ational Expectations in Microeconomic odels: An...

JOURNAL OF ECONOMIC THEORY 26, 201-223 (1982)

ational Expectations in Microeconomic odels: An Overview *

JAMES S.JORDAN

Department of Economics, University of Minnesota, Minneapolis, Minnesota 5545s

AND

ROY RADNER

Bell Laboratories, Murray Hil!. New Jersey 07374

This paper is an expository introduction to several topics of current research in the general equilibrium theory of rational expectations. More specifically, we discuss the existence of exact and approximate rational expectations equiiibria, the implementation of equilibria, the behavior of learning and smoothing processes by which traders construct expectations from repeated observations of the market, and the lagged use of the information revealed by prices in an intertemporal sequence of markets. The purpose of this discussion is to introduce papers on these topics appearing in the Journal of Economic Theory Symposium on Rational Expectations in Microeconomic Models. Journal of Economic Literafure Classification Numbers: 021, 022. 026.

1. TNTRODUCTI~N

In a market for commodities whose future utility is uncertain, the equilibrium prices will reflect the information and beliefs that the traders bring to the market, as well as their tastes and endowments. If the traders have different nonprice information, this situation presents an opportunity for each trader to make inferences from the market prices about other traders’ information. An example of this phenomenon is recognized by the everyday expression, “judging quality by price.” The term rationaE expectations equilibrium is applied to a model of market equilibrium that takes account of this potential informational feedback from market prices.

* The partial support of the National Science Foundation and of the University of

Minnesota School of Management is gratefully acknowledged. The views expressed are those of the authors and do not necessarily reflect those of Bell Laboratories or the Bell System.

201 0022-~53~~82~02020~-23~02.~0~~

Copyright ;B 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.

202 JORDAN AND RADNER

We may make the convention that the future utility of the commodities to each trader depends on the state of the environment. With this convention, we can model the inferences that a trader makes from the market prices and his own nonprice information signal by a family of conditional probability distributions of the environment given the market prices and his own nonprice information. We shall call such a family of conditional distributions the trader’s market model. Given such a market model, the market prices will influence a trader’s demand in two ways: first, through his budget constraint, and second, through his conditional expected utility function. It is this second feature, of course, that distinguishes theories of rational expectations equilibrium from earlier models of market equilibrium.

Given the traders’ market models, the equilibrium prices will be determined by the equality of supply and demand in the usual way, and thus will be a deterministic function of the joint nonprice information that the traders bring to the market. In order for the market models of the traders to be “rational,” they must be consistent with that function. To make this idea precise, it will be useful to have some formal notation. Let p denote the vector of market prices, e denote the (utility-relevant) state of the environment, and si denote trader i’s nonprice information signal (i = l,..., 1). The joint nonprice information of all traders together will be denoted by s = (s, )...) sl). We shall call s the “joint signal.” (The term “state of information” is also commonly applied to this array.) Trader i’s market model, say mi, is a family of conditional probability distributions of e, given si and p. Given the traders’ market models, the equilibrium price vector will be some (measurable) function of the joint nonprice information, say p = 4(s).

To model the required rationality of the traders’ models, suppose that, for each i, trader i has (subjective) prior beliefs about the environment and the information signals that are expressed by a joint probability distribution, say Qi, of e and s. These prior beliefs need not, of course, be the same for all traders. Given the price function 4, a rational market model for trader i would be the family of conditional probability distributions of e, given si and p, that are derived from the distribution Qi and the price function 4; thus (supposing e and s to be discrete variables),

mi(e' 1 s;, p’) = Probci(e = e’ 1 si = sf and 4(s) = p’). (1.1)

A given price function 4, together with the rationality condition (l.l), would determine the total market excess supply for each price vector p and each joint information signal s, say Z(p, s, 4). Note that the excess supply for any p and s depends also on the price function 4, since (in principle) the entire price function is used to calculate the conditional distribution in (1.1). We

RATIONAL EXPECTATIONS IN MICROECONOMIC MQDELS

can glow define a rational expectations equilibrium (REE) to be a price function #* such that, for (almost) every s, excess supply is zero at the price vector 4*(s)> i.e.,

The formal study of rational expectations equilibrium was introduced in Radner (1967); it was taken up independently by Lucas (1972) and Green (1973), and further investigated by Grossman, Jordan, and others. We shah make no attempt here to provide complete bibliographic notes on the subject; for this the reader is referred to Radner (19 2, Sects. 7.1, 7.4). The particular definition given above can be criticized on several grounds, and we shall return to this point below.

We should emphasize that we are concerned here with that aspect of “rational expectations” in which traders make inferences from market prices about other traders’ information, a phenomenon that is onEy of interest when traders do not all have the same nonprice information. The term “rational expectations equilibrium” has also been used to describe a sitution in which traders correctly forecast (in some sense or other) the probability distribution of future prices. (See Radner (1982) for references to the work of Muth and others on this topic.)

The concept of REE has been used to make a number of interesting predictions about the behavior of markets (see, for example, Futia (1981b) and the references cited there). A sound foundation for such applications requires the investigation of conditions that would ensure the existence and stability of REE, and this investigation has revealed a set of problems that

are more difficult and more subtle than those encountered in ordinary equilibrium analysis. The papers in this issue are all concerned w this

range of problems. First, if markets are incomplete, the existence of E is not assured by the “classical” conditions of ordinary general equilibrium analysis. Even under such conditions, if traders condition their expected utilities on market prices, then their demands can be discontinuous in the price function, Specific examples Qf the nonexistence of discontinuities were given by Kreps (1977), Green (1977), and others. (A fairly general existence theorem for the complete+markets case is given by Grossman 1981.)

Second, the definition of REE that was given above--we shall sometimes refer to it as REE in Ihe wide sense-is probably in need of some refinement. One direction of refinement would reflect a requirement that the REE be implemented by some specific class of institutional ‘“mechanisms”; such requirements would probably imply some further restrictions on the price function 4, beyond the condition of measurability. Another direction of refinement would recognize that market transactions are tjrpically spaced out

204 JORDANANDRADNER

in time, so that the information revealed by prices in any one transaction can only be used in subsequent transactions.

