HIAS-E-58

The Role of Inflation Target Adjustment in
Stabilization Policy
Yunjong Eo and Denny Lie
School of Economics, The University of Sydney, NSW, Australia

October 2017

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The Role of Inflation Target Adjustment in Stabilization Policy⇤
Yunjong Eo†

Denny Lie‡This version: October 31, 2017

Abstract
How and under what circumstances can adjusting the inflation target serve as astabilization-policy tool and contribute to welfare improvement? We answer thesequestions quantitatively with a standard New Keynesian model that includes cost-push
type shocks which create a trade-o↵ between inflation and output gap stabilization. We
show that this trade-o↵ leads to a non-trivial welfare cost under a standard Taylor rule,even with optimized policy coefficients. We then propose an additional policy tool of
an inflation target rule and find that the optimal target needs to be adjusted in apersistent manner and in the opposite direction to the realization of a cost-push shock.
The inflation target rule, combined with a Taylor rule, significantly reduces fluctuationsin inflation originating from the cost-push shocks and mitigates the policy trade-o↵,resulting in a similar level of welfare to that associated with the Ramsey optimal policy.
The welfare implications of the inflation target rule are more pronounced under a flatter
Phillips curve.
JEL Classification: E12; E32; E58; E61;
Keywords: Welfare analysis; Monetary policy; Cost-push shocks; Medium-run inflation targeting; Flat Phillips curve;⇤
We thank James Bullard, Efrem Castelnuovo, Chris Edmond, Nicolas Groshenny, Joonyoung Hur, Jinill
Kim, Mariano Kulish, Jae Won Lee, James Morley, Ed Nelson, Adrian Pagan, Bruce Preston, Jae W. Sim,Inhwan So, Willem Van Zandweghe, Toshiaki Watanabe, Yuzhe Zhang, and seminar and conference participants at Australian National University, Bank for International Settlements (Hong Kong), Bank of Korea,
Korea University, Seoul National University, Sogang University, University of Adelaide, University of Melbourne, University of New South Wales, University of Queensland, AJRC and HIAS Joint Conference on
Recent Issues in Finance and Macroeconomics, Hitotsubashi Summer Institute, KAEA-KEA International
Conference, KAEA-KIF-KAFA Joint Conference, SNU Economics International Workshop, Asian Meetingof Econometric Society, Society for Nonlinear Dynamics and Econometrics Conference, Sydney Macroeconomics Workshop, Sydney University Min-Macroeconomics Workshop, and Workshop of the Australasian
Macroeconomic Society for providing useful feedback. All errors are our own.
†
Corresponding author School of Economics, The University of Sydney, NSW, Australia; Email: yunjong.eo@sydney.edu.au‡
School of Economics, The University of Sydney, NSW, Australia; Email: denny.lie@sydney.edu.au

1Introduction

The constancy of the central bank’s inflation target has been one of the most enduringfeatures in stabilization policy studies in the literature. These studies, whether they focus
on positive or normative aspects of monetary policy, assume that the central bank stabilizes
inflation around a constant, long-run target. From an empirical standpoint, however, several
studies have documented that historically trend inflation has not been constant, see e.g.
Kozicki and Tinsley (2003), Ireland (2007), and Cogley, Primiceri and Sargent (2010) for
studies using the U.S. data. Interpreting time-varying trend inflation as adjustments in thecentral bank’s implicit inflation target (or in the public’s inflation-target expectations), thisobservation raises the question of whether there is an important role of such adjustments in acentral bank’s stabilization policy. How and under what circumstances are these adjustments
warranted and possibly welfare-improving?
In this paper we investigate the role of inflation target adjustment in central banks’ stabilization policy. We propose a scenario, as in Ireland (2007), where a central bank’s inflation
target is endogenous and depends on the state of the economy. In particular, the adjustmentof the target is conditional on the realization of cost-push shocks.1 It is well known that
cost-push shocks create a trade-o↵ between inflation and output-gap stabilization. In thispaper, we refer to this type of shocks as “cost-push shocks” but they potentially represent
any shock that creates such a trade-o↵ including variations in tax changes, changes in desired
price markups by firms and wage markups by households, and oil price shocks.2
We show that adjustment of the inflation target, done properly, improves the central
bank’s policy-stabilization trade-o↵ and can lead to a significant welfare improvement. In
fact, despite the existence of the cost-push shocks, an interest-rate rule with an optimal
inflation target adjustment is able to closely replicate the optimal Ramsey allocation. The
extent of the improvement in the policy trade-o↵ and the welfare improvement are above
1
Ireland (2007) assumes that adjustment of the inflation target is conditional on the realization of costpush shocks as well as technology shocks and that the central bank stabilizes inflation around the time-varying
target. Unlike Ireland (2007), however, we focus on the normative aspects of such adjustments, instead of
the positive aspects.
2
See Clarida, Galı́ and Gertler (1999); Erceg, Henderson and Levin (2000); Steinsson (2003); Smets andWouters (2007); Natal (2012) for details. Blanchard and Galı́ (2007) call such a trade-o↵ that the “divine
coincidence” no longer holds with respect to these shocks.

1and beyond what is achievable in the conventional Taylor-type-rule policy environment witha constant, long-run inflation target.
The theoretical framework used for our analysis is a simple, microfounded New Keynesianmodel, along the lines of Rotemberg and Woodford (1997) and Steinsson (2003). The modelhas two standard market distortions: the relative-price distortion arising from nominal pricerigidity and the average markup distortion due to firms’ monopoly power. Aggregate fluctuations are driven by productivity, government spending, and cost-push (markup) shocks.
Despite its simplicity, our model is rich enough to capture important implications of largerscale, empirically-driven models used to analyze the e↵ects of monetary policy. Our main
finding should hold true in those models, as long as there exists a trade-o↵ between the twostabilization goals. The cost-push shock itself in our model can be treated as a familiar andconvenient proxy for any shock that creates such a trade-o↵.3
We first show that in the face of cost-push shocks, it is not possible to achieve a similar
welfare level to that associated with the Ramsey policy in an environment in which themonetary authority conducts policy through a standard Taylor-type rule with a constantinflation target. In particular, there still exists a non-trivial welfare cost, relative to the
optimal Ramsey allocation, even under an optimized, implementable Taylor-type rule. Notethat if only productivity and government spending shocks exist, there is no stabilization
trade-o↵ and the central bank can stabilize inflation and output at the same time. As shownby Schmitt-Grohé and Uribe (2007), in such an environment an optimized Taylor rule witha constant inflation target can mimic the Ramsey allocation quite well.
We then consider our proposed policy rule, which in addition to the nominal interest-rate
adjustment using a Taylor rule, the monetary authority also adjusts the inflation target in
response to a cost-push shock. We allow this adjustment to be temporal using a persistent
inflation target that follows an autoregressive process. This implies that the nominal interest
rate now responds to the inflation gap, defined by the di↵erence between actual inflation andthe medium-run inflation target, as well as the output gap.
The optimal target adjustment calls for changing the inflation target in the opposite
3

Smets and Wouters (2007) find that cost-push shocks in their model, in the form of wage markup andprice markup shocks, are largely responsible for inflation fluctuations in the postwar U.S. economy.

2direction in response to cost-push shocks. That is, we find that when there is a positive
realization of a cost-push shock that raises inflation and leads to a negative output gap on
impact, it is optimal for the monetary authority to decrease the inflation target. For any
inflation rate above the initial target, a decreased target in turn leads to a bigger inflation gap
compared to that in the standard Taylor rule and translates to a more aggressive responseto the cost-push shock through a larger increase in the nominal interest rate. This policy
combination leads to a significant welfare gain where the inflation variability is sufficiently
reduced, without leading to too large increase in the variability of the output gap. Several
studies, (e.g. Bomfim and Rudebusch (2000), Orphanides and Wilcox (2002), and Ireland(2007)), discuss the possibility that over the postwar period the Federal Reserve consistently
translated adverse supply shocks (positive cost-push shocks) into more persistent inflation(a higher inflation target).4 Our finding implies that such an action is unwarranted andwelfare-reducing.
We also note that this more-aggressive response to a realization of a cost-push shockin our target rule is not equivalent to simply increasing the inflation feedback coefficientin the Taylor-type rule from a welfare perspective. Even when the cost-push shock is thedominant driving process in the economy, it may not be optimal to increase the inflation
feedback coefficient beyond a certain value because (i) the variability of the output gap isalso a relevant determinant of welfare and (ii) the welfare loss from increasing the variationin the output gap may dominate the welfare gain from reducing the variation in inflation
for too large values of the inflation coefficient in the Taylor-type rule.5 In particular, weshow that even when we allow for a wider range of possible values of the Taylor-type rule
feedback coefficients, it is not possible to achieve a similar welfare level to that associatedwith our optimized inflation target rule. Thus, the apparent improvement in a trade-o↵
between inflation and output-gap stabilization is mainly attributable to the adjustment of theinflation target, and not because of any implementability restriction on the policy coefficients.
4

Ireland (2007), however, finds that in the postwar U.S. economy a model with the endogenous movementsin the inflation target is statistically indistinguishable from the exogenous inflation target model. Garnier,Mertens and Nelson (2015) obtain the same result using unobserved components models of trend inflationand the inflation gap in the U.S. economy.
5
Woodford (2002) and Benigno and Woodford (2005) derive a welfare-based loss function and show thatit depends on the variability of inflation and an output-gap measure. See also our discussion in Section 4.

