The Taylor principle is valid under wage stickiness

Alexis Blasselle and Aurélien Poissonnier

Abstract

We consider the textbook neo-Keynesian model with staggered prices and wages in discrete time. We prove analytically that the Taylor principle holds in this case. When both contracts exhibit sluggish adjustment to market conditions, the policy maker faces a trade-off between stabilizing three welfare relevant variables: output, price inflation and wage inflation. We consider a monetary policy rule designed accordingly: the central banker can react to both inflations and the output gap. In addition to generalizing the Taylor principle we show that the frontier of determinacy embeds the frontier derived with staggered prices only, generalizes the frontier of determinacy in the limit case of continuous time and is symmetric in price and wage inflations.

1 Introduction

In (Taylor 1993), John Taylor advocates the use of monetary policy rules where the central banker reacts to both price inflation and output as a benchmark to be used judgementally.

In the simplest neo-Keynesian model with staggered prices only, this Taylor rule is key in ruling out sun-spot fluctuations. The Taylor principle associated to such rules states that the central banker should overreact to inflation to ensure the uniqueness of the solution under rational expectations.

Extensions to models with both staggered prices and wages have emphasized the welfare optimization problem while assuming, backed on numerical simulations, that the Taylor principle still holds in these cases (Erceg, Henderson, and Levin 2000; Galí 2008).

Flaschel, Franke, and Proaño (2008) derive analytically the frontier of determinacy in the same model but on the limit case of continuous time using a simpler strategy. Although the authors expect their demonstration in continuous time to be informative of the discrete time case, based on the precept that such a property should not depend on the time length of the period, they acknowledge in their concluding remarks that their intuition is “not unchallenged”. In this paper we prove them right: the Taylor principle does hold in the discrete time version of the model and is a generalization of their result.

We consider the same model as (Galí 2008, chapter 6) or (Erceg, Henderson, and Levin 2000) and a monetary policy rule in line with Erceg et al.’s results: the central banker can react to both inflations and the output gap. With straightforward notations, the Taylor rule takes the following form:

it=Φpπtp+Φwπtw+Φyyt (1)

We show that the necessary and sufficient condition to rule out sun-spot equilibria is symmetric in inflations:

Φp+Φw+1βκ˜Φy>1 (2)

with κ˜ a coefficient depending symmetrically on both slopes of the prices and wages Phillips curves and β the discount factor. This frontier, taking the form of a condition on Φpw eased as Φy increases, is in line with numerical investigations (Galí 2008, chapter 6).

Though the symmetry may not appear straightforward, it is deeply rooted in the model. In the simple model with staggered prices only, the Phillips curve implies that stabilizing price inflation is equivalent to stabilizing the output gap, a result (Blanchard and Galí 2007) present as a divine coincidence because it allows the central banker to enforce the social optimum. Blanchard and Galí (2007) note a weaker form of the divine coincidence in Erceg et al.’s model: combining the two Phillips curves yields that stabilizing the output gap is equivalent to stabilizing a weighted average of price and wage inflation (with the weight on each inflation being the slope of the other Phillips curve). Similar symmetry arises when studying the optimal monetary policy (see the functional form of the welfare criterion derived by Galí and Erceg et al.) as well as in our demonstration.

Our result generalizes that of Flaschel, Franke, and Proaño (2008) in the limit case of continuous time.

It is also a direct generalization of the frontier derived in the simpler case of staggered prices only (Woodford 2001). Moreover it has an identical interpretation in the long run: the central banker should react more than one for one to permanent changes in inflation (Woodford 2011, chapter 4).

In the remainder of this paper, Section 2 recalls the model, Section 3 outlines the proof detailed in Appendix A and Section 4 concludes.

2 A monetary model with sticky wages and prices

We study the model exposed in (Galí 2008, chapter 6) and (Erceg, Henderson, and Levin 2000). This model extends the standard neo-Keynesian model for monetary policy analysis which consist of an IS curve relating the output gap to the expected real interest rate, a Phillips curve relating inflation, expected inflation and output gap and a monetary policy rule describing how the interest rate is set by the central banker. The present extension of the model considers wage rigidities under the form of Calvo contracts. It follows from this rigidity that real wages may deviate from their flexible equivalent due to exogenous disturbances.

