Research InstituteDiscussion Paper Series№ 20Title：

Robust control and delegating optimal monetary policy
under incomplete exchange rate pass-through

Daisuke Ida
ida-dai@andrew.ac.jp

2022 年 4 月

http://www.andrew.ac.jp/soken/

Robust control and delegating optimal monetary policy
under incomplete exchange rate pass-through∗Daisuke Ida†
April 11, 2022

Abstract
This study examines the effect of model uncertainty on the performance of delegating
optimal monetary policy inertia in a small open economy with imperfect exchange ratepass-through. It addresses the role of nominal income growth targeting with the existenceof the law of one price gap under model uncertainty.

JEL codes: E52; E58; F41
Keywords : Optimal monetary policy; Targeting regime; Incomplete exchange ratepass-through; Robust control; small open economy∗

This work was supported by JSPS KAKENHI Grant Number JP20K01784. I would like to thank Mitsuhiro

Okano for his useful comments and suggestions. All remaining errors are my own.
†Faculty of Economics, Momoyama Gakuin University, 1-1, Manabino, Izumi, Osaka 594-1198, Japan. Tel.:+81-725-54-3131. E-mail: ida-dai@andrew.ac.jp

1Introduction

This study examines whether monetary policy objectives with lagged endogenous variables canovercome the stabilization bias caused by discretionary policies in a small open economy withincomplete exchange rate pass-through in the presence of model uncertainty. The stabilization
bias arises from inadequate policy responses to structural shocks that generate policy trade-offs
under a discretionary policy. The central banker implementing a discretionary policy suffersfrom stabilization bias even in an open economy because the small open new Keynesian (NK)model is isomorphic to the closed one (Clarida et al., 2001). Thus, the problem of stabilization
bias remains an important issue even in an open economy.
Following Bilbiie (2014), Ida and Okano (2021) analytically examined the role of delegating
optimal monetary policy inertia to answer the aforementioned problem. Although Ida andOkano (2021) showed the effectiveness of speed limit targeting suggested by Walsh (2003),
they did not consider both cases of robust control and imperfect exchange rate pass-through.
Ida and Okano (2022) examined the effectiveness of targeting regimes with lagged endogenous
variables in a model with complete exchange rate pass-through under robust control. However,they did not consider the case for incomplete pass-through of the exchange rate. Imperfectexchange rate pass-through exacerbates the trade-offs more within policy objectives than in astandard small open economy model.
This study aims to fill such a gap in the existing literature. More concretely, following
Monacelli (2005), I consider the role of incomplete exchange rate pass-through in the small
open NK model. The exchange rate’s incomplete pass-through would complicate more theproblem of robust control than in the case of complete pass-through because the law of one
price (LOP) no longer holds in the former case. Therefore, we can consider that the stabilization
bias will be more severe under the case of incomplete exchange rate pass-through. This paper
compares the performance of a commitment policy with several alternative targeting regimesthat have different monetary policy objectives from the social loss function. To the best of myknowledge, no study has attempted to investigate this issue.
The main results are summarized as follows. First, the commitment policy can lead topreferable outcomes to all policy regimes with the disappearance of robustness concerns. Second, the policy regimes with a lagged output gap are superior to inflation targeting. Particularly, I emphasize that when robustness considerably matters, nominal income growth targeting1

(NIT) outperforms all the regimes considered in this paper.

2Model description

This paper considers the role of incomplete pass-through of the exchange rate in a small open
NK model with robust control. I adopt the small open NK model developed by Monacelli(2005).1 The log-linearized dynamic system is summarized as follows:
πt = βEt πt+1 + κy xt + κψ ψt + ut ,(1)

fπtf = βEt πt+1+ λψ ψt ,(2)

xt = Et xt+1 −γ(σa − ωψ )ωs(it − Et πt+1 ) +Et ∆ψt+1 ,
σσ(3)

f
∗
Et ψt+1 = ψt + it − i∗t + Et πt+1− Et πt+1,(4)qt = ψt + (1 − γ)st ,(5)

