GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.34

Inflation Inertia and Optimal Delegation of
Monetary Policy

Keiichi MorimotoFebruary 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Inflation Inertia and Optimal Delegation of
Monetary Policy∗
Keiichi Morimoto†February, 2009

Abstract
This paper analyzes the relationship between the optimal weight on output gap
in the central bank’s loss function and the degree of inertia in a hybrid version ofNew Keynesian model with a pure discretionary inflation targeting. I present thepolicy recommendations as to the weight on output gap in the presence of endogenous persistence in inflation dynamics. Especially, I show that under endogenous
persistence of inflation dynamics, even in discretionary monetary policy regime,a Rogoff’s (1985) conservative central banker does not necessarily improve socialwelfare.

Keywords: hybrid New Keynesian model; inflation targeting; policy weight
JEL Classification: E52; E58∗I am very grateful to Kazuo Mino for his invaluable advice and encouragement. I also thank Yuzo
Honda and Yuichi Fukuta for their helpful comments.
†Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail address: gge014mk@mail2.econ.osaka-u.ac.jp

1Introduction

In recent years, hybrid New Keynesian models have been widely used for macroeconomic
analyses. Departing from the standard New Keynesian models, the hybrid New Keynesianmodels introduce inertia into dynamics of aggregate demand (AD) and aggregate supply(AS) relations. In this paper, I investigate the relationship between the optimal weight on
output gap in the central bank’s loss function and the parameters contained in a hybrid
New Keynesian model when the central bank conducts a discretionary monetary policy.
This issue has not been fully discussed in the existing literatures. Among the parameters,
I focus on the degree of inertia in a hybrid New Keynesian Phillips curve. This is because
the recent empirical studies such as Galı́ et al. (2005) support the presence of inertial
inflation dynamics.
In the literature, the seminal work of Rogoff (1985) suggests that to reduce the additional social loss generated by the inflation bias under discretionary optimal monetary
policy, it is optimal to appoint a central banker who places a higher weight on inflation
than the society. While Rogoff (1985) uses a traditional Lucas type of Phillips curve,
Clarida et al. (1999) use a basic New Keynesian model and shows that the Rogoff’s(1985) result holds under exogenous inflation persistence which a serial correlation of
cost shocks gives rise to. 1
In this paper, I analyse the relation ship between the degree of inflation inertia andthe optimal policy weight on output gap in the central bank’s loss function and showthat under endogenous persistence of inflation dynamics, even in discretionary monetary
policy regime, a Rogoff’s (1985) conservative central banker does not necessarily improvesocial welfare. 21

For details, see section 2.4.
Along the line of Rogoff (1985), a central banker is called conservative if λc is small, that is, sheplaces a large relative weight on inflation.
21

2

Inflation Inertia and Optimal Policy Weight

2.1

Hybrid New Keynesian Model

To analyze the relation between inflation inertia and optimal policy weight, I use thehybrid version of New Keynesian models introduced by Galı́ and Gertler (1999). Themodel does not have any theoretically rigorous micro foundations but it is very simple,useful and sufficient for the purpose of the paper.
A hybrid New Keynesian model consists of the hybrid versions of IS relation and New
Keynesian Phillips curve (HNKPC) respectively given byxt = (1 − φ)Et xt+1 + φxt−1 − σ(it − Et πt+1 ) + ut ,(1)πt = (1 − ψ)βEt πt+1 + ψπt−1 + κxt + vt ,(2)together with a monetary policy rule. Here, xt , it , πt , ut and vt denote output gap, nominal interest rate, inflation rate, demand shock and cost shock in period t, respectively.
Parameters σ, β and κ are positive constants, where β is the discount factor, σ is theintertemporal elasticity of substitution and κ is the impact of one unit of output gap oninflation. Constant parameters φ ∈ [0, 1] and ψ ∈ [0, 1] represent the degrees of inertia in
AD and AS relations. If φ = ψ = 0, then the model is identical to a basic New Keynesian∞model. I assume that {ut }∞t=0 and {vt }t=0 follow AR(1) processes. That is, dynamics ofut and vt are given by
ut+1 = ρu ut + εut+1 ,vt+1 = ρv vt + εvt+1 ,where ρu ∈ [0, 1), ρv ∈ [0, 1), εut+1 ∼ N (0, θu2 ) and εvt+1 ∼ N (0, θv2 ).
To measure social welfare, I adopt the traditional social loss function such that∞
1 X
L = E0β t (πt2 + λs x2t ),2 t=0(3)where parameter λs is the relative weight that the representative household places on
output gap relative to inflation.
22.2

Inflation Targeting under Discretion (ITD)

Suppose that the central bank pursues an inflation targeting under pure discretion. In
this regime, in each period, the central bank minimizes a discounted sum of the current
and future loss with future inflaiton and output gap given. I set the period-loss function
in such a way that1
Lt = (πt2 + λc x2t ), ∀t ≥ 0,2

where λc is the relative weight selected by the central bank that may be different fromλs . Since the state variables in period t are πt−1 and vt , the Bellman equation for thecentral bank’s optimization problem is½V (πt−1 , vt ) =¾

1 2(π + λc x2t ) + βEt V (πt , vt+1 ) ,2 txt = (1 − φ)Et xt+1 + φxt−1 − σ(it − Et πt+1 ) + ut ,min

it ,πt ,xts.t.

