Quasi-rational expectations

Economics Letters North-Holland

QUASI-RATIONAL Christophe

EXPECTATIONS

HENOCQ

University of Paris-I,

Hubert

93

30 (1989) 93-96

*

75231 Paris Cedex, France

KEMPF

Uniuersity of Caen, 14032 Caen Cedew, France Received Accepted

15 December 21 February

1988 1989

Using a convergence approach in decentralized models of the economy, it is shown that agents are able to form an expectation of an endogenous aggregate variable which is almost equal to the rational expectation, with minimal information only.

1. Introduction The rational expectations hypothesis commonly used in macroeconomics assumes that the economic agents forming the expectations have full knowledge of the true model of the economy. Only the exogenous variables and, more precisely, the random shocks which affect the economy at a given period are unknown or imperfectly known. But the validity of this hypothesis seems doubtful in ‘decentralized’ models of the economy which consists of a dispersion of numerous local markets. This dilemma has been pointed out by Frydman (1983) based on an example derived from Lucas (1973). The computation of rational expectations requires an amount of information about every local market clearly beyond the capabilities of dispersed, uncoordinated agents. The aim of this paper is to offer a solution to this informational problem and to show that an assumption on the heterogeneity of local markets allows one to escape this dilemma. With minimal information only, agents are able to compute an expectation almost equal to the true rational expectation. This assumption is to consider that the local structures are different but converge in mean towards a ‘representative’ structure. 2. Structural heterogeneity

and the convergence

approach

Let us first establish a critical proposition in regard to aggregation theory, formalizing an intuition made by Theil (1971). Assume a probability space (52, A, P) characterizing the individual agents of the constitutive units of the economy. Micro-exogenous variables are random vectors, defined in this space with values in I%“. There are N agents (or units) characterized by N vectors of random variables with values in Rx and N microeconomic functions fl, . . . , f,v from R” to R. For all xi,-..,x,V i, i = 1,. . .) N, we denote by y, the corresponding micro endogenous variable: Y! =f,(x,),

Aggregate 0165-1765/89/$3.50

i=l

,...> N.

variables

are defined

(1) as the means

0 1989, Elsevier Science

Publishers

of the corresponding

B.V. (North-Holland)

microeconomic

variables.

C. Henocq, H. Kempf / Quasi-ratmnal

94

Assuming tions:

that the number

of agents

increases

expectations

indefinitely,

we make

Hypothesis I. The vectors x1, . . . , xN are independent from each other, is a compact K of Rx.

the two following

of same law

assump-

the support of which

Hypothesis 2. The functions f,, . . . , fN are uniformly bounded on K. The series of functions converges uniformly on the compacts of R A towards a linear function f from R A to R.

N-’

C;“,l f,

Both assumptions formalize the idea of a representative (maybe fictitious) agent, characterized by an exogenous vector which is the common mean of all the variables x,, and the function f around which ‘oscillate’ the individual functions S,. With both assumptions, the following proposition can be proved: Proposition 1. When Hypotheses I and 2 are satisfied, the macro endogenous as a linear function of the macro exogenous vector of the form Y,=f(X,.)

+eN,

(2)

where eh’ is a random indefinitely. Proof Utilizing deduced that

lim N-m

E(Y,)

=

(vector)

Hypotheses

variable

=f(e)

the limit of which is almost

1 and 2, the theorem

lim N-’ E E(f,(x,)) N+ Cc r=l

The usual law of large numbers dFmf(XN)

variable can be expressed

of limited

= E(f(x,))

and the continuity

and the linearity

of f, it is

(3)

of f allow to write that

(4)

a.s.

lim Y, - E( Y,) = 0 Ndrn

3. Heterogeneous

convergence

=f(e).

The set of f,(x), for x describing K and i describing large numbers of Kolmogorov type 2. Hence,

Then, according same limit f(e)

surely zero when N increases

N being

bounded,

we can apply

as.

to eqs. (3), (4) and (5) the variables Y,, E( Y&) and f ( X,) have almost q when N increases indefinitely. Proposition 1 follows immediately.

the law of

(5) surely the

decentralized models

The preceding proposition may be utilized in order to reconcile the rational expectations concept and the heterogeneity of local structures in a decentralized model, when each market is affected by a local shock.

