Research InstituteDiscussion Paper Series№05
Title：

Delegating optimal monetary policy inertia in asmall open economy

Daisuke Ida／Mitsuhiro Okano
ida-dai@andrew.ac.jp

2017 年 11 月

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

Delegating optimal monetary policy inertia in asmall open economy∗
Daisuke Ida†and Mitsuhiro Okano‡November 15, 2017

Abstract
This paper explores the delegation of targeting regimes in a small
open economy model. It examines the eﬀects of an open economy on thecoeﬃcients of stabilization weights in the delegated objective function of
the central bank. We address the role of the exchange rate in delegating
optimal monetary policy. We analytically derive the condition whereby
each of the considered targeting regimes coincides with a commitment
policy. The results suggest the presence of non-negligible open economy
eﬀects on each coeﬃcient in delegated targeting regimes.

JEL codes: E52; E58; F41
Keywords : Optimal monetary policy; Targeting regime; Commitment; Discretion∗

We would like to thank Ryo Kato, Takeki Sunakawa, and Kenya Takaku for helpful comments and suggestions. This paper was supported by Grant-in-Aid for Scientific Research(16H03618). All remaining errors are our own.
†
Department of Economics, Momoyama Gakuin University, 1-1, Manabino, Izumi, Osaka594-1198, Japan. Tel.: +81-725-54-3131. E-mail: ida-dai@andrew.ac.jp‡
Department of Economics, Osaka Gakuin University, 2-36-1 Kishibeminami, Suita-City,
Osaka 564-8511, Japan. Tel.: +81-6-6381-8434. E-mail: okano@ogu.ac.jp

1

Introduction

It has been argued that a commitment policy leads to preferable outcomescompared with a discretionary policy in the standard new Keynesian model.1
As noted by Woodford (1999, 2003), the central bank that commits its future
policy stances to the public sector can conduct its monetary policy through the
management of expectations of the private sector. Such an approach can leadto economic policy inertia because a commitment policy is characterized by agradual change of policy stances in response to economic shocks. In contrast,
a discretionary policy that treats the future expectations of macro-variables asgiven cannot induce policy inertia in the economy. This diﬀerence is referred
to as stabilization bias, which is the main source by which commitment policy
can be superior to discretionary policy.2
Several studies have shown that in order for a discretionary policy to overcome the problem of stabilization bias, the government delegates the objectivefunction, which is diﬀerent from the true social objective function, to the central bank (Jensen, 2002; Walsh; 2003; Nessen and Vestin, 2005; Vestin, 2006).3
Recently, Bilbiie (2014) analytically derived the condition whereby a delegatedpolicy regime coincides with a commitment policy and showed that each coefficient for the stabilization terms in the delegated loss function is characterized
by deep parameters in an economy.
This paper analytically examines the role of delegated policy regimes ina small-open economy. Previous studies focused on delegated policy schemes
in a closed-economy setting, but as noted by various authors, optimal monetary policy should consider open-economy eﬀects. Clarida, Gali and Gertler(2001) and Gali and Monacelli (2005) investigated optimal monetary policy ina small open economy. 4 Furthermore, several studies examined optimal monetary policy in a two-country economy framework (Clarida, Gali and Gertler,
2002; Pappa, 2004; Benigno and Benigno, 2006). However, to the best of ourknowledge, no studies appear to focus on delegating optimal monetary policy
regimes in an open economy framework regardless of the importance of the1

This paper regards a commitment policy as a timeless perspective approach; see Woodford (2003) and McCallum and Nelson (2004) for further details.
2
See Loisel (2008) and Walsh (2010) for a detailed discussion of stabilization bias in a
forward-looking model.
3
In addition, to eliminate inflation bias, several studies have argued for the superiority ofdelegated policy schemes over discretionary alternatives in backward-looking model. Thesestudies numerically illustrate the superiority of delegated targeting regimes over purely
discretionary policy (Rogoﬀ, 1985; Walsh, 1995; Svensson, 1997)
4
Their approach is based on a producer currency pricing (PCP) model. Monacelli (2005)
explored optimal monetary policy when exchange rate pass-through is incomplete.

1

open-economy eﬀect on macro variables.
The objective of this paper is to analytically show the condition that delegated monetary policy regimes coincide with a commitment solution in asmall-open new Keynesian model. In such a sense, our study extends Bilbiie(2014)’s idea to a small-open new Keynesian model. To do this, we use theframework of a small open economy on the basis of Gali (2015). In contrast toa closed economy model, the degree of openness and the intratemporal elasticity of substitution between home and foreign goods plays an important rolein a small-open economy.
The findings of this paper are summarized as follows. First, in contrast to
previous studies that focus on delegating policy regimes in a closed economy
model, we consider the role of the real exchange rate in delegating monetary
policy schemes. This paper analytically derives the condition that the optimal
policy under the central bank’s loss function that incorporates the stabilizationof the real exchange rate corresponds to that under a commitment policy. We
show that the presence of the real exchange rate is crucial when we consider
delegated monetary policy schemes in a small-open economy.
Second, this paper explores whether several policy regimes correspond to
the solution of a commitment policy. More specifically, we consider the following policy regimes: speed limit policy, nominal income growth targeting,and consumer price index (CPI) inflation targeting. The results suggest thatspeed limit policy can also achieve the same outcomes as those under a commitment policy in a small-open economies. The diﬀerence between closed andsmall open economy is captured by an indirect real exchange rate channel, in
addition to the conventional role of the speed limit target for the output gap.
According to the standard new Keynesian model, the output gap aﬀects the
inflation rate through the new Keynesian Phillips Curve (NKPC). In addition to traditional channel, the output gap aﬀects the inflation rate through achange the terms of trade, resulting in a movement in the real exchange rate.
Hence, the speed limit policy contains an indirect mechanism for stabilizing
the output gap through the real exchange rate.
Third, in contrast, targeting regimes such as CPI inflation and nominal
income growth could lead to the same outcomes as an optimal commitment
policy if the relative weight of the output gap to inflation takes a considerably
smaller value in the social loss function.
The reminder of the paper is structured as follows. Section 2 briefly describes the characteristics of a small open economy new Keynesian model. We
specify targeting regimes in a small open economy in Section 3, considering
the following regimes: real exchange rate targeting, speed limit policy, CPI inflation targeting, and nominal income growth targeting. Section 4 analytically2derives the condition under which optimal delegation schemes coincide with acommitment policy and shows how the open economy aﬀects each delegatedmonetary policy regime. Section 5 briefly concludes. Appendix A provides
proofs of the propositions derived in Section 4 and Appendix B derives theproof of the proposition based on Bilbiie’s (2014) general quadratic method.

2

Model

The model used herein is based on Gali (2015). Section 2.1 provides a loglinearized system of structural equations in a small open new Keynesian model.
We explain the central bank’s loss function in Section 2.2 and discuss thecharacteristics of a commitment policy. In sum, we put forward the eﬀect anopen economy has on optimal monetary policy.

