《投资学原理 Investments》课程教学资源（阅读资料）Market_Efficiency in a Market with Heterogeneous Information.pdf_大学文库

Market""in a Market with Heterogeneous Information Stephen Figlewski New York Unicersity It is commonly felt that a financial market achieves informational efficiency as traders with the best information and the most skill make profits at the expense of those with inferior information or ability and come to dominate the market. This paper develops a model of a specula- tive market in which this redistribution of wealth among traders with different information and ability can be studied. In the short run the mar ket tends toward increased efficiency, but in neither the short nor the ong run is full efficiency likely. The average deviation from efficiency is shown to depend on traders' characteristics such as the quality and diversity of their information and their risk aversion From the time of Adam Smith, economists have extolled the virtues of the competitive price system as a mechanism for allocating scarce resources among competing use en the proper assumptions, the free operation of the competitive market can be shown to result in a Pareto-optimal allo- cation of goods. But for certain goods whose characteristics are not com- pletely known, the market has the additional role of aggregating the avai able information about these characteristics. a share of ibm stock represents a claim on future earnings whose total value cannot be known n the present. The market price for IBM represents an aggregate opinion about the company's future prospects based on whatever information may be currently available to the participants in the market. And in general if we define a speculative market broadly as a market for a good demande ot(entirely) for its own sake but for resale(or potential resale) in th future, the current price in a speculative market will always have at least a component which is the market's estimate of the future price. In most to Glenn Loury, Steven Sheffrin, Jerry low for an extraordinarily in- Hausman and iversity of Ch:ica:. 6022- 3808/78/8604-004501 4 Sep201303:16AM I use subject to

Market "Efficiency" in a Market with Heterogeneous Information Stephen Figlewski New York University It is commonly felt that a financial market achieves informational efficiency as traders with the best information and the most skill make profits at the expense of those with inferior information or ability and come to dominate the market. This paper develops a model of a speculative market in which this redistribution of wealth among traders with different information and ability can be studied. In the short run the market tends toward increased efficiency, but in neither the short nor the long run is full efficiency likely. The average deviation from efficiency is shown to depend on traders' characteristics such as the quality and diversity of their information and their risk aversion. From the time of Adam Smith, economists have extolled the virtues of the competitive price system as a mechanism for allocating scarce resources among competing uses. Given the proper assumptions, the free operation of the competitive market can be shown to result in a Pareto-optimal allocation of goods. But for certain goods whose characteristics are not completely known, the market has the additional role of aggregating the available information about these characteristics. A share of IBM stock represents a claim on future earnings whose total value cannot be known in the present. The market price for IBM represents an aggregate opinion about the company's future prospects based on whatever information may be currently available to the participants in the market. And in general, if we define a speculative market broadly as a market for a good demanded not (entirely) for its own sake but for resale (or potential resale) in the future, the current price in a speculative market will always have at least a component which is the market's estimate of the future price. In most I am indebted to Glenn Loury, Steven Sheffrin, Jerry Hausman, and an anonymous referee for many valuable suggestions and to Robert Solow for an extraordinarily insightful conjecture which proved to be the basis of this essay. [Journal of Political Economy, 1978, vol. 86, no. 4] (? 1978 by The University of Chicago. 0022-3808/78/8604-0004$01.44 58i This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:02:16 AM All use subject to JSTOR Terms and Conditions