Third, a thorough analysis of learning and stability needs to take account of the way that traders modify their market models in the light of experience, as well as of the usual problems of price adjustment in disequilibrium. Finally, since the acquisition of information is an activity that may require the. expenditure of economic resources, the study of REE would ideally treat the traders’ nonprice information as endogenously determined, and would take account of the costs and benefits of information acquisition.

In the remainder of this introductory paper we shall discuss these various matters in more detail, and attempt to indicate how the other articles in this issue fit into this framework.

2. EXISTENCE OF RATIONAL EXPECTATIONS EQUILIBRIUM IN THE WIDE SENSE

The examples of Green (1977) and Kreps (1977) naturally led theorists to question whether the absence of REE in the wide sense is pervasive or is confined to a “negligible” set of such examples. Indeed, this question was already anticipated by Green, whose example is robust to perturbations of the density function describing the uncertainty in the environment, but not to perturbations of traders’ characteristics. The paper by Jordan in this issue, in conjunction with the earlier results of Radner (1979), Allen (1981a) and (1981b), and Jordan and Radner (1979), provides an essentially complete answer. The answer can be loosely summarized in the statement that equilibria in the wide sense exist generically except when the dimension of the space of private information is equal to the dimension of the price space.’

We now give a brief survey of the results which comprise this answer. In view of the rather technical nature of generic existence theory, we shall motivate the results by extending a simple example taken from Jordan and Radner (1979). In this example there are two traders and two commodities. Trader 1 is fully informed before the market opens, and trader 2 has no private information. Hence trader l’s signal s, is also the joint signal s as well as the environment e. We shall use the symbol s. Suppose for the moment that the signal is an element of the unit interval, and define the traders’ utility functions as

’ It will be seen that the analysis leads one to question whether the definition of REE in the wide sense is not too weak. Thus the question of existence of suitably defined REEs may not yet be properly posed. See the remarks at the end of Section 2 and in Section 6 of this paper, and the papers by Anderson and Sonnenschein and by Jordan (Sect. 4) in this issue.

RATIONAL EXPECTATIONS IN MICROECONOMIC MODELS 205

u’(x,y;s)=(l +s)lnx+(2-s)Iny,

Each trader has the endowment (1, l), which is fixed independently of s. Let A = (0, 1) denote the space of prices, where each p in A represents a price of commodity x, the price of y being 1 -p. Under these specifications, one can show that a rational expectations equilibrium would have to be a function 4: s + p that satisfies, for almost every s,

4(s) = (1/6)[1 + s +E{(2 - st> /4(s)}], or

s = 64(s) - 3 + E{s2 j $(s)], (2.1)

where E{sZ j 4(s)} denotes trader 2’s conditional expectation of s2 given the observed price 4(s). (This follows from a straightforward calculation of the traders’ demand functions, and the condition that excess demand be zero.)

It is immediate from (2.1) that 4 must be 1-l. The qualitative reason for this is that at any given price trader l’s demand for x is a strictly increasmg function of S. Since trader 2’s demand is a function of price alone, market equilibrium requires that any change in s be accompanied by a change in p.

ence we can replace E{s* 1 i(s)} with s2 a.e., which yields

4(s) = (l/6)(3 + s - s’). (2.2)

However (2.2) implies that for any s < l/2, gi(s) = #(l - s), so 4 cannot be 1-1, at least not on the entire unit interval.

Up to this point we have not specified the probability First suppose that s has the uniform distribution on (0, 1). Then (2.2) implies that E(s / 4(s)] = l/2 a.e. which contradicts the ~rnpli~at~o~ of (2.1) that E(s 1 4(s)} = s a.e., so no equilibrium exists Existence fails in this example because $ must reveal s to trader 2, but if trader 2 knows s, the function @ intersects itself on a set of positive probability. These properties are sufficiently robust that perturbations of the example cannot resolve the problem. More precisely, let (cJ.I’~, ~‘~)f=~ be an endowment-uti

lose to (1, 1) for each i, and u’~ close to tii in a C* et Q be a probability distribution on (0, 1) close e

uniform distribution so that any event with positive probability under the Iatter distribution has positive probability under Q. Then it is proved in Jordan and Radner (19’79) that the economy {(w”, u’j)f=, 3 Q} does not have an equilibrium. Thus if the dimension of the signal space equals the dimension of the price space, the generic existence of equilibrium cannot be obtained.

Now suppose that the dimension of the signal space is less than the dimension of A, With only two commodities this requires the signal space to


be zero-dimensional, so suppose that the probability distribution of s is concentrated on a finite set {sj}. Equations (2.1) and (2.2) continue to hold for each sj, so equilibrium will still fail to exist if for somej, j’, sj = 1 - sj,. However in this case a slight perturbation of the utility function ~‘(a, sj) will perturb the price #(sj) to break the equality #(sj) = #(sj(). Hence 4 must still reveal s to trader 2, but a slight perturbation of the economy can move the self-intersections dictated by (2.2) away from the restricted set {sj}.

The first result in generic existence theory was Radner’s application of this reasoning to general financial asset markets (Radner, 1979). It was proved in that context that fully revealing equilibria exist generically. Beth Allen subse- quently generalized the finiteness condition to a condition on the dimension of the signal space relative to that of the price space.2 In Allen (1981a), it was proved that if the dimension of the signal space is less than half the dimension of the price space then fully revealing equilibria exist generically. In Allen (1981b) this result was extended to the case in which the dimension of the signal space is merely less than the dimension of the price space under the additional assumption that the probability distribution gives probability zero to events of Lebesgue measure zero. In the former case, Allen showed that perturbations of the economy can eliminate self-intersections of the price function, and in the latter case that perturbations can reduce the self- intersections to a set of Lebesgue measure zero, and hence probability zero. (Strictly speaking, since Allen does not parametrize signals, she uses a concept of a set of zero measure that is approapriate for general manifolds; e.g., Hirsh, 1976, p. 68.)

