ECONOMETRICA IOURV4I DF TIr TCNOMTERIr 54CItTY Rational Expectations and the Theory of Price Movements Author(s): John F. Muth Source: Econometrica, Vol 29, No. 3(JuL, 1961), pp 315-335 Published by: The Econometric Society StableUrl:http://www.jstor.org/stable/1909635 Accessed:197097200806:53 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jspJstOr'sTermsandConditionsofUseprovidesinpartthatunless may use content in the JSTOR archive only for your personal, non-commercial use al or multiple copies of articles, and you you have obtained prior permission, you may not download an entire issue of a jour Please contact the publisher regarding any further use of this work, Publisher contact information may be obtained at http://www.jstor.org/action/showpublisher?publishercode=ecoNosoc Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmIssion JStOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the cholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about jSTOR, please contact support(@jstor. org 图小 ittp://www.jstor.org
Rational Expectations and the Theory of Price Movements Author(s): John F. Muth Source: Econometrica, Vol. 29, No. 3 (Jul., 1961), pp. 315-335 Published by: The Econometric Society Stable URL: http://www.jstor.org/stable/1909635 Accessed: 19/09/2008 06:53 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=econosoc. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org. The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica. http://www.jstor.org
Econometrica, Vol, 29, No. 3 (July 1961) RATIONAL EXPECTATIONS AND THE THEORY OF PRICE MOVEMENTS BY JOHN F. MUTH In order to explain fairly simply how expectations are formed, we advance the hypothesis that they are essentially the same as the predictions of the relevant economic theory. In particular, the hypothesis asserts that the economy generally does not waste information, and that expectations depend specifically on the structure of the entire system. Methods of analysis, which hypothesis is illustrated by introducing commodity speculation into the 1. INTRODUCTION THAT EXPECTATIONS of economic variables may be subject to error has for some time, been recognized as an important part of most explanations of changes in the level of business activity. The"ex ante"'analysis of the tockholm School--although it has created its fair share of confusion--is a highly suggestive approach to short-run problems. It has undoubtedly been a major motivation for studies of business expectations and intentions data As a systematic theory of fluctuations in markets or in the economy, he approach is limited, however, because it does not include an explanation of the way expectations are formed. To make dynamic economic models mplete, various expectations formulas have been used. There is, however, little evidence to suggest that the presumed relations bear a resemblance to the way the economy works. 2 What kind of information is used and how it is put together to an estimate of future conditions is important to understand becaus te me character of dynamic processes is typically very sensitive to the way ex- pectations are influenced by the actual course of events. Furthermore it is often necessary to make sensible predictions about the way expectations would change when either the amount of available information or the struc 1 Research undertaken for the project, Planning and Control of Industrial operations under contract with the Office of Naval Research. Contract N-onr-760-(01), Project NR047011. Reproduction of this paper in whole or in part is permitted for any purpose of the united states Government An earlier version of this paper was presented at the winter Meeting of the E nometric Society, Washington, D. C, December 30, 1959 I am indebted to Z, Griliches, A. G. Hart, M. H. Miller, F. Modigliani, M. Ne and H. White for their comments 2 This comment also applies to dynamic theories in which expectations do not explicitly appear. See, for example, Arrow, Block, and Hurwicz [3, 4
Econometrica, Vol. 29, No. 3 (July 1961) RATIONAL EXPECTATIONS AND THE THEORY OF PRICE MOVEMENTS1 BY JOHN F. MUTH In order to explain fairly simply how expectations are formed, we advance the hypothesis that they are essentially the same as the predictions of the relevant economic theory. In particular, the hypothesis asserts that the economy generally does not waste information, and that expectations depend specifically on the structure of the entire system. Methods of analysis, which are appropriate under special conditions, are described in the context of an isolated market with a fixed production lag. The interpretative value of the hypothesis is illustrated by introducing commodity speculation into the system. 