*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 grantdSES-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.
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, USAAccepted 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 thelarge 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
1Evans and Honkapohja (1994a,b,1995) have written extensively on learning rules as selection criteria for rational expectations equilibria.
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 experi-ence; 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, how-ever, 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 expecta-tions 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.1Others, 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:
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) summar-izes 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 conver-gence 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 experi-ment 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 ofpredictorsof 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 theintensity of choice.
2Brock (1997) sketches the above approach in the context of"nancial modeling.
3See, for example, Lucas (1986), Arifovic (1995), and Lim et al. (1994), respectively.
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 calledpredictor sensitivityand whose variance is calledpredictorvariance.2In 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.3When the predictor sensitiv-ity is high, the monetary steady state is unstable, as in the original perfect foresight versions of the OLG model (Samuelson, 1958; Wallace, 1980); equilib-rium 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):
4Of particular relevance in this volume are the papers by P. Allen, J. Conlisk, and R. Day.
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;4LeBaron, 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 con-sists 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
cytwhen young andc0t when old. The amount of "at money that government supplies is denoted M, which is constant over time. The non-negative time
tprice level is denotedp
t. All agents have identical preferences, so that young
agentisolves the following maximization problem:
max
si,t
subject to
c:
i,t"w:!si,t, (2)
c0i,t"w0#(p
t/p%i,t`1)si,t, (3)
wheres
i,t denotes the real timetsavings of agenti, andp%i,t`1denotesi's time
t expectation of p
t`1. We make the following assumptions about the utility 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) u1/u2PR as
c:i,t/c0i,tP0, and u
1/u2P0 as c:i,t/c0i,tPR, whereu1and u2denote the partial derivatives ofu()) with respect to c:i,t and c0i,t, respectively. (d) c:i,t and c0i,t are
normal goods.
Assumption A2. u
1(w:,w0)(u2(w:,w0).
Assumption A3. c:i,tandc0i,tare gross substitutes.
Assumption A1 contains standard (Sargent, 1987, p. 232) restrictions on the utility function, which guarantee that for each expected price ratiop
t/p%i,t`1there is a unique value of savingss
i,t: si,t"s(pt/p%i,t`1). Assumption A2 implies that
s(1)'0. Assumption A3 implies that savings is an increasing function ofp
t/p%i,t`1 (Varian, 1984):s@(x)'0 for allx. 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
t~1"Mpt~1,2,pt~LN.
The expected price of agent i is a continuously di!erentiable function of her information set:
p%
i,t`1"hjt"hj(pLt~1). (4) 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
dynamics implyp
t`1"f(plt~1,h), wherehis a vector of (unknown) structural parameters andl4¸. Each period, agents can use some econometric procedure to obtain an estimate h%45(pLt
~1). They can then form their expectations by replacing the true value of h with the estimate h%45(pLt
~1), to obtain
p%t`1"f(plt
~1,h%45(pLt~1)).
Let the fraction of agents in generationtwho choose predictorhjat timetbe a continuously di!erentiable function ofpL`N
t~1:
njt"nj(pL`N
t~1). (5)
The parameterNindexes the number of extra lagged prices taken as arguments of the functionsnj()). At this level of generality, eachnj()) should be thought of
as some past performance metric for the predictor hj()). Since hj(pLt
~3), the predicted value of p
t~1, depends onpLt~3, it is clear that, for the OLG model,
Nmust 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:
We begin by reviewing the model's dynamics under two special cases. First, suppose that all agents have perfect foresight:p%i,t`1"h(pLt
~1)"pt`1for alli. The equilibrium condition (6) reduces to
M"p
ts(pt/pt`1). (7)
There is exactly one steady-state equilibrium where money has value. We refer to this equilibrium, given byp6"M/s(1), as themonetary 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
0e(p6,R), and all converge to autarky. 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 agents have naive expectations:p%i,t`1"h(pLt
~1)"pt~1for alli. In this case, (6) reduces to
M"p
5The 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 homo-geneous expectations.