95

C. Henocq, H. Kempf / Quasi-rational expectations

The economy is assumed characterized by the following

to be formed by N markets S, N being large. Each market is linear relationship, generated by an implicit specific structural model:

xrv is a random exogenous (vector) variable, representing the local shock which strikes the market s at time 1. Y, is the mean of the endogenous variables y,, and X, is the mean of the exogenous variables. SY,e is the expectation of Y, made by agents on market S, given their information. This information consists in the current value y,, and of all the past values of any endogenous or exogenous variables in the economy. The rational expectation of z, when using the information set I, is noted E(z,/Z,). For simplifying reasons, there are no lagged terms in the local markets. To form a rational expectation of Y,, the agents in each local market should know every local structure, i.e., every function f,. This is clearly an enormous amount of information and it is implausible that the agents can handle it. On the other hand, when assuming the existence of a ‘representative’ structure around which the local structures are distributed, it can be shown that the agents in each local market are able to form an expectation of the aggregate endogenous variable which is almost equal to the (true but unattainable) rational expectation when N is large. This is the main result of this note and is stated in the following proposition. Proposition 2. When Hypotheses 1 and 2 are verified and the function f is known by all agents but not the different functions f,, the agents in market s (for any s) are able to form an expectation of the aggregate endogenous variable ,Yre such that

E(Y,/Z,.s>- , Y = 6 it

(7)

3

where c,,,., is a random variable the limit of which is almost surely zero when N increases indefinitely. ProoJ: If agents in market s perfectly function used by them is of the form

know

the whole

structure

of the economy,

the prediction

E(Y,/Zm>= a.syu + P,. (Remember there are no lagged terms in the economy.) of local shocks which are to be anticipated by their reduced form for market s, Y,, =

(P,x,.~+ 4,

The constant term P, is due to the existence mean e. This prediction function induces a

with

+.s= a,(1 - bs)-l,

4, = b,P.s(l -b,)-‘.

Consider now the case where agents in market s only know that Hypotheses 1 and 2 are verified and know the representative function f in addition to their own function f,. Assuming that the reduced form coefficients ‘p,, 4, converge in mean towards $,, 4 and that this convergence of the reduced forms is compatible with the convergence of functions f,, it is deduced from Proposition 1 that Y, =

lim N+@Z

+x, + I) cNt = 0

=

ENI

a.s.

with

(II)

C. Henocq, H. Kempf / Quasr-rational expectntrons

96

Let us show now that this assumption is not arbitrary but is implied structure of the economy. According to (11) for very large N,

E(Y,/Z,,) =N-‘(%,+G(NIdentifying $., =

= a, + b,sqbN-‘,

Ic/,,= h,+ + N-‘( N - l)$e.

converging

l>e+$> =.TY’.

(11) and (8) combined

$( Nay)-'

by the whole

with (9)

N being in the neighborhood

(12) of cc, it is seen that (13) (14)

It is now possible to study the convergence (assumed until now) of & and $,Y. From (13) it is seen that C#B,converges to a. Identically, from (14) +,Y converges to +e(l - b)-‘. The convergence assumption made on the reduced forms is then consistent and confirmed. Then, the agents are able to know $ and 1+5which are necessary to compute ,y. Notice the self-fulfilling character of this assumption which is clearly in the spirit of the rational expectations concept.

4. Conclusion The heterogeneity of an economic system gives rise to a challenging problem for the rational expectations concept. This concept, in its traditional form, implies that the agents who rationally form their expectations know perfectly the complete structure of the economic system. In a ‘decentralized’ economy, such a claim is unrealistic. Nevertheless, it is possible to solve this difficulty by using a convergence approach and defining a quasi-perfect aggregation. This approach assumes that individual behaviors are different but are distributed around a ‘representative’ behavior. The assumptions of the convergence approach, applied to decentralized models, ensure to the agents in a local market that they can form an expectation of the aggregate endogenous variable, when they only know the representative function, which is almost equal to the rational expectation corresponding to the perfect knowledge of every local structure. That is why it appears to us that the obtained expectations have to be qualified as quasi-perfectly rational expectations or, more simply, as quasi-rational expectations.

References Frydman, R., 1983. Individual rationality, decentralisation and the rational expectations hypothesis, in: R. Frydman and E.S. Phelps, eds., Individual forecasting and aggregate outcomes (Cambridge University Press, Cambridge). Lucas, R.E., Jr., 1973. Some international evidence on output-inflation trade-offs, American Economic Review 63, 326-334. Theil, H.. 1971, Principles of econometrics (University of Chicago Press, Chicago. IL).