching to the optimal discretionary
policy Wd(u0 ), and the sustainability constraint in Period 0 is slack (V (u0 , x−1 ) > W d (u0 )).

In Period 1, the markup shock reverts to the mean (u1= 0), but the lagged output gap

remains negative (x0 < 0). Therefore, V (u1 , x0 ) is lower than W
constraint in Period 1 is binding (V (u1 , x0 ) = Wdd(u1 ), and the sustainability(u1 )). This makes the coecient on thelagged output gap less than 1 (z(u1 , x0 ) = Ψ0 /Ψ1 < 1).
The dierence between the conditional expected utility induced by the optimal timelesscommitment policy Vcdiscretionary policy W(ut , xt−1 ) and the conditional expected utility induced by the optimald(ut ) is analytically obtained (Figure 3.b ).14 − max{V c (ut , xt−1 )/W d (ut )−1, 0} is used for a measure of the dierence. If V c (ut , xt−1 ) < W d (ut ), it takes a value lessthan 0.

a shows the region wherein the sustainability constraint is binding and the policy

Figure 3.

function of z(ut , xt−1 ) takes a value less than 1. It largely corresponds to the region where theoptimal discretionary policy is superior to the optimal timeless commitment policy in termsbof the social welfare in Period t (Figure 3. ). When the absolute value of ut is smaller and/or
the absolute value of xt−1 is larger, z(ut , xt−1 ) < 1 and Vc(ut , xt−1 ) < W d (ut ) are more likely

to hold. This is because the value of policymakers' commitment is lowerthe outside option
is betterwhen ut is smaller, whereas there is more incentive for the policymaker to abandon
the current policy when xt−1 is larger.

14 See also Appendix A.1.

Sauer (2010a) shows a similar gure to argue that the optimal discretionary
policy can be superior to the optimal commitment policy under the timeless perspective, when the economy
is currently far from the steady state.
13

[Figure 3 is inserted here]

Note that the optimal quasi-sustainable policy is also intermediate between the optimal
time-0 commitment policy and the optimal discretionary policy, even in Period 0 when thesustainability constraint is slack. In Figure 4, the policy functions of π(ut , 0) and x(ut , 0) areplotted. When the lagged output gap xt−1 equals 0, the sustainability constraint in Period

t is slack (z(ut , 0) = 1) and V c (ut , 0) > W d (ut ) holds for every ut ∈ U , as Figure 3 shows.
When the markup shock is large, however, the optimal quasi-sustainable policy diers fromthe optimal time-0 commitment policy and the optimal discretionary policy.

This occurs

because rational expectations take into account the future possibility of binding constraints.
The markup shock reduces the output gap in Period 0, and the sustainability constraint
to bind via the dropped lagged output gap in Period 1.

Thus, the policymaker weakens

stabilizing ination and the output gap in Period 0.

[Figure 4 is inserted here]

3.2

Stochastic simulations

Stochastic simulations with a reasonably sized markup shock examine how the time-inconsistency
problem is relevant to actual conduct of monetary policy. The same exercise is performed
for each of the optimal quasi-sustainable, commitment, and discretionary policies using thenumerically obtained policy functions.
Using the sequence obtained by stochastic simulations of the policy functions, the probability of binding sustainability constraint P r{z < 1} is calculated, which exceeds 0 when
the optimal commitment policy is not sustainable. The unconditional expected (discounted
sum of ) utility V̄

= EV (xt−1 , ut ) = (1 − β)−1 (σπ2 + λσx2 ) is also calculated, where σπ2 and σx2

are the unconditional variances of ination and the output gap. The conditional expectedutility V (0, 0) is obtained by evaluating the value function at (u, x−1 ) = (0, 0).
The optimal time-0 commitment policy always surpasses the conditional expected utilityof the optimal quasi-sustainable policy because the policymaker maximizes the conditional
expected utility itself in Period 0 given (u, x−1 ) = (0, 0). A less constrained problem without
the sustainability constraint is solved for the optimal time-0 commitment policy than the
problem with the constraint for the optimal quasi-sustainable policy. By contrast, the unconditional expected utility of the optimal quasi-sustainable policy can surpass that of the
optimal timeless commitment policy, for it reects the variation of the lagged output gap
in the initial period and diminishes the steadfastness of commitment via the lagged output

14

gap.

