Stability, Multiplicity, and Sun Spots (Blanchard-Kahn...
Transcript of Stability, Multiplicity, and Sun Spots (Blanchard-Kahn...
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Stability, Multiplicity, and Sun Spots(Blanchard-Kahn conditions)
Wouter J. Den HaanLondon School of Economics
c© 2011 by Wouter J. Den Haan
August 29, 2011
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Multiplicity Getting started General Derivation Examples
Introduction
• What do we mean with non-unique solutions?• multiple solution versus multiple steady states
• What are sun spots?• Are models with sun spots scientific?
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Multiplicity Getting started General Derivation Examples
Terminology
• Definitions are very clear• (use in practice can be sloppy)
Model:
H(p+1, p) = 0
Solution:p+1 = f (p)
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Multiplicity Getting started General Derivation Examples
Unique solution & multiple steady states
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
0.5
1
1.5
2
p
p+1
45o
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Multiplicity Getting started General Derivation Examples
Multiple solutions & unique (non-zero) steady state
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
p
p +1
45o
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Multiplicity Getting started General Derivation Examples
Multiple steady states & sometimes multiple solutions
34
“Traditional” Pic
ut
ut+1
45o
positive expectations
negative expectations
From Den Haan (2007)
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Multiplicity Getting started General Derivation Examples
Large sun spots (around 2000 at the peak)
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Multiplicity Getting started General Derivation Examples
Sun spot cycle (almost at peak again)
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Multiplicity Getting started General Derivation Examples
Sun spots in economics
• Definition: a solution is a sun spot solution if it depends on astochastic variable from outside system
• Model:
0 = EH(pt+1, pt, dt+1, dt)
dt : exogenous random variable
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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
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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-fulfilling expectations don’t seem that unreasonable
• sun spots provide many sources of shocks• number of sizable fundamental shocks small
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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 notoverwhelming
• you need suffi ciently large increasing returns to scale orexternality
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Multiplicity Getting started General Derivation Examples
Overview
1 Getting started
• simple examples
2 General derivation of Blanchard-Kahn solution
• When unique solution?• When multiple solution?• When no (stable) solution?
3 When do sun spots occur?
4 Numerical algorithms and sun spots
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Multiplicity Getting started General Derivation Examples
Getting started
•Model: yt = ρyt−1
• infinite number of solutions, independent of the value of ρ
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Multiplicity Getting started General Derivation Examples
Getting started
•Model: yt = ρyt−1
• infinite number of solutions, independent of the value of ρ
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Multiplicity Getting started General Derivation Examples
Getting started
•Model: yt+1 = ρyt
y0 is given
• unique solution, independent of the value of ρ
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Multiplicity Getting started General Derivation Examples
Getting started
•Model: yt+1 = ρyt
y0 is given
• unique solution, independent of the value of ρ
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Multiplicity Getting started General Derivation Examples
Getting started
• Blanchard-Kahn conditions apply to models that add as arequirement that the series do not explode
yt+1 = ρytModel:
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
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Multiplicity Getting started General Derivation Examples
State-space representation
Ayt+1 + Byt = εt+1
E [εt+1|It] = 0
yt : is an n× 1 vectorm ≤ n elements are not determined
some elements of εt+1 are not exogenous shocks but prediction errors
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Multiplicity Getting started General Derivation Examples
Neoclassical growth model and state space representation
(exp(zt)kα
t−1 + (1− δ)kt−1 − kt)−γ
=
E
[β (exp(zt+1)kα
t + (1− δ)kt − kt+1)−γ
×(
α exp (zt+1) kα−1t + 1− δ
) ∣∣∣∣∣ It
]
or equivalently without E[·](exp(zt)kα
t−1 + (1− δ)kt−1 − kt)−γ
=
β (exp(zt+1)kαt + (1− δ)kt − kt+1)
−γ
×(
α exp (zt+1) kα−1t + 1− δ
)+eE,t+1
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Multiplicity Getting started General Derivation Examples
Neoclassical growth model and state space representation
Linearized model:
kt+1 = a1kt + a2kt−1 + a3zt+1 + a4zt + eE,t+1
zt+1 = ρzt + ez,t+1
k0 is given
• kt is end-of-period t capital• =⇒ kt is chosen in t
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Multiplicity Getting started General Derivation Examples
Neoclassical growth model and state space representation
1 0 a30 1 00 0 1
kt+1kt
zt+1
+ a1 a2 a4−1 0 00 0 −ρ
ktkt−1
zt
= eE,t+1
0ez,t+1
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Multiplicity Getting started General Derivation Examples
Dynamics of the state-space system
Ayt+1 + Byt = εt+1
yt+1 = −A−1Byt +A−1εt+1
= Dyt +A−1εt+1
Thus
yt+1 = Dty1 +t+1
∑l=1
Dt+1−lA−1εl
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Multiplicity Getting started General Derivation Examples
Jordan matrix decomposition
D = PΛP−1
• Λ is a diagonal matrix with the eigen values of D• without loss of generality assume that |λ1| ≥ |λ2| ≥ · · · |λn|
Let
P−1 =
p̃1...
