Research InstituteDiscussion Paper Series№08
Title：

Cross-checking monetary policy rule andEquilibrium determinacy

Daisuke Ida
ida-dai@andrew.ac.jp

2018 年 12 月

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

Does a cross-checking monetary policy rule uniquely
lead to equilibrium determinacy?∗
Daisuke Ida†
December 7, 2018

Abstract
This paper examines equilibrium determinacy under cross-checking optimalmonetary policy. The result shows that an optimal rule derived from cross-checking
optimal policy easily expands the regions of the unique rational expectations equilibrium (REE). Interestingly, this result is turned over in the case of the costchannel. This paper shows that introducing a cost channel is likely to make the
REE indeterminate under cross-checking optimal monetary policy.

Keywords: Cross-checking monetary policy; Taylor principle; Determinacy; Cost channel;
JEL classification: E52; E58∗

I thank Yasuo Hirose and Kohei Hasui for their constructive comments and suggestions. This paper

was supported by JSPS KAKENHI Grant Number 17K13766. All remaining errors are my own.
†

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

1

Introduction

How should the central bank implement optimal monetary policy considering the robustness of the model? According to Tillmann (2012), the central bank refers to monetary
policy rules that are robust to the misspecification of the model because it faces themodel uncertainty—the true model is not known with certainty. In such a situation,
Tillmann (2012) shows that it is desirable for the central bank to attach some weights tothe deviation of the policy rate from the prescription suggested by the Taylor rule when
the central bank follows a discretionary policy in a forward-looking economy. This gainis associated with the fact that such a policy can reduce stabilization bias generated bya discretionary policy in terms of the traditional loss function that contains inflation andoutput stabilization.1
Ilbas, Roisland and Sveen (2012) also argue the importance of the deviation term of
the policy rate from the prescription suggested by the Taylor rule under optimal monetary
policy. Walsh (2017b) considers the role of rule-based policy (RBP) regimes when thecentral bank follows a discretionary policy. The cross-checking optimal monetary policy
can be regarded as optimal with the information from the policy rule. Moreover, Taylor(2009) also argues that the deviation of the policy rate from the Taylor rule induced
the housing boom from 2000 to 2006 in the United States. It appears, therefore, thatthese assertions justify the significance of cross-checking optimal monetary policy. They
may also address the eﬀectiveness of cross-checking optimal monetary policy, whereas
it is unclear whether such a rule can lead to a unique rational expectations equilibrium(REE).
It has been argued that a non-fundamental shock can create multiple equilibria in a
forward-looking sticky price model. Hence, it is very important how policymakers make
REE determinate. In the standard new Keynesian model, Bullard and Mitra (2002)
argue that the REE is uniquely determinate when the traditional Taylor principle issatisfied, which requires that the central bank should respond with a more than one-forone increase in the interest rate when inflation rate increases. If a rise in the nominal
1

See Walsh (2017a) for a detailed explanation of the stabilization bias.

1interest rate cannot produce an increase in the real interest rate, both expected inflationand output increase in response to a sun-spot shock regardless of monetary tightening.
As a result, the economy reaches a non-fundamental equilibrium.2
The objective of this paper is to examine equilibrium determinacy under crosschecking optimal monetary policy with discretion. To the best of my knowledge, nostudies show how a unique REE is achieved in the case where the central bank implements a discretionary policy by using the information from the Taylor rule. This is thefirst study to show the condition under which the cross-checking rule achieves a unique
REE.
This paper contributes to the literature in the following ways. First, this paper examines how the cross-checking optimal policy aﬀects the determinacy condition underwhich the REE is uniquely determinate. It shows the condition for a unique REE to
be achieved under a cross-checking optimal rule in the standard new Keynesian model.
Interestingly, a decrease in a positive weight on a cross-checking rule expands the determinacy regions. Indeed, when the central bank puts a much smaller weight on theinformation from the Taylor rule, the REE is always determinate when it employs someweight in response to the output gap in a cross-checking optimal rule. Notably, there
are cases that the REE is always determinate even if the central bank sets the coeﬃcientfor inflation stabilization to zero in the Taylor rule. This is the natural extension of
Tillmann (2012) and Walsh (2017b).
The second contribution of this paper is to show that this result is easily turned overin the case of the cost channel. A cost channel implies that a change in the nominalinterest rate directly aﬀects the inflation rate through an increase in the firm’s working
capital. Previous studies have addressed the importance of a cost channel of monetary
policy.3 This paper shows that the central bank, which employs cross-checking monetary
2This is the case where the standard Taylor rule is employed in the model. When the forward-looking

Taylor rule is assumed, the upper bounds for the response to a change in inflation exist in addition to
the Taylor principle. See Bullard and Mitra (2002) for a detailed discussion of this issue.
3

See Barth and Ramey (2001), Ravenna and Walsh (2006), Chowdhury et al. (2006), and Tillmann(2008) for a detailed discussion of the cost channel. See also Section 4.