2.1

Linearized system of a small open new Keynesian
modelThis section briefly describes the small open economy new Keynesian model
based on Gali (2015). The home country is infinitesimally small relative to therest of the world. Representative households in the home country purchase domestic and foreign goods. Households in the home country can have access to
a complete set of state-contingent securities that are traded both domesticallyand internationally. Firms face both monopolistically competitive environments and nominal staggered-price rigidities as specified by Calvo (1983). As
shown in Gali (2015), based on the foregoing, a log-linearized system is given
as follows:5

πt = βEt πt+1 + κν xt + ut ,xt = Et xt+1 − σν−1 (it − Et πt+1 − rr¯ t ),cπt = πt + ν(st − st−1 ),qt = (1 − ν)sts t = σ ν xt .

(1)
(2)
(3)
(4)
(5)

where πt is producer currency inflation, xt is the output gap, it is the nominalinterest rate, πtc is CPI inflation, st is the terms of trade gap, and qt is the realexchange rate gap. Gap variables are expressed by the log deviation of the5

See Gali (2015) for a detailed derivation of a small open new Keynesian model.

3

endogenous variables from the eﬃcient level of their variables. In addition, rr¯tdenotes the natural rate of interest, which holds the real interest rate under
flexible price equilibrium, and ut is the exogenous cost-push shock, which
follows an AR(1) process. Finally, the coeﬃcients for each structural equation
are defined as follows:σ,
1 + ν[ση + (1 − ν)(ση − 1) − 1]()
1+ψκν = δ σν +1−α(1 − ω)(1 − ωβ)
δ=
Θ.
ω
1−α
Θ=
1 − α + αϵσν =The parameters β, σ, η, and ψ represent the discount factor, the relative risk
aversion coeﬃcient, the substitutability between domestic and foreign goods,
and the inverse labor supply elasticity, respectively. The parameter ν is thedegree of openness and α is the degree of diminishing returns to scale for labor
supply. The parameter ω characterizes the degree of nominal price rigidities(i.e. Calvo’s lottery), and ϵ denotes the elasticity of substitution for individual
goods.
Equation (1) represents the small open economy NKPC, which is derived
from the firm’s profit maximization problem subject to Calvo pricing. Equation (2) is a dynamic IS equation, which results from solving the household’sintertemporal optimization problem. Equation (3) specifies the relationshipbetween CPI and producer price index (PPI) inflation. Equation (4) statesthat the real exchange rate proportionally changes in response to a change inthe terms of trade. Finally, Equation (5) represents the relationship between
the terms of trade and the output gap.
The eﬀect of the open economy is characterized by changes by both ν andση. Here, the parameter ν is the degree of openness. These parameters aﬀect
the coeﬃcient of the σν , which is included in κν and σν . Where ση > 1, an
increase in ν induces a decline in σν . Thus, changes in ν and ση alter theslopes of the IS curve and the NKPC. In this case, through the risk-sharing
condition, the domestic output gap changes in accordance with changes in theterms of trade, which implies a fluctuating real exchange rate. This mechanism
can be depicted in the IS equation.
In addition, changes in the terms of trade aﬀect the sensitivity of inflation
to the real marginal cost in the NKPC through two channels. First, domestic
4

inflation reacts positively to a change in the terms of trade by a change in thereal exchange rate through international consumption risk sharing. Second,changes in the terms of trade produce fluctuations in domestic inflation because
they induce changes in the real marginal cost. Whether domestic inflation
increases depends on the movement of ση. In the case of σ = η = 1, bothκν and σν decrease to δ(1 + 1+ψ) ≡ κ and 1, respectively. Thus, the open
1−αeconomy eﬀect disappears in this case.
Finally, because we assume that the home country is infinitesimally small
relative to the rest of the world, note that the other variables for the rest of
the world (πt∗ , i∗t , x∗t ) are exogenous in terms of the home country. A variablewith (∗) denotes a foreign country.

2.2

Objective of the central bank and optimal monetary
policyThe central bank conducts its monetary policy following a targeting rule derived from the minimization problem of the central bank’s loss function. As
shown in Woodford (2003), the central bank’s loss function is derived fromthe second-order approximation of the household’s utility function. In thestandard new Keynesian model, the central bank’s loss function includes thestabilization of inflation and the output gap.
However, we often have diﬃculty deriving well-defined loss function in an
open economy model. In particular, we face this problem in the case of ση ̸= 1.
To overcome this shortcoming, Gali (2015) derived the central bank’s loss
function in the special case of σ = η = 1.
This paper assumes the standard loss function that stabilizes both inflationand the output gap in the case of ση > 1. We address the advantage of using
this loss function. The reason is twofold. First, we can focus on the open
economy eﬀect when considering delegating monetary policy regimes in a small
open economy. Monacelli (2005) used the standard loss function to consider
optimal monetary policy in cases where diﬃculties exist in deriving the loss
function in a small open economy with LCP. Second, this enables us to analyze
optimal delegation regimes in the case where ση is greater than unity.
More specifically, the central bank minimizes the following social loss function subject to structural equations:

Lt = Et∞∑

β j (πt2 + λx2t ).

j=0

5

(6)

where λ denotes the stabilization weight on the output gap relative to inflation
stabilization.
Herein, the central bank minimizes the loss function (6) subject to theNKPC under the presence of the cost-push shock. A commitment solution
produces the following targeting rule:

πt = −

λ(xt − xt−1 )κν(7)

Except for the structural equation coeﬃcients, a targeting rule with commitment is the same shape as the variant obtained in a closed economy. The
central bank that conducts a commitment policy can manipulate the expectations of the private sector by gradually changing its policy variable. Thus,
the presence of a lagged output gap in the targeting rule allows the central
bank to employ an inertial behavior vis-à-vis its policy stance. Therefore acommitment policy is superior to a purely discretionary policy (McCallum and
Nelson, 2004). Finally, note that the meaning of commitment policy as appliedin this paper implies a timeless perspective as proposed by Woodford (2003).6
From the optimization problem of the central bank, we can obtain the
reduced form of endogenous variables, solved by the standard factorization
method. More specifically, the reduced forms of both inflation and the output
gap are solved as follows:cπt = ψπc xt−1 + ψuπut ,cc
xt = ψx xt−1 + ψux ut ,

(8)
(9)

where
λ
λ c
c
(1 − µ1 ); ψuπ= − ψxu
,
κν
κν
1
κν
c
.
ψxc = µ1 ; ψux=−
βλ µ2 − ρu

ψπc =Here, µ1 and µ2 are eigenvalues obtained from the solution of the factorization
method in a commitment policy.7 The parameter ρu denotes the degree of
persistency of the cost-push shock.
6

See Woodford (2003) for a detailed discussion of a timeless perspective on optimalmonetary policy
7
See Bilbiie (2014) for a detailed calculation of µ1 and µ2 .

6

As we can see in Equations (8) and (9), notwithstanding the forwardlooking model, the reduced forms of endogenous variables depends on thelagged variable xt−1 . Thus, if the central bank can proﬀer a credible commitment to the private sector, such a commitment policy enables endogenous
variables to gradually change in response to a cost-push shock. In other words,
this implies that the central bank can alleviate a policy trade-oﬀ generated bya cost-push shock through management of private sector expectations.
Importantly, the optimal targeting rule (7) includes the stabilization ofboth the terms of trade eﬀect and the risk-sharing eﬀect. Thus, in contrast to
the closed economy model, the optimal targeting rule replaces κ with κν . This
modification indicates that the central bank considers not only the stabilizationof the domestic macroeconomy also the eﬀect of the open economy on thehome country. When the parameter ν takes zero, the optimal targeting rule(7) decreases to that of a closed economy.