πt = (1 − ψ)βEt πt+1 + ψπt−1 + κxt + vt .

(4)
(5)
(6)

Assuming that the nonnegativity constraint of nominal interest rate it is not binding,the equation (5) does not bind as a constraint since the Lagrangian for the optimization
problem of the right-hand-side of (4) is linear in it . 3 Thus, the necessary condition for
an optimum can be obtained as 4πt =λc(βψEt xt+1 − xt ), ∀t ≥ 0.
κ(7)The equations (6) and (7), the hybrid New Keynesian Phillips curve and the optimalmonetary policy rule, determine the equilibrium dynamics of the model economy as thesequences of inflation and output gap.
3As usual, the nonnegativity constraint of nominal interest rate is ignoired in optimal monetary policy
analyses in the New Keynesian framework.
4Substituting (6) into the right-hand-side of (4), the first order condition with respect to xt is
απt + λc xt + βEV1 (πt , vt+1 )κ = 0.
The envelope condition isλc
ψxt .
κEliminating the value function, we have the first order condition (7).
V1 (πt−1 , vt ) = −3

Table 1: Baseline Parameter Value
λs

2.3

βρu

ρv

κ

φσθu

θv

0.25 0.99

000.050.5 0.67

0.15

0.15

SimulationcSince the optimal sequences of inflation and output gap, {(πt , xt )}∞t=0 , depend on λ , inequilibrium, the social loss (3) can be expressed as a function of parameter λc . Therefore,
the optimal policy weight (denoted by λ∗ ) can be selected by minimizing the social loss
in equilibrium with respect to λc .
Moreover, note that from (7) the social loss in equilibrium depends on ψ, so thatthe selected λ∗ depends on ψ as well. Hence, I can obtain the relationship between thedegree of inertia, ψ, and the optimal policy weight, λ∗ . In a mathematical formation, theoptimal policy weight is∞³´1 Xλ (ψ) ∈ argmin E0β t πt (λc , ψ)2 + λs xt (λc , ψ)2 ,2 t=0λc∗where πt (λc , ψ), xt (λc , ψ) are equilibrium inflation rate and output gap in period t when
the policy weight and degree of inflation inertia are λc and ψ respectively.
To calculate the optimal policy weight on output gap, I reset the social loss function(3) aslim 2(1 − β)L = V [π] + λs V [x],β→1where V [π] and V [x] are the asymptotic variances of inflation and output gap.
I analyze the relation between λ∗ and ψ numerically. In so doing, I set the baseline
pameter value in Table 1.5
The numerical analysis reveals the following fact:
Result 1 For a set of parameters with plausible magnitudes, there is ψ ∗ such that λ∗decreases with ψ for ψ ∈ [0, ψ ∗ ] and increases with ψ for ψ ∈ [ψ ∗ , 1].
5

The values of parameters are the same as in Jensen (2002). I conduct the numerical calculation byuse of the algorithm in Söderlind (1999).

4

Figure 1 illustrates Result 1 based on our numerical example. Result 1 is intuitively
plausible. A conservative central banker tries to stabilize inflation actively. Endogenous
persistence in inflation dynamics helps such a stabilizing action directly, because a partof future inflation is controlled by current inflation through the economic agents’ partially backward-looking behaviors. Thus, appointing a more conservative central banker
improves a trade-off between inflation and output gap as long as the degree of inflation
inertia is not too high. This is because the cost of stabilizing inflation generated by anexpansion of output gap is relatively small. However, when the degree of inertia is sufficiently high, stabilizing inflation yields a large output gap. Hence, in this case stronger
conservatism leads to a worse trade-off between inflation and output gap. As a consequence, there is a turning point ψ ∗ in the relation between the degree of inflation inertia
ψ and the optimal weight λ∗ .

2.4Discussion

The results of Clarida et al. (1999) claim that in a basic New Keynesian model withoutinertia, if the cost shock {vt }∞t=0 is not serially correlated as related literatures often
assumes, then the optimal weight on output gap in ITD is identical to the social preference
λs . In other words, appointing a central banker sharing the social preference is optimal.
In fact, since that optimal weight λ∗ is given by6λ∗ = (1 − βρ)λs ,(8)

ρ = 0 implies that λ∗ = λs . However, Fact 1 demonstrates that this policy implication
does not hold when inflation dynamics has endogenous persistence. In this case, it raises
social welfare to appoint a more conservative central banker. This is because, in addition
to the expectations effect, there is an inertia effect mentioned in the previous section.7
6

