CHAPTER ONE

Expectations Before the RationalExpectations Revolution

HINTS OR even statements about time, uncertainty, and expecta-tions are present in the writings of early economists. In 1755, Cantillon,for example, defined the farmer as ‘‘an undertaker who promises to payto the Landowner . . . a fixed sum of money . . . without assurance ofthe profit he will derive from this enterprise.’’1

He reinforced his point by saying that ‘‘the price of the Farmer’sproduce depends naturally upon . . . unforeseen circumstances, andconsequently he conducts the enterprise of his farm at an uncertainty.’’

In the same vein, Cantillon defined merchants as ‘‘undertakers[who] pay a certain price following that of the place where they pur-chase it, to resell wholesale or retail at an uncertain price’’ and statedthat competition between them is such that it is impossible for them toever know ‘‘how long their customers will buy of them.’’

As Cantillon sought to refute the already widely held view accord-ing to which low interest rates are the consequence of an abundantsupply of money and argued, instead, that what matters is the balancebetween the supply of and the demand for loanable funds, he explicitlyreferred to the profit expectations held by the borrowers:2 ‘‘ever ybodyhad become an Undertaker in the South Sea scheme and wanted toborrow money to buy Shares, expecting to make an immense profitout of which it would be easy to pay this high rate of Interest rate.’’3

But it is certainly in his chapter on the augmentation and diminutionof coins in denomination that Cantillon proved the most farsighted andperceptive as regards the role of expectations in economics. Decided bythe Sovereign, changes in the nominal denomination of species were allmonetary policy was about in those days and had actually been since

4 The Progressive Emergence of Expectations in Economic Theory

Solon and the Roman Republic. Contracts---like loans---were denomi-nated in money of account, but payments were made in species, whichwere defined by their weight: for example, in 1714, in France, 1 ecu,that is, 1 ounce of silver, was worth 5 Livres Tournois. Resources-strapped states had found very early on that an increase in the nominalvalue of coins, that is, a devaluation of the money of account (whatwe would call inflation nowadays), was a way for them to ease theirfinancial strains.

In this fascinating chapter, Cantillon discusses both the short- andlong-term effects of decreases and increases in the nominal denomina-tion of coins, depending on whether these changes are made ‘‘suddenlywithout warning’’ or phased in over a period of time. His discus-sion implicitly considers there is a diversity of expectations in themarketplace, some undertakers being less ‘‘able and accredited’’ thanothers.

However perceptive Cantillon’s insights may be, they do notconstitute per se a conscious theory of time, uncertainty, and expecta-tions. For quite a long time---that is, almost a hundred and fifty years---economists continued to mention expectations in passing, as if they didnot fully realize the central role of expectations in dynamic economicanalysis, which involves the discussion of how economic adjustmentsare brought about through time.

Until the formulation of the rational expectations hypothesis be-tween 1960 and 1975, there was no formal academic debate about thenature and role of expectations. As a result, the major contributionsof the first half of the twentieth century often give the impression thatthey were presented without any reference to one another. Despite thisapparent lack of structured debate, three trends emerged:

• Some authors---Wicksell, Knight, Keynes, Hayek, Hicks---progres-sively put expectations at the center of the stage, implicitly con-sidered them to be adaptive, while suggesting at the same timethat expectations belonged to the realm of the incalculable, namely,human psychology.

• Other authors---Fisher, Cagan, Allais---dedicated their efforts toquantify the supposedly incalculable by proposing expectationsmodels liable to be confronted with empirical data.

• Inspired by an analysis of the shortcomings and inconsistencies ofthese efforts, the rational expectations hypothesis proposed that

Expectations Before the Rational Expectations Revolution 5

economic agents form their expectations rationally, that is, by using‘‘the” one and only correct (neoclassical) model of the economy.While this hypothesis has always failed to convince a large body ofacademics and practitioners, it has become the paradigm in policy-making circles.

1.1 Expectations Are Important, but They Belong to theRealm of the Incalculable

Knut Wicksell, John Maynard Keynes, Frank Knight, and FriedrichHayek have been the first economists to stress the importance of ex-pectations explicitly, as regards cyclical fluctuations, be it in economicactivity or in financial markets.

