taylor principle is valid under wage stickiness
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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
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model
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.
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model
The model
Plan
1 Introduction
2 The model
3 The result
4 The demonstration strategy
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model
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)
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model
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
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model
The result
Plan
1 Introduction
2 The model
3 The result
4 The demonstration strategy
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model
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
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model
The demonstration strategy
Plan
1 Introduction
2 The model
3 The result
4 The demonstration strategy
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model
The demonstration strategy
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
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model
The demonstration strategy
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
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model
The demonstration strategy
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
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model
The demonstration strategy
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
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model
The demonstration strategy
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)
Alexis Blasselle & Aurélien Poissonnier Uniqueness of the solution to a monetary policy model