p̃n
where p̃i is a (1× n) vector
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Multiplicity Getting started General Derivation Examples
Dynamics of the state-space system
yt+1 = Dty1 +t+1
∑l=1
Dt+1−lA−1εl
= PΛtP−1y1 +t+1
∑l=1
PΛt+1−lP−1A−1εl
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Multiplicity Getting started General Derivation Examples
Dynamics of the state-space system
multiplying dynamic state-space system with P−1 gives
P−1yt+1 = ΛtP−1y1 +t+1
∑l=1
Λt+1−lP−1A−1εl
or
p̃iyt+1 = λti p̃iy1 +
t+1
∑l=1
λt+1−li p̃iA−1εl
recall that yt is n× 1 and p̃i is 1× n. Thus, p̃iyt is a scalar
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Multiplicity Getting started General Derivation Examples
Model
1 p̃iyt+1 = λti p̃iy1 +∑t+1
l=1 λt+1−li p̃iA−1εl
2 E[εt+1|It] = 03 m elements of y1 are not determined
4 yt cannot explode
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Multiplicity Getting started General Derivation Examples
Reasons for multiplicity
1 There are free elements in y1
2 The only constraint on eE,t+1 is that it is a prediction error.
• This leaves lots of freedom
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Multiplicity Getting started General Derivation Examples
Eigen values and multiplicity
• Suppose that |λ1| > 1• To avoid explosive behavior it must be the case that
1 p̃1y1 = 0 and
2 p̃1A−1εl = 0 ∀l
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Multiplicity Getting started General Derivation Examples
How to think about #1?
p̃1y1 = 0
• Simply an additional equation to pin down some of the freeelements
• Much better: This is the policy rule in the first period
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Multiplicity Getting started General Derivation Examples
How to think about #1?
p̃1y1 = 0
Neoclassical growth model:
• y1 = [k1, k0, z1]T
• |λ1| > 1, |λ2| < 1, λ3 = ρ < 1• p̃1y1 pins down k1 as a function of k0 and z1
• this is the policy function in the first period
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Multiplicity Getting started General Derivation Examples
How to think about #2?
p̃1A−1εl = 0 ∀l
• This pins down eE,t as a function of εz,t
• That is, the prediction error must be a function of thestructural shock, εz,t, and cannot be a function of other shocks,
• i.e., there are no sunspots
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Multiplicity Getting started General Derivation Examples
How to think about #2?
p̃1A−1εl = 0 ∀l
Neoclassical growth model:
• p̃1A−1εt says that the prediction error eE,t of period t is a fixedfunction of the innovation in period t of the exogenous process,ez,t
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Multiplicity Getting started General Derivation Examples
How to think about #1 combined with #2?
p̃1yt = 0 ∀t
• Without sun spots• i.e. with p̃1A−1εt = 0 ∀t
• kt is pinned down by kt−1 and zt in every period.
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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
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Multiplicity Getting started General Derivation Examples
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−1Byt +A−1εt+1
• Bad news: A is often not invertible• Good news: Same set of results can be derived
• Schur decomposition (See Klein 2000)
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Multiplicity Getting started General Derivation Examples
How to check in Dynare
Use the following command after the model & initial conditions part
check;
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Multiplicity Getting started General Derivation Examples
Example - x predetermined - 1st order
xt−1 = Et [φxt + zt+1]
zt = 0.9zt−1 + εt
• |φ| > 1 : Unique stable fixed point• |φ| < 1 : No stable solutions; too many eigenvalues > 1
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Multiplicity Getting started General Derivation Examples
Example - x predetermined - 2nd order
φ2xt−1 = Et [φ1xt + xt+1 + zt+1]
zt = 0.9zt−1 + εt
• φ1 = −2.25, φ2 = −0.5 : Unique stable fixed point(1+ φ1L− φ2L2)x = (1− 2L)(1− 1
4L)• φ1 = −3.5, φ2 = −3 : No stable solution; too manyeigenvalues > 1(1+ φ1L− φ2L2)x = (1− 2L)(1− 1.5L)
• φ1 = −1, φ2 = −0.25 : Multiple stable solutions; too feweigenvalues > 1(1+ φ1L− φ2L2)x = (1− 0.5L)(1− 0.5L)
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Multiplicity Getting started General Derivation Examples
Example - x not predetermined - 1st order
xt = Et [φxt+1 + zt+1]
zt = 0.9zt−1 + εt
• |φ| < 1 : Unique stable fixed point• |φ| > 1 : Multiple stable solutions; too few eigenvalues > 1
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Multiplicity Getting started General Derivation Examples
References• Blanchard, Olivier and Charles M. Kahn, 1980,The Solution ofLinear Difference Models under Rational Expectations,Econometrica, 1305-1313.
• Den Haan, Wouter J., 2007, Shocks and the Unavoidable Road toHigher Taxes and Higher Unemployment, Review of EconomicDynamics, 348-366.• simple model in which the size of the shocks has long-termconsequences
• Farmer, Roger, 1993, The Macroeconomics of Self-FulfillingProphecies, The MIT Press.• textbook by the pioneer
• Klein, Paul, 2000, Using the Generalized Schur form to Solve aMultivariate Linear Rational Expectations Model• in case you want to do the analysis without the simplifyingassumption that A is invertible