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 speci"cationp%t`1"p
t~1 is the simplest and most ad hoc possible. We thus 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 expectationsp%t`1"h(pLt
~1),5the equi-librium condition (6) becomes
M"p
ts(pt/h(pLt~1)). (9)
The monetary steady statep6 is the solution to M"p6s(p6/h(p6,2,p6)). The
exist-ence of such ap6 is guaranteed by the following assumption:
Assumption A4. The predictor h()) satis"es the following conditions: (a)
lim
p?0p/h(p,2,p)(R, (b) limp?=p/h(p,2,p)'r!65, where the autarkic inter-est rater
!65is de"ned implicitly bys(r!65)"0.
Eq. (9) implicitly de"nes a nonlinear function and a corresponding system given by
p
t"/H(pLt~1), (10)
pLt"UH(pLt~1)"M/H(pLt
~1),pt~1,2,pt~L`1N. (11) Rewriting the equilibrium condition (9) in terms of/H()), we obtain
M"/H(pLt
~1)s(/H(pLt~1)/h(pLt~1)). (12) Di!erentiating (12) with respect top
t~q, evaluating at the monetary steady state
p6, and solving for/Hq yields
/Hq"lhq,
l" (p6/h(p6,2,p6))2s@(p6/h(p6,2,p6))
s(p6/h(p6,2,p6))#p6/h(p6,2,p6)s@(p6/h(p6,2,p6)),
where/Hq andhqdenote the partial derivatives of/H()) andh()) with respect to
p
state can then be written:
J"DUH(p6)"
C
lh
1 lh2 lh3 2 lhL~1 lhL
1 0 0 2 0 0
0 1 0 2 0 0
0 0 1 2 0 0
F F F F F
0 0 0 2 1 0
D
We show that the monetary steady statep6 is asymptotically stable provided that expectations are not too sensitive to past#uctuations in price level:
l+L q/1
DhqD(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(pLt
~1)"a1pt~1#2#aLpt~L.
If the absolute values of the coe$cientsa
lsum to some value less than 1/l, then
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
Dexnition. Let p6 denote the price level at the monetary steady state. If
hj(p6,2,p6)"p6, we say that the predictorhj()) has no steady-state bias.
Other-wise, we say thathj()) has steady-state bias.
In other words, a predictor has no steady-state bias if it predicts next period's price level to bep6 whenever the price level has beenp6 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 allj.
In the case of heterogeneous expectations, equilibrium condition (6) clearly implies thatp6"M/s(1) is the monetary steady state. As before, (6) implicitly de"nes a nonlinear function and a corresponding system given by:
p
t"/(pL`Nt~1), (14)
pL`N
t "U(pL`Nt~1)"M/(pL`Nt~1),pt~1,2,pt~L~N`1N. (15) Note that allowing for heterogeneity means that (14) and (15) incorporate
N more lagged values of p
t than do (10) and (11). (See also the discussion
immediately below Eq. (5).) Rewriting (6) in terms of/()), we obtain
M"/(pL`N
Di!erentiating with respect top
t~q, and evaluating at the steady state yields
0"+K
At the steady state, the multi-predictor system behaves exactly like the one predictor system with the one representative predictor given by
6The temporal independence assumption can be relaxed without a!ecting the model's aggregate dynamics (de Fontnouvelle, 1996).
7Brock (1993) shows how to use interacting particle systems theory to relax the assumption of independence across agents.
This system's stability properties can be analyzed by using the previous section's results:
Proposition 2. When none of the predictorshj())has steady-state bias,the system
(15)is asymptotically stable provided thatl+Lq/1Dh3%1q D(1.
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 spaceHis speci"ed. Any substantive conclusions we draw based upon local
stability at the steady state are thus contingent on how we specify the represen-tative predictor.
5. Heterogeneous expectations with bias
When steady-state bias is allowed, (16) cannot be analyzed unless we specify how the fractionsnjtare determined. Suppose that for each predictorheH, agent
i's associated utility is composed of a deterministic and a stochastic component:
;I
ji,t";jt#iji,t/b. (18)
The deterministic component;j
t represents the realized utility of an agent in
generationt!2 who predicted the time t!1 price level to bep%t~1"hjt~2:
;jt";(p
t~2,pt~1,hjt~2)
"u[w:!s(p
t~2/hjt~2)),w0#(pt~2/pt~1)s(pt~2/hjt~2))]. (19) The stochastic components iji,t are i.i.d. across time,6 across predictors, and across agents.7 The intensity of choice b speci"es how good a measure the deterministic component;jtis of the choice utility;I ji
,t. It regulates how sensitive
agents are to di!erences in realized utility between predictors. WhenbP0, the random component is very large, so that variation in;
jthas almost no e!ect on agents'choice of predictor. WhenbPR, all agents choose the predictor with the highest realization of ;
8The 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).