2

policy, can achieve the unique REE if the degree of a cost channel is small. On thecontrary, when a strong cost channel is introduced into the economy, a cross-checking
optimal policy rule fails to attain the unique REE. Here, this paper presumes that thestrength of the cost channel is measured by the degree of loan rate pass-through.
Interestingly, in the case of the cost channel, the REE is likely to be indeterminate as
the central bank puts more weight on the stabilization of the output gap. The indeterminacy problem is more severe when the central bank puts less weight on the cross-checking
term. The result implies that in the case of a strong cost channel, a smaller weight on the
RBP policy term, which leads to an attenuated response to the inflation rate, results inequilibrium indeterminacy even if the instrument rule satisfies the Taylor principle. This
result still holds in the forward-looking Taylor rule. Under that rule, the indeterminacy
problem is also more severe when the weight on the output gap stabilization takes alarger value.
The case of the cost channel in this paper is the natural extension of Llosa andTuesta (2009) for the case of cross-checking optimal monetary policy.4 First, this paper
diﬀers from Llosa and Tuesta (2009) in that it assumes that the central bank follows adiscretionary policy with the information from the Taylor rule. The paper shows thatthe determinacy condition derived by Llosa and Tuesta (2009) is strongly aﬀected by
the degree of the cross-checking parameter. Second, they assume that the cost-channel
parameter takes a range from 0 to 1, whereas this paper employs a wider range ofparameters pointed out by the existing literature. This is important because a higher
value of the cost-channel parameter implies the amplification of financial instability.
Indeed, larger values for a cost channel parameter are reported by the existing literature(Ravenna and Walsh, 2006; Castelnuovo, 2007). Therefore, in contrast to Llosa andTuesta (2009), this paper both analytically and numerically studies how the degree ofthe cost channel changes the region where the REE is determinate.
The rest of this paper is organized as follows. Section 2 briefly explains the standard

4

Note that in contrast to Llosa and Tuesta (2009), this paper abstracts the E-stability condition

under adoptive learning when the central bank cross-checks its monetary policy.

3new Keynesian model, deriving the cross-checking optimal monetary policy. Section 3
shows the condition under which a cross-checking optimal policy rule can uniquely attain
the REE in the standard new Keynesian model. Section 4 explores how the condition isaltered in the new Keynesian model with a cost-channel. Section 5 briefly concludes.

2

Model

The model description in this paper is based on Tillmann (2012) and Walsh (2017b). The
model is a standard new Keynesian model, which comprises three equations: a dynamic
IS equation, a new Keynesian Phillips curve (NKPC), and monetary policy.
The first equation is a dynamic IS curve derived from the intertemporal optimal
conditions from households. The dynamic IS curve is given by
xt = Et xt+1 − σ −1 (rt − Et πt+1 − rtn ),(1)

where xt is the output gap, πt is the inflation rate, and rt is the nominal interest rate.
Variable rtn denotes the natural rate of interest that stands for the real interest rate in
the flexible price equilibrium. This shock can be regarded as a demand shock in thispaper. Et is the expectations operator conditional on the information about period t.
The parameter σ is the relative risk aversion coeﬃcient for consumption.
The second equation is a NKPC, which is derived from the optimal condition of thefirms that are subject to monopolistic competition and Calvo (1983) type nominal pricerigidities. The NKPC is given by
πt = βEt πt+1 + κxt ,(2)

where the parameter κ is the sensitivity of inflation to a fluctuation of the output gap
and β denotes the discount factor.55This paper abstracts the cost-push shock in the NKPC. The reason is that the presence of the cost-

push shock generates the stabilization bias associated with a discretionary policymaking of the central
bank. Tillmann (2012) argues that cross-checking optimal policy help alleviate the stabilization bias.
In contrast to Tillmann (2012), this paper focuses on an alternative gain from cross-checking optimal

4

Finally, this paper describes monetary policy of the central bank. The central bankaims at stabilizing quadratic in inflation and the output gap and thus minimizes the loss
function, which is given as follows:Lstab= πt2 + λx2t ,t(3)

where the parameter λ represents the weight on the stabilization of the output gap
relative to inflation stabilization. As in Ilbas, Roisland and Sveen (2012), Tillmann(2012), and Walsh (2017b), this paper assumes that the central bank cross-checks its
monetary policy by referring to the information of the instrument rule suggested by
Taylor (1993). In other words, the central bank cares about the robustness of its interest
rate policy. Like Tillmann (2012), this paper also assume that squared deviation of
the policy rate from the Taylor rule describes the concern about the robustness of theinterest rate policy. This assumption can be regarded as the RBP term in the crosschecking monetary policy labeled by Walsh (2017b). More specifically, such a central
bank’s concern is captured by
Lrbp= (rt − rtT )2 ,t

(4)where rtT is the policy rate indicated by the Taylor rule, which is given as follows:
rtT = ϕπ πt + ϕx xt ,(5)

where ϕπ is the coeﬃcient of inflation stabilization and ϕx is the coeﬃcient of output gap
stabilization. The central bank, which is concerned about the robustness of its interest
rate policy, attempts to minimize equation (4).
In the case of cross-checking optimal monetary policy, the central bank minimizes
weighted by γ.
and Lrob
the loss function that is a convex combination between LstabttConcretely, such a loss function is given as follows:
Lt = Et∞∑β t {(1 − γ)Lstab+ γLrbp
t },
t

(6)

t=0

policy in terms of the determinacy problem. Therefore, this paper discards the presence of the cost-pushshock in the NKPC.

