Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue5-7.Jul2000:
Journal of Economic Dynamics & Control
24 (2000) 725}759
Expectational diversity in monetary economies
William A. Brock!,*, Patrick de Fontnouvelle"
!Department of Economics, University of Wisconsin, Madison, WI 53706, USA
"U.S. Securities and Exchange Commission, 450 5th Street N.W., Washington, DC 20549, USA
Accepted 30 April 1999
Abstract
We investigate an overlapping generations monetary economy in which expectations
depend upon backward looking predictors of the future price level. We use discrete
choice theory to model how agents select a predictor based on its past forecast error.
Letting the number of available predictors tend to in"nity, we obtain the large type limit
of the system. Taking the large type limit dramatically reduces the number of free
parameters, while maintaining the expectational diversity which we argue is necessary for
constructing plausible learning-based models. The model's dynamics are strongly in#uenced by the intensity of choice, which measures how sensitive an agent's predictor
choice is to di!erences in forecast errors across predictors. When the intensity of choice is
low, the monetary steady state is stable. As the intensity of choice increases (and if certain
parametric restrictions are met) the system undergoes a Hopf bifurcation, in which case
we document the existence of highly irregular equilibrium price paths. ( 2000
Published by Elsevier Science B.V. All rights reserved.
Keywords: Discrete choice; Endogenous #uctuations; Fiat money; Hopf bifurcation,
Learning; Overlapping generations
* Corresponding author. We thank Cars Hommes for helpful discussions and suggestions, and
also W. Davis Dechert for suggesting the reference by E. Jury. All errors are the authors' own
responsibility. W.A. Brock thanks the NSF under grant d SES-9122344, and the Vilas Trust for
"nancial support. P. de Fontnouvelle thanks the Iowa State University Department of Economics
for supporting this research while he was an Assistant Professor there.
The Securities and Exchange Commission, as a matter of policy, disclaims responsibility for any
private publication or statement by any of its employees. The views expressed herein are those of the
author and do not necessarily re#ect the view of the Commission or the author's colleagues upon the
sta! of the Commission.
0165-1889/00/$ - see front matter ( 2000 Published by Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 2 4 - X
726 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
1. Introduction
Overlapping generations (OLG) models of money (Samuelson, 1958; Wallace,
1980) typically display a continuum of deterministic rational expectations
equilibria. One of these equilibria is the monetary steady state, in which "at
money retains a constant value forever. The other equilibria all converge to
autarky, so that money gradually becomes worthless. The monetary steady state
is appealing to economists largely because it corresponds to everyday experience; the other equilibria seem implausible because they do not. Agents in
modern market economies take it for granted that, barring gross government
misconduct, currency will retain its role as medium of exchange inde"nitely. No
U.S. resident places any serious probability on the economy degenerating into
barter within the foreseeable future. The rational expectations hypothesis, however, provides no guidance as to which equilibrium path should prevail (Boldrin
and Woodford (1990)). The problem (often called &indeterminacy') is that each of
these paths corresponds to some consistent set of expectations about future
prices. This problem is serious because it implies that under rational expectations the OLG model cannot explain the existence of valued "at money, which
we have argued is a main &fact' of everyday life.
Some have suggested resolving the indeterminacy problem by noting that
when perfect foresight is replaced by learning rules, the OLG model often
converges to the monetary steady state (Lucas, 1986; Marcet and Sargent,
1989a,b). Since learning rules provide a plausible approximation of how people
actually behave, this convergence suggests that the monetary steady state is in
fact the most reasonable long-run outcome.1 Others, however, have shown that
such answers to the indeterminacy problem su!er from an inherent lack of
generality: while certain learning rules in certain models single out economically
appealing equilibria, other rules in other models actually increase the number of
possible equilibrium paths. In addition to generating explosive paths similar to
those which emerge under perfect foresight (Evans and Honkapohja, 1994a,b),
learning may lead to complex price paths which neither explode nor converge to
the monetary steady state (Bullard, 1994; Grandmont and Laroque, 1991). In
addition, Du!y (1994) shows that if agents use an adaptive rule to form
expectations about in#ation (rather than price level), then the economy can
converge to a continuum of nonstationary equilibria.
Because of this lack of generality, Lucas (1986) believes it will be impossible to
address fully the problem of indeterminacy via &purely mathematical' methods:
&It is hard to see what can advance the discussion short of assembling a collection of people, putting them in the situation of interest, and observing what they
1 Evans and Honkapohja (1994a,b,1995) have written extensively on learning rules as selection
criteria for rational expectations equilibria.
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 727
do.' This approach is pursued by Lim et al. (1994), who use experimental
methods to examine the OLG model with constant money supply; their results
lend strong support to the monetary steady-state equilibrium. Marimon and
Sunder (1993) examine a related experimental OLG economy in which the
money supply is allowed to grow over time, and "nd that observed price paths
tend to converge to the low in#ation steady state. Ochs (1995, p. 205) summarizes these results: &There is one characteristic common to all of these sessions
that is of direct interest to monetary economists. In none of these sessions is
there any evidence that individuals have the `foresighta to follow rational
expectations equilibrium paths that generate hyperin#ation.'
The experimental approach, however, will always leave some important
questions unanswered. Lim et al. (1994, p. 267) write, &With replication there is
marked convergence towards stationary equilibrium though the convergence is
not precise. To what extent further replications may a!ect the nature of convergence is an open question'. Given enough time, one would like to know if the
experiment would converge precisely to the steady state, or if small price
#uctuations would persist inde"nitely. Would further replications of the experiment a!ect the results? Might there be other parameter values for which the
results would be qualitatively di!erent?
In order to address these questions, we must accept Lucas's implicit challenge
of "nding a &purely mathematical' method of exploring learning dynamics which
also retains a high degree of generality. We believe that the necessary ingredient
for achieving such generality is expectational diversity: by allowing many
di!erent expectations (rules) to be nested within one model, one can greatly
reduce the danger that the model's stability properties depend on one particular
rule. The idea of agents choosing between several competing expectations dates
at least as far back as Arthur (1994a,b) and Conlisk (1980).
In this paper (Sections 2}5), we introduce expectational diversity by allowing
agents to choose among a large number of predictors of the future price level.
These predictors are "nitely parameterized functions of past prices. We model
the distribution of predictors across agents in two steps. First, we use discrete
choice tools to construct a tractable form of natural selection dynamics over the
space of predictors. The fraction of agents adopting a particular predictor is
determined by that predictor's past squared forecast error (SFE), relative to the
SFE of other predictors. The sensitivity of this fraction to variations in SFE is
called the intensity of choice.
The result is the Adaptive Rational Equilibrium Dynamics (ARED) of Brock
and Hommes (1997), in which the dynamics of predictor choice is determined by
the equilibrium dynamics of endogenous variables. These authors emphasize
that most economic models displaying complicated dynamics can be expressed
as one-dimensional di!erence equations, such as Benhabib and Day (1982) and
Grandmont (1985). Although there are some two-dimensional models (Benhabib and Farmer, 1994; Cazzavillan et al., 1996; de Vilder, 1995; Grandmont
728 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
et al. (1996); Woodford, 1986) ARED introduces entirely new state variables
corresponding to the fractions of agents using each predictor. As we shall see,
the extra dimensions added by these state variables create the possibility of very
rich dynamic behavior.
The second step in modeling the distribution of predictors across agents is to
let the number of available predictors tend to in"nity, obtaining what we call the
large type limit (LTL) of the ARED. In the large type limit, the parameters of
each predictor are drawn at random from continuous distributions, whose mean
is called predictor sensitivity and whose variance is called predictor variance.2 In
Section 6, we provide an analytical exploration of a simple version of the model,
in which predictors depend on only one lag of past price. The model's dynamics
are determined jointly by the predictor sensitivity, the predictor variance and
the intensity of choice. When the predictor sensitivity is low, the monetary
steady state is asymptotically stable. This stability is in accord with previous
analytic, computational, and experimental results.3 When the predictor sensitivity is high, the monetary steady state is unstable, as in the original perfect
foresight versions of the OLG model (Samuelson, 1958; Wallace, 1980); equilibrium paths not originating at the steady state are explosive.
The most unusual behavior occurs at intermediate levels of sensitivity. We
show that when predictor variance and intensity of choice are low, the monetary
steady state is asymptotically stable. As each of these parameters increases, the
monetary steady state undergoes a Hopf bifurcation, near which a closed
periodic orbit exists. Periodic orbits, however, are not economically plausible
because they imply simple forecast errors that never vanish (Hommes, 1998). In
Section 7, we use numerical simulations to investigate the global dynamics of
a higher dimensional version of the model, in which predictors depend on two
lags of the price level. We show that as the intensity of choice increases, the time
series behavior of the price level can become increasingly erratic. The structure
of the associated forecast errors can become complicated enough that linear
methods cannot distinguish them from white noise.
While it is true that nonlinear methods (such as those discussed in LeBaron,
1994) may detect some predictability in the forecast errors, it is possible that
other versions of the model may generate forecast errors that appear random
even to nonlinear tests. Although such work is beyond the scope of the current
paper, it would mesh nicely with the agenda outlined by Grandmont (1992):
&One might envision for instance a situation in which traders attribute their
forecasting mistakes to noise, although the observed dynamics are actually
deterministic but chaotic. As I said, this is largely unknown territory and should
be the subject of future research'.
2 Brock (1997) sketches the above approach in the context of "nancial modeling.
3 See, for example, Lucas (1986), Arifovic (1995), and Lim et al. (1994), respectively.
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 729
1.1. Related literature
Economic models of learning have received an enormous amount of recent
attention: Marimon (1997) provides a comprehensive survey. The original
rational expectations criticism of such models was that agents continue using
the same learning rule even after it has been found to be systematically wrong.
There have been two strands of work addressing this criticism. The "rst strand
proposes analytical models of expectational diversity (Bray and Savin, 1986;
Evans et al., 1995; Marcet and Sargent, 1989a,b). As we argued previously,
expectational diversity reduces the danger that agents continue using rules that
perform poorly.
The second strand is the evolutionary approach (Arifovic, 1995; Arthur,
1994a; Arthur et al., 1997; Bullard and Du!y, 1995a,b; Darley and Kau!man,
1997; Day and Chen, 1993;4 LeBaron, 1995; Weidlich, 1991), which handles the
rational expectations criticism very elegantly; natural selection serves to remove
rules that perform poorly. Arifovic (1995), for example, studies the same OLG
model as the current paper, but uses genetic algorithms to model how agents
update their price level forecast rules. Because these evolutionary approaches
have traditionally been highly computational, they have not been amenable to
direct analysis. Our methods can be viewed as o!ering analytic results for this
computational literature. The discrete choice approach to natural selection over
rules plus the notion of Large Type Limit is a novel combination of technique that
allows us to bridge these two strands. We believe our methods allow us to push
analytic results on evolution signi"cantly farther than the current literature.
2. The economic model
Our analysis is based upon a standard overlapping generations model of
a monetary economy (Samuelson, 1958; Wallace, 1980). Each generation consists of an equal number of agents. Agents live for two periods, and are endowed
with w: when young and w0 when old. An agent in generation t consumes
cy when young and c0 when old. The amount of "at money that government
t
t
supplies is denoted M, which is constant over time. The non-negative time
t price level is denoted p . All agents have identical preferences, so that young
t
agent i solves the following maximization problem:
max u(c: , c0 )
i,t i,t
si,t
4 Of particular relevance in this volume are the papers by P. Allen, J. Conlisk, and R. Day.
(1)
730 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
subject to
(2)
c: "w:!s ,
i,t
i,t
c0 "w0#(p /p% )s ,
(3)
i,t
t i,t`1 i,t
where s denotes the real time t savings of agent i, and p%
denotes i's time
i,t
i,t`1
t expectation of p . We make the following assumptions about the utility
t`1
function and endowments:
Assumption A1. (a) Indi!erence curves are convex to the origin. (b) The
utility function is strictly increasing in both arguments. (c) u /u PR as
1 2
c: /c0 P0, and u /u P0 as c: /c0 PR, where u and u denote the partial
i,t i,t
1 2
i,t i,t
1
2
derivatives of u( ) ) with respect to c: and c0 , respectively. (d) c: and c0 are
i,t
i,t
i,t
i,t
normal goods.