The model takes the following linear form:

πtp=βE(πt+1p|t)+κpyt+λpωt (3)

πtw=βE(πt+1w|t)+κwytλwωt (4)

ωt1=ωtπtw+πtp+Δωtn (5)

yt=E(yt+1|t)1σ(itE(πt+1p|t)rtn) (6)

it=Φpπtp+Φwπtw+Φyyt+vt (7)

with πp price inflation, πw wage inflation, ω the real wage gap, y output and i the nominal interest rate. Δωn is an exogenous wage gap shock, rn the natural real interest rate and v a monetary policy shock.

The complete derivation of the model is exposed in full details in (Galí 2008, chapter 6) with the same notations.

Denoting xt=[yt,πtp,πtw,ωt1]T, the endogenous variables, and zt=[rtnvt,Δωtn]T, the exogenous variables, the equations (3) to (7) can be written in the form:

xt=A1(E(xt+1|t)+Bzt) (8)

In equation (8), the matrix of interest A is:

A=[1+κpσβ+ΦyσβΦp1λpσββΦw+λpσβλpσβκpβ1+λpβλpβλpβκwβλwβ1+λwβλwβ0111] (9)

There are three forward looking variables in this model: [yt,πtp,πtw]. According to (Blanchard and Kahn 1980), the system (8) has a unique solution if and only if the matrix A defined by (9) has three eigenvalues strictly larger than one in modulus and one eigenvalue smaller than one in modulus. In addition, this solution has no unit root if this last eigenvalue is strictly smaller than one in modulus.

3 Outline of the proof

Defining the frontier of indeterminacy is based on the study of the roots of the characteristic polynomial of matrix A, a fourth degree polynomial. Though it is not complex mathematics, it is rather cumbersome.

We are particularly grateful to Yvon Maday and other mathematicians at Laboratoire Jacques-Louis Lions for proofreading and comments.

In Appendix A.1 we develop the proof in the case Φy=0. We use the intuition that in this case the frontier of determinacy is Φpw=1 and decompose the polynomial as a fourth degree polynomial corresponding to this case plus deviations from this case in both directions (Φp, Φw). In the limit case Φpw=1, the model is at the limit of solution determinacy: 1 is a root of its characteristic polynomial; its real roots are non-negative and at most one real root is in ]0, 1[; its complex roots have a modulus strictly greater than one. The deviation from this limit case ensures the uniqueness of the model’s solution if and only if the deviation from Φpw=1 is positive: the root 1 moves in the direction ensuring the uniqueness of the model’s solution (depending on the existence of another root smaller than one), the other roots are kept outside or inside the unit disk depending on their initial position.

In Appendix A.2 we show that the case Φy≠0 can be treated identically to the case Φy=0. We consider the frontier of indeterminacy under the form Φpw=1–θ and show that setting θ=Φy(1β)(λp+λw)κwλp+κpλw allows a decomposition of the characteristic polynomial which has the same properties as in the case Φy=0. We can conclude that equation (10) generalizes the frontier of indeterminacy.

4 Conclusion

In the textbook neo-Keynesian model with both staggered prices and wages, we show that any monetary policy rule satisfying

Φp+Φw+Φy(1β)(λp+λw)κwλp+κpλw>1 (10)

rules out sunspot fluctuations.

Using Dynare (Adjemian et al. 2011), it is possible to verify numerically frontier (10); code available upon request.

The admissibility of a policy rule symmetrically depends on wage inflation and price inflation: when the central bank does not respond to changes in output, the condition on the monetary policy parameters comes down to Φpw>1 in line with Galí’s numerical investigations. Also in line with Galí’s numerical investigations, when the central bank reacts to changes in output, doing so relaxes the constraint above, proportionally to Φy with a factor (1β)(λp+λw)κwλp+κpλw. This coefficient crucially and symmetrically depends on the Phillips curves of prices and wages: more impatient agents (smaller β) or flatter Phillips curves (smaller λ or κ), facilitate the task of the central banker to prevent sun spot fluctuations.

