Coordination on Saddle-Path Solutions: the
Eductive Viewpoint - Linear Multivariate
Modelsasteriskmath
George W. Evans
Department of Economics, University of Oregon
Roger Guesnerie
DELTAandCollègedeFrance
October 10, 2003.
Abstract
W examine lo cal strong rationalit (LSR) in multiviate mo dels
with b oth forward-lo oking exp ectations and predetermined viables.
Givn h othetical common kno dge restrictions that the dynamics
will b e close to those of a sp ecified minimal state viable solution,
we obtain eductiv stabilit conditions for the solution to b e LSR.
In the saddlep oin stable case the saddle-path solution is LSR pro-
vided the mo del is structurally homogeneous across agens. Hoevr,
the eductiv stabilit conditions are strictly more demanding when
heterogeneit is presen, as can b e exp ected in mltisectoral mo dels.
Heterogeneiy is th s a p otenialy i p ortan source of i stabiiy evn
in the saddlep oin stable case.
Key wor : Co ordi ati n, structural heterogeneiy strong rati -
nalit eductiv stabilit m ltisectoral mo dels.
JEL classificion : C72, C62.
asteriskmath We are indebted to Gabriel Desgranges and Stephane Gauthier for valuable com-
mens. This pap er also benefited f m discussions during the Decem er 2002 Univ sit
of Heidelberg conf ence on “Belief Fmation and Fluctuations in Economic and Financial
Markts,” sp onsored b Deutsche Fscungsgemeinschaft. This material is based up on
wrk supp orted b the National Science F tion under Gran No. 0136848.
1
1Introduction
This paper comes as a continuation of a previous paper entitled “Coordi-
nation on saddle path solutions : the eductive viewpoint — linear univariate
models”, by G. Evans and R. Guesnerie (2003). The purpose of both papers
is to revisit the justifications of the saddle path stable solution by taking
the somewhat more basic perspective of “eductive learning,” which refers to
considerations that have a game-theoretical flavour and explicitly refer to
Common Knowledge considerations.1
Specifically, the viewpoint we take, the “Strong Rationality viewpoint”2 ,
proceeds as follows. We start from restrictions on the possible paths of the
system, which themselves reflect restrictions on individual strategies. These
restrictions, tentatively supposed to be Common Knowledge (from now on
CK) trigger a mental process, which, when rationality is itself “commonly
known,” mimics the process of determination of rationalizable strategies
(from the initial set of restricted strategies). When such a process converges
to the candidate equilibrium, the equilibrium is said to be Strongly Rational.
Actually, as in the following, the CK initial restrictions will always be taken
locally, so that we shall only be concerned with a weaker variant of the test
that selects Locally Strongly Rational Equilibria. The question treated in
this paper, as well as in the companion paper, can then be more compactly
reformulated: when is it the case that the saddle path stable solution of a
dynamical system is a good candidate for expectational coordination, in the
sense just introduced of being Locally Strongly Rational, for restrictions to
be made precise?
At this stage, two different points are in order.
First, it is useful to stress the relevance of the question: economic mod-
elling routinely assumes that saddle-path like stable solutions, or stable man-
ifold solutions provide the appropriate “rational expectations solutions” even
when they compete with many others. Convenience pleads in favour of such
a practice but it is not as such a fully convincing intellectual argument. De-
terminacy considerations, which point out that such solutions are “locally
isolated” rational expectations equilibria, even when the broader concept of
1 A related but distinct approach is “adapti ve learning, ” e.g. Evans and Honkap ohja
(2001). F a comparison of eductive and adaptive learning see Evns (2001).
2 This could b e called the “lo cal unique rationalizability” viewp oint, in the terminology
of Bernheim (1984) or P arce (1984), or the “lo cal dominance solvbil it”viewp oi n in the
terminology of Fqhason (1969) and Mouli n (1979).
2
sunspot equilibria is envisaged, provide better intellectual arguments for se-
rious foundations but do not exhaust the question. The present approach
proposes an alternative, and in our view more basic, view on the problem.
As the reader will easily guess, our methodology to approach the question,
as briefly sketched above, is almost meaningless if we refer to standard re-
duced forms of dynamical systems. In order to make sense of the question we
raise concerning Common Knowledge, we must, as we did in Evans-Guesnerie
(1993) in a different context, imbed the model in a framework where agents
and their strategies are well defined. This is indeed what we did in the
previous paper and we repeat this set-up here in next Section.
2 The framework.
2.1 Dynamic expectations models
We are interested in models of the following kind:
Q(yt - 1 ,yt ,yet +1 )=O,
where t is a time index, y is a finite dimensional vector, and Q is a temporary
equilibrium map that relates yt to its lagged values and to expectations. The
quantity yet +1 denotes the expectation of yt +1 formed by agents at time t. In
this formulation we assume that agents are able to observe yt when forming
their expectations or, if not, that they can condition their actions on the
values yt that are realized.
We need to be more precise on the strategic aspects of the coordination
problem. To do so we will adopt a very simple strategic interpretation of the
model which makes explicit the decision theoretic aspects of the model and
the aggregation of these decisions into a temporary equilibrium map.
2.2 Strategic expectations model.
2.2.1 The basic structure
We thus embed the dynamic model in a dynamic game, along lines that are
somewhat similar to those of Evans-Guesnerie (1993). We assume that, at
each period t, there exists a continuum of agents, a part of whose strategies
are not reactive to expectations (in an OLG context, these are the agents,
3
who are at the last period of their lives), and a part of which “react to expec-
tations”. The latter agents are denoted omegat and belong to a convex segment
of R, endowed with Lebesgue measure domegat . It is assumed that an agent of
period t is different from any other agent of period tprime ,tprime negationslash= t. 3 More precisely,
agent omegat has a (possibly indirect) utility function that depends upon
1) his own strategy s(omegat ),
2) sufficient statistics of the strategies played by others i.e. on yt =
F(Piomega t {s(omegat )} ,asteriskmath), where F in turn depends first, upon the strategies of all
agents who at time t react to expectations, and second, upon (asteriskmath),which
is here supposed to be sufficient statistics of the strategies played by those
who do not react to expectations, and that includes but is not necessarily
identified with — see below — yt - 1 ,
3) finally upon the sufficient statistics for time t+1, as perceived at time t:
i.e. on yt +1 (omegat ),whichmay be random and, now directly, upon the sufficient
statistics yt - 1 .
We assume that the strategies played at time t can be made conditional
on the equilibrium value of the of the t sufficient statistics yt . Now, let (•)
denotes both (the product of) yt - 1 and the probability distribution of the
random variable yt +1 (omegat ), (the random expectation held by omegat of yt +1 ). Let
then G(omegat ,yt ,•) be the best response function of agent omegat . Under these as-
sumptions, the sufficient statistics for the strategies of agents who do not
react to expectations is (asteriskmath)=(yt - 1 ,yt ).
