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Trend inflation and monetary policy rules: determinacy analysis in New Keynesian model with capital accumulation

  • Elena Gerko und Kirill Sossounov EMAIL logo
Veröffentlicht/Copyright: 18. Dezember 2014
The B.E. Journal of Macroeconomics
Aus der Zeitschrift The B.E. Journal of Macroeconomics Band 15 Heft 1

Abstract

The paper analyzes the effect of positive trend inflation in the framework of a standard New Keynesian model with Calvo price setting and capital accumulation. We are building on the work of Carlstrom and Fuerst (Carlstrom, Charles T., and Timothy S. Fuerst. 2005. “Investment and Interest Rate Policy: A Discrete-Time Analysis.” Journal of Economic Theory 123: 4–20.) and Ascari and Ropele (Ascari, Guido, and Tiziano Ropele. 2007. “Optimal Monetary Policy under Low Trend Inflation.” Journal of Monetary Economics 54 (8): 2568–2583., Ascari, Guido, and Tiziano Ropele. 2009. “Trend Inflation, Taylor Principle, and Indeterminacy.” Journal of Money, Credit and Banking 48 (1): 1557–1584.) who separately considered effects of capital accumulation and trend inflation in a similar context. It is shown that the simultaneous presence of positive trend inflation and capital accumulation greatly affect the determinacy property of equilibrium under this setup. Namely, in the presence of positive trend inflation the determinacy region shrinks, and it is virtually impossible to produce a determinate equilibrium with the Taylor-type rule given a steady state of inflation of more than 5%. Even for a moderate value of 2.5%, the design of the rule that ensures the uniqueness of the equilibrium requires detailed knowledge of the parameters of an economy. We also show that for a large set of plausible parameters, the standard Taylor rule leads to indeterminacy. Alternative monetary policy rules such as interest rate smoothing, output growth targeting and price level targeting are also analyzed. It is shown that the latter improves the determinacy of the model solution, and the best way to guarantee the determinate equilibrium is to use price level targeting in the policy rule.

Keywords: monetary policy; multiple equilibria; Taylor rule
Corresponding author: Kirill Sossounov, National Research University Higher School of Economics (HSE), Russian Presidential Academy of National Economy and Public Administration (RANEPA), e-mail:

Acknowledgments

We are grateful to Yuriy Gorodnichenko, Konstantin Styrin and Oleg Zamulin for helpful comments and suggestions. We would also like to thank two anonymous referees, the editor Luisa Lambertini and seminar participants at New Economic School, European Economic Association meeting 2010, Asian Meeting of Econometric Society 2011 and Econometric Society Australasian meeting 2011.

Appendix

New-Keynesian Phillips curve

Given the aggregate production function the demand for the specific good Yi,t is given by Yi, tYt=[Pi, tPt]−η and the aggregate price level is Pt=[∫01Pi, t1−η]11−η. Let ψ be the probability of changing price by the representative firm, then

(46)Pt1−η=ψ[∑∞k=0(1−ψ)kXt−k1−η] (46)

where Xt stands for the reset price. Let P˜t=Pt(1+π∗)−t and X˜t=Xt(1+π∗)−t be detrended values, then

(47)P˜t1−η=(1+π∗)−t(1−η)ψ∑k=0∞[(1−ψ)k(1+π∗)(t−k)(1−η)X˜t−k1−η]=ψ∑k=0∞[(1−ψ)k(1+π∗)k(η−1)X˜t−k1−η] (47)

or equivalently in logs: pt=(1−μ)∑∞k=0μkxt−k or pt=(1–μ)xt+μpt–1, where μ=(1–ψ)(1+π*)η–1.

The reset price Xt is the solution of the following maximization problem:

maxXt∗Et∑k=0∞[((1−ψ)β)kU′(Ct+k)U′(Ct)Yi, t+kXt−C(Yi, t+k)Pt+kPt+k]

subject to the sequence of demand constraints: Yt+k=(XtPt+k)−ηCt+k.

where C(Yt,i)=MCtYt is a real cost function of the firm i and MCt are real marginal costs which are the same for all firms due to constant returns to scale property of the production.

