Research InstituteDiscussion Paper Series№6
Title：

Delegating nominal income growth targeting in asmall-open economy

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

2018 年 4 月

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

Delegating nominal income growth targetingin a small-open economy∗
Daisuke Ida†
Mitsuhiro Okano‡
April 9, 2018

Abstract
The present study examines the role of nominal income growth targeting (NIGT)
in a small-open economy and shows that the central bank’s selection of a price index crucially aﬀects the performance of NIGT in a small open economy. NIGT
based on a producer price index can always achieve the same outcome as the commitment policy. However, NIGT based on a consumer price index fails to create
the same outcome as the commitment policy unless the stabilisation weight on the
output gap is considerably small in the true loss function.

Keywords: Optimal monetary policy; Delegation; Nominal income growth targeting;Price index;
JEL classification: E52; E58; F41∗

We thank Kenya Takaku for his helpful comments and suggestions. All remaining errors are ourown.
†

Department 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‡

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 the outcomes of a commitment policy may be preferable tothose of a discretionary policy in the standard new Keynesian model. A central bankthat commits its future policy stances to the public sector can operate monetary policyby managing the expectations of the private sector, which can induce policy inertia into
the economy. In contrast, a discretionary policy, which treats the future expectations
as given, cannot induce policy inertia into the economy. This diﬀerence is referred to as
stabilisation bias.
Discretionary policy suﬀers from stabilization bias, which can be overcome by a government’s delegation to the central bank an objective that diﬀers from the true objectivefunction. Bilbiie (2014) analytically derived a condition by which a delegated policy
regime corresponds with a commitment policy. The analysis shows that each coeﬃcientfor stabilisation terms in the delegated loss function is characterised by deep parameters.
Following Bilbiie (2014), Ida and Okano (2017) analytically examined how the effect of an open economy changes the characteristics of delegated policy regimes, using
a framework of a small-open economy based on Gali (2015). They also analyticallyshowed the condition under which delegated monetary policy regimes correspond with acommitment solution in a small-open new Keynesian model.
Considering the monetary policy analysis in a closed economy framework, there isno need for diﬀerentiation between the consumer price index (CPI) and producer priceindex (PPI) inflation. However, extending the analysis to the small-open economy yields
an important distinction between CPI and PPI inflation. Indeed, as Linnemann andSchabert (2006) noted, when the central bank responds to changes in CPI inflation,
the Taylor principle may not apply, leading to the indeterminacy problem. Moreover,Gali (2015) emphasized the distinction between CPI and PPI inflation from a welfare
analysis perspective, indicating that in a canonical representation of the small open new
Keynesian model, the central bank should stabilise the PPI inflation for maximisinghousehold’s welfare.
We show that the diﬀerence between PPI and CPI inflation changes the properties1

of delegating the optimal monetary policy by especially focusing on nominal income
growth targeting (NIGT), which may use either PPI or CPI as the price index. First,
a nominal income growth target based on PPI inflation coincides with the commitment
solution. This policy regime can achieve the same outcome as the commitment policy. In
contrast, a CPI-based nominal income growth target fails to achieve the same outcome as
the commitment policy unless the stabilisation weight on the output gap is considerably
small in the true loss function.
The paper is organised as follows. Section 2 describes the characteristics of a smallopen economy new Keynesian model. Section 3 considers PPI and CPI-based NIGT
regimes. Analytical derivations of the conditions show that these optimal delegation
schemes coincide with a commitment policy, and show the properties of each policy
regimes. Section 4 briefly concludes.

2

Model

The model is based on Gali (2015). The home country is infinitesimally small relative tothe rest of the world. Representative households in the home country purchase both homeand foreign goods. They can have access to a complete set of state-contingent securities
that are traded both domestically and internationally. Firms face a monopolisticallycompetitive environment and nominal price rigidities specified by Calvo (1983). As
shown in Gali (2015), such situations leads to the following log-linearized system:
πt = βEt πt+1 + κν xt + ut ,(1)

xt = Et xt+1 − σν−1 (it − Et πt+1 − rr¯ t ),(2)

πtc = πt + ν(st − st−1 ),(3)

qt = (1 − ν)st(4)

s t = σ ν xt .