3

Targeting regimes in a small open economy

This section examines delegating optimal monetary policy inertia in a smallopen forward-looking model. As mentioned in many previous studies thatinvoke the standard new Keynesian model, as long as the central bank cancommit its future monetary policy stance and never renege on this commitment policy, the commitment solution is superior to the alternative under adiscretionary policy8 .
Assuming this is the case and as noted in the extant literature, the central
bank, which conducts a discretionary policy, can enhance social welfare when
the government delegates the loss function with lagged endogenous variables
to that bank. Jensen (2002) suggests the policy regime that the government
delegates the objective function that includes stabilization of nominal income
growth to the central bank. Obviously, nominal income growth possesses policy
inertia through lagged output. Further, Walsh (2003) asserted the eﬀectivenessof the speed limit policy that stabilizes a change in the output gap.9
These studies point out that the presence of the lagged variable in thecentral bank’s objective function allows the central bank to impart policy
inertia into the economy even when the central bank discretionally conducts
its monetary policy. Therefore, the central bank that employs these targeting8

For instance, see Kydland and Prescott (1977) for a detailed discussion of the timeinconsistency problem in optimal policy.
9
Nessen and Vestin (2005) and Vestin (2006) considers the eﬀectiveness considered average inflation and price level targeting, respectively, as delegating monetary policy regimes.

7

regimes can improve social welfare to a greater extent compared with the caseof pure discretionary monetary policy.
As mentioned earlier, this paper analytically explores whether delegating
optimal monetary policy inertia can coincide with a commitment solution ina small-open economy model. In contrast to a closed economy, we conjecturethat the presence of an open economy eﬀect should significantly change thecondition that delegating optimal monetary policy corresponds to a commitment policy. Bilbiie (2014) derived the condition that delegating monetary
policy regimes coincides with a commitment solution, whereas this paper extends that work to the small open new Keynesian model. Accordingly, theshape of the delegated objective function is based on Bilbiie (2014).
The merit of considering the optimal delegation problem in a small-open
economy is to explore whether the policy inertia from the real exchange rate
can achieve the same outcome as a commitment policy. Therefore, we consider
real exchange rate targeting as the first policy regime.10 More specifically, thereal exchange rate targeting regime is defined as follows:
πt2 + λx x2t + λq ∆qt2 + 2cπ xt−1 πt .

(10)

As explained in Bilbiie (2014), as with all policy regimes in this paper, technically we add a fourth term to the objective function to obtain the analyticalsolution that corresponds to the solution under a commitment policy. In otherwords, this term reflects the linear inflation contract term proposed by Walsh(1995) and Svensson (1997).
Clearly, the policy inertia is introduced through a change in the real exchange rate. According to Equation (5), since the output gap is proportional
to the terms of trade, we can replace the term for the real exchange rate withthat for the output gap in the targeting regime (10):πt2 + λx x2t + σν2 λq ∆x2t + 2cπ xt−1 πt .

(11)

One may suppose that as suggested by Walsh (2003), the central bank can
attain the same outcomes as a commitment policy if the government delegatesthe objective function to the central bank that directly includes a change inthe output gap. Introducing the lagged output gap into the objective function
allows the central bank to gradually respond to economic shocks even if thecentral bank conducts a discretionary policy. We consider the speed limit
policy, which is defined as follows:
πt2 + λx2t + λS ∆x2t + λπ (πtc )2 + 2cπ xt−1 πt .
10

(12)

See also Taylor (2001) for a detailed discussion of the role of real exchange rate targetingin an open economy.

8As explained in Walsh (2003), the central bank can achieve preferable outcomesover and above a pure discretionary policy.
Moreover, we consider whether the delegated objective function with policy inertia through CPI inflation coincides with a commitment solution. In
contrast to a closed economy model, we can diﬀerentiate CPI inflation and
PPI inflation. The inflationary diﬀerence between both metrics is reflected
in a change in the terms of trade gap. More specifically, we define the CPI
inflation targeting regime as follows:
πt2 + λx x2t + λπ (πtc )2 + 2cπ xt−1 πt(13)Assuming this formulation, one might speculate that this policy regime does
not include policy inertia. According to Equation (3), however, the CPI inflation rate is aﬀected by a change in the terms of trade. Thus, this policy
scheme can introduce policy inertia into the economy through a change in theterms of trade, which implies a change in the output gap. Hence, it is possiblethat the performance of CPI inflation targeting in a small open economy issuperior to that in a closed economy model.
Finally, as suggested by Jensen (2002), we consider the targeting regime
including a nominal income growth, which is defined as follows:
πt2 + λx x2t + λN (πtc + ∆xt )2 + 2cπ xt−1 πt .

(14)

We assume that nominal income growth is defined in terms of CPI because
the private sector may be interested in CPI fluctuations in terms of social
welfare.11 It follows from Equation (14) that policy inertia is incorporated into
the objective function through the nominal income growth rate. In addition,
nominal income growth includes policy inertia through the CPI inflation rate.
For these four regimes, we allow λx to diverge from λ.

4Optimal delegation problem in a small open
economyThis section details our results. To begin, we derive the condition whereby adiscretionary policy under real exchange rate targeting corresponds to a commitment solution. Then, we compare the results of real exchange rate targeting
11

Ida and Okano (2017) examines the role of nominal income growth targeting in a smallopen economy. They explore how the diﬀerence in between PPI and CPI inflation aﬀects
the performance of nominal income growth targeting.

9with those of a speed limit policy. Moreover, we examine the condition undernominal income growth targeting and CPI inflation targeting.
First, consider the optimal delegation parameters under real exchange rate
targeting. Solving the optimal monetary policy with discretion under thisregime leads to the following proposition:
Proposition 1. (Real exchange rate targeting) The Markov-perfect equilibrium value of domestic inflation and output gap that occur if the central bankminimizes the delegated loss function (10) are identical to the timeless-optimalcommitment solutions (8) and (9) if and only if the delegation parameters are
given by the following:λγ
κν γ + λ
(
)( )
λγ
λ
1
∗
λq =
(1 − ν 2 σν2 ) κν γ + λ
κν
2
λγ
λ∗x =
κν γ + λc∗π = −