Wicksell used the word ‘‘expectation” as early as 1898, as he madea distinction between ‘‘the natural rate of interest on capital” and ‘‘therate of interest on loans’’:4

For reasons connected with the conception of subjective value, theprobability that an entrepreneur will make a profit must always besomewhat greater than the probability that he will make a loss. Forotherwise his ‘‘moral expectation” would be negative. In many cases,however, the entrepreneurs’ gambling spirit will prevent this rulefrom applying to their behavior.5

Wicksell pushed expectations at the forefront of dynamic economicanalysis in discussing further the cumulative inflationary (or deflation-ary) effect of a discrepancy between the market rate of interest and thenatural rate of interest.6 He starts by distinguishing ‘‘present prices”and ‘‘future prices calculated at present [expected prices].” He alsomakes a clear distinction between ‘‘situations in which a rise in pricecan be foreseen with more or less certainty’’ and ‘‘the element of spec-ulation which necessarily enters into all business transactions.” In thatsecond situation, ‘‘the normal assumption is that present prices willremain constant.’’ But according to Wicksell,

if the market rate [of interest] falls below the marginal productiv-ity of waiting [the natural rate of interest], entrepreneurs, even withcurrent prices as the foundation of their calculation of future prices,

6 The Progressive Emergence of Expectations in Economic Theory

will be able to pay a somewhat higher price. In this way, the presentprice level will be raised indirectly and therefore the future price levelalso.7

However concise this very last sentence may be, it is an expectationmodel, albeit a very crude one. In Wicksell’s model, expectations aredetermined by the outcome observed in the previous period, and achange in current prices is assumed to change expected prices in thesame direction and in the same proportion.8 Wicksell’s central propo-sition of cumulative instability is entirely dependent on his assumptionthat people expect future prices to rise at least as fast as current prices.

Nowadays, Knight is essentially remembered for having forcefullyshown that uncertainty is the source of entrepreneurial profit inan economy where entrepreneurs compete for acquiring productiveservices.9 In chapter 8 of Risk, Uncertainty and Profit, there is, how-ever, much more than this fundamental contribution. This chapter ispresented by Knight as an ‘‘inquir y into the nature and function ofknowledge.” In a very Socratic tone, Knight claims that ‘‘we mustknow ourselves as well as the world.” Knight reinforces his point witha provocative statement: ‘‘In spite of rash statements by over-ardentdevotees of the new science of ‘behavior,’ it is preposterous to supposethat it will ever supersede psychology or the theory of knowledge.’’

However, says Knight, and this is the whole problem, ‘‘the universemay not be ultimately knowable.” Yet, we have to act and to make de-cisions. The question is then: on the basis of which knowledge? Fore-shadowing Keynes by a few years, Knight is keen to stress how limitedor fragile our knowledge is and how little we know about the way weacquire it:

At present we are concerned only to emphasize the fact that knowl-edge is in a sense variable in degree and that the practical problemmay relate to the degree of knowledge rather than to its presence orabsence in toto . . . The essence of the situation is action according toopinion, of greater or less foundation and value, neither entire igno-rance nor complete and perfect information, but partial knowledge.

Of course, successful adaptations on our part to a ‘‘world of change”require some forward-looking consciousness. But these adjustmentstake time, and this is precisely where the difficulties lie:

Expectations Before the Rational Expectations Revolution 7

An explanation of the readjustment necessarily runs in terms of stim-ulus and reaction, in this temporal order. Yet in our own experiencewe know that we do not react to past stimulus, but to the ‘‘image”of a future state of affairs . . . However successful mechanistic sci-ence may be in explaining the reaction in terms of a past cause, itwill still be irresistibly convenient for common sense to think of it asprompted by a future situation present to consciousness. The role ofconsciousness is to give the organism this ‘‘knowledge” of the future.

Needless to say, it is not only convenient, but also reassuring for us,confronted as we are with uncertainty, to think of our response to paststimulus as if it were a forward-looking consciousness. But how doesconsciousness work? Frustration is palatable in Knight’s answer:

The ordinary decisions of life are made on the basis of ‘‘estimates”of a crude and superficial character. It is only in very special casesthat anything like a mathematical study can be made . . . The mentaloperations by which ordinary practical decisions are made are veryobscure, and it is a matter of surprise that neither logicians nor psy-chologists have shown much interest in them. Perhaps . . . it is be-cause there is really very little to say about the subject . . . The reallogic or psychology of ordinary conduct is rather a neglected branchof inquiry. The processes of intuition or judgment, being uncon-scious, are inaccessible to study.

