Taylor principle is valid under wage stickiness

Uniqueness of the solution to a monetary policy model

Alexis Blasselle & Aurélien Poissonnier

laboratoire Jacques-Louis Lions & Crest-Insee

October 2010

Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model

Introduction

Plan

1 Introduction

2 The model

3 The result

4 The demonstration strategy


Introduction

Purpose

Working on a DSGE model by Jordi Galí, we answer his question on the

condition on the monetary policy parameters that validates Blanchard and

Kahn condition.

This is a joint work with Alexis Blasselle, Phd student in applied

mathematics, to be published in Maths In Action.


The model

Plan

1 Introduction

2 The model

3 The result



The model

πpt = βE (πpt+1|t) + κpyt + λpωt (1)

πwt = βE (πwt+1|t) + κwyt − λwωt (2)

ωt−1 = ωt − πwt + πpt + ∆ωt (3)

yt = E (yt+1|t)− 1

σ(it − E (πp

t+1|t)− rnt ) (4)

it = Φpπp

t + Φwπwt + Φyyt + vt (5)


The model

λp =(1−θp)(1−βθp)

θp1−α

1−α+αεp, with

0 < θp < 1, is the Calvo parameter on prices

0 < α < 1, is the Cobb-Douglas parameter

0 < εp ≤ 1, is the mark-up on goods

⇒ 0 < λp

λw = (1−θw )(1−βθw )θw (1+ϕεw ) , with

0 < θw < 1, is the Calvo parameter on wages

0 < ϕ, is the Frisch elasticity

0 < εw ≤ 1, is the mark-up on wage setting

⇒ 0 < λw

κp =αλp1−α , we will later denote λpnp = κp with np > 0

κw = λw (σ + ϕ1−α), we will later denote λwnw = κw with nw > 0


The result

Plan

1 Introduction

2 The model

3 The result



The result

The Blanchard and Kahn condition is veri�ed if and only if

Φp + Φw + Φy

(1− β)

(nw + np)

(1

λp+

1

λw

)> 1 (6)

This result is symmetric in price and wages in�ation and depends only on

the Phillips curve parameters


The demonstration strategy

Plan

1 Introduction

2 The model

3 The result




We �rst study the limit case Φp + Φw = 1 and Φy = 0.

The characteristic polynomial of the matrice is :

Pγ,ξ(t) = at4 − bt3 + cγ,ξt2 − dγ,ξt + eγ,ξ or

Pγ,ξ(t) = P0(t) + γ λw Q(t) + ξ λp S(t) where

Q(t) = βnw t2 − [κp + nw (1 + β + λp)]t + nw

S(t) = βnp t2 − [κw + np(1 + β + λw )]t + np



The model has one predetermined variable.

We want to know under which conditions three roots of this polynomial are

strictly larger than on in modulus



1 is a root of the characteristic polynomial and at least two others are

outside the unit circle

Then we study how the eigenvalues of the model are modi�ed by changes

in Φp or Φw .

Doing so, we �nd some speci�c conditions veri�ed by the parameters that

leads us to conclude that :

Blanchard and Kahn condition is veri�ed if and only if Φp + Φw > 1



Then we study the case where Φy 6= 0

We �nd that if Φp + Φw + Φy(1−β)

(nw+np)

(1λp

+ 1λw

)= 1 1 is a root of the

characteristic polynomial

We then �nd that only departing positively from this condition ensures

Blanchard and Kahn condition



The Blanchard and Kahn condition is veri�ed if and only if

Φp + Φw + Φy

(1− β)

(nw + np)

(1

λp+

1

λw

)> 1 (7)


Taylor principle is valid under wage stickiness

Economy & Finance

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