The example in Jordan and Radner (1979), which we described above, shows that if the signal space and price space have equal dimension then self-intersections of the price function cannot be perturbed away or reduced to an event of probability zero. Of course, this continues to be the case when the signal space has higher dimension than the price space. However, in the latter case the necessity of full revelation, as exemplified in Eq. (2.1) above, is no longer robust. This leads to the result [Jordan, 1979b, Theorem 2.41 that in the higher dimensional case, existence of equilibrium is again generic. Moreover, although full revelation is no longer generically possible (in fact it is generically impossible [Jordan (1979c, Theorem 2.1 l)]), approximately fully revealing equilibria exist generically.

To motivate the higher dimensional case we shall add a second dimension to the signal space in the above example and show that a slight perturbation of the example admits an approximately fully revealing equilibrium. Expand the signal space to the unit square, (0, 1) x (0, l), again with the uniform distribution, and embed the previous example by defining the utility

* Strictly speaking, Allen did not parameterize signals but dealt directly with the manifold of endowment and utility profiles that would obtain if all information were revealed.


functions ur(x, y; s,, s2) = u’(x, y; sr), and v2(x,y; s,, s,) = u*(x, y; s:). continue to assume that trader 1 observes the entire joint signal and trader 2 observes only the price, so we have essentially the same example since the new dimension is irrelevant. In particular, no equilibrium exists. Now perturb the example by redefining trader 2’s utility function as

vgx, y; S1) s2) = (2 - ST + as,) In X + (1 + ST - Ss,) In>‘,

where 6 is a small positive number. Then

so $ reveals sr but not necessarily s2. Hence

e shall demonstrate that for any F > 0 the final term on the right-band side of (2.2,) permits the existence of a price function $ that reveals s1 and also reveals s2 within E. Partition the signal space into rectangles (a, b) x (c, d) with d - c < E and b - a < (d - c)(6/8), neglecting the boundaries of such rectangles as a zero probability event. We shall construct # on a typical rectangle. Suppose that 4 has already been constructed on some of the (finitely many) other rectangles and that the image of # on these rectangles is contained in a closed nowhere dense subset of A. This leaves an open and dense subset G c A of unused price values. Given s, E (a, mi) and s; E (c, d), suppose we set #(s,, sJ =pl if s2 E (c, s;*) and #(s, 9 s2) =p2 if s2 E (~4, d), where p1 and p2 are distinct elements of G. This is consistent with (2.2,) if and only if

~1 = (l/6)(3 + s1 -s:) i- (6/6)(s; + c)/2,

and p2 = (l/6)(3 + sr - s;) + (6/6)(d + sf)/2. (2.4)

As s: varies in the interval I= [(3c + d)/4, (c $ 3d)/4], the right-hand side of (2.4) generates two disjoint intervals of length (d - c)(6/24). Call the first

such interval J,(s,), and let J, be the intersection of the intervals J,(s,) as s1 varies in the interval (a, b). Since

sup{/@, - s:> - (s; + s;“)l: sr, s; E (a, b)} < (b - a),

the set J, contains an open interval I, of length (6/48)(d - c)~ If we change the definition of this set by replacing SF + c with d + s:, it slated interval I, = I, + {(6/12)(d - c)}, with I, n 1, = 0. necessary we can ensure that both I, and I, are contained in the set G of unused prices. Now let ~~ be a l-l Bore1 measurable function on (a, b) in&~

208 JORDAN ANDRADNER

a closed nowhere dense (Cantor-like) subset of I,, and let 4; = 4; + (d/12) (d-c). Then we can use (2.4) to determine the appropriate sl for each s1 E (a, b) so that

and

$l*(sJ = (I/6)(3 + Sl -s:> + @/12)($%,) + c),

&%A = (I/6)(3 + s1 -ST> + (d/12)& + sf(s,)). (2.5)

Define 4 on the rectangle (a, b) X (c, d) by

Since 4:: and $t are l-l, 4 reveals s,, so (2.5) and (2.6) imply that 4 is an equilibrium on (a, b) x (c, d> provided that its image on this rectangle does not intersect the image of any other rectangle. Since I, U I, c G, the latter condition is satisfied. Also, since the image of # on (a, b) X (c, a) is contained in a closed nowhere dense subset of d, we can complete the definition of 4 by repeating this construction on each remaining rectangle. Since 4 reveals s, and d - c < E, # reveals (sl, s2) within E.

This type of construction is possible because in the higher dimensional case the informed trader’s demand at a fixed price is no longer even locally a l-l function of the signal. Hence the signal is no longer necessarily revealed to the uninformed trader, and the freedom to adjust the information revealed by the price, at least generically, creates an additional degree of freedom in the price function. In particular, the price function can be discontinuous. Since discontinuous functions are capable of packing a high dimensional signal space into a low dimensional price space, the self-intersections of 4 can be reduced to sets of arbitrarily small diameter. The only reason they cannot be removed entirely is that if the signal is revealed exactly, 4 must be continuous because the traders’ utilities are continuous in the signal. In terms of our example, (2.2,) becomes

@(sI, ~2) = (l/6)(3 + ~1 - s: + Js,), v-w

which, like (2.2), has self-intersections that conflict with the necessary revelation of s, .

The equilibrium constructed above seems rather artificial as a description of market behavior. This is partly due to the fact that although the coor- dinate s2 is irrelevant to the informed trader, the equilibrium reveals s2 within E to the uninformed trader. This feature of the example results from the use of log linear utilities with state-dependent coefficients. With this

RATIONALEXPECTATIONS INMlCROECONOMlCMODELS 209

parameterization it seems unreasonable for an equihbrium to enable the uninformed trader to distinguish between any two signals in which the informed trader has the same coefficient, and thus the same demand at every price. The presence of fixed level surfaces, in particular the sets {s,} x (c, k), on which the informed trader’s demand is the same at every price was central to our construction of equilibrium. For “generic” utility hmctions, the level surfaces of constant demand differ as prices differ. Hence in the generic case considered in Jordan’s paper, an objectionable feature of the above equilibrium is removed but the construction of equilibrium is severely complicated.

The equilibrium in the higher dimensional case may also be criticized for its reliance on contrived discontinuities to squeeze a high d~rne~sio~a~ information space into a low dimensional price space. This contrivance could of course be prohibited by requiring continuity as a additional condition of equilibrium. However, the direct imposition of mathematical regularity conditions is not likely to answer the underlying question of what sort of equilibrium could be the outcome of a natural market adjustment process.