Assumption A2. u (w:, w0)(u (w:, w0).
1
2
Assumption A3. c: and c0 are gross substitutes.
i,t
i,t
Assumption A1 contains standard (Sargent, 1987, p. 232) restrictions on the
utility function, which guarantee that for each expected price ratio p /p%
there
t i,t`1
is a unique value of savings s : s "s(p /p% ). Assumption A2 implies that
i,t i,t
t i,t`1
s(1)'0. Assumption A3 implies that savings is an increasing function of p /p%
t i,t`1
(Varian, 1984): s@(x)'0 for all x. This rules out the possibility of complicated
dynamics when all agents have perfect foresight (Kehoe et al., 1986), thus
emphasizing that our results depend more on how expectational diversity is
modeled than on a particular choice of savings function.
Our modi"cation to the standard OLG model is to allow agents to have
heterogeneous expectations about the future price level. Assume "nite memory,
so that agents' common information set consists of a "nite length vector of past
prices:
pL "Mp ,2, p N.
t~1
t~1
t~L
The expected price of agent i is a continuously di!erentiable function of her
information set:
(4)
p% "hj"hj( pL ).
t~1
t
i,t`1
Each hj( ) ) is called a predictor, and the space of predictors is denoted
H:H"Mh1,2, hKN. To make notation clearer, we always index agents with the
subscript &i' and predictors with the superscript &j'. The speci"cation (4) allows
for a wide range of expectations, including learning (Grandmont and Laroque,
1986, p. 140). Suppose, for instance, that agents believe the economy's actual
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 731
dynamics imply p "f ( pl , h), where h is a vector of (unknown) structural
t~1
t`1
parameters and l4¸. Each period, agents can use some econometric procedure
to obtain an estimate h%45( pL ). They can then form their expectations
t~1
by replacing the true value of h with the estimate h%45( pL ), to obtain
t~1
p% "f ( pl ,h%45( pL )).
t~1
t~1
t`1
Let the fraction of agents in generation t who choose predictor hj at time t be
a continuously di!erentiable function of pL`N:
t~1
nj"nj( pL`N).
(5)
t
t~1
The parameter N indexes the number of extra lagged prices taken as arguments
of the functions nj( ) ). At this level of generality, each nj( ) ) should be thought of
as some past performance metric for the predictor hj( ) ). Since hj( pL ), the
t~3
predicted value of p , depends on pL , it is clear that, for the OLG model,
t~1
t~3
N must be at least 2. In Section 5, we propose a speci"cation for the functions
nj( ) ) based on discrete choice theory. For the moment, leaving this speci"cation
open will emphasize the generality of Propositions 1 and 2.
Equilibrium requires that nominal aggregate savings be equal to the money
supply:
K
M"p + nj( pL`N)s(p /hj( pL )).
t~1
t
t~1
t
j/1
(6)
3. Homogeneous expectations
We begin by reviewing the model's dynamics under two special cases. First,
for all i.
suppose that all agents have perfect foresight: p% "h( pL )"p
t~1
t`1
i,t`1
The equilibrium condition (6) reduces to
M"p s(p /p ).
(7)
t t t`1
There is exactly one steady-state equilibrium where money has value. We refer
to this equilibrium, given by p6 "M/s(1), as the monetary steady state. There is
also exactly one steady state (autarky) where money has no value, as well as
a continuum of nonstationary equilibria where money has value. The latter
are indexed by the initial price level p e(p6 ,R), and all converge to autarky.
0
The coexistence of many possible equilibrium price paths is referred to as
indeterminacy.
As mentioned previously, the problem of indeterminacy has led researchers to
consider various types of backward looking expectations as selection criteria for
plausible long-run equilibria. We begin with a simple example, in which all
for all i. In this case, (6)
agents have naive expectations: p% "h( pL )"p
t~1
t~1
i,t`1
reduces to
M"p s(p /p ).
t t t~1
(8)
732 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
It is easy to verify that under (8), all equilibrium paths converge to the monetary
steady state p6 "M/s(1). Naive expectations thus support the claim that the
monetary steady state is a reasonable long-run equilibrium for (7). Of course, the
is the simplest and most ad hoc possible. We thus
speci"cation p% "p
t`1
t~1
examine the stability properties of the monetary steady state under a more
general speci"cation of expectations.
3.1. General backward expectations
If agents have general homogeneous expectations p% "h( pL ),5 the equit~1
t`1
librium condition (6) becomes
(9)
M"p s(p /h( pL )).
t~1
t t
The monetary steady state p6 is the solution to M"p6 s(p6 /h(p6 ,2, p6 )). The existence of such a p6 is guaranteed by the following assumption:
Assumption A4. The predictor h( ) ) satis"es the following conditions: (a)
lim p/h(p,2, p)(R, (b) lim
p/h(p,2, p)'r , where the autarkic interp?0
p?=
!65
est rate r is de"ned implicitly by s(r )"0.
!65
!65
Eq. (9) implicitly de"nes a nonlinear function and a corresponding system
given by
p "/H( pL ),
t~1
t
N.
pL"UH( pL )"M/H( pL ), p ,2, p
t~1 t~1
t~L`1
t~1
t
Rewriting the equilibrium condition (9) in terms of /H( ) ), we obtain
(10)
(11)
(12)
M"/H( pL )s( /H( pL )/h( pL )).
t~1
t~1
t~1
Di!erentiating (12) with respect to p , evaluating at the monetary steady state
t~q
p6 , and solving for /H yields
q
/H"lh ,
q
q
(p6 /h(p6 ,2, p6 ))2s@(p6 /h(p6 ,2, p6 ))
l"
,
s(p6 /h(p6 ,2, p6 ))#p6 /h(p6 ,2, p6 )s@(p6 /h(p6 ,2, p6 ))
where /H and h denote the partial derivatives of /H( ) ) and h( ) ) with respect to
q
q
p , evaluated at the steady state pL "p6 . The Jacobian of UH( ) ) at the steady
t~1
t~q
5 The analysis in this section is related to Grandmont and Laroque (1986), who explore the
relationship between perfect foresight and backward looking dynamics when agents have homogeneous expectations.
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 733
state can then be written:
C
J"DUH(p6 )"
lh
1
1
lh
2
0
lh
3
0
2
2
lh
L~1
0
lh
L
0
0
1
0
2
0
0
0
0
1
2
0
0
F
F
F
F
F
0
0
0
1
0
2
D
We show that the monetary steady state p6 is asymptotically stable provided that
expectations are not too sensitive to past #uctuations in price level:
L
l + Dh D(1,
q
q/1
(13)
Proposition 1. Condition (13) is a suzcient condition for the monetary steady state
p6 to be asymptotically stable.
Proof. See appendix. h
Although the stability criterion (13) seems like a plausible conjecture, there is
no a priori way of verifying how sensitive agents' expectations are to past price
#uctuations. As in the simple case of naive expectations, the classi"cation of
equilibria as stable or unstable must ultimately rest on an ad hoc assumption.
Consider the following example, in which expectations are an average of past
prices:
h( pL )"a p #2#a p .
t~1
1 t~1
L t~L
If the absolute values of the coe$cients a sum to some value less than 1/l, then
l
the monetary steady state is asymptotically stable. If, on the other hand, these
absolute values sum to some number larger than 1/l, then (13) does not hold and
the monetary steady state may be unstable.
4. Heterogeneous expectations without bias
We reduce the dependence of our analysis on ad hoc assumptions by allowing
agents to have heterogeneous expectations; when there are many predictors, the
model's dynamics will depend less on the speci"cation of any one predictor.
Before considering the case of heterogeneous expectations, however, we
introduce a useful property called steady-state bias:
734 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
Dexnition. Let p6 denote the price level at the monetary steady state. If
hj(p6 ,2,p6 )"p6 , we say that the predictor hj( ) ) has no steady-state bias. Otherwise, we say that hj( ) ) has steady-state bias.
In other words, a predictor has no steady-state bias if it predicts next period's
price level to be p6 whenever the price level has been p6 for the past ¸ periods.
Suppose there is no steady-state bias:
Assumption A4@. Each predictor hj( ) ) has no steady-state bias, so that
hj(p6 ,2,p6 )"p6 for all j.
In the case of heterogeneous expectations, equilibrium condition (6) clearly
implies that p6 "M/s(1) is the monetary steady state. As before, (6) implicitly
de"nes a nonlinear function and a corresponding system given by:
(14)
p "/( pL`N),
t~1
t
N.
(15)
pL`N"U( pL`N)"M/( pL`N), p ,2, p
t~1 t~1
t~L~N`1
t~1
t
Note that allowing for heterogeneity means that (14) and (15) incorporate
N more lagged values of p than do (10) and (11). (See also the discussion
t
immediately below Eq. (5).) Rewriting (6) in terms of /( ) ), we obtain
K
M"/( pL`N) + nj( pL`N)s(/( pL`N)/hj( pL )).
t~1
t~1
t~1
t~1
j/1
Di!erentiating with respect to p , and evaluating at the steady state yields
t~q
K
(16)
0" + Mnjp6 s(1)#n6 j/ s(1)#n6 js@(1)[/ !hj]N
q
q
q
q
j/1
where nj and / denote the partial derivative of nj( ) ) and /( ) ) with respect to
q
q
p , and n6 j"nj(p6 ,2, p6 ). Now +K nj( ) )"1 implies that the summation
t~q
j/1
+K nj"0, so that solving (16) for / yields
j/1 q
q
K
/ "l + n6 jhj,
q
q
j/1
s@(1)
l"
.
s(1)#s@(1)
At the steady state, the multi-predictor system behaves exactly like the one
predictor system with the one representative predictor given by
K
h3%1( pL )" + n6 jhj( pL ).
t~1
t~1
j/1
(17)
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 735
This system's stability properties can be analyzed by using the previous section's
results:
Proposition 2. When none of the predictors hj( ) ) has steady-state bias, the system
(15) is asymptotically stable provided that l+L Dh3%1D(1.
q/1 q
The proof of Proposition 2 is identical to that of Proposition 1. As in the one
predictor case, the system's local stability depends upon how sensitive the
representative predictor is to past price #uctuations, not on how the predictor
space H is speci"ed. Any substantive conclusions we draw based upon local
stability at the steady state are thus contingent on how we specify the representative predictor.
5. Heterogeneous expectations with bias
When steady-state bias is allowed, (16) cannot be analyzed unless we specify
how the fractions nj are determined. Suppose that for each predictor heH, agent
t
i's associated utility is composed of a deterministic and a stochastic component:
(18)
;I j ";j#ij /b.
i,t
t
i,t
The deterministic component ;j represents the realized utility of an agent in
t
generation t!2 who predicted the time t!1 price level to be p% "hj :
t~2
t~1
;j";(p ,p ,hj )
t
t~2 t~1 t~2
(19)
"u[w:!s(p /hj )),w0#(p /p )s(p /hj ))].
t~2 t~1 t~2 t~2
t~2 t~2
The stochastic components ij are i.i.d. across time,6 across predictors, and
i,t
across agents.7 The intensity of choice b speci"es how good a measure the
deterministic component ;j is of the choice utility ;I j . It regulates how sensitive
i,t
t
agents are to di!erences in realized utility between predictors. When bP0, the
random component is very large, so that variation in ;j has almost no e!ect on
t
agents' choice of predictor. When bPR, all agents choose the predictor with
the highest realization of ;j. It is clear from the above discussion that the
t
random component ij together with the intensity of choice b will be crucial in
i,t
determining the model's dynamics. We thus present in some detail three possible
sources of the uncertainty represented by the ij .
i,t
6 The temporal independence assumption can be relaxed without a!ecting the model's aggregate
dynamics (de Fontnouvelle, 1996).