OURNAL OF POLITICAL ECONOMY cases,of course, speculative prices serve as signals which affect produc decisions in the rest of the economy. Equity prices affect firms tment decisions, commodity futures prices determine storage and production, ar so on. We would like to know how information is actually incorporated into a market price and especially whether the information-processing function of a competitive speculative market has optimality properties like those which apply to the allocative function. Is there formational invisible hand" which leads a competitive speculative market to make the optimal use of the information that society has available? One answer to this question comes in the form of the"efficient-markets hypothesis. Under this hypothesis, a competitive speculative market is typically asserted to be"efficient"in the sense that the current marke price always fully reflects"all available information or the current price, plus normal profits, is the "best estimate 'of the future price. Although there is clearly some kind of optimality property involved, such terms as fully reflects"or"best estimate "are sufficiently imprecise that there is a wide latitude among economists on what"efficiency"should mean exactly Fama(1970) distinguishes three degrees of market efficiency,"weak semistrong, 'and"strong, " according to what type of information is fully reflected in the market price. A market is weakly efficient if the current price always completely discounts the information contained in the history of past market prices. The semistrong form of efficiency widens the scope to include all publicly available information. In addition to the history of past prices, the market accurately evaluates such things as dividend decla- rations, crop reports, and, we might expect, Wall Street Journal articles Finally, the strong form of efficiency occurs when the market accurately discounts all information, including that held only by small numbers of market participants Obviously there is a large difference between weak and strong efficiency in terms of the optimality properties they imply for information processing in a decentralized market. The social value of a mechanism for aggregat- g information depends on its ability to generate price signals that accu rately reflect all of society s information. Thus only a market that is effi- cient in the strong sense can really be said to have the optimality properties we would like. Throughout this paper, "efficiency"will be taken to mean strong efficiency. ine the mechanism by which a speculative market would achieve and maintain informational efficiency, we are led, like Cootner, to the following kind of story: Given the uncertainty of the real world, the many actual and virtual investors will have many, perhaps equally many, price If any group of consistently]better than average in forecasting stock price, they would accumulate Sep201303:16AM I use subject to

582 JOURNAL OF POLITICAL ECONOMY cases, of course, speculative prices serve as signals which affect productive decisions in the rest of the economy. Equity prices affect firms' investment decisions, commodity futures prices determine storage and production, and so on. We would like to know how information is actually incorporated into a market price and especially whether the information-processing function of a competitive speculative market has optimality properties like those which apply to the allocative function. Is there an informational "invisible hand" which leads a competitive speculative market to make the optimal use of the information that society has available? One answer to this question comes in the form of the "efficient-markets hypothesis." Under this hypothesis, a competitive speculative market is typically asserted to be "efficient" in the sense that the current market price always "fully reflects" all available information or the current price, plus normal profits, is the "best estimate" of the future price. Although there is clearly some kind of optimality property involved, such terms as "fully reflects" or "best estimate" are sufficiently imprecise that there is a wide latitude among economists on what "efficiency" should mean exactly. Fama (1970) distinguishes three degrees of market efficiency, "weak," "semistrong," and "strong," according to what type of information is fully reflected in the market price. A market is weakly efficient if the current price always completely discounts the information contained in the history of past market prices. The semistrong form of efficiency widens the scope to include all publicly available information. In addition to the history of past prices, the market accurately evaluates such things as dividend declarations, crop reports, and, we might expect, Wall Street Journal articles. Finally, the strong form of efficiency occurs when the market accurately discounts all information, including that held only by small numbers of market participants. Obviously there is a large difference between weak and strong efficiency in terms of the optimality properties they imply for information processing in a decentralized market. The social value of a mechanism for aggregating information depends on its ability to generate price signals that accurately reflect all of society's information. Thus only a market that is efficient in the strong sense can really be said to have the optimality properties we would like. Throughout this paper, "efficiency" will be taken to mean strong efficiency. If we try to imagine the mechanism by which a speculative market would achieve and maintain informational efficiency, we are led, like Cootner, to the following kind of story: Given the uncertainty of the real world, the many actual and virtual investors will have many, perhaps equally many, price forecasts. . . . If any group of investors was consistently better than average in forecasting stock price, they would accumulate This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:02:16 AM All use subject to JSTOR Terms and Conditions