(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
componentsiji,
tas coming from three sources, all of which represent a lack of
knowledge on the part of the econometrician. (a)Measurement error: the eco-nomist has no direct knowledge of the choice utilities;I
ji,t. (b)Specixcation error:
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! er-ent decisions.
(C)Random information: Suppose that;jtis unknown to agents at the
begin-ning of periodt, 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(hjt), the probability that an agent chooses predictorhj()) at timet, will depend on the time t!1 squared forecast error
associated with predictorhj()).
Dexnition. The forecast errorejtis the di!erence between the actual price at time
tand that predicted byhj()): ejt"ej(pL`2
t )"pt!hj(pLt~2). Begin by taking the following second-order Taylor expansion:
;(p
t~2,pt~1,hjt~2)";(pt~2,pt~1,pt~1)#ejt~1;3(pt~2,pt~1,pt~1)
#(ejt~1)2;
33(pt~2,pt~1,pt~1)#o((ejt~1)2), where;
3()) and;33()) denote, respectively, the"rst and second partial
9;
3(pt~2,pt~1,pt~1)"[u1())!u2())pt~2/pt~1]pt~2s@(pt~2/pt~1)/(pt~1)2 is the "rst-order term. The"rst-order condition implied by (1) isu
1())!u2())pt~2/pt~1"0.
10It is not necessary to assume that/
1,tis constant over time, since/1,t~1drops out of the choice
probabilities (20) and (21).
the original maximization problem (1) implies that the "rst-order term
;
3(pt~2,pt~1,pt~1) is zero,9so that
;jt"/
1,t~1!/2,t~1(ejt~1)2#o((ejt~1)2) where/1,
t~1and/2,t~1are appropriately de"ned non-negative constants. The deterministic utility ;
jt can thus be approximated with a mean-squared error based performance measure, where the constants are time-varying. In order to ensure tractability, we make the approximation that/
2,t"/2is constant over time.10Consideration of the more general case is deferred to future research.
Following Anderson et al. (1992), assume that theiji,
thave an extreme value
distribution. One then obtains a logit model for the distribution of predictors across agents, with choice probabilities given by
P(hjt)"e~b(ejt~1)2/z
where we have combined the two constants b and /
2 into one by renaming
b"b/
2. Because the number of agents is in"nite and theiji,tare independent
across agents, the law of large numbers implies that the fraction of agents choosing predictor hj()) equals the probability that any individual agent
chooseshj()):
njt"P(hjt)"e~b(ejt~1)2/z
t. (22)
Or in the notation of (5), we writenjt"n(pL`2
t~1), noting thatN"2. One may 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
t~2,pt~1,pjt~1), and natural selection by Eq. (22).
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 predictorhj())
at timetconverges tonjt"e~b(ejt~1)2/z
11This point can be seen explicitly in Eq. (30).
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 predictorsK(I) be a function ofIsuch thatKandItend to in"nity at the same rate. Suppose that each predictor in the predictor space can be expressed in the formhj(pLt
~1)"h(pLt~1,hjI), wherehjIis a"nite dimensional 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 ash
t(hjI)"h(pLt~1,hjI) andet(hjI)"e(pL`t 2,hjI).