3. APPROXIMATE REE

When a rational expectations equilibrium fails to exist it is natural to ask whether there is a price function that is in some sense an approximate equilibrium. The two equilibrium conditions that define a rational expectations equilibrium are: (a) prices must clear markets, and (b) traders must have the correct expectations conditional on prices and their private information. In her paper in this Symposium, Allen demonstrates the general existence of price functions that satisfy (a) approximately and (b) exactly.

Alien’s approximate equilibria can be illustrated in the context of the example described above by equations (2.1) and (2.2). For any E > 0, we can partition the interval [O, 1) into subintervals [sj, sj+ ,) so that for each j, if pj = (l/6)(3 + sj - sf) then for any s E [sj, sj+ 1)

If1 f S)/3Pj + [2 - (Sj+, -s3)/3(Sj+, -Sj)]/3P,j- 2/ <E/4. (3.1)

The inequahty (3.1) states that if pj is the price that clears markets when both traders know the signal is sj, then for any s E [sj, sj+ I), the aggregate excess demand for commodity 1 is less that s/4 in absolute value when the uninformed trader knows only that the signal is in the interval [sj, sj+ L). Since pj < 2/3 for all j, (3.1) and Walras’ Law imply that the excess demand for commodity 2 is less than s/2 in absolute value. Define a price function 4

bY

$Cs) = Pj if sE [s~,s,/+ 1). (3 2)


Then 4 will satisfy the equilibrium condition (b) if pj # pj, whenever j #j’, and the equilibrium condition (a) is violated by less than a/2. If for somej, j’, pj =pj, (that is, sj = 1 - sj, a slight perturbation of Pj will ensure that pj #pj, without increasing the absolute value of aggregate excess demand above E.

The price function $ is typical of the approximate equilibria given by Allen’s Theorem 1. Since 4 is constant on intervals [sj, sj+ ,), it does not fully reveal the signal. Allen’s Theorem 2 essentially states that if the dimension of the signal space is less than the dimension of the price space, fully revealing approximate equilibria exist. In the context of our two commodity example, the dimension hypothesis corresponds to restricting the signal space to a discrete subset (sk} of (0, 1). Then if we set #‘(sk) = (l/6)(3 + sk - s:) for. each k, we may encounter the problem that #‘(s,) = #‘(sk,) for some k # k’. However, a slight perturbation of #‘(sk) will recover the full revelation property, and thus ensure equilibrium condition (b), while violating the market clearing condition (a) by an arbitrarily small amount.

Relaxing the market clearing condition by requiring the magnitude of aggregate excess demand to be less than E gives an additional degree of freedom, loosely speaking, by permitting the price for each signal to be chosen from an interval of admissible prices. It seems likely that even in the absence of dimensionality restrictions, one could construct a l-l price function that weaves the signal space into these intervals. However, such price functions, like the price functions constructed in Jordan’s paper, necessarily have a dense set of discontinuities. Hence it is natural to ask how much revelation can be obtained without violating the regularity that one intuitively associates with a “well-behaved” price function. This question is addressed by Allen’s Theorem 3, which states essentially that an economy that is close enough to an economy satisfying the dimensionality condition has an approximate equilibrium price function whose level sets are surfaces of codimension equal to the dimension of the nearby low dimensional economy. In terms of our two-commodity example, the price function 4 constructed in the second paragraph of this section has level sets of this description, where the low dimensional economy is the economy with the discrete signal space {sj}.

4. EQUILIBRIUM WITH IMPERFECT PRICE MODELS~

The nonexistence of rational expectations equilibrium illustrated in Section 2 can be traced to discontinuities in the demand function with respect to the price function. In a sense, these discontinuities arise because the theory

3 This section is based on Radner (1978).

RATIONAL EXPECTATIONS 1N MICROECONOMIC MODELS 211

postulates that, in equilibrium, the traders klzow the price function perfectly? and the price function is perfectly accurate. To see this, consider the example of Section 2, in which there are two traders, one who is initially fuiiy informed about the environment and one who is inltially uninformed. Suppose that there are two possible information signals, s’ and s”, let #, be a sequence of price functions, and define

P:, = h?(f)3 P:: = d,(s”).

Suppose that, for every n, p; #pL, but that the two sequences, (p;) and (pi), both converge to a common value, say TV. Finally, defme the price function lo by #,,(s’) = #,,(s”) =pO. Thus each price function in the sequence is revealing, but the limit price function is not. Therefore, for each n > 0 the second trader’s demands at the two prices will be different, and these differences wili not tend to zero as n increases because although the differences between p; and pi are tending to zero the two prices reveal the respective signals, s’ and s”, as long as the prices are different. Gn the other hand, in the limit trader 2 cannot infer the information signal from the price, since the limit price function is not revealing. Hence, in the limit, trader 2’s expected utility will not be conditioned on the signal s, and so his demand will (typically) have a discontinuity at the limit price function. This follows from the (implicit) assumtion that, no matter how close #(s’) and #(s”) are, if they are different then trader 2 can infer the signal s from the price.

Let us now modify the description of the market, by replacing the assumption that each trader knows a (perfectly accurate) forecast function with the assumption that each trader has an “econometric” model of how equilibrium prices are determined. Let E denote the set of environments e, let S denote the set of pooled information signals s, and iet A denote the set of all nonnegative price vectors: normalized so that the sum of all prices is unity. Trader i’s econometric model of price determination, which we shall call his price model, is charcterized by a family, ~,(pis)~ of strictly positive conditional probability density functions on the set A of possible price vectors. (This includes, as a special case, the typical econometric mode! in which a “disturbance” with a probability density function is added to a deterministic relationship.) The joint probability distribution, for trader i3 of the variables e, s, and p is determined by his probability distribution Qj on E x S together with his price model, vi.

One can show that, for each trader i, maximizing conditional expected utility given si and p is equivalent to maximizing

E~~[ui(Xi~ e) Wi(Pls)lsi19 (4.1)

where EQi[. lsi] denotes conditional expectation, with respect to the probability measure Qi, given si. Given s, an equilibrium is characterized by


a price vectorp* and an I-tuple (xi*) of demand vectors such that, for each i, x7 maximizes

EQi[ui(xiY 9 WitP” Is>lsil (4.2)

subject to xi > 0 and p*xi <p*wi, and such that excess demand is zero, i.e.,

c (xj” - Wi) = 0. I (4.3)

One can show that, with suitable assumptions, an equilibrium exists for almost every s. The assumptions are in two sets.