MARKET“ EFFICIENCY wealth and give their forecasts greater and greater weight In the process, they would bring the present price closer to the true value. Conversely, investors who were worse than average in forecasting ability would carry less and less weight. If this process orked well enough, the present price would reflect the best information about the future in the sense that the present price, plus normal profits, would be the best estimate of the future price [967,P.80] In this view market efficiency is a condition that is achieved in the long run as wealth is redistributed from investors with inferior information to hose with better information. Of course in the short run. before this has a chance to happen, the distribution of wealth and the distribution of ormation quality may be very different. Since the market weights traders'information by"dollar votes, " not quality, a trader with superior information but little wealth may have his information undervalued in the market price, and the market will be inefficient. However, there will exist some distribution of wealth--we might call it the efficient-market distribu tion-for which the dollar- vote weights are identical with information quality weights and each trader's information is accurately reflected in the market price. If the distribution of wealth ultimately converges to thi efficient-market distribution, in the long run the market does become informationally efficient. The purpose of this paper is to develop a model of a speculative market in which the convergence can be analyzed. We find that in the short run the distribution of wealth tends to move toward the efficient-market distribution. a trader whose information was under- valued has an expected profit, and one whose information was overvalued has an expected loss. But in the longer run, random fluctuations in wealth sulting from the inability st prices perfectly lead to deviatio from the efficient-market wealth distribution and consequently from market efficiency. In general the market price is not the best estimate of the future price, given the currently available information. From the long run orsteady-state distribution of wealth we can calculate the average fficiency of the market, as measured by the average variance of the market's forecast error, and compare it with the efficient-market vari ance. The difference can be thought of as the efficiency cost of processing information through a decentralized market rather than a centralized hority. A series of examples will give some idea of how this cost depends on the underlying parameters of the market such as the traders' risk aversion, the disparity in their forecasting abilities, and so on The existence of the efficient-market wealth distribution undoubtedly requires some on traders'demands In the model presented below, these are satis- fied. and market distribution exists and is unique. Sep201303:16AM I use subject to

MARKET EFFICIENCY 1 583 wealth and give their forecasts greater and greater weight. In the process, they would bring the present price closer to the true value. Conversely, investors who were worse than average in forecasting ability would carry less and less weight. If this process worked well enough, the present price would reflect the best information about the future in the sense that the present price, plus normal profits, would be the best estimate of the future price. [1967, p. 80] In this view market efficiency is a condition that is achieved in the long run as wealth is redistributed from investors with inferior information to those with better information. Of course in the short run, before this has a chance to happen, the distribution of wealth and the distribution of information quality may be very different. Since the market weights traders' information by "dollar votes," not quality, a trader with superior information but little wealth may have his information undervalued in the market price, and the market will be inefficient. However, there will exist some distribution of wealth we might call it the efficient-market distribution-for which the dollar-vote weights are identical with informationquality weights and each trader's information is accurately reflected in the market price.1 If the distribution of wealth ultimately converges to this efficient-market distribution, in the long run the market does become informationally efficient. The purpose of this paper is to develop a model of a speculative market in which the convergence can be analyzed. We find that in the short run the distribution of wealth tends to move toward the efficient-market distribution. A trader whose information was undervalued has an expected profit, and one whose information was overvalued has an expected loss. But in the longer run, random fluctuations in wealth resulting from the inability to forecast prices perfectly lead to deviations from the efficient-market wealth distribution and consequently from market efficiency. In general the market price is not the best estimate of the future price, given the currently available information. From the long run or "steady-state" distribution of wealth we can calculate the average efficiency of the market, as measured by the average variance of the market's forecast error, and compare it with the efficient-market variance. The difference can be thought of as the efficiency cost of processing information through a decentralized market rather than a centralized authority. A series of examples will give some idea of how this cost depends on the underlying parameters of the market such as the traders' risk aversion, the disparity in their forecasting abilities, and so on. I The existence of the efficient-market wealth distribution undoubtedly requires some regularity conditions on traders' demands. In the model presented below, these are satisfied, and the efficient market distribution exists and is unique. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:02:16 AM All use subject to JSTOR Terms and Conditions