Suppose further that instead of the predictor spaceH
I"Mh1,2,hK(I)Nbeing
deterministic, eachhjIis drawn at random from a multivariate densityf(h). Note
that this density does not depend on eitherjorI. The densityf(h) represents the
distribution of all the di!erent predictors agents may choose from. In practice
N
f, the number of parameters required to specifyf()) will always be small. For
the normal distribution, for example, N
f"2]Dim(h). The advantage of the
LTL approach is that it reduces the number of parameters in the model from
K(I)]Dim(h) toN in"nite sequence of these arrays. Corresponding toHwill 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 ofH. For any number of agentsI, Eq. (18) implies that each agent chooses (the parameter vector associated with) her predictor randomly fromH
I. For any"nite number of agentsI, the fraction
of agents choosing predictorh
t(hjI) will thus also be random. Let 1t(i,j,HI) be the
indicator function of the event that agentichooses predictorh
The fraction of agents choosing predictorh
For eachI, equilibrium condition (6) becomes
M"p
Taking expectations of both sides of (23) yields
E[n8
t(hjI)]"E[1t(i,j,HI)] (25)
and from (20) it follows that
E[1
By (22), right-hand side of (26) equalsn
t(hjI). Combining this with (25) and (26)
yields
E[n8t(hjI)]"n
t(hjI). (27)
Inspection of (27) suggests that the limits (as I tends to in"nity) of +K(I)
The argument in the above paragraph is formally demonstrated in Appendix. Now use (22) to expand (28), obtaining
M"p
The numerator and denominator of (29) approach
E use (29) to obtain the large type limit equilibrium condition:
12Recall that without bias, each predictor has an equal weightn6jat the steady state.
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:i,t,c0i,t)"ln(c:i,t)#ln(c0i,t) (31) so that the savings function is
s(p
t/h(pLt~1))"(w:!w0h(pLt~1)/pt)/2.
Substituting into the equilibrium condition (30) yields
p
t"/(ptL`~12)"2M/w:#(w0/w:)h!''(pL`t~12) (32) whereh!''()) represents an aggregate predictor of the future price:
h!''(pL`2
t~1)"Eh[n(pL`2
t~1,h)h(pLt~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.12Rather 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 makeh!''()) much richer thanh3%1()). The latter depends only
on¸ lags of p
t; the former depends on ¸#2 lags. So while h3%1()) could be
interpreted as the predictor of an individual agent,h!''()) cannot:h!''()) depends
13A steady state may exist under weaker but less compact assumptions. The subsequent results would still hold.
depend on additional lags. In addition,h!''()) can be a much more complicated
function than any of its component predictors h(pLt
~1,h), so that in general
h!''()) will not be linear even if h(pLt
~1,h) is linear for all h. The aggregate predictor can thus induce much richer behavior than any of the component predictorsh(pLt
~1,h).
Assumption A4@@. The aggregate predictorh!''()) has no steady-state bias, so that
h!''(p6,2,p6)"p6.
Assumption A4Aguarantees the existence of a monetary steady state.13Note we require only that the aggregate predictorh!''()) 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! erenti-ating (32) with respect top
t~qand evaluating at the monetary steady state yields
/q"wz6Eh[hq(h)e~be62(h
)]!2bwz6E
h[hM(h)e6(h)eq(h)e~be62(h)]
#2bwz62Eh[hM(h)e~be62(h
)]Eh[e6(h)e
q(h)e~be62(h)] (33)
where w"w0/w: is the ratio of old age to youthful endowment,
hM(h)"h(p6,2,p6,h), and hq(h) denotes the partial derivative of h(pLt
~1,h) with respect top
t~q. The termse6(h) andeq(h) are de"ned similarly. By the de"nition of the forecast error, it follows that
e1(h)"1, (34)
e2(h)"0, (35)
eq(h)"!hq
~2(h) forq53. (36) So in order to evaluate (33), we need not specify the entire distributionf(h); we
need only specify the implied distributions for the functionse6(h) andh
q(h). We do so via the following assumption:
Assumption A5. (a) hq(h) and e6(h) each have a normal distribution:
hq(h)&N(kq,p2
h,q) ande6(h)&N(0,p2e) (b) hq(h) ande6(h) are independent of each other. (c)hq(h) ande6(h) are each independently distributed.
14The stability of the steady state depends on the values ofM/
1,/2,2N, which by (37)}(39) do not
depend onM.
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))(0Qh(p6)/p6'w:/w0Q(M#2e)/M'w:/w0.