(15)
(16)
(17)

where γ = κν + βψπc .
The proof is shown in Appendix A. As we will show, under real exchange
rate targeting and the speed limit policy, the policy implications are diﬀerent
although both solutions coincide with the solution under commitment.
An important characteristic of real exchange rate targeting is the direct
stabilization of the exchange rate. Since an increase in ν induces exchange
rate fluctuations, the central bank needs to respond by militating against instability. This is reflected as 1/(1 − ν 2 σν2 ) in (16). Note that λq ̸= 0 even when
ν = 0. Not surprisingly, this is because the central bank employs the delegatedloss function that directly includes the stabilization term for the real exchange
rate.
[Figure 1 around here]
Figure 1 illustrates the optimal delegation parameters when the degree of
openness ν moves from 0.0 (which is reduced to the closed economy case) to 0.4(which represents a typical small open economy,) under alternative regimes:
real exchange rate targeting and the speed limit policy. Other deep parameters
are based on the calibration used in the standard new Keynesian model. Other
calibrated deep parameters are displayed in Table 1. Note that an increase inν lowers σν since ση = 2.0 > 1.0.
10

[Table 1 around here]
The parameters under real exchange rate targeting are shown in Figure1(a). λq is a real exchange rate stabilization weight, and it plays a key rolein the regime: hence, the magnitude of λq dominates the other parameters
unless the degree of openness is too restricted (ν < 0.03, approximately).
The reason is that λq not only stabilizes the output gap through its influenceon the real exchange rate but also generates inertia from the real exchange
rate. Thus, a weight on the stabilization of the output gap is smaller, whereas
optimal inertia can be replicated by placing a larger weight on real exchange
rate stabilization. Note that λq is increasing in openness ν because an increasein openness lowers σν , thus increasing output gap volatility due to the larger
eﬀect of the real exchange rate on the output gap. The other parameters λx
and cπ , however, cannot fully stabilize this volatility since they cannot capture
the eﬀect of the real exchange rate on the output gap. Therefore, λq has anadvantage of compensating for the inferior stabilization properties of both λx
and cπ .
The contract term cπ represents a penalty/reward for a change in inflation,and the marginal penalty depends upon the past values of the output gap.12
This parameter appears under all regimes as an assumption. cπ is negative forall ν; hence, for the same given prior contraction in output gap, future contractions should be rewarded under these regimes. These results are consistent
with those in previous studies. As shown in Figure 1, the absolute value ofcπ is smaller under the real exchange rate targeting compared with its value
under the speed limit policy; the weight is smaller (larger) under the former(latter). This result suggests that under real exchange rate targeting, the required inertia has been provided by the real exchange rate, as measured bylarge λq , resulting in the smaller dependency on cπ . The reward for additional
inflation increases in ν under both regimes, although the change in cπ is small.
λx is the output gap stabilization weight. Note that these regimes allow
λx to diﬀer from λ, which represents the relative weight of output gap stabilization to inflation in the social loss function. λx is positive for all ν. Thismeans stabilizing the output gap is rewarded for replicating the optimal outcome. As noted earlier, the parameter λx is slightly smaller than that in thesocial loss function because the positive weight on λq means the central bankdirectly stabilizes the output gap through the real exchange rate channel. The
penalty for output gap volatility decreases in ν, although the degree of change
is negligible.
12

See Bilbiie (2014) for a detailed discussion about this issue.

11Next, we derive the optimal delegation parameters under a speed limit
policy. Then, we compare the properties of the delegation parameters underreal exchange rate targeting with that under the speed limit policy. Before
doing this, we summarize the properties of the optimal delegation parameters
under the speed limit policy.
Proposition 2. (Speed limit policy) The Markov-perfect equilibrium values ofdomestic inflation and output gap, which occur if the central bank minimizes
the delegated loss function (12), are identical to the timeless-optimal commitment solutions (8) and (9) if and only if the delegation parameters are givenby the following:λ∗S =λγ 2 κν

c∗π = −

(νσν (γκν + λ) + γλ)

λ
γ

(18)
(19)

λ(γκν + λ)λ∗π = − 2γ (κν νσν − λ)(20)

The proof is shown in Appendix A.
The diﬀerence between closed and small open economy appears in ν, σν andκν . In addition to the speed limit channel, the terms of trade (the real exchangerate) channel is also used for stabilization. In other words, the real exchange
rate is indirectly stabilized under the speed limit policy. The intuition of thisresult is as follows. The output gap aﬀects the inflation rate through theNKPC in the standard new Keynesian model. In addition to this traditional
channel, the output gap aﬀects the inflation rate through a change in the termsof trade, resulting in a movement in the real exchange rate. Therefore, thespeed limit policy contains an indirect stabilization of the output gap throughthe real exchange rate. Note that when openness ν is increasing, the increasein λS∗ is smaller than that in λ∗q .
Parameters under the speed limit policy are shown in Figure 1(b). λS plays
a key role in the regime, and this parameter value never takes negative values
for all ν. Comparing λS in the speed limit policy with λq under real exchange
rate targeting, both parameters increase in openness, whereas those levels andthe degree of changes are diﬀerent as mentioned above. When degree of the
openness is small, λq is smaller than λS since for the purpose of stabilizing
the output gap λq need not to be weighted heavily. A smaller value of ν
weakens the direct eﬀect of the real exchange rate of the output gap. Since
the speed limit target directly aims at stabilizing a change of the output gap.
12

On the contrary, λS must be weighted heavily. Moreover, while the marginalchange of λq increases with respect to openness, the marginal change of λS
decreases. This implies that under real exchange rate targeting there are noother parameters generating real exchange rate inertia except for λq ; under
the speed limit policy, λS indirectly alleviates a fluctuation of the output gap
produced by a change in the real exchange rate through its eﬀects on the
CPI.13The optimal delegation parameter λπ is the CPI inflation stabilization
weight. Note that the figure indicates λπ > 1 since λπ deals with both domestic inflation stabilization—on which the weight equals 1 in general—andreal exchange rate stabilization through its influence on CPI inflation. The
absolute value of the contract term cπ is larger under the speed limit policy
than under real exchange rate targeting. This result implies that the parameter cπ under the speed limit policy fills the gap between optimal inertia andthe inertia provided by λS and λπ .
As already noted, we can also consider alternative targeting regimes. First,
we consider the optimal delegation problem under CPI inflation targeting.
Then, we consider the role of nominal income growth targeting. The result ofsolving a discretionary policy under CPI inflation targeting is summarized in
Proposition 3.
Proposition 3. (CPI inflation targeting) The Markov-perfect equilibrium values of domestic inflation and output gap which occur if the central bank minimizes the delegated loss function (13), are identical to the timeless-optimalcommitment solutions (8) and (9) if and only if the delegation parameters are
given by the following:
λ∗π =

γ
χπ

(21)

)( γ )
κνΨπ − 1(22)λχπ)γ 2 ( κνΨπ − 1
(23)
λ∗x =
χπ
λ
(
)
where Ψπ = νσν and χπ = κλν Ψ2π κλν γ + 1 − γ − Ψπ .In particular, thesedelegation parameters under CPI inflation targeting lead to the same outcome
as commitment if and only if the parameter λ takes a smaller value.
c∗π = −Ψπ(13

If ση > 1, openness influences the sensitivity of the output gap to a change in the realexchange rate, as measured by 1/(σν (1 − ν)); the marginal weight on λq also increases withrespect to openness.