In continuous time, Flaschel, Franke, and Proaño (2008) derive an identical result; determinacy holds under the following condition:

Φp+Φw+Φyρ(λp+λw)κwλp+κpλw>1 (11)

with ρ related to the discount factor as follows 1/β(t)=1+ρt. In (11) ρ is the limit of (1–β(t))/t when the time step t tends to zero, while the time step is one in (10). Hence the combination of these results proves, as expected by Flaschel et al., that the determinacy condition of this model can be taken to the limit of continuous time.

In a model with staggered prices only, the frontier of determinacy is (12) (Woodford 2001), of which our result (10) is a direct generalization.

Φp+1βκpΦy>1 (12)

With both rigidities, a permanent shift in price inflation (π˜) implies an identical permanent shift in wage inflation [equation (5)]. The Phillips curves [equations (3) and (4)] imply a proportional shift in output gap y˜=(1β)(λp+λw)κwλp+κpλwπ˜. In turn, the Taylor rule (7) implies that the reaction of the central banker is to raise the nominal interest rate by i˜=[Φp+Φw+Φy(1β)(λp+λw)κwλp+κpλw]π˜. Thus, as in the standard neo-Keynesian model without wage rigidities (Woodford 2011, chapter 4), our frontier of indeterminacy can also be interpreted as a strict Taylor principle applied to permanent shocks: the central banker should react more than one for one to permanent changes in inflation.

Finally, if wage inflation and price inflation play similar roles for the design of the optimal monetary policy (Erceg, Henderson, and Levin 2000; Galí 2008), we show that they also play symmetric roles for eliminating sun-spot fluctuations. This extended conclusion remains “at odds with the practice of most central banks, which seem to attach little weight to wage inflation as a target variable” (Galí 2008).

Acknowledgments

We are grateful to Jordi Galí for raising this problem during the 2009 Barcelona Macroeconomic Summer School; to Michel Juillard and Benoit Mojon for the opportunity to attend this summer school; to Yvon Maday (LJLL) and other mathematicians at LJLL for proofreading; to Olivier Loisel for insightful suggestions regarding the interpretation of the result; to Jean-Michel Grandmont and Edouard Challe for their improving comments and advice.

Appendix

A.1 The case for Φy=0

Based on numerical evidence (Galí 2008), we study the problem in deviation from the limit case Φwp=1, we introduce a new parameter ϕp and use the following parametrization:

Φp=ϕp+ξ   Φw=ϕw+γ=1ϕp+γ (13)

0<ϕp<1,   ξ s.t.   Φp>0,   γ s.t.   Φw>0 (14)

The domain of interest, Dp,w, is displayed on Figure 1. This parametrization of Dp,w is not injective, as three parameters (ϕp, ξ, γ) describe a two dimensional domain, but this choice makes the study easier. In particular, it allows to use deviations of the same sign in both directions to describe the two domains of solution determinacy and indeterminacy.

Figure 1:
Figure 1:

Domain of interest (solution determinacy dashed, indeterminacy dotted) and parametrization.

Citation: The B.E. Journal of Macroeconomics 16, 2; 10.1515/bejm-2014-0160

Let X denote the vector of parameters:

X=[β,ϕp,κp,κwλp,λw,σ]   ]0,1[×]0,1[×(+)5=DX. (15)

from which γ and ξ are excluded and hereafter treated as special parameters.