By making the money supplyM large, we can make the mass of agents with negative savings arbitrarily small. Notice too thatMdoes not a!ect the stability properties of the steady statep6.14We 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:
/
1"w(k1#c(b,p2e)), (37)
/
2"wk2, (38)
/q"w(kq!k
q~2c(b,p2e)), (39)
c(b,p2e)" 2bp2e
1#2bp2e.
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 seth
q"0 for all q'1. By (38)}(39), it follows that /
2"0,/3"!uk1c(b,p2e), and /q"0 for q53. This leads to a three-dimensional system whose Jacobian at the steady statep6 has eigenvalues which are given by the roots to the following character-istic polynomial:
p(j)"j3!j2w(k1#c(b,p2e))#wk1c(b,p2e).
To economize on notation, setc"c(b,p2e). We letwbe"xed, and analyze the stability properties of the systempL`2
t "U(pL`t~12) as k1andcvary.
Theorem. (Jury, 1974, p. 29).A necessary and suzcient condition for the roots of
p(j)"0to lie inside the unit circle is that both of the following criteria be met:
15It is easy to verify thatp(!1)(0 andp(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.
16Application of this theorem to mappings of the formfk:RnPRpis achieved via center manifold reduction, as discussed in Guckenheimer and Holmes (1983), Section 3.2.
2. Determinant criterion: The following determinants are positive:
D
B"
KC
1 !w(k1#c)
0 1
D
$C
0 wk1c wk1c 0
DK
.It is clear that for k1'0, p(!1) is always negative, and D
` is positive
wheneverD
~is. We can thus analyze the stability of the system by plotting the two curves corresponding top(1)"0 andD~"0:
p(1)"0 Q c"(wk1!1)/(wk1!w), (40)
D~"0 Q (wk1c)2#w2k1c(k1#c)"1. (41) Values of c satisfying the root criterion (40) can be parameterized as an increasing concave functionc
rc(k1) withcrc(1/w)"0. Values of csatisfying the
determinant criterion can be parameterized as a decreasing convex function
c
dc(k1) withcdc(k-)"1. The two functionscrc(k1) andcdc(k1) intersect at a point
k6. The values fork-andk6are 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
k:R2PR2be a one-parameter family of mappings which has a smooth family of
xxed pointsx(k)at which the eigenvalues are complex conjugatesj(k),j(k6).Assume
Dj(k0)D"1butjj(k0)O1 for j"1, 2, 3, 4. (44) d
Fig. 1. Bifurcation analysis.
17This nondegeneracy assumption is generic in the sense of Kuznetsov (1995, p. 124).
Then there is a smooth change of coordinateshso that the expression ofhfkh~1in
polar coordinates has the form
hf
kh~1(r,h)"(r(1#d(k!k0)#ar2),h#c#br2)#higher-order terms.
If,in addition
aO0. (46)
Then there is a two-dimensional surfaceR inR2]Rhaving quadratic tangency
with the planeR2]Mk
0Nwhich is invariant for f.If RW(R2]MkN)is larger than
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 curveHOPF. In the Appendix, we verify the second part of condition (44), as
Proposition 3. If k-(k
1(k6, wherek- andk6are given in (42) and (43), then
a closed orbit exists near the curve (labeledHOPF) 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 top2e"0 and hence to c(b,p2e)"0. The system loses stability only when k1'1/w, which 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, p2e'0 so that c(b,p2e) ranges between 0 and 1 as b increases. This allows for a Hopf bifurcation ask1andccross theHOPFcurve, so that prices #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 enrich-ing the basic model is to determine whether doenrich-ing 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:
u(c:i,t,ci0,t)"(c:i,t)c`1/(c#1)#(ci0,t)c`1/(c#1). (47) The preferences (31) are a special case of (47) with the risk aversion parameter set toc"!1. Settingw"0.5 andM"100, we use numerical methods detailed in the Appendix to calculate the eigenvalues ofDU()) at a discrete grid of points in
Mk1,c(b,p2)Nspace for varying degrees of risk aversion. The results are displayed 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
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, forc"!0.1 a Hopf bifurcation occurs for values ofk1as low 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 timetutility associated with each predictor depended only upon the time t!1 performance of that pre-dictor. 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:
; jt"+=
q/0
aq;(p
t~2~q,pt~1~q,hjt~2~q) (48)
for somea3[0,1). Again settingw"0.5 andM"100, we use numerical calcu-lations detailed in the Appendix to explore the stability of the system generated by (48) at a discrete grid of points inMk1,c(b,p2e)Nspace. The results are displayed in Figs. 3 and 4:
Fig. 3. Persistence in predictor selection:a"0.3.
