trend inflation, sticky prices, and expectational stability
TRANSCRIPT
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Journal of Economic Dynamics & Control
Journal of Economic Dynamics & Control 42 (2014) 175–187
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journal homepage: www.elsevier.com/locate/jedc
Trend inflation, sticky prices, and expectational stability
Takushi Kurozumi n
Bank of Japan, 2-1-1 Nihonbashi Hongokucho, Chuo-ku, Tokyo103-8660, Japan
a r t i c l e i n f o
Article history:Received 14 August 2010Received in revised form23 June 2013Accepted 30 March 2014Available online 4 April 2014
JEL classification:E31E52
Keywords:Trend inflationSticky priceIndeterminacyExpectational stabilityLeast-squares learnability
x.doi.org/10.1016/j.jedc.2014.04.00189/& 2014 Elsevier B.V. All rights reserved.
þ81 3 3279 1111.ail address: [email protected] Ascari and Sbordone (2013) for a review or recent micro evidence on price changes, snother paper of Ascari and Ropele (2007) exry authority seeks to minimize a loss functio
ley (2007) obtains a similar result of equilib
a b s t r a c t
Micro evidence indicates that each period a fraction of prices is kept unchanged under apositive trend inflation rate. In a sticky price model based on this evidence, recentresearch shows that high trend inflation is a serious cause for indeterminacy of rationalexpectations equilibrium under the Taylor rule. This paper examines implications of trendinflation for expectational stability of the equilibrium. An empirically plausible calibrationof the model demonstrates that a fundamental rational expectations equilibrium islikely to be expectationally stable even in cases of indeterminacy induced by high trendinflation.
& 2014 Elsevier B.V. All rights reserved.
1. Introduction
Recent studies have investigated implications of trend inflation for sticky price models.1 Ascari (2004),Bakhshi et al. (2007), and Cogley and Sbordone (2008), for instance, derive a generalized New Keynesian Phillips curvefrom a Calvo (1983)-style sticky price model on the basis of micro evidence that each period a fraction of prices is keptunchanged under a positive trend inflation rate.2 These studies find that higher trend inflation reduces the slope of thegeneralized New Keynesian Phillips curve. Focusing on this finding, Ascari and Ropele (2009) analyze its implications fordeterminacy of equilibrium under the Taylor (1993) rule.3 Their main conclusion is that high trend inflation is a seriouscause for indeterminacy.4
This paper examines implications of trend inflation for expectational stability (or E-stability) of rational expectationsequilibrium (REE) under the Taylor rule in a Calvo-style sticky price model based on the micro evidence. As McCallum(2007) indicates, E-stability is very closely linked with least-squares learnability (i.e., stability under least-squares learning),
f this strand of literature.ee, e.g., Bils and Klenow (2004), Klenow and Kryvtsov (2008), and Nakamura and Steinsson (2008).amines determinacy of equilibrium under an inflation-targeting monetary policy regime in which an given by a weighted sum of expected squared deviations of inflation and output from their trend
rium indeterminacy in a Taylor (1980)-style sticky price model.
T. Kurozumi / Journal of Economic Dynamics & Control 42 (2014) 175–187176
and this learnability is arguably a necessary property for an REE to be plausible as an equilibrium for the model at hand.In a broad class of linear models with expectations (including the log-linearized model of the present paper), a non-explosive fundamental REE is least-squares learnable if it is E-stable; otherwise, it is not least-squares learnable (Evans andHonkapohja, 2001).5 Therefore, E-stability is an essential condition for an REE to be regarded as plausible.
In the paper an empirically plausible calibration of the model demonstrates that a fundamental REE is likely to beE-stable even in cases of indeterminacy induced by high trend inflation.6 The model and the calibration of model parametersare based on Ascari and Ropele (2009), and high trend inflation then induces indeterminacy of REE in line with their resultand generates at least one E-stable fundamental REE within a sufficiently wide range of the Taylor rule's coefficients.This E-stability is obtained because a finite elasticity of labor supply causes price distortion to affect inflation dynamicsrepresented by a generalized New Keynesian Phillips curve, and the persistence of price distortion then generatesendogenously persistent inflation dynamics. For the REE in question, E-stability examines whether an associatedequilibrium in which agents form expectations under adaptive learning reaches over time the REE. Under such expectationformation, the endogenous persistence of inflation dynamics through price distortion helps agents form inflationexpectations. Consequently, a fundamental REE is likely to be E-stable.7 Moreover, a higher probability of price adjustment,a lower price elasticity of demand for differentiated goods, and a lower elasticity of labor supply are all more likely to ensureE-stability of a non-explosive fundamental REE. With a higher probability of price adjustment or a lower price elasticity ofdemand, REE is more likely to be determinate and E-stable, since a key condition for determinacy and E-stability is morelikely to be satisfied. This condition is the long-run version of the Taylor principle: in the long run the interest rate should beraised by more than the increase in inflation. By contrast, a lower elasticity of labor supply makes this Taylor principle lesslikely to be met and hence leaves REE indeterminate or explosive, while it generates at least one E-stable fundamental REEin cases of indeterminacy induced by high trend inflation. This is because it strengthens the influence of price distortion oninflation dynamics and hence the endogenous persistence of inflation dynamics through price distortion, which furtherhelps agents form inflation expectations under adaptive learning. Therefore, a lower elasticity of labor supply makes a non-explosive fundamental REE more likely to be E-stable.