1. INTRODUCTION THAT EXPECTATIONS ofeconomic variables may be subject to error has, for some time, been recognized as an important part of most explanations of changes in the level of business activity. The "ex ante" analysis of the Stockholm School-although it has created its fair share of confusion-is a highly suggestive approach to short-run problems. It has undoubtedly been a major motivation for studies of business expectations and intentions data. As a systematic theory of fluctuations in markets or in the economy, the approach is limited, however, because it does not include an explanation of the way expectations are formed. To make dynamic economic models complete, various expectations formulas have been used. There is, however, little evidence to suggest that the presumed relations bear a resemblance to the way the economy works.2 What kind of information is used and how it is put together to frame an estimate of future conditions is important to understand because the character of dynamic processes is typically very sensitive to the way expectations are influenced by the actual course of events. Furthermore, it is often necessary to make sensible predictions about the way expectations would change when either the amount of available information or the struc- 1 Research undertaken for the project, Planning and Control of Industrial Operations, under contract with the Office of Naval Research. Contract N-onr-760-(01), Project NR 04701 1. Reproduction of this paper in whole or in part is permitted for any purpose of the United States Government. An earlier version of this paper was presented at the Winter Meeting of the Econometric Society, Washington, D.C., December 30, 1959. I am indebted to Z. Griliches, A. G. Hart, M. H. Miller, F. Modigliani, M. Nerlove, and H. White for their comments. 2 This comment also applies to dynamic theories in which expectations do not explicitly appear. See, for example, Arrow, Block, and Hurwicz [3, 4]. 315
316 JoHN F. MUTH ture of the system is changed. (This point is similar to the reason we are curious about demand functions, consumption functions, and the like instead of only the reduced form"predictors"in a simultaneous equatio system. )The area is important from a statistical standpoint as well, because parameter estimates are likely to be seriously biased towards zero if the wrong variable is used as the expectation The objective of this paper is to outline a theory of expectations and to show that the implications are-as a first approximation--consistent with the relevant data THE“ RATION AL EXPECTATIONS’ HYPOTHESIS Two major conclusions from studies of expectations data are the following 1. Averages of expectations in an industry are more accurate than naive models and as accurate as elaborate equation systems, although there are considerable cross-sectional differences of opinion 2. Reported expectations generally underestimate the extent of changes that actually take place In order to explain these phenomena, I should like to suggest that expectations, since they are informed predictions of future events, are essentially the same as the predictions of the relevant economic theory. At the risk of confusing this purely descriptive hypothesis with a pronounce- ment as to what firms ought to do, we call such expectations" rational. It is sometimes argued that the assumption of rationality in economics leads to theories inconsistent with, or inadequate to explain, observed phenomena, especially changes over time(e.g, Simon [29]). Our hypothesis is based on exactly the opposite point of view: that dynamic economic models do not assume enough rationality The hypothesis can be rephrased a little more precisely as follows that expectations of firms (or, more generally, the subjective probability distribution of outcomes) tend to be distributed, for the same information set, about the prediction of the theory or the"objective probability distributions of outcomes The hypothesis asserts three things:(1)Information is scarce, and the conomic system generally does not waste it.(2) The way expectations are formed depends specifically on the structure of the relevant system describin the economy. (3)A public prediction, 'in the sense of Grunberg and modi gliani [14], will have no substantial effect on the operation of the economic system(unless it is based on inside information). This is not quite the same thing as stating that the marginal revenue product of economics is zero 3 We show in Section 5 that the hypothesis is consistent with these two phenomena
316 JOHN F. MUTH ture of the system is changed. (This point is similar to the reason we are curious about demand functions, consumption functions, and the like, instead of only the reduced form "predictors" in a simultaneous equation system.) The area is important from a statistical standpoint as well, because parameter estimates are likely to be seriously biased towards zero if the wrong variable is used as the expectation. The objective of this paper is to outline a theory of expectations and to show that the implications are-as a first approximation-consistent with the relevant data. 2. THE "RATIONAL EXPECTATIONS" HYPOTHESIS Two major conclusions from studies of expectations data are the following: 1. Averages of expectations in an industry are more accurate than naive models and as accurate as elaborate equation systems, although there are considerable cross-sectional differences of opinion. 