(5)

where πt is producer currency inflation, xt is the output gap, it is the nominal interest
rate, πtc is CPI inflation, st is the terms of trade gap, and qt is the real exchange rate2

gap. Gap variables are expressed by the log-deviation of endogenous variables from theeﬃcient level of their variables. In addition, rr¯ t denotes the natural rate of interest,
which holds the real interest rate under flexible price equilibrium. ut is the exogenous
cost-push shock, which follows an AR(1) process. Finally, coeﬃcients for each structural
equation are defined as follows:σ,
1 + ν[ση + (1 − ν)(ση − 1) − 1]
)
(
1+ψ
,
κν = δ σν +1−α(1 − ω)(1 − ωβ) 1 − αδ=.
ω
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
labour supply elasticity, respectively. ν is the degree of openness, and α is the degree ofdiminishing return to scale for labour supply. ω characterises the degree of nominal pricerigidities (i.e. Calvo’s lottery), and ϵ denotes the elasticity of substitution for individual
goods.
Equation (1) represents the small-open economy new Keynesian Phillips curve (NKPC)
which is derived from the firm’s profit maximisation problem subject to Calvo pricing.
Equation (2) is a dynamic IS equation resulting from a household’s intertemporal optimisation problem. Equation (3) is the relationship between CPI and PPI inflation.
Equation (4) states that the real exchange rate proportionally changes in response tochanges in the 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 characterised by changes in both ν and ση, where
the parameter ν is the degree of openness. For a given ση > 1, through the risk-sharing
condition, the home output gap changes proportionally to a change in the terms of trade,which implies real exchange rate fluctuations.
In addition, the change in the terms of trade aﬀects the sensitivity of inflation to
the real marginal cost in the NKPC through two channels. First, home inflation reacts
positively to improvement in the terms of trade by a change in the real exchange rate3

through international consumption risk sharing. Second, the terms of trade produces achange in home inflation because it induces a change in the real marginal cost. Whether
home inflation increases depends on the movement of the value of ση. In the case ofση = 1, both κν and σν reduce to δ(1 + 1+ψ) ≡ κ and 1, respectively. Thus, the open
1−αeconomy eﬀect disappears in this case.
The central bank conducts its monetary policy following a targeting rule derived
from the minimization problem of the central bank’s loss function. More specifically, thecentral bank minimises the following social loss function subject to structural equations:
Lt = Et∞∑2
β j (πt+j+ λx2t+j ).

(6)

j=0

where the parameter λ denotes the stabilisation weight on the output gap relative to
inflation stabilisation.
In this paper, the central bank minimises the loss function (6) subject to NKPC
under cost-push shock. A commitment solution produces the following targeting rule:
πt = −

λ(xt − xt−1 ).
κν(7)

Except for structural equations coeﬃcients, a targeting rule with commitment is thesame shape as the one obtained in a closed economy. The central bank that conducts a
commitment policy can manipulate the private sector expectations by gradually changing
its policy variable. Thus, the presence of the lagged output gap in the targeting rule
allows the central bank to employ an inertial behaviour of policy stance; therefore, acommitment policy is superior to a purely discretionary policy (e.g., Woodford, 2003;
McCallum and Nelson, 2004). The term of ’commitment policy’ in this paper implies atimeless perspective proposed by Woodford (2003).
The optimisation problem of the central bank leads to the reduced form of endogenous
variables, which is solved by the standard factorisation method. Specifically, the reduced
form of both inflation and the output gap are solved as follows:coπt = ψπco xt−1 + ψuπ
ut ,

(8)

co
xt = ψxco xt−1 + ψux
ut ,

(9)

4

whereλλ c
co(1 − µ1 ); ψuπ= − ψxu
,
κν
κν
κν
1
co
=−
ψxco = µ1 ; ψux.
βλ µ2 − ρuψπco =

Here, µ1 and µ2 are the eigenvalues obtained from the solution of that factorization
method. The parameter ρu denotes the degree of persistency of the cost-push shock.
The superscript co denotes the commitment solution.
Equations (8) and (9) present the reduced forms of endogenous variables dependingon the lagged variable xt−1 . Thus, if the central bank can make a credible commitment to
the private sector, such a commitment policy enables endogenous variables to graduallychange in response to a cost-push shock. In other words, the central bank can alleviate a policy trade-oﬀ generated by a cost-push shock by managing the private sector’s
expectations.