7 Brock (1993) shows how to use interacting particle systems theory to relax the assumption of
independence across agents.
736 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
(A) Random preferences: Agents' behavior may be inherently unpredictable, so
that an agent facing repeated instances of the same decision problem (and
observing the same information concerning the problem) varies his choices over
time.
(B) Random characteristics: Manski (1977, p. 235) interprets the stochastic
components ij as coming from three sources, all of which represent a lack of
i,t
knowledge on the part of the econometrician. (a) Measurement error: the economist has no direct knowledge of the choice utilities ;I j . (b) Specixcation error:
i,t
the utilities given in (19) may be of the wrong functional form. (c) Unobservable
characteristics: parameters such as risk aversions or endowments might be
unobserved or only partially observed. Faced with the same decision problem
repeatedly, each individual agent will always make the same decision. Two
agents who appear identical to the economist, however, may make quite di!erent decisions.
(C) Random information: Suppose that ;j is unknown to agents at the begint
ning of period t, but that they learn about it via a Bayesian updating strategy.8
Under the random information motivation for heterogeneity, an agent facing
repeated instances of the same decision problem will vary his choices over time
* not because his preferences are random but because his information is.
5.1. Calculating choice probabilities
In this section, we show how to calculate from (18) an expression for agents'
choice probabilities. In particular, P(hj), the probability that an agent chooses
t
predictor hj( ) ) at time t, will depend on the time t!1 squared forecast error
associated with predictor hj( ) ).
Dexnition. The forecast error ej is the di!erence between the actual price at time
t
t and that predicted by hj( ) ): ej"ej( pL`2)"p !hj( pL ).
t~2
t
t
t
Begin by taking the following second-order Taylor expansion:
;(p , p , hj )";(p , p , p )#ej ; (p , p , p )
t~1 3 t~2 t~1 t~1
t~2 t~1 t~1
t~2 t~1 t~2
#(ej )2; (p , p , p )#o((ej )2),
t~1
33 t~2 t~1 t~1
t~1
where ; ( ) ) and ; ( ) ) denote, respectively, the "rst and second partial deriva3
33
tives of ;( ) ) with respect to its third argument. The "rst-order condition to
8 The unabridged version of this paper (Brock and de Fontnouvelle, 1996) contains further details
concerning the random information motivation for heterogeneity. The approach is closely inspired
by Chamberlain and Imbens (1996).
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 737
the original maximization problem (1) implies that the "rst-order term
; (p , p , p ) is zero,9 so that
3 t~2 t~1 t~1
!/
(ej )2#o((ej )2)
;j"/
t~1
t
1,t~1
2,t~1 t~1
where /
and /
are appropriately de"ned non-negative constants. The
1,t~1
2,t~1
deterministic utility ;j can thus be approximated with a mean-squared error
t
based performance measure, where the constants are time-varying. In order to
ensure tractability, we make the approximation that / "/ is constant over
2,t
2
time.10 Consideration of the more general case is deferred to future research.
Following Anderson et al. (1992), assume that the ij have an extreme value
i,t
distribution. One then obtains a logit model for the distribution of predictors
across agents, with choice probabilities given by
P(hj)"e~b(ejt~1)2/z ,
(20)
t
t
K
(21)
z " + e~b(ekt~1)2,
t
k/1
where we have combined the two constants b and / into one by renaming
2
b"b/ . Because the number of agents is in"nite and the ij are independent
2
i,t
across agents, the law of large numbers implies that the fraction of agents
choosing predictor hj( ) ) equals the probability that any individual agent
chooses hj( ) ):
nj"P(hj)"e~b(ejt~1)2/z .
(22)
t
t
t
Or in the notation of (5), we write nj"n( pL`2), noting that N"2. One may
t~1
t
think of discrete choice as a tractable way of modeling predictor selection in an
evolutionary context, in which each predictor competes against the others for
survival. Survival consists of a predictor being used by agents. In our model, the
population of predictors is represented by H, each predictor's "tness by
;(p , p , pj ), and natural selection by Eq. (22).
t~2 t~1 t~1
5.2. The large type limit
In the previous section, we used standard discrete choice theory to show that
for a "xed "nite number of predictors, the fraction of agents using predictor hj( ) )
at time t converges to nj"e~b(ejt~1)2/z as the number of agents I tends to in"nity.
t
t
9 ; (p , p , p )"[u ( ) )!u ( ) )p /p ]p s@(p /p )/(p )2 is the "rst-order term.
3 t~2 t~1 t~1
1
2
t~2 t~1 t~2 t~2 t~1 t~1
The "rst-order condition implied by (1) is u ( ) )!u ( ) )p /p "0.
1
2
t~2 t~1
10 It is not necessary to assume that / is constant over time, since /
drops out of the choice
1,t
1,t~1
probabilities (20) and (21).
738 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
In this section, we explore the case where the number of predictors also tends to
in"nity. Our "rst task will thus be to extend the discrete choice results to this
limit case. In particular, we must check that some version of equilibrium
condition (6) carries through. To our knowledge, the following large type limit
(LTL) arguments are new, and may be useful beyond the current context. In
particular, they could be applied in any situation in which one wishes to model
choice with observational error across a large (in"nite) number of alternatives.
Let the number of predictors K(I) be a function of I such that K and I tend to
in"nity at the same rate. Suppose that each predictor in the predictor space can
be expressed in the form hj( pL )"h( pL , hj ), where hj is a "nite dimensional
I
t~1 I
t~1
parameter vector that completely describes the predictor hj( ) ). Then the time
t values of the predictor and forecast error for the I agent economy can be
rewritten as h (hj )"h( pL , hj ) and e (hj )"e( pL`2, hj ).
I
t
t I
t~1 I
t I
Suppose further that instead of the predictor space H "Mh1,2, hK(I)N being
I
deterministic, each hj is drawn at random from a multivariate density f (h). Note
I
that this density does not depend on either j or I. The density f (h) represents the
distribution of all the di!erent predictors agents may choose from. In practice
N , the number of parameters required to specify f ( ) ) will always be small. For
f
the normal distribution, for example, N "2]Dim(h). The advantage of the
f
LTL approach is that it reduces the number of parameters in the model from
K(I)]Dim(h) to N . When K(I) is large, this reduction can be dramatic.
f
Now for each value of I, let
H "Mh1,2, hK(I)N,
I
I
I
be a K(I) dimensional array of parameter vectors hj , and let H"MH N= be an
I
I I/1
in"nite sequence of these arrays. Corresponding to H will be an in"nite sequence
of monetary economies, so that each random draw of H will correspond to
a di!erent sequence of economies. By construction,11 all such sequences will
converge to the same large type limit economy.
Consider now a particular random draw of H. For any number of agents I,
Eq. (18) implies that each agent chooses (the parameter vector associated with)
her predictor randomly from H . For any "nite number of agents I, the fraction
I
of agents choosing predictor h (hj ) will thus also be random. Let 1 (i, j, H ) be the
t
I
t I
indicator function of the event that agent i chooses predictor h (hj ):
t I
G
1
1 (i, j, H )"
t
I
0
if ; (hj )#ij /b'max M; (hk)#ik /bN,
i,t
i,t
kEj t I
t I
otherwise.
11 This point can be seen explicitly in Eq. (30).
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 739
The fraction of agents choosing predictor h (hj ) is then given by
t I
I
n8 (hj )"I~1 + 1 (i, j, H ).
t
I
t I
i/1
For each I, equilibrium condition (6) becomes
K(I)
M"p + n8 (hj )s(p /h (hj )).
t
t I t t I
j/1
Taking expectations of both sides of (23) yields
E[n8 (hj )]"E[1 (i, j, H )]
t
I
t I
and from (20) it follows that
(23)
(24)
(25)
e~be2t~1(hjI)
E[1 (i, j, H )]"
.
(26)
t
I
+K(I) e~be2t~1(hkI)
k/1
By (22), right-hand side of (26) equals n (hj ). Combining this with (25) and (26)
t I
yields
(27)
E[n8 (hj )]"n (hj ).
t I
t I
Inspection of (27) suggests that the limits (as I tends to in"nity) of
+K(I) n8 (hj )s(p /h (hj )) and +K(I) n (hj )s(p /h (hj )) should be identical, so that taking
j/1 t I t t I
j/1 t I t t I
he limit of (24) as I tends to in"nity yields
K(I)
(28)
M"p lim + n (hj )s(p /h (hj )).
t I t t I
t
I?= j/1
The argument in the above paragraph is formally demonstrated in Appendix.
Now use (22) to expand (28), obtaining
lim
K~1+K(I) s(p /h (hj ))e~be2t~1(hjI)
j/1 t t I
I?=
.
lim
K~1+K(I) e~be2t~1(hjI)
j/1
I?=
The numerator and denominator of (29) approach
M"p
t
P
E [s(p /h (h))e~be2t~1(h)]"
h
t t
2 h
t~1
P
E [e~be ( )]"
h
h
(29)
s(p /h (h))e~be2t~1(h)f (h) dh,
t t
2 h
t~1
e~be ( )f (h) dh,
h
respectively. Substituting in h ( h)"h( pL , h) and e (h)"e( pL`2, h), we can
t~1
t~1
t~1
t
use (29) to obtain the large type limit equilibrium condition:
M"p E [s(p /h( pL , h))n( pL`2, h)],
t h
t
t~1
t~1
h
t~1 , )/z ,
n( pL`2, h)"e~be2( pL`2
t~1
t
h
t~1 , )].
z "E [e~be2(pL`2
t
h
(30)
740 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
Taking the limit of an economy as some index (typically the number of agents)
tends to in"nity has a long history in economics (Aumann, 1964; Debreu and
Scarf, 1973). A chief motivation for doing so has been to provide a justi"cation
for modeling agents as price takers. Hildenbrand and Kirman (1975, p. 13) write,
&The only situation where accepting prices as beyond one's in#uence seems
reasonable is in a large market. If there are many participants it becomes less
plausible that one individual can have a signi"cant impact on the market price
of a commodity'. The motivation behind the LTL limit economy is much the
same: to ensure that no individual agent (or predictor) can a!ect aggregate
behavior.
6. A log linear utility example
In order to evaluate the properties of the system de"ned by this equilibrium
condition, we specify a utility function so that the Jacobian at the steady state
can be calculated analytically. Let agents' preferences be given by
u(c: , c0 )"ln(c: )#ln(c0 )
i,t
i,t
i,t i,t
so that the savings function is
(31)
s(p /h( pL ))"(w:!w0h( pL )/p )/2.
t~1 t
t~1
t
Substituting into the equilibrium condition (30) yields
p "/( pL`2)"2M/w:#(w0/w:)h!''( pL`2)
t~1
t~1
t
where h!''( ) ) represents an aggregate predictor of the future price:
(32)
h!''( pL`2)"E [n( pL`2, h)h( pL , h)].
t~1
t~1
t~1
h
Allowing for predictors to display steady-state bias has reduced the dependence
of our substantive conclusions on ad hoc assumptions in two ways. First,
steady-state bias allows a form of natural selection to operate at the steady
state.12 Rather than the system being exogenously determined by how we chose
our predictors, its behavior is in part endogenously determined by the forces of
natural selection, which reduce the e!ects of poor predictors.