Despite this avowed frustration, Knight cannot conceal a clear con-viction:

We perceive the world before we react to it, and we react not towhat we perceived, but always to what we infer . . . The universalform of conscious behavior is thus action designed to change a futuresituation inferred from a present one . . . We know the absent fromthe present, the future from the now, by assuming that connectionsor associations among phenomena which have been valid will be so;we judge the future by the past . . . Prophecy seems to be a gooddeal like memory, on which it is based. There is very little that wecan tell about the operation of memory, very little ‘‘technique” . . .

In the main, it seems that we ‘‘infer” largely from our experience ofthe past as a whole.

8 The Progressive Emergence of Expectations in Economic Theory

Knight’s contention is actually quite strong: if we are to understandhow ‘‘expectations” are formed and to explain economic behavior,then we have to describe the operations of memory, we have to under-stand how memory establishes a link between the past and the future,and we have to evaluate the extent to which the past is present in ourcurrent decisions. This sounds more as a research program for logiciansand psychologists than for economists.

Keynes, too, suggested that expectations are rooted in memory andtherefore adaptive, but---like Knight---he did not use this very word.Keynes’s insight is focused on uncertainty and the influence of short-period changes on what he called the state of long-term expectations.In a well-known chapter of The General Theory, Keynes wrote thefollowing:10

The considerations upon which expectations of prospective yields arebased are partly future events which can only be forecasted with moreor less confidence.

It would be foolish, in forming our expectations, to attach greatweight to matters which are very uncertain (by very ‘‘uncertain” I donot mean the same thing as very improbable). It is reasonable, there-fore, to be guided to a considerable degree by the facts about whichwe feel somewhat confident. For this reason the facts of the existingsituation enter, in a sense disproportionately, into the formation ofour long-term expectations; our usual practice being to take the exist-ing situation and to project it into the future, modified only to the ex-tent that we have more or less definite reasons for expecting a change.

In practice we have tacitly agreed, as a rule, to fall back on what is,in truth, a convention. The essence of this convention lies in assum-ing that the existing state of affairs will continue indefinitely, exceptin so far as we have specific reasons to expect a change. This doesnot mean that we really believe that the existing state of affairs willcontinue indefinitely. We know from extensive experience that this ismost unlikely.

We are assuming, in effect, that the existing market valuation,however arrived at, is uniquely correct in relation to our existingknowledge of the facts which will influence the yield of the invest-ment, and that it will only change in proportion to changes in thisknowledge.

For an investor can legitimately encourage himself with the ideathat the only risk he runs is that of a genuine change in the news

Expectations Before the Rational Expectations Revolution 9

over the near future, as to the likelihood of which he can attempt toform his own judgement and which is unlikely to be very large. Forassuming that the convention holds good, it is only these changeswhich can affect the value of his investment.

However, most of these persons are, in fact, largely concerned,not with making superior long-term forecasts of the probable yieldof an investment over its whole life, but with foreseeing changes inthe conventional basis of valuation a short-time ahead of the generalpublic.

Appealing ideas, brilliant exposition, but not very helpful, as Keynesstopped short of defining precisely what he meant by the existing stateof affairs and short, too, of proposing any operational model of long-term expectations.11 Keynes’s essential intuition is the inherent insta-bility of ‘‘long-term” expectations and their sensitivity to changes inthe existing situation. But the concept of elasticity of expectations andits quantification are both absent from Keynes’s writings.

Keynes’s chapter 12 has become an inevitable and revered referencein any contemporary discussion about the role of expectations in eco-nomics. Yet, it is not without criticisms, some of them discretely ironic,that economists concerned with dynamic economic analysis promptlygreeted Keynes’s ‘‘novel” emphasis on expectations. These criticismsare still worth reading, for they highlight the issue of how economicagents learn, an issue that we shall encounter later in this book. In hisBusiness Cycles, Schumpeter does not mention Keynes explicitly, butclearly he has Keynes in mind with the following words:12

We have admitted any expectations and we have taken them as given.. . . We have ourselves to blame if with such tremendous generality wedo not get any results. . . . We have emptied the schema of everythingthat matters. In other words, if we discontinue the practice of treatingexpectations as if they were ultimate data, and treat them as what theyare---variable which it is our task to explain---properly linking themwith the business situations that give rise to them, we shall succeedin restricting expectations to those which we actually observe andnot only reduce their influence to its proper proportions but alsounderstand how the course of events moulds them and at certaintimes so turns them as to make them work toward equilibrium.