The characteristic polynomial of the matrix defined in (9) can be expressed as follows, with implicit dependency on X:

Pγ,ξ(t)=at4bt3+cγ,ξt2dγ,ξt+eγ,ξ. (16)

with the coefficients:

a=σβ2b=β[κp+σ(2+2β+λp+λw)]cγ,ξ=κp[1+β+λw+βξ]+κw[λp+βγ]+σ[1+4β+β2+(λp+λw)(1+β)]+βϕpκp+β(1ϕp)κwdγ,ξ=κp[1+λw(1+γ+ξ)+(1+β)ξ]+κw[λp(1+γ+ξ)+(1+β)γ]+σ[2(1+β)+λp+λw]+(1+β)κpϕp+(1+β)κw(1ϕp)eγ,ξ=σ+κpξ+κwγ+κpϕp+κw(1ϕp)

To study the eigenvalues of this polynomial in deviation from the limit case ϕp+ϕw=1, we use the simplifying notations P0=P0,0, c=c0,0, d=d0,0 and e=e0,0 and the following decomposition:

Pγ,ξ(t)=P0(t)+γκwQ(t)+ξκpS(t)   where (17)

Q(t)=βt2[λwκpκw+(1+β+λp)]t+1 (18)

S(t)=βt2[λpκwκp+(1+β+λw)]t+1 (19)

It is noteworthy that the deviation in direction Φp (ξκpS(t)) is exactly symmetric to the deviation in direction Φw (γκwQ(t)).

Property A.1:For every vector of parameters XDX, (a,b)(+)2 andγ≥–ϕp, ∀ξϕp–1, (cγ,ξ,dγ,ξ,eγ,ξ)(+)3. This implies thatt≤0, Pγ,ξ(t)>0.

Proof: From their definition and the definition (15) of DX, coefficients a to eγ,ξ are all positive. The sign of the polynomial Pγ,ξ(t) derives from (16).■

Property A.2:The limit case polynomial P0verifies the following properties:

  • 1 is a root of P0which can be writtenP0(t)=(t–1)R0(t) withR0(t)=at3(ba)t2+(ab+c)t+ab+cd
  • P0has no negative root,
  • P0has at most two complex roots, these roots are conjugate and their modulus is strictly larger than one
  • ifP0has a multiple root in 1, it is a double one and the other roots are real and larger than one
  • P0has at most one root in ]0, 1[

Proof:

  • From the definition of P0 and its coefficients, one can check that ab+cd+e= P0(1)=0 ♦
  • This property derives directly from Property A.1 ♦
  • P0 is a fourth degree polynomial with real coefficients and one real root (1). It follows that it has at most two complex roots which are conjugate. These roots are also the roots of polynomial R0 which we denote z and z̅ and let r denote the remaining real root. We can write:R0(t)=at3(ba)t2+(ab+c)t+ab+cd=a(tr)(tz)(tz¯)=a(tr)(t2+|z|2)=at3art2+a|z|2tr|z|2propertyA.6r=baa>3&|z|2=ab+ca>3Thus z, z̅ are outside the unit disk. ♦
  • We showed above that if P0 has complex roots, the two remaining real roots are 1 and r>3. So 1 can not be a double root of P0 in this case. Property A.8 implies that if P0(1)=0 then P0(1)>0, thus 1 can not be a triple root of P0. If 1 is a double root of P0, the two other roots, necessarily real and positive, can not be both smaller than one as the product of the four is larger than one. If we assume that one of them is smaller than one, a table of variation of P0 shows that it implies that P″(1)<0, which contradicts Property A.8. ♦
  • We showed above that if P0 has complex roots, the two remaining real roots are 1 and r>3.If P0 has four real roots, one of them is 1 and their product is ea>1. Thus at least one of its roots is strictly larger than 1.

Property A.7 also implies if P0 has four real roots, that 3b2>8ac and P0 has two real roots, denoted t and t+. As P0(t)=12at26bt+2c, we obtain that:

t±=b4a±((b4a)2c6a)12.

We proceed by contradiction and assume that P0 has two roots ∈]0, 1[. A table of variations of P0 and its derivatives show that this assumption implies P0(1)<0 and t<1.

0<b4a1<((b4a)2c6a)121b2a<c6a6a3b+c<0P0(1)<0.