See Vestin (2006).
If ρ > 0, the future values of the cost shocks can be partially forecast. Hence, the rational agents, who
know that a conservative central banker react to the cost shocks harder, expect stable future inflation.
This behavior contributes to stabilizing current inflation, which may be called the expectations effect.
It disappears if ρ = 0. Note that the expectations effect is generated through a mechanism which isdifferent from the inertia effects.
75Equation (8) also means that in the absence of endogenous persistence of inflation
dynamics, if {vt }∞t=0 is serially correlated, the optimal weight on output gap is lower thanλs and it monotonically decreases with the degree of exogenous persistence ρ. However,when the inflation persistence is endogenous, by the mechanism mentioned above, there
is a critical value ψ ∗ after which λ∗ increases with ψ: see Figure 1. That is, underinertial inflation dynamics, the behavior of the optimal policy weight is not monotone,so that stronger inflation persistence does not necessarily require a more conservative
central banker: the central bank should place a higher weight on the loss from income
fluctuation when inflation inertia is intense enough. Besides, the former result, λ∗ < λs ,can be reversed if inflation dynamics exhibits very strong inertia. In this case, since
inflation behaves stably by itself, the gain of stabilizing income flactuation more actively
is relatively large. Hence, the central bank should place a higher weight on output gap
than society, that is, λ∗ > λs .
It is important to study how the critical point ψ ∗ varies as the other parameterschange. Figures 2 and 3 respectively depict the relations between ψ ∗ and two key parameters, ρv and κ. It is easy to interpret those graphs: a rise in ρv increases both theexpectations effect explained in footnote 6 and the relevance of reducing income fluctuation so that ψ ∗ is lowered, while a larger κ requires to reduce output gap and, hence, ψ ∗again decreases.
Result 2 For a set of parameters with plausible magnitudes, the critical point ψ ∗ decreases with ρv and κ.
Result 1 and 2 suggests that under inertial inflation dynamics, the policy implicationon the optimal weight on output gap in the central bank’s loss function is not simple as
the literature claims. The parameter value should be considered more carefully in theface of monetary policy delegation problem.

63Concluding Remark

One of the remaining problems is to find the degree of inertia in inflation dynamics onwhich researchers majoring in monetary economics reach a consensus. According to Fact
1, the concrete policy implication about the optimal weight for the real economy depends
mainly on the true degree of inertia. Rudebusch (2002) estimates ψ = 0.71 for the U.S.
data. Galı́ et al. (2005) estimate ψ by three methods and the values of the estimators
are 0.349, 0.374 and 0.260.8 Fuhrer (1997) demonstrates that the case ψ = 1 can not berejected. Thus, there has not been a general agreement with the value of ψ.

8In Galı́ et al. (2005), they use real marginal costs in place of output gaps of HNKPC.

7References
Clarida, R., Galı́, J. and M. Gertler (1999), ”The Science of Monetary Policy: A New
Keynesian Perspective”, Journal of Economic Literature, 37, 1661-1707.
Fuhrer, J.,C. (1997), ”The (Un)Importance of Forward-Looking Behavior in Price Specifications”, Journal of Money, Credit and Banking, 29, 338-350.
Galı́, J. and M. Gertler (1999), ”Inflation dynamics: A structural econometric analysis”,
Journal of Monetary Economics, 44, 195-222.
Galı́, J., Gertler, M. and J.D. López-Salido (2005), ”Robustness of the estimates of thehybrid New Keynesian Phillips curve”, Journal of Monetary Economics, 52, 1107-1118.
Jensen, H. (2002), ”Targeting Nominal Income Growth or Inflation ?”, American Economic Review, 94, 928-956.
Rogoff, K. (1985), ”The Optimal Degree of Commitment to an Intermediate MonetaryTarget”, Quarterly Journal of Economics, 100, 1169-1189.
Rudebusch, G., D. (2002), ”Assessing Nominal Income Rules for Monetary Policy with
Model and Data Uncertainty”, Economic Journal, 112, 402-432.
Söderlind, P. (1999), ”Solution and estimation of RE macromodels with optimal policy”,
European Economic Review, 43, 813-823.
Vestin, D. (2006), ”Price-level versus inflation targeting”, Journal of Monetary Economics, 53, 1361-1376.

8

0.5
0.45
0.4
0.35

λ∗

0.3
0.25
0.2
0.15
0.1
0.05
0ψ∗
0

0.1

0.2

0.3

0.4

0.5

ψ

0.6

0.7

0.8

0.9

1

Figure 1: relation between ψ and λ∗ (ρ = 0, κ = 0.05, λs = 0.25)

9

0.52

0.5

ψ∗

0.48

0.46

0.44

0.42

00.1

0.2

ρv

0.3

0.4

0.5

Figure 2: relation between ρv and ψ ∗ in Fig.1 (κ = 0.05)10

0.53

0.52

ψ∗0.51

0.5

0.49

0.48

0.47

00.05

0.1

0.15

κ

Figure 3: relation between κ and ψ ∗ in Fig.1 (ρ = 0)110.2