Those in the first set concern the traders’ models of price determination. They express the two ideas that (I) each trader’s price model is appropriately continuous in the price vector p, and (2) no trader’s price model could predict the equilibrium price perfectly from the joint signal S. Formally, we assume that, for every trader i:

wi(+ IS) is continuous on A, and strictly positive on its interior, for almost every s; (4.4a)

yi(pI .) is majorized in absolute value by an integrable function on S, uniform& in p. (4.4b)

The assumptions in the second set are standard in the theory of exchange equilibrium and in expected utility theory. (They have been chosen more for reasons of familiarity than to maximize generality.)

For every trader i,

(a) ui(., e) is continuous, strictly concave, and increasing on RT , for almost every e;

(b) ui(xi, .) is measurable on E, for every xi;

(c) Ui is bounded on RI: X E;

(d) his endowment, wi, is nonnegative. (4.5)

To prove the existence of an equilibrium one wants to establish the continuity of a trader’s conditional expected utility in the variables x and p. We shall only indicate here how this is done. To show that the conditional expectation (4.1) is continuous jointly in xi and p, it is sufficient to show (1) that the family of functions

ui(xiY *I WdPI ‘> (4.7)

RATIONALEXPECTATIONS INMICROECONOMIC MODELS 213

is majorized in absolute value by an integrable function on .E x in xi and p, and that (2) the family of functions

ui(e2 4 Wi(’ I $> (4.8)

is continuous on RH, X d for almost every (e, s). One can then apply the Lebesgue Dominated Convergence Theorem, as generahzed to ~ond~tio~~~ expectation. Property (l), of (4.7), follows from (44b), (4.5b), and (4.5~). Property (2). of (4.8), follows from (4.4a) and (4.5a).

With only the assumption stated in the previous subsection, the resulting equilibrium has no obvious “self-fulfilling” or “rational” expectations property. A minimal requirement along these lines is that an eq~il~~~ium price p* not be “inconsistent” with any trader’s model of price determination. One way to express this formally is to require that p* be in the support of u/;(- Is), considered as a probability density on d. But this follows from the assumption that ~~(~1s) is strictly positive on the interior of ri. This expresses the idea that, given any exogenous signal si, trader i’s model vi does not exclude as impossible any open set of e~~i~i~~i~~ price vectors.

A more stringent “rational expectations” requirement would concern the o~~ort~~it~es that traders might have for learning from experience, For example, suppose that the sequence {e(t), s(t)] were independent and identically distributed, with each {e(t), s(t)} being generated by a ~robabi~~t~ measure Q on E X S, and let (p*(t)} be a corresponding sequence of equilibrium price vectors. Suppose further that every trader would then have available at least the sequence of observations (s,(t),p*(t)]; indeed, he might have, with some lag, more complete information on the sequence {s(t)] of joint signals (e.g., through published statistics). Traders would then be expected to modify their models I+Y~ in the light of these observations, For a given learning process, one might ask whether the process converges in any useful sense, and if so, whether the models vi are asymptotically consistent with the (endogenously determined) actual relationship between signals and equilibrium prices. We shall take up this topic in the next section.

5. LEARNING

We begin this section with an example to illustrate the relationship between the market models of Section 1 and the price models introduced in Section 4. This example will also be used to illustrate some of the ideas discussed in this and the next sections.

Consider a market for a single risky asset (and money), with I ‘“active” traders and one “passive” trader (i = 0). The environment, e, is interpreted as the future value of the risky asset, and each active trader’s nonprice infor-

214 JORDANANDRADNER

mation signal, si, differs from e by a random error, yi, so that si = e +yi, The money price of the asset is denoted by p. The market model, mi, of active trader i is that, given si and p, e is normally distributed with mean Ci = b,p + cisi, and constant variance. Given pi andp, i’s demand is assumed to be

Di(PISi) = diIgi - &p

= (6ij ci> si + (6{1 bi - 6iZ) P9 i = I,..., I. (5.1)

The supply of the passive trader is assumed to be exogenous, and is denoted by x. It is common knowledge that the random variables e, y, ,..., yr, x are independent and normally distributed, with (say) means equal to zero and strictly positive variances 6, yI, ,..., nr, [, respectively. The joint information signal is s = (x, s r ,...,sI). Thus the traders all have the same prior distribution, Qi = Q, on E X S (cf. Section 1).

The condition that demand equal supply is expressed by

C 6i, CjSi + C (Si, bi - 6i,)p = X, I t

so that the unique market clearing price is

P = 4(s) = x - xi ai, cisi

Ci C6il bi - ‘i2> ’

(5.2)

(5.3)

provided that the denominator of the right-hand side is not zero. We shall call (5.3) a temporary expectations equilibrium (TEE). We emphasize that the price function 4 has among its parameters the parameters (bi, ci) of the traders’ models m,.

The prior distribution Q and the price function 4 of (5.3) imply that, given si and p, the environment e is normally dstributed with conditional mean e”’ = hip + cfsi. For each i, the “new” regression coefficients, b; and ci, are complicated nonlinear functions of all of the parameters (bj, cj) in all the traders’ models mj, say

(b;, c;) = zi(b,, c1 ,..., b,, cI). (5.4)

(For explicit formulas for the functions ri see Radner, 1982). A rational expectations equilibrium is a TEE for which the “new” regression coefficients are the same as those in the traders”models, i.e., for which

b; = bi, c; = ci, i = I,..., I. (5.5)

One can show that, in this model, a REE does exist (see Hellwig, 1980); any such REE will not be revealing.

RATIONAL EXPECTATIONS IN MICROECONOMIC MODELS 2:s

In the TEE (5.3) the price is a linear function of s, say 4(s) = f. s, whereJ is a vector of coefficients that depend on the various parameters of the model, including the parameters (bi, ci) of the traders market models. If a trader i did not know the vector h he might replace the function 4 by an “econometric” model of the form

where f is a vector of unknown parameters, and u is a “disturbance” term. An econometric specification of (5.6) together with a prior distribution of parameters would constitute a price model vi as described in Section 4.