This is in contradiction with Property A.8, both the first and second derivatives of P0 can not be negative in 1. Thus P0 has at most one root in ]0, 1[.■

Property A.3:There exist a neighborhood of 1 where both Q(t)<0 and S(t)<0

Proof: From their definition, it follows that both Q and S are strictly negative in 1, so there is a neighborhood of 1 were each of them is negative. On the intersection of these neighborhoods they are both negative.■

Property A.4:ξ>0, γ≥0 orγ>0, ξ≥0 (that is on any point in the dashed domain on Figure 1), Pγ,ξhas only one root within the unit disk. This root is in ]0, 1[and if it has complex roots they remain outside the unit disk.

Proof: We consider the dynamic of the roots of Pγ,ξ starting from the limit case P0. Due to Property A.1, the real roots of Pγ,ξ can not be negative. Property A.2 implies that if one is a single root of P0, there is at most one root of P0 in ]0, 1[, 1 is another of its roots and the remaining real roots, if any, are strictly greater than one. The other possible configuration is, 1 is a double root of P0, the other roots are real and strictly greater than one. Applying Lemma A.1 to this configuration shows that:

  • if P0 has a root strictly smaller than one, then ∀ξ>0, γ≥0 or ∀γ>0, ξ≥0 the root 1 will shift to the right,
  • if P0 has no root strictly smaller than one, then ∀ξ>0, γ≥0 or ∀γ>0, ξ≥0 the root 1 will shift to the left,
  • if 1 is a double root of P0, then ∀ξ>0, γ≥0 or ∀γ>0, ξ≥0, the root 1 will split on both sides of the unit disk.

Still in application of Lemma A.1, 1 is a repulsive point for the dynamic of the roots of Pγ,ξ: with γ and ξ increasing, if a root of Pγ,ξ enters the neighborhood of 1 were both Q and S are negative, this root will be pushed back to the left (resp. right) if it is smaller (resp. larger) than one.

Thus ∀ξ>0, γ≥0 or ∀γ>0, ξ≥0, Pγ,ξ has one real root in ]0, 1[ and its other real roots are larger than 1.

If Pγ,ξ has complex roots, it can have only two complex conjugate roots, since Pγ,ξ(0)>0, limt→+∞Pγ,ξ(t)=+∞ and Pγ,ξ(1)<0. Let r± denote the real roots of Pγ,ξ and z, z̅ its complex roots. A fortiorir<1<r+.

Pγ,ξ(t)=a(tr+)(tr)(t2+|z|2) (20)

=a(t4(r++r)t3+(|z|2+r+r)t2(r++r)|z|2t+r+r|z|2) (21)

By identification we get

r++r=ba|z|2+r+r=cγ,ξa(r++r)|z|2=dγ,ξar+r|z|2=eγ,ξar<1r+<baa|z|2+r+r=cγ,ξa<baa+|z|2cγ,ξb+aa<|z|2PropertyA.63<cγ,ξb+aa<|z|2

Hence, if Pγ,ξ has complex roots, their modulus is strictly larger than one.■

Property A.5:ξ<0 orγ<0 (that is on any point in the dotted domain on Figure 1), Pγ,ξhas two or zero real roots and two or zero complex roots within the unit disk.

Proof: Symmetrically to the previous one, this property derives from the application of Lemma A.1 with ξ<0 or γ<0 in the neighborhood of one.

  • if P0 has a root strictly smaller than one, then with ξ<0 or γ<0 the root 1 shifts to the left,
  • if P0 has no root strictly smaller than one, then with ξ<0 or γ<0 the root 1 shifts to the right,
  • if 1 is a double root of P0, then with ξ<0 or γ<0 the root 1 becomes two complex conjugate roots.

From this configuration, 1 is a repulsive point for the dynamic of the roots of Pγ,ξ which implies that this configuration remains with γ or ξ decreasing.