The second way in which steady state-bias reduces dependence on ad hoc
assumptions is to make h!''( ) ) much richer than h3%1( ) ). The latter depends only
on ¸ lags of p ; the former depends on ¸#2 lags. So while h3%1( ) ) could be
t
interpreted as the predictor of an individual agent, h!''( ) ) cannot: h!''( ) ) depends
not only on ¸ lagged prices, but also on the previous forecasting error(s), which
12 Recall that without bias, each predictor has an equal weight n6 j at the steady state.
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 741
depend on additional lags. In addition, h!''( ) ) can be a much more complicated
function than any of its component predictors h( pL , h), so that in general
t~1
h!''( ) ) will not be linear even if h( pL , h) is linear for all h. The aggregate
t~1
predictor can thus induce much richer behavior than any of the component
predictors h( pL , h).
t~1
Assumption A4@@. The aggregate predictor h!''( ) ) has no steady-state bias, so that
h!''(p6 ,2, p6 )"p6 .
Assumption A4A guarantees the existence of a monetary steady state.13 Note
we require only that the aggregate predictor h!''( ) ) have no steady-state bias,
whereas in Section 4 we required that none of the individual predictors hj( ) )
have steady-state bias. This di!erence has signi"cant consequences. Di!erentiating (32) with respect to p and evaluating at the monetary steady state yields
t~q
/ "wz6 E [h (h)e~be6 2(h)]!2bwz6 E [hM (h)e6 (h)e (h)e~be6 2(h)]
q
h q
h
q
#2bwz6 2E [hM (h)e~be6 2(h)]E [e6 (h)e (h)e~be6 2(h)]
(33)
h
h
q
where w"w0/w: is the ratio of old age to youthful endowment,
hM (h)"h(p6 ,2, p6 , h), and h (h) denotes the partial derivative of h( pL , h) with
t~1
q
respect to p . The terms e6(h) and e (h) are de"ned similarly. By the de"nition of
t~q
q
the forecast error, it follows that
e (h)"1,
(34)
1
e (h)"0,
(35)
2
e (h)"!h (h) for q53.
(36)
q
q~2
So in order to evaluate (33), we need not specify the entire distribution f (h); we
need only specify the implied distributions for the functions e6 (h) and h (h). We do
q
so via the following assumption:
Assumption A5. (a) h (h) and e6 (h) each have a normal distribution:
q
h (h)&N(k , p2 ) and e6 (h)&N(0, p2) (b) h (h) and e6 (h) are independent of each
e
q
q
q h,q
other. (c) h (h) and e6 (h) are each independently distributed.
q
Two points are in order. First, the gaussian speci"cation together with linear
utility implies that savings will be negative for some agents, which does not
13 A steady state may exist under weaker but less compact assumptions. The subsequent results
would still hold.
742 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
make sense in the monetary OLG model. In practice, however, the mass of these
agents may be negligible. At the steady state, for example,
s(p6 /h(p6 ))(0 Q h(p6 )/p6 'w:/w0 Q (M#2e)/M'w:/w0.
By making the money supply M large, we can make the mass of agents with
negative savings arbitrarily small. Notice too that M does not a!ect the stability
properties of the steady state p6 .14 We also explored this issue numerically: for
each numerical simulation of our model, we searched for instances where an
individual savings function was negative. Since no such instances were found
over numerous simulations, we conclude that the negative savings problem is of
little practical importance.
The second point to be made in connection with Assumption A5 is that it
may not be immediately clear whether it is possible to satisfy the independence
Assumption A5(b). An example of how to do so is provided in Section 7.2
(Eq. (49)). In the Appendix we show the following:
(37)
/ "w(k #c(b, p2)),
e
1
1
/ "wk ,
(38)
2
2
(39)
/ "w(k !k c(b, p2)),
e
q
q
q~2
2bp2
e .
c(b, p2)"
e
1#2bp2
e
The appendix also veri"es that Assumption A5 implies (and is thus consistent
with) the previously made Assumption A4A.
We begin by analyzing a small dimensional system, and thus set h "0 for all
q
q'1. By (38) }(39), it follows that / "0, / "!uk c(b, p2), and / "0 for
e
q
2
3
1
q53. This leads to a three-dimensional system whose Jacobian at the steady
state p6 has eigenvalues which are given by the roots to the following characteristic polynomial:
p(j)"j3!j2w(k #c(b,p2))#wk c(b, p2).
e
e
1
1
To economize on notation, set c"c(b, p2). We let w be "xed, and analyze the
e
stability properties of the system pL`2"U(pL`2) as k and c vary.
t~1
1
t
Theorem. (Jury, 1974, p. 29). A necessary and suzcient condition for the roots of
p(j)"0 to lie inside the unit circle is that both of the following criteria be met:
1. Root criterion: p(1)'0 and p(!1)(0.
14 The stability of the steady state depends on the values of M/ , / ,2N, which by (37)}(39) do not
1 2
depend on M.
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 743
2. Determinant criterion: The following determinants are positive:
KC
1
D "
B
0
D C
!w(k #c)
0
1
$
1
wk c
1
DK
wk c
1 .
0
It is clear that for k '0, p(!1) is always negative, and D is positive
1
`
whenever D is. We can thus analyze the stability of the system by plotting the
~
two curves corresponding to p(1)"0 and D "0:
~
p(1)"0 Q c"(wk !1)/(wk !w),
(40)
1
1
D "0 Q (wk c)2#w2k c(k #c)"1.
(41)
~
1
1 1
Values of c satisfying the root criterion (40) can be parameterized as an
increasing concave function c (k ) with c (1/w)"0. Values of c satisfying the
rc 1
rc
determinant criterion can be parameterized as a decreasing convex function
c (k ) with c (k-)"1. The two functions c (k ) and c (k ) intersect at a point
dc 1
dc
rc 1
dc 1
k6. The values for k- and k6 are given by
k-(2k-#1)"1/w2,
(42)
k6"(3#J9!8w)/(4w).
(43)
The above discussion is summarized in Fig. 1. It is clear from Fig. 1 that the root
and determinant criteria divide the plane into four regions; one stable and three
unstable. The boundary between the stable and unstable regions is comprised of
two curves, one of which is marked HOPF. Along this curve, there is one real
eigenvalue inside the unit circle and two complex eigenvalues on the border of
the unit circle.15 We have just demonstrated that the main conditions for the
Hopf bifurcation theorem hold along HOPF.
Hopf bifurcation theorem16 (Guckenheimer and Holmes, 1983, p. 162). Let
f :R2PR2 be a one-parameter family of mappings which has a smooth family of
k
xxed points x(k) at which the eigenvalues are complex conjugates j(k), j(k6). Assume
Dj(k )D"1 but jj(k )O1 for j"1, 2, 3, 4.
0
0
d
(Dj(k )D)"dO0.
0
dk
(44)
(45)
15 It is easy to verify that p(!1)(0 and p(0)'0, so that there is always one real eigenvalue in the
interval (!1,0). Because HOPF de"nes a border between the stable and unstable regions, we know
that at least one eigenvalue must be on the unit circle. Because (along HOPF) p(!1)O0 and
p(1)O0, this eigenvalue cannot be real. Hence there must be a conjugate pair of complex eigenvalues
on the unit circle.
16 Application of this theorem to mappings of the form f : RnPRp is achieved via center manifold
k
reduction, as discussed in Guckenheimer and Holmes (1983), Section 3.2.
744 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
Fig. 1. Bifurcation analysis.
Then there is a smooth change of coordinates h so that the expression of hf h~1 in
k
polar coordinates has the form
hf h~1(r, h)"(r(1#d(k!k )#ar2), h#c#br2)# higher-order terms.
k
0
If, in addition
aO0.
(46)
Then there is a two-dimensional surface R in R2]R having quadratic tangency
with the plane R2]Mk N which is invariant for f. If RW(R2]MkN) is larger than
0
a point, then it is a simple closed curve.
We have already shown that the "rst part of condition (44) is satis"ed along
the curve HOPF. In the Appendix, we verify the second part of condition (44), as
well as condition (45). Assuming nondegeneracy with respect to condition (46),17
we can summarize via the following proposition:
17 This nondegeneracy assumption is generic in the sense of Kuznetsov (1995, p. 124).
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 745
Proposition 3. If k-(k (k6, where k- and k6 are given in (42) and (43), then
1
a closed orbit exists near the curve (labeled HOPF) satisfying Eq. (40).
Fig. 1 also provides a clear demonstration of the e!ects of steady-state bias.
The case where no predictor has steady-state bias corresponds to p2"0 and
e
hence to c(b, p2)"0. The system loses stability only when k '1/w, which
e
1
corresponds to agents' expectations being highly sensitive to past variations in
prices. This is exactly the same result as in Proposition 2: the system is unstable
and the dynamics resemble those under perfect foresight. When there is steady
state bias, p2'0 so that c(b,p2) ranges between 0 and 1 as b increases. This
e
e
allows for a Hopf bifurcation as k and c cross the HOPF curve, so that prices
1
#uctuate periodically.
Periodically #uctuating prices are not economically plausible because they
imply that agents make predictable forecast errors which do not vanish over
time. We conjecture, however, that in higher dimensional systems, the price
paths resulting from the Hopf bifurcation become increasingly complex. We
take up this issue in the next section.
7. Numerical results
The basic model explored in Section 6 was deliberately chosen to be the
simplest that would display the qualitative behavior described in Fig. 1. We
explore various alterations to this basic model with two purposes in mind. The
"rst is to explore the qualitative robustness of our results to variations in
preferences and predictor selection mechanisms. The second purpose for enriching the basic model is to determine whether doing so may result in more
economically plausible behavior (in terms of forecast error predictability).
7.1. Qualitative robustness
Our "rst check for qualitative robustness is to vary the agents' risk aversion.
We model preferences by the following CES class of utility functions:
(47)
u(c: ,c0 )"(c: )c`1/(c#1)#(c0 )c`1/(c#1).
i,t
i,t
i,t i,t
The preferences (31) are a special case of (47) with the risk aversion parameter set
to c"!1. Setting w"0.5 and M"100, we use numerical methods detailed in
the Appendix to calculate the eigenvalues of DU( ) ) at a discrete grid of points in
Mk ,c(b,p2)N space for varying degrees of risk aversion. The results are displayed
1
in Fig. 2.
The model's qualitative behavior is invariant to reasonable changes in the
degree of risk aversion. In particular, there exists for each value of c a pair
Mk-, k6N such that whenever k e(k-, k6), two complex conjugate eigenvalues cross
1
746 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
Fig. 2. Stability for varying risk aversion.
the unit circle as c(b, p2) increases from zero to one. Quantitatively, the
results make the occurrence of complicated dynamics seem more likely.
In particular, for c"!0.1 a Hopf bifurcation occurs for values of k as low
1
as 0.6.
Our second check for qualitative robustness is to allow agents' choice of
predictors to depend on a larger set of past squared forecast errors. In Section
5.1, we assumed that the deterministic part of the time t utility associated with
each predictor depended only upon the time t!1 performance of that predictor. It seems reasonable that when choosing predictors, agents may consider
a longer run performance measure. In particular, suppose that performance is
a distributed lag of past realized utilities:
=
)
,p
, hj
;j" + aq;(p
t
t~2~q t~1~q t~2~q
q/0
(48)
for some a3[0,1). Again setting w"0.5 and M"100, we use numerical calculations detailed in the Appendix to explore the stability of the system generated
by (48) at a discrete grid of points in Mk , c(b, p2)N space. The results are displayed
e
1
in Figs. 3 and 4:
The model's behavior (both qualitative and quantitative) is
24 (2000) 725}759
Expectational diversity in monetary economies
William A. Brock!,*, Patrick de Fontnouvelle"
!Department of Economics, University of Wisconsin, Madison, WI 53706, USA
"U.S. Securities and Exchange Commission, 450 5th Street N.W., Washington, DC 20549, USA
Accepted 30 April 1999
Abstract
We investigate an overlapping generations monetary economy in which expectations
depend upon backward looking predictors of the future price level. We use discrete
choice theory to model how agents select a predictor based on its past forecast error.
