SUNSPOTS:
pt = a
mt
Let zt = mt
pt ]
b [Ept+1
pt: Then
b [Ept+1 Emt+1]
b [Emt+1 mt]
zt = mt pt = a
+b [pt mt]
or:
zt = a + bEzt+1
bzt
where t+1 = mt+1 mt and
Therefore,
zt = a + b(Ezt+1
Let z = a
bE
t+1
t
N
zt)
;
2
b
b : Then,
zt = z + b (Ezt+1
zt)
Note that zt = z is one solution, which
implies constant real money balances. No
signal extraction problem here.
1
:
Let qt = zt
z: Then, we have
qt = b(Ezt+1
= b[Ezt+1
= b(Eqt+1
zt)
z (zt
qt )
z)]
Hence,
1+b
qt
Eqt+1 =
b
As before, qt = 0 is a solution. There are
other solutions:
1+b
qt+1 =
qt + t+1
b
where
N 0; 2 and q0 is arbitrary.
Real balances are not constant, presumably causing output e¤ects. But are such
solutions reasonable? Let us require bounded
variance of q, the deviations of real bal-
2
ances from their mean.
2
1+b
V ar(qt+1) =
V ar (qt) + V ar (
b
2
1+b
V ar (qt)
>
b
> V ar (qt)
f or b > 0:5
If b < 0:5; 1+b
< 1; V ar(q) does
b
not blow up. Note that b < 0:5 implies
that the demand for real balances is positively related to the in‡ation rate. Is this
possible? Reasonable?
3
t+1 )
REDUCED MODEL LINEARIZED AROUND
STEADY STATE - Deterministic
kt+1
k
= [A] t
pt+1
pt
Let [V ] be the characteristic vector matrix of [V ] : Let
k~t+1
p~t+1
kt+1
= [V ]
pt+1
:
k~t+1
p~t+1
k~t
=[ ]
p~t
REDUCED MODEL LINEARIZED AROUND
STEADY STATE - Stochastic
2
3
2
3
2
3
kt+1
kt
0
4 pt+1 5 = [A] 4 pt 5 + [B] 4 st+1 5
zt+1
zt
et+1
4
2
3
2 3
2
3
~
~
0
kt+1
kt
4 p~t+1 5 = [ ] 4 p~t 5 + [ ] 4 st+1 5
et+1
z~t
z~t+1
5
0
1
2
30
1
y1;t+1
y1;t
1 0 0 0
B y2;t+1 C 6 0 2 0 0 7 B y2;t C
B
C=6
7B
C+[ ] s^t+1:
@ y3;t+1 A 4 0 0 3 0 5 @ y3;t A
y4;t+1
0 0 0 4
y4;t
(1)
Suppose 1
2
3 < 1 < 4:
First consider the deterministic case,
i.e., suppose s^t+1 = 0. Then it is easy
to see that the stable manifold is characterized by the equation
y4;t = 0:
(2)
(Note that y4;t is a linear function of the
components of vt.) Hence a deterministic
equilibrium path must satisfy (2) for all
t. Now consider the stochastic case; st =
(0; 0; s^x;t; s^y;t)0. Then for a path fvtg
to be nonexplosive, it must satisfy
4 y4;t
+ a^
sx;t+1 + b^
sy;t+1 = 0;
(3)
where a and b are the appropriate entries
of the matrix . To see (3), note that if
6
y4;t 6= 0 for some t, then jEty4;t+j j ! 1
as j ! 1.
Now the problem is that as long as (a^
sx +
b^
sy ) is stochastic (and i.i.d. over t), (3)
almost never holds. Thus the sunspot
shocks must satisfy
a^
sx;t + b^
sy;t = 0;
(4)
i.e., a sunspot shock can be added to one
of the Euler equations but the shock to
the other Euler equation is determined
by (4).
7