As for the complex roots, they are conjugate thus either both inside or outside the unit disk.■

Theorem A.1:For every XDX, if Φy=0, for all (Φp,Φw)(+)2, model (3) to (7) has a unique and stable solution if and only if Φpw>1

Proof: Properties A.4 and A.5 imply that for all admissible values of γ and ξ (namely such that Φp>0 and Φw>0), Pγ,ξ has only one root smaller than one in modulus and three roots larger than one in modulus if and only ifγ+ξ>0, i.e. Φpw>1. The Blanchard and Kahn (1980) condition implies that this configuration of the roots of Pγ,ξ is equivalent to the model’s solution determinacy.■

A.2 The case for Φy≠0

A priori the frontier of indeterminacy in this case can be written ϕp+ϕw=1–θ, with θ positive, decreasing with Φy and to be determined (Galí 2008). We introduce the following parametrization:

Φp=ϕp+ξ   Φw=1ϕpθ+γ   Φy=ϕy+ζ (22)

0<ϕp<1   0<θ<1ϕp (23)

ξ s.t   Φp>0,   γ s.t.   Φw>0,   ζ s.t.   Φy>0 (24)

The characteristic polynomial P of A is now defined by:

Pγ,ξ,ζ(t)=at4bζt3+cγ,ξ,ζt2dγ,ξ,ζt+eγ,ξ,ζ.

It is equivalent to (16) with augmented coefficients:

a=σβ2bζ=β[κp+Φyβ+σ(2+2β+λp+λw)]cγ,ξ,ζ=κp[1+β+λw+βξ]+κw[λp+βγ]+σ[1+4β+β2+(λp+λw)(1+β)]+βϕpκp+β(1θϕp)κw+Φyβ[λp+λw+2+β]dγ,ξ,ζ=κp[1+λw(1+γ+ξ)+(1+β)ξ]+κw[λp(1+γ+ξ)+(1+β)γ]+σ[2(1+β)+λp+λw]+(1+β)κpϕp+(1+β)κw(1θϕp)+Φy[1+λw+λp+2β](κpλw+κwλp)θeγ,ξ,ζ=σ+κpξ+κwγ+κpϕp+κw(1θϕp)+Φy.

We define the baseline polynomial by choosing {θ, ϕy} such that P0(1)=0, which implies that θ is proportional to ϕy:

θ=ϕy(1β)(λp+λw)κwλp+κpλw. (25)

Our limit case, i.e. with ζ=γ=ξ=0, is:

ϕp+ϕw+ϕy(1β)(λp+λw)κwλp+κpλw=1 (26)

Let now X denote the extended vector of parameters:

X=[β,ϕp,θ,κp,κwλp,λw,σ]]0,1[3×(+)5=DX. (27)

We can decompose polynomial Pγ,ξ,ζ similarly to (17):

Pγ,ξ(t)=P0(t)+γκwQ(t)+ξκpS(t)+ζT(t)   where (28)

Q(t)=βt2[λwκpκw+(1+β+λp)]t+1 (29)

S(t)=βt2[λpκwκp+(1+β+λw)]t+1 (30)

T(t)=βt3+β(λp+λw+2+β)t2(1+λp+λw+2β)t+1 (31)

From the definition of Pγ,ξ,ζ and its coefficients, we can generalize Property A.1 to the case Φy≠0. The limit case polynomial also verifies Property A.2 (the proof is identical). Since T(1)<0, we can generalize Property A.3 as well. All the developments derived in the case Φy≠0 apply and Theorem A.1 can be generalized:

Theorem A.2:For every XDX, for all (Φp,Φw,Φy)(+)3, model (3) to (7) has a unique and stable solution if and only ifΦp+Φw+Φy(1β)(λp+λw)κwλp+κpλw>1

A.3 Some useful properties

A.3.1 On the polynomial coefficients if Φy≠0

Property A.6:The following inequalities hold:

XDX   γ0   ξ0   b>4a   cγ,ξ>2a+b>6a   eγ,ξ>a

The same inequalities hold for the coefficients of Pγ,ξ,ζwhen Φy≠0.

Proof: These inequalities derive directly for the definitions of the coefficients.■

Property A.7:XDX, ∀γ≥0, ∀ξ≥0 Pγ,ξhas two conjugate complex roots if and only if 3b2<8acγ,ξ

The same property applies toPγ,ξ,ζwhen Φy≠0.