In the literature, the most closely related studies have been done by Bullard and Mitra (2002) and Kobayashi and Muto(2013). In the case of the zero trend inflation rate, Bullard and Mitra analyze E-stability of fundamental REE under fourspecifications of the Taylor rule, i.e., forward expectations, contemporaneous expectations, and lagged data, in addition tocontemporaneous data examined in the present paper. They show that a non-explosive fundamental REE is likely to beE-stable under all the four specifications if the long-run version of the Taylor principle is satisfied. Like Bullard and Mitra, thepresent paper shows that for low trend inflation, the long-run version of the Taylor principle is the relevant, necessary andsufficient condition for E-stability.8 For high trend inflation, however, it is not necessarily an E-stability condition. Indeed,the paper finds a range of the Taylor rule's coefficients that do not satisfy the long-run version of the Taylor principle butgenerate at least one E-stable fundamental REE. Kobayashi and Muto (2013) use a Calvo-style sticky price model basedon Sbordone (2007) and Cogley and Sbordone (2008) and investigate implications of trend inflation for E-stability offundamental REE under the four specifications of the Taylor rule. The analysis of Kobayashi and Muto is complementary tothe present paper in that they examine the other three specifications of the Taylor rule as well as the contemporaneous-dataone but their model follows Sbordone (2007) to assume that real marginal cost does not reflect cost arising from pricedistortion.9 Due to this assumption, price distortion never affects inflation dynamics represented by their generalized NewKeynesian Phillips curve. Consequently, Kobayashi and Muto reach the conclusion that when trend inflation is high,all fundamental REE are likely to be E-unstable under each of the four specifications of the Taylor rule.
The remainder of the paper proceeds as follows. Section 2 presents a Calvo-style sticky price model with a non-negativetrend inflation rate. In this model, Section 3 analyzes implications of trend inflation for E-stability of fundamental REE.Section 4 concludes.
2. The sticky price model with a non-negative trend inflation rate
The model is a Calvo-style sticky price model based on Ascari and Ropele (2009). The economy consists of a central bank,a representative household, a representative final-good firm, and a continuum of intermediate-good firms. In each period afraction αAð0;1Þ of intermediate-good firms keeps prices of their differentiated products unchanged. The trend inflation
5 Throughout the paper, the term “ fundamental” refers to Evans and Honkapohja's (2001) minimal-state-variable (MSV) solutions to linear rationalexpectations models to distinguish them from McCallum's (1983) original MSV solution.
6 This paper does not examine E-stability of non-fundamental REE (e.g., sunspot equilibrium), which exists in cases of indeterminacy. For E-stabilityanalysis of non-fundamental REE, see, e.g., Honkapohja and Mitra (2004), Carlstrom and Fuerst (2004), and Evans and McGough (2005). There are severalpossible reactions to the result of the present paper. One reaction would be that indeterminacy induced by high trend inflation is not important because afundamental REE is likely to be E-stable. Another would be that there may exist an E-stable non-fundamental REE in addition to the E-stable fundamentalREE. The result of the paper thus indicates that E-stability of non-fundamental REE should be investigated in future research.
7 When the elasticity of labor supply is infinity, high trend inflation is a serious cause for not only indeterminacy but also E-instability.8 The long-run version of the Taylor principle is a key necessary condition for determinacy of equilibrium, regardless of the level of trend inflation, as
shown by Ascari and Ropele (2009) and Kurozumi (2011).9 This assumption holds for the model of the present paper only in the empirically implausible case of an infinite elasticity of labor supply.
Table 1Benchmark calibration of parameters for the quarterly model.
β Subjective discount factor 0.99α Probability of no price adjustment 0.75θ Price elasticity of demand for differentiated goods 11sn Inverse of elasticity of labor supply 1ρ Shock persistence 0.8
T. Kurozumi / Journal of Economic Dynamics & Control 42 (2014) 175–187 177
rate of the final-good price is assumed to be non-negative, i.e., ΠZ1, where Π denotes the gross trend inflation rate.Moreover, the following assumption is imposed so that intermediate-good firms' profit functions are well-defined.
Assumption 1. The inequality αΠθo1 holds, where θ41 is the price elasticity of demand for differentiated goods.
2.1. Log-linearized equilibrium conditions
Under Assumption 1, log-linearizing equilibrium conditions of the model and rearranging the resulting equations lead to
ıt ¼ ϕΠΠ tþϕY Y t ; ð1Þ
Y t ¼ EtY tþ1�ðıt�EtΠ tþ1Þ; ð2Þ
Π t�αβΠθEtΠ tþ1 ¼ βðEtΠ tþ1�αβΠ
θEtΠ tþ2ÞþβθðΠ�1Þð1�αΠθ�1ÞEtΠ tþ1
þλ½ð1þsnÞðY t�αβΠθ�1EtY tþ1Þþsnðst�αβΠ
θ�1Et stþ1Þ�þut�αβΠθ�1Etutþ1; ð3Þ
st ¼αθΠ
θ�1ðΠ�1Þ1�αΠ
θ�1 Π tþαΠθ st�1; ð4Þ
where all hatted variables denote the log-deviations from trend levels, Et is the rational expectation (RE) operatorconditional on information available in period t, and λ¼ ð1�αΠ
θ�1Þð1�αβΠθÞ=ðαΠθ�1Þ.