15

The result of stochastic simulations appears in Table 2. When β = 0.9, the probability ofbinding constraints P r{z < 1} is approximately 6.8%. This gure seems small (once in ve
years in a quarterly model), but it slightly aects the optimal volatilities of ination and the22
output gap (σπ and σx ) and hence unconditional expected utility (V̄= (1 − β)−1 (σπ2 + λσx2 )).

The optimal quasi-sustainable policy yields higher social welfare than the optimal timelesscommitment policy with regard to unconditional expected utility because it diminishes thesteadfastness of commitment and reduces volatility of ination and the output gap.

16

However, when β = 0.9913, the dierence between the optimal quasi-sustainable policy
and the optimal commitment policy istional expected utility.

17

practically zero in terms of conditional or uncondi-

In other words, policymakers have no incentive to deviate fromthe Ramsey-optimal commitment policy under the parameter set commonly used in theliterature.

18[Table 2 is inserted here]

4 Dierence from Kurozumi (2008)
The framework presented in Sections 2.1 and 2.2 originates in Kurozumi (2008). This study
uses a method dierent from Kurozumi, inducing dierences in the simulation result. Thissection explains such dierences.
To evaluate the sustainability constraint, Kurozumi (2008) assumes that disturbance ε
is bounded by mσε where m is an integer, and the upper bounds on the markup shockand lagged output gap of the optimal commitment policy are ut ≤ B = mσε /(1 + |ρ|) and

xt−1 ≤ |ax |B/(1 − bx ), where ax and bx are the coecients of the policy function of theoutput gap (xt = ax ut + bx xt−1 ). Then Kurozumi calculates a lower bound on the discount

15 This is consistent with the argument that the optimal discretionary policy can be superior to the optimal

timeless commitment policy when the lagged output gap is large (Dennis, 2010; Sauer, 2010a,b).
16 Jensen and McCallum (2002) point out that other policy rules are better than the optimal timelesscommitment policy to maximize conditional or unconditional expected utility. However, rules proposed in
Jensen and McCallum (2002) are not time-consistent, whereas the optimal quasi-sustainable policy is.
17 This result is only up to a precision of numerical solutions. Approximation errors are very small especiallywhen β = 0.9913, which are computed in Appendix A.2.
18 As Section 2.2.1 explained, it is assumed that if the policymaker abandons the previously adopted policy,
he/she and all private agents switch to the optimal discretionary policy indenitely. If other equilibria werechosen with a less harsh trigger strategy, the policymaker may have incentive to deviate from the Ramseyoptimal commitment policy.

15

19

factor for which the optimal commitment policy is sustainable. This is given by

β = inf{β ∈ (0, 1) : F (β̃) ≥ 0∀β̃ ∈ (β, 1)},
where

F (β) = V c (B, |ax |B/(1 − bx )) − W d (B).
This approach, however, can nd only whether the sustainability constraint is binding forthe maximum size of the markup shock and a particular value of β .

The policy function

approach solves for the policy function instead, and calculates the probability of a binding
constraint with a reasonably sized markup shock.
Table 3.

a shows the probability of binding constraints for parameter values (κ, λ, ρ) con20

sidered in Kurozumi (2008).

The probability of a binding constraint is 0 except whenλ/κ is high. The optimal commitment policy is sustainable for some plausible parameters,
including the baseline parameters in Section 3. This nding contrasts with Kurozumi's nding. In Table 2 of Kurozumi (2008), the optimal commitment policy is not sustainable with

β = 0.99i.e., β = 0.99 < β for most parameter values of (κ, λ, ρ).21 Also, Table 3.a showsthat the optimal commitment policy is less likely to be sustainable when λ/κ or ρ is higher,
i.e., the probability of binding constraints is higher. This nding coincides with Kurozumi's
nding for λ/κ, but not for ρ.
The variation of the lagged output gap, which Kurozumi's analysis disregards, is the
key to understand the dierences. As Section 3.1 and Figure 3 explain, two opposite forces
threaten the sustainability of the optimal commitment policy. First, the smaller the absolutevalue ofut is, the better is the outside option of the optimal discretionary policy, andpolicymakers have greater incentive to abandon their current policy.
the absolute value ofSecond, the larger

xt−1 is, the more likely policymakers are to abandon its previous

commitment. When ρ is higher, ut is larger, which also increases the lagged output gap in
the next period xt .

The eect of the latter dominates that of the former.