3

In addition, we consider an interest rate smoothing rule but it does not alter these policy
implications and our findings are robust.
Our investigation also reveals that the extent of the welfare gain from the target adjustment depends on the slope of the Phillips curve. Various studies in the literature havedocumented a flattening of the Phillips curve for the U.S. and other advanced economies inrecent years.6 One possible reason for this flattening is the increase in the degree of nominal
rigidity, e.g. the probability of price fixity in the familiar Calvo (1983) model, as pointed by
Blanchard (2016). A flatter Phillips curve makes the task of an inflation-targeting monetary
authority more difficult, as inflation becomes less responsive to output-gap fluctuations. We
show that the welfare gain from adjusting the inflation target is larger the flatter the slopeof the Phillips curve is. Our proposed policy rule is thus even more appealing in such anenvironment.
Our finding is also potentially relevant to the current economic environment of low inflation rates under a flat Phillips curve, especially in advanced economies such as the U.S. andthe Euro area. Blanchard (2016) argues that the flat Phillips curve raises serious challenges
for monetary policy and may require very flexible inflation targeting. However, he does notpropose how to implement flexible inflation targeting in practice and its welfare implication.
If a negative realization of cost-push shocks or similar shocks contributes to the low-inflation
environment, our finding suggests that an appropriate policy action is to just simply increase
the medium-run inflation target. In addition, in a broader context, the finding in our paper can also serve as a justification for the practice of central banks in several countries ofregularly readjusting and announcing their medium-run inflation targets.7 For example, in
September 2016, the Bank of Japan introduced an inflation-overshooting commitment, under which it aims to exceed the inflation target of 2% and stay above the target in a stablemanner. The Bank of Japan plans to make policy adjustments conditional on developmentsin economic activity and prices, as well as financial conditions toward achieving the inflation
target of 2% in the long run. Though motivated by a di↵erent set of objectives, our finding
6

The flattening of the Phillips curve in the U.S. and other advanced economies has been documented invarious studies, e.g. Roberts et al. (2006), Kuttner and Robinson (2010), and Blanchard (2016).
7
A non-exhaustive list of countries currently practising this policy includes Brazil, India, Indonesia,
Nigeria, and South Korea.

4

suggests that such a conditional adjustment might be welfare-improving.
The rest of this paper is organized as follows. Section 2 presents the model used forwelfare analysis and its calibration. Section 3 introduces monetary policy rules and discusses
a measure of household welfare. Section 4 conducts welfare analysis for several alternative
policy rules, including our proposed policy of adjusting the inflation target. Section 5 presents
sensitivity analysis involving the flattening of the Phillips curve and interest-rate smoothing.
Section 6 concludes.

2

Model and Calibration

We consider a New Keynesian model along the lines of Rotemberg and Woodford (1997)
and Steinsson (2003). The model consists of a representative household, a continuum ofmonopolistically-competitive firms producing di↵erentiated varieties, and a monetary policy
authority.8 Our model is closest to the cashless model in Schmitt-Grohé and Uribe (2007),
although we abstract from capital accumulation and fiscal policy.9 Aggregate fluctuations
are driven by three exogenous shocks: productivity, government purchase, and cost-pushshocks. As we discussed, the inclusion of a cost-push shock is important because it creates a
trade-o↵ between inflation stabilization and output-gap stabilization, as in Steinsson (2003)
and is consistent with empirical findings in the literature.

2.1

Households

The representative household maximizes a discounted sum of utilities of the form
Et

1
X

Nt+s ) ]1
1

s [Ct+s (1

s=0

where

1

,

(1)

2 (0, 1) is the discount factor and Nt denotes the household’s labor supply. The

consumption index Ct is a Dixit-Stiglitz CES aggregator of di↵erentiated consumption goods
8
We abstract from fiscal policy dynamics by assuming that it follows a passive policy, in the sense of
Leeper (1991).
9Schmitt-Grohé and Uribe (2007) find that monetary distortions are not quantitatively important incomparison to cashless economy and passive fiscal policy is optimal.

5

or varieties, given by
Ct =

Z 1

1+✓t

Ct (i)

1/(1+✓t )

di

,

(2)

0

where ✓t = 1/(⌘t1) is the firms’ average markup at time t and ⌘t is the elasticity of

substitution across varieties. The average markup shock follows
log(✓t ) = (1with "✓,t ⇠ i.i.d.N (0,¯ + ⇢✓ log(✓t 1 ) + "✓,t ,⇢✓ ) log(✓)

(3)

2
✓ ).

Households earn the nominal wage rate Wt by supplying Nt and have access to a domestic
bond market where the riskless one-period nominal government bonds, Bt , are traded. Thesebonds pay the gross interest rate Rt . Households also receive firms’ profits, ⇧prof, andt
government transfers or taxes, Tt . Thus, the one-period budget constraint is given by
Z 1
0

Pt (i)Ct (i)di + Bt  Rt 1 Bt 1 + Wt Nt + ⇧prof+ Tt ,t(4)

where Pt (i) denotes the nominal price of variety i. Solving the household’s problem andtaking the first-order approximations of the resulting efficiency conditions around the longrun steady-state equilibrium yield a standard consumption Euler equation. Additional details
on the households’ optimality conditions are contained in the Appendix.

2.2

Firms

Each monopolistically-competitive firm produces a di↵erentiated variety i using labor witha production function
Yt (i) = zt Nt (i),(5)

where Yt (i) is the production of good i and zt is the aggregate productivity shock, which isassumed to follow a univariate autoregressive processlog(zt ) = ⇢z log(zt 1 ) + "z,t ,with "z,t ⇠ i.i.d.N (0,

2
z ).

6

(6)

Following Calvo (1983) and Yun (1996), only a (1↵) 2 [0, 1) fraction of firms areallowed to optimally adjust their prices at any given time period. We assume that firmsthat are not allowed to optimally reset their prices, with probability ↵, simply index theirprices to the steady-state gross inflation, ⇧, which coincides with the monetary authority’stime-invariant long-run inflation target. Thus, each optimizing firm i chooses an identicaloptimal nominal price, P̃t (i) = P̃t , to maximize the expected discounted sum of profits1X
s=0

where Qt,t+s =ands

t+s /Pt+st /Pth¯ s Yt+s (i)↵s Qt,t+s P̃t (i)⇧i
Wt+s (i)Nt+s (i) ,(7)is the nominal stochastic discount factor between time t and t + s

t is the marginal utility of consumption.

The resulting first-order condition of the firms’ optimal pricing problem and the associated aggregate-price level equationhPt = (11↵)(P̃t ) ✓t + ↵ ⇧Pt 1

1
✓t

i ✓t

make up the pricing block of the model. Taking the first-order approximations of theseequations around the long-run steady-state equilibrium leads to the following New KeynesianPhillips curve (NKPC) equation (see Appendix for more details):

where

⇡
ˆ t = Et ⇡
ˆt+1 + mc
c t + ût ,(8)⌘ (1 ↵ ↵)(1 ↵) , ût is the reduced-form cost-push shock, which is a function of theshock to the average markup (✓ˆt ),10 and mc
c t is log deviation of the real marginal cost.11
Alternatively, we can write the NKPC as a function of the output gap, Yt /Yt⇤ :⇣⇡ˆ t = Et ⇡
ˆt+1 +  Ŷt⌘
Ŷt⇤ + ût .

hThe coefficient on the output gap, i.e. the slope of the NKPC, is  = (1That is, ût ⌘ (1 ↵ ⇢↵✓ )(1 ↵) ⌘¯ 1 1 ✓ˆt .
11
See Clarida, Galı́ and Gertler (1999) for interpretation of the cost-push shock.
10

7

(9)
ḡ) 1 + 1 N̄N̄i,

with ḡ and N̄ denoting the steady-state government spending-output ratio and steady-statelabor, respectively. Following Woodford (2003), we define potential output, Yt⇤ , as the output
level under the flexible-price equilibrium with constant average markup.