The equilibrium equations at time t are written:
yt = F [Piomega t {G(omegat ,yt ,yt - 1 , ˜t +1 (omegat ))} ,yt - 1 ,yt ]. (1)
Note that when all agents have the same point expectations denoted yet +1 ,
the equilibrium equations determine what we called earlier the temporary
equilibrium mapping
Q(yt - 1 ,yt ,yet +1 )=yt -F bracketleftbigPiomega t braceleftbigG(omegat ,yt ,yt - 1 ,yet +1 )bracerightbig,yt - 1 ,yt bracketrightbig .
3 This means either that each agent is “physically” different or that the agents have
strategi es that are indep enden from p erio d to p erio d. In an OLG in rpretation of the
mo del, eac agen liv f r t perio ds but only reacts to expectations in the first perio d
of his lif
4
2.2.2 Linearization
The right hand side of (1) is a rather complex term, but under regularity
assumptions4 , it has, through two different channels, derivatives with respect
to yt , and with respect to yt - 1 . Also assuming that all ˜t +1 have a very small
common support “around” some given yet +1 , decision theory suggests that
G, to the first order, depends on the expectation5 of the random variable
˜t +1 (omegat ) thatisdenotedyet +1 (omegat ) (and is close to yet +1 )
Taking into account the previous remark, the heterogeneity of expecta-
tions across agents, and assuming again the existence of adequate derivatives,
it is reasonable to linearize (1), around any initially given situation, denoted
(0), as follows6 :
yt = U(0)yt + V (0)yt - 1 +
integraldisplay
W(0,omegat )yet +1 (omegat )domegat ,
where yt ,yt - 1 ,yet +1 (omegat ) now denote small deviations from the initial values of
yt ,yt - 1 ,yet +1 ,andU(0),V(0),W(0,omegat ) are n ×n square matrices.
Such a linearization is valid everywhere, but we will consider it only
around a steady state of the system. Hereafter, yt ,yt - 1 ,etc denote devia-
tions from the steady state and U(0),V(0),W(0,omegat ) are simply U, V, W(omegat ).
Supposing I -U is invertible, we have
yt =((I -U)- 1 V )yt - 1 +(I -U)- 1
integraldisplay
W(omegat )yet +1 (omegat )domegat .
When expectations are homogenous, yet +1 (omegat )=yet +1 , the system becomes
yt = Byet +1 + Dyt - 1 , with B =(I -U)- 1 W,whereW =
integraldisplay
W(omegat )domegat . (2)
With the new notation, assuming W invertible, the initial system can
also be written
yt = Dyt - 1 + BW- 1
integraldisplay
W(omegat )yet +1 (omegat )domegat .
4 For a less sketchy discussion, see Evans-Guesnerie (1993), p.637.
5 This could b e formalized along lines similar to those taken in Chiapp ori-Guesnerie
(1989), who also argue that the prop ert is general in economic mo dels that adopt the
Basian view of uncertain
6 As in Guesnerie (2002), this can b e viewed as an “axiom”, whose field of validity is
ve r y l a r g e .
5
Putting Z(omegat )=W- 1 W(omegat ), we rewrite this as
yt = Dyt - 1 + B
integraldisplay
Z(omegat )yet +1 (omegat )domegat , (3)
where integraltext Z(omegat )domegat = I. This will be the basic equation of our study. We
assume that (3) holds for t =1,2,3,..., and that initial conditions y0 are
given.
3 An Economic Example
To illustrate our results we develop a two-sector version of the overlapping
generations model with production that was introduced and analyzed by Re-
ichlin (1986). In Reichlin’s model there is a single perishable output (“corn”),
which can either be consumed or set aside as capital (“seed corn”) for use in
production the following period. Capital is combined with labor according
to a Leontief fixed coefficients technology to produce output and the capital
is fully used up in production. We develop a two-sector competitive version
of this model in which there is trade in goods but no labor or capital flow
permitted between sectors. That is, in the version we set forth here (other
formulations would of course be possible), labor is immobile and agents can
only invest in capital in their own sector. However, there is trade in goods,
and households in both sectors consume both goods.
For ease of presentation we begin with the perfect foresight version. Pop-
ulation, which is normalized to one in each sector, is stationary and composed
of identical consumers living for two periods. Households work when young
and consume when old. For convenience we assume that all agents in both
sectors have the same utility functions. In each sector i =1,2, the household
problem is thus
max u(c1t +1 ,c2t +1 ) -v(lit ),
subject to Nit = Wit lit and c1t +1 + pt +1 c2t +1 = Nit Rit +1 ,
where lit is labor supply, cit +1 is the consumption of sector i goods, Nit is
investment in sector i capital carried into the following period, Wit is the
wage rate in sector i units, Rit +1 is the real interest factor in sector i units,
and pt +1 is the relative price of sector 2 goods in terms of sector 1 goods in
period t +1. We make the standard assumptions that vprime (l) > 0,vprimeprime (l) > 0
6
with liml arrowrightinfinity vprime (l)=+infinity and liml arrowright 0 vprime (l)=0,andu(c1 ,c2 ) is assumed to
be concave with positive first partial derivatives. We additionally require
that u(c1 ,c2 ) is homogeneous of degree lambda <=< 1; thus its partial derivatives are
homogeneous of degree 1 -lambda and partialdiffpartialdiffc i u(c1 ,c2 )=(c2 )lambda - 1 partialdiffpartialdiffc i u(c1 /c2 ,1).
The first-order conditions for the household include the Euler equations
vprime (lit )=Wit Rit +1 ui parenleftbigc1t +1 ,c2t +1 parenrightbig or
lit vprime (lit )=Wit lit Rit +1 parenleftbigc2t +1 parenrightbiglambda - 1 ui parenleftbigc1t +1 /c2t +1 ,1parenrightbig
where ui (c1 ,c2 ) equivalence partialdiffpartialdiffc i u(c1 ,c2 ). In addition we have the static condition
pt +1 = u2 parenleftbigc1t +1 /c2t +1 parenrightbig /u1 parenleftbigc1t +1 /c2t +1 parenrightbig equivalence phiparenleftbigc1t +1 /c2t +1 parenrightbig .
Inserting into the budget constraints yields
parenleftbigc1
t +1 /c
2
t +1 + phi(c
1
t +1 /c
2
t +1 )
parenrightbig c2
t +1 = W
i
t l
i
t R
i
t +1 ,
which, when combined with the Euler equations, leads to
lit vprime (lit )=parenleftbigWit lit Rit +1 parenrightbiglambda zetai (c1t +1 /c2t +1 ), (4)
where
zetai (z)=(z + phi(z))1 - lambda ui (z,1).