18Recall that in Section 6, it was shown for the case ofa"0 that the stability properties of the steady state depend only onc(b,p2e), so that varyingbwithc(b,p2e) held"xed did not alter these properties. For aO0, however, varying b with c(b,p2e) held "xed does alter the steady-state properties. Lettingbvary within the setM0.01,0.1,1,10Nis meant to represent a reasonable range of possible values of the intensity of choice.
increases from zero to one. This was true for a"M0.3, 0.6N and b ranging between 0.01 and 10.18
7.2. Complicated forecast errors
In this section, we explore the four dimensional system created by allowingk2
(see Assumption A5) to be nonzero. In particular, we investigate the predictabil-ity of the associated forecast errors. To do so, we must simulate the system's time series behavior rather than merely exploring its steady-state stability properties. We thus need to specify the model (in particular the predictor space) more completely than in Section 6. We continue to assume log linear utility:
u(c:
i,t,c0i,t)"ln(c:i,t)#ln(c0i,t). We also assume that each predictor is linear:
h(p2
t~1,h)"h1(pt~1!p6)#h2(pt~2!p6)#p6#h3,
Fig. 4. Persistence in predictor selection:a"0.6.
where as before the coe$cients are distributed:
h1&N(k1,p2
h,1),
h2&N(k2,p2h,2),
h3&N(0,p2e).
Although we cannot analytically evaluateh!''()) away from the steady state, we
can approximate it via numerical techniques as detailed in the Appendix. We then explore equilibrium sample paths generated by (32). We do this for the following parameter values: w"0.5, M"100, b"50, k1"0.5, k2"!1,
p21"p22"2,p2e"30.
The results are summarized in Figs. 5}8. Fig. 7 displays a time series of 200 simulated prices, and suggests that price moves in an unsystematic manner which may be di$cult for agents to predict. This is emphasized in Fig. 5, which plots 25,000 simulated prices inMp
t~1,ptNspace. To better evaluate how much
predictability there is in this price series, de"ne the mean forecast error (MFE
t)
to be the average forecast error over all predictors:
MFE
t"Eh[et(h)].
Fig. 6 plots 25,000 simulated MFE's inMMFE
t~1,MFEtNspace. Fig. 8 displays
Fig. 5. 25,000 points inMp
t~1,ptNspace.
Fig. 7. Time series forp
t.
19Although nonlinear methods may detect some predictability in the forecast errors, it is possible, as discussed previously, that alternate versions of the model may generate forecast errors that look random even to nonlinear tests.
20The unabridged version of this paper (Brock and de Fontnouvelle, 1996) contains some preliminary work towards this agenda.
21When the model is close to the REE, it is no longer worth paying the cost associated with perfect foresight. Brock and Hommes call this theexpectational free riderproblem since agents can free ride once someone else has invested in perfect foresight. The expectational free rider problem creates an economically based dynamical tension between those agents who invest the extra resources to acquire rational expectations and those who free ride on the expectational e!orts of others. This tension creates additional complicated dynamics if agents'choices are responsive enough to di!erences in"tness measures.
forecast errors display no apparent linear structure, so it seems unlikely that agents could easily improve their forecasts. We can follow Hommes (1998) in saying that on average, agents haveconsistentexpectations: ones for which the associated predictors have zero autocorrelations at all lags.19Hommes writes,
&Simple habitual rule of thumb expectational rules which are consistent would not be inconsistent with rational behavior'.
8. Conclusion
In conclusion, we provide a brief motivation for our long-term research agenda,20 which is to explore the issues discussed in Robert Lucas's Nobel lecture (Lucas, 1996). In particular, we wish to construct models where learning plays a role in generating long-run monetary neutrality while allowing money to have real e!ects in the short run.