2.3

The monetary authority and government

The monetary authority follows a standard Taylor-type rule in which the authority engages
in interest-rate smoothing and responds to deviations of inflation from an inflation target,
as well as the output gap. We first present our benchmark rule in which the inflation target
is defined by the long-run inflation target (i.e. the steady-state inflation rate). We thenintroduce our additional policy tool in which the inflation target is adjusted in response tocost-push shocks.
• Benchmark: the long-run inflation targetinglog(Rt /R̄) =¯⇡ log(⇧t /⇧) +⇤Y log(Yt /Yt )(10)

where ⇧t is the quarterly rate of inflation and R̄ is the steady state nominal interest rate.
Alternatively, as in Ireland (2007), we consider that instead of the long-run constantinflation target, the monetary authority reacts to the inflation deviation from the potentially time-varying, medium-run inflation target, which is adjusted in response to cost-pushshocks.12• Proposed policy: the medium-run inflation targetinglog(Rt /R̄) =⇤⇡ log(⇧t /⇧t ) +¯ + µ⇡⇤ ⇡⇤ ,tlog(⇧⇤t ) = log(⇧)

where

⇤
Y log(Yt /Yt )

(11)
(12)

⇡ ⇤ ,t = ⇢⇡ ⇤ ⇡ ⇤ ,t 1 +"✓,t and "✓,t is the innovation in the markup shock, as previously de-fined in (3). The medium-run inflation targeting (MRIT) in (12) shows that the medium-run
12

Ireland (2007) also considers the inflation target’s response to a technology shock, but we do not take itinto account in our policy framework because this shock does not create the trade-o↵ between inflation andthe output gap stabilization.

8

inflation target, ⇡t⇤ = log(⇧⇤t ), is the sum of two distinct components. The first component,¯ is the long-run inflation target. The second component µ⇡⇤ ⇡⇤ ,t , that consti⇡¯ = log(⇧),
tutes the endogenous adjustment of the target, i.e. its evolution is directly controlled by themonetary authority. The dynamics of the adjustment is controlled through the coefficientsµ⇡⇤ and ⇢⇡⇤ — the former can be interpreted as the instantaneous response to the markup
shock, while the latter can be thought as the smoothness parameter, i.e. how the monetary
authority spreads the target adjustment across multiple periods.
The government issues one-period nominal risk-free bonds, makes transfers, imposes
taxes, and faces an exogenous expenditure stream, Gt . Thus, the one-period government
budget constraint is given by
B t = R t 1 B t 1 + Pt G t + T t .

(13)Government spending is assumed to follow a univariate autoregressive process of the formlog(Gt /Ḡ) = ⇢g log(Gt 1 /Ḡ) + "g,t ,with "g,t ⇠ i.i.d.N (0,

2.4

(14)

2
g ).

Competitive equilibrium

The stationary equilibrium in our economy is characterized by prices and quantities thatsatisfy the optimality conditions of the households and firms, in addition to a monetary
policy rule and the aggregate market clearing condition for goods, labor, and assets. The
aggregate employment is given by the sum of employment across firms:
Nt =

Z 1

Nt (i)di.

(15)

0

We assume that the government minimizes the cost of producing Gt . Thus, the public good
⇣
⌘ 1+✓
t
✓t
Pt (i)
demand for each intermediate good i is given by Gt (i) = Pt
Gt . Thus, aggregate
demand is given by

Yt = C t + G t
9(16)

Table 1: Model Calibration
Parameter

✓¯
Ḡ
↵
¯
⇧
⇢z
⇢✓
⇢g
z
✓
g

where Yt (i) =

⇣

Value
2
3.613
1.04 1/4
0.25
0.055
0.80
1
0.95
0.60
0.50
0.007
0.130
0.160

Pt (i)
Pt

⌘ 1+✓
t
✓
t

Description
preference parameter
preference parameter
quarterly discount rate
steady-state price markup 25%; price elasticity of demand 5
steady-state level of government spending
share of firms that cannot change their price each period
steady-state gross inflation or long-run inflation target
persistence of productivity shock
persistence of cost-push shock
persistence of government spending
standard deviation of productivity innovation
standard deviation of cost-push innovation
standard deviation of government spending innovation

Yt and Yt (i) = Ct (i) + Gt (i). We provide the complete equilibrium

equations in the Appendix.

2.5

Calibration

We calibrate our model based on the values of structural parameters from Schmitt-Grohéand Uribe (2007) and present these values in Table 1. The choice of

implies that thesteady-state labor N̄ is 0.23 and the steady-state labor-to-leisure ratio (N̄ /(1N̄ )) of 0.29.

⌘¯ = 5 corresponds to the steady-state average price markup of 25%. The choice of Ḡ = 0.055
implies that steady-state ratio of government spending to output (ḡ ⌘ Ḡ/Ȳ ) is 24%. The¯ = 1. This zero net inflation
steady-state gross inflation, or the long-run inflation target, is ⇧coincides with the (Ramsey) optimal steady-state inflation in our model.13
A key parameter for the welfare analysis in the next section is the Calvo parameter, which
determines the slope of the NKPC. The Calvo parameter ↵, which denotes the probability
that a firm cannot change its price optimally in any given period. Following Schmitt-Grohé
13
The model contains two market distortions: the relative-price distortion due to sticky prices and averagemarkup distortion arising from firms’ monopoly power. As shown in various studies, e.g. Benigno andWoodford (2005), Khan, King and Wolman (2003), King and Wolman (1999), and Woodford (2002), zeroinflation minimizes both distortions and constitutes the optimal policy.

10and Uribe (2007), we set the Calvo parameter to ↵ = 0.8.14 The average duration ofprice fixity is about 5 quarters for ↵ = 0.8. In addition, this calibrated Calvo parametercorresponds to the NKPC slope of= 0.052 in (8) when marginal cost is used as thedriving process for inflation and the NKPC slope of  = 0.084 in (9) when the output
gap is used. Following the business cycle literature, the persistence parameter ⇢z and thestandard deviationz for the productivity shock are taken to be 0.95 and 0.007, respectively.

The remaining parameters for the structural shock processes, ⇢g , ⇢✓ ,g , and✓ are set toclosely match the standard deviations of inflation and output for the postwar U.S. economy
from 1947:Q1 to 2008:Q4.15 To obtain these values, we assume the standard Taylor-rule
coefficients of3⇡ = 1.5 andY= 0.5.

Monetary policy and welfare

We conduct welfare analysis mainly for three di↵erent policies in comparison to the Ramsey
optimal policy:(i) Taylor rule: the Taylor rule ( ⇡ = 1.5,

Y

= 0.5) without the MRIT;(ii) Taylor-MRIT rule: the optimal MRIT conditional on the Taylor rule;(iii) Optimized Taylor rule: the optimal implementable Taylor rule without the MRIT.
The Taylor rule (i) sets the policy parameters to⇡ = 1.5 andy = 0.5 in (10), which

are quite close to those reported in the literature for the post-1984 U.S. economy. The
optimized Taylor-MRIT rule (ii) entails a monetary authority that adjusts the medium-run
inflation target by choosing the parameter values µ⇡⇤ and ⇢⇡⇤ in (12) to maximize welfareof the representative household, conditional on the Taylor rule coefficients⇡= 1.5 andy = 0.5. Under the optimized Taylor rule (iii), the monetary authority simply chooses thewelfare-maximizing values of⇡ andy in (10), without adjusting the medium-run inflation

14

Estimates in the literature of the Calvo parameter range from 0.72 to 0.83 for the postwar U.S. economy.
For example, they are found to be 0.83 in Levin et al. (2005), 0.79 for a DSGE-VECM model, 0.83 for a
DSGE model in Del Negro et al. (2007), and 0.72 in Herbst and Schorfheide (2014). The average of theseestimates is 0.79, which is quite close to our calibrated parameter value of ↵ = 0.8.
15
We use CPI inflation and detrended output.

11

target. Formally, for policy rules (ii) and (iii), we search for the relevant policy parameters,{µ⇡⇤ , ⇢⇡⇤ } in (ii) and { ⇡ ,y } in (iii), that maximize the unconditional expectation of lifetime

utility, E(Vt ), where
Vt = Et

1Xs

U (Ct+s , Nt+s )

(17)

s=0

and E denotes the unconditional expectation operator.16 For the welfare measure in (17) toadequately capture the e↵ect of uncertainties, we compute the second-order accurate solution
to the equilibrium policy functions, including for Vt .17
Following Schmitt-Grohé and Uribe (2007), in searching for the optimal values of the
Taylor parameters, we restrict⇡ 2 (0, 3] andy 2 (0, 3] for the optimized Taylor rule (iii).