Eq. (1) is the Taylor rule, where ϕΠ ;ϕY Z0 represent the degrees of monetary policy responses of the interest rate ıt tothe inflation rate Π t and output Y t . Eq. (2) is obtained from an Euler equation for optimal spending decisions of therepresentative household whose consumption preferences are represented by a log utility function, along with a final-goodmarket clearing condition. Combining intermediate-good firms' Calvo-style staggered price-setting equation and a final-good price equation yields Eq. (3), where st denotes price distortion arising from the staggered price setting and evolvesaccording to Eq. (4), βAð0;1Þ is the subjective discount factor, snZ0 is the inverse of the elasticity of labor supply, and ut isan exogenous stochastic disturbance related to the household's labor disutility shock and follows a first-order autoregressiveprocess with the persistence parameter ρAð�1;1Þ.
Eq. (3) is a generalized New Keynesian Phillips curve, since under the zero trend inflation rate (i.e., Π ¼ 1), Eq. (4)becomes st ¼ 0 and hence Eq. (3) is rewritten as
Π t�αβEtΠ tþ1 ¼ βðEtΠ tþ1�αβEtΠ tþ2Þþλð1þsnÞðY t�αβEtY tþ1Þþut�αβEtutþ1;
which can be reduced to
Π t ¼ βEtΠ tþ1þλð1þsnÞY tþut :
2.2. The benchmark calibration of model parameters
The ensuing analysis uses an empirically plausible calibration of model parameters to illustrate conditions for E-stability.The benchmark calibration for the quarterly model is summarized in Table 1.10 In line with Ascari and Ropele (2009), thepresent paper sets the subjective discount factor at β¼ 0:99, the probability of no price adjustment at α¼ 0:75, the priceelasticity of demand for differentiated intermediate goods at θ¼ 11, and the inverse of the elasticity of labor supply atsn ¼ 1. The shock persistence is set at ρ¼ 0:8 as in Woodford (2003). Note that to meet Assumption 1 under the benchmarkcalibration, the annualized trend inflation rate must not exceed 11%.
10 The robustness exercises presented later demonstrate that the result obtained with the benchmark calibration of model parameters still hold foralternative, empirically plausible calibrations.
T. Kurozumi / Journal of Economic Dynamics & Control 42 (2014) 175–187178
3. The analysis of expectational stability
In the log-linearized model presented above, this section examines implications of trend inflation for E-stability offundamental REE.
3.1. The methodology
Following the literature on learning in macroeconomics (e.g., Evans and Honkapohja, 2001), the present paper uses theso-called “ Euler equation” approach suggested by Honkapohja et al. (2011): the RE operator Et is replaced with a possiblynon-RE operator E t in the system of Eqs. (1)–(4). Then, the system can be rewritten as
zt ¼ AzEtztþ1þBz½1 0 0�E tztþ2þCzst�1þDzut ; ð5Þ
where zt ¼ ½Π t Y t st �0 and the coefficient matrix Az and the coefficient vectors Bz, Cz are given in Appendix A.11 In thissystem, fundamental REE are given by
zt ¼ czþΦzst�1þΓ zut : ð6Þ
Here, the coefficient vectors cz, Φz, Γ z are determined by
cz ¼ 03�1; Bz½1 0 0�Φz½0 0 1�Φz½0 0 1�ΦzþAzΦz½0 0 1�Φz ¼Φz�Cz;
Γ z ¼ fI�AzðρIþΦz½0 0 1�Þ�Bz½1 0 0�ðρ2IþρΦz½0 0 1�þΦz½0 0 1�Φz½0 0 1�Þg�1Dz;
where I denotes a conformable identity matrix. Note that the vector Γ z is uniquely determined given a vector Φz, but thevector Φz is not generally uniquely determined, which induces multiplicity of the fundamental REE (6).