5

As Tillmann (2012) explains, the parameter γ can be regarded as measuring the shareof members of the monetary policy committee referring to the Taylor rule. It turns outthat Equation (6) reduces to (3) when γ = 0.
The central bank that implements a discretionary policy minimizes equation (6) subject to equations (1) and (2). As shown in Tillmann (2012), from the first order conditionsof the central bank’s optimization problem, the following optimal monetary policy ruleis derived:
rt = rtT +1−γ(κπt + λxt ),γA(7)

where A = κϕπ + ϕx + σ.6 When γ = 1, the policy rule completely coincides with the
implied rate suggested by the Taylor rule, whereas the central bank implements thetargeting rule by referring to the implied policy rate from the Taylor rule as long as
γ = 1. It follows that the smaller value of the γ, the stronger is the response to inflationand the output gap. Actually, since a positive demand shock causes the positive responseof both πt and xt , we can easily confirm ∂rt /∂γ = −(κπt + λxt )/γ 2 A < 0.

3

Benchmark results

This section examines how cross-checking optimal monetary policy changes the condition
to attain the REE. While previous studies argue that the Taylor principle must be
satisfied to guarantee the unique REE, it remains unclear how the cross-checking optimal
rule achieves the unique REE. For instance, is the REE uniquely pinned down when themonetary policy committee decides to refer to the information from the Taylor rule
under a discretionary policy? How does the degree of the committee’s preferences to thecross-checking policy γ aﬀect the uniqueness of the REE?
To this end, the paper examines the case of the optimal monetary policy with theinformation from the standard Taylor rule (5). Substituting the optimal policy rule (7)6

See Tillmann (2012) for a detailed derivation of this equation.

6

into equation (1), the system of two endogenous variables xt and πt are written as follows:
Xt = M Et Xt+1 + µrtn ,(8)′

where Xt = [xt πt ] and


M=

−1

−1



σ
1 − β(ϕπ + (1 − γ)γ A κ)
1

.
−1
−1
2
σ + κϕπ + ϕx + (1 − γ)γ A (κ + λ) σκ κ + β(σ + ϕx + (1 − γ)γ −1 A−1 λ)Note that the form of µ is omitted because it does not aﬀect the following discussion.
Since xt and πt are endogenous jump variables, both eigenvalues of M should be insidethe unit circle for determinacy. Thus, we have the following result:
Proposition 1 Under the optimal monetary policy rule (7) the necessary and suﬃcientcondition for a unique rational expectations equilibrium is given as follows:κ(ϕπ − 1) + (1 − β)ϕx +1−γ 2[κ + (1 − β)λ] > 0,γA

(9)Proof. The proof of this proposition is provided in Appendix A.

This proposition provides the modified Taylor principle. The first and second terms of
the left-hand side corresponds to the Taylor rule. This is associated with the informationfrom the Taylor rule. On the other hand, the third term is associated with the condition
derived from a discretionary policy. Third term never takes a value less than zero. In
other words, condition (9) reinforces the condition of the standard Taylor principle, which
is given as follows:κ(ϕπ − 1) + (1 − β)ϕx > 0.

(10)This is the case of γ = 1.0, which corresponds to that of Bullard and Mitra (2002). The
Taylor principle requires that the central bank should aggressively raise its interest rate
in response to a change in inflation unless it does not consider the stabilization of theoutput gap (i.e., ϕx = 0).
Interestingly, the analytical result shows that in the case of ϕx = 0, if the central
bank puts some weights on the RBP term, the condition (9) is satisfied even when the7coeﬃcient ϕπ is less than unity. As an extreme case, the condition is satisfied even if
the central bank does not employ inflation stabilization in the Taylor rule (i.e., ϕπ = 0).
Therefore, the result implies that a combination of targeting rule and an RBP leads toa preferable outcome.
How does the strengthened Taylor rule aﬀect the determinacy regions? The findingsof this paper are illustrated using numerical methods, mainly by employing the deepparameters calibrated in Tillmann (2012). The discount factor (β) is set to 0.99, andthe inverse of the intertemporal elasticity of consumption (σ) is set to 1.0. The slope
coeﬃcient of the NKPC (κ) is set to 0.024 and set the weight on output gap stabilization
relative to inflation stabilization in the loss function (λ) to 0.048.
Next, four values of the parameter γ are chosen. First, the case of γ = 1.0 is set,
which corresponds to the determinacy condition under the standard Taylor rule shown in
Bullard and Mitra (2002). Second, the parameter γ is set to 0.25.7 As shown in Tillmann(2012), this value can produce the smallest welfare loss in the model with cross-checking
optimal policy. Third, in the case of γ = 0.5, half of committee members refer to theinformation from the Taylor rule. Fourth, the parameter γ is set to 0.1.
[Insert Figure 1 around here]
Figure 1 portrays the determinacy condition under a cross-checking optimal monetary
policy rule. In an extreme case, regardless of values of both ϕπ and ϕx , the REE is always
determinate when γ = 0.01. This is not a surprising result. When γ = 0.01, the central
bank can respond aggressively to inflation and the output gap because this value suﬃces
to satisfy the condition (9). In other words, this might indicate that when γ = 0.01, thecentral bank can almost generate a unique REE through targeting rule without ex post
facto referring to the information from the Taylor rule.
When the central bank puts a smaller weight on the information from the Taylor rule
(i.e., the case of γ = 0.1), the REE is always determinate in the case where it employs a7Note that in Tillmann (2012), γ = 0.25 minimizes the welfare loss when both demand and supply

shocks coexist.