Therefore, thehigher the ρ, the less likely is the optimal commitment policy sustained. Kurozumi disregards
the latter eectthe variation of the lagged output gap.
Table 3.

b compares the ratio of the unconditional mean of the output gap under thecdoptimal commitment policy to that under the optimal discretionary policy (σx /σx ), which
19 The dierence in social welfare between the optimal commitment and discretionary policies V c (u , xtW d (ut ) is analytically obtained in Appendix A.1.

20 The other parameters are xed (β, σ , m) = (0.99, 0.154, 6) . The standard deviation σt−1 )−

ε
ε is irrelevant to
the probability of a binding constraint because we can rewrite F (β) = σε2 F̃ (β).
21 The best Markov quasi-sustainable policy is also calculated in Table 3 of Kurozumi (2008); it is alinear function of the markup shock. This study calculates the optimal quasi-sustainable policy, which is a
non-linear function of the markup shock and lagged output gap.

16

can be calculated analytically. When λ/κ and/or ρ is higher, the output gap is relatively morevolatile under the optimal commitment policy. The sustainability constraint is more likely to
be binding, and the optimal sustainable policy diminishes the steadfastness of commitment
via the lagged output gap.
If the lower bound of discount factor β is calculated using the method in Kurozumi (2008)
but with the unconditional variances of the markup shock and lagged output gap, also used in
Loisel (2008), the result obtained is consistent with that using the policy function iterationcmethod (Table 3. ).

The higher the ρ, the less likely is the optimal commitment policy

sustained (i.e., the higher the β ) because the variation of the lagged output gap is taken into
account.
[Table 3 is inserted here]

5 Conclusion
This study solved for the policy functions of the optimal quasi-sustainable monetary policy to invite a quantitative analysis of optimal sustainable monetary policies as proposedby Kurozumi (2008). Stochastic simulations show that the dierence between the optimal
quasi-sustainable policy and the optimal commitment policy ispractically zero for a pa-

rameter set calibrated using U.S. data. In an economy with some alternative parameters,
the optimal quasi-sustainable policy yields higher social welfare in terms of unconditional
expected utility than the optimal commitment policy. The dynamics of the lagged output
gap provide the key to such dierences. When the lagged output gap, which measures policymakers' previous commitment, is large, the optimal quasi-sustainable policy endogenously
diminishes the steadfastness of policymakers' commitment via the lagged output gap.
One interesting direction for future research is to introduce a ZLB on the nominal interest
rate into the model. Following a sizable economic shock, the presence of ZLB is also crucial
for conduct of monetary policy.

22

When a ZLB is explicitly considered, markup shocks andreal rate shocks aect the short-term dynamics of ination and the output gap because
the ZLB prevents policymakers from controlling the real economy in the short run.

For

quantitative analysis, the optimization problem is solved with the sustainability constraint
and the ZLB, both of them occasionally binding constraints, by applying techniques used in23

this study.

22 See Adam and Billi (2006; 2007), Nakov (2008) and Bodenstein, Hebden, and Nunes (2012).

Nakata(2014) examines sustainable plans in a NK model with ZLB, but he checks only whether the sustainability
constraint is binding, like Kurozumi (2008) and Loisel (2008).
23 However, the optimal discretionary policy might not induce the worst sustainable equilibrium value,
which is on the right-hand side of the sustainability constraint, when ZLB is imposed.
17

24

Solving a non-linear version of the model also remains for further research.

This study

considers the non-linearity induced by the sustainability constraint; hence, considering theoriginal non-linear model is a logical extension of the linear-quadratic framework consideredhere.

Acknowledgement

An earlier version of this paper is circulated titled Computing Op-

timal Sustainable Monetary Policy.

The author is especially grateful to Bill Dupor for

valuable guidance and support. The author appreciates the comments and discussions fromanonymous referees, Klaus Adam, Martin Bodenstein, Richard Dennis, Paul Evans, Ippei
Fujiwara, Marc Giannoni, Hirokazu Ishise, Kenneth Judd, Aubhik Khan, Paul Klein (editor), Tomiyuki Kitamura, Takushi Kurozumi, Bin Li, Mario Miranda, Taisuke Nakata,James Peck, Fabrizio Perri, Paulina Restrepo-Echavarria, Katsuyuki Shibayama, Etsuro Shioji, Tamon Takamura, Julia Thomas, and Yuichiro Waki, as well as seminar participants at
The Ohio State University, University of Kent, Hitotsubashi University, The University ot
Tokyo, the Far East and South Asian Meeting of the Econometric Society 2009, International
Conference on Computing in Economics and Finance 2010, the Econometric Society World
Congress 2010, and the Royal Economic Society Annual Conference 2012. Errors remaining
are the author's.