. . . Another such deus ex machina . . . is ‘‘anticipation” . . . Un-less we know why people expect what they expect, any argument is

10 The Progressive Emergence of Expectations in Economic Theory

completely valueless which appeals to them as causae efficientes. Suchappeals enter into the class of pseudo-explanations.

Just one year after the General Theory, Haberler had already madethe same point, but much more forcefully and directly:13

In recent years, it has become fashionable to lay stress on the elementof expectations. It should not, however, be forgotten that even thetheories of authors who do not usually refer explicitly to expectationsand anticipations can, and should, be interpreted in terms of expec-tations, as the authors in question [Lindahl, Myrdal, Ohlin, Hicks,Hayek, Morgenstern] are themselves often well aware.

There is . . . one feature about Mr. Keynes’ system, which hasgiven the impression to many readers that the General Theory . . .

is a dynamic theory---namely, the fact that . . . it runs in terms of ex-pectations. [But] if we confine ourselves to saying that it is not actualprices . . . but expected prices . . . which induce an entrepreneur toproduce and to invest, we do not say very much, unless we give somehint as to how these expectations are determined.

A theory which takes expectations as given at any point of time,and does not say anything on how they grow out of past experiences,is of very little value; for such a theory would still be static, and it isalmost impossible to determine expectations as such [expectationsmay almost be called ‘‘nonoperational concepts”]. Only if it is possi-ble to give some hypotheses about how expectations are formed onthe basis of past experience can a really useful and verifiable theorybe evolved. And such a theory is evidently dynamic . . . for it linksthe past with the present. [From a strictly logical point of view, thepsychological link between the past and the present consisting of ex-pectations may be dropped and the theory stated in terms of a directrelationship between observable phenomena at different points intime. Psychologically it may, however, be useful to retain the word‘‘expectations,” or a similar concept, as a link, because it reminds usof the fact that those dynamic ‘‘laws” (relationships) are nothing buthypotheses.]

Not content with this, Haberler rams home his critique of Keynesby quoting Lundberg:14

Expectations Before the Rational Expectations Revolution 11

The introduction of expectations can, in a way, be said to mark thestepping-stone to dynamic analysis, because they express the connec-tion linking present plans and activities with future events. However,economists have indulged in too much purely formal exercise withthis term . . . It is sensible to link actions with expectations, only ifthe latter can be explained on the basis of past and present economicevents. Total lack of correlation here would mean the complete liq-uidation of economics as a science.

In the year when the General Theory was published, 1936, Hayekmade a contribution that reads very much as a research program onexpectations.15 If, Hayek said, economic theory is to be valued for itsability to explain the real world rather than for its internal (logical)consistency, then dynamic analysis, that part of economic theory whichpurports to account for the workings of a monetary economy throughtime, has no choice but to make explicit assumptions about foresightsand anticipations (which are only part of the broader problem of theacquisition of knowledge). The logical conclusions reached in the con-text of dynamic analysis depend in fact on the assumptions made aboutforesight and anticipations. These assumptions are the only part of eco-nomic theory which can be proven wrong by observed facts: they makeeconomics falsifiable. No one before Hayek had so clearly stated thestrategic importance of the theory of expectations.

Hayek goes on by distinguishing two kinds of economic data: objec-tive data, as given ex post to the ‘‘observing economist,’’ and subjectivedata. Economic agents draw their individual plans on the basis of sub-jective data, that is, what they have in mind. Knight would say that ‘‘theessence of the situation [an uncertain one to be sure] is action accord-ing to opinion.” With this, Hayek defines a dynamic equilibrium as asituation where individual plans are compatible and outcomes (objec-tive data) are in line with expectations (subjective data).

The next decisive step is then to understand the interaction be-tween objective and subjective data, for there is no a priori reason topostulate that they should converge. There are actually two types ofchange in the objective data: those which are ‘‘as expected” (whichdoes not mean the objective data should be constant through time) andthose which are unexpected and which cause a change in the subjectivedata.

12 The Progressive Emergence of Expectations in Economic Theory

For Hayek, it is all too easy to assume that every economic agentknows everything instantly. Experience is the only source of knowledge.There is no such thing as a timeless learning process:

We must explain by what process people will acquire the necessaryknowledge. . . . it is these apparently subsidiary hypotheses or as-sumptions that people do learn from experience, and about how theyacquire knowledge, which constitute the empirical content of ourpropositions about what happens in the real world. . . . And theseassumptions must necessarily run in terms of assertions about causalconnections, about how experience creates knowledge.