Following Section 10.5 of Evans and Honkapohja (2001), the present analysis investigates E-stability of the fundamentalREE (6).12 Corresponding to this REE, all agents are assumed to be endowed with a perceived law of motion (PLM) of zt
zt ¼ czþΦzst�1þΓzut : ð7Þ
Using a forecast from this PLM and the relation st ¼ ½0 0 1�zt to substitute E tztþ1 and E tztþ2 out of the system (5) leads to theactual law of motion (ALM) of zt
zt ¼ fAzðIþΦz½0 0 1�ÞþBz½1 0 0�ðIþΦz½0 0 1�þΦz½0 0 1�Φz½0 0 1�ÞgczþfðAzþBz½1 0 0�Φz½0 0 1�ÞΦz½0 0 1�ΦzþCzgst�1
þfAzðρIþΦz½0 0 1�ÞΓzþBz½1 0 0�ðρ2IþρΦz½0 0 1�þΦz½0 0 1�Φz½0 0 1�ÞΓzþDzgut : ð8Þ
Then, the mapping T from the PLM (7) to the ALM (8) can be defined by
T
czΦz
Γz
0B@
1CA
0
¼fAzðIþΦz½0 0 1�ÞþBz½1 0 0�ðIþΦz½0 0 1�þΦz½0 0 1�Φz½0 0 1�Þgcz
ðAzþBz½1 0 0�Φz½0 0 1�ÞΦz½0 0 1�ΦzþCz
AzðρIþΦz½0 0 1�ÞΓzþBz½1 0 0�ðρ2IþρΦz½0 0 1�þΦz½0 0 1�Φz½0 0 1�ÞΓzþDz
0B@
1CA
0
: ð9Þ
For a fundamental REE ðcz;Φz;Γ zÞ to be E-stable, the matrix differential equation
ddτ
cz;Φz;Γzð Þ ¼ T cz;Φz;Γzð Þ� cz;Φz;Γzð Þ
must have local asymptotic stability at the REE, where τ denotes a notional time. Hence, a fundamental REE ðcz;Φz;Γ zÞ isE-stable if and only if all eigenvalues of the three matrices DTcðcz;Φz;Γ zÞ, DTΦðcz;Φz;Γ zÞ, DTΓðcz;Φz;Γ zÞ have real parts lessthan unity.
E-stability of the fundamental REE (6) is numerically investigated using the three matrices DTcðcz;Φz;Γ zÞ, DTΦðcz;Φz;Γ zÞ,DTΓðcz;Φz;Γ zÞ given in Appendix B, since it seems to be impossible to analytically solve the matrix equation for Φz in theREE. As McCallum (1998) indicates, distinct fundamental REE are obtained for different orderings of stable generalizedeigenvalues of the matrix pencil for the system (5). Indeed, in cases of indeterminacy, the benchmark calibration of modelparameters shows that there are two or three distinct fundamental REE of the form (6).
11 The form of the coefficient vector Dz is omitted, since it is not needed in what follows.12 The system (5) contains the predetermined variable s t�1 and thus it is possible to consider two learning environments, which are studied
respectively in Sections 10.3 and 10.5 of Evans and Honkapohja (2001). One environment allows agents to use current endogenous variables in expectationformation, whereas the other does not. The present paper shows only E-stability analysis under the latter environment as in Bullard and Mitra (2002),Kurozumi (2006), and Kurozumi and Van Zandweghe (2008), Kurozumi and Van Zandweghe (2012). This is because the former induces a problemwith thesimultaneous determination of expectations and current endogenous variables, which is critical to equilibrium under non-RE as indicated by Evans andHonkapohja (2001) and Bullard and Mitra (2002).
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8%
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0% 2%
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�Π �Π
�Π �Π
�Y
�Y
�Y
�Y
Fig. 1. Regions of the Taylor rule's coefficients ðϕΠ ;ϕY Þ that guarantee E-stability of a fundamental REE as well as determinacy of REE: Benchmarkcalibration of model parameters. Note: In each panel, the mark “� ” shows Taylor's (1993) estimates and the thick solid line, the thin solid line, and thedotted line represent the boundary of E-stability, that of determinacy, and that given by the long-run version of the Taylor principle (10), respectively.
T. Kurozumi / Journal of Economic Dynamics & Control 42 (2014) 175–187 179
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3.2. Main results
Under the benchmark calibration of model parameters, Fig. 1 presents the regions of the Taylor rule's coefficients ðϕΠ ;ϕY Þthat guarantee E-stability of a fundamental REE as well as determinacy of REE in the cases of the annualized trend inflationrate of zero, two, four, six, and eight percent. The present paper focuses on the range of the Taylor rule's coefficients givenby 0rϕΠr4:5 and 0rϕY r1:5=4¼ 0:375. Because Taylor's (1993) estimates of these coefficients are ϕΠ ¼ 1:5 andϕY ¼ 0:5=4¼ 0:125 (marked by “ � ” in each panel of Fig. 1), it is reasonable to set upper bounds on the coefficients atthe values three times larger than the estimates.
Fig. 1 demonstrates that a fundamental REE is likely to be E-stable even in cases of indeterminacy induced by high trendinflation. In the cases of the annualized trend inflation rate of zero and two percent, Fig. 1a shows one region of the Taylorrule's coefficients that generate determinacy and E-stability of REE and the other region of the coefficients that induceindeterminacy of REE and E-instability of all fundamental REE. The former region is fairly wide and contains Taylor's (1993)estimates of the coefficients. The boundary between the two regions is given by ϕΠþϵYϕY ¼ 1, where
ϵY ¼αΠ
θ�1½ð1�βÞð1�αΠθÞð1�αβΠ
θÞ�θðΠ�1Þfβð1�αΠθ�1Þð1�αΠ
θÞþsnð1�αβΠθ�1Þð1�αβΠ
θÞg�ð1þsnÞð1�αΠ
θ�1Þð1�αΠθÞð1�αβΠ
θ�1Þð1�αβΠθÞ
:
Then, the condition
ϕΠþϵYϕY 41 ð10Þcharacterizes the region of the Taylor rule's coefficients that ensure determinacy and E-stability.
