Research InstituteDiscussion Paper Series№02
Title：On the determinacy condition in the new Keynesian model:
The role of deposit rate pass-throughIda Daisuke／ 井田

大輔

ida-dai@andrew.ac.jp

2015 年 12 月

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

On the determinacy condition in the new Keynesian model:
The role of deposit rate pass-through☆
Daisuke Ida*
December 2, 2015

Abstract
This paper studies the role of deposit rate pass-through in the new Keynesian model with a costchannel. It shows that the degree of deposit rate pass-through crucially changes the determinacy
regions. Incompleteness of deposit rate pass-through narrows the determinacy regions. On the other
hand, a higher pass-through of the deposit rate expands the determinacy regions. In particular,
compared to a current-looking policy rule, a forward-looking policy rule predominately shrinks theregions where the rational expectations equilibrium (REE) is determinate. Thus, this paper addresses
the importance how the central bank conducts its monetary policy when the degree of deposit ratepass-through significantly affects the relative dominance of the demand channel of monetary policy
over the cost channel.

JEL classifications: E52, E58
Keywords: Determinacy; Monetary policy rule; Deposit rate pass-through; Cost channel☆

I would like to thank Kohei Hasui for his helpful suggestions. All remaining errors are my own.

*Department of Economics at Momoyama Gakuin University, 1-1, Manabino, Izumi, Osaka 599-8251, Japan. Tel.:+81-725-54-3131. E-mail: ida-dai@andrew.ac.jp

11. Introduction
The role of financial frictions in the new Keynesian model has been emphasised in the literature(Bernanke, Gertler and Gilchrist, 1999; Ravenna and Walsh, 2006; Chowdhury, Hoffmann andSchabert, 2006). Chowdhury, Hoffmann and Schabert (2006) focus on the degree of financial market
imperfection that captures the impact of a monetary policy of the cost channel on inflation. A costchannel is a mechanism wherein an increase in the nominal interest rate induces a rise in inflation
through an increase in firms’ working capital (Ravenna and Walsh, 2006).
Financial market imperfections, which are generated by the presence of a cost channel, are also
related to the degree of pass-through between the lending and policy rates. Chowdhury, Hoffmann
and Schabert (2006) argue that an imperfect lending rate pass-through is observed in developed
countries. A change in the policy rate has the supply side of monetary policy in a model with a costchannel (Choudhury, Hoffmann, and Schabert, 2006; Surico, 2008; Llosa and Tuesta, 2009).
However, these studies assume that the deposit rate pass-through is complete. Indeed, Kwapil and
Schaler (2010) stress that considering the incompleteness of deposit rate pass-through is important.1
Why is pass-through of the deposit rate important in a sticky price model where the cost channel
matters?
In the standard new Keynesian model without cost channel, the Taylor principle, which meansthat the central bank should react more aggressively to inflation, is required to attain a unique REE(Bullard and Mitra, 2002). As long as the central bank conducts its monetary policy by following the
Taylor principle, the demand channel of monetary policy stabilises the expectations of inflation as
follows. A sufficient increase in the policy rate in response to unanticipated inflation results in an
increase in the real interest rate, which leads to a decline in the output gap through the dynamic IS
equation. Hence, actual inflation declines through the new Keynesian Phillips curve (NKPC).
However, this mechanism is not applicable to the cost channel case. In particular, an increase in
the policy rate induces an increase in the inflation rate when the supply side of monetary policy
dominates the demand channel. In this case, the satisfaction of the Taylor principle may not sufficeto achieve a unique REE. Consequently, the expectations of inflation are self-fulfilling becausemonetary tightening results in a decline in the real interest rate. Thus, whether the real interest rate
declines in response to a rise in the policy rate crucially depends on the relative strength of thedemand channel to the cost channel.
The reason that pass-through of the deposit rate is important is as follows. As I will show,whether the cost channel dominates the demand channel significantly depends on the degree ofdeposit rate pass-through. A low pass-through of the deposit rate weakens the demand channel.

1

See also De Bondt and Mojon (2005), Scharler (2008) and Ida (2014) for a detailed explanation of thepass-through of retail rates.