This being said, Hayek acknowledges that ‘‘we are pretty much inthe dark about the nature of the process by which individual knowl-edge is changed.” One very last word on Hayek. All through Economicsand Knowledge, he emphasizes the need for every agent to form ex-pectations about other people’s actions. While Hayek does not explic-itly refer to policymakers’ actions, nothing in his words precludes fromthinking that he also had them in mind. Should we therefore see inHayek a precursor of the rational expectations hypothesis, which, as weshall soon see, claims that people know the probabilities of the variouscourses liable to be taken by policymakers and factor in these probablepolicy responses in their own behavior, thus making policy ineffective?Nothing seems further away from the thinking of Hayek, who nevermisses an opportunity to remind us that any learning process takes time:‘‘But, as all other people will change their decisions as they gain experi-ence about the external facts and about other people’s actions, there isno reason why theses processes of successive changes should ever cometo an end.’’

Without saying it, it is clearly adaptive expectations that Hayek hasin mind. Such is also the case of Hicks.

In Value and Capital, John Hicks introduced the concept of elas-ticity of expectations: ‘‘I define the elasticity of a particular person’sexpectations of the price of commodity X as the ratio of the propor-tional rise in expected future prices of X to the proportional rise in itscurrent price.’’

Hicks suggested that in most circumstances the elasticity of expec-tations should be anywhere between 0 and 1, which he regarded asthe two ‘‘pivotal cases.” And he went on to analyze the impact of the

Expectations Before the Rational Expectations Revolution 13

elasticity of price expectations on the stability of equilibrium. When ex-pectations are inelastic (elasticity = 0), ‘‘all changes in current pricesare regarded as being temporary changes, any change in current priceswill induce very large substitution effects, . . . subsitution over time willbe strongly stabilizing.’’

When expectations are elastic (elasticity equal to or greater than 1),a rise in prices is interpreted as an ‘‘indication that [prices] will go onchanging in the same direction,’’ and it does foster excess demand,which in turn feeds back into higher prices; if the rate of interest remainsconstant, such a system is prone to cumulative instability. However, asHicks did not formulate any quantitative theory of expectations for-mation, his perceptive analysis of the implications of various elasticitiesof price expectations did not lend itself to some form of confrontationwith observed data.

In particular, Hicks does not seem to have contemplated the pos-sibility that price expectations for a given commodity might be ratherinelastic at some times and very elastic at other times. As a consequence,he did not give any hint as to what might cause the elasticity of expecta-tions to be variable over time. But the mere introduction of the conceptof elasticity of expectations was nevertheless a great step forward, for itclearly showed that different regimes of expectations were theoreticallypossible. This is clearly revealed by the following sentence:

The elasticity of expectations will be greater than unity, if a changein current prices makes people feel that they can recognize a trend,so that they try to extrapolate; it will be negative if they make theopposite kind of guess, interpreting the change as the culminatingpoint of a fluctuation.

The different regimes of expectations are not given their currentnames---adaptive, extrapolative, or rational---but the idea is clearlypresent.

1.2 Early Attempts to Quantify Expectations

The distinctive feature of the approach presented by Fisher as early as1896 is its focus on empirical data, that is, the observed relationshipbetween price movements and interest rate fluctuations.16 From thisempirical starting point, Fisher was led to introduce key concepts:

14 The Progressive Emergence of Expectations in Economic Theory

• The distinction between foreseen and unforeseen changes in thevalue of money.

• The distinction between imperfection of forecast, which causeswealth transfer between creditors and debtors, on one hand and,on the other hand, inequality of foresight, which causes cyclicalfluctuations.17

Fisher’s crucial empirical observation is twofold. First, it takes timefor interest rates to adjust to price movements. Second, this adjustmentis generally inadequate (partial). According to Fisher, the ‘‘borrowingclass” (or the ‘‘managers of capital”) tends to have better foresightsthan lenders. This is so because these managers are in a position wherethey are the first to know of pressures on prices. Hence, they have anadvantage over lenders when it comes to assessing the return on capitalemployed. In short, rising prices tend to inflate profits. Thanks to theseelevated profits, entrepreneurs can afford to pay higher interest rates.But, as the ‘‘ignorant lenders” do not demand an immediate upwardadjustment in interest rates, the borrowing class is enticed to developexpectations of even higher profits and to increase the volume of in-vestment.

