130 M.C. Chang et al. International Review of Economics and Finance 9 2000 123–137
3 starting from D
A
b
3 5
1
will eventually fall inside {b
f
, R:R . 0}. Once this happens, it is easy to see that bt, Rt → b
f
, R
f
as t → ∞
; QED.
3. Deriving the maximum stable set: Some examples
We show in Proposition 2 that the maximum ODE-stable set is in fact the intersection of three regions. Specifically, we need to see 1 when the autoregressive coefficient
matrix A· in Eq. 1 have all its eigenvalues less than one in modulus condition a5 in Appendix 1; 2 when the parameters of the model can bound the state variables
infinitely often with probability one condition c1 in Proposition 1, and 3 when the differential equations which govern the original stochastic difference equations are
stable implicit in the condition D , D
A
. Each of these three conditions imposes restrictions on initial beliefs on relevant parameters in agents’ forecasts. In what
follows we show how to derive the ODE-stable set in some well-known models in the learning literature.
3.1. Cagan’s model: A case of an endogenous variable as one of regressors We first study a stationary version of Cagan’s 1956 inflation model with endoge-
nous money supply, which consists of:
5
p
t
5 l E
t
p
t
1
1
1 m
t
, 5
m
t
5 r m
t
2
1
2 g p
t
2
1
1 u
t
. 6
Eq. 5 and 6 are the demand function for real cash balance and the money supply rule, respectively. l . 0 implies that the rising opportunity cost of holding cash due
to rising E
t
p
t
1
1
decreases agents’ demand for real cash balance. The information set at time t in Eq. 5 includes m
t
, and all lagged values of m
t
and p
t
. When g . 0, the monetary authorities adopt a counter-cyclical monetary policy: the authorities decrease
current money supply as a response to an increase in price level in the previous period.
6
Suppose agent formulates his forecast of p
t
1
1
at time t according to E
t
p
t
1
1
5 b
t
m
t
. Substituting b
t
m
t
into Eqs. 5 and 6 yields the following actual law of motion for p
t
: p
t
5 r lb
t
1 1 m
t
2
1
2 g lb
t
1 1 p
t
2
1
1 lb
t
1 1u
t
. For a given estimate of b, we can use the actual law of motion of p
t
to deduce the linear least squares projection of p
t
1
1
: E ˆ [p
t
1
1
|m
t
] 5 lb 1 1 [r 2 g lb 1 1] m
t
, which can be expressed as the operator for the learning mechanism:
T b 5 lb 1 1[r 2 glb 11].
Substituting the perceived law of motion of p
t
into Eq. 6 for p
t
2
1
yields m
t
5 [r 2 glb
t
2
1
1 1]m
t
2
1
1 u
t
. As shown in Appendix 2, if b
t
P {b||r 2 glb 1 1| , 1} and u
t
is a white noise process, then m
t
is bounded infinitely often with probability one, and hence so is p
t
. Thus,
M.C. Chang et al. International Review of Economics and Finance 9 2000 123–137 131
Fig. 2. The maximum domain of attraction of b
f
.
{b||r 2 glb 1 1| , 1} 7
is one of the three regions we are looking for. Notice that the two characteristic roots of Ab in Eq. 3 are 0 and r 2 glb 2 g. Let D
d
b
be D
d
b
;
5
b|
1
b 2 r 2 g 2
1 lg
2 1
b 2 r 2 g 1
1 lg
2
,
6
. It is easy to see that if g . 0 and b P D
d
b
, then the regularity condition a5 in Appendix 1 is satisfied. Further, in view of Eq. 7, b P D
d
b
also guarantees the boundedness of m
t
. Next we need to find the domain of attraction for the corresponding ordinary
differential equation. For a scalar case of b
t
, Proposition 3 in Section 2 shows that the stability is governed by the following ordinary differential equation:
b˙ 5 T
b 2 b 5 lb 1 1[r 2 glb 1 1] 2 b. It is easy to verify that T
″ ,
0, and therefore Tb 2 b should look like Fig. 2. There are two stationary points of the mapping Tb → b, but we find that only
b
f
g, r, l 5
√
lr 2 1
2
1 4lg 1 lr 2 2lg 2 1
2gl
2
, is the possible candidate of equilibrium.