3

Nominal income growth targeting in a small-open
economy3.1

Two specifications of nominal income growth targeting

If the central bank cannot commit to its monetary policy, it naturally conducts a discretionary policy. However, a pure discretionary policy may produce a worse outcome thanthe commitment policy. Several studies argue that the discretionary policy can achievea preferable outcomes if the government delegates the objective function with policy
inertia, which is diﬀerent from the social welfare criteria, to the central bank (Jensen,
2002; Walsh, 2003; Vestin, 2006). Jensen (2002) showed that performance under NIGT
is similar to that under the commitment policy in a closed economy model.
However, in an open economy framework, PPI inflation does not correspond with
CPI inflation (e.g, Equation (3)). How do these diﬀerences change the condition thatthe delegation problem under NIGT coincides with the optimal policy?
5

First, consider NIGT based on PPI. Specifically, the objective function under PPIbased NIGT is defined as follows:
LP,t = πt2 + λx x2t + λN (πt + ∆xt )2 + 2cπ xt−1 πt .

(10)This policy regime imparts inertia through a change in the output gap. The subscript
P denotes PPI-based NIGT.
In contrast, as shown in Ida and Okano (2017), the targeting regime under CPI-based
NIGT leads to policy inertia through a change in not only the output gap but also theterms of trade. In particular, CPI-based NIGT is defined as follows:
2
C
c
2
C
LC,t = πt2 + λC
x xt + λN (πt + ∆xt ) + 2cπ xt−1 πt ,(11)where the subscript C is the delegated parameter under NIGT based on CPI.
Finally, for both regimes, as explained in Bilbiie (2014), a fourth term is technically
added to the objective function for an analytical solution corresponding to the solution
under a commitment policy. In other words, this term reflects the linear inflation contract
term proposed by Walsh (1995) and Svensson (1997).

3.2

The properties of nominal income growth targeting

We show that PPI-based NIGT can always achieve the same outcome as the commitment policy, whereas CPI-based NIGT can do so if and only if the parameter λ takes aconsiderably small value. The delegation problem under PPI-based NIGT leads to thefollowing result:
Proposition 1. (PPI-based NIGT) The Markov-perfect equilibrium values of domestic
inflation and output gap, which occur if the central bank minimises the delegated loss
function (10), are identical to the timeless-optimal commitment solutions (8) and (9) if

6and only if the delegation parameters are given as follows:
γλ2,(κν − λ)(γκν + γλ + λ)γλcπ = −,γκν + γλ + λ
γ 2λ
λx =
.
γκν + γλ + λ

λN =

(12)
(13)
(14)

The proof of this proposition is given in the Appendix. This regime satisfies the properties
of NIGT suggested by Jensen (2008) and Billbie (2014). In particular, it is possible that
λN takes a negative value when κν > λ. In contrast to Billbie (2014), λN is now aﬀectedby ν and ση which specify the open economy eﬀect. In other words, unsurprisingly,
except for the eﬀect of the open economy, this regime can completely replicate thecommitment solution because cπ never takes any positive values. As shown in Billbie(2014), a negative value of cπ is required to satisfy the second-order condition of the
optimal delegation problem under NIGT.
This condition only holds in the case of PPI. If this second-order condition is violated,
the solution for NIGT does not lead to the same outcome as the commitment policy.
This is possible if cπ takes a positive value. Under CPI-based NIGT, the second-ordercondition is invalid when the parameter λ does not take a small value. This is because
cπ may take a positive value when λ takes a larger value.
Indeed, as shown in Ida and Okano (2017), the optimal delegated parameters under
CPI-based NIGT are summarised as follows:
Proposition 2. (CPI-based NIGT) The Markov-perfect equilibrium values of domestic
inflation and output gap, which occur if the central bank minimises the delegated loss
function (10), are identical to the timeless-optimal commitment solutions (8) and (9) if
and only if the delegation parameters are given as follows:
λC
N =

γ
,
χN

(15)

)( γ )
κν
−1
,
ΨN
λ
χN
(
)
γ2
κν
λC
=
Ψ
Ψ
−
1
,
N
N
x
χN
λ
cC
π = −ΨN

(

7

(16)
(17)

where ΨN = 1 + νσν and χN = κλν Ψ2N

( κν
λ

)
γ + 1 − γ − ΨN . In particular, these delegation

parameters under nominal income targeting lead to the same outcome as commitment if
and only if λ takes a small value.
Ida and Okano (2017) provide the proof of this proposition. In contrast to the case of
PPI-based NIGT, the optimal delegation parameters are more complicated. As mentioned in Ida and Okano (2017), CPI-based NIGT does not lead to the same outcome
as commitment unless the parameter λ takes a considerably small value. As noted earlier, the parameter cC
π takes a negative value to satisfy the second-order condition of the
optimal delegation problem. It follows from Equation (16) that this only holds whenνσν κν > λ. Indeed, Ida and Okano (2017) showed that this condition is violated when
the parameter λ takes a value above 0.2.
Why does this diﬀerence occur in open economies? In contrast to the closed economy,a change in the terms of trade adjusts the diﬀerence between PPI and CPI inflation. In
the case of CPI-based NIGT, the objective function under NIGT can be rewritten as
follows:
1
C
LC,t = {Ht + 2[cC
π − (1 + νσν )λN ]πt xt−1 },
2