13

The proof is shown in Appendix A. Like in the case of both real exchange rate
targeting and speed limit targeting, one may conjecture that this targeting
regime also corresponds to the commitment equilibrium. However, we showthat this regime cannot lead to the same outcome as commitment because
the second-order condition of this problem is not satisfied unless λ takes asmaller value. As shown in Bilbiie (2014), while the parameter θπs must takea negative value to satisfy the second-order condition,14 that condition is notsatisfied if λ takes a higher value in this study.
Figure 3 shows θπs satisfying the second-order conditions of the optimization problem under the delegated regime. Figure 3(a) illustrates the case of
CPI inflation targeting. We observe that the second-order condition is notsatisfied when λ > 0.3. This implies that the CPI inflation targeting regimes
cannot lead to the same solution as commitment when the private sector requires the policymaker to put a higher weight on the stabilization of the output
gap relative to inflation.
(B.11 ) in Appendix B shows the loss functions of the nominal income
growth targeting. Consider the coeﬃcients of x2t , which is given by (λx − λ +2). As shown in Fig 2(a), the delegation parameter λx never takes negaλπ ν 2 σnu
tive values, whereas λπ necessarily takes negative values. Therefore, if λ takes
a higher value, the stabilization term of the output gap might be negative.
This implies that the welfare would be improved when the output gap fluctuates. However this contradicts a traditional assertion that the central bankfaces the trade-oﬀ between output gap stabilization and inflation stabilizationin the presence a the cost-push shock.
[Figure 2 around here]
Now, we derive the optimal delegation parameters under nominal income
growth targeting, which are summarized in the following proposition:
Proposition 4. (Nominal income growth targeting) The Markov-perfect equilibrium value of domestic inflation and output gap, which occur if the central
bank minimizes the delegated loss function (14) are identical to the timelessoptimal commitment solutions (8) and (9) if and only if the delegation param-

14

See Appendix B for the definition of parameter θπs .

14

eters are given by the following:
γ
λ∗N =
χN

(24)

)( γ )
κν
ΨN
−1
(25)
λ
χN
)
(
γ2
κν
∗
ΨN ΨN
−1
(26)
λx =
χN
λ
(
)
where ΨN = 1 + νσν and χN = κλν Ψ2N κλν γ + 1 − γ − ΨN . In particular,
these delegation parameters under nominal income targeting lead to the sameoutcome as commitment if and only if λ takes a smaller value.
c∗π = −ΨN(

The proof is shown in Appendix A. As in the case of CPI inflation targeting, this regime fails to satisfy the condition that θπs is negative. Nominal
income growth targeting requires a smaller value of λ for the same reason as
in the CPI inflation targeting case. Thus, (B.11 ) in Appendix B presents
loss functions of nominal income growth targeting. Consider the coeﬃcientsof x2t , which is given by (λx − λ + (1 + νσν )2 λN ). As shown in Figure 2(b), thedelegation parameter λx never takes negative values, whereas λN necessarily
takes negative values. Therefore, if λ takes a higher value, the stabilization
term of the output gap might be negative. This implies that welfare would
be improved when the output gap fluctuates. Again, this contradicts the traditional assertion that the central bank faces a trade-oﬀ between output andinflation stabilization in the presence of a cost-push shock.
Figure 3(b) illustrates the case of nominal income growth targeting. We
observe that the second-order condition is satisfied provided λ < 0.1. In contrast to the case of CPI inflation targeting, this requires considerably smaller
values compared with the calibrated value of λ in the standard new Keynesianmodel. For instance, Jensen (2002) and Walsh (2003) considered the range of
λ between 0.1 and 1.0. However, we show that this parameter value is invalid
in that the targeting regime that stabilizes nominal income growth cannot
achieve the same solution as a commitment policy.

5

Conclusions

This paper explores the delegation of targeting regimes in a small open economy model. We examined how the presence of an open economy aﬀects thecoeﬃcients in the delegated objective function of the central bank. We analytically derived the condition whereby each targeting regime considered herein
coincides with a commitment policy.
15

The findings of this paper are summarized as follows. First, in contrast to
previous studies that focused on delegating policy regimes in a closed economy
model, we consider the role of the real exchange rate in delegating monetary
policy schemes. We analytically derived the condition whereby the optimal
policy under the central bank’s loss function that incorporates the stabilizationof the real exchange rate corresponds to that under a commitment policy. We
showed that the presence of the real exchange rate is crucial when considering
delegated monetary policy schemes in a small open economy.
Second, we explored whether several policy regimes corresponds to thesolution of a commitment policy. More specifically, we considered the following
policy regimes: speed limit policy, nominal income growth targeting, and CPI
inflation targeting. Based on this, the speed limit policy can also achieve thesame outcomes as that of a commitment policy in a small-open economy. The
diﬀerence between a closed and small open economy is captured by an indirect
real exchange rate channel in addition to the conventional role of the speed
limit target for the output gap. According to the standard new Keynesianmodel, the output gap aﬀects the inflation rate through the NKPC. In addition
to the traditional channel, the output gap aﬀects the inflation rate througha change in the terms of trade, resulting in a movement in the real exchange
rate. Hence, the speed limit policy contains an indirect stabilization of theoutput gap through the real exchange rate.
Third, in contrast, targeting regimes such as CPI inflation and nominal
income growth could lead to the same outcomes as an optimal commitment
policy if the relative weight on the output gap to inflation takes a considerably
smaller value in the social loss function.
Finally, we suggest possible extensions to our study. First, we examined the
optimal delegation problem in the case of perfect exchange rate pass-through.
As pointed out by Monacelli (2005), that assumption may not be realistic.
Therefore, extending our study to the case of incomplete exchange rate passthrough could generate results with greater empirical credibility. Second, ourstudy is also applicable to the case of a two-large country model (Clarida, Gali
and Gertler, 2002; Benigno and Benigno, 2006): as such, this provides anotheravenue that could be explored in the future.

References
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Bilbiie, F.O. (2014) “Delegating optimal monetary policy inertia,”Journal of16Economic Dynamics and Control 48, 63-78.
Clarida, R., Gali, J., and Gertler, M. (2001) “Optimal monetary policy inopen vs. closed economies,” American Economic Review 91, 248-252.
Clarida, R., Gali, J., and Gertler, M. (2002) “ A simple framework for international monetary policy analysis,” Journal of Monetary Economics 49, 879-904.
Gali, J. (2015) Money, inflation and the business cycles, Second edition, Princeton University Press, Princeton.
Gali, J. and Monacelli, T. (2005) “Monetary policy and exchange rate volatility in a small open economy,” Review of Economic Studies 72, 707-734.
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Jensen, H. (2002) “Targeting nominal income growth or inflation?” American
Economic Review 92, 928-956.
Kydland, F. and Prescott, E. (1977) “Rules rather than discretion: the inconsistency of optimal plans,” Journal of Political Economy 85, 473-490.
Loisel, O. (2008) “Central bank reputation in a forward-looking model,” Journal Economic Dynamics and Control 32, 3718-3742.
McCallum, B.T. and Nelson, E. (2004) “Timeless perspective vs discretionary
monetary policy in forward-looking models,” NBER Working Paper No.7915.
Monacelli, T. (2005) “Monetary policy in a low pass-through environment,”
Journal of Money Credit and Banking 37, 1047-1066.
Nessen M. and Vestin, D. (2005) “Average inflation targeting.” Journal of
Money, Credit and Banking 37, 837-863.
Pappa, E. (2004) “Do the ECB and Fed really need to cooperate? Optimalmonetary policy in a two-country world,” Journal of Monetary Economics 51,
753-779.
Rogoﬀ, K. (1985) “The optimal degree of commitment to a monetary target,”Quarterly Journal of Economics 100, 1169-1190.
Svensson, L.E.O. (1997) “Optimal inflation targets, conservative central bankers
and linear inflation contracts,” American Economic Review 87-1, 99-115.
Vestin, D. (2006) “Price-level versus inflation targeting,” Journal of Monetary
Economics 53, 1361-1376.
Walsh, C.E. (1995) “Optimal contracts for central bankers,” American Economic Review 85, 150-167.
Walsh, C.E. (2003) “Speed limit policies; The output gap and optimal monetary policy,” American Economic Review 93, 265-278.
Walsh, C.E. (2010) Monetary theory and policy, Third edition, MIT Press,
Cambridge.
Woodford, M. (1999) “Optimal monetary policy inertia,” National Bureau of
Economic Research Working Paper No.7261.
17