24 See Anderson, Kim, and Yun (2010) and Van Zandweghe and Wolman (2010) for the Calvo-type (1983)

price adjustment model.

18

A Appendix
A.1

Analytical solutions

The policy rules for the optimal commitment policy and the optimal discretionary policy are
given by:xct = ax ut + bx xct−1 , πtc = aπ ut + bπ xct−1 ,πtd = cπ ut ,xdt = cx ut ,where

ax = −(κ/λ)/[β(b+ − ρ)] < 0,aπ = (1/[β(b+ − ρ)]) > 0,bx = b− ∈ (0, 1),bπ = (λ/κ)(1 − b− ) > 0,cx = −(κ/λ)/(1 − βρ + κ2 /λ) < 0, cπ = 1/(1 − βρ + κ2 /λ) > 0,
and b

±

= [(1 + β + κ2 /λ) ± ((1 + β + κ2 /λ)2 − 4β).5 ]/(2β) is the solution of a quadratic

equation f (b) ≡ βb2

− (1 + β + κ2 /λ)b + 1 = 0. Note that b+ > 1 and b− < 1.

For measures of social welfare, expected utilities conditioned on initial conditions (ut , xt−1 )are given by25
c2π + λc2xβσε22
W (ut ) = −
ut +
,
(9)
1 − βρ21−βb2π + λb2x 2
βa2x (b2π + λb2x )
1
βσε2
c
2
2
2
V (ut , xt−1 ) = −x − aπ + λax +
ut +
(10)
.
1 − βb2x −1
1 − βb2x
1 − βρ21−βdAlso, unconditional variances of the markup shock and output gap are given by

σu2 =
A.2

2 (1+b ρ)σ 2
σε2x
ε, (σxc )2 = (1−ba2x)(1−b(σxd )2 = c2x σu2 .
2 ,2x ρ)(1−ρ )x
1−ρComputation

A.2.1 Numerical algorithm
To compute the policy function, a version of the policy function iteration method withoccasionally binding constraints is used, as in Kehoe and Perri (2002). Let s = (u, x−1 ) ∈25 The form of conditional expected utility for the optimal commitment policy is dierent from Equation(6) in Kurozumi (2008), because (b2π + λb2x )(1 − βb2x )−1 βax bx + aπ bπ + λax bx = 0 is utilized.
19

U × X where U and X are closed sets. A state-space representation is written as:
V (s) = −([π(s)]2 + λ[x(s)]2 ) + βXp(u0 |u)V (u0 , x(s)),u0κx(s) = − π(s) + z(s)x−1 ,λXπ(s) = κx(s) + u + βp(u0 |u)π(u0 , x(s)),
u0

V (s) ≥ Ŵ d (u),where V (s), π(s), x(s) and z(s) are the policy functions and Ŵd(u) is the worst sustainableequilibrium value induced by the discretionary policy, which is computed numerically. The
algorithm is as follows:
1. Set the initial guess of the functions V(0)(s) and π (0) (s) where s = (u, x−1 ) is each grid

point on U × X .
2. Solve the relevant equations for the values (Vgiven the functions V
3. Set new functions V

(i−1)

(i)

i,s

, π i,s , xi,s , z i,s ) for every grid point s,(u0 , x) and π (i−1) (u0 , x).

(s) = {V i,s }s∈U ×X and π (i) (s) = {π i,s }s∈U ×X .

4. Iterate 2-3 until the functions V(i)(s) and π (i) (s) converge at each grid point.

In Step 2, there are two possible binding patterns of the sustainability constraint: (i) the
constraint is slack or (ii) the constraint binds.
(i) z

i,s

= 1. Then I can solve
V i,s = −([π i,s ]2 + λ[xi,s ]2 ) + βXp(u0 |u)V (i−1) (u0 , xi,s ),u0

κ
xi,s = − π i,s + x−1 ,λXi,sp(u0 |u)π (i−1) (u0 , xi,s ),π = κxi,s + u + β
u0
for the values of (x(ii) zi,si,s, π i,s , V i,s ).