We now turn explicitly to the question of learning. Suppose that there is a market at each of a succession of dates t, and that the successive exogenous vectors (e,, s,) are independent and identically distributed. Suppose further that at the beginning of date t trader i knows t past history of environments, prices, and his own nonprice information. n the basis of this history he updates his initial market model, say mi, form a current market model mi,. These current market models, together with the non information signals at date t, then determine a TEE price at date t, say

For example, in the linear asset-market model described above, each trader j might form m,, by calculating the ordinary least-squares estimates bj, and tit in the regression of e on p and si, using the past values of these variables. In the context of a linear model much like this one (but somewhat simpler), M. Bray has shown in her paper in this Symposium that, for each trader, the sequence of least-squares regression coefficients converges almost surely to their corresponding (unique) REE values. It follows, of course:, that the corresponding sequence of TEE price functions converges almost surely to the REE.

We emphasize that although the successive exogenous vectors (e,, s,) are independent and identically distributed, the triples (e,, s,,p,) are not, because the traders are modifying their market models from one date to the next. Thus the Bray result cannot be derived from the standard statistical theory of estimation from random samples. Bray also examines a case i.n which the exogenous supply, x, is correlated with the other exogenous variables. In this case it appears that for some parameter values the sequence of TEES does not converge to the REE, but the complete analysis of this case has not yet been carried out.

The paper by Blume and Easley in this Symposium presents a somewhat less optimistic view of the possibility of learning rational expectations. and Easely define a class of learning procedures by which traders use successive observations to form their subjective models, where the term model for trader i means a conditional distribution of s, given si and pi They show that rational expectations equilibria are at least ‘“Iocally stable” under

64212612.2


learning, but that learning processes may also get stuck at a profile of subjective models that is not an REE. The learning procedures defined by Blume and Easely are applied to a fairly general class of stochastic exchange environments that do not possess the special linear structure of the above example. However, to accommodate this additional generality, Blume and Easely constrain traders to choose their subjective models from a fixed finite set of models and convex combinations thereof. Hence for some profiles of subjective models, market clearing may result in a “true” model that lies outside the admissible set. It is then intuitively plausible that a natural learning procedure could get stuck at a profile of subjective models that differs from the resulting true model but is in some sense the best admissible approximation to the true model, even if the admissible set contains an REE model. This phenomenon is illustrated in Section 5 of their paper.

6. ALTERNATIVE DEFINITIONS OF REE

The concept of rational expectations equilibrium that was defined in Section 1 (REE in the wide sense) has been criticized on several grounds, and these criticisms have inspired alternative definitions of REE. In this section we shall review three of these criticisms, and the corresponding alternative definitions, under the abbreviated headings, (1) implementation, (2) lagged use of information revealed by prices, and (3) smoothed models, Each of these three issues is addressed by papers in this Symposium.

Implementation

Recall that in REE an equilibrium price function 4 determines a market clearing price $(s) for each joint information signal s. One might imagine that once the several traders obtain their respective nonprice information signals, si, the corresponding market clearing price, 4(s), would be determined by some “institutional mechanism,” although the definition given in Section 1 is silent about this. As Beja (1976) and others have pointed out, one can describe “reasonable” mechanisms such that there exist price functions that are REEs in the wide sense that cannot be implemented by the mechanism in question. For example (without attempting to be completely precise), suppose that each trader observes his nonprice signal si and then communicates to a central computer a demand function, say Dj. From the I- tuple (Dl ,..., DI) the computer calculates a price for which total excess demand is zero. We shall call this the C-mechanism.4 Consider a completely revealing REE price function; in such an REE, for each si and pricep trader

4 Beja (1976) calls such a mechanism a “genuine trading process.” We are using the symbol C for “computer.”


i’s demand function Di that he transmits to the computer can be made independent of si. In this case the computer will receive the same 1-tupie of demand functions for each joint signal s, and thus will be unable to compute the values of the REE price function.

This phenomenon is not confined to the case of completely reveaiing REEs. The linear example of Section 5 can be used to iilustrate this point, as well as to give a simple and precise description of the C-mechanism for this special case. In that model, suppose that the variance of the exogenous supply x is zero (i.e., x is identically zero); for simplicity of calculation suppose further that the parameters of the several traders are identical, so that di, = 6, I) ai, = 6,) and vi = q. We shall show that for a suitable constant K the function do(s) =IG is an REE in the wide sense, where I is the arithmetic mean of the signals si. Recalling that all of the random variables are normally distributed, one can verify that

which is the same for all traders. If each trader i uses the regression function (6. I), then this corresponds to taking

ci = 0, b:=:(f) (&))

and from (5.2) the corresponding total excess demand given s is

This last can be made zero for every s by taking

K= (2) (z%b (6.4)

Hence, with K specified by (6.4), q& is an REE price function. On the other hand, given si, the demand function Di(. ] sj) of trader i is, from (5. I),

which is zero for all si and p. Hence, if for each s the computer receives only the message ]Dy(. IsI),..., D,“(. ]s1)], t i is unable to compute the price (b,(s) for each s. Hence #O is an REE in the wide sense but is not implementable by


the C-mechanism. Indeed, in this example there is no REE in the wide sense that is implementable by the C-mechanism.

In the Anderson-Sonnenschein model (see their paper in this symposium) prices are selected by the C-mechanism, but only after the aggregate demand function of the traders has been perturbed by a “random disturbance” (p. 267). Their model is applied to general stochastic exchange environments and the disturbance in demand is introduced to smooth out the discontinuity in the classical Walrasian equilibrium correspondence.’ Hellwig (1980) has shown in a more general version of the preceding linear model that if demand is perturbed by a random disturbance then an REE exists in which the price is not a sufficient statistic, so that traders’ demands depend on their private nonprice information. The disturbance term is introduced by Hellwig as a random exogeneous supply, corresponding to x in the above example. If the variance of x is positive, our example appears to have an REE which is implementable by the C-mechanism. That is, the equations

Zi=bp+csi, (6.6)

x-6,c~isi ‘= I(d,b-6,) ’ (6.7)

have a solution (b*, c*), which specifies an REE.6 Since (6.7) is simply the rule used by the computer to determine p, the REE is obviously C- implementable. (If the variance of x is zero (6.2) and (6.4) imply that the denominator in (6.7) is zero.)