Firms produce output under conditions of perfect competition. In each
sector output is given by xit =min(alphai Nit - 1 ,betai lit ), where alphai ,betai > 0.Profit
maximization gives
xit = alphai Nit - 1 = betai lit ,
and goods market implies that xit = cit + Nit . It follows that
lit =(alphai /betai )Nit - 1 and cit +1 = alphai Nit -Nit +1
Finally, from the zero profit condition
Wit lit + Rit Nit - 1 = xit ,
together with Nit = Wit lit and xit = alphai Nit - 1 ,weobtain
Rit +1 = alphai - N
it +1
Nit .
7
Substituting the preceding relationships into the Euler equation we arrive
at the equation that specifies the perfect foresight dynamics, namely
V
parenleftbiggalpha
i
betai N
i
t - 1
parenrightbigg
= parenleftbigalphai Nit -Nit +1 parenrightbiglambda zetai
parenleftbiggalpha
1 N1t -N1t +1
alpha2 N2t -N2t +1
parenrightbigg
(5)
for i =1,2,where
V (z) equivalence zvprime (z)
Note that V (z),Vprime (z) > 0 for all z>0. Existence of a steady state requires
alpha1 ,alpha2 > 1. When linearized this yields a perfect foresight dynamic equation
of the form yt = Byt +1 +Dyt - 1 where yt =(N1t ,N2t )prime with variables expressed
as deviations from steady state values.
The strategic form of the model is obtained by dropping perfect fore-
sight and allowing for heterogeneous expectations across individual agents.
Equation (4) becomes
(lit (omegait ))1 - lambda vprime (lit (omegait )) = parenleftbigWit Ri,et +1 (omegait )parenrightbiglambda zetai parenleftbigc1 ,et +1 (omegait )/c2 ,et +1 (omegait )parenrightbig ,
where omegait denotes an agent in sector i, lit (omegait ) is the agent’s labor supply
and a superscript e denotes the expectation of a variable. Because every
agent in every period will equate its marginal rate of substitution between
the two goods to the relative price, we have c1t +1 (omegait )/c2t +1 (omegait )=c1t +1 /c2t +1 ,
where now variables without omegait denote aggregate quantities. Furthermore
the aggregate relationships Nit = Wit lit , lit =(alphai /betai )Nit - 1 , cit +1 = alphai Nit -Nit +1
and Rit +1 = alphai - Nit +1 /Nit continue to hold. It follows that the individual’s
labor supply can be rewritten as
lit (omegait )=˜- 1
braceleftBiggbracketleftbig
Wit parenleftbigalphai -Ni,et +1 (omegait )/Nit parenrightbigbracketrightbiglambda zetai
parenleftBigg
alpha1 N1t -N1 ,et +1 (omegait )
alpha2 N2t -N2 ,et +1 (omegait )
parenrightBiggbracerightBigg
,
where ˜(z) equivalence z1 - lambda vprime (z).
In equilibrium we have lit =
integraldisplay
lit (omegait )domegait , so that using again the above
aggregate relationships we arrive at
alphai
betai N
i
t - 1 =
integraldisplay
˜V - 1
braceleftBiggparenleftbigg
betai
alphai Nit - 1
parenrightbigglambda parenleftbig
alphai Nit -Ni,et +1 parenrightbiglambda zetai
parenleftBigg
alpha1 N1t -N1 ,et +1 (omegait )
alpha2 N2t -N2 ,et +1 (omegait )
parenrightBiggbracerightBigg
domegait ,
8
for i =1,2, which fits the framework of Section 2 with
yt =(N1t ,N2t )prime and yet +1 (omegat )=(N1 ,et +1 (omegait ),N2 ,et +1 (omegait ))prime .
Linearization around a steady state then yields a reduced form that can be
written as (3).
The above formulation allows for heterogeneous expectations within each
sector, but it is revealing to consider the special case in which expectations
within each sector are homogeneous. In this case the linearization (3) reduces
to
yt = Dyt - 1 + B braceleftbigZ(1)yet +1 (1) + Z(2)yet +1 (2)bracerightbig (6)
where now yet +1 (i)=(N1 ,et +1 (i),N2 ,et +1 (i))prime .HereNj,et +1 (i) denotes the expec-
tations held at time t by agents in sector i concerning the future capital
expenditure in sector j. Despite the simplification of assuming homogeneous
within-sector expectations, this formulation of the model retains the key fea-
ture of intersectoral expectational heterogeneity combined with differential
impacts of these expectations. Furthermore, and this is a point we stress
below, it is apparent from the form of the nonlinear model that Z(1) and
Z(2) are not identical (or proportional), even in the symmetric case in which
alpha1 = alpha2 and beta1 = beta2 .
4 Perfect Foresight.
We begin our analysis of the linearized model (3) with a discussion of the
perfect foresight solutions.
4.1 Perfect Foresight Paths.
A Perfect Foresight path is a sequence of n-dimensional vectors, yt ,t=
1,..... + infinity, starting from y0 , and such that :
yt = Byt +1 + Dyt - 1 . (7)
We begin with a review of the standard methodology of the study of such
paths. We assume throughout that B is nonsingular.
Defining X(t)=
braceleftbigg y
t
yt - 1
bracerightbigg
, we write
X(t +1)=PhiX(t),
9
where Phi is the 2n ×2n matrix
Phi=
bracketleftbigg B- 1 -B- 1 D
IO
bracketrightbigg
.
Such a matrix has 2n eigenvalues, lambdai ,i=1,....2n, associated with eigenvec-
tors of the form
braceleftbigg lambda
i xi
xi
bracerightbigg
. We will now remind the reader of the solutions
to this dynamical system.
Associated with the equilibrium point 0 of the dynamical system is a
stable linear manifold, generated by all eigenvectors associated with eigen-
values of modulus strictly smaller than one and an unstable linear manifold
generated by all eigenvectors associated with eigenvalues of modulus strictly
greater than one. Throughout the paper we assume that Phi is diagonalizable
in the set of complex matrices and that it has no eigenvalue with modulus
equal to one.
We further assume that we are in the so-called “saddle-path” case in
which the stable manifold has dimension n andisin“generalposition.” It
follows that for any given initial y0 there is a unique y1 on the stable manifold
and a trajectory converging to the equilibrium. It also follows that any other
trajectory starting from y0 has at least one component going to infinity.
Thus we have exactly n eigenvalues of modulus strictly smaller than one.