Woodford, M. (2003) Interest and prices: Foundations of a theory of monetary
policy, Princeton University Press, Princeton.

A

Appendix A: Proof of Proposition

A.1

Proof of Proposition 1

We define the Bellman equation as follows:
]
[
1
2
2
2
V (xt−1 ; ut ) = min λx xt + πt + λq ∆qt + 2cπ πt xt + βEt V (xt ; ut+1 ) .
2
Substituting the NKPC into the above Bellman equation, we obtain{
1
V (xt−1 ; ut ) = min
λx x2t + [βEt πt+1 + κν xt + ut ]2 + λq ∆qt2 + 2cπ πt xt
2}+ βEt V (xt ; ut+1 ) .
The first-order condition with respect to xt is as follows:[λx + λq (1 − ν)2 σν2 ]xt + γπt + cπ γxt−1 − λq (1 − ν)2 σν xt−1 + βEt∂V (xt ; ut+1 )
= 0.
∂xt
(A.1)

In addition, from the envelop theorem, we obtain∂V (xt−1; ut )= −λq (1 − ν)2 σν ∆xt + cπ πt .
∂xt−1

(A.2)

Substituting Equation (A.2) and Equations (8) and (9) into Equation(A.1) and then considering the optimal targeting rule, we obtain the following
second-order diﬀerence equation:λλγ
λ− λq (1 − ν)2 σν ]Et xt+1 + [λx + (1 + β)λq (1 − ν)2 σν2 + βcπ −
]xt
κν
κν
κν
γλ
+ [cπ γ +− λq (1 − ν)2 σν ] = 0.
(A.3)
κν

β[−cπ

18

As explained in Bilbiie (2014), in this equation, the optimal delegation parameter is found by noticing that Equation (A.3) evaluated at the conjectured
solution (xct , πtc ) should be identity. Therefore, to identify the optimal delegation parameter, all coeﬃcients should be zero. This leads to the following
relationship:cπλ+ λq (1 − ν)2 σν = 0,κνλx + (1 + β)λq (1 − ν)2 σν2 + βcπcπ γ +

(A.4)
λ
λγ
−
= 0,
κν
κν

γλ
− λq (1 − ν)2 σν = 0.
κν

(A.5)
(A.6)

Solving Equations (A.4)–(A.6), we obtain the optimal delegation parametersin Proposition 1.

A.2

Proof of Proposition 2

Before defining the Bellman equation, we arrange the objective function. Substituting the definition of CPI inflation into the objective function, we obtainthe following objective function:(λ + λS + λπ ν 2 σν2 )x2t + (λS + λπ ν 2 σν2 )x2t−1 − 2(λS + λπ ν 2 σν2 )xt xt−1
+ 2λπ νσν πt ∆xt + 2cπ xt−1 πt + λπ πt2 .
(A.7)
Then we define the Bellman equation as follows:{1(λ + λS + λπ ν 2 σν2 )x2t + (λS + λπ ν 2 σν2 )x2t−1
V (xt−1 ; ut ) = min2− 2(λS + λπ ν 2 σν2 )xt xt−1 + 2λπ νσν πt ∆xt + 2cπ xt−1 πt + λπ πt2 + βEt V (xt ; ut+1 )}.

The first-order condition with respect to xt is as follows:(λ + λS + λπ ν 2 σν2 )xt − (λS + λπ ν 2 σν2 + λπ νσν γ − cπ γ)xt−1 + λπ (γ + νσν )πt∂V (xt ; ut+1 )
= 0.
(A.8)
+ βEt
∂xt
19

In addition, from the envelop theorem, we obtain∂V (xt−1; ut )= (λS + λπ ν 2 σν2 )∆xt + (cπ − λπ νσν )πt .
∂xt−1

(A.9)

Substituting (A.9) into (A.8) and considering the optimal targeting rule,
we obtain the following second-order diﬀerence equation:
]
[(
)
λ
2 2
− (λS + λπ ν σν ) Et xt+1
β λπ νσν − cπκν[]λπ (γ + νσν )λβ(cπ − λπ νσν )λ2 2
2 2+ λ + λS + λπ ν σν + λπ νσν γ −+ β(λS + λπ ν σν ) +
xt
κν
κν
[
]
λπ (γ + νσν )λ2 2− (λ + λS + λπ ν σν + λπ νσν γ) + cπ γ xt−1 = 0.
(A.10)+
κνAs explained in Proposition 1, in this equation, the optimal delegation parameter is founded by noticing that Equation (A.10) evaluated at the conjectured
solution (xct , πtc ) should be identity. Therefore, to observe the optimal delegation parameter, all coeﬃcients should be zero. This leads to the following
relationship:
λ− (λS + λπ ν 2 σν2 ) = 0(A.11)κν
β(cπ − λπ νσν )λλπ (γ + νσν )λ+ β(λS + λπ ν 2 σν2 ) +=0
λ + λS + λπ ν 2 σν2 + λπ νσν γ −κνκν(A.12)λπ (γ + νσν )λ− (λ + λS + λπ ν 2 σν2 + λπ νσν γ) + cπ γ = 0(A.13)κν(λπ νσν − cπ )Solving Equations (A.11)–(A.13), we obtain the optimal delegation coeﬃcients
under the speed limit policy.