8

stronger response to the output gap. When the value of ϕx is above approximately 1.8,
the REE is always determinate even if the central bank sets ϕπ to zero.
Unfortunately, even when γ = 0.1, the central bank needs to put some weight on
inflation stabilization in the case of a smaller value of ϕx . In contrast to γ = 0.01, theresult shows that even if the central bank employs the cross-checking optimal policy
rule, it is possible that the REE is indeterminate if the central bank does not target
both inflation and output stabilization.
It follows from Figure 1 that the indeterminacy regions expands as γ approaches unity.
The case of the parameter γ = 1 corresponds to the determinacy regions generated by
the standard Taylor rule. This line is satisfied as long as the standard Taylor principle(10) holds. To sum up, a decrease in γ expands the determinacy regions. In other words,a some weight on γ can reinforce the modified Taylor principle.

4

Determinacy and cross-checking monetary policy
in an economy with a cost channel

This section extends the benchmark model to the case of a cost channel. The benchmark
result in Section 3 is likely to overturn in an economy with a cost channel.8 Section 4.1
briefly explains the introduction of a cost channel into the standard new Keynesianmodel. Section 4.2 explores the condition that the REE is uniquely determinate in thecase of the cost channel.

4.1

An introduction of the cost channel

Several studies incorporate the role of a cost channel into the standard new Keynesianmodel. Ravenna and Walsh (2006) show that the presence of a cost channel generates a
trade-oﬀ between inflation and output stabilization when the central bank implements
optimal monetary policy. Chowdhury, Hoﬀmann and Schabert (2006) argue that the8

See Barth and Ramey (2000) and Rabanal (2007) for a detailed discussion of empirical studies of

the cost channel.

9inflation rate increases in response to a rise in the policy rate after monetary tightening
in a model in which cost channels matter.
In a model with a cost channel, the firms borrow the working capital from financial
intermediaries to finance the wage bill. In an economy with a cost channel, the term forthe nominal interest rate is augmented in the NKPC (e.g., Ravenna and Walsh, 2006;
Chowdhury, et al., 2006; Tillmann, 2008). The NKPC in an economy with a cost channel
is given by
πt = βEt πt+1 + κ(xt + δrtl ),(11)

where rtl is the lending rate and δ is the presence of the cost channel. When δ = 0,
equation (11) reduces to the standard one. As in previous studies for the cost channel,
we assume that δ = 1.9 As shown in Chowdhury et al. (2006) and Tillmann (2009), thelending rate is related to the nominal interest rate:
rtl = ψr rt ,(12)

where ψr is the degree of incomplete pass-through of the lending rate. In other words,
the parameter ψr represents the strength of the cost channel (e.g., Chowdhury, et al.
2006; Castelnuovo, 2007; Tillmann, 2009).
The cost channel increases inflation through the supply side eﬀect of monetary policy.
If the cost channel dominates the demand channel that monetary tightening induces adecline in inflation rate, an increase in the policy rate generated by monetary tightening
results in an increase in inflation. In this case, should the central bank more aggressively
raise its policy rate to stabilize the inflation rate? The question is whether a crosschecking monetary policy helps achieve the unique REE in an economy with a costchannel.
In the recent literature, several studies show the determinacy condition in a forwardlooking sticky price model with a cost channel. Bruckner and Schabert (2003) findthat the cost channel produces an additional upper bound to the coeﬃcient for inflation
9

See Ravenna and Walsh (2006) and Tillmann (2009) for a detailed discussion of the parameter δ.

10stabilization in the Taylor rule. Surico (2008) also finds that a positive weight on thestabilization of the output gap in a monetary policy rule is likely to cause indeterminacy
in a model with a cost channel. Furthermore, Llosa and Tuesta (2009) show that standard policies can induce equilibrium indeterminacy and expectational instability under
adaptive learning when a cost channel is present.
Finally, in this study the standard loss function is used in the case of the cost channel.
In fact, Ravenna and Walsh (2006) show that the central bank’s loss function contains
the standard policy objectives of both inflation and output gap stabilization.10 Tillmann(2009) also uses the standard loss function to examine optimal monetary policy with themodel uncertainty in a model with a cost-channel. Therefore, the standard loss function
is applicable to the case of cross-checking optimal monetary policy in an economy witha cost-channel.
Note that as shown in Woodford (2003), the term of the interest rate stabilization
term is include in the central bank’s loss function as an additional target variable when
the household’s utility function depends on real money balances.11 Therefore, it may beinteresting how alternative specifications of the central bank’s loss function aﬀect thedeterminacy condition under cross-checking optimal policy.

