Stability, Multiplicity, and Sunspots
(deriving solutions to linearized system
&
Blanchard-Kahn conditions)
Wouter J. Den Haan
London School of Economics
c by Wouter J. Den Haan
September 2, 2015
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Content
1
A lot on sunspots
2
A simple way to get policy rules in a linearized framework
and an even simpler way based on time iteration
(an idea of Pontus Rendahl)
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Introduction
What do we mean with non-unique solutions?
multiple solution versus multiple steady states
What are sunspots?
Are models with sunspots scienti…c?
Multiplicity
Getting started
General Derivation
Examples
Terminology
De…nitions are very clear
(use in practice can be sloppy)
Model:
H (p+1 , p ) = 0
Solution:
p+1 = f (p )
Time iteration and linear solutions
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
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
Time iteration and linear solutions
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
Time iteration and linear solutions
“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
Time iteration and linear solutions
Large sunspots (around 2000 at the peak)
Multiplicity
Getting started
General Derivation
Examples
Past Sun Spot Cycles
Sun spots even had a "Great Moderation"
Time iteration and linear solutions
Multiplicity
Getting started
General Derivation
Examples
Current cycle (at peak again)
Time iteration and linear solutions
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Cute NASA video
https://www.youtube.com/watch?v=UD5VViT08ME
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Sunspots in economics
De…nition: a solution is a sunspot 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
Time iteration and linear solutions
Sunspots in economics (Cass & Shell 1983)
Non-sunspot solution:
pt = f (pt
1 , pt 2 ,
, dt , dt
1,
)
Sunspot:
pt = f (pt 1 , pt 2 ,
, dt , dt 1 ,
, st )
st : random variable with E [st+1 ] = 0
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Origin of sunspots in economics
William Stanley Jevons (1835-82) explored empirical
relationship between sunspot activity (that is, the real thing!!!)
and the price of corn.
Fortunately, Jevons had some other contributions as well, such
as "Jevons Paradox". His work is considered to be the start of
mathematical economics.
Multiplicity
Getting started
General Derivation
Jevons Paradox
Examples
Time iteration and linear solutions
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Sunspots and science
Why are sunspots attractive
sunspots: st matters, just because agents believe this
self-ful…lling expectations don’t seem that unreasonable
sunspots provide many sources of shocks
number of sizable fundamental shocks small
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Sunspots and science
Why are sunspots not so attractive
Purpose of science is to come up with predictions
If there is one sunspot 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
Examples
Time iteration and linear solutions
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 sunspots occur?
Numerical algorithms and sunspots
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Getting started
Model: yt = ρyt
1
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Getting started
Model: yt = ρyt
1
in…nite number of solutions, independent of the value of ρ
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Getting started
Model:
yt+1 = ρyt
y0 is given
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Getting started
Model:
yt+1 = ρyt
y0 is given
unique solution, independent of the value of ρ
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
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: unique 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
Time iteration and linear solutions
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
Time iteration and linear solutions
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
Time iteration and linear solutions
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
Time iteration and linear solutions
Neoclassical growth model and state space representation
3
3 2
32
3 2
32
eE,t+1
kt
-a1 -a2 -a4
kt+1
1 0 -a3
4 0 1 0 5 4 kt 5 + 4 -1 0
0 5 4 kt 1 5 = 4 0 5
ez,t+1
0
0 -ρ
zt
zt+1
0 0 1
2
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Dynamics of the state-space system
Ayt+1 + Byt = εt+1
yt + 1 =
A
1
Byt + A
= Dyt + A
Thus
t
1
ε t+1
ε t+1
yt+1 = Dt y1 + ∑ Dt l A
l=1
1
1
ε l+1
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
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
Time iteration and linear solutions
Dynamics of the state-space system
t
yt+1 = Dt y1 + ∑ Dt l A
1
ε l+1
l=1
= PΛt P
1
t
y1 + ∑ PΛt l P
l=1
1
A
1
ε l+1
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Dynamics of the state-space system
1
multiplying dynamic state-space system with P
P
1
yt+1 = Λt P
1
t
y1 + ∑ Λt l P
1
A
gives
1
ε l+1
l=1
or
t
p̃i yt+1 = λti p̃i y1 + ∑ λti l p̃i A
1
ε l+1
l=1
recall that yt is n
1 and p̃i is 1
n. Thus, p̃i yt is a scalar
Multiplicity
Getting started
General Derivation
Examples
Model
1
2
3
4
p̃i yt+1 = λti p̃i y1 + ∑tl=1 λti l p̃i A 1 εl+1
E[εt+1 jIt ] = 0
m elements of y1 are not determined
yt cannot explode
Time iteration and linear solutions
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
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
Examples
Time iteration and linear solutions
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
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
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
Examples
Time iteration and linear solutions
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
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
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
Time iteration and linear solutions
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
Time iteration and linear solutions
How to think about #1 combined with #2?