Suppose that in addition to backward-looking rules, agents can also choose a perfect foresight predictor. This predictor will require extremely complex calculations because it must take into account the equilibrium e!ects of all the other predictors. It thus seems reasonable to introduce an explicit computation cost associated with perfect foresight, as in Evans and Ramey (1992). This is the scenario explored by Brock and Hommes (1997), who show explicitly how the perfect foresight predictor keeps the model centered around the rational expec-tations equilibrium. When dynamics stray too far from the REE, many agents will incur the cost of acquiring perfect foresight and bring the model closer to the REE.21 We can thus obtain analytically tractable models that allow us to uncover economic forces that push the system towards or away from the rational expectations equilibrium under scrutiny.
to observe high quality information concerning the merits of di!erent predictors in the predictor spaceH. Using the logic of the&random information'part of
Section 5, this high precision of information should translate into less random-ness in the choice of expectations, i.e. a higher intensity of choice. By making it easier to be certain about future policy, credibility should also make perfect foresight fairly inexpensive to acquire. Because of the low cost and high intensity of choice, we would expect most agents to use perfect foresight, and would thus predict a small real e!ect as in conventional rational expectations analysis.
Conversely, when there is no credibility, one might expect a lower intensity of choice and a higher cost to acquisition of sophisticated expectations relative to crude backward looking ones. One would thus expect monetary contractions to have large real e!ects. But it is not so simple as this. Brock and Hommes (1997) show that it may be more valuable to acquire perfect foresight in such a noisy setting; what matters is the net bene"t of each predictor. Rule based learning models may thus also play a role in a future explanation of the failure of measures of unexpected price movements to explain output variability (Lucas, 1996, p. 679). In any event, one would expect the dynamics of agent drift across a population of forecasting rules to be very di!erent in the case of credible and/or transparent policy than in the case of incredible and/or opaque policy, and it is di$cult to imagine modeling such e!ects without some framework resembling the rule based approach advocated in this paper.
Appendix
A.1. Proof of Proposition 1
The eigenvalues of J are the zeros of p(j)"DJ!jID, which can also be written as
p(j)"(!1)LMf(j)#g(j)N,
f(j)"lh
L#jlhL~1#2#jL~2lh2#jL~1lh1,
g(j)"!jL.
The triangle inequality implies that
Df(j)D4Dlh
LD#DjD DlhL~1D#2#DjL~2D Dlh2D#D jL~1DD lh1D. (50) On the boundary of the unit circle, the right-hand side of (50) is less than or equal tol+Lq/1Dh
A.2. Proof of(28)
t(HI)]. To evaluate this last quantity, begin with the conditional expectation
This tends to zero because the numerator is"nite and the denominator tends to in"nity. So the law of iterated expectations implies that E[A2
t(HI)]P0 and thus
<ar[A
t(HI)]P0. The desired result follows from Tchebychev's inequality, which
says thatProbMDA
t(HI)D'eN4<ar[At(HI)]/e2. h
A.3. Derivation of Eqs.(37)}(39)
Ifxis distributed N(0,p2), andcis any positive constant then:
E[e~cx2
]"(2p2c#1)~1@2, (51)
E[xe~cx2
]"0, (52)
E[x2e~cx2
]"p2(2p2c#1)~3@2. (53) To derive the above equations, recall that the normal density function is given by f(x)"(2pp2)~1@2expM!x2/(2p2)N, and that :f(x)dx"1, E[x]"
:xf(x) dx"0, Var[x]":x2f(x) dx"p2.
From (51), we can immediately calculatez6"(Eh[e~be62(h)])~1"J
2p2eb#1, and Assumption A5 implies E
h[hq(h)e~be62(h)]"kq/z6. By de"nition it follows that in equilibriumhM(h)"p6!e6(h), and so using (34)}(36) yields
!E
h[hM(h)e6(h)e1(h)e~be6 2(h)]
"E
h[e62(h)e~be62(h)]"p2ez6~3,
!E
h[hM(h)e6(h)e2(h)e~be6 2(h)]
"0,
!E
h[hM(h)e6(h)eq(h)e~be62(h)]"!kq~2p2ez6~3 forq53. From (52) it follows that Eh[e6(h)e
q(h)e~be62(h)]"0. Eqs. (37)}(38) follow immedi-ately. Note that (52) also implies thatz6Eh[hM(h)e~be62(h)]"p6, so that Assumption A4Ais satis"ed.