4.2

Determinacy and cross-checking optimal rule in the case ofa cost channel

In the case of the cost channel, the cross-checking optimal policy rule is derived as follows:
rt = rtT +1−γ[κ(1 − σψr )πt + λxt ],
γ Ã

(13)

where Ã = κ(1 − ψr σ)ϕπ + ϕx + σ.
In the benchmark new Keynesian model, a smaller value of γ strengthens the response10

See Ravenna and Walsh (2006) and Demirel (2013) for a detailed derivation of the central bank’s

loss function in an economy with a cost channel.
11

Ida (2018) derives the cross-checking optimal policy rule under interest rate smoothing and examines

the determinacy condition under that rule11

of the nominal interest rate to fluctuations in endogenous variables. How does thepresence of the cost channel aﬀects a change in the nominal interest rate in response
to fluctuations in macro-variables? Following equation (13), when both πt and xt are
positive responses to the positive demand shock, we obtain the following results:
Proposition 2 Under a positive demand shock, a decline in γ leads to the attenuatedresponse of the nominal interest rate if 1/σ < ψr .
Proof. Diﬀerentiating (13) with respect to γ leads to the following result:∂rtκ(1 − σψr )πt + λxt=−.
∂γ
γ 2 Ã

(14)

Under a positive demand shock, since a numerator takes a positive value as long as
1/σ > ψr , this condition should take a negative value. Therefore, it is possible that adecline in γ induces an attenuated response of the policy rate to endogenous variables
for larger values of ψr such that 1/σ < ψr .

This proposition states that if a strong cost-channel is introduced into the economy, adecline in γ counteracts the response of the nominal interest rate under cross-checking
policy rule. This is the contrast result to the benchmark case where ψr = 0. Additionally,
this never occurs in Llosa and Tuesta (2009). How does this modification of the policy
rule aﬀect equilibrium determinacy?
To answer this question, substituting the optimal policy rule (7) into equation (1)and using equation (11), the system of two endogenous variables xt and πt are written
as follows:
Xt = M̃ Et Xt+1 + µrtn ,
where

(15)



−1 −1
−1 −1
1 − (β + κψr )(ϕπ + (1 − γ)γ Ã κ)
1 σ(1 − κψr )(ϕπ + (1 − γ)γ Ã κ)),M̃ = Γ σκ(1 + ψr )(ϕx + (1 − γ)γ −1 Ã−1 λ) κ + σβ + (1 + β)(ϕx + (1 − γ)γ −1 A−1 λ)withΓ = σ + κ(1 − ψr σ)ϕπ + ϕx + (1 − γ)γ −1 A−1 [κ2 (1 − ψr σ) + λ].
12Since xt and πt are endogenous jump variables, both eigenvalues of M̃ should be
inside the unit circle for determinacy. Thus, we have the following result:
Proposition 3 Let 1/σ > 2ψr . In the case of the cost channel, under the optimalmonetary policy rule (13) the necessary and suﬃcient condition for a unique rational
expectations equilibrium is given as follows:
[
]
1−γ
2
ϕx + κ(1 − ψr σ)ϕπ + (1 − β)σ +(16)(1 − ψr σ)κ + λ > 0,
γ Ã
1−γ 2
(1 − β − κψr )ϕx + κϕπ +(17)[κ + (1 − β − κψr )λ] > 0,
γ Ã
(
)
(
)
(1 − γ)λ(1 − γ)κ
2(1 + β)σ + κ + (1 + β + κψr ) ϕx ++ κ(1 − 2ψr σ) ϕπ +
> 0.
γ Ã
γ Ã
(18)
Proof. The proof of this proposition is provided in Appendix B.

Equation (17) is the modified Taylor principle derived Llosa and Tuesta (2009) who show
the determinacy condition in the case of the cost channel. According to this condition,
if the RBP term is absent, equilibrium indeterminacy might occur when ϕx takes alarger value in accordance with an increase in the value of ψr . Thus, when γ = 1.0, in
the case where incomplete pass-through of the lending rate to the policy rate is very
severe, the REE is indeterminate when the central bank takes a larger weight on output
stabilization in the Taylor rule, given the value of ϕπ . This is consistent with the result of
Llosa and Tuesta (2009).12 Can the third-term derived from the targeting rule ease thisproblem? In particular, as the benchmark result suggests, does the cross-checking rule
help alleviate the indeterminacy problem? Does a larger weight on output stabilization
lead to indeterminacy as in the case of Llosa and Tuesta (2009)?
Equation (18) is related to the condition derived by Surico (2008) and Llosa andTuesta (2009). If the cross-checking optimal monetary policy is absent, as shown by
Surico (2008) and Llosa and Tuesta (2009), the condition (18) imposes the upper boundfor the response of inflation in the Taylor rule. Under the cost channel, responding too
12

In their study, the cost channel parameter ψr is set to 1.0.