We rank the eigenvalues in the order of increasing modulus, so that i <=< j,arrowdblboth
|lambdai |<=<|lambdaj |.ItiswellknownthatwhenPhi is “semisimple”, i.e. diagonalizable
in the set of complex matrices, it has a real factorization of the form
Phi=P?P- 1 ,
where ? is a 2n ×2n block diagonal matrix, in which each block is either a
single element lambdaj ,whenlambdaj isreal,orisa2×2 block
parenleftbigg a
j -bj
bj aj
parenrightbigg
correspond-
ing to non-real eigenvalues lambdaj = aj ±ibj . We order the elements and blocks
in terms of increasing eigenvalue modulus. The nonsingular 2n ×2n matrix
P has columns given by the coordinates of the eigenvectors in the canonical
basis of R2 n if the corresponding eigenvector is real. In the case of nonreal
eigenvalues the corresponding pair of columns of P are given by the coor-
dinates of the imaginary and real parts, respectively, of the corresponding
eigenvectors.
With this factorization we can obtain
X(t +1)=P?t P- 1 X(1).
10
Partitioning P, and calling P- 1 X(1) =
braceleftbigg I
1
I2
bracerightbigg
, the dynamics of the system
can be written, with straightforward notation,
braceleftbigg y
t +1
yt
bracerightbigg
=
parenleftbigg P
11 P12
P21 P22
parenrightbiggparenleftbigg ?t
1 0
0?t2
parenrightbiggbraceleftbigg I
1
I2
bracerightbigg
.
Here the submatrices Pij and ?i are n ×n and note that P11 = P21 ?1 , and
P12 = P22 ?2 . It follows, in particular, that
yt = P21 ?t1 I1 + P22 ?t2 I2 . (8)
We assume that P21 is nonsingular.
Let XS denote the (n dimensional) subspace of R2 n generated by the
eigenvectors associated with the n eigenvalues of smallest modulus, lambda1 ,....lambdan .
XS is a solution subspace, i.e. X(t-1) element XS and X(t)=PhiX(t-1) implies
X(t) element XS . AvectorX(t)=(yprimet ,yprimet - 1 )prime belongs to XS if and only if, in the
basis of eigenvectors, it can be written as
braceleftbigg eta
0
bracerightbigg
, i.e. in the canonical basis it
is of the form
braceleftbigg P
11 eta
P21 eta
bracerightbigg
. Hence X(t) element XS if and only if yt = P11 (P21 )- 1 yt - 1 ,
i.e.
yt = Syt - 1 , where S = P21 ?1 (P21 )- 1 . (9)
This solution corresponds to (8) with I2 =0, i.e. yt = P21 ?t1 I1 .
We have shown the following:
Proposition 1 A saddle-path solution (y0 ,y1 ,...,yt ,...) satisfies yt = Sasteriskmath yt - 1 ,
where Sasteriskmath = P21 ?1 (P21 )- 1 and where P21 and ?1 are the matrices just defined.
The Proposition describes the unique nonexplosive solution in the “saddle-
point stable case,” for given initial y0 . Any other perfect foresight solution
satisfies (8) with P22 ?t2 I2 negationslash=0, which implies that at least one component of
yt tends to ±infinity as t arrowrightinfinity.
We now make a parenthetical digression. Economists have long been in-
terested in solutions of the dynamical system that are of the form yt = Syt - 1 .
These are called by McCallum (1983) “minimal state variable” solutions and
more recently by Gauthier “equilibrium extended growth rate” solutions. If
such a solution exists then from (7) we have
yt = Byt - 1 + DS- 1 yt ,
11
provided S is nonsingular. Thus B- 1 (I -DS- 1 )yt = yt +1 and
S = B- 1 (I -DS- 1 ), (10)
which can be rewritten as the matrix quadratic equation
S2 -B- 1 S + B- 1 D = O,
so that minimal state variable solutions are solutions of this equation. We
note that obviously, in our case, Sasteriskmath is a solution of this equation, but there
are others and we outline how these can be constructed.
Consider a set of n vectors of Rn , parenleftbig .zi ,.zj, .parenrightbig where the n vectors
zi ,zj ,... are associated with a subset K of n of the 2n eigenvalues of Phi,pro-
vided that if a nonreal eigenvalue is included in K then so is its complex
conjugate. If lambdaj element K is real then zj is taken to be the n-dimensional restric-
tion of the corresponding eigenvector
braceleftbigg lambda
j xj
xj
bracerightbigg
, i.e. zj = xj .Iflambdaj ,lambdaj +1 element K
are not real and equal aj ±ibj the eigenvectors take the same form but with
xj ,xj +1 = uj ±ivj . In this case the corresponding vectors zj ,zj +1 are taken
to be vj and uj . Consider the case where the n vectors under consideration
form a basis of Rn . The matrix, denoted SK , which transforms zj to lambdaj zj for
real lambdaj element K and analogously for nonreal conjugate members of K,isafixed
point of (10). Indeed, in the basis consisting of the zj , for lambdaj element K,avectorbraceleftbigg
yt
yt - 1
bracerightbigg
=
braceleftbigg ?alpha
alpha
bracerightbigg
is transformed into yt +1 =?2 alpha ,sothatyt +1 =?yt .
In the canonical basis of Rn ,
SK = PK ?K P- 1K ,
wherewehavenowfactoredPhi=P?P- 1 as
P =
parenleftbigg P
K ?K PL ?L
PK PL
parenrightbigg
and ?=
parenleftbigg ?
K 0
0?L
parenrightbigg
The solution of the previous section, Sasteriskmath = P21 ?1 P- 121 , corresponds to the
choice ?K =?1 . We remark that there can be up to C2 nn distinct solutions
SK , with exactly C2 nn such solutions when all roots are real and all subsets
of n vectors parenleftbig .zi ,.zj, .parenrightbig yield linearly independent sets.
It is straightforward to verify algebraically that, in the canonical basis
of Rn , SK = PK ?K P- 1K satisfies the fixed point equation (10). This follows
12
immediately from partitioned matrix multiplication of the equation PhiP =
P?, using the above partition. Therefore yt = SK yt - 1 is a solution for any
initial condition y0 . The converse can also be shown, i.e. every S that
provides a solution of the form yt = Syt - 1 for every initial condition y0 can
be expressed as PK ?K P- 1K .
5 Eductive Learning.
5.1 Iterative Expectational Stability.
In developing the Strong Rationality conditions we will examine their connec-
tion to the conditions for Iterative Expectational Stability (or IE-Stability).