2

Therefore, where pass-through of the deposit rate is predominately low, the central bank may notachieve a unique REE even if it satisfies the Taylor principle. This is because the real interest rate
declines regardless of whether the central bank reacts more than one-for-one to an increase in the
inflation rate. Accordingly, as the supply side of monetary policy works in the presence of the costchannel, sunspot equilibria are likely to occur regardless of monetary tightening. On the other hand,when deposit rate pass-through is considerably high, the central bank can stabilise a
non-fundamental shock even if it does not satisfy the Taylor principle. Indeed, Kwapil and Schaler(2010) report several estimated values of the degree of deposit rate pass-through.
This paper is related to the study of Llosa and Tuesta (2009). They show that the cost channel
induces equilibrium indeterminacy and expectational instability under adaptive learning in thestandard new Keynesian model. Their study assumes, however, that both lending and deposit rates
experience complete pass-through. Kwapil and Schaler (2010) emphasize the importance of deposit
rate pass-through, and also study whether the deposit rate pass-through affects the determinacy
condition in the new Keynesian model. However, Kwapil and Schaler (2010) abstract the role offinancial intermediaries. A notable contribution of this paper is to show that pass-through of thedeposit rate is crucially important in a sticky price model with financial market imperfections.
The main findings of this paper are summarised as follows. First, imperfect deposit ratepass-through shrinks the determinacy region under a contemporaneous policy rule. In such a case,the central bank cannot retain equilibrium determinacy even if it satisfies the Taylor principle. On
the other hand, the determinacy region expands when the increase in the deposit rate is more than theincrease in the policy rate. In this case, it is possible that the central bank can achieve determinate
equilibrium even if the Taylor principle is not guaranteed. Second, in contrast to the case of acontemporaneous rule, employing a forward-looking policy rule shrinks the determinacy region. In
particular, this problem is more severe in cases where an incomplete deposit rate pass-through ispresent. Third, in the case where the central bank follows a current-looking policy rule, a higher
pass-through of the deposit rate can reinforce the demand channel even if financial market
imperfections are considerably severe. When the central bank employs a forward-looking policy rule,
however, a higher pass-through of the deposit rate cannot strengthen the demand channel if financial
market imperfections are considerably severe.
The present study is designed as follows. Section 2 provides the model description. Section 3
derives the determinacy condition in a new Keynesian model in which the pass-through of both
lending and deposit rates is incomplete. Section 4 briefly concludes.

2. Model Description
With the exception of the assumption that both lending and deposit rates face incomplete
3

pass-through, the model description in this paper is a standard new Keynesian model2.
The first equation is the NKPC, which is derived from the optimal condition of firms that are
subject to monopolistic competition and Calvo pricing. In a model with a cost channel, firms borrow
the working capital from financial intermediaries to finance the wage bill. As shown in Ravenna andWalsh (2006) and Chowdhury et al. (2006), in this case, the term for the nominal interest rate isaugmented in the NKPC, which is given t  Et t 1  xt   r Rt ,(1)

where κ = λ(σ + η). 𝑥𝑡 is the output gap, 𝜋𝑡 is the inflation rate and 𝑅𝑡 is the nominal interest
rate set by the central bank. The parameter κ is the slope of the NKPC and λ is the sensitivity of
inflation to the real marginal cost. η is the inverse of the elasticity of labor supply and β denotes
the discount factor. 𝐸𝑡 is the expectations operator. 𝜓𝑅 represents the degree of a cost channel.3
The parameter 𝜓𝑅 is associated with asymmetrical information in the financial market (Chowdhury,
Hoffmann and Schabert, 2006). Hence, a higher value of 𝜓𝑅 implies that financial market
imperfections are severe.
The second equation is a dynamic IS curve, which is derived from the inter-temporal optimal
conditions from households. It is given byxt  Et xt 1   1 ( RtD  Et t 1 )  rt n ,(2)where 𝑅𝑡𝐷 is the deposit rate. The parameter σ is the relative risk aversion coefficient for
consumption and 𝑟𝑡𝑛 denotes the natural rate of interest. Following Kwapil and Schalar (2010), thedeposit rate evolves as follows:

RtD   D Rt ,(3)

where 𝜓𝐷 denotes the pass-through of the deposit rate.4 A key parameter in this paper is the degreeof deposit rate pass-through 𝜓𝐷 . The pass-through is incomplete when 𝜓𝐷 ≤ 1. A smaller value of𝜓𝐷 dampens the demand channel of monetary policy. Also, the deposit rate increases more than anincrease in the policy rate when 𝜓𝐷 > 1. In this case, a higher pass-through of the deposit rate
reinforces the demand channel.
Finally, to close the model, the model requires a monetary policy rule, which is specified asfollows:

Rt   Et t k  x Et xt k , for k = 0,1.

(4)

2

See Woodford (2003) for a detailed derivation of the basic new Keynesian model.
See Chowdhury et al. (2006) for a detailed discussion of the presence of the parameter 𝜓𝑅 in the NKPC.
4 In contrast to their model, this paper abstracts the presence of the lagged deposit rate because this paper focuses on
the effect of immediate pass-through of the deposit rate on equilibrium determinacy.
34In this rule, k = 0 corresponds to a contemporaneous policy rule, whereas k = 1 implies aforward-looking policy rule. 𝜙𝜋 is the coefficient for inflation stabilisation and 𝜙𝑥 is thecoefficient for the stabilisation for the output gap.

3. Determinacy Analysis
This section provides how the deposit rate pass-through is important in that the REE is uniquelydeterminate in an economy with a cost channel. Section 3.1 examines the determinacy condition
under a contemporaneous monetary policy rule. In Section 3.2, I consider the determinacy condition
under a forward-looking policy rule.

3.1. A current-looking monetary policy rule
First, I consider the determinacy condition when the central bank follows a current-looking monetary
policy rule. To examine how the degree of deposit rate pass-through affects determinacy of the REE,
I construct the system that consists of two endogenous variables 𝑥𝑡 and 𝜋𝑡 .
Substituting a contemporaneous rule into Equations (1) and (2), the system of two endogenous
variables 𝑥𝑡 and 𝜋𝑡 is written as follows:

X t  M Et X t 1   rt n ,(5)where 𝑋𝑡 = [𝑥𝑡 , 𝜋𝑡 ]′ andM

1   R   D   (1   R )1.
(  D x )(1   R )  D (   R x )  (   R x )    R x   (  D x )

(6)

Note that the form of μ is ignored because it does not affect the following discussion. Since 𝑥𝑡 and𝜋𝑡 are endogenous jump variables, both of the eigenvalues of M should be inside the unit circle fordeterminacy.5 Thus, we have the following proposition:

Proposition 1. Under the contemporaneous policy rule, the necessary and sufficient conditions for aunique REE are given as follows: Dx  ( D   R )  (1   )  0(7)

 Dx  (1   )  ( R   D )
 ( D  1)  [(1   ) D   R ]x  0

(8)
(9)

 ( D  2 R  )  [(1   ) D   R ]x  2 (1   )    0 .

(10)

1

5

See Bullard and Mitra (2002) for a detailed discussion of this condition.

5

Proof. See Appendix A. ∎As shown in Llosa and Tuesta (2009), when 𝜓𝐷 = 𝜓𝑅 = 1, the necessary and sufficient conditions
for a unique REE need to satisfy Equations (9) and (10). Equation (9) is regarded as the modified
Taylor principle. The Conditions (7) and (8) are redundant as long as the pass-through of the lending
and deposit rates is complete. However, this paper underlines that unless 𝜓𝐷 = 𝜓𝑅 = 1, Conditions(7)–(10) should guarantee a unique REE. In particular, equation (7) implies that
 D   R .