13

strongly to inflation leads to real indeterminacy. Finally, Equation (16) is also satisfied
as long as 1/σ > ψr .
It remains unclear how the presence of γ aﬀects the conditions (16) - (18) under
several parameterizations of ψr and ϕπ given the value of γ. Numerically showing theregions where the REE is uniquely determinate, since the conditions (16) - (18) arecomplicated, will answer these questions.
Figure 2 shows the determinacy/indeterminacy regions when the central bank crosschecks its monetary policy in the presence of the cost channel. Surprisingly, in contrast to
the benchmark case, when γ = 0.01, unless the degree of the cost channel is considerably
weak, the REE is always indeterminate even if the central bank reacts more aggressively
to inflation given the stabilization of the output gap. Thus, as Proposition 2 states, thesmaller γ is, the more attenuated is the response of the policy rate in the cross-checking
optimal policy rule. Therefore, an extremely small value of γ is likely to violate theconditions (16) - (18) compared to the case where the cross-checking policy is absent.
If the central bank sets γ to 0.1, an aggressive response of the policy rate to inflation
leads to the unique REE even when ψr takes a larger value. Therefore, compared to thecase of γ = 0.01, γ = 0.1 expands the determinacy regions. However, this prescription is
overturned when γ = 0.25. Thus, as shown in Figure 2, the upper bound for the responseof inflation in the Taylor rule is required to guarantee the unique REE as ψr takes above
3.0. This upper bound is more severer as the parameter γ takes a larger value, whereasa larger value of γ forces the central bank to employ the Taylor rule that ϕπ is set toabove unity. When γ = 1.0, the REE is always indeterminate when ψr takes a value
around 5.0.
[Insert Figure 2 around here]
This result is intuitive. As shown in Proposition 2, a change in γ weakens the responseof the interest rate to inflation and the output gap once a strong cost channel is introduced
into the economy. Therefore, if the central bank cross-checks its monetary policy, given
the output gap stabilization a higher response to inflation rate may be required to attainthe unique REE. In other words, a larger value of ϕπ compensates the attenuated response14

of the interest rate induced from the cross-checking monetary policy in the case of thecost channel. Tillmann (2012) argues that a smaller weight on γ can improve social
welfare, whereas our result shows that such a parameterization can easily make the REE
indeterminate. In particular, unless the degree of the cost channel is weak, the central
bank that employs γ = 0.01 can be prone to cause multiple equilibria even though it
employs any value of ϕπ .
[Insert Figure 3 around here]
As shown by Surico (2008) and Llosa and Tuesta (2009), in a model with the costchannel, a larger value of the output gap stabilization in the Taylor rule is likely to induce
equilibrium indeterminacy. Does their result hold in the case of a cross-checking optimalmonetary policy? Figure 3 portrays how determinacy/indeterminacy regions change as
the parameter ϕx increases in the case of γ = 0.1.13 ϕx is allowed to to take negativevalues. This is because the condition (17) may be easily satisfied if the central bank sets
ϕx to negative values for any large values of ψr . In this case, the REE is determinate if
ϕπ is above 1.0 in the case of ψr < 2.0. The REE is indeterminate, however, even if ϕπtakes larger values in the case of ψr > 3.0.
Compared to the case where the central bank employs the negative response to the
output gap (i.e., ϕx = −0.5), the central bank that sets ϕx to zero can expand theregions where the REE is uniquely determinate. However, the upper bound for inflation
response is required as ψr takes a larger value. When ϕx = 0.5, for any larger value ofψr , the upper bound for inflation stabilization disappears, whereas the lower bound for it
increases. Interestingly, the indeterminacy regions expands again when the central bankputs a larger weight on the output gap stabilization. In particular, when ϕx = 2.0, theREE is always indeterminate even if the central bank tries to respond more aggressively
to inflation in the case where ψr is above around 2.0.
[Insert Figure 4 around here]13

I abstract the result in the case of γ = 0.01. This is because we can easily reach the inference that

under this value the REE is almost all indeterminate.

15

Figure 4 shows the eﬀect of a change of ϕx on the determinacy area where ϕπ and ψrchange in the case of γ = 0.5. In contrast to the case of γ = 0.1, a negative response of
the output gap almost results in multiple equilibria. In addition, even in the case of no
output gap response in the Taylor rule, the determinacy regions are smaller in the caseof γ = 0.5 than in the case of γ = 0.1.
Furthermore, in the case of ϕx = 0.5, the severe upper bound for inflation stabilization
is introduced when γ = 0.5. On the other hand, it follows from Figure 4 that the lower
bound for it seems to be eased in that case. Interestingly, compared to the case ofγ = 0.1, the determinacy regions expand as the parameter ϕx takes a larger value. Evenin the case of a larger value of ψr , the unique REE is guaranteed if the central bank puts
a suﬃciently larger weight on inflation stabilization in the Taylor rule.
In sum, in contrast to the benchmark case, a smaller weight on the cross-check monetary policy is likely to generate equilibrium indeterminacy in the case of the cost channel.
Regardless of the value of γ, in contrast to the result of Llosa and Tuesta (2009), for agiven value of ϕx , an increase in ϕπ expands the determinacy regions as long as the costchannel has a negligible eﬀect on the economy (i.e., a smaller value of ψr ). The parameter γ matters considerably, however, because even for a strong response of inflation in
the Taylor rule a larger value of ϕx is likely to make the REE indeterminate as ψr takesa larger value. This result appears to be more severe in the case of a smaller value of γ.

5Forward-looking instrument rule and cross-checking
optimal monetary policy

This section examines how the determinacy of REE is aﬀected when the central bank,which cross-checks its monetary policy, employs the following forward-looking Taylor
rule:
rtT = ϕπ Et πt+1 + ϕx Et xt+1 .