p̃1 yt = 0 8t
Without sunspots
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
Time iteration and linear solutions
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
Examples
Time iteration and linear solutions
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 and Soderlind 1999)
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
How to check in Dynare
Use the following command after the model & initial conditions part
check;
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
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
Time iteration and linear solutions
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 )xt = (1 2L)(1 14 L)xt
φ1 = 3.5, φ2 = 3 : No stable solution; too many
eigenvalues > 1
(1 + φ1 L φ2 L2 )xt = (1 2L)(1 1.5L)xt
φ1 = 1, φ2 = 0.25 : Multiple stable solutions; too few
eigenvalues > 1
(1 + φ1 L φ2 L2 )xt = (1 0.5L)(1 0.5L)xt
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
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
Solutions to linear systems
1
2
3
The analysis outlined above
(requires A to be invertible)
Generalized version of analysis above
(see Klein 2000)
Apply time iteration to linearized system
(I learned this from Pontus Rendahl)
Time iteration and linear solutions
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Solutions to linear systems
Model:
Γ2 kt+1 + Γ1 kt + Γ0 kt
or
Γ2 0
0 1
kt+1
kt
+
Γ1 Γ0
1 0
1
=0
kt+1
kt
=0
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Standard approaches
1
The method outlined above
=) a unique solution of the form
kt = akt
1
if BK conditions are satis…ed
2
Impose that the solution is of the form
kt = akt
1
and solve for a from
Γ2 a2 kt
1
+ Γ1 akt
1
+ Γ0 kt
1
= 0 8kt
1
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Time iteration
Impose that the solution is of the form
kt = akt
1
Use time iteration scheme, starting with a[1]
Recall that time iteration means using the guess for tomorrows
behavior and then solve for todays behavior
(This simple procedure was pointed out to me by Pontus Rendahl)
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Time iteration
Follow the following iteration scheme, starting with a[1]
Use a[i] to describe next period’s behavior. That is,
Γ2 a[i] kt + Γ1 kt + Γ0 kt
1
=0
(note the di¤erence with last approach on previous slide)
Obtain a[i+1] from
( Γ 2 a [ i ] + Γ 1 ) kt + Γ 0 kt
kt =
a[i+1] =
Γ 2 a[i] + Γ 1
1
1
Γ 2 a[i] + Γ 1
=0
Γ0 kt
1
1
Γ0
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Advantages of time iteration
It is simple even if the "A matrix" is not invertible.
(the inversion required by time iteration seems less problematic
in practice)
Since time iteration is linked to value function iteration, it has
nice convergence properties
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
Example
kt+1
2kt + 0.75kt
1
=0
The two solutions are
kt = 0.5kt
1
& kt = 1.5kt
1
Time iteration on kt = a[i] kt 1 converges to stable solution for
all initial values of a[i] except 1.5.
Multiplicity
Getting started
General Derivation
Examples
Time iteration and linear solutions
References and Acknowledgements
Larry Christiano taught me (a long time ago) this simple way of deriving the BK
conditions and I think that I did not even change the notation.
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, Journal of Economic Dynamics and Control, 1405-1423.
in case you want to do the analysis without the simplifying assumption that A is
invertible
Soderlind, Paul, 1999, Solution and estimation of RE macromodels with optimal policy,
European Economic Review, 813-823
also doesn’t assume that A is invertible; possibly a more accessible paper