(11)

As mentioned earlier, the higher value of 𝜓𝑅 amplifies the cost channel. In addition, a higher valueof 𝜓𝐷 reinforces the demand channel that implies the effect of the policy rate on the output gap
through a change in the real interest rate. Monetary tightening leads to a decline in inflation when the
demand channel dominates the cost channel. Given the value of σλ/κ, Condition (11) requires alower bound for the parameter 𝜓𝐷 to guarantee the unique REE if financial market imperfections
are present. To obtain an intuition of the above conditions, I numerically show determinacy andindeterminacy regions under a contemporaneous monetary policy rule.6
Figure 1 shows how determinacy regions are altered when the parameters 𝜙𝜋 and 𝜙𝑥 change.
Figure 1 (a) is the determinacy region in the case of 𝜓𝐷 = 1.0. This case corresponds to the result of
Llosa and Tuesta (2009). In contrast to the standard new Keynesian model, as pointed out by Surico(2008) and Llosa and Tuesta (2009), as long as the central bank does not aggressively raise thepolicy rate in response to a change in inflation, a larger response to the output gap results inindeterminacy. According to Surico (2008), the reason that a rule that contains a larger response to
the output gap leads to indeterminacy is that a larger value of 𝜙𝑥 produces a situation that the costchannel dominates the demand channel. In that case, therefore, inflation expectations become
self-fulfilling.

[Figure 1 around here]
Figure 1(b) depicts determinacy and indeterminacy regions in the case of 𝜓𝐷 = 0.5. In this case,importantly, the central bank cannot achieve a unique REE even if it satisfies the Taylor principle. In
addition, the figure reveals that the central bank should respond more aggressively to inflation as it
employs a larger value of the parameter 𝜙𝑥 . Intuitively, a smaller value of the parameter 𝜓𝐷
counteracts the demand channel of monetary policy. In addition, a larger value of 𝜙𝑥 makes the6

The parameter values used in this simulation are as follows. Based on the existing literature, the calibration isstandard. β is set to 0.99. The parameter σ is set to 2.0. λ and η are set to 0.08 and 1.0, respectively. 𝜙𝑥 is set to0.125.

6demand channel weaker. Hence, monetary tightening in response to a sunspot shock induces adecline in the real interest rate, which may make the expectations of inflation self-fulfilling. In sucha case, the central bank should react more aggressively to a change in inflation. Another way tostabilise a sunspot shock is for the central bank to employ a smaller response to the output gap.
On the other hand, in the case of 𝜓𝐷 = 1.5, the central bank can attain a unique REE even if it
does not react more than one-for-one to a change in the inflation rate. Indeed, even if 𝜙𝜋 is lessthan unity, Equation (9), which is the modified Taylor principle, is satisfied as long as 𝜓𝐷 𝜙𝜋 isabove unity. Intuitively, due to a higher pass-through of the deposit rate, the real interest rate may
increase even if the central bank does not satisfy the Taylor principle. Such an increase in the realinterest rate prevents self-fulfilling movements of inflation.

[Figure 2 around here]

Figure 2 plots the determinacy and indeterminacy regions in the case of a contemporaneous policy
rule. Compared with the case of 𝜓𝐷 = 1, the incomplete deposit rate pass-through shrinks theregion where the REE is determinate. Surprisingly, even when financial market imperfections are notsevere, the central bank cannot achieve the unique REE unless 𝜙𝜋 ≥ 2. Intuitively, when the deposit
rate pass-through is low, the central bank that employs a weak response to inflation cannot achieve
the unique REE even if financial market imperfections are not severe. This is because low deposit
rate pass-through simply counteracts the demand channel of monetary policy. On the contrary, thecase of 𝜓𝐷 > 1 expands the determinacy regions unless financial market imperfections are
predominately severe. The simple reason is that a larger value of 𝜓𝐷 strengthens the demand
channel.
Figure 3 portrays determinacy and indeterminacy regions under a contemporaneous monetary
policy rule when the parameters 𝜓𝑅 and 𝜓𝐷 change. When the parameter 𝜓𝑅 is less than 1.5, theREE is determinate as long as the parameter 𝜓𝐷 is above than 0.6. As the parameter 𝜓𝑅 takes ahigher value than 1.5, indeterminacy regions expand if the parameter 𝜓𝐷 is smaller values. If
pass-through of the deposit rate is above 1.5, however, the REE is determinate even if the costchannel parameter takes a higher value7. Thus, in the case where the central bank employs acurrent-looking policy rule, a higher pass-through of the deposit rate can reinforce the demand
channel even if financial market imperfections are considerably severe.