Unfortunately the general relation between demand “noise” and C- implementability is not as simple as this example might suggest. For general stochastic exchange economies there remains the prior problem of the nonexistence of an REE, and Anderson and Sonnenschein show that this cannot be resolved simply by adding a random disturbance to demand, at least if the disturbance term has a small support (Note 1, p. 275).

An alternative implementation scheme has been studied by one of the present authors (Jordan, 1979b), following a suggestion by Reiter (1976), and independently by Kobayashi (1977). Suppose that traders initially condition their expectations only on their private nonprice information. Market clearing will then determine a price function #i. If traders then refine expectations by conditioning on $1 as well as on their initial information, their demands will change so market clearing will result in a new price

’ This discontinuity commonly occurs when the aggregate demand function has zero Jacobian at an equilibrium price. The constant demand function that prevents implementation in the preceding example is an extreme case of this.

6 After some tedious manipulations, (6.6) and (6.7) yield a cubic equation in c whose solution then determines b. A more general treatment is given in Hellwig (1980).


function, &. This process continues, with traders retaining the information they acquire at each stage, until no new information is revealed. It is shown in Jordan (1979b) that this process will converge, in a particul~ sense, for general stochastic exchange economies. Furthermore the limit is an REE provided that traders’ expectations are conditioned on their limiting trades as well as on the limiting prices.’ It is assumed that the ‘6pre-equ~libri~m9~ trades corresponding to the price functions 4, are not actually consumated, because otherwise the capital gains and losses generated by the adjustment process would, if rationally anticipated, prevent the existence of the price functions (s, (Jordan, 1979b, Sect. 4). Kobayshi (1977) studies this adjustment process for a “noiseless” version of the linear model described above. Pre-equilibrium trading occurs but traders are assumed not to anticipate interim capital gains or losses. In this setting Kobayashi shows that the process will stop at a conventional FLEE after N+ 1 iterations, where N is the number of traders. IIence this process implements the WEE, which is not C-implementable.

These are only two of probably many “reasonable” implementation schemes. A natural question is whether the axiomatic approach to implementation theory that has recently been developed in a nonstocbastic setting can be extended to stochastic economies with endogenous expectations. Such an accomplishment would ideally provide a characterization of those equilibria that could be implemented by a natural allocation process.

Lagged Use of Information Revealed by Prices

In markets in which transactions are spaced out in time, traders typically submit orders that are simpler than the complete specification of their actual demand functions. The information revealed by the prices in one transaction can also be fully utilized only in subsequent transactions. In principle, this should allow private information to be more valuable than in the model of REE defined previously, since a trader with private information can utilize it in one transaction before the prices reveal (or partially reveal) this information to other traders. In M. Hellwig’s paper in this Symposium, this consideration leads to a model of a sequence of markets, with a corresponding sequence of price functions. In equilibrium, these price functions are determined simultaneously, and t e price function at date i cleass the market at date t conditional on the history of nonpric up through date t and the history of prices before date t. himself to a linear-Gaussian model in the spirit of the example of Section 5, IIellwig is able to derive considerable information about the nature of the equilibrium price process, and to analyze the effect of changing important parameters such as the length of time between successive markets.

’ A general treatment of the existence of equilibrium with alternative market information is given by Jordan (1979a).

220 JORDANANDRADNER

We should note that the analysis of REE with lagged use of information revealed by prices has not yet been done for more general models, but the work of Futia (1981a) and Border and Jordan (1979) indicates that the problems of existence of this type of REE may be of the same qualitative nature as those that have been discovered in the case of the REE of Sections 1 and 2.

Smoothed Models

If an REE price function were implementable by some reasonable mechanism, in the sense we have discussed above, and were also the limit of a reasonable learning process, in the spirit of Section 5, then in our view it would be a strong candidate to be regarded as a “reasonable” equilibrium. However, learning process of the type described in Section 5 would typically not converge in a finite number of dates, which suggests that one might want to formulate a definition of REE that more appropriately describes the behavior of traders who have available to them only a finite set of observations on the working of the market.

For example, suppose again (as in Section 5) that the successive pairs (e,, st) are independent and identically distributed, with probability distribution Q, and suppose further that each trader i uses his initial market model, mi, for T dates, without making any adjustments in his market model during this time. If p1 denotes the corresponding TEE price vector at date t, then the successive triples (e,, st,pt) will be independent and identically distributed, and a fortiori so will the triples (e,, sit,pJ, for each trader i. Let pi denote the joint distribution of e, si, and p that is implied by mi and Q, and let ,u; denote the joint distribution of (e,, si,,p,). These two probability distributions will of course not be the same for each trader except in REE. Call r the mapping that takes fi = (u, ,..., ul) into p’ = (u; ,..., pi). (In the linear model of Section 5, this mapping is illustrated by Eq. (5.4).) To simplify the notation, let xit denote (for the time being) the triple (e,, sit,pf).

At the end of date T, trader i will have available to him the T observations xjl ,*-., xfT, which, since they are independent and identically distributed, can be summarized by their empirical distribution, say Fi. Suppose that trader i estimates ,uU~ from F, by smoothing Fi in some prescribed way, and denote the smoothed distribution by &’ = ai( (A particular family of smoothing operations will be described in a moment.) We shall say that the original I- tuple ,U is a smoothed expectations equilibrium (SEE) if for each trader i the expected value of his estimate of ,uU~ is the same as his original pi, that is to say,

E,ui” = ,ui. (6.6)

In the absence of smoothing, i.e., if the “smoothing” operation were the


identity (l), then an SEE would of course be an REE, since the expectation of Fi is &. On the other hand, for a particular (nontrivial) smoothing operation, the corresponding SEE will typically be different from the and may exist even when no REE exists.