Proof: The discriminant of the second order derivative Pγ,ξ is Δ(Pγ,ξ)=12[3b28acγ,ξ]; if it is negative, this polynomial is positive for every t. In this case, Pγ,ξ is strictly increasing and has only one real root, and Pγ,ξ is strictly decreasing and then strictly increasing. As we already know that it has two real roots, the two others are complex.■

Property A.8:Whether Φy≠0 or not, with the respective definitions of polynomial P0and the parameters domain DX: for all XDX, P0(1)=4a3b+2cd and P0(1)=12a6b+2c cannot be both negative or null.

Proof:If Φy≠0:

P0(1)=4a3b+2cd=κpλw+κwλp(1β)((1ϕp)(κwκp)σ(λp+λw))

12P0(1)=6a3b+c=κp[1β+λw]+κwλp+σ[(1β)2+(λp+λw)(1β)]+β((1ϕp)(κwκp)σ(λp+λw))

It is a necessary condition for P0(1) to be negative or zero that:

(1ϕp)(κwκp)σ(λp+λw)>0 (32)

However, if this is the case, then

12P0(1)>κp[1β+λw]+κwλp+σ[(1β)2+(λp+λw)(1β)]>0 (33)

If Φy≠0:

P0(1)=4a3b+2cd=κpλw+κwλp(1β)((1ϕp)(κwκp)+ϕy(1β)σ(λp+λw)θκw)

12P0(1)=6a3b+c=κp[1β+λw]+κwλp+σ[(1β)2+(λp+λw)(1β)]+ϕyβ(λp+λw)+β((1ϕp)(κwκp)+ϕy(1β)σ(λp+λw)θκw)

It is a necessary condition for P0(1) to be negative or zero that:

(1ϕp)(κwκp)+ϕy(1β)σ(λp+λw)θκw>0 (34)

However, if this is the case, then

12P0(1)>κp[1β+λw]+κwλp+σ[(1β)2+(λp+λw)(1β)]+ϕyβ(λp+λw)>0 (35)

A.3.2 Real roots shift

Lemma A.1:Let P be a polynomial and λ a simple or double root of this polynomial.

If the root is simple (P(λ)=0 butP′(λ)≠0) and we add a real quantity qsufficiently small to the polynomial, it translates the root λin the direction defined by the sign ofP′(λ)q, + defining a translation to the right, – to the left.

If the root is double (P(λ)=0, P′(λ)=0 butP″(λ)≠0) and we add a real quantity q sufficiently small to the polynomial, the root becomes two distinctive roots on both sides of λifP′(λ)q>0 and become complex roots otherwise.

Proof: If P(λ)=0 but P′(λ)≠0, the property is a direct application of the implicit function theorem to f(t, q)=P(t)+q in the neighborhood of (t, q)=(λ, 0).

Let’s assume that P(λ)=0, P′(λ)=0 and P″(λ)>0. For regularity, we consider a neighborhood V of λ where P″>0. In V, a table of variation of P and its derivatives shows that P is strictly positive except in λ. If we add a positive q, arbitrarily small, to polynomial P, P+q becomes strictly positive on V. By a continuity argument the double root λ became two conjugate complex roots.

Also, there exists δ>0, arbitrarily small, and λ<λ<λ+ such that P(λ)=P(λ+)=δ. If we subtract q, arbitrarily small, to polynomial P, P(λ)–q is strictly negative while P(λ)–q=P(λ+)–q=δq can be set strictly positive. Since P′ is strictly monotonous on V, Pq has two roots in V, one on each side of the original root λ.

All in all, if P″(λ)>0, the double root become complex if we add a positive quantity, arbitrarily small, to P and two real roots on each side of the initial one if this quantity is negative. A symmetric argument completes the proof for the case if P″(λ)<0.