(19)

Bullard and Mitra (2002) show that in the standard new Keynesian model, ϕπ > 1,
whereas too large a value of ϕπ makes the REE indeterminate. In addition, as shown by16

Llosa and Tuesta (2009), an introduction of the cost channel makes the determinacy conditions under a forward-looking Taylor rule more complicated. Our question is whether
the presence of the forward-looking Taylor rule when the central bank cross-checks its
monetary policy helps ease the indeterminacy problem.
The cross-checking optimal monetary policy rule becomes14
rt =

1−γ
[κ(1 − σψr )πt + λxt ] + ϕπ Et πt+1 + ϕx Et xt+1 .
γ(20)Compared to the rule (13), the coeﬃcient Ã disappears because the forward-looking
macro-variables in equation (19) are given when the central bank implements a discretionary policy. In contrast to rules suggested in numerous previous studies, this rule
contains not only the contemporaneous response of endogenous variables but also theforward-looking response of those. To best of my knowledge, none of the studies attempts to assess this type of the rule (7). Our question is whether this type of the rulecan attain a desirable outcome in that it uniquely achieves the REE.
To answer this question, substituting the optimal policy rule (20) into equation (1)and using equation (11), the system of two endogenous variables xt and πt are written
as follows:
Xt = P Et Xt+1 + µrtn ,
where



σ(1−κψr )(1−γ)κ2 (1−σψr )2
− ϕx
1 
γ
P =
Γf σκ(1+ψr )(1−γ)λ − κ(1 − σψr )ϕx
γ

(21)

2

2

(1−σψr )1 − (β+κψr )(1−γ)κ
− ϕπγ

κ + σβ + (κψr +β)(1−γ)λ− κ(1 − σψr )ϕπ
γ


,

with
(1 − γ)λ (1 − γ)κ2 (1 − σψr )2+.
Γf = σ +γγAs mentioned repeatedly, since xt and πt are endogenous jump variables, both eigenvaluesof P should be inside the unit circle for determinacy.15
14This paper examines the determinacy condition under the forward-looking rule in the case of thecost channel. The result of this rule in the benchmark case is available on request.
15

Unfortunately, this paper has diﬃculty in analytically deriving the determinacy condition underEquation (21). Deriving the analytical expressions will be future works.

17Figure 5 shows the the eﬀect of a change of ϕx on the determinacy area where ϕπand ψr change in the case of γ = 0.1. Compared to the case of the contemporaneous
Taylor rule, the result appears to remain unchanged when ϕx = 0. However, as ϕx takesa larger value, the determinacy regions shrink as the stronger cost channel is introduced
into the economy.
[Insert Figure 5 around here]
Figure 6 shows the the eﬀect of a change of ϕx on the determinacy area where ϕπ andψr change in the case of γ = 0.5. In contrast to the case of Figure 4, the determinacy
regions shrink when ϕx = 0. This is also comparable with the case of γ = 0.1. In thecase of the forward-looking rule under cross-checking policy, for given ϕx = 0, a larger
value of γ drastically narrows the determinacy regions for a larger value of ψr .
[Insert Figure 6 around here]
Compared to the cases of Figures 4 and 5, the determinacy regions expands again
when the central bank employs a stronger response to inflation as ψr takes a larger value.
This prescription no longer holds, however, when ϕx takes a larger value. As shown in
Figures 3 and 4, even under the forward-looking cross-checking rule, a larger value of ϕx
is likely to attain the real indeterminacy as the degree of the cost channel is amplified.

6

Conclusions

How should the central bank implement optimal monetary policy with the considerationof the robustness of the model? As suggested by Walsh (2017b) and Tillmann (2012), thecentral bank may refer to monetary policy rules that are robust to the misspecification of
the model because it faces the model uncertainty that the true model is not known withcertainty. This assertion justifies the significance of cross-checking optimal monetary
policy. These studies may address the eﬀectiveness of cross-checking optimal monetary
policy, whereas it is unclear whether such a rule can lead to a unique REE.

18

This paper has studied equilibrium determinacy under cross-checking optimal monetary policy with discretion. The main message of this paper is twofold. First, thispaper shows the condition that a unique REE is achieved under a cross-checking optimal
policy rule in the standard new Keynesian model. Interestingly, a decrease in a positive
weight on a cross-checking rule expands the determinacy regions. Surprisingly, when thecentral bank puts a considerably smaller weight on the information from the Taylor rule,
the REE is always determinate when it employs a stronger response to the output gap.
There are cases that the REE is always determinate even if the central bank sets thecoeﬃcient for inflation stabilization to zero in the Taylor rule. This result is the naturalextension of Tillmann (2012) and Walsh (2017b).
Second, this paper has extended the determinacy condition of cross-checking optimalmonetary policy rule in the case of the cost channel. The central bank, which employs
cross-checking optimal monetary policy rule, can achieve the unique REE so long as thedegree of a cost channel is small. Surprisingly, this result is overturned when a strong
cost channel is introduced into the economy, a cross-checking optimal rule fails to attainthe unique REE. In the case of the cost channel the REE is likely to be indeterminate
as the central bank puts on a higher weight on the stabilization of the output gap. This
result is more severe when the central bank puts a smaller weight on the cross-checking
term. The result of this paper implies that as a smaller weight on the RBP term leads
to an attenuated response to the inflation rate, such a policy results in equilibrium
indeterminacy even if the instrument rule satisfies the Taylor principle.
Finally, suggestions for future research are included. This paper has examined the
optimal monetary policy rule with cross-checking in a closed economy. We may consider
the role of cross-checking optimal monetary policy in a small-open economy (c.f., Clarida,
Gali, and Gertler , 2002; Gali and Monacelli, 2005). If the cross-checking rule contains therules, such as exchange rate targeting, how does the determinacy condition change (c.f.,
Linnemann and Schabart, 2006)? While the determinacy analyses under cross-checking
optimal monetary policy have been the focus, it is unclear whether the expectations of
the private sector, which can be regarded as the so-called E-stability condition, is also19