∈ (0, 1) and V i,s = Ŵ d (u). I can solveŴ d (u) = −([π i,s ]2 + λ[xi,s ]2 ) + βXp(u0 |u)V (i−1) (u0 , xi,s ),u0

κ
xi,s = − π i,s + z i,s x−1 ,λXi,sπ = κxi,s + u + βp(u0 |u)π (i−1) (u0 , xi,s ),u020

for the values of (xi,s, π i,s , z i,s ).

xi,s may not be on the grid points. Therefore, the functions need to be approximated.
Spline interpolation is used between the grid points X , and linear outerpolation is usedoutside of X .

P

26

More specically, the conditional expectations are approximated as hvi (x) =uj p(uj |ui )V (uj , x) and hπi (x) =P

uj p(uj |ui )π(uj , x) for each i = 1, ..., nu , by cubic splines.
d
The worst sustainable equilibrium value Ŵ (u) is computed easier. From Equations (1)and (4):



κ2
1+
λ


πt = βEt πt+1 + ut .

It can be written as



X
κ2
1+
π(ui ) = βp(uj |ui )π(uj ) + ui ,λjThis is linear in π(ui ) for each grid u1 , u2 , ..., unu . Equivalently,2

1 + κλ − βp(u1 |u1 )







−βp(u1 |u2 )···−βp(unu |u1 )1 + κλ − βp(u2 |u2 ) · · ·−βp(unu |u2 )−βp(u2 |u1 )
2

.
.
.

π(u1 )u1−βp(u2 |unu )which can be easily solved for π(ui ). Having π(ui ), substituting (4) into (3) yields:κ2Ŵ (ui ) = − 1 +λdwhich can be solved similarly for Ŵ

d




 

  π(u2 )   u2  =  . ,
.
.
..

  . 
.
.
.
.
.

  . 
κ2
π(unu )
· · · 1 + λ − βp(unu |unu )
unu

.
.
.

−βp(u1 |unu )[π(ui )]2 + βXp(uj |ui )Ŵ d (uj ),j(ui ).

A.2.2 Approximation errors
The optimal quasi-sustainable policy is obtained by numerical method, therefore, the accuracy of the solution is needed to be assessed. Following Adam and Billi (2006), the residualfunction is computed:R(s̃) = −V (s̃) − ([π(s̃)]2 + λ[x(s̃)]2 ) + βhvi (x(s̃)),26 It may be justied by observing that the policy function in the optimal commitment policy is linear,which is obtained by solving the problem without the sustainability constraint.

21

where s̃ ∈ X̃ × U has ner grid points than the original ones to solve for the policy function.
A total of 201 grids are used for X̃ compared to the original 15 grids on X

27

. Note thatR(s) = 0 exactly holds on s ∈ X × U . Linear interpolation is used to evaluate V (s̃), π(s̃),
and x(s̃), which are o the original grids.
relative errors are calculated as e

abs

For the baseline parameters, the absolute and= kR(s̃)k∞ and erel = kR(s̃)/V (s̃)k∞ . When β = 0.9,

eabs = 0.0013 and erel = 0.0079. When β = 0.9913, eabs = 0.0016 and erel = 0.0008. If thesustainability constraints never bind, the errors are almost zero.

A.3

Further numerical results

Figure 5 shows a sample simulated path with baseline parameters and β = 0.9. A total of
51 periods (from period 500 to 550) are selected from a simulated sample of 1000 periods, tosee the dynamics with reasonably-sized shocks. At period 510, the sustainability constraint
is binding.

Before that, negative disturbances sequentially hit the markup shock and theoutput gap rises. After the markup shock reverts to the mean, a remaining large output gap
triggers the constraint to bind, and then the output gap in the optimal quasi-sustainable
policy drops more than the output gap in the optimal commitment policy does. At periods
505, 546 and 548, the sustainability constraint is also binding, but the eect of binding
constraints on the output gap is much smaller.

[Figure 5 is inserted here]

Table 4 looks at the unconditional means of ination and the output gap, as well as theunconditional and conditional expected utilities. The unconditional means of ination σπ is
not signicantly aected by binding constraints. Those in the optimal quasi-sustainable andcommitment policies are almost the same. When λ/κ is higher, the unconditional expectedutility V̄ in the optimal quasi-sustainable policy is higher than the unconditional expectedutility in the optimal commitment and discretionary policies. As predicted, the conditional
expected utility V (0, 0) takes the highest value in the optimal commitment policy for allparameter sets considered here, and it takes the same value in the optimal quasi-sustainable
policy when the probability of binding constraints is 0.