st =1(σxt − ωψ ψt ),ωs(6)where πt is producer currency inflation, xt is the output gap, it is the nominal interest rate,and πtf is import inflation. Moreover, ψt is the LOP gap, which represents the deviation of law
of one price in terms of import prices. Note that ψt becomes zero under complete exchange
rate pass-through. πt∗ and i∗t are the foreign inflation and the foreign interest rate, respectively.
Meanwhile, qt , st , and ut denote the real exchange rate, the terms of trade, and the pricemark-up (cost-push) shock, respectively. The coefficients of this system are defined as follows:ωψ
σ(1 − αh )(1 − αh β)(1 − αh )(1 − αh β)κy =φ+; κψ =1−;αhωsαhωs(1 − αf )(1 − αf β),ωs = 1 + γ(2 − γ)(σa − 1); ωψ = 1 + γ(σa − 1); λψ =αfwhere σ is the relative risk aversion for the utility of consumption, φ is the inverse of labor
supply elasticity, a is the elasticity of substitution between home and foreign goods. αh andαf are Calvo lottery for home-goods firms and import-goods firms, respectively. γ denotes thedegree of openness. ut and ϵt denote the demand and supply shocks captured by the standard
AR(1) process.
1See Monacelli (2005) for a detailed derivation of a small open NK model with incomplete path-through ofthe exchange rate.

2Equation (1) represents the price new Keynesian Phillips curve (NKPC) derived from thepresence of nominal price rigidity. Unlike complete exchange rate pass-through, the LOP
gap affects price inflation dynamics. Equation (2) is the import-price NKPC derived by theincomplete pass-through of the exchange rate. Equation (3) represents the dynamic aggregate
demand curve, which is now affected by the LOP gap that evolves in accordance with Equation(4). Equations (5) and (6) determine the real exchange rate and the terms of trade, respectively.
Next, to consider optimal monetary policy in this model, we must specify the central bank’s
loss function. Following Monacelli (2005), I use the following standard loss function:
Lt = E0∞Xβt

[(1 − γ)πt2 + γ(πtf )2 ] + λx2t.

(7)

t=0The loss function (7) is the social true loss function. The parameter λ denotes the stabilization
weight on the output gap relative to inflation.

3

Solution methods: Robust control approach

Following Giordani and Söderlind (2004) and Tillmann (2009), I briefly describe the robust
control approach in this section.2 In this approach, the central bank considers the model
presented in the previous section as a reference model that suggests the most likely model
description of the economic structure. However, the central bank also knows that this structuralmodel faces various distortions associated with the possibility of model misspecification. In thiscase, the central bank sets the policy instrument rt to minimize the loss function. However, it
is concerned about the existence of model misspecification and attempts to be robust to such
misspecification. To examine this robust control problem, previous studies have considered anoptimization problem in which the central bank minimizes the value of the loss function, but theevil agent maximizes the loss. In this way, the optimization problem faced by the central bankunder model uncertainty is transformed into a min–max problem. In the following analysis,
we focus on the worst-case distortion defined as the economy’s behavior when the pessimismof the planners is fully warranted.
Under model uncertainty, the central bank faces the following distorted model that incor2

See Giordani and Söderlind (2004) for a detailed explanation of robust control approach in linear-quadratic

optimization problem.

3

porates the distortions associated with possible model misspecification:
Xt+1 = AXt + Brt + C(εt+1 + νt+1 ),(8)

where Xt+1 = [y1t+1 Et y2t+1 ]′ . y1t and y2t denote vectors of predetermined and forwardlooking variables, respectively. rt denotes the vector of policy instruments. Meanwhile, A and
B are coefficient matrices constructed by structural parameters. Distortion νt+1 denotes theworst-case distortions chosen by evil agent, and εt+1 is a vector caused by structural shocks.
The additional term νt+1 + εt+1 is multiplied by matrix C. Evil agents who maximize thecentral bank’s loss function chose the worst-case distortions, subject to the following budget
constraints:E0∞X′β t νt+1 νt+1 ≤ η,(9)t=0where η denotes the measure of the central bank’s preference for robustness. Accordingly,
in our model the central bank implements optimal monetary policy by solving the followingoptimization problem:
min max E0{it } {νt+1 }