2. Reported expectations generally underestimate the extent of changes that actually take place. In order to explain these phenomena, I should like to suggest that expectations, since they are informed predictions of future events, are essentially the same as the predictions of the relevant economic theory.3 At the risk of confusing this purely descriptive hypothesis with a pronouncement as to what firms ought to do, we call such expectations "rational." It is sometimes argued that the assumption of rationality in economics leads to theories inconsistent with, or inadequate to explain, observed phenomena, especially changes over time (e.g., Simon [29]). Our hypothesis is based on exactly the opposite point of view: that dynamic economic models do not assume enough rationality. The hypothesis can be rephrased a little more precisely as follows: that expectations of firms (or, more generally, the subjective probability distribution of outcomes) tend to be distributed, for the same information set, about the prediction of the theory (or the "objective" probability distributions of outcomes). The hypothesis asserts three things: (1) Information is scarce, and the economic system generally does not waste it. (2) The way expectations are formed depends specifically on the structure of the relevant system describing the economy. (3) A "public prediction," in the sense of Grunberg and Modigliani [14], will have no substantial effect on the operation of the economic system (unless it is based on inside information). This is not quite the same thing as stating that the marginal revenue product of economics is zero, 3 We show in Section 5 that the hypothesis is consistent with these two phenomena
RATIONAL EXPECTATIONS 317 because expectations of a single firm may still be subject to greater error than the theory It does not assert that the scratch work of entrepreneurs resembles the trepreneurs are perfect or that their expectations are all the samg ns of en- system of equations in any way; nor does it state that predictic For purposes of analysis, we shall use a specialized form of the hypothesis In particular, we assume 1. The random disturbances are normally distributed 2. Certainty equivalents exist for the variables to be predicte 3. The equations of the system, including the expectations formulas, are These assumptions are not quite so strong as may appear at first because any one of them virtually implies the other two 3. PRICE FLUCTUATIONS IN AN ISOLATED MARKET We can best explain what the hypothesis is all about by starting the analysis in a rather simple setting: short-period price variations in an isolated market with a fixed production lag of a commodity which cannot be stored. 5 The market equations take the form Ct=-Bpe (Demand) Pt= ypi +u Pe Market equilibrium where: Pt represents the number of units produced in a period lasting as long as the production lag Ct is the amount consumed pt is the market price in the tth period, is the market price expected to prevail during the tth period on the basis of information available through the(t-1)'st period, is an error term--representing, say, variations in yields due to weather Au the variables used are deviations from equilibrium values 4 As long as the variates have a finite variance, a linear regression function exists and only if the variates are normally distributed. (See Allen [2] and Ferguson [12]. The certainty-equivalence property follows from the linearity of the derivative of the appropriate quadratic profit or utility function. (See Simon [28] and Theil [32].) 5 It is possible to allow both short- and long-run supply relations on the basis of dynamic costs. See Holt et al. [17, esp. Chapters 2-4, 19]). More difficult are the supply effects of changes in the number of firms. The relevance of the cost effects has bee emphasized by Buchanan [7] and Akerman [1]. To include them at this point would however, take us away from the main objective of the paper
RATIONAL EXPECTATIONS 317 because expectations of a single firm may still be subject to greater error than the theory. It does not assert that the scratch work of entrepreneurs resembles the system of equations in any way; nor does it state that predictions of entrepreneurs are perfect or that their expectations are all the same. For purposes of analysis, we shall use a specialized form of the hypothesis. In particular, we assume: 1. The random disturbances are normally distributed. 2. Certainty equivalents exist for the variables to be predicted. 3. The equations of the system, including the expectations formulas, are linear. These assumptions are not quite so strong as may appear at first because any one of them virtually implies the other two.4 3. PRICE FLUCTUATIONS IN AN ISOLATED MARKET We can best explain what the hypothesis is all about by starting the analysis in a rather simple setting: short-period price variations in an isolated market with a fixed production lag of a commodity which cannot be stored.5 The market equations take the form Ct -AfiPt (Demand), (3. 1) P=t -yIP + ut, (Supply), Pt Ct (Market equilibrium), where: Pt represents the number of units produced in a period lasting as long as the production lag, Ct is the amount consumed, Pt is the market price in the tth period, pe is the market price expected to prevail during the tth period on the basis of information available through the (t -1)'st period, ut is an error term-representing, say, variations in yields due to weather. All the variables used are deviations from equilibrisui3 values. 4 As long as the variates have a finite variance, a linear regression function exists if and only if the variates are normally distributed. (See Allen [2] and Ferguson [12].) The certainty-equivalence property follows from the linearity of the derivative of the appropriate quadratic profit or utility function. (See Simon [28] and Theil [32].) 5 It is possible to allow both short- and long-run supply relations on the basis of dynamic costs. (See Holt et al. [17, esp. Chapters 2-4, 19]). More difficult are the supply effects of changes in the number of firms. The relevance of the cost effects has been emphasized by Buchanan [7] and Akerman [1]. To include them at this point would, however, take us away from the main objective of the paper