[Table 4 is inserted here]

27 The same grids are used as the original ones for U because Tauchen's method is used.

22

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25

Figure 1: Stabilization bias and time inconsistency.

a. Period 0଴ ൌ ଴ ൅ ଴଴ ൌ ଴ ൅ ଴ ൅ ଵ

‫ܦ‬଴
‫ܥ‬଴


଴ ൌ െሺ⁄ሻ଴b. Period 1ଵ ൌ ଵ ൅ ଵ
‫ܦ‬ଵ



‫ܥ‬ଵ

ଵ ൌ െሺ⁄ሻଵଵ ൌ െሺ⁄ሻሺଵ െ ଴ ሻ

26

Figure 2: Impulse response functions: β = 0.9.

Ψ

Output gap

/Ψ

t−1

0

t

1
0.9

−2

0.8
0.7

−4

−6

quasi−sustainable
time−0 commitment
discretion
0

2

4

0.6
0.5

6

0

Inflation rate
4

2

3

1

2

0

1

0

2

4

6

Mark−up shock

3

−1

2

4

0

6

27

0

2

4

6

Figure 3: Policy function of z(u, x−1 ) and dierence in social welfare: β = 0.9.

a. z(u, x−1 )z(u,x−1)

1

t=−1
0.5
0

t=0

−0.8
−0.6
−0.4
−0.2

t=1
6
4

0

2

0.2

0

0.4

−2

0.6
0.8

u

−4
−6

x

−1

−max(Vc(u,x−1)/Wd(u)−1,0)b. dierence in social welfare (− max{V (u, x−1 )/W d (u) − 1, 0})

0

t=−1

−0.5
−1
−1.5

t=0

t=1

−0.8
−0.6
−0.4
−0.2

6
4

0

2

0.2

0

0.4

−2

0.6
u

0.8

−4
−6

x−1

Notes: The state space consists of the markup shock U and the lagged output gap X . The solidline with the circle indicates responses of the optimal quasi-sustainable policy with the initialvalues (x−1 , u0 ) = (0, 5σ).
28

Figure 4: Policy functions of π(u, 0) and x(u, 0): β = 0.9.

quasi−sustainable
time−0 commitment
discretion

π(u,0)

0.5

0

−0.5
−0.8

−0.6

−0.4

−0.2

0
u

0.2

0.4

0.6

0.8

−0.8

−0.6

−0.4

−0.2

0
u

0.2

0.4

0.6

0.8

x(u,0)

5

0

−5

29

Figure 5: Sample simulated path: β = 0.9.

1

z

0.9
0.8
0.7
500

505

510

515

520

525

530

535

540

545

550

505

510

515

520

525

530

535

540

545

550

505

510

515

520

525
time

530

535

540

545

550

x

5

0

−5
500

u

1

0

−1
500

30

Table 1: Parameter values.
Parameters

Values

SourcesβDiscount factor

0.9 or 0.9913

3.5% annual real rate

κ
λ
ρ
σε

Slope of the Phillips curve

0.024
0.048/42
0.0
0.154(for β = 0.9913)
Weight to the output gap
AR(1) parameter of markup shock
Std. deviation of markup shock (%, q/q)

31

Woodford (2003)
Woodford (2003)
Adam and Billi (2006)
Adam and Billi (2006)

Table 2: Simulation results with baseline parameters.

β = 0.9σπσxV̄V (0, 0)P{z<1}

Sustainable

0.4556

1.1228-0.1675-0.1456

6.79

Commitment

0.45611.1289-0.1682-0.1454



Discretion

0.52031.0405-0.2017-0.1815

β = 0.9913
Sustainable

σπ

σx

V̄

V (0, 0)P{z<1}

0.4435

1.0612-1.8013-1.7778

0.00

Commitment

0.4435

1.0612-1.8013-1.7778Discretion

0.52031.0405-2.3179-2.2972Notes: 100 simulations are perfomed to calculate the mean of standard deviations in simulations foreach policy. The length of periods in each simulation is 1,000, and 50 initial periods are discarded
to eliminate the eect of initial values. The same pseudo-random numbers are used in simulations
for each policy.