The analysis focuses on minimal state variables solutions, i.e. perfect fore-
sight solutions of the form yt = Syt - 1 for all t. IE-Stability can be viewed
as a process, taking place in virtual or notional time tau, that works as follows
(see, for example, Evans (1985)). Economic agents posit a conjectured or
“perceived” law of motion, consistent with a minimal state variable solution,
in which yt evolves in accordance with some arbitrary fixed coefficient matrix
Stau . That is, all agents believe that yt = Stau yt - 1 for all t,whereStau is some
fixed matrix. From this PLM (perceived law of motion) Stau , one can obtain
the actual law of motion and show that the actual dynamics take the same
form, but with a fixed coefficient matrix T(Stau ), which is in general different
from Stau . IE-stability then considers the iterative revision, in notional time tau,
given by Stau +1 = T(Stau ). If this sequence converges to a fixed point ¯ = T(¯),
from all initial S in a neighborhood of ¯, then we say that ¯ is locally IE-
stable (or LIE-stable). The sequence Stau +1 = T(Stau ) can be thought of as a
stylized notional time learning rule in which the PLM coefficient matrices
used to make forecasts are updated to the actual coefficient matrices implied
by those forecasts.
More specifically, for the case at hand, consider forecasts
yet +1 = Stau yt ,universalt.
Inserting these (homogeneous) expectations into the model (2) the actual
dynamics between t -1 and t will be yt = Dyt - 1 + BStau yt ,sothat
yt =(I -BStau )- 1 Dyt - 1 ,universalt,
13
provided I - BStau is invertible. Thus constant PLM coefficients Stau would
lead to constant actual coefficients T(Stau )=(I -BStau )- 1 D. The IE-Stability
(or “IE-learning”) dynamics are therefore given by
Stau +1 =(I -BStau )- 1 D. (11)
Fixed points of the IE learning process S =(I -BS)- 1 D satisfy (10).
We are now going to prove the following:
Proposition 2 Assume that we are in the saddle-point case |lambdan | < 1 <
|lambdan +1 | with solution yt = Sasteriskmath yt - 1 ,whereSasteriskmath = P21 ?1 P- 121 . Then this solution
is LIE-Stable.
One possible proof of this is would be to show that all eigenvalues of the
linear mapping tangent ( at Sasteriskmath ) to the mapping S arrowright(I-BS)- 1 D are smaller
that one in modulus. However, a more interesting proof obtains by making
use of the concept of perfect foresight extended growth rates proposed by
Gauthier (2002, 2003), which is defined as follows.
Suppose that
yt = St yt - 1
for some given n × n invertible matrix St . Then the perfect foresight “fol-
lower” of yt is the yt +1 that satisfies, assuming B invertible,
yt +1 = B- 1 (I -DS- 1t )yt .
Writing yt +1 = St +1 yt it follows that
St +1 = B- 1 (I -DS- 1t ). (12)
Choosing S1 arbitrarily, (12) generates an infinite sequence of matrices, St ,
t =1,2,3,... with the property that for arbitrary y0 the sequence y1 =
S1 y0 ,...,yt = St yt - 1 ,...is a perfect foresight path. Following Gauthier, one
may call the sequence St a sequence of “extended growth rates,”7 or an
“EGR sequence.” A limit point S of a sequence of extended growth rates
must satisfy 10).
Comparing the dynamics (11) of IE-stability with the EGR dynamics
(12) we immediately see that they are governed by inverse mappings. A
fixed point is therefore locally a sink under IE dynamics if and only if it is a
source under EGR dynamics. This observation immediately yields:
7 cf the one dimensional case for the terminology
14
Lemma 3 Sasteriskmath is LIE stable if and only if it is locally determinate under EGR
dynamics.
Note that this fact, here obvious, obtains under more general models and is
stressed by Gauthier as a general “equivalence principle.” We are now in a
position to complete the proof of Proposition 2.
Proof. From the Lemma it is enough to show that the equilibrium Sasteriskmath is
locally determinate, i.e. locally divergent, under the EGR dynamics. Assume
the contrary. Then in every neighborhood of Sasteriskmath there exists initial S2 negationslash= Sasteriskmath
such that St arrowrightSasteriskmath under the perfect foresight dynamics (12). Then take y0 negationslash=
0 and y1 negationslash= Sasteriskmath y0 ,sothat(yprime1 ,yprime0 )prime does not belong to the stable subspace. The
sequence y2 = S2 y1 ,...,yt = parenleftbigproducttextti =2 Si parenrightbig y1 ,... is a perfect foresight sequence
in states yt . But producttextti =2 Si arrowright0 as t arrowrightinfinity,sinceSt arrowrightSasteriskmath and all eigenvalues of
Sasteriskmath are smaller than one in modulus. Hence yt arrowright 0. This is a contradiction
since we know that there does not exist a perfect foresight sequence in states
that starts outside the stable subspace and converges to zero. Q.E.D.
LIE-stability plays a role in the theory of strong rationality, to which we
now turn.
5.2 Strong Rationality.
5.2.1 Preliminaries on “eductive learning”
We now develop the eductive learning argument and the criterion, called in
Guesnerie (1992) Strong Rationality, under which eductive learning will lead
to coordination on an equilibrium path. We therefore return to the “strategic
reduced form” of the model (3) developed in Section 2.2.2 and reproduced
here for convenience:
yt = Dyt - 1 + B
integraldisplay
Z(omegat )yet +1 (omegat )domegat .
We apply the strong rationality test to the saddle-point solution Sasteriskmath ,butit
could also be applied to any ¯ that satisfies ¯ =(I -B ¯)- 1 D.
We develop the argument as follows.
We first note that the subjective expected value yet +1 (omegat ) of agent omegat can
be viewed as S(omegat )yt where S(omegat ) is the “expected” matrix S that agent omegat ’s
subjective probability distribution on V ( ¯) generates. Then, for this state
15
of beliefs the value of yt is given by
yt = Dyt - 1 +
parenleftbigg
B
integraldisplay
Z(omegat )S(omegat )domegat
parenrightbigg
yt .
This is our basic relationship, which we rewrite :
yt =
parenleftbigg
I -B
integraldisplay
Z(omegat )S(omegat )domegat
parenrightbigg- 1
Dyt - 1
We will consider a neighborhood V ( ¯) of the form:
vextenddoublevextenddoubleS - ¯Svextenddoublevextenddouble <=< epsilon1,
where bardbl.bardbl is a Euclidean vector norm (discussed further below) on Rn 2 ,which
is identified with the space of n ×n matrices.
Next consider the map
G :
productdisplay
omega t (increment(S(omegat ))) arrowrightincrement(
bracketleftbigg
I -B
integraldisplay
Z(omegat )S(omegat )domegat
bracketrightbigg- 1
D),
where increment denotes deviations from ¯,e.g.increment(S(omegat )) = S(omegat ) - ¯.