References

  • Adjemian, S., H. Bastani, M. Juillard, F. Karamé, F. Mihoubi, G. Perendia, J. Pfeifer, M. Ratto, and S. Villemot. 2011. “Dynare: Reference Manual, Version 4.” Dynare Working Papers, CEPREMAP, http://www.dynare.org.

  • Blanchard, O. and C. Kahn. 1980. “The Solution of Linear Difference Models under Rational Expectations.” Econometrica: Journal of the Econometric Society 48: 1305–1311.

  • Blanchard, O. and J. Galí. 2007. “Real Wage Rigidities and the New Keynesian Model.” Journal of Money, Credit and Banking 39: 35–65.

  • Erceg, C., D. Henderson, and A. Levin. 2000. “Optimal Monetary Policy with Staggered Wage and Price Contracts.” Journal of monetary Economics 46: 281–313.

  • Flaschel, P., R. Franke, and C. R. Proaño. 2008. “On Equilibrium Determinacy in New Keynesian Models with Staggered Wage and Price Setting.” The B.E. Journal of Macroeconomics 8. http://www.degruyter.com/view/j/bejm.2008.8.1/bejm.2008.8.1.1802/bejm.2008.8.1.1802.xml?format=INT.

  • Galí, J. 2008. Monetary Policy, inflation, and the Business Cycle: An introduction to the new Keynesian Framework and its applications. Princeton, NJ: Princeton University Press, http://press.princeton.edu/titles/10495.html.

  • Taylor, J. 1993. “Discretion versus Policy Rules in Practice.” Carnegie-Rochester Conference Series on Public Policy 39: 195–214.

  • Woodford, M. 2001. “The Taylor Rule and Optimal Monetary Policy.” The American Economic Review 91: 232–237.

  • Woodford, M. 2011. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton, NJ: Princeton University Press, http://press.princeton.edu/titles/7603.html.

Footnotes

1

We are particularly grateful to Yvon Maday and other mathematicians at Laboratoire Jacques-Louis Lions for proofreading and comments.

2

Using Dynare (Adjemian et al. 2011), it is possible to verify numerically frontier (10); code available upon request.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • Adjemian, S., H. Bastani, M. Juillard, F. Karamé, F. Mihoubi, G. Perendia, J. Pfeifer, M. Ratto, and S. Villemot. 2011. “Dynare: Reference Manual, Version 4.” Dynare Working Papers, CEPREMAP, http://www.dynare.org.

  • Blanchard, O. and C. Kahn. 1980. “The Solution of Linear Difference Models under Rational Expectations.” Econometrica: Journal of the Econometric Society 48: 1305–1311.

  • Blanchard, O. and J. Galí. 2007. “Real Wage Rigidities and the New Keynesian Model.” Journal of Money, Credit and Banking 39: 35–65.

  • Erceg, C., D. Henderson, and A. Levin. 2000. “Optimal Monetary Policy with Staggered Wage and Price Contracts.” Journal of monetary Economics 46: 281–313.

  • Flaschel, P., R. Franke, and C. R. Proaño. 2008. “On Equilibrium Determinacy in New Keynesian Models with Staggered Wage and Price Setting.” The B.E. Journal of Macroeconomics 8. http://www.degruyter.com/view/j/bejm.2008.8.1/bejm.2008.8.1.1802/bejm.2008.8.1.1802.xml?format=INT.

  • Galí, J. 2008. Monetary Policy, inflation, and the Business Cycle: An introduction to the new Keynesian Framework and its applications. Princeton, NJ: Princeton University Press, http://press.princeton.edu/titles/10495.html.

  • Taylor, J. 1993. “Discretion versus Policy Rules in Practice.” Carnegie-Rochester Conference Series on Public Policy 39: 195–214.

  • Woodford, M. 2001. “The Taylor Rule and Optimal Monetary Policy.” The American Economic Review 91: 232–237.

  • Woodford, M. 2011. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton, NJ: Princeton University Press, http://press.princeton.edu/titles/7603.html.

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    Domain of interest (solution determinacy dashed, indeterminacy dotted) and parametrization.