Stability, Multiplicity, and Sun Spots
(Blanchard-Kahn conditions)
Wouter J. Den Haan
London School of Economics
c 2011 by Wouter J. Den Haan
August 29, 2011
Multiplicity
Getting started
General Derivation
Introduction
What do we mean with non-unique solutions?
multiple solution versus multiple steady states
What are sun spots?
Are models with sun spots scienti…c?
Examples
Multiplicity
Getting started
General Derivation
Terminology
De…nitions are very clear
(use in practice can be sloppy)
Model:
H (p+1 , p ) = 0
Solution:
p+1 = f (p )
Examples
Multiplicity
Getting started
General Derivation
Examples
Unique solution & multiple steady states
2
p+1
1.5
1
0.5
45o
0
0
0.2
0.4
0.6
0.8
p
1
1.2
1.4
1.6
Multiplicity
Getting started
General Derivation
Examples
2
Multiple
solutions & unique (non-zero) steady state
1.8
1.6
1.4
p+1
1.2
1
0.8
0.6
0.4
0.2
45o
0
0
0.2
0.4
0.6
0.8
1
p
1.2
1.4
1.6
1.8
2
Multiplicity
Getting started
General Derivation
Examples
“Traditional” Pic
Multiple steady states & sometimes multiple solutions
ut+1
negative expectations
positive expectations
45o
ut
34
From Den Haan (2007)
Multiplicity
Getting started
General Derivation
Examples
Large sun spots (around 2000 at the peak)
Multiplicity
Getting started
General Derivation
Sun spot cycle (almost at peak again)
Examples
Multiplicity
Getting started
General Derivation
Examples
Sun spots in economics
De…nition: a solution is a sun spot solution if it depends on a
stochastic variable from outside system
Model:
0 = EH (pt+1 , pt , dt+1 , dt )
dt : exogenous random variable
Multiplicity
Getting started
General Derivation
Examples
Sun spots in economics
Non-sun-spot solution:
pt = f (pt
1 , pt 2 ,
, dt , dt
1,
)
Sun spot:
pt = f (pt 1 , pt 2 ,
, dt , dt 1 ,
, st )
st : random variable with E [st+1 ] = 0
Multiplicity
Getting started
General Derivation
Examples
Sun spots and science
Why are sun spots attractive
sun spots: st matters, just because agents believe this
self-ful…lling expectations don’t seem that unreasonable
sun spots provide many sources of shocks
number of sizable fundamental shocks small
Multiplicity
Getting started
General Derivation
Examples
Sun spots and science
Why are sun spots not so attractive
Purpose of science is to come up with predictions
If there is one sun spot solution, there are zillion others as well
Support for the conditions that make them happen not
overwhelming
you need su¢ ciently large increasing returns to scale or
externality
Multiplicity
Getting started
General Derivation
Overview
1
Getting started
simple examples
2
General derivation of Blanchard-Kahn solution
When unique solution?
When multiple solution?
When no (stable) solution?
3
4
When do sun spots occur?