We can now introduce our definition of local strong rationality.
Definition 4 ¯ is said to be LSR (Locally Strongly Rational) if there exists
eta>0,0 <µ<1 and a Euclidean vector norm bardbl.bardbl on Rn 2 such that for all
0 <epsilon1<eta, bardblincrement(S(omegat ))bardbl<=<epsilon1 for all omegat implies
vextenddoublevextenddouble
vextenddoubleG
parenleftBigproductdisplay
omega t increment(S(omegat ))
parenrightBigvextenddoublevextenddouble
vextenddouble <µepsilon1.
We remark that each choice of basis for Rn 2 yields a different vector norm,
namely the Euclidean norm computed using the coordinates of the vector in
that basis.
Applied to Sasteriskmath , the definition of LSR states that if beliefs of the agents,
concerning the law of motion of the system, are close enough to the saddle
path beliefs, then the actual law of motion is even closer. We also say then
that the equilibrium is “eductively stable.”
Alternatively, we can refer to the standard procedure of starting from a
Common Knowledge restriction, i.e. to assume first:
CK Assumption: It is CK that, for all t, yt = St yt - 1 with St element V ( ¯),
where V is a small (enough) neighborhood of ¯.
Then:
16
Lemma 5 If Sasteriskmath is LSR in the sense of the previous definition, and if the
above CK assumption holds for V (Sasteriskmath )=[S/bardblS -Sasteriskmath bardbl<=<epsilon1], for some epsilon1<eta,
then it is CK that S = Sasteriskmath
Proof. Given the CK assumption bardblincrementSt bardbl = bardblSt -Sasteriskmath bardbl<=<epsilon1 the expectations
of all agents omegat must satisfy bardblincrement(S(omegat )bardbl<=<epsilon1 for all t. LSR according to our
definition implies that it is CK, after a first step of ”mental process” that
bardblincrementSt bardbl<=<µepsilon1 for all t, which tightens the CK assumption The above process
can then be iterated and, after n stages it is CK that bardblincrementSt bardbl<=<µn epsilon1 for all
t. Since this holds for all positive integers n it follows that it is CK that
bardblincrementSt bardbl =0for all t and hence that yt = Sasteriskmath yt - 1
Hence the definition of LSR given here is closely connected to previous
definitions that directly refer to a CK restriction. It is indeed quite similar,
although not exactly identical: for further discussion, the reader will refer to
Guesnerie (2002)
5.2.2 Characterizing Local Strong Rationality
The next step of our investigation relies on the study of the linear approxi-
mation to the above map G. We call this linear map G,andwriteit:
G:
productdisplay
omega t (increment(S(omegat ))) arrowright
integraldisplay
L(omegat )increment(S(omegat ))domegat
where L(omegat ) is a linear mapping from Rn 2 into Rn 2 , given explicitly later
below.
Hence
vextenddoublevextenddouble
vextenddoubleincrement(bracketleftbigI -B integraltext Z(omegat )S(omegat )domegat bracketrightbig- 1 D)
vextenddoublevextenddouble
vextenddouble is approximately equal to
vextenddoublevextenddouble
vextenddoublevextenddouble
integraldisplay
L(omegat )increment(S(omegat ))domegat
vextenddoublevextenddouble
vextenddoublevextenddouble <=<
integraldisplay
bardblL(omegat )increment(S(omegat )bardbl domegat .
Let rho(omegat )=bardblL(omegat )bardbl be the norm induced8 (on linear mappings from Rn 2 to
Rn 2 ) by the initial vector norm (on Rn 2 ). Then
bardblincrement(S(omegat ))bardbl = vextenddoublevextenddouble(S(omegat )) - ¯vextenddoublevextenddouble <=< epsilon1 universalomegat
8 For the theory of matrix norms, see Horn and Johnson (1985). We shall refer here
only to induced matrix norms.
G i ve n a ve c t o r n o r m | . | , the matrix norm bardbl . bardbl induced b the vctor norm is defined as
f llo s: bardbl A bardbl =max | x | =1 | Ax |
17
implies that to a first approximation
vextenddoublevextenddouble
vextenddoublevextenddouble
vextenddoubleincrement(
bracketleftbigg
I -B
integraldisplay
Z(omegat )S(omegat )domegat
bracketrightbigg- 1
D)
vextenddoublevextenddouble
vextenddoublevextenddouble
vextenddouble <=< (
integraldisplay
rho(omegat )domegat )epsilon1.
We can also define bardblGbardbl asthenormofthelinearmappingG(Piomega t (increment(S(omegat ))))
induced by the norm on Piomega t (increment(S(omegat ))), whichwetaketobesupomega t bardblincrement(S(omegat ))bardbl.
Then
bardblGbardbl<=<
integraldisplay
rho(omegat )domegat .
Consider now the homogenous expectations case and introduce the map
g (from Rn 2 to Rn 2 )
g : S arrowright(I -BS)- 1 D.
Call gamma the linear map tangent, at ¯ to the map g.
Note, that, since integraltext Z(omegat )domegat = I whenever (S(omegat ) - ¯)=incrementS indepen-
dently of omegat , the map G,actingonPiomega t (incrementS ), takes the same value as the
map gamma acting on incrementS.
In other words,
gamma =
integraldisplay
L(omegat )domegat
Hence, it must be the case that
bardblgammabardbl<=<bardblGbardbl ,
where bardblgammabardbl is the norm induced by the vector norm in Rn 2 previously intro-
duced.
Finally, this implies, to a first order approximation, that
bardblgammabardbl epsilon1 <=< sup
bardbl increment( S ( omega t ) bardbl<=< epsilon1
vextenddoublevextenddouble
vextenddoublevextenddouble
vextenddoubleincrement(
bracketleftbigg
I -B
integraldisplay
Z(omegat )S(omegat )domegat
bracketrightbigg- 1
D)
vextenddoublevextenddouble
vextenddoublevextenddouble
vextenddouble <=<
parenleftbiggintegraldisplay
rho(omegat )domegat
parenrightbigg
epsilon1
It follows that the condition integraltext rho(omegat )domegat < 1 is sufficient for LSR of ¯.
We can translate this analysis into a formal proposition that will provide
a basic reference for further reflection.
18
Theorem 6 i) LSR =arrowdblright LIE-Stability
ii) If agents are homogeneous, LSR is identical to LIE-Stability
iii) A sufficient condition for LSR is
integraldisplay
rho(omegat )domegat < 1,
where rho(omegat ) is the norm of L(omegat ),induced by a vector norm on Rn 2 , and where
L(omegat ) describes the approximate change on the aggregate state variable, trig-
gered by a change in expected EGR of agent omegat .