The condition (10) can be interpreted as the long-run version of the Taylor principle. From the generalized NewKeynesian Phillips curve (3) and the law of motion of price distortion (4), each percentage point of permanently higherinflation implies ϵY percentage points of permanently higher output. Thus, ϵY represents the long-run inflation elasticity ofoutput. Then, ϕΠþϵYϕY shows the permanent increase in the interest rate by the Taylor rule (1) in response to each unitpermanent increase in inflation. Therefore, the condition (10) suggests that in the long run the interest rate should be raisedby more than the increase in inflation. This Taylor principle (10) restricts the size of the output coefficient more severelyunder higher trend inflation, since the value of the elasticity ϵY decreases to become negative and further declines as trendinflation rises. For the annualized trend inflation rate less than a threshold (e.g., 2.3% under the benchmark calibration), theTaylor principle (10) is the relevant, necessary and sufficient condition for determinacy and E-stability, as is similar to theresult of Bullard and Mitra (2002) who study the case of the zero trend inflation rate.
As trend inflation increases beyond the threshold, the region of the Taylor rule's coefficients that ensure determinacy ofREE narrows remarkably, in line with the result of Ascari and Ropele (2009), whereas the region of the coefficients thatguarantee E-stability of a fundamental REE becomes slightly narrower. In Fig. 1b, where the annualized trend inflation rate isfour percent, the thick solid line, the thin solid line, and the dotted line represent the boundary of E-stability, that ofdeterminacy, and that given by the long-run version of the Taylor principle (10), respectively. Note that in the region of E-stability at least one E-stable fundamental REE is generated, while in the region of E-instability no fundamental REE is E-stable. The Taylor principle (10) is no longer a sufficient condition for E-stability or determinacy. Even when the Taylorprinciple (10) is satisfied, there are small coefficients on inflation and output in the Taylor rule (1) that induce not onlyindeterminacy of REE but also E-instability of all fundamental REE. One point to be emphasized is that the Taylor principle(10) is not even a necessary condition for E-stability. There is a region of the Taylor rule's coefficients that do not meet theTaylor principle (10) but ensure E-stability of a fundamental REE (1:05rϕΠr1:40). These properties of E-stability also holdin the cases of the annualized trend inflation rate of six and eight percent, as can be seen in Fig. 1c and d. In these cases,there appears a region of the Taylor rule's coefficients that induce explosive REE, which Ascari and Ropele (2009) fail topoint out.
Under the benchmark calibration of model parameters, what makes a fundamental REE likely to be E-stable even in cases ofindeterminacy induced by high trend inflation? A finite elasticity of labor supply causes price distortion to affect inflation dynamicsrepresented by the generalized New Keynesian Phillips curve (3), and the persistence of price distortion in Eq. (4) then generatesendogenously persistent inflation dynamics. For the REE in question, E-stability examines whether an associated equilibrium inwhich agents form expectations based on a PLM reaches over time the REE. Under such expectation formation (i.e., the presence oflagged price distortion st�1 in the PLM (7)), the endogenous persistence of inflation dynamics through price distortion helps agentsform inflation expectations. Consequently, E-stability of a fundamental REE is likely.13
3.3. Robustness exercises
This subsection evaluates the robustness of the result obtained with the benchmark calibration of model parameterswith respect to the probability of no price adjustment, the price elasticity of demand for differentiated goods, and theelasticity of labor supply.
13 See Appendix C for the case of an infinite elasticity of labor supply (i.e., sn ¼ 0). In this case, high trend inflation is a serious cause of E-instability offundamental REE as well as indeterminacy of REE.
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Fig. 2. Regions of the Taylor rule's coefficients ðϕΠ ;ϕY Þ that guarantee E-stability of a fundamental REE as well as determinacy of REE: Robustness exerciseswith respect to the probability of no price adjustment α and the price elasticity of demand θ. Note: in each panel, the mark “� ” shows Taylor's (1993)estimates and the thick solid line, the thin solid line, and the dotted line represent the boundary of E-stability, that of determinacy, and that given by thelong-run version of the Taylor principle (10), respectively.
T. Kurozumi / Journal of Economic Dynamics & Control 42 (2014) 175–187 181
T. Kurozumi / Journal of Economic Dynamics & Control 42 (2014) 175–187182
3.3.1. On the probability of no price adjustmentThe first exercise examines the robustness of the result presented above using an alternative calibration of the
probability of no price adjustment α. The benchmark calibration of α¼ 0:75 implies that the average frequency of priceadjustment is 1 year, which is somewhat longer than micro evidence on price changes (e.g., Bils and Klenow, 2004; Klenowand Kryvtsov (2008); Nakamura and Steinsson, 2008) although several macroeconomic studies use the calibration. Thus, thesmaller calibration of α¼ 0:5 is employed. Its implied average frequency of price adjustment is half a year.
The result obtained with the benchmark calibration of model parameters still holds for the calibration of α¼ 0:5. Underthis calibration, Fig. 2a shows how each two percentage point increases in the annualized trend inflation rate from zero toeight percent alters the region of the Taylor rule's coefficients that ensure E-stability as well as determinacy. In all the casesof the annualized trend inflation rate of zero, two, four, six, and eight percent, the region of E-stability and the determinacyis fairly wide and contains Taylor's (1993) estimates of the coefficients, and the long-run version of the Taylor principle (10)is the relevant, necessary and sufficient condition for E-stability and determinacy. Compared with the benchmark one, thecalibration of α¼ 0:5 enlarges the long-run inflation elasticity of output ϵY in each case of trend inflation, thereby makingthe Taylor principle (10) more likely to be satisfied. Therefore, E-stability as well as determinacy is more likely.