stable even when a cross-checking optimal policy rule leads to the unique REE (c.f.,
Evans and Honkaphoja, 2001; Evans and Honkapohja, 2003; Llosa and Tuesta, 2009).
Moreover, we may also apply the cross-checking monetary policy to several cases, suchas asset price fluctuations, interest rate smoothing, and so on. Does the cross-checking
optimal monetary policy always achieve the unique REE under such conditions? Does
the cross-checking optimal policy leads to the unique REE even in the medium-scale new
Keynesian model (Christiano, Eichenbaum and Evans, 2005; Smets and Wouters, 2007)?
These topics may be interesting subjects for future research.

A

Appendix

A.1

Appendix A: Proof of Proposition 1

The characteristic equation of polynomial is given by P (λ) = λ2 − tr(M )λ + det(M ),where
σ + κ + β(σ + ϕx + (1 − γ)γ −1 A−1 λ),σ + κϕπ + ϕx + (1 − γ)γ −1 A−1 (κ2 + λ)σβdet(M ) =,σ + κϕπ + ϕx + (1 − γ)γ −1 A−1 (κ2 + λ)tr(M ) =Both eigenvalues of M are inside the unit circle if both of the following conditions hold:(i) |det(M )| < 1,(ii) |tr(M )| < 1 + det(M ).
Condition (i) leads to the following inequality:
κϕπ + ϕx +1−γ 2
(κ + λ) + (1 − β)σ > 0,γA

which is trivially satisfied since β < 1. Condition (ii) implies equation (9).

20

A.2

Proof of Proposition 3

The characteristic equation of polynomial is given by P (λ) = λ2 − tr(M̃ )λ + det(M̃ ),where
σ[1 − κψr (ϕπ + (1 − γ)γ −1 Ã−1 κ)] + κ(1 + ψr )(ϕx + (1 − γ)γ −1 Ã−1 λ) + κ + σβ,
σ + κ(1 − ψr σ)(ϕπ + (1 − γ)γ −1 Ã−1 κ) + ϕx + (1 − γ)γ −1 Ã−1 λσβ,det(M̃ ) =σ + ϕx + κ(1 − ψr σ)ϕπ + (1 − γ)γ −1 Ã−1 [κ(1 − ψr σ) + λ]tr(M̃ ) =Both eigenvalues of M̃ are inside the unit circle if both of the following conditions hold:(i) |det(M̃ )| < 1,(ii) |tr(M̃ )| < 1 + det(M̃ ).
Condition (i) leads to equation (16). Since β < 1 and 0 < γ < 1, this condition is always
satisfied as long as 1/σ > ψr .
To check whether the condition (ii) is guaranteed, we must confirm it under thefollowing cases:(Case 1) σ + κ + σβ + (β + κψr )(ϕx + (1 − γ)γ −1 Ã−1 λ) > σκψr (ϕπ + (1 − γ)γ −1 Ã−1 κ)Under this inequality, the condition (ii) leads to equation (17).
(Case 2) σ + κ + σβ + (β + κψr )(ϕx + (1 − γ)γ −1 Ã−1 λ) < σκψr (ϕπ + (1 − γ)γ −1 Ã−1 κ)Under this inequality, the condition (ii) leads to equation (18), which is always satisfied
as long as 1/2σ > ψr .

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23

Figure 1: Determinacy and cross-checking monetary policy

Note: Dark-shaded areas: determinacy regions; Light-gray areas: indeterminacy regions

24

Figure 2: Determinacy and cross-checking monetary policy in the case of a cost channel

Note: Dark-shaded areas: determinacy regions; Light-gray areas: indeterminacy regions

25

Figure 3: Determinacy and cross-checking monetary policy in the case of the cost channel:
A change of ϕx when γ = 0.1

Note: Dark-shaded areas: determinacy regions; Light-gray areas: indeterminacy regions

26

Figure 4: Determinacy and cross-checking monetary policy in the case of the cost channel:
A change of ϕx when γ = 0.5

Note: Dark-shaded areas: determinacy regions; Light-gray areas: indeterminacy regions

27

Figure 5: Determinacy and cross-checking monetary policy in the case of the cost channel:
Forward-looking Taylor rule when γ = 0.1

Note: Dark-shaded areas: determinacy regions; Light-gray areas: indeterminacy regions

28

Figure 6: Determinacy and cross-checking monetary policy in the case of the cost channel:
Forward-looking Taylor rule when γ = 0.5

Note: Dark-shaded areas: determinacy regions; Light-gray areas: indeterminacy regions29