Indeed, in their paper in this Symposium, Anderson and ~o~~enschei~ give conditions for the existence of SEE that is implementable by a mechanism like the C-mechanism described above, They do this in the context of a general-equilibrium model of exchange, and for a particular family of smoothing operations called convolutions. In addition, they use a condition that there be a random disturbance in the total excess demand (cf* the remarks in the above subsection on implementation).

The concept of convolution can be illustrated in the context of the Iinear example of Section 5. In that example, xi = (e, s,,p) is a three~dime~sio~a~ vector. Let vi be a three-dimensional random vector with a probability density function&. If xi has a probability distribution P, then define o,(P) to be the probability distribution of (xi + vi), with xi and ui independent. The mapping c‘i is called a “smoothing operator” because a,(P) will be absolutely continuous (with respect to Lebesgue measure), even if P is not. This smoothing operator also has a convenient “iinearity” property. Let P be a probability distribution of xi, and let F denote the empirical distribution function of the observations in a random sample from P, i.e., for any set A, let F(A) denote the relative frequency of sample points in A. (Note that F is a random function.) It is straightforward to show that (with suitable regularity conditions),

En,(F) = oj(B). (6 .7f

The significance of this linearity property for SEE is as follows. Recall that r is the mapping that takes .D into ,u’ (see above), and suppose that i(l” is a fixed point of the mapping crz, i.e.,

p* = crr(u”), (6.8)

where S(B) = (G@~),..., aI(u If P = xi@*), and F is the em distribution function corresponding to P, then (5.7) and (6.8) imply that EF = a,(r,(u *)) = ,U *, so that p * is an SEE. The problem of demonstrating the existence of an SEE is thus transformed into the problem of demonstrating the existence of a fixed point of the mapping or. Finally, we should point out that the concept of convolution can be made more general than the illustration given above, and in particular is not confined to smoothing by the addition of independent random variables.

The smoothed models described above correspond to the absolutely co~ti~~~~s “imperfect” market models of Section 4. The “‘imperfection” consists in the fact that, if the smoothing is nontrivial, then an SEE will not

222 JORDANANDRADNER

be an REE in the sense of Section 1. On the other hand, the “imperfection” has the virtue of removing potential discontinuities that contribute to the possible nonexistence of REE.

Note that the definition of SEE does not in fact involve any specification of sample size, and an SEE model is deterministic, not random. The discussion of the empirical distribution functions and the linearity property of the smoothing operators serves only to motivate the definition of SEE.

7. CONCLUDING REMARKS

The papers in this symposium will themselves suggest numerous problems for further research, so we close this introduction merely by mentioning two general goals for the future: (1) a general theory of mechanisms for implementing rational expectations equilibria; and (2) a general characterization of environments in which rational expectations can be learned by inference from repeated observations of market data. We would also like to call attention to a large body of experimental evidence being developed by Charles Plott and Shyam Sunder (e.g., Plott and Sunder, 1980), which should stimulate future theoretical developments.

REFERENCES

ALLEN, B. (198 la), Generic existence of completely revealing equilibria for economies with uncertainty when prices convey information, Econometrica 49, 1173-l 199.

ALLEN, B. (1981 b), Strict rational expectations equilibria with diffuseness, J. Econ. Theory, in press.

BEJA, A. (1976), “The Limited Information Efficiency of Market Processes,” Working Paper No. 43, Research Program in Finance, University of California, Berkeley (unpublished).

BORDER, K. AND J. JORDAN (1979), Expectations equilibrium with expectations conditioned on past data, J. Econ. Theory 22, 395406.

FUTIA, C. A. (1981a), Rational expectations in stationary linear models, Econometrica 49, 171-192.

FUTIA, C. A. (1981b), “Information and Macroeconomics,” Bell Laboratories Discussion Paper, 1981.

GREEN, J. R. (1973), “Information, Efficiency, and Equilibrium,” Discussion Paper 284, Harvard Institute of Economic Research, Harvard University (unpublished).

GREEN, J. R. (1977), The nonexistence of informational equilibria, Rev. Econ. Studies 44, 45 l-463.

GROSSMAN, S. J. (1981), An introduction to the theory of rational expectations under asym- metric information, Rev. Econ. Studies 48, 541-559.

HELLWIG, M. (1980) On the aggregation of information in capita1 markets, J. Econ. Theory 22, 477-498.

HIRSH, M. (1976), “Differential Topology,” Springer-Verlag, New York. JORDAN, J. (1979a), Admissible market data structures: A complete characterization, J. Econ.

Theory, in press.


JORDAN, J. (1979b), A dynamic model of expectations equilibrium, J. Econ. Theory, in press. JORDAN, J. (1979c), On the efficient markets hypothesis, Ecommetrica, in press. JORDAN, 9. AND R. RADNER (1979), “The Nonexistence of Rational Expectations

Equilibrium: A Robust Example,” Department of Economics, University of Minnesota, (unpublished).

KOBAYASHI, T. (1977) “A Convergence Theorem on Rational ’ Expectations Equilibrium With Price Information,” Working Paper No. 79, the Economic Series, Institute for Mathematical Studies in the Social Sciences, Stanford University.

PREPS. D. M. (1977) A note on “fulfilled expectations” equilibria, 1, Econ. Theory 14, 3243.

LUCAS. R. E. (1972), Expectations and the neutrality of money. J. &on. Theor.% 4. 103-124 PLQTT, C. AND S. SUNDAR (1980) “Efftciency of Experimental Security Markets With Insider

Information: An Application of Rational Expectations Models,” Social Science Working Paper No. 33 1, California Institute of Technology, J. Pol. Econ., in press.

RADNER, R. (1967), Equilibre des marches a terme at au comptant en cas d’iacertitude. Cahiers Econ. C.N.R.S. Paris 4, 35-52.

R.~~NER, R. (1978), “Rational Expectations With Differential Information,” Technical Report OW-12, Center for Research in Management Science, University of California, Serkeiey (unpub!ished).

RADNER, R. (1979), Rational expectations equilibrium: Generic existence and the information revealed by prices, Econometrica 41, 655-678.

RADNER, R. (1982), Equilibrium under uncertainty, ivr Handbook of Mathematical Economics (K. J. Arrow and M. D. Intriligator, Eds.), Vol. 2, Chap. 20, North-Holiand, Amsterdam.

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