318 JoHN F.MUT直 The quantity variables may be eliminated from(3. 1)to give (32) =-h The error term is unknown at the time the production decisions are made but it is known-and relevant--at the time the commodity is purchased in th The prediction of the model is found by replacing the error term by its expected value, conditional on past events. If the errors have no serial correlation and Ent=0, we obtain (33) If the prediction of the theory were substantially better than the ex- pectations of the firms, then there would be opportunities for the"insider to profit from the knowledge--by inventory speculation if possible, by operating a firm, or by selling a price forecasting service to the firms. The profit opportunities would no longer exist if the aggregate expectation of the firms is the same as the prediction of the theor (34) Ept= pi Referring to (3. 3)we see that if y/B =-1 the rationality assumption (3. 4) implies that Pt 0, or that the expected price equals the equilibrium price As long as the disturbances occur only in the supply function, price and quantity movements from one period to the next would be entirely along the The problem we have been discussing so far is of little empirical interest because the shocks were assumed to be completely unpredictable. For most markets it is desirable to allow for income effects in demand and alternative costs in supply, with the assumption that part of the shock variable may be predicted on the basis of prior information. By retracing our steps from (3.2), we see that the expected price would be (35) E B+y ervable, then the conditional expected value or its egression estimate may be found directly. If the shock is not observable it must be estimated from the past history of variables that can be measured Expectations weith Serially Correlated Disturbances. We shall write the u's as a linear combination of the past history of normally and independentl
318 JOHN F. MUTH The quantity variables may be eliminated from (3.1) to give (3.2) Pt== - -et The error term is unknown at the time the production decisions are made, but it is known-and relevant-at the time the commodity is purchased in the market. The prediction of the model is found by replacing the error term by its expected value, conditional on past events. If the errors have no serial correlation and Eut = 0. we obtain (3.3) Ept AfptA If the prediction of the theory were substantially better than the expectations of the firms, then there would be opportunities for the "insider" to profit from the knowledge-by inventory speculation if possible, by operating a firm, or by selling a price forecasting service to the firms. The profit opportunities would no longer exist if the aggregate expectation of the firms is the same as the prediction of the theory: (3.4) EPt=Pt . Referring to (3.3) we see that if y//3 - 1 the rationality assumption (3.4) implies that =0, or that the expected price equals the equilibrium price. As long as the disturbances occur only in the supply function, price and quantity movements from one period to the next would be entirely along the demand curve. The problem we have been discussing so far is of little empirical interest, because the shocks were assumed to be completely unpredictable. For most markets it is desirable to allow for income effects in demand and alternative costs in supply, with the assumption that part of the shock variable may be predicted on the basis of prior information. By retracing our steps from (3.2), we see that the expected price would be (3.5) Pt e Eut . If the shock is observable, then the conditional expected value or its regression estimate may be found directly. If the shock is not observable, it must be estimated from the past history of variables that can be measured. Expectations zwith Serially Correlated Distuyrbances. We shall write the u's as a linear combination of the past history of normally and independently
RATIONAL EXPECTATIONS 319 distributed random variables et with zero mean and variance o2 Ee=0, a2 if i=i Ee=10近i≠ Any desired correlogram in the us may be obtained by an appropriate hoice of the ghts wet The price will be a linear function of the same independent disturbances thus (37) 巾=∑W d-0 2E-i The expected price given only information through the (t-1)'st period has the same form as that in(3.7), with the exception that et is replaced by its expected value(which is zero). We therefore have (38) p=WEet+∑Wtet=∑We- If, in general, we let pu, L be the price expected in period t+L on the basis of information available through the tth period, the formula becomes (3.9) p ∑Wtet-t Substituting for the price and the expected price into(3. 1), which reflect the market equilibrium conditions, we obtain (310) Woet a"e=1 wet Et-t Equation(3. 10) is an identity in the es; that is, it must hold whatever values of e, happen to occur. Therefore, the coefficients of the correspond The weights Wi are therefore the followin (3.11a) W 8+y (=1,2,3,…) and price expectations functions in terms of the past history of independent shocks. The problem remains of writing the results in terms of the history of observable variables. We wish to find a relation of the form (312) p=∑Vpt