Numerical algorithms and sun spots
Examples
Multiplicity
Getting started
General Derivation
Getting started
Model: yt = ρyt
1
Examples
Multiplicity
Getting started
General Derivation
Examples
Getting started
Model: yt = ρyt
1
in…nite number of solutions, independent of the value of ρ
Multiplicity
Getting started
General Derivation
Getting started
Model:
yt+1 = ρyt
y0 is given
Examples
Multiplicity
Getting started
General Derivation
Getting started
Model:
yt+1 = ρyt
y0 is given
unique solution, independent of the value of ρ
Examples
Multiplicity
Getting started
General Derivation
Examples
Getting started
Blanchard-Kahn conditions apply to models that add as a
requirement that the series do not explode
yt+1 = ρyt
Model:
yt cannot explode
ρ > 1: nique solution, namely yt = 0 for all t
ρ < 1: many solutions
ρ = 1: many solutions
be careful with ρ = 1, uncertainty matters
Multiplicity
Getting started
General Derivation
Examples
State-space representation
Ayt+1 + Byt = εt+1
E [εt+1 jIt ] = 0
yt :
is an n 1 vector
m n elements are not determined
some elements of εt+1 are not exogenous shocks but prediction errors
Multiplicity
Getting started
General Derivation
Examples
Neoclassical growth model and state space representation
E
"
exp(zt )ktα 1 + (1 δ)kt 1
β (exp(zt+1 )ktα + (1 δ)kt
α exp (zt+1 ) ktα
1
+1
γ
kt
=
kt+1 ) γ
δ
or equivalently without E[ ]
exp(zt )ktα 1 + (1 δ)kt 1
β (exp(zt+1 )ktα + (1 δ)kt
α exp (zt+1 ) ktα
+eE,t+1
1
+1
γ
kt
=
kt+1 ) γ
δ
It
#
Multiplicity
Getting started
General Derivation
Examples
Neoclassical growth model and state space representation
Linearized model:
kt+1 = a1 kt + a2 kt
1
+ a3 zt+1 + a4 zt + eE,t+1
zt+1 = ρzt + ez,t+1
k0 is given
kt is end-of-period t capital
=) kt is chosen in t
Multiplicity
Getting started
General Derivation
Examples
Neoclassical growth model and state space representation
3 2
32
a1 a2
kt+1
1 0 a3
4 0 1 0 5 4 kt 5 + 4 1 0
0 0
zt+1
0 0 1
2
3
3 2
32
eE,t+1
kt
a4
0 5 4 kt 1 5 = 4 0 5
ez,t+1
ρ
zt
Multiplicity
Getting started
General Derivation
Examples
Dynamics of the state-space system
Ayt+1 + Byt = εt+1
yt + 1 =
A
1
Byt + A
= Dyt + A
Thus
t+1
1
1
ε t+1
ε t+1
yt+1 = Dt y1 + ∑ Dt+1 l A
l=1
1
εl
Multiplicity
Getting started
General Derivation
Examples
Jordan matrix decomposition
D = PΛP
Let
1
Λ is a diagonal matrix with the eigen values of D
without loss of generality assume that jλ1 j jλ2 j
P
where p̃i is a (1
n) vector
1
2
3
p̃1
6
7
= 4 ... 5
p̃n
j λn j
Multiplicity
Getting started
General Derivation
Examples
Dynamics of the state-space system
t+1
yt+1 = Dt y1 + ∑ Dt+1 l A
1
εl
l=1
= PΛt P
1
t+1
y1 + ∑ PΛt+1 l P
l=1
1
A
1
εl
Multiplicity
Getting started
General Derivation
Examples
Dynamics of the state-space system
multiplying dynamic state-space system with P
P
1
yt + 1 = Λ t P
1
1
t+1
y1 + ∑ Λ t + 1 l P
gives
1
A
1
εl
l=1
or
t+1
p̃i yt+1 = λti p̃i y1 + ∑ λti +1 l p̃i A
1
εl
l=1
recall that yt is n
1 and p̃i is 1
n. Thus, p̃i yt is a scalar
Multiplicity
Getting started
General Derivation
Model
1
2
3
4
+1 t+1 l
p̃i yt+1 = λti p̃i y1 + ∑tl=
p̃i A 1 εl
1 λi
E[εt+1 jIt ] = 0
m elements of y1 are not determined
yt cannot explode
Examples
Multiplicity
Getting started
General Derivation
Examples
Reasons for multiplicity
1
2
There are free elements in y1
The only constraint on eE,t+1 is that it is a prediction error.