Proof. i) bardblgammabardbl is greater than the modulus of the eigenvalue of maximal
modulus of gamma,andbardblgammabardbl<=<bardblGbardbl, which is smaller than one because of LSR.
That is enough for LIE-Stability, which requires that the maximal modulus
of gamma be less than one.9
ii) Take Z(omegat )=sigma(omegat )I,sigma(omegat ) greaterequal 0, and integraltext sigma(omegat )domegat =1. Then G “coin-
cides” with gamma, and bardblgammabardbl can be made arbitrarily close to the maximal modulus
of gamma,for an appropriate choice of norms, (See Horn and Johnson (1991)).
iii) already shown. Q.E.D.
We remark that this proposition implies the corollary that, in the saddle
point case, if agents are homogeneous then the equilibrium Sasteriskmath is LSR since it
isLIE-StablebyProposition2.Infact,ascanbeseenfromtheproofofii),
some (mild) deviations from complete homogeneity leave this result intact.
5.2.3 Further characterization
In the Proposition, iii) captures the idea of heterogeneity: it is powerful
but abstract. In order to make it more intuitive, we have to specialize the
statement by choosing some special vector norms in Rn 2 .
First Specialization:
Consider the mapping gamma, the derivative of (I -BS)- 1 D, at Sasteriskmath .Thiscan
be written, in matrix form10 :
gamma :incrementS arrowright(I -BSasteriskmath )- 1 B(incrementS)(I -BSasteriskmath )- 1 D = Sasteriskmath D- 1 B(incrementS)Sasteriskmath .
9 The result is a priori obvious, but it is worth deriving from our inequalities.
10 Taking the differential of ( I - BS ) - 1 ( I - BS )= I it follows that the differential of
( I - BS ) - 1 is givn b d ( I - BS ) - 1 =( I - BS ) - 1 B ( dS )( I - BS ) - 1 . See Magn and
Neudeckr (1988) for these and related matrix results.
19
or, after “vectorization,”
gamma :vecincrementS arrowright(Sprimeasteriskmath circlemultiplySasteriskmath D- 1 B)(vecincrementS)
where circlemultiply designates the Kronecker product11 , Sprime is the transpose of S and
vec(incrementS) denotes the vector obtained by stacking in order the columns of incrementS.
We have here used the relationship vec(ABC)=(Cprime circlemultiplyA)vec(B).
Similarly the mappings L(omegat ) can be written in matrix form as12
L(omegat ):incrementS arrowrightSasteriskmath D- 1 BZ(omegat )(incrementS)Sasteriskmath ,or
L(omegat ):vecincrementS arrowrightparenleftbigSprimeasteriskmath circlemultiplyparenleftbigSasteriskmath D- 1 BZ(omegat )parenrightbigparenrightbig (vecincrementS).
Again, one can check that :
gamma =
integraldisplay
L(omegat )domegat
Now, take as vector norm for Rn 2 ,theEuclideannormintheeigenvector
basis of gamma.13 For this vector norm, the induced matrix norm for Sprimeasteriskmath circlemultiplySasteriskmath D- 1 B
is the modulus of its eigenvalue of highest modulus.
Now rho(omegat ) is the norm of the matrix Sprimeasteriskmath circlemultiply(Sasteriskmath D- 1 BZ(omegat )) induced by the
just defined Euclidean norm.(It must be at least as large as the modulus of
the eigenvalue of highest modulus of the considered matrix).
Hence, the next Theorem specializes conditions i) and iii) of Theorem 6 :
Theorem 7 A sufficient condition for the saddle path solution, Sasteriskmath ,tobe
LSR is that: integraldisplay
rho(omegat )domegat < 1
where rho(omegat ) is the norm of the matrix Sprimeasteriskmath circlemultiply(Sasteriskmath D- 1 B)(Z(omegat )) induced by the
Euclidean norm of the eigenvector basis of the matrix (Sprimeasteriskmath circlemultiplySasteriskmath D- 1 B).
11 If A is an m × n matrix with elements a ij and B is a p × q matrix, then the Kronecker
pro duct A circlemultiply B is the mp × nq matrix obtained b replacing eac elemen a ij of A with
the p × q blo c a ij B .
12 The differential of the map parenleftbig I - B integraltext Z ( omega t ) S ( omega t ) domega t parenrightbig - 1 D at S ( omega t )= ¯S universal omega t is
d parenleftbig I - B integraltext Z ( omega t ) S ( omega t ) domega t parenrightbig - 1 D =( I - B ¯ ) - 1 B parenleftbig d parenleftbigintegraltext Z ( omega t ) S ( omega t ) domega t parenrightbigparenrightbig ( I - B ¯ ) - 1 D
= ¯ - 1 B parenleftbig d parenleftbigintegraltext Z ( omega t ) S ( omega t ) domega t parenrightbigparenrightbig ¯ = integraltext ¯ - 1 BZ ( omega t )( dS ( omega t )) ¯ t
= integraltext L ( omega t )( dS ( omega t )) domega t ,with L ( omega t ) givnasstatedand ¯ = S asteriskmath .
13 which exists if gamma is semi simple, as we are assuming.
20
A necessary condition is that
alpha <=< 1,
where alpha is the modulus of the eigenvalue of highest modulus of the matrix
(Sprimeasteriskmath circlemultiplySasteriskmath D- 1 B)
Note that what has been done applies to saddle path solution, but may
also apply to other “minimal state variables,” (or to equilibrium “extended
growth rates”) solutions.
The next Theorem gives alternative sufficient conditions (the proof is
similar):
Theorem 8 A sufficient condition for the saddle path stable solution Sasteriskmath , to
be LSR is that: integraldisplay
rho(omegat )domegat < 1
where rho(omegat ) is the norm of the matrix Sprimeasteriskmath circlemultiply(Sasteriskmath D- 1 B)(Z(omegat )) induced by any
vector norm, and in particular the standard Euclidean norm, on Rn 2 .
In other words, alternative sufficient conditions may be obtained by al-
ternative choices of norms. This may prove useful in applications.
We show, in particular that the just stated theorem, applied to the one
dimensional case that we have studied earlier (Evans-Guesnerie (2003)), gives
the sharpest14 statement that can be then obtained :
Corollary 9 Let us consider the one dimensional version of our problem
(n=1). Then a sufficient condition for the saddle path solution to be LSR is
-1/[2(Ohm -1)] <BD<1/[2(Ohm + 1)]
where Ohm=Z+ -Z- , with Z+ = integraltextomega t /Z ( omega t ) > 0 Z(omegat )domegat ,Z- = integraltextomega t /Z ( omega t ) < 0 Z(omegat )domegat
Proof. Thenormofthe(nowonedimensional)matrixofTheorem4isintegraltext
(Sasteriskmath )2 |D|- 1 |B||Z(omegat )| domegat . But, here Sasteriskmath satisfies B(Sasteriskmath )2 -Sasteriskmath + D =0, so
that the condition becomes |D||B| (1-BSasteriskmath )- 2 Ohm < 1. This is the intermediate
inequality from which we obtain the above condition (see Evans-Guesnerie
(2003), footnote 14).