32

Table 3: Results with alternative parameters.

a. Probability of binding constraints, P{z<1} (%)κ

ρ=0ρ = 0.35

λ
0.001

0.01

0.0625

0.005

0.00

0.53

1.35

0.01

0.00

0.01

0.36

0.05

0.00

0.00

0.00

κ

λ
0.001

0.01

0.0625

0.005

0.00

0.57

5.45

0.01

0.00

0.00

0.33

0.05

0.00

0.00

0.00

b. Ratio of unconditional means of the output gap, σxc /σxdκ

ρ=0

ρ = 0.35

λ
0.001

0.01

0.0625

0.005

1.71

3.25

5.61

0.01

1.19

2.20

3.70

0.05

0.85

0.97

1.50

κ

λ
0.001

0.01

0.0625

0.005

2.20

4.51

7.97

0.01

1.41

2.94

5.17

0.05

0.90

1.10

1.89

c. Lower bounds of discount factor (with unconditional means), βκ

ρ=0ρ = 0.35

λ
0.001

0.01

0.0625

0.005

0.838

0.944

0.977

0.01

0.719

0.893

0.955

0.05

0.493

0.626

0.802

κ

λ
0.001

0.01

0.0625

0.005

0.844

0.945

0.977

0.01

0.734

0.895

0.956

0.05

0.511

0.649

0.810

33

34

0.81100.65550.18150.81100.65550.18150.97820.88080.2105

0.005
0.01
0.05

0.005
0.01
0.05

0.005
0.01
0.05

Sustainable

Commitment

Discretionary

1.01171.00030.7346

0.94880.8812
0.52250.94880.8812
0.52251.01501.0131
0.95700.99270.96320.7651

0.99290.96310.76511.22272.20212.63082.6696
3.10862.37092.6696
3.10862.37090.001

0.06250.7563
1.40952.21491.29081.6713
1.8760

0.001

0.01

0.61990.61920.59630.61320.60410.53451.29081.6713
1.8760λσπ0.6186
0.61400.49610.5996
0.57720.42250.61340.60410.5345λDiscretionaryκ0.60500.56380.1772

0.005
0.01
0.05

Commitment(ii) ρ = 0.35

0.5521
0.48830.1849

0.005
0.01
0.05

Sustainable

0.5996
0.57720.42250.001

0.0625

0.001

0.01

λ

σπ
λ

0.5521
0.48830.1849

κ

0.005
0.01
0.05

(i) ρ = 0.0

0.12650.25010.91820.55470.72701.01180.55030.72691.01180.01

σx0.07730.15350.62020.2467
0.33630.60070.2442
0.33630.60070.01

σx0.02030.04050.19140.14750.20180.35980.13100.19810.35980.06250.01240.02480.11930.06420.08910.17930.05710.08730.17930.0625-6.1298-5.3339-0.9690-4.8230-3.6521-0.7681-4.8230-3.6521-0.76810.001

λ-2.3452-2.1853-0.6868-2.0717-1.7695-0.5657-2.0717-1.7695-0.5657

0.001

λ-6.4136-6.3163-4.2158-5.9347-5.3822-2.7298-5.9287-5.3821-2.7298

0.01

V̄-2.3978-2.3800-1.9230-2.3078-2.1957-1.4767-2.3064-2.1956-1.4767

0.01

V̄

Table 4: Simulation results with alternative parameters.

-6.4413-6.4254-5.9534-6.2952-6.0534-4.4679-6.2686-6.0421-4.4679

0.0625-2.4028-2.3999-2.3113-2.3762-2.3308-1.9867-2.3720-2.3285-1.9867

0.0625-6.0394-5.2553-0.9547-4.7101-3.5709-0.7540-4.7109-3.5709-0.75400.001

λ-2.3212-2.1629-0.6798-2.0398-1.7428-0.5578-2.0408-1.7428-0.55780.001

λ-6.3190-6.2232-4.1536-5.7934-5.2543-2.6723-5.7939-5.2551-2.6723

0.01

V (0, 0)-2.3733-2.3556-1.9034-2.2727-2.1619-1.4549-2.2729-2.1623-1.4549

0.01

V (0, 0)-6.3463-6.3307-5.8656-6.1503-5.9098-4.3647-6.1621-5.9105-4.3647

0.0625-2.3782-2.3754-2.2877-2.3413-2.2955-1.9563-2.3437-2.2958-1.9563

0.0625