7
It is clear from Figure 2 that the maximum domain of attraction of b
f
is D
A
b
5 {b|b . b9
f
g, r, l}, which, by Proposition 3, implies that D
A
b
3 5
1
is the maximum domain of attraction. Hence, the maximum stable set is the intersection of {b, R|b P D
d
b
} and D
A
b
3 5
1
. Finally, it is interesting to see that the range of the open interval D
d
b
is decreasing with g, and hence the maximum stable set decreases with g. Specifically, it is possible
that a very sensitive counter-cyclical monetary policy could lead to an ODE-unstable learning mechanism.
8
132 M.C. Chang et al. International Review of Economics and Finance 9 2000 123–137
3.2. Frydman’s model: A case of global stability Here we study a Marcet and Sargent’s 1986 two-firm version of Frydman’s 1982
model. This example illustrates that the learning mechanism can be easily verified as globally stable when agents use only exogenous variables to forecast equilibrium price
level. Suppose there are two competitive firms indexed by i 5 1, 2. Demand for the firm’s output is assumed to be p
t
5 a 2 b
o
2 i
5
1
h
it
1 e
t
, in which a, b . 0, h
it
is the output of firm i, and e
t
is an i.i.d. random variable. Firm i’s cost function at time t is C
h
it
5 s2h
it 2
1 k
it
h
it
1 c
, where c and s are positive parameters, and k
it
; dt 1 e
it
, i 5 1, 2, in which d
t
and e
it
are i.i.d. random variables. Firm i observes k
it
at time t
and believes that the expectation of p
t
evolves according to Ep
t
|1, k
it
5 b
i 1
1 b
i 2
k
it
, i 5
1, 2. Let b
i t
; b
i 1t
, b
i 2t
be the least squares estimate of b
i
; b
i 1
, b
i 2
at time t with b
t
; b
t 1
, b
t 2
, z9
t
5 p
t
, k
1t
, k
2t
be the state variable vector, and u9
t
5 1, d
t
, e
t
, e
1t
, e
2t
be the error term vector. It can be shown that the actual law of motion for price can be
expressed in the form of Eq. 1
9
, where Ab
t
is a 3 3 3 zero matrix, and Bb
t
is
3
a 2 bs
o
2 i
5
1
b
i 1t
2 bs
[
o
2 i
5
1
b
i 2t
2 1
]
1 2
bs b
1 2t
2 1
2 bs
b
2 2t
2 1
1 1
1 1
4
. Because Ab
t
is a zero matrix and Bb
t
is bounded for b
t
in any compact set, we see that the characteristic roots of Ab
t
are all zeros for all possible values of b, and that z
t
is bounded infinitely often with probability one under the i.i.d. assumption of u
t
.
10
As such, the stability property of this model hinges upon the domain of attraction of the differential Eq. 3.
For a given value of b, the least squares projection of p
t
is given by E ˆ [p
t
|1, k
it
] 5 z9
i ,2t
T
i
b, in which the regressor z
i ,2t
; 1, k
it
9, and T
i
b 5
3
a 2 bs
o
2 j
5
1
b
j 1
, 2 bsb
i 2
2 1 1 l
o
j?i
b
j 2
2 1
4
9 ,
with l 5 [Var d
t
][Var d
t
1 Var e
t
]. b
f
is a rational expectations equilibrium if b
1 f
5 T
1
b
f
and b
2 f
5 T
2
b
f
. After some algebra, we derive the stationary point of b
f
:
b
i f
5
3
a 1 1 2bs
bs 1 1 l
1 1 bs 1 1 l
4
, i 5 1, 2.
Because z
i , 2t
; 1, k
it
9 in Frydman’s model, Mb ; E z
i ,2t
z9
i , 2t
in Eq. 3 is clearly independent of b. Thus, the stability property of Eq. 3 is this model can be derived
by studying the simpler differential equation b˙ t 5 Tb t 2 bt, where Tb 5 T
1
b, T
2
b. We find that the four characteristic roots of the derivative of Tb 2 b
with respect to b evaluated at b 5 b
f
are 22bs 2 1, 22bsl 2 1, 21 and 2bs1 2 l
2 1. Because bs . 0, these characteristic roots are all less than zero. The differential equations are globally stable.