Letting the number of available predictors tend to in"nity, we obtain the large type limit
of the system. Taking the large type limit dramatically reduces the number of free
parameters, while maintaining the expectational diversity which we argue is necessary for
constructing plausible learning-based models. The model's dynamics are strongly in#uenced by the intensity of choice, which measures how sensitive an agent's predictor
choice is to di!erences in forecast errors across predictors. When the intensity of choice is
low, the monetary steady state is stable. As the intensity of choice increases (and if certain
parametric restrictions are met) the system undergoes a Hopf bifurcation, in which case
we document the existence of highly irregular equilibrium price paths. ( 2000
Published by Elsevier Science B.V. All rights reserved.
Keywords: Discrete choice; Endogenous #uctuations; Fiat money; Hopf bifurcation,
Learning; Overlapping generations
* Corresponding author. We thank Cars Hommes for helpful discussions and suggestions, and
also W. Davis Dechert for suggesting the reference by E. Jury. All errors are the authors' own
responsibility. W.A. Brock thanks the NSF under grant d SES-9122344, and the Vilas Trust for
"nancial support. P. de Fontnouvelle thanks the Iowa State University Department of Economics
for supporting this research while he was an Assistant Professor there.
The Securities and Exchange Commission, as a matter of policy, disclaims responsibility for any
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0165-1889/00/$ - see front matter ( 2000 Published by Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 2 4 - X
726 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
1. Introduction
Overlapping generations (OLG) models of money (Samuelson, 1958; Wallace,
1980) typically display a continuum of deterministic rational expectations
equilibria. One of these equilibria is the monetary steady state, in which "at
money retains a constant value forever. The other equilibria all converge to
autarky, so that money gradually becomes worthless. The monetary steady state
is appealing to economists largely because it corresponds to everyday experience; the other equilibria seem implausible because they do not. Agents in
modern market economies take it for granted that, barring gross government
misconduct, currency will retain its role as medium of exchange inde"nitely. No
U.S. resident places any serious probability on the economy degenerating into
barter within the foreseeable future. The rational expectations hypothesis, however, provides no guidance as to which equilibrium path should prevail (Boldrin
and Woodford (1990)). The problem (often called &indeterminacy') is that each of
these paths corresponds to some consistent set of expectations about future
prices. This problem is serious because it implies that under rational expectations the OLG model cannot explain the existence of valued "at money, which
we have argued is a main &fact' of everyday life.
Some have suggested resolving the indeterminacy problem by noting that
when perfect foresight is replaced by learning rules, the OLG model often
converges to the monetary steady state (Lucas, 1986; Marcet and Sargent,
1989a,b). Since learning rules provide a plausible approximation of how people
actually behave, this convergence suggests that the monetary steady state is in
fact the most reasonable long-run outcome.1 Others, however, have shown that
such answers to the indeterminacy problem su!er from an inherent lack of
generality: while certain learning rules in certain models single out economically
appealing equilibria, other rules in other models actually increase the number of
possible equilibrium paths. In addition to generating explosive paths similar to
those which emerge under perfect foresight (Evans and Honkapohja, 1994a,b),
learning may lead to complex price paths which neither explode nor converge to
the monetary steady state (Bullard, 1994; Grandmont and Laroque, 1991). In
addition, Du!y (1994) shows that if agents use an adaptive rule to form
expectations about in#ation (rather than price level), then the economy can
converge to a continuum of nonstationary equilibria.
Because of this lack of generality, Lucas (1986) believes it will be impossible to
address fully the problem of indeterminacy via &purely mathematical' methods:
&It is hard to see what can advance the discussion short of assembling a collection of people, putting them in the situation of interest, and observing what they
1 Evans and Honkapohja (1994a,b,1995) have written extensively on learning rules as selection
criteria for rational expectations equilibria.
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 727
do.' This approach is pursued by Lim et al. (1994), who use experimental
methods to examine the OLG model with constant money supply; their results
lend strong support to the monetary steady-state equilibrium. Marimon and
Sunder (1993) examine a related experimental OLG economy in which the
money supply is allowed to grow over time, and "nd that observed price paths
tend to converge to the low in#ation steady state. Ochs (1995, p. 205) summarizes these results: &There is one characteristic common to all of these sessions
that is of direct interest to monetary economists. In none of these sessions is
there any evidence that individuals have the `foresighta to follow rational
expectations equilibrium paths that generate hyperin#ation.'
The experimental approach, however, will always leave some important
questions unanswered. Lim et al. (1994, p. 267) write, &With replication there is
marked convergence towards stationary equilibrium though the convergence is
not precise. To what extent further replications may a!ect the nature of convergence is an open question'. Given enough time, one would like to know if the
experiment would converge precisely to the steady state, or if small price
#uctuations would persist inde"nitely. Would further replications of the experiment a!ect the results? Might there be other parameter values for which the
results would be qualitatively di!erent?
In order to address these questions, we must accept Lucas's implicit challenge
of "nding a &purely mathematical' method of exploring learning dynamics which
also retains a high degree of generality. We believe that the necessary ingredient
for achieving such generality is expectational diversity: by allowing many
di!erent expectations (rules) to be nested within one model, one can greatly
reduce the danger that the model's stability properties depend on one particular
rule. The idea of agents choosing between several competing expectations dates
at least as far back as Arthur (1994a,b) and Conlisk (1980).
In this paper (Sections 2}5), we introduce expectational diversity by allowing
agents to choose among a large number of predictors of the future price level.
These predictors are "nitely parameterized functions of past prices. We model
the distribution of predictors across agents in two steps. First, we use discrete
choice tools to construct a tractable form of natural selection dynamics over the
space of predictors. The fraction of agents adopting a particular predictor is
determined by that predictor's past squared forecast error (SFE), relative to the
SFE of other predictors. The sensitivity of this fraction to variations in SFE is
called the intensity of choice.
The result is the Adaptive Rational Equilibrium Dynamics (ARED) of Brock
and Hommes (1997), in which the dynamics of predictor choice is determined by
the equilibrium dynamics of endogenous variables. These authors emphasize
that most economic models displaying complicated dynamics can be expressed
as one-dimensional di!erence equations, such as Benhabib and Day (1982) and
Grandmont (1985). Although there are some two-dimensional models (Benhabib and Farmer, 1994; Cazzavillan et al., 1996; de Vilder, 1995; Grandmont
728 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
et al. (1996); Woodford, 1986) ARED introduces entirely new state variables
corresponding to the fractions of agents using each predictor. As we shall see,
the extra dimensions added by these state variables create the possibility of very
rich dynamic behavior.
The second step in modeling the distribution of predictors across agents is to
let the number of available predictors tend to in"nity, obtaining what we call the
large type limit (LTL) of the ARED. In the large type limit, the parameters of
each predictor are drawn at random from continuous distributions, whose mean
is called predictor sensitivity and whose variance is called predictor variance.2 In
Section 6, we provide an analytical exploration of a simple version of the model,
in which predictors depend on only one lag of past price. The model's dynamics
are determined jointly by the predictor sensitivity, the predictor variance and
the intensity of choice. When the predictor sensitivity is low, the monetary
steady state is asymptotically stable. This stability is in accord with previous
analytic, computational, and experimental results.3 When the predictor sensitivity is high, the monetary steady state is unstable, as in the original perfect
foresight versions of the OLG model (Samuelson, 1958; Wallace, 1980); equilibrium paths not originating at the steady state are explosive.
The most unusual behavior occurs at intermediate levels of sensitivity. We
show that when predictor variance and intensity of choice are low, the monetary
steady state is asymptotically stable. As each of these parameters increases, the
monetary steady state undergoes a Hopf bifurcation, near which a closed
periodic orbit exists. Periodic orbits, however, are not economically plausible
because they imply simple forecast errors that never vanish (Hommes, 1998). In
Section 7, we use numerical simulations to investigate the global dynamics of
a higher dimensional version of the model, in which predictors depend on two
lags of the price level. We show that as the intensity of choice increases, the time
series behavior of the price level can become increasingly erratic. The structure
of the associated forecast errors can become complicated enough that linear
methods cannot distinguish them from white noise.
While it is true that nonlinear methods (such as those discussed in LeBaron,
1994) may detect some predictability in the forecast errors, it is possible that
other versions of the model may generate forecast errors that appear random
even to nonlinear tests. Although such work is beyond the scope of the current
paper, it would mesh nicely with the agenda outlined by Grandmont (1992):
&One might envision for instance a situation in which traders attribute their
forecasting mistakes to noise, although the observed dynamics are actually
deterministic but chaotic. As I said, this is largely unknown territory and should
be the subject of future research'.
2 Brock (1997) sketches the above approach in the context of "nancial modeling.
3 See, for example, Lucas (1986), Arifovic (1995), and Lim et al. (1994), respectively.
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 729
1.1. Related literature
Economic models of learning have received an enormous amount of recent
attention: Marimon (1997) provides a comprehensive survey. The original
rational expectations criticism of such models was that agents continue using
the same learning rule even after it has been found to be systematically wrong.
There have been two strands of work addressing this criticism. The "rst strand
proposes analytical models of expectational diversity (Bray and Savin, 1986;
Evans et al., 1995; Marcet and Sargent, 1989a,b). As we argued previously,
expectational diversity reduces the danger that agents continue using rules that
perform poorly.
The second strand is the evolutionary approach (Arifovic, 1995; Arthur,
1994a; Arthur et al., 1997; Bullard and Du!y, 1995a,b; Darley and Kau!man,
1997; Day and Chen, 1993;4 LeBaron, 1995; Weidlich, 1991), which handles the
rational expectations criticism very elegantly; natural selection serves to remove
rules that perform poorly. Arifovic (1995), for example, studies the same OLG
model as the current paper, but uses genetic algorithms to model how agents
update their price level forecast rules. Because these evolutionary approaches
have traditionally been highly computational, they have not been amenable to
direct analysis. Our methods can be viewed as o!ering analytic results for this
computational literature. The discrete choice approach to natural selection over
rules plus the notion of Large Type Limit is a novel combination of technique that
allows us to bridge these two strands. We believe our methods allow us to push
analytic results on evolution signi"cantly farther than the current literature.
2. The economic model
Our analysis is based upon a standard overlapping generations model of
a monetary economy (Samuelson, 1958; Wallace, 1980). Each generation consists of an equal number of agents. Agents live for two periods, and are endowed
with w: when young and w0 when old. An agent in generation t consumes
cy when young and c0 when old. The amount of "at money that government
t
t
supplies is denoted M, which is constant over time. The non-negative time
t price level is denoted p . All agents have identical preferences, so that young
t
agent i solves the following maximization problem:
max u(c: , c0 )
i,t i,t
si,t
4 Of particular relevance in this volume are the papers by P. Allen, J. Conlisk, and R. Day.
(1)
730 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
subject to
(2)
c: "w:!s ,
i,t
i,t
c0 "w0#(p /p% )s ,
(3)
i,t
t i,t`1 i,t
where s denotes the real time t savings of agent i, and p%
denotes i's time
i,t
i,t`1
t expectation of p . We make the following assumptions about the utility
t`1
function and endowments:
Assumption A1. (a) Indi!erence curves are convex to the origin. (b) The
utility function is strictly increasing in both arguments. (c) u /u PR as
1 2
c: /c0 P0, and u /u P0 as c: /c0 PR, where u and u denote the partial
i,t i,t
1 2
i,t i,t
1
2
derivatives of u( ) ) with respect to c: and c0 , respectively. (d) c: and c0 are
i,t
i,t
i,t
i,t
normal goods.