Schmitt-Grohé and Uribe (2007) argue that policy coefficients larger than 3 (or negative)
would be difficult for policymakers to communicate to the public. We also rule out thoseparameter values that yield an indeterminate equilibrium. In addition, we restrict |µ⇡⇤ | <0.10 and ⇢⇡⇤ 2 [0, 1) when searching for the Taylor-MRIT rule (ii). The restriction on thevalue of µ⇡⇤ is somewhat ad-hoc. However, for the same reason as with regard to the Taylortype rule parameters, an excessively large value of µ⇡⇤ may undermine the credibility of themonetary authority in delivering the stated objectives.18

3.1

The Ramsey policy

To evaluate and compare various policies above we use the time-invariant (timeless perspective) stochastic Ramsey optimal policy as the benchmark. Under the Ramsey policy, thepolicy authority acts benevolently and searches for the allocation that maximizes the welfareof the representative agent. Following the standard approach in the literature, we abstractfrom any specific form of the policy rule and search instead for the optimal Ramsey alloca16

Our results continue to hold if, instead, the policy parameters used are those that maximize the conditional expectation, V0 , i.e. conditional on the initial state of the economy being the nonstochastic steadystate.
17
See Schmitt-Grohé and Uribe (2004) for a discussion on why a second-order approximation to theequilibrium solution is needed for an accurate welfare computation. Also see Johnston, King and Lie (2014)
for the detail of the solution method that we use.
18
We sidestep this possible credibility issue in this paper and leave it for future research.

12

tion.19 In addition, as in the competitive equilibrium, the Ramsey equilibrium is solved up
to a second-order approximation. The Appendix contains additional details on the derivation and computation of the Ramsey policy. Armed with the Ramsey policy, we can thencalculate the conditional and unconditional welfare costs of any alternative policy relative
to the Ramsey policy, which is described in the following section.

3.2

Welfare cost measure

As in Schmitt-Grohé and Uribe (2007), we define the welfare cost of implementing an alternative policy as the fraction of consumption that the representative household would be
willing to give up under the benchmark Ramsey policy environment to be equally well o↵,as under the alternative policy environment. Specifically, let {Ctr , Ntr } and {Cta , Nta } be thestate-contingent plans for consumption and labor under the Ramsey policy and under thealternative policy, respectively. The conditional welfare cost,
E0

1
X

t

U (Cta , Nta ) = E0

t=0

1
X

t

U ((1

c , is implicit in the expressionrrc )Ct , Nt ).

t=0

The expectation operator above makes clear that the welfare cost is conditional on theinitial state at time t = 0, which we assume to be the deterministic steady state under theRamsey policy. The deterministic steady state under the alternative policy is assumed to be
identical to that under the Ramsey policy. Similarly, the unconditional welfare cost,u , canbe obtained from
E

1
X
t=0

t

U (Cta , Nta ) = E

1
X

t

U ((1

r
r
u )Ct , Nt ).

t=0

Here, E is the unconditional expectation operator. For the specific form of the utility
function and the calibrated parameters in Table 1, we can then calculate the second-orderapproximations to

c and

20
u for any alternative policy rule.

19

See, for example, Erceg, Henderson and Levin (2000), Khan, King and Wolman (2003), Lie (2015), andSchmitt-Grohé and Uribe (2007).
20
In fact, since our utility function in (1) is identical to that in Schmitt-Grohé and Uribe (2007), we obtainthe same expressions for both c and u — see equations (38) and (39) in the expanded version of theirpaper, Schmitt-Grohé and Uribe (2006).

13

Table 2: Welfare cost of various policies(i) Taylor rule(ii) Taylor-MRIT rule(iii) Optimized Taylor rule(iv) Optimized Taylor-MRIT rule

Policy Parameters
µ⇡ ⇤
⇢⇡⇤
⇡
Y
1.50 0.50––
1.50 0.50 -0.0731 0.77
3.00 0.38––
3.00 2.22
-0.10
0.77

Conditional
Welfare Costc ⇥ 100
0.507
0.186
0.310
0.044

Unconditional
Welfare Costu ⇥ 100
0.531
0.192
0.321
0.044

(⇡t

⇡
¯)

4.015
2.186
3.123
1.556

(ŷt

ŷt⇤ )

8.819
13.55810.17010.727

Note: The conditional and unconditional welfare costs are in terms of percent per annum consumption loss relative to theRamsey allocation. The standard deviation (.) is expressed in percent per year. For the Taylor-MRIT rule (ii), we search forµ⇡⇤ and ⇢⇡⇤ that maximize the unconditional welfare, while fixing ⇡ = 1.5 and Y = 0.5. For the optimized Taylor-MRIT
rule, we search for optimal ⇡ , Y , and µ⇡⇤ , while fixing ⇢⇡⇤ = 0.77. We restrict | ⇡ | < 3, | Y | < 3, and |µ⇡⇤ | < 0.10 whensearching for optimized coefficients.

4

Optimized interest rate rules

4.1

Welfare under the medium-run inflation targeting

We evaluate various policies discussed in the previous section. Table 2 reports the conditional
and unconditional welfare cost measures,c andu , in comparison to the Ramsey policy.

We find that the Taylor rule (i) yields conditional and unconditional welfare costs of 0.507%
and 0.531% per annum of consumption relative to the Ramsey policy. To express the cost inmonetary value, we could use U.S. annual personal consumption expenditure per capita asa proxy for real consumption. In 2015, the figure is $38,183 in current U.S. dollars. Hence,
a 0.531% consumption cost is equivalent to $204 per annum per capita. This is in fact quitea sizable business-cycle cost and is comparable to the finding in Schmitt-Grohé and Uribe(2007).
We next study the implication of our proposed policy, the Taylor-MRIT rule (ii), where
the monetary authority can additionally adjust the medium-run inflation target in response
to cost-push shocks over time. Our numerical search yields the optimal MRIT coefficients ofµ⇡ ⇤ =

0.073 and ⇢⇡⇤ = 0.77. The negative value of µ⇡⇤ implies that when inflation increases

because of a positive realization of cost-push shock, the monetary authority decreases themedium-run inflation target. All else equal, this policy action leads to a bigger inflation gap(⇧t /⇧⇤t ) compared to the Taylor rule (i) in (10) without the MRIT adjustment. It follows
that a bigger inflation gap translates to a more aggressive response to the cost-push shock14

through a larger increase in the nominal interest rate, as evident from the Taylor-MRIT
rule in (11). In addition, the fact that the optimized MRIT rule features substantial inertia
means that the monetary authority needs to react persistently to the cost-push shocks givena realization of the cost-push shock. This smoothing adjustment can be explained by thepersistent e↵ect of the cost-push shocks on inflation.
We find that the welfare gain from the MRIT adjustment is significant. The TaylorMRIT rule (ii) yields conditional and unconditional welfare costs of 0.186% and 0.192% ofconsumption relative to the Ramsey policy. When compared to the welfare costs associatedwith the Taylor rule (i), the welfare gain implies that agents would be willing to give up morethan 32 basis points of their consumption per annum stream under the Taylor-MRIT rule(ii) to be as well o↵ as under the Taylor rule (i). Or alternatively, using the unconditional
cost measure, we can say that the welfare gain from adopting the Taylor-MRIT rule policy
over the Taylor rule policy is 64%.21
To better understand the reason for the welfare gain we compute the standard deviationsof inflation and the output gap. The standard deviations of inflation and the output gap for
the Taylor rule (i) are 4.02% and 8.82%, respectively, while they are 2.19% and 13.56% for
the optimized Taylor-MRIT (ii).22 These calculations show that the MRIT, which allows formore aggressive reaction to inflation due to the cost-push shocks, reduces inflation volatility
by a factor of two, but only increases the volatility of the output gap by about 65%. This,
coupled with the fact that inflation variation carries a much higher weight in the utility-based
welfare measure relative to the output-gap variation, is the source of the significant welfare
gain.23
We next investigate whether the optimized Taylor rule (iii) can produce a comparable
welfare gain as the Taylor-MRIT rule (ii). As in Schmitt-Grohé and Uribe (2007), we focus
on an implementable rule and limit | ⇡ ,y |  3. We find that the best implementable Taylor-type rule requires an aggressive response to inflation and a somewhat muted response to the
output gap fluctuation. The optimized Taylor rule needs the policy coefficients of21⇡ = 3,