Translated into the current terminology of economics, Fisher’s in-sights are surprisingly modern. What Fisher described as early as 1896 isa model of cyclical fluctuations in which a cumulative process dependsupon

• The heterogeneity of agents.• Information asymmetry.• A blend of adaptive and nonadaptive expectations.

More than thirty years later, in The Theory of Interest, Irving Fishercame back on the same issue, but with additional insights.18 As in hisearlier work, Fisher’s starting point is an empirical observation: the‘‘delayed, but accumulated, adjustment” of interest rates to prices.Fisher did not deny that economic agents are making their best effortsto ‘‘look ahead” as he studied the relation between prices and interestrates, but he also recognized that

if perfect foresight existed, continuously rising prices would be as-sociated not with a continuously rising rate of interest but with a

Expectations Before the Rational Expectations Revolution 15

continuing high rate of interest . . . , and a constant price level wouldbe associated with a constant rate of interest---assuming in each case,that other influences than price change remained the same.

If this perfect foresight correctly described reality, nominal interestrates should be more volatile than actually observed, with ‘‘real interest. . . comparatively steady.’’ This is exactly the reverse of what we canactually see.

This latter observation led Fisher to successfully test the assumptionthat the influence of past price changes is distributed in time, ‘‘as, infact, must evidently be true of any influence.’’ In other words, ‘‘theassumption that a change in prices occurring during one year exhaustsits influence upon interest rates in the same year or in another singleyear is shown to be quite wrong.’’

Fisher added the following crucial remarks: ‘‘when we make themuch more reasonable supposition that price changes . . . manifest theirinfluence with diminishing intensity, over long periods which vary inlength with the conditions, we find a very significant relationship, es-pecially . . . when prices were subject to violent fluctuations.’’

Nobody before Fisher had ever formulated this assumption of thediminishing intensity of the influence of the past in current affairs.Nobody before Fisher had suggested that the speed of this diffusionprocess could vary through time, and be context-dependent. Findingthe law of diminishing intensity of the influence of the past was tobecome the focus of much of subsequent research.

Fisher used a very crude method to distribute the effect of past infla-tion rates over time: instead of using constant weights---as would havebeen the case with a geometric average of past inflation rates computedover a certain arbitrary number of years---he assigned to each of themcoefficients that decline linearly with the lag of actual inflation (i.e., aweighted average of current and past values with weights that declinelinearly with time into the past until the weight reaches zero).

Cagan and Allais refined this lag distribution in 1953--54.19 Al-though they worked independently of each other, used different nota-tions, and gave different interpretations of their respective work, theyboth used the same mathematical formulation, namely, exponential av-erages, to model inflation expectations.20 In an exponential average,the weighting coefficients decline exponentially at a constant rate withthe lag of past inflation rates.

16 The Progressive Emergence of Expectations in Economic Theory

There are different ways of defining the exponential average y of avariable x. The simplest one is the following recurrence relationship indiscrete time:

yn = yn−1 + k(xn − yn−1) with 0 < k < 1 (1.1)

which, after rearranging the terms in the right member, is equivalentto

yn = kxn + (1 − k)yn−1 (1.2)

By recurrence, and with y0 = kx0, one can easily demonstrate thatrelationship 1.2 is equivalent to

yn = k[xn + (1 − k)xn−1 + (1 − k)2xn−2 + · · · + (1 − k)nx0] (1.3)

or, in a more concise form,

yn = kn∑

i=0

(1 − k)i xn−i (1.4)

The sequence of exponentially declining coefficients 1,1−k, (1−k)2,(1−k)3, . . . , (1−k)n in relationship 1.3 is a geometric sequence having1 − k as common ratio and 1/k for sum when n → +∞, since 0 <

1 − k < 1. Hence, the weighted sum of coefficients

k + (1 − k) + (1 − k)2 + · · · + (1 − k)n

1/k

= k[k + (1 − k) + (1 − k)2 + · · · + (1 − k)n] (1.5)

is equal to 1, which justifies calling y an average.As exponential averages foreshadowed Allais’s HRL formulation,

chapters 2 and 4 of this book will deal with them in great detail. There-fore, at this stage, there is no need to describe this mathematical toolfurther.

Adaptive expectations came under fire not so much because theyfailed to deliver good econometric results, but rather because they ap-peared logically flawed.