∞
Xβt



[(1 − γ)πt2 + γ(πtf )2 ] + λx2t

,

t=0

subject to Equations (8) and (9). Following Giordani and Söderlind (2004), we can rewrite thementioned optimization problem as the following equivalent multiplier problem:3
∞X′
min max E0β t [(1 − γ)πt2 + γ(πtf )2 ] + λx2t − θνt+1 νt+1 ,{it } {νt+1 }

t=0

subject to Equation (8). We define θ as 1/η; therefore, θ ∈ (0, ∞) means a set of models
available to the evil agent.
Let us now consider the optimal delegation problem in the presence of imperfect exchange
rate pass-through. As described by Walsh (2003), the alternative targeting regime can becharacterized by the policy objectives delegated to the central bank and the stabilization weights
for each objective in the loss function. Following Walsh (2003), this study defined the targeting
rule as (a) the variables in the central bank’s objective function and (b) the stabilization weights
for the policy objectives in the loss function chosen to minimize the expected discounted valueof the social loss function (7).
3See Giordani and Söderlind (2004) for a detailed discussion of this transformation.

4

The targeting rules in this study are specified in Table 1. Except for the definition ofinflation in this paper, Table 1 defines the first three regimes, following Walsh (2003). Accordingto Jensen (2002) and Walsh (2003), the performance of speed limit policy (SLP) and NIT issuperior to that of inflation targeting (IT). The fourth regime is suggested by Ida and Okano(2021) in a model with a complete exchange rate pass-through. Then, we consider the policy
regime targeting a change in the LOP gap under a discretionary policy. Note again that thestabilization weight λi (for i = IT, SLP, N IT, LOP, REX) is chosen so that the central bankminimizes the social loss function under each delegated objective function, indicating that λi
does not generally coincide with λ.
[Table 1 around here]
I numerically solve the optimal delegation problem under robust control using the solution
algorithm developed by Giordani and Söderlind (2004).

4

Quantitative results

This section provides the quantitative results of the performance of targeting regimes underrobust control in a model with incomplete exchange rate pass-through. The calibrated values
in this paper are used in the standard NK model (Table 2). Before reporting the simulation
result, I briefly mention the selection of the parameter θ representing the degree of robustness.
I assume that the value of θ, which represents the degree of robustness, ranges from 30 to
500 to obtain stable equilibrium. My calibration strategy is based on Tillmann (2009). Thus,
this paper considers the values of θ that the detection error probability ranges from 0 to 0.5.
Moreover, it assumes that the selected values of θ satisfy the range of detection error probability
suggested in previous studies.
[Table 2 around here]
Table 3 reports the welfare losses under several values of robustness parameter θ. The
results are summarized as follows. First, the precommitment policy can lead to preferable
outcomes to all policy regimes as long as the parameter θ is above 75. Second, the policy
regimes with a lagged output gap are superior to IT. Third, the LOP targeting induces theworst welfare losses of all regimes. Finally and importantly, NIT becomes the most effective
5

tool of all policy regimes when θ < 75. In this case, the performance of NIT outperforms that ofa precommitment policy. Moreover, it follows from Table 3 that within the parameter range of
θ, NIT produces the most stable outcomes for all policy regimes. Ida and Okano (2022) showedthat the performance of NIT dominates that of a commitment policy in the case of a smaller
value of θ under complete exchange rate pass-through. In contrast to their study, the presentstudy addresses the role of NIT when the central banker worries about model uncertainty inthe presence of ψt .
[Table 3 around here]
Figure 1 shows that a price mark-up shock induces policy trade-off between PPI inflationand the output gap in a worst-case equilibrium. Also, it causes the trade-off between priceand import inflation stabilization. Thus, in contrast to a standard open economy model, an
introduction of incomplete exchange rate pass-through causes a worse policy trade-off between
inflation and the output gap. More concretely, stabilizing PPI inflation creates a fluctuationin the output gap following the movements in ψt .
[Figure 1 around here]
As shown in Figure 1, unlike discretion, a policy target with a lagged output gap imparts
policy inertia into the economy. This policy inertia can ease a policy trade-off in the presenceof ψt . Jensen (2002) and Walsh (2003) suggested that this indicates the gain from adopting
delegated optimal monetary policy inertia to overcome the stabilization bias. In particular, I
address that NIT containing the stabilization of both PPI inflation and a change in the output
can alleviate such a trade-off caused by incomplete exchange rate pass-through. The gain fromemploying NIT might be justified by the dynamics of the nominal exchange rate. Indeed,
Figure 2 illustrates that the exchange rate under NIT retains the stationarity compared withother discretionary policy regimes.
[Figure 2 around here]

5Concluding remarks

This paper examined the effect of model uncertainty on the performance of delegating optimalmonetary policy inertia in a small open economy with imperfect exchange rate pass-through.
6

It showed the effectiveness of NIT under model uncertainty in the presence of the LOP gap.