3.3.2. On the price elasticity of demand for differentiated goodsThe second exercise examines the robustness of the result with respect to the price elasticity of demand for
differentiated goods θ. The benchmark calibration of θ¼ 11 implies that the price markup is 10 percent. In the literature,the price markup of 20 percent is also frequently used. This exercise thus sets the price elasticity at θ¼ 6.
The result still holds for the calibration of θ¼ 6. Under this calibration, Fig. 2b illustrates the cases of the annualized trendinflation rate of zero, two, and four percent and shows that the region of E-stability and determinacy, which is completelycharacterized by the long-run version of the Taylor principle (10), is fairly wide and contains Taylor's (1993) estimates of thecoefficients. Fig. 2c and d presents the cases of the annualized trend inflation rate of six and eight percent, respectively. Theregion of E-stability of a fundamental REE is wide enough to contain realistic coefficients of the Taylor rule including Taylor'sestimates, and this region is almost featured by the Taylor principle (10). As is the case with the first robustness exerciseusing a lower probability of no price adjustment, a lower price elasticity of demand increases the long-run inflationelasticity of output ϵY and hence makes the Taylor principle (10) more likely to be satisfied. Therefore, E-stability as well asdeterminacy is more likely.
3.3.3. On the elasticity of labor supplyThe third exercise investigates the robustness of the result with respect to the elasticity of labor supply. The benchmark
calibration of the inverse of this elasticity is set at sn ¼ 1, that is, the elasticity of labor supply is set at unity. Yet microevidence (e.g., Altonji, 1986; Ball, 1990; Card, 1994) indicates that its value is less than unity. Thus, this exercise employs thecalibration of sn ¼ 2, that is, the elasticity of labor supply of 0.5.14
For the calibration of sn ¼ 2, the result still holds. Under this calibration, Fig. 3 illustrates the regions of the Taylor rule'scoefficients that generate E-stability as well as determinacy. Compared with Fig. 1 under the benchmark calibration ofsn ¼ 1, Fig. 3 shows that the region of indeterminacy of REE remains, whereas the region of E-stability of a non-explosivefundamental REE widens, particularly in the cases of the higher annualized trend inflation rate of six and eight percentshown respectively in Fig. 3c and d. Indeterminacy remains because a lower elasticity of labor supply (i.e., a higher sn)reduces the long-run inflation elasticity of output ϵY and hence makes the Taylor principle (10) less likely to be met.E-stability is more likely, since a lower elasticity of labor supply strengthens the influence of price distortion on inflationdynamics represented by the generalized New Keynesian Phillips curve (3) and hence the endogenous persistence ofinflation dynamics through price distortion, which further helps agents form inflation expectations based on the PLM (7).
4. Concluding remarks
This paper has examined implications of trend inflation for E-stability of fundamental REE under the Taylor rule ina Calvo-style sticky price model based on micro evidence that each period a fraction of prices is kept unchanged undera positive trend inflation rate. An empirically plausible calibration of the model has demonstrated that at least onefundamental REE is likely to be E-stable even in cases of indeterminacy induced by high trend inflation.
The robustness exercise has shown that a lower probability of no price adjustment makes E-stability of a fundamentalREE more likely even under high trend inflation. Previous macroeconomic studies on price stickiness, such as Ball et al.(1988), Romer (1990), Kiley (2000), Devereux and Yetman (2002), and Levin and Yun (2007), show that when firms choosethe probability of price adjustment in a Calvo-style sticky price model, higher trend inflation leads to a lower probability ofno price adjustment. Along the lines of these previous studies, Kurozumi (2011) introduces a framework of endogenousprice stickiness in the analysis of determinacy of REE under the Taylor rule and demonstrates that indeterminacy caused by
14 An infinite elasticity of labor supply (i.e., sn ¼ 0), although it is not empirically plausible, is used in some theoretical macroeconomic studies for thesake of simplicity. The exercise using this calibration is presented in Appendix C.
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
0.250
0.275
0.300
0.325
0.350
0.375
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
0%, 2%
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
0.250
0.275
0.300
0.325
0.350
0.3754%
Indeterminateand
E-unstable
Determinateand
E-stable
Indeterminatebut
E-stable
Explosive
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
0.250
0.275
0.300
0.325
0.350
0.3756%
Indeterminateand
E-unstable
Determinateand
E-stable
Indeterminatebut
E-stable
Explosive
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
0.250
0.275
0.300
0.325
0.350
0.3758%
Indeterminateand
E-unstable Determinateand
E-stable
Indeterminatebut
E-stable
Explosive
0% 2%
Indeterminateand
E-unstable
Determinateand
E-stable
�Π �Π
�Π �Π
�Y
�Y
�Y
�Y
Fig. 3. Regions of the Taylor rule's coefficients ðϕΠ ;ϕY Þ that guarantee E-stability of a fundamental REE as well as determinacy of REE: Robustness exercisewith respect to the elasticity of labor supply (sn ¼ 2). Note: in each panel, the mark “� ” shows Taylor's (1993) estimates and the thick solid line, the thinsolid line, and the dotted line represent the boundary of E-stability, that of determinacy, and that given by the long-run version of the Taylor principle (10),respectively.