Assumption A2. u (w:, w0)(u (w:, w0).
1
2
Assumption A3. c: and c0 are gross substitutes.
i,t
i,t
Assumption A1 contains standard (Sargent, 1987, p. 232) restrictions on the
utility function, which guarantee that for each expected price ratio p /p%
there
t i,t`1
is a unique value of savings s : s "s(p /p% ). Assumption A2 implies that
i,t i,t
t i,t`1
s(1)'0. Assumption A3 implies that savings is an increasing function of p /p%
t i,t`1
(Varian, 1984): s@(x)'0 for all x. This rules out the possibility of complicated
dynamics when all agents have perfect foresight (Kehoe et al., 1986), thus
emphasizing that our results depend more on how expectational diversity is
modeled than on a particular choice of savings function.
Our modi"cation to the standard OLG model is to allow agents to have
heterogeneous expectations about the future price level. Assume "nite memory,
so that agents' common information set consists of a "nite length vector of past
prices:
pL "Mp ,2, p N.
t~1
t~1
t~L
The expected price of agent i is a continuously di!erentiable function of her
information set:
(4)
p% "hj"hj( pL ).
t~1
t
i,t`1
Each hj( ) ) is called a predictor, and the space of predictors is denoted
H:H"Mh1,2, hKN. To make notation clearer, we always index agents with the
subscript &i' and predictors with the superscript &j'. The speci"cation (4) allows
for a wide range of expectations, including learning (Grandmont and Laroque,
1986, p. 140). Suppose, for instance, that agents believe the economy's actual
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 731
dynamics imply p "f ( pl , h), where h is a vector of (unknown) structural
t~1
t`1
parameters and l4¸. Each period, agents can use some econometric procedure
to obtain an estimate h%45( pL ). They can then form their expectations
t~1
by replacing the true value of h with the estimate h%45( pL ), to obtain
t~1
p% "f ( pl ,h%45( pL )).
t~1
t~1
t`1
Let the fraction of agents in generation t who choose predictor hj at time t be
a continuously di!erentiable function of pL`N:
t~1
nj"nj( pL`N).
(5)
t
t~1
The parameter N indexes the number of extra lagged prices taken as arguments
of the functions nj( ) ). At this level of generality, each nj( ) ) should be thought of
as some past performance metric for the predictor hj( ) ). Since hj( pL ), the
t~3
predicted value of p , depends on pL , it is clear that, for the OLG model,
t~1
t~3
N must be at least 2. In Section 5, we propose a speci"cation for the functions
nj( ) ) based on discrete choice theory. For the moment, leaving this speci"cation
open will emphasize the generality of Propositions 1 and 2.
Equilibrium requires that nominal aggregate savings be equal to the money
supply:
K
M"p + nj( pL`N)s(p /hj( pL )).
t~1
t
t~1
t
j/1
(6)
3. Homogeneous expectations
We begin by reviewing the model's dynamics under two special cases. First,
for all i.
suppose that all agents have perfect foresight: p% "h( pL )"p
t~1
t`1
i,t`1
The equilibrium condition (6) reduces to
M"p s(p /p ).
(7)
t t t`1
There is exactly one steady-state equilibrium where money has value. We refer
to this equilibrium, given by p6 "M/s(1), as the monetary steady state. There is
also exactly one steady state (autarky) where money has no value, as well as
a continuum of nonstationary equilibria where money has value. The latter
are indexed by the initial price level p e(p6 ,R), and all converge to autarky.
0
The coexistence of many possible equilibrium price paths is referred to as
indeterminacy.
As mentioned previously, the problem of indeterminacy has led researchers to
consider various types of backward looking expectations as selection criteria for
plausible long-run equilibria. We begin with a simple example, in which all
for all i. In this case, (6)
agents have naive expectations: p% "h( pL )"p
t~1
t~1
i,t`1
reduces to
M"p s(p /p ).
t t t~1
(8)
732 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
It is easy to verify that under (8), all equilibrium paths converge to the monetary
steady state p6 "M/s(1). Naive expectations thus support the claim that the
monetary steady state is a reasonable long-run equilibrium for (7). Of course, the
is the simplest and most ad hoc possible. We thus
speci"cation p% "p
t`1
t~1
examine the stability properties of the monetary steady state under a more
general speci"cation of expectations.
3.1. General backward expectations
If agents have general homogeneous expectations p% "h( pL ),5 the equit~1
t`1
librium condition (6) becomes
(9)
M"p s(p /h( pL )).
t~1
t t
The monetary steady state p6 is the solution to M"p6 s(p6 /h(p6 ,2, p6 )). The existence of such a p6 is guaranteed by the following assumption:
Assumption A4. The predictor h( ) ) satis"es the following conditions: (a)
lim p/h(p,2, p)(R, (b) lim
p/h(p,2, p)'r , where the autarkic interp?0
p?=
!65
est rate r is de"ned implicitly by s(r )"0.
!65
!65
Eq. (9) implicitly de"nes a nonlinear function and a corresponding system
given by
p "/H( pL ),
t~1
t
N.
pL"UH( pL )"M/H( pL ), p ,2, p
t~1 t~1
t~L`1
t~1
t
Rewriting the equilibrium condition (9) in terms of /H( ) ), we obtain
(10)
(11)
(12)
M"/H( pL )s( /H( pL )/h( pL )).
t~1
t~1
t~1
Di!erentiating (12) with respect to p , evaluating at the monetary steady state
t~q
p6 , and solving for /H yields
q
/H"lh ,
q
q
(p6 /h(p6 ,2, p6 ))2s@(p6 /h(p6 ,2, p6 ))
l"
,
s(p6 /h(p6 ,2, p6 ))#p6 /h(p6 ,2, p6 )s@(p6 /h(p6 ,2, p6 ))
where /H and h denote the partial derivatives of /H( ) ) and h( ) ) with respect to
q
q
p , evaluated at the steady state pL "p6 . The Jacobian of UH( ) ) at the steady
t~1
t~q
5 The analysis in this section is related to Grandmont and Laroque (1986), who explore the
relationship between perfect foresight and backward looking dynamics when agents have homogeneous expectations.
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 733
state can then be written:
C
J"DUH(p6 )"
lh
1
1
lh
2
0
lh
3
0
2
2
lh
L~1
0
lh
L
0
0
1
0
2
0
0
0
0
1
2
0
0
F
F
F
F
F
0
0
0
1
0
2
D
We show that the monetary steady state p6 is asymptotically stable provided that
expectations are not too sensitive to past #uctuations in price level:
L
l + Dh D(1,
q
q/1
(13)
Proposition 1. Condition (13) is a suzcient condition for the monetary steady state
p6 to be asymptotically stable.
Proof. See appendix. h
Although the stability criterion (13) seems like a plausible conjecture, there is
no a priori way of verifying how sensitive agents' expectations are to past price
#uctuations. As in the simple case of naive expectations, the classi"cation of
equilibria as stable or unstable must ultimately rest on an ad hoc assumption.
Consider the following example, in which expectations are an average of past
prices:
h( pL )"a p #2#a p .
t~1
1 t~1
L t~L
If the absolute values of the coe$cients a sum to some value less than 1/l, then
l
the monetary steady state is asymptotically stable. If, on the other hand, these
absolute values sum to some number larger than 1/l, then (13) does not hold and
the monetary steady state may be unstable.
4. Heterogeneous expectations without bias
We reduce the dependence of our analysis on ad hoc assumptions by allowing
agents to have heterogeneous expectations; when there are many predictors, the
model's dynamics will depend less on the speci"cation of any one predictor.
Before considering the case of heterogeneous expectations, however, we
introduce a useful property called steady-state bias:
734 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
Dexnition. Let p6 denote the price level at the monetary steady state. If
hj(p6 ,2,p6 )"p6 , we say that the predictor hj( ) ) has no steady-state bias. Otherwise, we say that hj( ) ) has steady-state bias.
In other words, a predictor has no steady-state bias if it predicts next period's
price level to be p6 whenever the price level has been p6 for the past ¸ periods.
Suppose there is no steady-state bias:
Assumption A4@. Each predictor hj( ) ) has no steady-state bias, so that
hj(p6 ,2,p6 )"p6 for all j.
In the case of heterogeneous expectations, equilibrium condition (6) clearly
implies that p6 "M/s(1) is the monetary steady state. As before, (6) implicitly
de"nes a nonlinear function and a corresponding system given by:
(14)
p "/( pL`N),
t~1
t
N.
(15)
pL`N"U( pL`N)"M/( pL`N), p ,2, p
t~1 t~1
t~L~N`1
t~1
t
Note that allowing for heterogeneity means that (14) and (15) incorporate
N more lagged values of p than do (10) and (11). (See also the discussion
t
immediately below Eq. (5).) Rewriting (6) in terms of /( ) ), we obtain
K
M"/( pL`N) + nj( pL`N)s(/( pL`N)/hj( pL )).
t~1
t~1
t~1
t~1
j/1
Di!erentiating with respect to p , and evaluating at the steady state yields
t~q
K
(16)
0" + Mnjp6 s(1)#n6 j/ s(1)#n6 js@(1)[/ !hj]N
q
q
q
q
j/1
where nj and / denote the partial derivative of nj( ) ) and /( ) ) with respect to
q
q
p , and n6 j"nj(p6 ,2, p6 ). Now +K nj( ) )"1 implies that the summation
t~q
j/1
+K nj"0, so that solving (16) for / yields
j/1 q
q
K
/ "l + n6 jhj,
q
q
j/1
s@(1)
l"
.
s(1)#s@(1)
At the steady state, the multi-predictor system behaves exactly like the one
predictor system with the one representative predictor given by
K
h3%1( pL )" + n6 jhj( pL ).
t~1
t~1
j/1
(17)
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 735
This system's stability properties can be analyzed by using the previous section's
results:
Proposition 2. When none of the predictors hj( ) ) has steady-state bias, the system
(15) is asymptotically stable provided that l+L Dh3%1D(1.
q/1 q
The proof of Proposition 2 is identical to that of Proposition 1. As in the one
predictor case, the system's local stability depends upon how sensitive the
representative predictor is to past price #uctuations, not on how the predictor
space H is speci"ed. Any substantive conclusions we draw based upon local
stability at the steady state are thus contingent on how we specify the representative predictor.
5. Heterogeneous expectations with bias
When steady-state bias is allowed, (16) cannot be analyzed unless we specify
how the fractions nj are determined. Suppose that for each predictor heH, agent
t
i's associated utility is composed of a deterministic and a stochastic component:
(18)
;I j ";j#ij /b.
i,t
t
i,t
The deterministic component ;j represents the realized utility of an agent in
t
generation t!2 who predicted the time t!1 price level to be p% "hj :
t~2
t~1
;j";(p ,p ,hj )
t
t~2 t~1 t~2
(19)
"u[w:!s(p /hj )),w0#(p /p )s(p /hj ))].
t~2 t~1 t~2 t~2
t~2 t~2
The stochastic components ij are i.i.d. across time,6 across predictors, and
i,t
across agents.7 The intensity of choice b speci"es how good a measure the
deterministic component ;j is of the choice utility ;I j . It regulates how sensitive
i,t
t
agents are to di!erences in realized utility between predictors. When bP0, the
random component is very large, so that variation in ;j has almost no e!ect on
t
agents' choice of predictor. When bPR, all agents choose the predictor with
the highest realization of ;j. It is clear from the above discussion that the
t
random component ij together with the intensity of choice b will be crucial in
i,t
determining the model's dynamics. We thus present in some detail three possible
sources of the uncertainty represented by the ij .
i,t
6 The temporal independence assumption can be relaxed without a!ecting the model's aggregate
dynamics (de Fontnouvelle, 1996).