[Figure 3 around here]

7

This value is considerably high. In the standard new Keynesian model with a cost channel, the price puzzle, which
is a situation where a rise in the policy rate increases the inflation rate, emerges when the parameter 𝜓𝑅 is above 1.5.
See Castelnuovo (2007) for a detailed discussion of this issue.

73.2. A forward-looking monetary policy rule
Next, this section studies determinacy and indeterminacy analysis under a forward-looking policy
rule. In a forward-looking policy rule, the matrix M is modified as follows:M  D x
1  D
1.
  (  D x )   R x  (1  D )   (    R )

(11)

In this case the necessary and sufficient conditions for a unique REE are given as follows:

Proposition 2. Under the forward-looking policy rule the necessary and sufficient conditions for aunique REE are given as follows: R  ( R   D )x   (1   )( R   D )x   R   (1   ) ( D  1)  [(1   ) D   R ]x  0(14) ( D   R 1 )  [(1   ) D   R ] x  2 (1   )   .

(15)

(12)
(13)

Proof. See Appendix B. ∎Equations (12) to (15) are the modified version of the conditions derived Llosa and Tuesta (2009).
Thus, when 𝜓𝐷 = 𝜓𝑅 = 1, these conditions reduce to those of Llosa and Tuesta (2009). Equation(12) shows that the response to inflation is bounded from above. Equation (14) is again the modified
Taylor principle. In contrast to Llosa and Tuesta (2009), in this model these conditions are affectedby the parameter 𝜓𝐷 . To intuitively understand this proposition, I again use numerical methods.
Figure 4 illustrates determinacy and indeterminacy regions under a forward-looking monetary
policy rule. First, consider the determinacy regions shown in Figure 4(a). This case corresponds to
Llosa and Tuesta (2009). As in the prescription derived in the standard new Keynesian model,
inflation reaction is bounded from above (Bullard and Mitra, 2002). As shown in Equation (13), theresponse to the output gap is also bounded from below. A larger value of the parameter 𝜙𝑥 leads toindeterminacy, even if the Taylor principle is satisfied.
As shown in Figure 4(b), the determinacy condition is severe in the case of incomplete
pass-through of the deposit rate. This case is more severe than the case of complete pass-through.
Thus, while the central bank faces the upper bound for 𝜙𝜋 and the lower bound for 𝜙𝑥 , it must
consider a larger response to both inflation and the output gap to achieve a unique REE. For instance,
for any value of the parameter 𝜙𝜋 , a REE is never determinate when the parameter 𝜙𝑥 is less thanabout 0.6. This is because the parameter 𝜙𝑥 violates a lower bound for it. Moreover, even if the8parameter 𝜙𝑥 is above 0.6, the REE is not determinate unless the central bank responds moreaggressively to the inflation rate.

[Figure 4 around here]
On the other hand, the determinacy regions expand in the case of 𝜓𝐷 = 1.5. Interestingly, in thiscase, the parameter 𝜙𝑥 faces both upper and lower bounds. In particular, the REE is always
determinate if the parameter 𝜙𝑥 is set by the central bank within such bounds. Thus, as long as theparameter 𝜙𝑥 satisfies lower bounds, the central bank can lead the economy to the determinacy
regions even if 𝜙𝜋 ≤ 1. Figure 5 shows the determinacy regions in the case of a forward-looking
policy rule. Compared with Figure 2, the determinacy regions shrinks in the case of aforward-looking rule. This problem is severer when 𝜓𝐷 = 0.5.
[Figure 5 around here][Figure 6 around here]

Figure 6 displays determinacy regions under a forward-looking monetary policy rule. In contrast to
the case of a contemporaneous rule, a forward-looking policy rule shrinks determinacy regions
further. As long as 𝜓𝑅 is less than 0.5, the REE is determinate if the parameter 𝜓𝐷 is above 0.5.
However, regions where the REE is indeterminate expand as the parameter 𝜓𝑅 is above 0.5. In
particular, the REE is never determinate in the case of 𝜓𝑅 ≥ 1.5. When the central bank employs aforward-looking policy rule, a higher pass-through of the deposit rate cannot enhance the demand
channel if financial market imperfections are considerably severe.