T. Kurozumi / Journal of Economic Dynamics & Control 42 (2014) 175–187 183
T. Kurozumi / Journal of Economic Dynamics & Control 42 (2014) 175–187184
higher trend inflation is less likely. Thus, one direction for future research would be to investigate whether the framework ofendogenous price stickiness leads to a stronger E-stability result than that of the present paper.
Another future research direction would be to analyze E-stability of non-fundamental REE in the model along the lines ofthe literature, such as Honkapohja and Mitra (2004), Carlstrom and Fuerst (2004), and Evans and McGough (2005). Theremay be several reactions to the result of the present paper. One reaction would be that indeterminacy caused by high trendinflation is not critical, since even in such indeterminacy cases a fundamental REE is likely to be E-stable. Another would bethat there may be an E-stable non-fundamental REE in addition to the E-stable fundamental REE. Thus, whether such anE-stable non-fundamental REE exists is an important topic for future research.
Acknowledgments
The author would like to thank the editor James Bullard, Teruyoshi Kobayashi, Bennett McCallum, Ichiro Muto, WillemVan Zandweghe, an associate editor, and an anonymous referee for their valuable comments. Any remaining errors are thesole responsibility of the author. The views expressed herein are those of the author and should not be interpreted as thoseof the Bank of Japan.
Appendix A. Coefficient matrix and vectors of the system (5)
In the system (5), the coefficient matrix Az and the coefficient vectors Bz;Cz are given by
Az ¼
A11 A12 A131�ϕΠA111þϕY
1�ϕΠA121þϕY
�ϕΠA131þϕY
αθΠθ� 1ðΠ �1ÞA11
1�αΠθ� 1
αθΠθ� 1ðΠ �1ÞA12
1�αΠθ� 1
αθΠθ� 1ðΠ �1ÞA13
1�αΠθ� 1
26664
37775;
Bz ¼
B1
�ϕΠB111þϕY
αθΠθ� 1ðΠ �1ÞB111�αΠ
θ� 1
26664
37775; Cz ¼
C1
� ϕΠC11þϕY
αΠθþαθΠ
θ� 1ðΠ �1ÞC1
1�αΠθ� 1
26664
37775;
where
A11 ¼αβΠ
θ�1ð1þϕY Þ½1þαΠθþθðΠ�1Þð1�αΠ
θ�1Þ�þð1�αΠθ�1Þð1�αβΠ
θÞð1þsnÞð1�αΠ
θ�1Þð1�αβΠθÞϕΠð1þsnÞþαΠ
θ�1ð1þϕY Þ½1�snθðΠ�1Þð1�αβΠθÞ�
;
A12 ¼ð1�αΠ
θ�1Þð1�αβΠθÞ½1�αβΠ
θ�1ð1þϕY Þ�ð1þsnÞð1�αΠ
θ�1Þð1�αβΠθÞϕΠð1þsnÞþαΠ
θ�1ð1þϕY Þ½1�snθðΠ�1Þð1�αβΠθÞ�
;
A13 ¼ � snαβΠθ�1ð1þϕY Þð1�αΠ
θ�1Þð1�αβΠθÞ
ð1�αΠθ�1Þð1�αβΠ
θÞϕΠð1þsnÞþαΠθ�1ð1þϕY Þ½1�snθðΠ�1Þð1�αβΠ
θÞ�;
B1 ¼ � αβΠθ�1
αβΠθð1þϕY Þ
ð1�αΠθ�1Þð1�αβΠ
θÞϕΠð1þsnÞþαΠθ�1ð1þϕY Þ½1�snθðΠ�1Þð1�αβΠ
θÞ�;
C1 ¼snαΠ
θð1þϕY Þð1�αΠθ�1Þð1�αβΠ
θÞð1�αΠ
θ�1Þð1�αβΠθÞϕΠð1þsnÞþαΠ
θ�1ð1þϕY Þ½1�snθðΠ�1Þð1�αβΠθÞ�
:
Appendix B. Matrices for the analysis of expectational stability
From the mapping T given by (9), we have
DTcðcz;Φz;ΓzÞ ¼ AzðIþΦz½0 0 1�ÞþBz½1 0 0�ðIþΦz½0 0 1�þΦz½0 0 1�Φz½0 0 1�Þ;DTΦðcz;Φz;ΓzÞ ¼ AzðΦz3IþΦz½0 0 1�ÞþBz½1 0 0�fðΦz3Þ2IþΦz3Φz½0 0 1�þΦz½0 0 1�Φz½0 0 1�g;DTΓðcz;Φz;ΓzÞ ¼ AzðρIþΦz½0 0 1�ÞþBz½1 0 0�ðρ2IþρΦz½0 0 1�þΦz½0 0 1�Φz½0 0 1�Þ:
Therefore, a fundamental REE ðcz;Φz;Γ zÞ is E-stable if and only if all eigenvalues of the three matrices AzðφIþΦz
½0 0 1�ÞþBz½1 0 0�ðφ2IþφΦz½0 0 1�þΦz½0 0 1�Φz½0 0 1�Þ, φAf1; ρ;Φz3g have real parts less than unity, where Φz3 is the thirdelement of the RE coefficient vector Φz.