We find the welfare gain based on (0.53-0.19)/0.53 = 64.
The standard deviations are expressed in percent per year.
23
This much-higher weight on inflation variation can be seen by deriving the welfare-theoretic loss function,
as in Woodford (2003) and Benigno and Woodford (2005). We also illustrate this in Section 5.1 in the paper.
22

15

which is the largest allowable value, and⇡Y= 0.38. Note that this inflation coefficient of= 3 in the optimized Taylor rule is consistent with that in Schmitt-Grohé and Uribe(2007). This finding about taking the largest value for the inflation coefficient reflects thefact that inflation variation is an important element in welfare. In contrast to their finding,however, we find that the optimal

Y is not zero. This discrepancy comes from two importantdi↵erences between our economy and the one considered in Schmitt-Grohé and Uribe (2007):the presence of the cost-push shock in our model; and the fact that we define the output
gap measure in the policy rule as the deviation of output from its flexible-price equilibrium
level, instead of from the constant steady-state level. Since the welfare-relevant output gap
is the measure of output gap that we use, the optimalY

may no longer be zero.24 The

existence of the policy stabilization trade-o↵ between inflation and output-gap due to thepresence of the cost-push shock is responsible for the low optimal value of⇡ — here, the trade-o↵ is resolved in favor of inflation stabilization.

Y

relative to

If there is no cost-

push shock, productivity and government spending shocks are the only source of aggregate
fluctuations as in Schmitt-Grohé and Uribe (2007) and there is no trade-o↵ between thetwo stabilization goals. Then it does not matter whether the monetary authority responds
strongly to inflation or output-gap fluctuations.25
In terms of the welfare implication, the optimized Taylor-rule (iii) still yields nonnegligible welfare costs of 0.310% and 0.321%, respectively. Thus, relative to the TaylorMRIT rule (ii), the welfare gain from the optimized Taylor-rule (iii) over the Taylor rule (i)is smaller. The standard deviation of inflation in the optimized Taylor-rule is about 50%
larger compared to that in the Taylor-MRIT rule, while the standard deviation of the output gap is only about 25% smaller. Hence, it appears that adjustment of the medium-run
inflation target leads to a better trade-o↵ between inflation and output-gap variations. We
investigate this important finding in the next section.
24

See Woodford (2003) for additional details on the welfare-relevant output gap measure.
In fact, without the cost-push shock ( ✓ = 0), we find that the optimized Taylor-type rule comprises⇡ = Y = 3.
25

16

4.2

The improvement in inflation-output gap trade-o↵

The extent of the inflation-output trade-o↵ is an important consideration when conducting
monetary policy. The greater the trade-o↵ is, the more difficulty the monetary authority
faces in stabilizing inflation and output-gap fluctuations. We argue that the ability of themonetary authority to adjust the medium-run inflation target in the face of cost-push shocks
leads to a more favorable inflation-output trade-o↵. This improvement in the trade-o↵ leads
to welfare gain, above and beyond what is possible in the standard Taylor-rule policy. This
source of inflation-output trade-o↵ improvement, to the best of our knowledge, is a new
result in the literature.26
We first show this improvement by searching jointly for optimal⇡,

Y , and µ⇡ ⇤ .

Thecoefficient ⇢⇡⇤ is fixed at 0.77, which is the optimal value under the Taylor-MRIT policy(ii).27 We call this policy rule the optimized Taylor-MRIT rule (iv). The last row of Table
2 reports the numerical results. Compared to the Taylor-MRIT rule (ii) in which we fix
the Taylor coefficients at⇡ = 1.5 andy = 0.5, the welfare costs are now much smaller at

0.04% of consumption per capita per annum. The optimized Taylor-MRIT rule thus appears
to be able to replicate the Ramsey allocation quite well. This policy calls for the maximum
allowable value for⇡ = 3, a much higher value ofY= 2.22 in comparison to the output-gap

coefficient of 0.38 in the optimized Taylor rule (iii), and the minimum allowable value forµ✓ =0.10. Thus, in the face of a positive realization of the cost-push shock, the monetary

authority chooses to respond more aggressively to output gap fluctuations. It is optimal to doso because the monetary authority can now respond to the increase in inflation by further
reducing the medium-run inflation target (a more negative µ⇡⇤ ). Without an improved
inflation-output trade-o↵ by adopting the MRIT rule, reacting strongly to the output gap
withY= 2.22 would not be optimal, as shown for the Taylor-MRIT rule (iii). Note that,in the standard environment with a fixed inflation target, a more aggressive response toinflation fluctuations originating from cost-push shocks (i.e. a higher value ofvalue of⇡ or a lowerY ) would be optimal because inflation fluctuations are much more important than26

Other sources have been identified in the literature. For example, Clarida, Galı́ and Gertler (1999) showthat adopting a precommitment policy leads to an improved inflation variability-output variability compared
to a discretionary policy.
27
The results do not materially change when we also numerically search for optimal ⇢⇡⇤ .

17

output gap fluctuations in the utility-based welfare measure. Finally, consistent with thewelfare cost results, the standard deviation of inflation is now only 1.56%, while the standard
deviation of the output gap is roughly the same as in the optimized Taylor rule (iii).
The results above raise the question whether the allocation under the optimized TaylorMRIT policy (iv) can be achieved solely with an “unconstrained” Taylor rule, i.e. when we remove the upper limit of⇡ and

Y coefficients. If so, the improvement in the inflation-output

trade-o↵ under the MRIT policy is just a mirage, arising because of the implementability
restriction. To investigate this, we look at the implications of various values of

⇡ and

Y

for the unconditional welfare cost.28
Figure 1 plots unconditional welfare costs for various combinations of policy coefficients
under the Taylor-type rule without the MRIT. The top panel of Figure 1 depicts the welfare
cost when we vary

⇡ , while fixing= 0.38 (the optimal value in the optimized Taylor-Y

MRIT policy (iii)). The panel shows that the unrestricted optimal value of⇡is 18 —

above this value, the welfare cost starts increasing again. Intuitively, a too-high value of⇡means that the monetary authority responds too aggressively to inflation fluctuations

and too timidly to output-gap fluctuations given the trade-o↵ originating from cost-pushshocks. The middle and bottom panels of Figure 1 depict the welfare cost when we varyY , given a value of⇡.

When the value of

The welfare cost is higher when 0 <Y⇡ is fixed to 3, the optimal

of

Y

⇡

value is 0.38.

< 0.38 since the monetary authority’s response to

output-gap fluctuations is too timid. On the contrary, whenaggressively. WhenYY

> 0.38 she responds too

is fixed at a higher value of 50 (bottom panel), the optimal value

becomes higher.29 All in all, the results presented in Figure 1 imply that the jointly

optimized values of

⇡ and

Y

should be finite and relatively small. More importantly, atleast for the range of coefficient values considered in Figure 1, the welfare costs appear tobe higher than 0.044%, which is the welfare cost from the optimized Taylor-MRIT (vi): thecombination of the optimized Taylor-type rule and the MRIT. Thus, the improvement in the
inflation-output trade-o↵ under the MRIT policy is clearly evident.
28

The results are unaltered when the conditional cost measure is used instead.
This makes sense since in the presence of a policy stabilization trade-o↵ between inflation and outputgap, a more aggressive response to inflation fluctuations calls for a more aggressive response to output-gap
fluctuations, albeit at di↵erent degrees.
29

18Figure 1: Unconditional welfare cost under the Taylor-type rule without the MRIT

Note: We fix one of policy coefficients,
R = 0 for all cases.

⇡ and

Y , and vary the other parameter. In addition, we set

19

Figure 2: Unconditional welfare cost under the Taylor-type rule for various values ofY ( R = 0 for all cases)⇡ andFigure 2 o↵ers more comprehensive, three-dimensional pictures of the unconditional welfare costs when we jointly vary
⇡ 2 (1, 100] and

Y

⇡

and

Y.

Panel A considers a wider range of values,

2 (0, 10], while panel B limits

the two panels show that the joint optimal values of
unconditional welfare cost is atY

2 (0, 3] for a clearer view. Again,
⇡

and

Y

u = 0.07%, which is achieved atare finite. The smallest⇡ = 100 andY= 8.5.30

We note that this combination of optimal parameter values is not unique. For example, one
can always achieve a similar optimal allocation by increasing both

⇡ andDespite this, it is not possible to reduce the welfare cost much beyondYappropriately.

u = 0.07%.