A.3

Proof of Proposition 3

Before defining the Bellman equation for solving this optimization problem,
we calculate the objective function under CPI inflation targeting as follows:
λx x2t + (1 + λπ )πt2 + 2λπ νσν πt ∆xt + λπ ν 2 σν2 ∆x2t + 2cπ xt−1 πt .
20

(A.14)

Using this objective function, we can define the Bellman equation as follows:{1
V (xt−1 ; ut ) = min
λx x2t + (1 + λπ )πt2 + 2λπ νσν πt ∆xt + λπ ν 2 σν2 ∆x2t + 2cπ xt−1 πt
2}+ βEt V (xt ; ut+1 ) .
The first-order condition of this optimization is given as follows:
λx xt + (1 + λπ )γπt + λνσν γ∆xt + λπ νσν πt + λπ ν 2 σν2 xt − λπ ν 2 σν2 xt−1
∂V (xt ; ut+1 )cπ γxt−1 + βEt= 0.
(A.15)∂xt
From the envelope theorem, we obtain the following equation:∂V (xt−1; ut )= −λπ νσν πt − λπ ν 2 σν2 ∆xt + cπ πt .
∂xt−1

(A.16)

Using this envelope theorem and the optimal targeting rule with commitment leads to the following second-order diﬀerence equation:λγ
λ− λπ ν 2 σν2 ]Et xt+1 + [λx −(1 + λπ )κνκν
λλπ νσνλ+ λπ νσν γ −+ λπ ν 2 σν2 + β(cπ − λπ νσν ) + βλπ ν 2 σν2 ]xtκνκνλγ[ (1 + λπ − λπ νσν γ + λπ νσν − λπ ν 2 σν2 + cπ γ]xt−1 = 0.
(A.17)κν
β[(λπ νσν − cπ )Again, as per the foregoing, the optimal delegation parameter is found by
noticing that Equation (A.24) evaluated at the conjectured solution (xct , πtc )should be identity. To identify the optimal delegation parameter, all coeﬃcients should be zero. This leads to the following relationship:
λ(A.18)− λπ ν 2 σν2 = 0,κνλγ
λλλπ νσνλx −(1 + λπ ) + λπ νσν γ −+ λπ ν 2 σν2 + β(cπ − λπ νσν ) + βλπ ν 2 σν2 = 0,κνκνκν(A.19)
λγ(A.20)(1 + λπ ) − λπ νσν γ + λπ νσν − λπ ν 2 σν2 + cπ γ = 0,κν(λπ νσν − cπ )21

Solving Equations (A.11)–(A.13) leads to the result of Proposition 3.
We now show that the delegation parameters under CPI inflation targeting
lead to the same outcome as commitment if and only if parameter λ takes asmaller value. As shown in Appendix B, θπs under nominal income targetingis given byθπs = −λπ νσν + cπ .
Using the delegation parameters, this equation can be written as()γκνθπs = −Ψπ
Ψπ − 1 .
χπλAs argued in Bilbiie (2014), the second-order condition is satisfied if θπs < 0.
Because Ψπ , γ, and γ are positive, the condition requires χπ and (Ψπ κλν − 1)are positive. These two conditions can be rewritten as)κν 2 ( κνΨγ + 1 > γ + Ψπ
λ π λ
κν
Ψπ
> 1.
λ
Both conditions are satisfied if and only if λ takes a smaller value.

A.4

Proof of Proposition 4

Before defining the Bellman equation, we arrange the objective function. Substituting the definition of CPI inflation into the objective function, we obtainthe following objective function:(1 + λN )πt2 + [λx + λN (1 + νσν )2 ]x2t + λN (1 + νσν )2 x2t−1 + 2λN (1 + νσν )πt xt+ 2[cπ − λN (1 + νσν )]πt xt−1 − 2λN (1 + νσν )2 xt xt−1
(A.21)
Then, we define the Bellman equation as follows:{1
V (xt−1 ; ut ) = min(1 + λN )πt2 + [λx + λN (1 + νσν )2 ]x2t + λN (1 + νσν )2 x2t−1
2+ 2λN (1 + νσν )πt xt + 2[cπ − λN (1 + νσν )]πt xt−1 − 2λN (1 + νσν )2 xt xt−1}
+ βEt V (xt ; ut+1 ) .
22

The first-order condition of this optimization is given as follows:(1 + λN )γπt + [λx + λN (1 + νσν )2 ]xt + λN (1 + νσν )γxt + λN (1 + νσν )πt∂V (xt ; ut+1 )= 0.
(A.22)+ [cπ − λN (1 + νσν )] + βEt∂xt
It follows from the envelope theorem that∂V (xt−1; ut )= −λN (1 + νσν )2 ∆xt + [cπ − λN (1 + νσν )]πt .
∂xt−1(A.23)

Using the envelope theorem and the optimal targeting rule, we obtain thefollowing second-order diﬀerence equation:
)
[(
]
λ
2
β λN (1 + νσν ) − cπ
− λN (1 + νσν ) Et xt+1
κν
[
λλN
γλ
+ λN (1 + νσν )γ −(1 + νσν )+ λx + λN (1 + νσν )2 − (1 + λN )
κν
κν
(
) ]
λ
+ βλN (1 + νσν ) + β cπ − λN (1 + νσν )
xt
κν
]
[
γλ λλN2+(1 + νσν ) + (cπ − λN (1 + νσν ))γ − λN (1 + νσν ) xt−1 = 0+ (1 + λN )κνκν
(A.24)
Once more, this equation the optimal delegation parameter requires noticing that Equation (A.24) evaluated at the conjectured solution (xct , πtc ) should
be identity, and all coeﬃcients should be zero. This leads to the following
relationship:
λ− λN (1 + νσν )2 = 0,
(A.25)
κνλλN
γλ+ λN (1 + νσν )γ −(1 + νσν )λx + λN (1 + νσν )2 − (1 + λN )κνκν
λ+ βλN (1 + νσν ) + β[cπ − λN (1 + νσν )] = 0,
(A.26)κν
γλ λλN(1 + λN )+(1 + νσν ) + (cπ − λN (1 + νσν ))γ − λN (1 + νσν )2 = 0.
κνκν
(A.27)[λN (1 + νσν ) − cπ ]23

Solving Equations (A.11)–(A.13) provides the result of Proposition 4.
We now show that the delegation parameters under nominal income growthtargeting lead to the same outcome as commitment if and only if λ takes asmaller value. As shown in Appendix B, θπs under nominal income targetingis given byθπs = −(1 + νσν )λN + cπ .
Using the delegation parameters, this equation can be written asθπs = −

γ
κν
ΨN .
χN
λ

As argued in Bilbiie (2014), the second-order condition is satisfied if θπS < 0.
Because ΨN κν /λ and γ are positive, the condition requires that χN is positive.
Thus, the following condition must be satisfied to guarantee χN > 0:
(
)
κν γ
κν
ΨN
+ 1 > γ + ΨN .
λλThis condition is satisfied if and only if λ is very small.

