Discussion of Patir’s Sync and Bias
NBER: Multiple Equilibria and Financial Crises
Bruce Mcgough, University of Oregon
May 14, 2015
Background
TM
Learning to believe in sunspots
• Indeterminacy and sunspot equilibria
• Equilibrium selection: stability under adaptive learning
• Correlated equilibria and sentiments
• Are sentiments stable? Good question!
A simplified model
Model
yit = β Eit yt + β0 Eit εit
Z
yt =
yit di
A simplified model
PLM (agent i)
yt = φ i + ξ i · zt
sit = λ εit + (1 − λ )(φ i + ξ i · zt )
εit ∼ N(0, σε2 ), zt ∼ N(0, I2 )
A simplified model
Expectations
Eit (?) = E(?|sit )
= −
(1 − λ )2 φ i kξ i k2
(1 − λ )kξ i k2
+
sit
λ 2 σε2 + (1 − λ )2 kξ i k2 λ 2 σε2 + (1 − λ )2 kξ i k2
Eit εit = −
λ σε2
(1 − λ )λ φ i σε2
+
sit
λ 2 σε2 + (1 − λ )2 kξ i k2 λ 2 σε2 + (1 − λ )2 kξ i k2
Eit ξ i · zt
A simplified model
ALM
sit = λ εit + (1 − λ )
Z
Z
i
φ di +
(ξ · zt )di
i
yit = β φ i + β Eit ξ i · zt + β0 Eit εit = T φ (φ i , ξ i ) + T ξ (ξ i )sit
Z
yt =
yit di
Z
=
T φ (φ i , ξ i )di + (1 − λ )
+ (1 − λ )
Z
T ξ (ξ i )di
Z
Z
T ξ (ξ i )di
(ξ i · zt )di
Z
φ i di
A simplified model
REE
• Homogeneity: ξ i = ξ j = ξ , φ i = φ j = 0
2
2
0 λ σε
• (1 − λ )T ξ (ξ )(ξ · zt ) = (1 − λ ) β kξ2 k2 (1−λ )+β
(ξ · zt )
λ σε +(1−λ )2 kξ k2
2
ε (β0 −(β0 +1)λ )
• Thus ξ = 0 and kξ k2 = λ σ(1−β
)(1−λ )2
• Circle of stochastic equilibria!
λ<
β0
1+β0
A simplified model
RTL
ẑt =
Z
yt =
1
zt
and
ζti
=
φti
ξti
i
i
T φ (φt−1
, ξt−1
)di + (1 − λ )
+ (1 − λ )
Z
i
T ξ (ξt−1
)di
Z
Z
i
T ξ (ξt−1
)di
i
(ξt−1
· zt )di
Rt = Rt−1 + γt (ẑt ẑ0t − Rt−1 )
i
i
i
ζti = ζt−1
+ γt R−1
t ẑt (yt + ∆φ − ζt−1 · ẑt )
Z
i
φt−1
di
A simplified model
Figure : RTL, 1000 agents, no bias
0.6
2
0.4
1
0.2
constant
phase
3
0
-1
0.0
-0.2
-0.4
-2
-0.6
-3
0
20
40
60
80
time
100
120
140
0
20
40
60
80
time
100
120
140
A simplified model
Figure : RTL, 1000 agents, no bias
3
2
2
constant
phase
1
0
-1
1
0
-1
-2
-2
-3
0
10
20
time
30
40
0
10
20
time
30
40
A simplified model
Figure : RTL, 1000 agents, normal bias, no constant
1.0
3
2
0.5
constant
phase
1
0
-1
0.0
-0.5
-2
-3
-1.0
0
50
100
150
time
200
250
300
A simplified model
Figure : RTL, 1000 agents, normal bias
3
0.4
2
0.2
constant
phase
1
0
-1
0.0
-0.2
-2
-0.4
-3
0
50
100
150
time
200
250
300
0
50
100
150
time
200
250
300
A simplified model
Figure : RTL, 1000 agents, normal bias
3
0.4
2
0.2
constant
phase
1
0
-1
0.0
-0.2
-2
-0.4
-3
0
10
20
30
time
40
50
0
10
20
30
time
40
50
A simplified model
Figure : RTL, 20 agents, normal bias
1.0
3
2
0.5
constant
phase
1
0
-1
0.0
-0.5
-2
-3
-1.0
0
100
200
300
time
400
500
0
100
200
300
time
400
500
A simplified model
Summary
• Patir’s model and analysis are quite interesting and thought
provoking.
• I have a lot to learn about coordination and sunspots