This leaves lots of freedom
Multiplicity
Getting started
General Derivation
Eigen values and multiplicity
Suppose that jλ1 j > 1
To avoid explosive behavior it must be the case that
1
2
p̃1 y1 = 0 and
p̃1 A 1 εl = 0 8l
Examples
Multiplicity
Getting started
General Derivation
Examples
How to think about #1?
p̃1 y1 = 0
Simply an additional equation to pin down some of the free
elements
Much better: This is the policy rule in the …rst period
Multiplicity
Getting started
General Derivation
How to think about #1?
p̃1 y1 = 0
Neoclassical growth model:
y1 = [k1 , k0 , z1 ]T
jλ1 j > 1, jλ2 j < 1, λ3 = ρ < 1
p̃1 y1 pins down k1 as a function of k0 and z1
this is the policy function in the …rst period
Examples
Multiplicity
Getting started
General Derivation
Examples
How to think about #2?
p̃1 A
1
εl = 0 8l
This pins down eE,t as a function of εz,t
That is, the prediction error must be a function of the
structural shock, εz,t , and cannot be a function of other shocks,
i.e., there are no sunspots
Multiplicity
Getting started
General Derivation
Examples
How to think about #2?
p̃1 A
1
εl = 0 8l
Neoclassical growth model:
p̃1 A 1 εt says that the prediction error eE,t of period t is a …xed
function of the innovation in period t of the exogenous process,
ez,t
Multiplicity
Getting started
General Derivation
Examples
How to think about #1 combined with #2?
p̃1 yt = 0 8t
Without sun spots
i.e. with p̃1 A
1ε
t
= 0 8t
kt is pinned down by kt
1
and zt in every period.
Multiplicity
Getting started
General Derivation
Examples
Blanchard-Kahn conditions
Uniqueness: For every free element in y1 , you need one λi > 1
Multiplicity: Not enough eigenvalues larger than one
No stable solution: Too many eigenvalues larger than one
Multiplicity
Getting started
General Derivation
How come this is so simple?
In practice, it is easy to get
Ayt+1 + Byt = εt+1
How about the next step?
yt + 1 =
A
1
Byt + A
1
ε t+1
Bad news: A is often not invertible
Good news: Same set of results can be derived
Schur decomposition (See Klein 2000)
Examples
Multiplicity
Getting started
General Derivation
Examples
How to check in Dynare
Use the following command after the model & initial conditions part
check;
Multiplicity
Getting started
General Derivation
Examples
Example - x predetermined - 1st order
xt
1
zt
= Et [φxt + zt+1 ]
= 0.9zt 1 + εt
jφj > 1 : Unique stable …xed point
jφj < 1 : No stable solutions; too many eigenvalues > 1
Multiplicity
Getting started
General Derivation
Examples
Example - x predetermined - 2nd order
φ 2 xt
1
zt
= Et [φ1 xt + xt+1 + zt+1 ]
= 0.9zt 1 + εt
φ1 = 2.25, φ2 = 0.5 : Unique stable …xed point
(1 + φ1 L φ2 L2 )x = (1 2L)(1 14 L)
φ1 = 3.5, φ2 = 3 : No stable solution; too many
eigenvalues > 1
(1 + φ1 L φ2 L2 )x = (1 2L)(1 1.5L)
φ1 = 1, φ2 = 0.25 : Multiple stable solutions; too few
eigenvalues > 1
(1 + φ1 L φ2 L2 )x = (1 0.5L)(1 0.5L)
Multiplicity
Getting started
General Derivation
Examples
Example - x not predetermined - 1st order
xt = Et [φxt+1 + zt+1 ]
zt = 0.9zt 1 + εt
jφj < 1 : Unique stable …xed point
jφj > 1 : Multiple stable solutions; too few eigenvalues > 1
Multiplicity
Getting started
General Derivation
Examples
References
Blanchard, Olivier and Charles M. Kahn, 1980,The Solution of
Linear Di¤erence Models under Rational Expectations,
Econometrica, 1305-1313.
Den Haan, Wouter J., 2007, Shocks and the Unavoidable Road to
Higher Taxes and Higher Unemployment, Review of Economic
Dynamics, 348-366.
simple model in which the size of the shocks has long-term
consequences
Farmer, Roger, 1993, The Macroeconomics of Self-Ful…lling
Prophecies, The MIT Press.
textbook by the pioneer
Klein, Paul, 2000, Using the Generalized Schur form to Solve a
Multivariate Linear Rational Expectations Model
in case you want to do the analysis without the simplifying
assumption that A is invertible