A.4. Technical conditions for the Hopf bifurcation theorem
Theorem. (Girko, 1990, p. 97). Let Cbe a measurable set of the complex plane
whose Lebesgue measure on this plane is zero. Suppose that the linear measure of
the intersection of Cwith the real direct is also zero. Then if the coezcients of
f(t)"tn`1#m
1tn#2#mnhave continuous joint density,the roots of the
22Recall also that we have sethq"0 for allq'1.
Girko's theorem says that complex numbers that are roots of one form a zero measure set on the complex plane. So if we view the parametersw, k1,p2e, and
b as being randomly selected, then there is zero probability of selecting para-meters which result in a violation of condition (44).
Now letj(c) denote the one real root of the characteristic polynomial, and di!erentiate with respect toc:
j@(c)" wk1!wj2(c)
2j(c)(wk1#wc)!3j2(c).
LetK(c) equal the modulus of the complex eigenvalues, which is also equal to their product. Because the determinant of the Jacobian equals the product of the eigenvalues,
K(c)"!wk1c/j(c),
K@(c)"!wk1/j(c)#wk1cj@(c)/j2(c)"wk1
j(c)
A
2j(c)!wk1!2wc
2wk1#2wc!3j(c)
B
. Becausep(!1)(0 andp(0)'0, it follows thatj(c)(0, and thusK@(c)'0 so that the eigenvalues cross the unit circle with positive speed, thus demonstrating condition (45).A.5. Numerical calculations forvarying risk aversion
The savings function obtained by substituting the utility function (47) into the maximization problem (1)}(3) is
s(p
t/h(pLt~1))"(w:!w0(pt/h(pLt~1))1@c)/(1#(pt/h(pLt~1))1`1@c). Substituting into (30) and di!erentiating with respect top
t~qyields
/q"M2bEh[s(r6(h))e6(h)e
q(h)e~be62(h)]#Eh[s@(r6(h))r62(h)hq(h)e~be62(h)]
!2bMEh[e6(h)e
q(h)e~be62(h)]N
]MEh[s(r6(h))e~be62(h)]#Eh[s@(r6(h))r6(h)e~be62(h)]N~1 (54)
wherer6(h)"p6/(p6!e6(h)). Although the expectations in (54) cannot be evaluated
analytically, one can obtain an approximation as follows.
Recalling Assumption A5,22 draw I (we used I"1000) values of h
1,i and e6
can then be approximated by
i). This approximation enables us to calculate the eigenvalues
of DU()) at a discrete grid of points in Mk1,c(b,p2e)N space. The values of
Mh
1,iN1000i/1 and Me6iN1000i/1 are drawn once at random, and then "xed during all simulations.
A.6. Numerical calculations for persistence in predictor selection
By a Taylor series argument similar to that given in Section 5.1, one can derive the approximation ;jt"+=
q/0aq(ejt~1~q)2#o(supqMaq(ejt~1~q)2N). Using
t depends on an in"nite number of lagged squared forecast
errors, the system (56) is in"nite dimensional. It is thus impossible to calculate the associated eigenvalues either analytically or numerically. As in Section 7.2, we assume that each predictor is linear:
h(p2
t~1,h)"h1(pt~1!p6)#p6#h2,
where as before the coe$cientsh1andh2are normally distributed. Although we cannot evaluateh!''()) analytically away from the steady state, we can
approx-imate it as follows.
We then use the computer to explore equilibrium sample paths generated by (57) at a grid of points inMk,c(b,p2e)Nspace. The values ofMh
A.7. Numerical approximation of the large type limit
The large type limit equilibrium condition given in (28) can be written as
M"p
Without loss of generality, assume thatK(I)"I. For a large but"nite value of
I(we usedI"1000), (58) can be well approximated by the following equilibrium condition:
where the parametershjIare drawn once at random, and then"xed during all
simulations. Eq. (59) serves as the basis for the evaluation of the system given in Section 7.2. The simulated mean forecast errors are given by
MFE
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