This

indicates that the improved inflation-output trade-o↵ under the MRIT policy is not due tothe implementability constraint. Treating the optimized Taylor-MRIT policy (iv) in Table 2
as the best, the monetary authority could further reduce the welfare cost by 43% using the
MRIT.31 In addition, we search over the policy coefficients under the Taylor-type rule, which
30

Unlike the result in Schmitt-Grohé and Uribe (2007), it is not possible in our environment to fully
replicate the Ramsey allocation (i.e. zero welfare cost) due to the existence of the cost-push shock.
31
That is, (0.07 0.04)/0.07 = 0.43.

20Figure 3: Impulse response to a 1% cost-push shock

Note: The evolution of the average markup cost-push shock is given by¯ + ⇢✓ log(✓t 1 ) + "✓,t , where ⇢✓ = 0.6. We calibrate the size of thelog(✓t ) = (1 ⇢✓ ) log(✓)
innovation, "✓,t , so that ✓t increases by one percent on impact relative to the steady-statevalue.

would lead to the same welfare level associated with the Taylor-MRIT rule. We find thatmultiple pairs of the policy coefficients — for example, ( ⇡ ,Y ) = (5.51, 0.00), (22.01, 8.20)— can achieve the welfare level, but they require implausibly high inflation coefficients.32
The inflation coefficient of⇡ = 5.51 is the lower bound.

The inflation-output trade-o↵ improvement under the MRIT policy is also visible from
impulse response functions, as depicted in Figure 3. We plot the impulse responses ofinflation, the nominal interest rate, the output gap, and the medium-run inflation target to32

The reason we find the multiple pairs of the coefficients is that the welfare level is not the optimum
under the unconstrained optimized Taylor rule.

21

one percent increase in the cost-push shock for the alternative policies in Table 2. Under the
Taylor rule (i), inflation increases with the biggest magnitude in comparison to other policy
rules and then dies out slowly. The optimized Taylor rule (iii) with

⇡ = 3 andY

= 0.38produces only marginally smaller inflation fluctuations than the Taylor rule (top left panel)— this, however, is enough to make the welfare costs smaller in the optimized Taylor rule,
despite larger output gap fluctuations (bottom left panel).
Under the Taylor-MRIT rule (ii), the impulse response function of inflation almost perfectly matches that under the Ramsey policy. This is a reflection of the fact that the welfare
cost of this policy is close to that under the Ramsey policy, shown in Table 2. The output-gap
fluctuations under this policy are larger than under the other policies considered, although
they mimic the Ramsey dynamics quite well in latter periods. But again, compared toinflation fluctuations, output-gap fluctuations matter much less in the welfare function.
The Taylor-type rules restrict the dynamics of the nominal interest rates as a function of
inflation and the output gap and make the impulse response functions die out smoothly overtime. However, the Ramsey policy optimally determines the nominal interest rates without
any restrictions and sets negative nominal interest rates in the first period as shown in Figure
3. This flexibility helps generate a smaller fluctuation in the output gap in the first period
and achieve a lower welfare cost in comparison to the Taylor-MRIT rule while two policy
rules produce very similar impulse response functions for inflation.
We note that under the benchmark Ramsey policy, complete price stability is not optimal.
This is true as long as the weight on the output-gap stabilization in the welfare function is notzero and there exists a trade-o↵ between inflation stabilization and output-gap stabilization,
due to the existence of cost-push shocks.
To measure the improvement in the inflation-output trade-o↵ by adopting the TaylorMRIT rule (ii) over the optimized Taylor rule (iii), we compute the cumulative deviations ofinflation and the output gap based on the impulse responses in Figure 3. In the optimized
Taylor rule, the ratio of cumulative inflation deviations to output-gap deviations is 0.42. The
ratio is 0.12 in the Taylor-MRIT rule. This means that to close the output gap by 1% in
response to a cost-push shock, the monetary authority has to increase the inflation gap on
average by 0.42% under the optimized Taylor rule, but only by 0.12% when she is allowed22

to optimally adjust the medium-run inflation target. By this measure, the inflation-output
trade-o↵ is improved by more than twofold.

5

The slope of the Phillips curve and interest-rate smoothing

We conduct two additional analyses on the welfare implications of the NKPC slope andinterest rate smoothing.
First, we establish the relationship between the welfare gains from our proposed MRIT
and the NKPC slope, which is governed by the Calvo parameter. As we discuss below, theCalvo parameter is an important parameter for welfare analysis as it a↵ects the propagationof cost-push shocks and the relative weights of various stabilization goals in the welfare
function.
Second, we introduce interest-rate smoothing in the Taylor rule. We investigate if thewelfare gain from ”interest-rate” smoothing is significant and if this alternative specificationa↵ects the magnitude of the welfare gain from our proposed MRIT we found in the previous
section.

5.1

The flattening (steepening) of the NKPC

5.1.1

Illustration with a prototype model

To illustrate the implications of a flatter (or a steeper) NKPC for monetary policy in the presence of cost-push shocks, we consider a prototypical three-equation New Keynesian model.
Despite its simplicity, the model structure is rich enough to understand the propagation
mechanism of the cost-push shock and its welfare implication for the conduct of monetary
policy. The log-linearized version of the prototype New Keynesian model yields the following
representation for the New Keynesian Phillips curve, the IS curve, and the monetary policy

23rule:⇡t =Et ⇡t+1 + yt + µt

(18)yt = Et yt+1

(Rt

Et ⇡t+1 )

(19)

Rt =

⇡⇤) +

y yt

(20)

⇡ (⇡t

where ⇡t is inflation, yt is the output gap, Rt is the short-term nominal interest rate, ⇡ ⇤ isthe (long-run) inflation target, and µt is a cost-push shock.
Now, we consider the policy response to cost-push shocks in this model. A positive
realization of these shocks creates an upsurge in inflation and a negative output gap. Iterating
the NKPC in (18) forward indicates that inflation is a function of the expected discounted
sum of the future output gap yt and cost-push shock µt :⇡t = E t

1Xs(yt+s + µt+s ) .

(21)

s=0

The strength of the link between inflation and the output gap is directly governed by theslope parameter  in (21). When  is small (i.e. the NKPC is flat), the output gap hardly
drives inflation dynamics and the variation in inflation would be mostly explained by the
cost-push shocks, µt . In addition, the IS curve in (19) shows that the output gap is a functionof the sum of expected future real interest rates by iterating the IS curve forward:
yt =

Et

1
X

(Rt+s

⇡t+s+1 ).

(22)

s=0

Thus, the monetary authority can influence the output gap by manipulating the real rateindirectly through adjustments of the short-term nominal interest rate given sticky prices
using the Taylor-type policy rule in (20). In response to the increase in inflation, the monetary authority raises the nominal interest rate by amounts greater than increases in inflation
so that the real interest rate goes up.33 The IS curve in (22) implies that the increase in thereal interest rate lowers the output gap further and in turn, inflation is reduced through the33

The increase in the nominal interest rate is due to ⇡ > 1. More precisely, we consider the determinate
regions of the parameter space for y , ⇡ , and . For more details, see Bullard and Mitra (2002).

24

NKPC relationship.
However, as pointed out in the cost-push channel for inflation, a flat NKPC implies
that the e↵ect of output gap on inflation fluctuations is limited so that monetary policy
is less e↵ective in controlling inflation in the face of cost-push shocks. Thus, the e↵ect of
cost-push shocks on inflation would last longer. It is also notable that the decrease of theoutput gap arising from the positive cost-push shocks lowers the nominal interest rate in the
opposite direction compared to the e↵ect of inflation gap on the adjustment of the nominalinterest rates, as noted in the monetary policy rule (20). In the presence of the trade-o↵
between inflation stabilization and output-gap stabilization, e.g. due to the existence of thecost-push shock, this o↵setting e↵ect weakens the monetary authority’s reaction to inflation
fluctuations and presents an additional difficulty in conducting monetary policy
A flat NKPC has also the important welfare implication that the inflation variation
becomes enormously more important than the output gap fluctuation. Following Rotemberg
and Woodford (1997, 1999), Woodford (2003), and Steinsson (2003), the utility-based welfare
loss function in period t can be represented by
Lt = ⇡t2 + yt2⌘(23)

where ⌘ is the steady-state elasticity of substitutions between intermediate-good varieties.
Woodford (2003) shows that the weight on the output gap /⌘ in the loss function (23) issmall using the conventional calibrated parameters so that inflation plays a primary role indetermining household welfare. In addition, we can see that as the NKPC becomes flatter(i.e.  ! 0), the inflation variation becomes even more important because the weight on the
output gap in the loss function gets smaller.
In short, the flattening of the NKPC results in the difficulty in improving household welfare through monetary policy from three perspectives: (i) the fraction of inflation variation
due to cost-push shocks becomes increasingly substantial; (ii) controlling inflation throughmonetary policy becomes much harder conditional on the cost-push shocks; and (iii) inflation
becomes completely dominant over the output gap in determining household welfare.