7 Brock (1993) shows how to use interacting particle systems theory to relax the assumption of
independence across agents.
736 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
(A) Random preferences: Agents' behavior may be inherently unpredictable, so
that an agent facing repeated instances of the same decision problem (and
observing the same information concerning the problem) varies his choices over
time.
(B) Random characteristics: Manski (1977, p. 235) interprets the stochastic
components ij as coming from three sources, all of which represent a lack of
i,t
knowledge on the part of the econometrician. (a) Measurement error: the economist has no direct knowledge of the choice utilities ;I j . (b) Specixcation error:
i,t
the utilities given in (19) may be of the wrong functional form. (c) Unobservable
characteristics: parameters such as risk aversions or endowments might be
unobserved or only partially observed. Faced with the same decision problem
repeatedly, each individual agent will always make the same decision. Two
agents who appear identical to the economist, however, may make quite di!erent decisions.
(C) Random information: Suppose that ;j is unknown to agents at the begint
ning of period t, but that they learn about it via a Bayesian updating strategy.8
Under the random information motivation for heterogeneity, an agent facing
repeated instances of the same decision problem will vary his choices over time
* not because his preferences are random but because his information is.
5.1. Calculating choice probabilities
In this section, we show how to calculate from (18) an expression for agents'
choice probabilities. In particular, P(hj), the probability that an agent chooses
t
predictor hj( ) ) at time t, will depend on the time t!1 squared forecast error
associated with predictor hj( ) ).
Dexnition. The forecast error ej is the di!erence between the actual price at time
t
t and that predicted by hj( ) ): ej"ej( pL`2)"p !hj( pL ).
t~2
t
t
t
Begin by taking the following second-order Taylor expansion:
;(p , p , hj )";(p , p , p )#ej ; (p , p , p )
t~1 3 t~2 t~1 t~1
t~2 t~1 t~1
t~2 t~1 t~2
#(ej )2; (p , p , p )#o((ej )2),
t~1
33 t~2 t~1 t~1
t~1
where ; ( ) ) and ; ( ) ) denote, respectively, the "rst and second partial deriva3
33
tives of ;( ) ) with respect to its third argument. The "rst-order condition to
8 The unabridged version of this paper (Brock and de Fontnouvelle, 1996) contains further details
concerning the random information motivation for heterogeneity. The approach is closely inspired
by Chamberlain and Imbens (1996).
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 737
the original maximization problem (1) implies that the "rst-order term
; (p , p , p ) is zero,9 so that
3 t~2 t~1 t~1
!/
(ej )2#o((ej )2)
;j"/
t~1
t
1,t~1
2,t~1 t~1
where /
and /
are appropriately de"ned non-negative constants. The
1,t~1
2,t~1
deterministic utility ;j can thus be approximated with a mean-squared error
t
based performance measure, where the constants are time-varying. In order to
ensure tractability, we make the approximation that / "/ is constant over
2,t
2
time.10 Consideration of the more general case is deferred to future research.
Following Anderson et al. (1992), assume that the ij have an extreme value
i,t
distribution. One then obtains a logit model for the distribution of predictors
across agents, with choice probabilities given by
P(hj)"e~b(ejt~1)2/z ,
(20)
t
t
K
(21)
z " + e~b(ekt~1)2,
t
k/1
where we have combined the two constants b and / into one by renaming
2
b"b/ . Because the number of agents is in"nite and the ij are independent
2
i,t
across agents, the law of large numbers implies that the fraction of agents
choosing predictor hj( ) ) equals the probability that any individual agent
chooses hj( ) ):
nj"P(hj)"e~b(ejt~1)2/z .
(22)
t
t
t
Or in the notation of (5), we write nj"n( pL`2), noting that N"2. One may
t~1
t
think of discrete choice as a tractable way of modeling predictor selection in an
evolutionary context, in which each predictor competes against the others for
survival. Survival consists of a predictor being used by agents. In our model, the
population of predictors is represented by H, each predictor's "tness by
;(p , p , pj ), and natural selection by Eq. (22).
t~2 t~1 t~1
5.2. The large type limit
In the previous section, we used standard discrete choice theory to show that
for a "xed "nite number of predictors, the fraction of agents using predictor hj( ) )
at time t converges to nj"e~b(ejt~1)2/z as the number of agents I tends to in"nity.
t
t
9 ; (p , p , p )"[u ( ) )!u ( ) )p /p ]p s@(p /p )/(p )2 is the "rst-order term.
3 t~2 t~1 t~1
1
2
t~2 t~1 t~2 t~2 t~1 t~1
The "rst-order condition implied by (1) is u ( ) )!u ( ) )p /p "0.
1
2
t~2 t~1
10 It is not necessary to assume that / is constant over time, since /
drops out of the choice
1,t
1,t~1
probabilities (20) and (21).
738 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
In this section, we explore the case where the number of predictors also tends to
in"nity. Our "rst task will thus be to extend the discrete choice results to this
limit case. In particular, we must check that some version of equilibrium
condition (6) carries through. To our knowledge, the following large type limit
(LTL) arguments are new, and may be useful beyond the current context. In
particular, they could be applied in any situation in which one wishes to model
choice with observational error across a large (in"nite) number of alternatives.
Let the number of predictors K(I) be a function of I such that K and I tend to
in"nity at the same rate. Suppose that each predictor in the predictor space can
be expressed in the form hj( pL )"h( pL , hj ), where hj is a "nite dimensional
I
t~1 I
t~1
parameter vector that completely describes the predictor hj( ) ). Then the time
t values of the predictor and forecast error for the I agent economy can be
rewritten as h (hj )"h( pL , hj ) and e (hj )"e( pL`2, hj ).
I
t
t I
t~1 I
t I
Suppose further that instead of the predictor space H "Mh1,2, hK(I)N being
I
deterministic, each hj is drawn at random from a multivariate density f (h). Note
I
that this density does not depend on either j or I. The density f (h) represents the
distribution of all the di!erent predictors agents may choose from. In practice
N , the number of parameters required to specify f ( ) ) will always be small. For
f
the normal distribution, for example, N "2]Dim(h). The advantage of the
f
LTL approach is that it reduces the number of parameters in the model from
K(I)]Dim(h) to N . When K(I) is large, this reduction can be dramatic.
f
Now for each value of I, let
H "Mh1,2, hK(I)N,
I
I
I
be a K(I) dimensional array of parameter vectors hj , and let H"MH N= be an
I
I I/1
in"nite sequence of these arrays. Corresponding to H will be an in"nite sequence
of monetary economies, so that each random draw of H will correspond to
a di!erent sequence of economies. By construction,11 all such sequences will
converge to the same large type limit economy.
Consider now a particular random draw of H. For any number of agents I,
Eq. (18) implies that each agent chooses (the parameter vector associated with)
her predictor randomly from H . For any "nite number of agents I, the fraction
I
of agents choosing predictor h (hj ) will thus also be random. Let 1 (i, j, H ) be the
t
I
t I
indicator function of the event that agent i chooses predictor h (hj ):
t I
G
1
1 (i, j, H )"
t
I
0
if ; (hj )#ij /b'max M; (hk)#ik /bN,
i,t
i,t
kEj t I
t I
otherwise.
11 This point can be seen explicitly in Eq. (30).
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 739
The fraction of agents choosing predictor h (hj ) is then given by
t I
I
n8 (hj )"I~1 + 1 (i, j, H ).
t
I
t I
i/1
For each I, equilibrium condition (6) becomes
K(I)
M"p + n8 (hj )s(p /h (hj )).
t
t I t t I
j/1
Taking expectations of both sides of (23) yields
E[n8 (hj )]"E[1 (i, j, H )]
t
I
t I
and from (20) it follows that
(23)
(24)
(25)
e~be2t~1(hjI)
E[1 (i, j, H )]"
.
(26)
t
I
+K(I) e~be2t~1(hkI)
k/1
By (22), right-hand side of (26) equals n (hj ). Combining this with (25) and (26)
t I
yields
(27)
E[n8 (hj )]"n (hj ).
t I
t I
Inspection of (27) suggests that the limits (as I tends to in"nity) of
+K(I) n8 (hj )s(p /h (hj )) and +K(I) n (hj )s(p /h (hj )) should be identical, so that taking
j/1 t I t t I
j/1 t I t t I
he limit of (24) as I tends to in"nity yields
K(I)
(28)
M"p lim + n (hj )s(p /h (hj )).
t I t t I
t
I?= j/1
The argument in the above paragraph is formally demonstrated in Appendix.
Now use (22) to expand (28), obtaining
lim
K~1+K(I) s(p /h (hj ))e~be2t~1(hjI)
j/1 t t I
I?=
.
lim
K~1+K(I) e~be2t~1(hjI)
j/1
I?=
The numerator and denominator of (29) approach
M"p
t
P
E [s(p /h (h))e~be2t~1(h)]"
h
t t
2 h
t~1
P
E [e~be ( )]"
h
h
(29)
s(p /h (h))e~be2t~1(h)f (h) dh,
t t
2 h
t~1
e~be ( )f (h) dh,
h
respectively. Substituting in h ( h)"h( pL , h) and e (h)"e( pL`2, h), we can
t~1
t~1
t~1
t
use (29) to obtain the large type limit equilibrium condition:
M"p E [s(p /h( pL , h))n( pL`2, h)],
t h
t
t~1
t~1
h
t~1 , )/z ,
n( pL`2, h)"e~be2( pL`2
t~1
t
h
t~1 , )].
z "E [e~be2(pL`2
t
h
(30)
740 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
Taking the limit of an economy as some index (typically the number of agents)
tends to in"nity has a long history in economics (Aumann, 1964; Debreu and
Scarf, 1973). A chief motivation for doing so has been to provide a justi"cation
for modeling agents as price takers. Hildenbrand and Kirman (1975, p. 13) write,
&The only situation where accepting prices as beyond one's in#uence seems
reasonable is in a large market. If there are many participants it becomes less
plausible that one individual can have a signi"cant impact on the market price
of a commodity'. The motivation behind the LTL limit economy is much the
same: to ensure that no individual agent (or predictor) can a!ect aggregate
behavior.
6. A log linear utility example
In order to evaluate the properties of the system de"ned by this equilibrium
condition, we specify a utility function so that the Jacobian at the steady state
can be calculated analytically. Let agents' preferences be given by
u(c: , c0 )"ln(c: )#ln(c0 )
i,t
i,t
i,t i,t
so that the savings function is
(31)
s(p /h( pL ))"(w:!w0h( pL )/p )/2.
t~1 t
t~1
t
Substituting into the equilibrium condition (30) yields
p "/( pL`2)"2M/w:#(w0/w:)h!''( pL`2)
t~1
t~1
t
where h!''( ) ) represents an aggregate predictor of the future price:
(32)
h!''( pL`2)"E [n( pL`2, h)h( pL , h)].
t~1
t~1
t~1
h
Allowing for predictors to display steady-state bias has reduced the dependence
of our substantive conclusions on ad hoc assumptions in two ways. First,
steady-state bias allows a form of natural selection to operate at the steady
state.12 Rather than the system being exogenously determined by how we chose
our predictors, its behavior is in part endogenously determined by the forces of
natural selection, which reduce the e!ects of poor predictors.