B

Appendix B: Proof of Proposition by GeneralQuadratic Method

This appendix shows that the proof provided in Appendix A can also be arrived
by general quadratic method. According to Bilbiie (2014), we consider thegeneral form of delegation by adding a quadratic form Zt′ ΘZt over all relevant
variables Zt = (πt , xt , xt−1 )′ to the social loss function:
Lbt =1[Lt + Zt′ ΘZt ] ,
2

(B.1)

where Θ is the symmetric matrix of all delegation parameters, Θ = {θij }, i, j =π, x, s, θij = θji . The script s denotes the term for lagged output gap.
Applying Bilbiie’s(2014) proposition to an open economy, the conditions
that need to be fulfilled by the delegation parameters to induce the optimal

24

amount of inertia areas follows:
∗
θxs
=

λ ∗
θ
κ (πs

)
λ λ
∗
γ + + β θπs
κ κ
(κ
)
∗
∗
=−
γ(1 + θππ )∗ + θπxγ + 1 θπs
λ

∗
∗
∗
=−
+ βθssλ + θxx+ γθπx

(B.2)

∗
< 0.
θπs

(B.3)

and

B.1

Real Exchange Rate Targeting

The policy regime is described by the following loss function:1
Lqt = [πt2 + λx x2t + λq ∆qt2 + 2cπ πt xt−1 ]
2

(B.4)

Substituting (4) and (5), we obtain
1
Lqt = [πt2 + λx x2t + (1 − ν)2 σν2 λq ∆x2t + 2cπ πt xt−1 ]2
1 2q⇒ Lt = [πt + λx2t + (λx − λ + λq (1 − ν)2 σν2 )x2t
2
+ (1 − ν)2 σν2 λq x2t−1 + 2cπ πt xt−1 − 2λq (1 − ν)2 σν2 xt xt−1 ].

(B.5)

For the loss function (B.5 ), we have the following mapping between itsdelegating parameters in the general case [the Θ matrix in (B.1 )]:θππ = 0θπx = 0

θxx = λx − λ + λq (1 − ν)2 σν2θπs = cπθss = (1 − ν)2 σν2 λq
θxs = −λq (1 − ν)2 σν2Substituting this in the parameter restriction for the optimal delegation(B.2 ), we obtain the optimal delegation parameters for our targeting regime
as in Proposition 1.

25

B.2

Speed Limit Policy

The policy regime is described by the following loss function:1
LSt = [λx2t + λS ∆x2t + λπ (πtc )2 + 2cπ πt xt−1 ]
2

(B.6)

Substituting (3) and (5) into the loss function, we obtain
1
LtS = [πt2 − πt2 + λx2t + λS ∆x2t + λπ (νσν ∆xt − πt )2 + 2cπ πt xt−1 ]2
1S⇒ Lt = [πt2 + λx2t + (−1 + λπ )πt2 + λπ ν 2 σν2 x2t + (λS + λπ ν 2 σν2 )x2t−12+ 2λπ νσν πt xt + 2(−λπ νσν + cπ )πt xt−1 + 2(−λS − λπ ν 2 σν2 )xt xt−1 ].
(B.7)
For the loss function (B.7 ), we have the following mapping between it’sdelegating parameters in the general case [the Θ matrix in (B.1 )]:θππ = −1 + λπθπx = λπ νσνθxx = λπ ν 2 σν2θπs = −λπ νσν + cπ

θss = λS + λπ ν 2 σν2θxs = −λS − λπ ν 2 σν2Substituting this in the parameter restriction for the optimal delegation(B.2 ), we obtain the optimal delegation parameters for our targeting regime
as in Proposition 2.

B.3

CPI Inflation Targeting

The policy regime is described by the following loss function:1
Lπt = [πt2 + λx x2t + λπ (πtc )2 + 2cπ πt xt−1 ]
2

(B.8)

Substituting (3) and (5) into the loss function, we obtain
1
Lπt = [πt2 + λx2t + (λx − λ)x2t + λπ (νσν ∆xt + πt )2 + 2cπ πt xt−1 ]2
1⇒ Lπt = [πt2 + λx2t + λπ πt2 + (λx − λ + λπ ν 2 σν2 )x2t + λπ ν 2 σν2 x2t−12+ 2λπ νσν πt xt + 2(−λπ νσν + cπ )πt xt−1 − 2λπ νσν xt xt−1 ].

(B.9)

For the loss function (B.9 ), we have the following mapping between it’sdelegating parameters in the general case [the Θ matrix in (B.1 )]:26θππ = λπθπx = λπ νσνθxx = λx − λ + λπ ν 2 σν2θπs = −λπ νσν + cπ

θss = λπ ν 2 σν2θxs = −λπ νσνSubstituting this in the parameter restriction for the optimal delegation(B.2 ), we obtain the optimal delegation parameters for our targeting regime
as in Proposition 3.

B.4

Nominal Income Growth Targeting

The policy regime is described by the following loss function:
1 2
2
c
2
LN
t = [πt + λx xt + λN (πt + ∆xt ) + 2cπ πt xt−1 ]2(B.10)

Substituting (3) and (5) into the loss function, we obtain
1 2
2
2
2
LN
t = [πt + λxt + (λx − λ)xt + λN ((1 + νσν )∆xt + πt ) + 2cπ πt xt−1 ]2
1 2
N⇒ Lt = [πt + λx2t + λN πt2 + (λx − λ + (1 + νσν )2 λN )x2t
2
+ (1 + νσν )2 λN x2t−1 + 2(1 + νσν )λN πt xt+ 2(−(1 + νσν )λN + cπ )πt xt−1 + 2(−(1 + νσν )2 λN )xt xt−1 ].
(B.11)For the loss function (B.11 ), we have the following mapping between itsdelegating parameters in the general case [the Θ matrix in (B.1 )]:θππ = λNθπx = (1 + νσν )λNθxx = λx − λ + (1 + νσν )2 λN
θπs = −(1 + νσν )λN + cπ

θss = (1 + νσν )2 λN
θxs = −(1 + νσν )2 λN

Substituting this in the parameter restriction for the optimal delegation(B.2 ), we obtain the optimal delegation parameters for our targeting regime
as in Proposition 4.

27

Table 1: Deep parameters
Parameters
Valuesβ Discount rate
0.99
σ Relative risk aversion coeﬃcient for consumption
2.0
φ Inverse of the elasticity of labor supply
5.0
α Degree of decreasing return on labor
0.25
ϵ Elasticity of substitution for individual intermediate goods
9.0θ Calvo lottery
0.75η Elasticity of substitution between domestic and foreign goods
1.0ν Degree of openness
0.4
λ Weight on the output gap in the true loss function
0.5

28

3.5
3.0
2.5

cπ
λx
λq

2.0
1.5
1.0
0.5
0.0
0.5
0.0

0.1

0.2

0.3

0.4

ν

(a) Real Exchange Rate Target
3.5
3.0
2.5

cπ
λπ
λS

2.0
1.5
1.0
0.5
0.0
0.5
1.0
0.0

0.1

0.2

0.3

0.4

ν

(b) Speed Limit Policy

Figure 1: Real Exchange Rate Target vs. Speed Limit Policy

29

0.2

0.0

0.2

0.4

cπ
λx
λπ

0.6

0.8

1.0
0.0

0.1

0.2

0.3

0.4

ν

(a) CPI Inflation Target
0.4

0.2

0.0

cπ
λx
λN

0.2

0.4

0.6

0.8
0.0

0.1

0.2

0.3

ν

(b) Nominal Income Growth Target

Figure 2: Other Regimes

30

0.4

1.0

0.5

0.0

0.5

θπs
1.0

1.5

2.0

2.5

0.2

0.4

0.6

0.8

λ

(a) CPI Inflation Target
10

0

10

θπs
20

30

40

0.2

0.4

0.6

0.8

λ

(b) Nominal Income Growth Target

Figure 3: Whether the Second-Order Condition θπs < 0 Holds With Respectto λ31