25

Table 3: Welfare cost of various policies with alternative Calvo parameter values⇡↵ = 0.85(i) Taylor rule(ii) Taylor-MRIT rule↵ = 0.70(i) Taylor rule(ii) Taylor-MRIT rule

Policy Parameters
µ✓
⇢v✓
Y

Conditional
Welfare Costc ⇥ 100

Unconditional
Welfare Costu ⇥ 100

(⇡t

⇡
¯)

(ŷt

ŷt⇤ )

1.50
1.50

0.50
0.50

–
-0.0641

–
0.79

0.783
0.171

0.832
0.176

4.020
2.013

9.346
18.031

1.50
1.50

0.50
0.50

–
-0.0817

–
0.73

0.311
0.215

0.321
0.221

4.452
3.414

7.466
8.993

Note: We conduct welfare analysis for policies (i) and (ii) as in Table 2, but with di↵erent values of the NKPC slope, ↵ = 0.85
and ↵ = 0.70. For each ↵ case, the standard deviation of the innovation in the markup shock "✓,t is rescaled so that thevariance of the reduced-form cost-push shock ût in the NKPC remains the same as in the benchmark case (↵ = 0.80) — allother parameter values are as in Table 1. The welfare costs are relative to the Ramsey allocation in each respective ↵ case, i.e.
we resolve the Ramsey policy equilibrium for each case. We restrict | ⇡ | < 3, | Y | < 3, and |µ✓ | < 0.10 when searching for
optimized coefficients. The standard deviation of inflation and the output gap rate is expressed in percent per year.

5.1.2

The slope of the Phillips curve and welfare costs

The flattening of the NKPC since the mid-1980s has been widely documented in the literature.34 The NKPC slope in (8) is given by

= (1

↵ )(1

↵)/↵ and is mainly governedby the Calvo parameter ↵. To establish the relationship between the NKPC slope and thee↵ectiveness of monetary policy, we set ↵ = 0.70 for a steep slope and ↵ = 0.85 for a flat
slope. These parameter values are closely matched to the pre- and post-1980 U.S. economies.
For example, Bhattarai, Lee and Park (2016) find that the Calvo parameter is estimated tobe 0.67 for 1960:Q1 to 1979:Q2 and 0.84 for 1982:Q4 to 2008:Q2. These Calvo parametervalues correspond to 0.166 for the pre-1980 period and 0.032 for the post-1980 period underthe marginal-cost-based NKPC in (8). In addition, the slope  in the output-gap-based
NKPC in (9) is equal to 0.268 and 0.052 for the same two sub-periods. Thus, the slopeof the NKPC appears to have flattened substantially, with the slope coefficient reduced toabout a fifth of its pre-1980 period size.
The discussion using a prototype model in the previous section implies that a flatter
NKPC due to a higher Calvo parameter should lead to a higher welfare cost under a standard Taylor rule but a higher welfare gain from adopting the MRIT policy — that is, theimprovement in the inflation-output trade-o↵ should be larger. Or conversely, we should34

See our discussion in introduction and references therein for more details.

26see a relatively smaller welfare gain from adopting the MRIT under a steeper NKPC. We
conduct the same welfare analysis as in Section 4 with di↵erent slopes of the NKPC. In eachcase, we rescale the standard deviation of the innovation in the markup shock "✓,t so thatthe variance of the reduced-form cost-push shock ût in the NKPC in (9) remains the sameas in the benchmark case of ↵ = 0.8. We do this to focus on the impact of the change in theslope of the NKPC on the e↵ectiveness of monetary policy rules.
Table 3 reports the welfare-cost results under the Taylor rule (i) and the Taylor-MRIT
rule (ii) for ↵ = 0.85 and ↵ = 0.70 as in Table 2. When the NKPC is flat (i.e. ↵ = 0.85),
the standard Taylor rule produces the conditional and unconditional welfare cost of 0.783%
and 0.832%, respectively. The larger welfare costs compared to those for ↵ = 0.70 are to
be expected, since as discussed above, the flatter NKPC due to a high Calvo parameter
makes it relatively harder for the monetary authority to stabilize inflation. However, themonetary authority can do better by adjusting the medium-run inflation target in the faceof a cost-push shock when the Phillips curve becomes flatter. Here, despite a marginallysmaller (absolute) value of µ✓ for ↵ = 0.85, the welfare gain, however, is significantly bigger
in comparison to the case of ↵ = 0.70. We calculate the welfare gain using the unconditional
cost criterion by ( u,iu,ii )/ u,i whereu,i andu,ii are the unconditional welfare costs for

the Taylor rule (i) and the Taylor-MRIT rule (ii), respectively. We find that the welfare gainfrom adopting the Taylor-MRIT rule over the Taylor rule increases from 0.31 for ↵ = 0.70 to
0.79 for ↵ = 0.85. In the benchmark case for ↵ = 0.8, reported in Table 2, the corresponding
welfare gain is 0.64.
Thus, the results in Table 3 confirm that the flattening of the NKPC, as evidenced in theliterature, makes our proposed policy rule of adjusting the medium-run inflation target under
an inflation-output trade-o↵ even more appealing. When considering the current economic
environment of low inflation rates under a flat Phillips curve in many developed economies,
our main findings imply that the monetary authority should increase the inflation target insuch an environment.

27

Table 4: Welfare cost of various policies with interest-rate smoothing(i) Taylor rule(ii) Taylor-MRIT rule(iii)Optimized Taylor rule(iv) Optimized Taylor-MRIT rule

Policy Parameters
µ✓
⇢v✓
⇡
Y
1.50 0.50––
1.50 0.50 -0.0767 0.73
3.00 0.13––
3.00 1.95
-0.10
0.73

Conditional
Welfare Costc ⇥ 100
0.404
0.146
0.239
0.050

Unconditional
Welfare Costu ⇥ 100
0.423
0.150
0.245
0.050

(⇡t

⇡
¯)

3.611
1.889
2.572
1.458

(ŷt

ŷt⇤ )

9.505
13.79012.417
11.595

Note: All the respective cases above replicate the results reported in Table 2, but with the Taylor rule now given by⇤¯log(Rt /R̄) = R log(Rt 1 /R̄) + (1R )( ⇡ log(⇧t /⇧) + Y log(Yt /Yt )), with the smoothing parameter R set to 0.65. We
restrict | ⇡ | < 3, | Y | < 3, and |µ✓ | < 0.10 when searching for optimized coefficients. The standard deviation of inflation andthe output gap rate is expressed in percent per year.

5.2

Interest-rate smoothing

We now introduce interest-rate smoothing and examine if our findings based on a simple
Taylor-type rule are robust to this alternative specification. The interest-rate smoothing rule
becomeslog(Rt /R̄) =where we set¯

R log(Rt 1 /R̄) + (1R )( ⇡ log(⇧t /⇧) +⇤Y log(Yt /Yt )),R = 0.65 and all other parameters are as in Table 1.

Table 4 reports the implication of introducing interest-rate smoothing in the Taylor-type
rule. The result under interest-rate smoothing with the Taylor rule coefficients (policy (i))
implies that there is a gain from smoothing the nominal interest rate movements. Both theconditional and unconditional welfare costs are now about 0.1 percentage points smaller compared to those without smoothing reported in Table 2. However, the results under policies(ii)-(iv) confirm that the interest rate smoothing does not alter our main findings qualitatively and they are robust. Even under interest-rate smoothing, the monetary authority canachieve a non-trivial improvement under the inflation-output trade-o↵ by optimally adjusting the medium-run inflation target in the face of a cost-push shock. The improved trade-o↵,in turn, leads to a significant welfare gain.

28

6Conclusion

In this paper we show that there is an important role of inflation target adjustment in central
bank’s stabilization policy. Our findings demonstrate that it is welfare-improving to adjust
the medium-run inflation target in the opposite direction in response to a realization of
cost-push shocks. That is, the target needs to be increased when negative cost-push shocks
contribute to a low-inflation environment. This additional policy tool improves the policy
stabilization trade-o↵ and leads to significant welfare improvement that is not achievable ina conventional interest-feedback rule with a long-run, constant inflation target. The welfare
implications are more pronounced under a flatter Phillips curve.
Our proposed scenario is relevant to the current U.S. low-inflation environment since therecovery from the Great Recession. The inflation rates have been persistently lower than the
Fed’s 2% target since the target was introduced in 2012 and the economy appears to approach
the long-run unemployment rate in 2017. This also implies that the Phillips curve appears
to be flatter than in the past as pointed out by Blanchard (2016). If inflation continues toundershoot the 2% target while job growth remains strong and the unemployment rate is
below the long-run unemployment rate, low inflation and low unemployment can be largely
attributable to negative “cost-push” shocks such as changes in price and wage markups andoil price shocks. If this is the case, our findings suggest that it may be prudent for theFederal Reserve to increase its target, at least in the medium run.