M.C. Chang et al. International Review of Economics and Finance 9 2000 123–137 133
In the above discussion, we find that the fulfillment of the sufficient conditions in Proposition 2 does not need any additional restrictions, so the maximum stable set
is a universal set and the learning mechanism is globally ODE-stable. 3.3. Bray’s model: Using Lyapunov function to find the maximum stable set
Bray 1982 studied a model in which there are N
i
. 0 informed traders and N
u
. 0 uninformed traders. Both types of traders observe market price p
t
at time t, and the return on the asset r
t
at a date after t but before t 1 1. Further, the informed traders are assumed to observe the exogenous supply of the asset, s
t
. Both r
t
and s
t
are i.i.d. random variables, satisfying the following relationship: r
t
5 E
[r
t
] 1 rs
t
2 E
[s
t
] 1 e
t
, in which E is the expectation operator, r 5 covr
t
, s
t
var s
t
, and e
t
is a white noise process orthogonal to s
t
. The demand for the asset by informed agents at time t is u
i
[E[r
t
|s
t
] 2 p
t
], u
i
. 0, in
which E[r
t
| s
t
] 5 E[r
t
] 1 r s
t
2 E
[s
t
]. The uninformed agents use p
t
in forming their expectations of r
t
at time t: E[r
t
|p
t
] 5 b
1
1 b
2
p
t
, which determine their demand for the asset: u
u
[E[r
t
|p
t
] 2 p
t
] with u
u
. 0. Let b 5 b
1
, b
2
9 and b
t
5 b
1t
, b
2t
}9. The estimate of b at time t b
t
is determined by Eqs. 2a–2b with z
2t
5 1, p
t
9 and z
1t
5 r
t
2
1
.
11
If var s
t
. 0, then the least squares projection of r
t
on z
2t
is given by E ˆ [r
t
|z
2t
] 5 z9
2t
T b, in which
T b 5
3
E [r
t
] 2 kN
i
u
i
E [r
t
] 2 E[s
t
]N
u
u
u
2 kb
1
k N
i
u
i
1 N
u
u
u
N
u
u
u
2 kb
2
4
, with k 5 N
u
u
u
N
i
u
i
2 r
2
1
. It is straightforward to show that if k ? 21, then there is a unique stationary point of Tb → b:
b
f
5
3
[E[r
t
] 2 kN
i
u
i
E [r
t
] 2 E[s
t
]N
u
u
u
]1 1 k
2
1
k N
i
u
i
1 N
u
u
u
1 1 kN
u
u
u
4
. It can be shown that the learning mechanism is locally stable unstable if k . 21
k , 21. Therefore, it is assumed that k . 21. Let z
t
5 p
t
, r
t
, s
t
9 and u
t
5 r
t
, s
t
, 19. It is easy to see that the coefficient matrix A
b
t
in Eq. 1 is a 3 3 3 zero matrix and hence the regularity condition a5 is satisfied in this model. Therefore, the limiting behavior of b
t
, R
t
can be approximated by that of bt, Rt. However, it is hard to study the stability of this learning mechanism
beyond the local sense since p
t
used as a regressor in forming the uninformed agents’ expectation of r
t
is an endogenous variable. In what follows we shall show that the global stability of the learning mechanism can be studied directly, without having to
analyze the simpler differential equation in Eq. 4 as a bridge. This is a step beyond the contribution of Marcet and Sargent 1989a.
We first show that expectation of the squared prediction error is a Lyapunov function for the differential equation system of Eq. 3. Consider Vbt 5 12E[e
t
bt]
2
, in which e
t
bt ; r
t
2 z
2t
bt9bt is the prediction error. When evaluated along the trajectory of bt and Rt, we have
134 M.C. Chang et al. International Review of Economics and Finance 9 2000 123–137
d dt
V bt 5 2E[lbtz
2t
bte
t
bt]9 Rt
2
1
E [z
2t
bte
t
bt], 8
in which l
b 5 N
u
u
u
1 N
i
u
i
N
u
u
u
1 2 b
2
1 N
i
u
i
. According to Eq. 8, if Rt is a positive definite matrix and lb . 0 that is, b
2
, N
i
u
i
1 N
u
u
u
N
u
u
u
, then ddtVbt 0 and the equality holds when bt 5 b
f
. Thus, the maximum domain of attraction of b
f
, R
f
is
12
D
A
5 {b, R|b
2
, N
i
u
i
1 N
u
u
u
N
u
u
u
, R is a positive definite matrix}. It is clear from the assumption of k . 21 that b
2
f , N
i
u
i
1 N
u
u
u
N
u
u
u
, which implies b
f
, R
f
P D
A
. Finally, because a1–a5 are always satisfied in Bray’s 1982 model, D
A
is also the maximum ODE-stable set for the learning mechanism.
4. Conclusions
