Real wage rigidity and the Taylor principle

Economics Letters 104 (2009) 46–48

Contents lists available at ScienceDirect

Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t

Real wage rigidity and the Taylor principle Eurilton Araújo ⁎ Ibmec Sao Paulo Rua Quata, 300. Zip Code: 04546-042. Sao Paulo, Brazil

a r t i c l e

i n f o

Article history: Received 10 April 2008 Received in revised form 17 March 2009 Accepted 31 March 2009 Available online 5 April 2009

a b s t r a c t This paper investigates conditions for equilibrium determinacy in the basic new Keynesian model with real wage rigidity. The determinacy region is increased under forward-looking rules and the Taylor principle continues to ensure equilibrium uniqueness under current-looking rules. © 2009 Elsevier B.V. All rights reserved.

Keywords: Real wage rigidity Determinacy Inflation targeting JEL Classification: E43 E52 E58

1. Introduction Monetary policy rules can be destabilizing, leading to multiple equilibria in which non-fundamental shocks can influence aggregate dynamics. In the basic new Keynesian model, the Taylor principle, which prescribes that the interest rate should increase by more than the increase in inflation, guarantees equilibrium determinacy. In some extensions of the basic model1, equilibrium determinacy imposes additional restrictions to the Taylor principle, shrinking the determinacy region. Therefore, it is important to know if this principle continues to ensure equilibrium determinacy in a model that incorporates additional market imperfections. This paper examines determinacy conditions in a model studied by Blanchard and Galí (2007), in which real wages respond sluggishly to economic conditions. The analysis focus on inflation-targeting interest-rate rules, without a feedback on the output gap. I show that the Taylor principle is a necessary condition for determinacy. If the central bank responds to current inflation, it is also a sufficient condition. Forward-looking rules impose an upper bound for the reactiveness of the interest rate. This upper bound is an increasing function of the index of real wage rigidity, allowing the central bank to react more ⁎ Tel.: +55 11 4504 2422; fax: +55 11 4504 2350. E-mail address: [email protected]. 1 Here is a list of papers that show how the Taylor principle needs to be modified when the basic model incorporates additional elements. Brückner and Schabert (2003) introduce working capital requirements. Røisland (2003) considers interest income taxation. De Fiore and Liu (2005) study the role of international trade and openness. Kurozumi (2006) adds monetary frictions. Finally, Wei (2008) investigates the implications of increasing returns. 0165-1765/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2009.03.022

strongly to expected inflation as real wage rigidity becomes more important. Thus, the introduction of real wage rigidity does not impose additional restrictions to ensure determinacy. 2. The model The model can be summarized by three equations: the aggregate demand, the new Keynesian Phillips curve, and the partial adjustment mechanism for real wages. The log-linear form of the aggregate demand equation is: yt = Et yt


+ 1

−

1 i − Et π t σ t

+ 1


 :

ð1Þ

where the positive parameter σ represents the inverse of the intertemporal elasticity of substitution. Inflation dynamics is described by the following new Keynesian Phillips curve: πt = βEt πt


+ 1

+ kwt :

ð2Þ

The positive parameter k is a measure of the speed of price adjustment, and β denotes the discount factor of the representative household, where 0 b β b 1. The partial adjustment mechanism for real wages is incorporated to introduce real wage rigidity. In log-linear form, the equation is: wt = γwt − 1 + ð1 − γ Þðσ + uÞyt :

ð3Þ

E. Araújo / Economics Letters 104 (2009) 46–48

The parameter 0 b γ b 1 is the index of real wage rigidity, and the positive parameter φ is the inverse of the elasticity of substitution between work and leisure. The variables yt, it, πt and wt are output, the nominal interest rate, inflation and real wages, expressed in log-deviations from their steady states. 3. Inflation targeting and determinacy I focus on pure inflation-targeting rules for two basic reasons. First, they are simple and transparent, acting as a benchmark for more sophisticated interest-rate rules. Second, there are problems in choosing an appropriate measure for the output gap, as explained by Orphanides (2003). It is assumed that the central bank follows two types of interestrate rules: it = α π π t ;

ð4Þ

it = α π Et π t

+ 1;