References
Bilbiie, F.O., 2014. Delegating optimal monetary policy inertia. Journal of Economic Dynamics
and Control 48, 63–78.
Clarida, R., Galı́, J., Gertler, M., 2001. Optimal monetary policy in open vs. closed economies.
American Economic Review 91, 248–252.
Giordani, P., Söderlind, P., 2004. Solution of macromodels with hansen–sargent robust policies:
Some extensions. Journal of Economic Dynamics and control 28, 2367–2397.
Ida, D., Okano, M., 2021. Delegating optimal monetary policy inertia in a small-open economy.
The BE Journal of Macroeconomics 21, 473–507.
Ida, D., Okano, M., 2022. Robust control and delegeted optimal monetary policy inertia insmall open new keynesian model. Momoyama Gakuin University Discussion Paper No.19 .
Jensen, H., 2002. Targeting nominal income growth or inflation? American Economic Review
92, 928–956.
Monacelli, T., 2005. Monetary policy in a low pass-through environment. Journal of Money,Credit, and Banking 37, 1047–1066.
Tillmann, P., 2009. Does model uncertainty justify conservatism? robustness and the delegationof monetary policy. The BE Journal of Macroeconomics 9.
Walsh, C.E., 2003. Speed limit policies: The output gap and optimal Monetary Policy. American Economic Review 93, 265–278.

7

Table 1: Alternative policy regimes
RegimeLoss function

Inflation targeting (IT)(1 − γ)πt2 + γ(πtf )2 + λIT x2t

Speed limit policy (SLP)(1 − γ)πt2 + γ(πtf )2 + λSLP (xt − xt−1 )2Nominal income growth targeting (NIT)(1 − γ)πt2 + γ(πtf )2 + λN IT (πt + xt − xt−1 )2

Real exchange rate targeting (REX)(1 − γ)πt2 + γ(πtf )2 + λREX (qt − qt−1 )2

LOP gap targeting (LOPG)(1 − γ)πt2 + γ(πtf )2 + λLOP (ψt − ψt−1 )2Table 2: Parameterization
ParameterValueβ

Discount rate

0.99

σ

Relative risk aversion coefficient for consumption

2.0φ

Inverse of the elasticity of labor supply

5.0αhCalvo lottery for home goods firms

0.75

αfCalvo lottery for import goods firms

0.75

aElasticity of substitution between home and foreign goods

1.0γ

Degree of openness

0.4

λWeight on the output gap in the true loss function

0.25

ρ

Persistency of a price mark-up shock

0.4

8

Table 3: Welfare losses under alternative delegated policy regimes: Worst-case equilibrium

PC

ITSLP

NIT

REX

LOPG

θ = 500

89.48

135.97

97.30

92.33

102.03

153.32

θ = 400

89.59

136.18

97.34

92.35

102.06

153.33

θ = 300

89.77

136.44

97.40

92.38

102.12

153.36

θ = 20090.14

137.06

97.53

92.43

102.25

153.41

θ = 100

91.27

138.87

97.93

92.59

102.62

153.57

θ = 7592.05

140.09

98.19

92.60

102.87

153.67

θ = 5093.71

142.60

98.72

92.91

103.37153.88

θ = 4095.04

144.49

99.13

93.07

103.74154.04

θ = 3097.41

147.70

99.80

93.35

104.38154.30(Note) PC: precommitment, IT: inflaiton targeting, SLP: speed limit policy, NIT: nominal
income growth targeting, REX: real exchange rate targeting, LOPG: LOP gap targeting. Eachloss is multiplied by 100.

9Figure 1: Impulse response to a cost-push shock under worst-case equilibrium

10Figure 2: Impulse response of the nominal exchange rate to a cost-push shock under worst-caseequilibrium

11