(18)

where
2
C
2 C 2
C
Ht = (1 + λCN )πt + [λx − λ + (1 + νσν ) λN ]xt + 2(1 + νσν )λN πt xt
2
2 C+ (1 + νσν )2 λC
N xt−1 − 2(1 + νσν ) λN )xt xt−1 .

Ht is the correction term excluding the inflation linear contract term πt xt−1 . Therefore,
the second-term of the right-hand side can be regarded as the linear inflation contract
term in an open economy. The term (1 + νσν )λCN crucially aﬀects the sign of this term.
This term indicates that policy inertia derived from the terms of trade might weaken theeﬀectiveness of the linear contract term, which guarantees that the parameter cC
π takesa negative value.

8

4Concluding remarks

This study has examined the role of NIGT in a small-open economy. We analyticallyshowed that the central bank’s selection of the price index crucially aﬀects the performance of NIGT in a small open economy. NIGT based on a PPI can always achieve
the same outcome as the commitment policy. However, NIGT based on a CPI fails tocreate the same outcome as the commitment policy unless the stabilisation weight on
the output gap is considerably small in the true loss function.

References
Bilbiie, F.O. (2014) “Delegating optimal monetary policy inertia,”Journal of Economic
Dynamics and Control 48, 63-78.
Calvo, G. (1983).“Staggered prices in a utility-maximizing framework.” Journal of Monetary Economics 12, pp.383-398.
Gali, J. (2015) Money, inflation and the business cycles, Second edition, Princeton University Press, Princeton.
Ida, D. and Okano, M. (2017) “Delegating optimal monetary policy inertia in a smallopen economy”, Momoyama Gakuin University Discussion paper series No.5.
(http://www.andrew.ac.jp/soken/pdf_5-5/DP_05.pdf)
Linnemann, L. and Schabert, A. (2006) “Monetary policy and the Taylor principle inopen economies,” International Finance 9, 343-367.
Svensson, L.E.O. (1997) “Optimal inflation targets, conservative central bankers andlinear 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.
9

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

10

A

Proof of Proposition 1 (Not for publication)

To solve a discretionary policy, we define the Bellman equation as follows:[]1
V (xt−1 ; ut ) = min LP,t + βEt V (xt ; ut+1 ) .
2The first-order condition with respect to xt is as follows:2(1 + λN )γπt + 2λN πt + 2λN (1 + γ)(xt − xt−1 ) + 2λx xt + 2cπ γxt−1∂V (xt ; ut+1 )
1
=0
+ βEt
2
∂xt

(19)

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

(20)

Substituting a one-period ahead Equation (20) and Equations (8) and (9) into Equation (19) and then considering the optimal targeting rule, we obtain the following secondorder diﬀerence equation:
[
]
[
λ
λ
β (λN − cπ ) − λN Et xt+1 + (1 + γ)λN + βλN − [γ + λN (1 + γ)] + λx
κν
κν
]
[
]
λ
λ
− β(λN − cπ )xt + cπ γ − (1 + γ)λN + (γ + λN (1 + γ))
xt−1 = 0.
(21)
κν
κν
As explained in Bilbiie (2014), in this equation, the optimal delegation parametercois found by noticing that Equation (21) evaluated at the conjectured solution (xcot , πt )should be identity. Therefore, to identify the optimal delegation parameter, all coeﬃcients should be zero. This leads to the following relationship:(λN − cπ )λ− λN = 0,
κν

(22)

λ
λ
+ λN − β(λN − cπ ) = 0,κνκνλcπ γ − (1 + γ)λN + (γ + λN (1 + γ)) = 0.
κν
(1 + γ)λN + βλN − [γ + λN (1 + γ)]

(23)
(24)

Solving Equations (22)–(24), we obtain the optimal delegation parameters in Proposition1.
11