14 It is sharp est in the sense that it is then a necessary and sufficient condition.
21
5.3 Discussion
The results of the previous section provide a defense, from basic principles,
of the saddle path solution in the “saddle-point stable” case. When there are
homogeneous agents, or if the degree of structural heterogeneity is not too
large, we have shown that the saddle-path solution is always LSR, so that
a process of eductive reasoning, beginning with a local CK restriction leads
ineluctably to coordination on this solution. Theorems 6 and 7 show that
when there is sufficient heterogeneity the saddle-point solution will no longer
invariably be LSR, and we provide convenient sufficient conditions for LSR
of the saddle-point solution. Thus these results highlight the importance
of heterogeneity, which might be overlooked in “representative agent” or
reduced form models.
In fact, coming back on the differences between LIE stability and LSR,
which coincide in the “representative agent” case, we see now that in the
case of heterogeneity they differ strictly. More precisely, considering the
linear map G:producttextomega t (increment(S(omegat ))) arrowright integraltext L(omegat )increment(S(omegat ))domega, whichallowsusto
check LSR, and the map gamma that is obtained by replacing increment(S(omegat ) with increment(S)
and which determines LIE, and assuming that both gamma = integraltext L(omegat )domegat and
(almost) all L(omegat ) have full rank, we can state:
Proposition 10 LSR is always strictly more demanding than LIE, unless
(except for a subset of measure zero of omegat ) all the linear maps L(omegat ) are
proportional. In the latter case agents can be said to be “essentially identical”.
Proof. We will only give here an incomplete proof with only “two” agents omegat
(as in the case of the specific illustrative model introduced above). Extending
the argument to a finite number of agents is straightforward; going to the
continuum requires more formal care.
In the case being considered, the assertion will hold if one can prove that
if A and B are two linear maps, here from Rn 2 into Rn 2 ,andifS is a closed
convex set with smooth boundary, here containing zero, then:
AS+BS =[y/y = Ax1 +Bx2 ,x1 element S,x2 element S] strictly contains (A+B)S =
[y/y = Ax + Bx,x element S].
The fact that AS + BS reflexsuperset (A + B)S is obvious. Assume now that the
inclusion is not strict so that AS + BS =(A + B)S .Considerx element FrS ,
where FrS denotes the frontier of S, and consider the tangent hyperplane
to S in x,denotedTgx.
22
Ax is on the (smooth) frontier of the (convex) set AS,withatangent
hyperplane ATgx. Bx is on the (smooth) frontier of the (convex) set BS,
with a tangent hyperplane BTgx. (A + B)x is on the (smooth) frontier of
the (convex) set (A + B)S.
Our assumption implies that Fr(AS + BS)=Fr((A + B)S),sothat
(A + B)x is on the frontier of (AS + BS). However, from the standard
theory of addition of convex sets, we know that latter fact is possible only if
AS and BS are associated with the same tangent hyperplane in Ax and Bx,
i.e. if ATgx = BTgx.
The argument just made holds for all x on FrS. In order to conclude
it remains only to note that as x varies, Tgx can be any hyperplane of the
initial space, so that ATgx = BTgx is possible only if A = µB,µ negationslash=0. It
is finally easy to show, using the intuition of the one dimensional case, that
µ>0. Q.E.D.
The above statement, together with the one dimensional result of the
previous Corollary, provides our sharpest abstract illustrations of the desta-
bilizing effect of heterogeneity.
The class of models we have considered, multivariate one-step ahead one-
step memory multivariate models, is quite general, but it is of course not fully
general.15 In particular, altering the information assumptions so that not all
time t information is available, when time t decisions are taken, can lead to
more restrictive LSR conditions that may not be met in the saddlepoint case,
even with homogeneous agents, as our earlier work has shown (Guesnerie
(1992), Evans and Guesnerie (1993, 2003)).
The multivariate framework of the current paper serves, however, to em-
phasize a crucial aspect of heterogeneity that can impede coordination of ex-
pectations. As our simple economic example illustrates, multisectoral models
will typically exhibit a dependence of the economic decisions in each sector
on expected future economic activity in both the same and in other sectors.
Furthermore, because of the interconnection of current economic variables,
the expectations of agents in other sectors will also matter. In consequence
economic activity in each sector will typically depend on the expectations of
agents in all sectors about future economic activity in all sectors.
We have seen that if expectations of agents are heterogenous in terms of
effects then this can impede coordination on rational expectations even in the
saddlepoint case. Our economic illustration shows how differential impacts
15 An imp ortant generalization will be to consider the mo del (3) with B singular.
23
are likely to arise even if the economic structure is symmetric. Intersectoral
and intrasectoral differences in production technologies and preferences can
be expected to magnify the structural heterogeneity, increasing further the
problem of rational agents coordinating on “rational expectations.”
6 Conclusions
In the saddle-path stable case, the unique non-explosive perfect foresight
solution provides a natural focus of attention for economic theorists, but there
remain deep questions about how this solution would attained by economic
agents. The eductive approach examines this issue from the viewpoint of full
rationality: would rational agents necessarily coordinate on the saddlepoint
solution if they knew the correct model, knew that other agents knew the
correct model and knew that other agents were rational?
In addressing this question we provide our agents with strong additional
common knowledge restrictions designed to facilitate this coordination: specif-
ically we assume that it is common knowledge that the state dynamics,
in every subsequent period, will be close to those followed by the perfect
foresight path. The economy is said to be Locally Strongly Rational, or
eductively stable, if these (hypothetical) restrictions are sufficient to imply
common knowledge of the perfect foresight path itself.
Our characterization provides alternative sufficient conditions for LSR of
a minimal state variable solution. Iterative expectational stability provides
simple necessary conditions, and these will be satisfied by the saddlepoint
solution in the saddlepoint stable case. However, we have shown that these
conditions are not in general sufficient. Our analysis has emphasized in par-
ticular the potential role of heterogeneity in destabilizing the economy. When
expectations of different agents have heterogeneous impacts on the economy,
as is natural even in symmetric multisectoral models, the required conditions
for eductive stability are tightened because of the potential interaction of this
structural heterogeneity with heterogeneous expectations. Sufficient struc-
tural heterogeneity can therefore render rational coordination of expectations
impossible even in the saddlepoint stable case.
24
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