ð5Þ

where απ is strictly positive. Eqs. (4) and (5) describe a current-looking and a forward-looking rule, respectively. By eliminating the interest rate using Eq. (4) or Eq. (5), Eqs. (1), (2) and (3) can be written as Et(xt + 1) = Axt, where xt is a vector containing πt, yt, and wt − 1. The characteristic equation of the matrix A is: P(λ) = λ3 + A2λ2 + A1λ + A0 = 0. Determinacy requires that one root of the equation is inside the unit circle and two roots are outside. Following Barbeau (1989, p. 171) and Woodford (2003, p. 672), this occurs if and only if the conditions stated in either case I, case II, or case III are satisfied. These three cases, based on the coefficients of P(λ), are: 1 + A2 + A1 + A0 b 0 and − 1 + A2 − A1 + A0 N 0. 1 + A2 + A1 + A0 N 0, 1 + A2 + A1 + A0 N 0, and A20 − A0 A2 + A1 − 1 N0. 1 + A2 + A1 + A 0 N 0, − 1 + A2 − A1 + A0 b 0, A20 − A0 A2 + A1 − 1 b0, and |A2| b 3

Case I Case II Case III

Equilibrium determinacy conditions are summarized in the following propositions. Proposition 1. Under current-looking rules, the necessary and sufficient condition for a rational-expectations equilibrium to be determinate is that:

Equilibrium is determinate if and only if Eq. (6) and either Eq. (7) or Eq. (8) are satisfied. I establish that Eq. (6) is both necessary and sufficient for determinacy, by showing that any set of parameters satisfying Eq. (6) but not Eq. (8) must necessarily satisfy Eq. (7). Since β1 + β − 2 N 0 and the second term is strictly positive, Eq. (8) will not hold only if β N γ. If β N γ, then βγ − 1 b 0 and γβ −    γ − β1 + 1 b 0 imply: βγ − 1 βγ − γ − β1 + 1 N 0.

ð6Þ

Proof. The coefficients of the characteristic equation are:  A2 = −



1 k γ 1 k +1+ ð1 − γ Þðσ + uÞ + γ ; A1 = γ + + + ð1 − γ Þðσ + uÞα π ; β σβ β β σβ γ and A0 = − : β

Since −1 + A2 − A1 + A0 b 0, case I can be ruled out. In cases II and III, 1 + A2 + A1 + A0 N 0. This condition can be reduced to Eq. (6), which is a necessary condition for equilibrium determinacy. The condition A20 − A0 A2 + A1 − 1 N0 leads to: 

γ −1 β

   γ 1 k γ −γ− +1 + ð1 − γÞðσ + uÞ α π − N 0; β β σβ β

b 1 imply that α π N 1 N

Proposition 2. Under forward-looking rules, the necessary and sufficient condition for a rational-expectations equilibrium to be determinate is that: 1 b απ b 1 +

  2ð1 + βÞ 1 + γ : kðσ + uÞ 1 − γ

ð9Þ

Proof. The coefficients of the characteristic equation are:   1 k γ 1 ð1 − γÞð1 − α π Þðσ + uÞ ; A1 = γ + +1+ + ; A2 = − γ + β σβ β β γ and A0 = − : β

Again, case I can be ruled out, since conditions 1 + A2 + A1 + A0 b 0 and − 1 + A2 − A1 + A0 N 0 lead to the following contradiction: 2ð1 + γÞð1 + β Þ b ð1 − γÞðα π − 1Þðσ + uÞ σkβ b 0, but 2ð1 + γβÞð1 + βÞ N 0. β In cases II and III, 1 +A2 +A1 +A0 N 0. This condition can be reduced to Eq. (9), which is a necessary condition for equilibrium determinacy. The condition A20 − A0 A2 + A1 − 1 N0 leads to: 

    γ γ 1 γ k −1 −γ− + 1 + ð1 − γÞðα π − 1Þðσ + uÞ N 0: β β β β σβ

ð10Þ

The condition |A2| b 3 leads to:  j ð1 − γÞðα π − 1Þðσ + uÞ

   k 1 − γ+ + 1 j b 3: σβ β

ð11Þ

Equilibrium is determinate if and only Eq. (9) and either Eq. (10) or Eq. (11) is satisfied. I establish that Eq. (9) is both necessary and sufficient for determinacy, by showing that any set of parameters satisfying Eq. (9) but not Eq. (11) must necessarily satisfy Eq. (10). Suppose Eq. (11) does not hold; then 