Solution of the problem is given by Xt=ηη−1∑k=0∞((1−ψ)β)kEt[MCt+kPt+kηYt+kU′(Ct+k)]∑k=0∞((1−ψ)β)kEt[Pt+kη−1Yt+kU′(Ct+k)] and, therefore,

(48)X˜t=ηη−1∑k=0∞((1−ψ)β(1+π∗)η)kEt[MCt+kP˜t+kηYt+kU′(Ct+k)]∑k=0∞((1−ψ)β(1+π∗)η−1)kEt[P˜t+kη−1Yt+kU′(Ct+k)] (48)

In logs xt=(1−λ)∑∞k=0λkEt(mct+k+ηpt+k+yt+k−ct+k)−(1−γ)∑∞k=0γkEt((η−1)pt+k+yt+k−ct+k) where λ=(1−ψ)(1+π∗)η1+θ and γ=(1−ψ)(1+π∗)η−11+θ.

Therefore

(49)xt=(1−λ)(mct+pt)+λEtxt+1−π∗(1−η)zt (49)

and

(50)zt+γη−1(yt−ct)=γEt[zt+1+πt+1+yt+1−ct+1η−1]. (50)

From equation (47) one gets πt=1−μμ(xt−pt) while equation (49) results in: xt–pt=(1–λ)mct+λEt(xt+1–pt+1+πt+1)–π*(1–η)zt. Therefore πt=1−μμ[(1−λ)mct−π∗(1−η)zt]+λμEtπt+1 or

(51)Etπt+1=1+θ1+π∗πt−(1−μ)(1−λ)λmct+π∗λ(1−η)(1−μ)zt (51)

where (51) is a New-Keynesian Phillips curve, which reduces to the standard one by setting π*=0.

Steady-state values

From (46) one can get P˜∗1−η=ψ1−μX˜∗1−η or

(52)[X˜P]∗=[1−μψ]1/(1−η). (52)

Equation (48) leads to X˜∗=ηη−11−γ1−λMC∗P˜∗, and, therefore,

(53)MC∗=η−1η1−λ1−γ[X˜P]∗=η−1η1+θ−(1−ψ)(1+π∗)η1+θ−(1−ψ)(1+π∗)η−1[X˜P]∗. (53)

The output-capital ratio can be reduced from the demand for capital equation:

(54)[YK]∗=θ+δα1−(1−ψ)(1+π∗)ηψ1MC∗[X˜P]∗η (54)

while the consumption-capital ratio is given by

(55)[CK]∗=[YK]∗−δ. (55)

Equilibrium system

The equilibrium is given by a system of 10 equations with 10 variables y˜, k, l, c, w, r, mc, π, r where y˜=y−ξ and ξ is a price dispersion wedge defined in the paper:

Production function

(56)y˜t=αkt+(1−α)lt, (56)

Relative price dispersion:

(57)ξt=(1−ψ)ξt−1−η1−μ−ψ1−μπt. (57)

Capital accumulation equation

(58)kt+1=(1−δ)kt+[YK]∗(y˜t+ξt)−[CK]∗ct, (58)

Labor demand

(59)y˜t−lt+mct=wt, (59)

Labor supply

(60)wt=ct+slt, (60)

Demand for capital

(61)y˜t−kt+mct=1+r∗δ+r∗rt, (61)

Euler equation

(62)−ct=Et[−ct+1+rt+1], (62)

Phillips curve

(63)Etπt+1=1+θ1+π∗πt−(1−μ)(1−λ)λmct+π∗λ(1−η)(1−μ)zt, (63)
(64)γEt[zt+1+πt+1+y˜t+1+ξt+1−ct+1η−1]=zt+γη−1(y˜t+ξt−ct). (64)

Monetary policy rule and Fisher’s parity

(65)it=Et[(1+a1)πt+l+a2(y˜t+l+ξt+l)]=Et[rt+πt+1] (65)

or

(66)it=Et[(1+a1)πt+l+a2(Δy˜t+l+Δξt+l)]=Et[rt+1+πt+1], l=0, 1. (66)

The system can be reduced to six intertemporal equations with two predetermined and four endogenous variables. Therefore, unique equilibrium requires that four eigenvalues of the resulting system be greater than one in absolute value. If the number of such eigenvalues is <4 then the equilibrium is indeterminant.