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
0.250
0.275
0.300
0.325
0.350
0.375
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
0%, 2%
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
0.250
0.275
0.300
0.325
0.350
0.375
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
4%
Indeterminateand
E-unstable
Determinateand
E-stable
Indeterminatebut
E-stable
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
0.250
0.275
0.300
0.325
0.350
0.375
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
6%
Indeterminateand
E-unstable
Determinateand
E-stable
Indeterminatebut
E-stable
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
0.250
0.275
0.300
0.325
0.350
0.375
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
8%
Indeterminateand
E-unstable
Determinateand
E-stable
Indeterminatebut
E-stable
0% 2%
Indeterminateand
E-unstable
Determinateand
E-stable
�Π �Π
�Π �Π
�Y
�Y
�Y
�Y
Fig. 4. Regions of the Taylor rule's coefficients ðϕΠ ;ϕY Þ that guarantee E-stability of the fundamental REE as well as determinacy of REE in the case of aninfinite elasticity of labor supply (i.e., sn ¼ 0). Note: in each panel, the mark “� ” shows Taylor's (1993) estimates and the thick solid line, the thin solid line,and the dotted line represent the boundary of E-stability, that of determinacy, and that given by the long-run version of the Taylor principle (10),respectively.
T. Kurozumi / Journal of Economic Dynamics & Control 42 (2014) 175–187 185
T. Kurozumi / Journal of Economic Dynamics & Control 42 (2014) 175–187186
Appendix C. The case of an infinite elasticity of labor supply
In the case of an infinite elasticity of labor supply (i.e., sn ¼ 0), E-stability of fundamental REE can be analyticallyexamined. This is because in that case the generalized New Keynesian Phillips curve (3) no longer depends on pricedistortion.
The system of Eqs. (1)–(4) with the possibly non-RE operator E t can be rewritten as
xt ¼ AxEtxtþ1þBx½1 0�E txtþ2þDxut ; ð11Þ
st ¼αθΠ
θ�1ðΠ�1Þ1�αΠ
θ�1 1 0½ �xtþαΠθ st�1; ð12Þ
where xt ¼ ½Π t Y t �0 and the coefficient matrix Ax and the coefficient vector Bx are given by
Ax ¼A11 A12
1�ϕΠA111þϕY
1�ϕΠA121þϕY
" #; Bx ¼
B1
� ϕΠB11þϕY
" #;
where A11, A12, B1 are given in Appendix A.15 The fundamental RE solution to the system (11) is given by
xt ¼ cxþΓxut ¼ 02�1þðI�ρAx�ρ2Bx½1 0�Þ�1Dxut : ð13ÞUsing this to substitute xt out of Eq. (12), the fundamental REE process of st can be obtained.
From a similar argument to that in Section 3.1, the necessary and sufficient condition for E-stability of the fundamentalREE as well as that for determinacy of the REE can be obtained.
Proposition 1. Suppose that Assumption1 holds and the elasticity of labor supply is infinity (i.e., sn ¼ 0). Then, the fundamentalREE is E-stable if and only if the long-run version of the Taylor principle (10), the inequality
ϕΠþαΠ
θ�1½2�β�αβΠθð1�βÞ�βθðΠ�1Þð1�αΠ
θ�1Þ�ð1�αΠ
θ�1Þð2�αβΠθ�1Þð1�αβΠ
θÞϕY þ1� �
41�αβΠ
θð1�αΠθ�1Þ
ð1�αΠθ�1Þð2�αβΠ
θ�1Þð1�αβΠθÞ; ð14Þ
and the other two conditions given in the proof of this proposition are satisfied. The REE is determinate if and only if the Taylorprinciple (10) and the other condition given in the proof are satisfied. Moreover, if the REE is determinate, it is E-stable.16
In the case of an infinite elasticity of labor supply (i.e., sn ¼ 0), the fundamental REE is likely to be E-unstable under hightrend inflation.17 Note that in that case the long-run version of the Taylor principle (10) is a necessary condition not only fordeterminacy of the REE but also for E-stability of the fundamental REE. Fig. 4 illustrates the regions of the Taylor rule'scoefficients that ensure E-stability as well as determinacy under the calibration of sn ¼ 0. When the annualized trendinflation rate is zero or two percent, the region of E-stability and determinacy, which is completely characterized by theTaylor principle (10), is fairly wide and contains Taylor's (1993) estimates of the coefficients, as can be seen in Fig. 4a.However, in the cases of the annualized trend inflation rate of four, six, and eight percent, Fig. 4b–d shows that as trendinflation increases, the Taylor principle (10) restricts the size of the output coefficient ϕY more severely. Besides, the otherdeterminacy condition induces much severer lower bounds on the inflation and output coefficients ϕΠ ;ϕY . Hence,indeterminacy is more likely. E-instability is also more likely, since the condition (14) is the other relevant condition forE-stability and brings about severer lower bounds on the inflation and output coefficients ϕΠ , ϕY .
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