The second way in which steady state-bias reduces dependence on ad hoc
assumptions is to make h!''( ) ) much richer than h3%1( ) ). The latter depends only
on ¸ lags of p ; the former depends on ¸#2 lags. So while h3%1( ) ) could be
t
interpreted as the predictor of an individual agent, h!''( ) ) cannot: h!''( ) ) depends
not only on ¸ lagged prices, but also on the previous forecasting error(s), which
12 Recall that without bias, each predictor has an equal weight n6 j at the steady state.
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 741
depend on additional lags. In addition, h!''( ) ) can be a much more complicated
function than any of its component predictors h( pL , h), so that in general
t~1
h!''( ) ) will not be linear even if h( pL , h) is linear for all h. The aggregate
t~1
predictor can thus induce much richer behavior than any of the component
predictors h( pL , h).
t~1
Assumption A4@@. The aggregate predictor h!''( ) ) has no steady-state bias, so that
h!''(p6 ,2, p6 )"p6 .
Assumption A4A guarantees the existence of a monetary steady state.13 Note
we require only that the aggregate predictor h!''( ) ) have no steady-state bias,
whereas in Section 4 we required that none of the individual predictors hj( ) )
have steady-state bias. This di!erence has signi"cant consequences. Di!erentiating (32) with respect to p and evaluating at the monetary steady state yields
t~q
/ "wz6 E [h (h)e~be6 2(h)]!2bwz6 E [hM (h)e6 (h)e (h)e~be6 2(h)]
q
h q
h
q
#2bwz6 2E [hM (h)e~be6 2(h)]E [e6 (h)e (h)e~be6 2(h)]
(33)
h
h
q
where w"w0/w: is the ratio of old age to youthful endowment,
hM (h)"h(p6 ,2, p6 , h), and h (h) denotes the partial derivative of h( pL , h) with
t~1
q
respect to p . The terms e6(h) and e (h) are de"ned similarly. By the de"nition of
t~q
q
the forecast error, it follows that
e (h)"1,
(34)
1
e (h)"0,
(35)
2
e (h)"!h (h) for q53.
(36)
q
q~2
So in order to evaluate (33), we need not specify the entire distribution f (h); we
need only specify the implied distributions for the functions e6 (h) and h (h). We do
q
so via the following assumption:
Assumption A5. (a) h (h) and e6 (h) each have a normal distribution:
q
h (h)&N(k , p2 ) and e6 (h)&N(0, p2) (b) h (h) and e6 (h) are independent of each
e
q
q
q h,q
other. (c) h (h) and e6 (h) are each independently distributed.
q
Two points are in order. First, the gaussian speci"cation together with linear
utility implies that savings will be negative for some agents, which does not
13 A steady state may exist under weaker but less compact assumptions. The subsequent results
would still hold.
742 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
make sense in the monetary OLG model. In practice, however, the mass of these
agents may be negligible. At the steady state, for example,
s(p6 /h(p6 ))(0 Q h(p6 )/p6 'w:/w0 Q (M#2e)/M'w:/w0.
By making the money supply M large, we can make the mass of agents with
negative savings arbitrarily small. Notice too that M does not a!ect the stability
properties of the steady state p6 .14 We also explored this issue numerically: for
each numerical simulation of our model, we searched for instances where an
individual savings function was negative. Since no such instances were found
over numerous simulations, we conclude that the negative savings problem is of
little practical importance.
The second point to be made in connection with Assumption A5 is that it
may not be immediately clear whether it is possible to satisfy the independence
Assumption A5(b). An example of how to do so is provided in Section 7.2
(Eq. (49)). In the Appendix we show the following:
(37)
/ "w(k #c(b, p2)),
e
1
1
/ "wk ,
(38)
2
2
(39)
/ "w(k !k c(b, p2)),
e
q
q
q~2
2bp2
e .
c(b, p2)"
e
1#2bp2
e
The appendix also veri"es that Assumption A5 implies (and is thus consistent
with) the previously made Assumption A4A.
We begin by analyzing a small dimensional system, and thus set h "0 for all
q
q'1. By (38) }(39), it follows that / "0, / "!uk c(b, p2), and / "0 for
e
q
2
3
1
q53. This leads to a three-dimensional system whose Jacobian at the steady
state p6 has eigenvalues which are given by the roots to the following characteristic polynomial:
p(j)"j3!j2w(k #c(b,p2))#wk c(b, p2).
e
e
1
1
To economize on notation, set c"c(b, p2). We let w be "xed, and analyze the
e
stability properties of the system pL`2"U(pL`2) as k and c vary.
t~1
1
t
Theorem. (Jury, 1974, p. 29). A necessary and suzcient condition for the roots of
p(j)"0 to lie inside the unit circle is that both of the following criteria be met:
1. Root criterion: p(1)'0 and p(!1)(0.
14 The stability of the steady state depends on the values of M/ , / ,2N, which by (37)}(39) do not
1 2
depend on M.
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 743
2. Determinant criterion: The following determinants are positive:
KC
1
D "
B
0
D C
!w(k #c)
0
1
$
1
wk c
1
DK
wk c
1 .
0
It is clear that for k '0, p(!1) is always negative, and D is positive
1
`
whenever D is. We can thus analyze the stability of the system by plotting the
~
two curves corresponding to p(1)"0 and D "0:
~
p(1)"0 Q c"(wk !1)/(wk !w),
(40)
1
1
D "0 Q (wk c)2#w2k c(k #c)"1.
(41)
~
1
1 1
Values of c satisfying the root criterion (40) can be parameterized as an
increasing concave function c (k ) with c (1/w)"0. Values of c satisfying the
rc 1
rc
determinant criterion can be parameterized as a decreasing convex function
c (k ) with c (k-)"1. The two functions c (k ) and c (k ) intersect at a point
dc 1
dc
rc 1
dc 1
k6. The values for k- and k6 are given by
k-(2k-#1)"1/w2,
(42)
k6"(3#J9!8w)/(4w).
(43)
The above discussion is summarized in Fig. 1. It is clear from Fig. 1 that the root
and determinant criteria divide the plane into four regions; one stable and three
unstable. The boundary between the stable and unstable regions is comprised of
two curves, one of which is marked HOPF. Along this curve, there is one real
eigenvalue inside the unit circle and two complex eigenvalues on the border of
the unit circle.15 We have just demonstrated that the main conditions for the
Hopf bifurcation theorem hold along HOPF.
Hopf bifurcation theorem16 (Guckenheimer and Holmes, 1983, p. 162). Let
f :R2PR2 be a one-parameter family of mappings which has a smooth family of
k
xxed points x(k) at which the eigenvalues are complex conjugates j(k), j(k6). Assume
Dj(k )D"1 but jj(k )O1 for j"1, 2, 3, 4.
0
0
d
(Dj(k )D)"dO0.
0
dk
(44)
(45)
15 It is easy to verify that p(!1)(0 and p(0)'0, so that there is always one real eigenvalue in the
interval (!1,0). Because HOPF de"nes a border between the stable and unstable regions, we know
that at least one eigenvalue must be on the unit circle. Because (along HOPF) p(!1)O0 and
p(1)O0, this eigenvalue cannot be real. Hence there must be a conjugate pair of complex eigenvalues
on the unit circle.
16 Application of this theorem to mappings of the form f : RnPRp is achieved via center manifold
k
reduction, as discussed in Guckenheimer and Holmes (1983), Section 3.2.
744 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
Fig. 1. Bifurcation analysis.
Then there is a smooth change of coordinates h so that the expression of hf h~1 in
k
polar coordinates has the form
hf h~1(r, h)"(r(1#d(k!k )#ar2), h#c#br2)# higher-order terms.
k
0
If, in addition
aO0.
(46)
Then there is a two-dimensional surface R in R2]R having quadratic tangency
with the plane R2]Mk N which is invariant for f. If RW(R2]MkN) is larger than
0
a point, then it is a simple closed curve.
We have already shown that the "rst part of condition (44) is satis"ed along
the curve HOPF. In the Appendix, we verify the second part of condition (44), as
well as condition (45). Assuming nondegeneracy with respect to condition (46),17
we can summarize via the following proposition:
17 This nondegeneracy assumption is generic in the sense of Kuznetsov (1995, p. 124).
W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759 745
Proposition 3. If k-(k (k6, where k- and k6 are given in (42) and (43), then
1
a closed orbit exists near the curve (labeled HOPF) satisfying Eq. (40).
Fig. 1 also provides a clear demonstration of the e!ects of steady-state bias.
The case where no predictor has steady-state bias corresponds to p2"0 and
e
hence to c(b, p2)"0. The system loses stability only when k '1/w, which
e
1
corresponds to agents' expectations being highly sensitive to past variations in
prices. This is exactly the same result as in Proposition 2: the system is unstable
and the dynamics resemble those under perfect foresight. When there is steady
state bias, p2'0 so that c(b,p2) ranges between 0 and 1 as b increases. This
e
e
allows for a Hopf bifurcation as k and c cross the HOPF curve, so that prices
1
#uctuate periodically.
Periodically #uctuating prices are not economically plausible because they
imply that agents make predictable forecast errors which do not vanish over
time. We conjecture, however, that in higher dimensional systems, the price
paths resulting from the Hopf bifurcation become increasingly complex. We
take up this issue in the next section.
7. Numerical results
The basic model explored in Section 6 was deliberately chosen to be the
simplest that would display the qualitative behavior described in Fig. 1. We
explore various alterations to this basic model with two purposes in mind. The
"rst is to explore the qualitative robustness of our results to variations in
preferences and predictor selection mechanisms. The second purpose for enriching the basic model is to determine whether doing so may result in more
economically plausible behavior (in terms of forecast error predictability).
7.1. Qualitative robustness
Our "rst check for qualitative robustness is to vary the agents' risk aversion.
We model preferences by the following CES class of utility functions:
(47)
u(c: ,c0 )"(c: )c`1/(c#1)#(c0 )c`1/(c#1).
i,t
i,t
i,t i,t
The preferences (31) are a special case of (47) with the risk aversion parameter set
to c"!1. Setting w"0.5 and M"100, we use numerical methods detailed in
the Appendix to calculate the eigenvalues of DU( ) ) at a discrete grid of points in
Mk ,c(b,p2)N space for varying degrees of risk aversion. The results are displayed
1
in Fig. 2.
The model's qualitative behavior is invariant to reasonable changes in the
degree of risk aversion. In particular, there exists for each value of c a pair
Mk-, k6N such that whenever k e(k-, k6), two complex conjugate eigenvalues cross
1
746 W.A. Brock, P. de Fontnouvelle / Journal of Economic Dynamics & Control 24 (2000) 725}759
Fig. 2. Stability for varying risk aversion.
the unit circle as c(b, p2) increases from zero to one. Quantitatively, the
results make the occurrence of complicated dynamics seem more likely.
In particular, for c"!0.1 a Hopf bifurcation occurs for values of k as low
1
as 0.6.
Our second check for qualitative robustness is to allow agents' choice of
predictors to depend on a larger set of past squared forecast errors. In Section
5.1, we assumed that the deterministic part of the time t utility associated with
each predictor depended only upon the time t!1 performance of that predictor. It seems reasonable that when choosing predictors, agents may consider
a longer run performance measure. In particular, suppose that performance is
a distributed lag of past realized utilities:
=
)
,p
, hj
;j" + aq;(p
t
t~2~q t~1~q t~2~q
q/0
(48)
for some a3[0,1). Again setting w"0.5 and M"100, we use numerical calculations detailed in the Appendix to explore the stability of the system generated
by (48) at a discrete grid of points in Mk , c(b, p2)N space. The results are displayed
e
1
in Figs. 3 and 4:
The model's behavior (both qualitative and quantitative) is