Matrix notation

Let X1=[k, c, r, π, z, ξ]′ and X2=[y, l, w, mc]′.

J1X1=J2X2

where J1=| 000000101+θδ+θ000010000α00000 | and J2=| 1−1−1110010−s101α−100 |,

The system of intertemporal equations can be written as Et[J4X1t+1+J5X2t+1]= J6X1t+J7X2t, where

J4=| 10000001−10000001−1−a1000001000−1η−10111η−1000η1−μ−ϕ1−μ01 |, J5=| 00000000−a200000001η−10000000 |,

J6=| 1−δ−[CK]∗000[YK]∗0100000−101+a10a20001+θ1+π∗λ100−1η−1001γ1η−1000001−ϕ |, J7=| [YK]∗0000000a2000000−λ21η−10000000 |,

where λ1=π∗λ(1−η)(1−μ) and λ2=(1−μ)(1−λ)λ.

Let J8=J4+J5J2−1J1 and J9=J6+J7J2−1J1, then the system could be summarized by

(67)EtX1t+1=JXt (67)

where J=J8−1J9.

References

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Published Online: 2014-12-18
Published in Print: 2015-1-1

©2015 by De Gruyter

Artikel in diesem Heft

  1. Frontmatter
  2. Advances
  3. Consumption composition and macroeconomic dynamics
  4. Evaluating linear approximations in a two-country model with occasionally binding borrowing constraints
  5. Contributions
  6. The bank lending channel and monetary policy rules for Eurozone banks: further extensions
  7. Investment lags and macroeconomic dynamics
  8. The zero lower bound: frequency, duration, and numerical convergence
  9. What drives endogenous growth in the United States?
  10. Environmental policy and economic growth: the macroeconomic implications of the health effect
  11. US household deleveraging following the Great Recession – a model-based estimate of equilibrium debt
  12. Price-level instability and international monetary policy coordination
  13. Households forming macroeconomic expectations: inattentive behavior with social learning
  14. Topics
  15. Complementarity and transition to modern economic growth
  16. Trend inflation and monetary policy rules: determinacy analysis in New Keynesian model with capital accumulation
  17. Discussions
  18. Preface to “Reflections on Macroeconometric Modeling” by Ray C. Fair
  19. Reflections on macroeconometric modeling
https://doi.org/10.1515/bejm-2012-0071
Schlagwörter für diesen Artikel
monetary policy; multiple equilibria; Taylor rule

Artikel in diesem Heft

  1. Frontmatter
  2. Advances
  3. Consumption composition and macroeconomic dynamics
  4. Evaluating linear approximations in a two-country model with occasionally binding borrowing constraints
  5. Contributions
  6. The bank lending channel and monetary policy rules for Eurozone banks: further extensions
  7. Investment lags and macroeconomic dynamics
  8. The zero lower bound: frequency, duration, and numerical convergence
  9. What drives endogenous growth in the United States?
  10. Environmental policy and economic growth: the macroeconomic implications of the health effect
  11. US household deleveraging following the Great Recession – a model-based estimate of equilibrium debt
  12. Price-level instability and international monetary policy coordination
  13. Households forming macroeconomic expectations: inattentive behavior with social learning
  14. Topics
  15. Complementarity and transition to modern economic growth
  16. Trend inflation and monetary policy rules: determinacy analysis in New Keynesian model with capital accumulation
  17. Discussions
  18. Preface to “Reflections on Macroeconometric Modeling” by Ray C. Fair
  19. Reflections on macroeconometric modeling
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