     1 k 1 + 1 − 3 b ð1 − γÞðα π − 1Þðσ + uÞ + 1 + 3: b γ+ β σβ β

  If γ + β1 + 1 − 3 b 0, then β1 + β − 2 + γ − β b 0. This inequality holds only if β N γ, given that β1 + β − 2 N 0. If β N γ,



γ β

 − 1 γβ − γ −

1 β

     + 1 = − βγ − 1 ð1 − γ Þ β1 − 1 N 0:Since

  k Eq. (9) is satisfied, 0 b ð1 − γ Þðα π − 1 Þðσ + uÞ σβ b implies

γ β ð1 −

γÞðα π − 1Þðσ + uÞ

k σβ

2ð1 + γ Þð1 + βÞ β

N 0. Therefore, Eq. (10) is

necessarily satisfied.      If γ + β1 + 1 − 3 N 0, then ð1 − γÞðα π − 1Þðσ + uÞ σkβ N γ + Using this last inequality, condition Eq. (10) becomes:

1 β

 + 1 − 3.



    γ γ 1 γ k −1 −γ− + 1 + ð1 − γ Þðα π − 1Þðσ + uÞ β β β β σβ      γ γ 1 γ 1 N −1 −γ− +1 + γ+ + 1− 3 : β β β β β



The condition |A3| b 3 leads to:  1 k +β−2 + ð1 − γÞðσ + uÞ + ðγ − βÞN 0: β σβ

γ β N 0. This is the case since, απ N 1 and γ . This last step completes the proof. □ β

Therefore, Eq. (7) holds if α π −

γ β

γ+

α π N 1:

47

ð7Þ

After some algebraic manipulations, the last term in the expression can be reduced to:



ð8Þ



     2   γ γ 1 γ 1 γ 1 2 −1 −γ− +1 + γ+ + 1− 3 = −1 + γ + N 0: β β β β β β β

48

E. Araújo / Economics Letters 104 (2009) 46–48

 The last inequality shows that condition Eq. (10) holds if □ γ + β1 + 1 − 3 N 0. This last step completes the proof. Since

1 + γ 1− γ

N 1 and increases in γ, the central bank can be more

aggressive as real wage rigidity becomes more important. If γ → 1, the Taylor principle is restored as a necessary and sufficient condition for determinacy. 4. Conclusion In a new Keynesian model with real wage rigidity, I show that an active monetary policy is a necessary condition for equilibrium determinacy under current-looking and forward-looking inflation-targeting rules. In contrast to some extensions of the new Keynesian model that incorporate additional market imperfections, the Taylor principle continues to be an important recommendation for the design of simple monetary policy rules. Furthermore, the determinacy region is increased under forwardlooking rules, allowing the central bank to respond more strongly to inflation expectations as the importance of real wage rigidity increases.

References Barbeau, E.J., 1989. Polynomials. Springer-Verlag, New York. Blanchard, O., Galí, J., 2007. Real wage rigidities and the new Keynesian model. Journal of Money, Credit and Banking 39, 35–65. Brückner, M., Schabert, A., 2003. Supply-side effects of monetary policy and equilibrium multiplicity. Economics Letters 79, 205–211. De Fiore, F., Liu, Z., 2005. Does trade openness matter for aggregate instability? Journal of Economic Dynamics and Control 29, 1165–1192. Kurozumi, T., 2006. Determinacy and expectational stability of equilibrium in a monetary sticky-price model with Taylor Rule. Journal of Monetary Economics 53, 827–846. Orphanides, A., 2003. The quest for prosperity without inflation. Journal of Monetary Economics 50, 633–663. Røisland, Ø., 2003. Capital income taxation, equilibrium determinacy and the Taylor principle. Economics Letters 81, 147–153. Wei, X., 2008. Increasing returns and the design of interest rate rules. Macroeconomic Dynamics 12, 22–49. Woodford, M., 2003. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton University Press, Princeton.