RATIONAL EXPECTATIONS 319 distributed random variables 8t with zero mean and variance a2: (3.6) co~0 r2 if ij (3.6) 6t =z Wi -Et-i, E8j = 0, E8j = (o ifi#j Any desired correlogram in the u's may be obtained by an appropriate choice of the weights wi. The price will be a linear function of the same independent disturbances; thus 00 (3.7) it- E wiet-iE i=0 The expected price given only information through the (t -1)'st period has the same form as that in (3.7), with the exception that 8t is replaced by its expected value (which is zero). We therefore have (3.800 pe O8 O0 (3.8) pt W0E6t + Wi t-i Wiet-i;E i=l1= If, in general, we let Pt,L be the price expected in period t +L on the basis of information available through the tth period, the formula becomes 00 (3.9) fit-L,L -E Wist-iE i=L Substituting for the price and the expected price into (3.1), which reflect the market equilibrium conditions, we obtain (3. 10) Wo E-t + 1 + )zwi Et-{ = - zSfet-z . A i=1 i{=0 Equation (3.10) is an identity in the e's; that is, it must hold whatever values of ej happen to occur. Therefore, the coefficients of the corresponding ej in the equation must be equal. The weights Wi are therefore the following: (3.1 la) p ze , (3.1 I1b) Wi -+w (i =1,2,3, *).. Equations (3.1 1) give the parameters of the relation between prices and price expectations functions in terms of the past history of independent shocks. The problem remains of writing the results in terms of the history of observable variables. We wish to find a relation of the form 00 (3.12) pt 1Vjfit-1
320 HN F MUTH We solve for the weights V, in terms of the weights W, in the following manner. Substituting from (3.7)and(3.8), we obtai 1+-=v,w+=(vW)4 Since the equality must hold for all shocks, the coefficients must satisfy the equations ViWi-g (=1,2,3, This is a system of equations with a triangular structure, so that it may be If the disturbances are independently distributed, as we assumed before, then we0=-1/B and all the others are zero. Equations(3. 14) therefore 少=0 (315b) pt=十 These are the results obtained before Suppose, at the other extreme, that an exogenous shock affects all future onditions of supply, instead of only the one period. This assumption would be appropriate if it represented how far technological change differed from its trend. Because ut is the sum of all the past es, ue=1(i=0, 1, 2,.) From(3.11) (316a) 1/B, (316b) From 3. 14)it can be seen that the expected price is a geometrically weighted moving average of past prices (3.17) 8 go p This prediction formula has been used by Nerlove [26] to estimate the supply elasticity of certain agricultural commodities. The only difference is that our analysis states that the coefficient of adjustment"in the ex pectations formula should depend on the demand and the supply coeffi- cients. The geometrically weighted moving average forecast is, in fact optimal under slightly more general conditions (when the disturbance is composed of both permanent and transitory components). In that case the coefficient will depend on the relative variances of the two components as well as the supply and demand coefficients. (See [24]
320 JOHN F. MUTH We solve for the weights V1 in terms of the weights Wj in the following manner. Substituting from (3.7) and (3.8), we obtain 00 00 00 00 t (3.13) WiVt- EV IWiet-i-i = V Wi 8t-ti. {=1 ?~=1 i=0 J5 =1 Since the equality must hold for all shocks, the coefficients must satisfy the equations (3.14) Wi VWiy (i = 1,2,3,...). 1=1 This is a system of equations with a triangular structure, so that it may be solved successively for the coefficients V1, V2, V3,.... If the disturbances are independently distributed, as we assumed before, then wO -1 /8 and all the others are zero. Equations (3.14) therefore imply (3.15a) t (3.15b) Pt = P+Wost - lete These are the results obtained before. Suppose, at the other extreme, that an exogenous shock affects all future conditions of supply, instead of only the one period. This assumption would be appropriate if it represented how far technological change differed from its trend. Because ut is the sum of all the past ej, wi 1 (i = 0,1,2,...). From (3.1 1), (3.16a) Wo -1/fl, (3.16b) Wi l/0 +y) From (3.14) it can be seen that the expected price is a geometrically weighted moving average of past prices: (3.17) ( ) . y pt y P t: + yJtThis prediction formula has been used by Nerlove [26] to estimate the supply elasticity of certain agricultural commodities. The only difference is that our analysis states that the "coefficient of adjustment" in the expectations formula should depend on the demand and the supply coefficients. The geometrically weighted moving average forecast is, in fact, optimal under slightly more general conditions (when the disturbance is composed of both permanent and transitory components). In that case the coefficient will depend on the relative variances of the two components as well as the supply and demand coefficients. (See [24].)