4. Concluding Remarks
This paper has shown the role of deposit rate pass-through in the new Keynesian model using asimple cost channel. The main findings of this paper are summarised as follows.
The main findings of this paper are summarised as follows. First, imperfect deposit ratepass-through shrinks the determinacy region under a contemporaneous policy rule. In such a case,the central bank cannot retain equilibrium determinacy even if it satisfies the Taylor principle. On
the other hand, the determinacy region expands when the increase in the deposit rate is more than theincrease in the policy rate. In this case, it is possible that the central bank can achieve determinate
equilibrium even if the Taylor principle is not guaranteed. Second, in contrast to the case of acontemporaneous rule, employing a forward-looking policy rule shrinks the determinacy region. In
particular, this problem is more severe in cases where an incomplete deposit rate pass-through is9present. Third, in the case where the central bank follows a current-looking policy rule, a higher
pass-through of the deposit rate can reinforce the demand channel even if financial market
imperfections are considerably severe. Furthermore, when the central bank employs aforward-looking policy rule, a higher pass-through of the deposit rate cannot strengthen the demand
channel if financial market imperfections are considerably severe.
Thus, this paper underlines the importance how the central bank conducts its monetary policy
when the degree of deposit rate pass-through significantly affects the relative dominance of thedemand channel of monetary policy over the cost channel.

Appendix A: Proof of Proposition 1
The characteristic polynomial of M is given by p(μ) = 𝜇2 − 𝑡𝑟(𝑀)𝜇 + det(𝑀) wheretr ( M )  (1   R )     R x   (   D x ),(   D x )(1   R )   D (   R x )det(M ) .
(   D x )(1   R )   D (   R x )Both eigenvalues of M are inside the unit circle if both of the following conditions hold:(i) |det(𝑀)| < 1,(A.1)(ii) |𝑡𝑟(𝑀)| < 1 + det(𝑀).

(A.2)

The first condition implies Equations (7) and (8) and the second equation implies Equations (9) and(10).

Appendix B: Proof of Proposition 2
The characteristic polynomial of M is given by p(μ) = 𝜇2 − 𝑡𝑟(𝑀)𝜇 + det(𝑀) where (1   )   R  ( D  x D )  ,   D x   R   R x.
det(M ) tr ( M ) 

The first condition (A.1) implies equations (12) and (13); on the other hand, the condition (A.2)
implies equations (14) and (15).

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11Figure 1: Determinacy and indeterminacy under a contemporaneous interest rate rule(a) 𝜓𝐷 = 1.0 (Standard new Keynesian model with a cost channel)(b) 𝜓𝐷 = 0.5 (Incomplete pass-through of deposit rates)(c) 𝜓𝐷 = 1.5 (Higher pass-through of deposit rates)

12Figure 2. Determinacy and indeterminacy under a contemporaneous interest rate rule(a) 𝜓𝐷 = 1.0(b) 𝜓𝐷 = 0.5(c) 𝜓𝐷 = 1.5

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Figure 3: Determinacy and indeterminacy when the parameters 𝜓𝑅 and 𝜓𝐷 change: The case of acontemporaneous rule14

Figure 4: Determinacy and indeterminacy under a forward-looking interest rate rule(a) 𝜓𝐷 = 1.0 (Standard new Keynesian model with a cost channel)(b) 𝜓𝐷 = 0.5 (Incomplete pass-through of deposit rates)(c) 𝜓𝐷 = 1.5 (Higher pass-through of deposit rates)

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Figure 5. Determinacy and indeterminacy under a forward-looking interest rate rule(a) 𝜓𝐷 = 1.0(b) 𝜓𝐷 = 0.5(c) 𝜓𝐷 = 1.5

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Figure 6: Determinacy and indeterminacy when the parameters 𝜓𝑅 and 𝜓𝐷 change: The case of aforward-looking rule17