29

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33

A

Appendix

This appendix describes additional details of the model and computational issues.
Our model consists of a representative household, a continuum of monopolisticallycompetitive firms producing di↵erentiated varieties, and a monetary policy authority. Our
adopted model is based upon the model used in Schmitt-Grohé and Uribe (2007), although
we abstract from monetary distortions, capital accumulation, and fiscal policy.

Households
Households choose the state-contingent consumption, Ct , labor service, Nt , and one-period
discount bond, Bt , to maximize the lifetime utility,
Et

1
X

Nt+s ) ]1
1

s [Ct+s (1

s=0

1

,

subject to the per-period nominal budget constraint
Pt Ct + Bt  Rt 1 Bt 1 + Wt Nt + ⇧prof+ Tt ,tor, in real terms,
C t + bt  Rt 1

bt 1
⇧prof
+ w t Nt + t+ ⌧t .
⇧tPt

bt ⌘ Bt /Pt is real bond, Rt is the nominal interest rate, wt is the real wage,⇧prof
tis the proceedPtof real profits from intermediate-goods firms (owned by households), and ⌧t is the real tax
or transfer. The consumption index Ct is a Dixit-Stiglitz CES aggregator of di↵erentiatedconsumption goods or varieties, given by

Ct =

Z 1

1+✓t

Ct (i)

1/(1+✓t )

di

(A.1)

0

where
✓t = 1/(⌘t

34

1)

(A.2)

is the firms’ stochastic average markup at time t and ⌘t is the elasticity of substitution acrossvarieties.
The resulting households’ efficiency conditions:

0=

Here,

0 = Ct(1

Nt ) (1

)

Ct1

(1

Nt ) (1) 1

0=Rt E t

t+1

t

(A.3)

t

+

t wt

⇧t+1

(A.4)
(A.5)

t is the Lagrange multiplier (the shadow cost of consumption).

Firms
Firms produce the di↵erentiated varieties using the production function
Yt (i) = zt Nt (i).
The labor market is global. Firms face infrequent opportunities to adjust their prices optimally in a Calvo (1983) manner, with probability 1↵ every period. When a firm i is notallowed to adjust optimally, with probability ↵, it simply indexes its current prices to theconstant long-run (steady-state) inflation target:
Pt (i) = Pt 1 (i)⇧.
Given the CES aggregation and the above structure, the demand for each variety i attime t + j for firms that last adjusted its price optimally at time t is"P̃t jtYt+j (i) =
Pt+j

# ⌘t

where, given the indexation scheme,jjt = ⇧

35

Yt+j ,

Yt is the aggregate output and P̃t is the common optimal price at t chosen by all optimizing
firms, satisfying
0 = Et1
X⌘t↵ Qt,t+j Yt+j Pt+j
j

1 ⌘t
jt

j=0

Qt,t+j ⌘

j



P̃t

⌘t
⌘t

1

M Ct+j

1
jt

.

/P

Et t+jt /Pt+j, M Ct , and Pt denoting the nominal stochastic discount factor (betweenttime t and t + j), nominal marginal cost, and the aggregate price level, respectively.
The aggregate price level is given by
Pt1 ⌘t = (1↵)(P̃t )1 ⌘t + ↵ ⇧Pt 1

1 ⌘t

Equating aggregate demand with aggregate supply, we obtain (after some rearranging)z t Nt = Yt

where

t =

R1

t,

(A.6)

Yt = C t + G t ,(A.7)(Pt (i)/Pt ) ⌘t di is a measure of price dispersion, i.e. the relative-price distor-

0tion. Gt is the aggregate government spending, aggregated the same way as in (A.1).

Recursive representations
The optimal price equation (A) and the aggregate price equation (A) can be expressedrecursively asP̃t⌘t K1t=Pt⌘t 1 K2thi⌘tt
K1t = t Yt mct + Et ↵ ⇧ ⇧⌘t+1
K1,t+1p̃t ⌘K2t =1 = (1

(A.8)
(A.9)

h
i
1 ⌘ t ⌘t 1
Y
+
E
↵
⇧
⇧
K
t t
t
2,t+1
t+1

(A.10).

(A.11)↵)(p̃t )1 ⌘t + ↵ ⇧⇧t 1

36

1 ⌘t

mct ⌘ M Ct /Pt is the average real marginal cost — given the production function,mct = wt /zt .

(A.12)

We can also recursively write the relative-price distortion ast = (1↵)(p̃t ) ⌘t + ↵ ⇧⇧t 1⌘tt 1(A.13)The monetary policy rule and the adjustment of the medium-run
inflation target
In the benchmark model without the adjustment of the medium-run inflation target (MRIT),
the monetary policy authority is assumed to follow a Taylor-type rule,log(Rt /R̄) =¯⇡ log(⇧t /⇧) +⇤Y log(Yt /Yt ),(A.14)

where Yt⇤ is the potential or natural level of output, defined as the level of output in theflexible-price equilibrium with constant markup, satisfying
Yt⇤ =

(¯
⌘ 1)
zt + Gt⌘¯.
+ (¯⌘ ⌘¯ 1)(A.15)In an alternative model with our proposed policy, the monetary authority employs themodified Taylor-type rule and adjusts the MRIT in response to the markup shock:log(Rt /R̄) =⇤⇡ log(⇧t /⇧t ) +⇤Y log(Yt /Yt )(A.16)¯ + µ⇡⇤ ⇡⇤ ,tlog(⇧⇤t ) = log(⇧)(A.17)= ⇢ ⇡⇤ ⇡⇤ ,t 1 + "✓,t

(A.18)⇡ ⇤ ,tHere, we assume that the adjustment of the MRIT is entirely endogenous, i.e. it evolves
entirely due to the monetary authority’s action, rather than exogenously. Setting µ✓ = 0
yields the standard assumption that the medium-run inflation target is equal to its constant37

long-run inflation target.

Evolution of exogenous variables
The model has three exogenous variables: productivity, zt , government spending, Gt , andaverage markup, ✓t . Their evolutions follow
log(zt ) = ⇢z log(zt 1 ) + "z,t ,(A.19)

log(Gt /Ḡ) = ⇢g log(Gt 1 /Ḡ) + "g,t ,(A.20)

log(✓t ) = (1¯ + ⇢✓ log(✓t 1 ) + "✓,t .
⇢✓ ) log(✓)(A.21)

Aggregate fluctuations are driven by three exogenous shocks: "z,t ⇠ i.i.d. N (0,i.i.d. N (0,2g ), and "✓,t ⇠ i.i.d. N (0,

2
✓ ).

2
z ), "g,t ⇠Complete equilibrium equations (competitive equilibrium)
In the standard model without MRIT, the complete equilibrium conditions (under competitive equilibrium) are given by equations (A.2), (A.3), (A.4), (A.5), (A.6), (A.7), (A.8), (A.9),(A.10), (A.11), (A.12), (A.13), (A.14), (A.15), and the evolution equations of exogenous variables (A.19)- (A.21). The corresponding endogenous variables are Ct , Yt , Nt ,p̃t , K1t , K2t ,t , w t , R t , ⇧t ,⇤t , mct , Yt , and ⌘t .

In the model with the MRIT adjustment, the policy rule (A.14) is replaced with (A.16)(A.18). The medium-run inflation target, ⇧⇤t , is now an endogenous variable. We solve forthe equilibrium using the perturbation method, up to a second-order approximation.

Ramsey policy
We derive the Ramsey equilibrium by formulating a recursive Lagrangian as in Marcet andMarimon (2011). The objective function is the households’ welfare as the Ramsey policy
authority is benevolent. The Ramsey authority conducts policy in a decentralized setting.
This means that all the private-sector efficiency conditions described above have to be re-38

spected, and becomes the constraint set in the optimal policy problem (the Lagrangian). As
is standard in the literature, we do not take a stand on the form of the optimal policy ruleand search instead for the equilibrium allocation under the Ramsey policy — in technicalterm, this means that policy rule (A.14) is not part of the Ramsey authority’s constraint set.
The Ramsey policy is solved under the assumption of a constant long-run inflation target,¯⇧⇤t = ⇧.

39