RATIONAL EXPECTATION: Deviations from Rationality. Certain imperfections and biases in the expectations may also be analyzed with the methods of this paper. Allowing for cross-sectional differences in expectations is a simple matter, because their aggregate effect is negligible as long as the deviation from the rational forecast for an individual firm is not strongly correlated with those of the others. Modifications are necessary only if the correlation of the errors is large and depends systematically on other explanatory variables. We shall examine the effect of over-discounting current information and of differences in the information possessed by various firms in the industry. Whether such biases in expectations are empirically important remains to be seen I wish only to emphasize that the methods are flexible enough to handle th Let us consider first what happens when expectations tently over- or under-discount the effect of current events. Equation(3.8), which gives the optimal price expectation, will then be replaced by (3.18) p=AW1et-1+∑Wet-4 In other words the weight attached to the most recent exogenous dis- turbance is multiplied by the factor fi, which would be greater than unity if current information is over-discounted and less than unity if it is under discounted If we use(3. 18) for the expected price instead of (3.8)to explain market orice movements, then (3. 11)is replaced by W0=--o (319b 8+fy l (3.19c) (=2,34…) The effect of the biased expectations on price movements depends on the statistical properties of the exogenous disturbances If the disturbances are independent(that is, wo= l and we=0 fori> 1) the biased expectations have no effect. The reason is that successive obser- vations provide no information about future fluctuations. On the other hand, if all the disturbances are of a permanent type(that =1), the properties of the expectations function are significantly affected. To illustrate the magnitude of the differences, the parameters of the function p=∑V
RATIONAL EXPECTATIONS 321 Deviations from Rationality. Certain imperfections and biases in the expectations may also be analyzed with the methods of this paper. Allowing for cross-sectional differences in expectations is a simple matter, because their aggregate effect is negligible as long as the deviation from the rational forecast for an individual firm is not strongly correlated with those of the others. Modifications are necessary only if the correlation of the errors is large and depends systematically on other explanatory variables. We shall examine the effect of over-discounting current information and of differences in the information possessed by various firms in the industry. Whether such biases in expectations are empirically important remains to be seen. I wish only to emphasize that the methods are flexible enough to handle them. Let us consider first what happens when expectations consistently overor under-discount the effect of current events. Equation (3.8), which gives the optimal price expectation, will then be replaced by 00 (3.18) Pt = fi Wiet-i + I Wi Et-i i=2 In other words the weight attached to the most recent exogenous disturbance is multiplied by the factor f1, which would be greater than unity if current information is over-discounted and less than unity if it is underdiscounted. If we use (3.18) for the expected price instead of (3.8) to explain market price movements, then (3.1 1) is replaced by (3.19a) Wo wo (3.19b) WW WI (3.19c) Wi Wi (i = 2,3,4,...). /3+y The effect of the biased expectations on price movements depends on the statistical properties of the exogenous disturbances. If the disturbances are independent (that is, wo =1 and wj = 0 for i > 1), the biased expectations have no effect. The reason is that successive observations provide no information about future fluctuations. On the other hand, if all the disturbances are of a permanent type (that is, w0 = w, = ... = 1), the properties of the expectations function are significantly affected. To illustrate the magnitude of the differences, the parameters of the function 00 pt - V}fit-
322 JoHN F. MUTH are compared in Figure 3. 1 for B= 2y and various values of fi. If current information is under-discounted (i= 1/2), the weight Vi attached to the latest observed price is very high. With over-discounting 1= 2),the weight for the first period is relatively low FIGURe 3. 1.Autoregression Coefficients of Expectations for Biased Use of Recent Information(o =Wn The model above can be interpreted in another way. Suppose that some of the firms have access to later information than the others. That is there is a lag of one period for some firms, which therefore form price expectations according to(3.8). The others, with a lag of two periods, can only use the following (320) ∑w Then the aggregate price expectations relation is the same as (3. 18), if fi represents the fraction of the firms having a lag of only one period in obtain ing market information(that is, the fraction of"insiders") 4. EFFECTS OF INVENTORY SPECULATION Some of the most interesting questions involve the economic effects of inventory storage and speculation. We can examine the effect by adjoining to(3. 1)an inventory demand equation depending on the difference between the expected future price and the current price. As we shall show, the
322 JOHN F. MUTH are compared in Figure 3.1 for ,B 2y and various values of fi. If current information is under-discounted (f= 1/2), the weight VI attached to the latest observed price is very high. With over-discounting (fi 2), the weight for the first period is relatively low. UNDERDI COUNTING Vk .4, ~~~~~~~RECENT INFORMATION a 2 3 4 5 6 k .8 ~~~~~~~~UN31A5ED USE oF RECENT INFORMATION Uk * Io i i 2 4 5 1 OVERD SCOUNTINGC , ~~~~~~~~RECENT INFORMATIOW 1 2 3 4 5 6 FIGURE 3.1.-Autoregression Coefficients of Expectations for Biased Use of Recent Information. (wo = w, = ... == 1). The model above can be interpreted in another way. Suppose that some of the firms have access to later information than the others. That is, there is a lag of one period for some firms, which therefore form price expectations according to (3.8). The others, with a lag of two periods, can only use the following: , 00 (3.20) pt t-= Ew i=2 Then the aggregate price expectations relation is the same as (3.18), if fi represents the fraction of the firms having a lag of only one period in obtaining market information (that is, the fraction of "insiders"). 4. EFFECTS OF INVENTORY SPECULATION Some of the most interesting questions involve the economic effects of inventory storage and speculation. We can examine the effect by adjoining to (3.1) an inventory demand equation depending on the difference between the expected future price and the current price. As we shall show, the
RATIONAL EXPECTATIONS 323 price expectation with independent disturbances in the supply function then turns out to have the form (4.1) p=1 where the parameter A would be somewhere between zero and one, its value depending on the demand, supply, and inventory demand parameters Speculation with moderately well-informed price expectations reduces the variance of prices by spreading the effect of a market disturbance over several time periods, thereby allowing shocks partially to cancel one another out. Speculation is profitable, although no speculative opportunities remain These propositions might appear obvious. Nevertheless, contrary views have been expressed in the literature. Before introducing inventories into the market conditions briefly examine the nature of speculative demand for a commodity. optimal speculation. We shall assume for the time being that storage interest, and transactions costs are negligible. Anindividual has an opportun ity to purchase at a known price in the tth period for sale in the succeeding period. The future price is, however, unknown. If we let It represent the speculative inventory at the end of the tth period, 7 then the profit to be realized is I(九+1-) Of course, the profit is unknown at the time the commitment is to be made There is, however, the expectation of gain. The individual demand for speculative inventories would presumably be based on reasoning of the following sort. The size of the commitment depends on the expectation of the utility of the profit. For a sufficiently small range of variation in profits, we can approximate the utility function by the first few terms of its Taylor's series expansion about the origin (4.3) 4=p()=0)+(0)x+"(0a+ The expected utility depends on the moments of the probability distribu Em=0)+(0)Em+2"(E2+ 6 See Baumol [5]. His conclusions depend on a nonspeculative demand such that prices would be a pure sine function, which may always be forecast perfectly 7 Speculative inventories may be either positive or negativ
RATIONAL EXPECTATIONS 323 price expectation with independent disturbances in the supply function then turns out to have the form (4.1) ft'At_i where the parameter A would be somewhere between zero and one, its value depending on the demand, supply, and inventory demand parameters. Speculation with moderately well-informed price expectations reduces the variance of prices by spreading the effect of a market disturbance over several time periods, thereby allowing shocks partially to cancel one another out. Speculation is profitable, although no speculative opportunities remain. These propositions might appear obvious. Nevertheless, contrary views have been expressed in the literature.6 Before introducing inventories into the market conditions, we shall briefly examine the nature of speculative demand for a commodity. Optimal Speculation. We shall assume for the time being that storage, interest, and transactions costs are negligible. An individual has an opportunity to purchase at a known price in the tth period for sale in the succeeding period. The future price is, however, unknown. If we let It represent the speculative inventory at the end of the tth period,7 then the profit to be realized is (4.2) at-It(pt+1-Pt). Of course, the profit is unknown at the time the commitment is to be made. There is, however, the expectation of gain. The individual demand for speculative inventories would presumably be based on reasoning of the following sort. The size of the commitment depends on the expectation of the utility of the profit. For a sufficiently small range of variation in profits, we can approximate the utility function by the first few terms of its Taylor's series expansion about the origin: (4.3) Ut - 0 (t) (O) + ?' (O) at +"2 O)' (t +... The expected utility depends on the moments of the probability distribution of a: (4.4) Eut - (0) + 0'(O) Ent + 0) Eat 6 See Baumol [5]. His conclusions depend on a nonspeculative demand such that prices would be a pure sine function, which may always be forecast perfectly. 7 Speculative inventories may be either positive or negative