JOURNAL OF POLITICAL ECONOMY The question of the informational efficiency of a decentralized financial market has already been raised in a different manner in a series of recent papers by Grossman(1975, 1976)and Grossman and Stiglitz(1975, 1976 They show that when information is costly to obtain it cannot be true that the market price will accurately reflect all available information. If the market price revealed all information for free, it would not pay anyone invest in information gathering individually, But if no one gathered information there would be none for the market to reveal. Thus an infor nationally efficient market is incompatible with costly information. Gross- man and Stiglitz 's solution to this difficulty is to expand the traditional concept of market equilibrium to one of"informational equilibrium In addition to the standard equilibrium condition that supply equals demand in every period, there is the further condition that the market price must reveal just enough of the costly information that participants are indifferent between becoming"informed" or remaining"uninformed In full equilibrium, the market price does not reveal all the information, nd traders who buy information do earn a higher return in the market But the extra return is just sufficient to offset the cost of the information and the expected return, including the cost of information, is equal for informed and uninformed traders An important feature of the Grossman and Stiglitz models is that the redistribution of wealth among bad and good forecasters which Cootner (1967)talks about is not a factor. In all of their models, investors are assumed to have constant absolute risk-aversion utility functions which have the property that the demand for a given risky asset is independent of wealth. Even if the good forecasters do accumulate wealth over time, this does not lead to a heavier weighting of their forecasts in the marke price. Adjustment to informational equilibrium occurs not because of re- distribution of wealth among traders but because of entry into and exit from the information-collection business. Thus deviation from market ciency arising from wealth redistribution, which will be a principal feature of the model I analyze below, is over and above the informational inefficiency that Grossman and Stiglitz discuss. The foregoing discussion has revolved around the question of how a ompetitive financial market processes information without considering specifically what"information"consists of. In the Grossman and Stiglitz framework, information is a datum which allows a trader to reduce h forecast variance of the next period price. If every trader possessed the formation, they would have identical expectations about the future price learly, this view of information is rather restrictive. We can easily think of things which not all market participants would consider even to be information at all. For example, the news that IBM had just completed a perfect"head and shoulders"top would be information to some, not others. More generally, even if all traders accept a certain piece of informa Sep201303:16AM I use subject to

584 JOURNAL OF POLITICAL ECONOMY The question of the informational efficiency of a decentralized financial market has already been raised in a different manner in a series of recent papers by Grossman (1975, 1976) and Grossman and Stiglitz (1975, 1976). They show that when information is costly to obtain it cannot be true that the market price will accurately reflect all available information. If the market price revealed all information for free, it would not pay anyone to invest in information gathering individually. But if no one gathered information, there would be none for the market to reveal. Thus an informationally efficient market is incompatible with costly information. Grossman and Stiglitz's solution to this difficulty is to expand the traditional concept of market equilibrium to one of "informational equilibrium." In addition to the standard equilibrium condition that supply equals demand in every period, there is the further condition that the market price must reveal just enough of the costly information that participants are indifferent between becoming "informed" or remaining "uninformed." In full equilibrium, the market price does not reveal all the information, and traders who buy information do earn a higher return in the market. But the extra return is just sufficient to offset the cost of the information, and the expected return, including the cost of information, is equal for informed and uninformed traders. An important feature of the Grossman and Stiglitz models is that the redistribution of wealth among bad and good forecasters which Clootner (1967) talks about is not a factor. In all of their models, investors are assumed to have constant absolute risk-aversion utility functions which have the property that the demand for a given risky asset is independent of wealth. Even if the good forecasters do accumulate wealth over time, this does not lead to a heavier weighting of their forecasts in the market price. Adjustment to informational equilibrium occurs not because of redistribution of wealth among traders but because of entry into and exit from the information-collection business. Thus deviation from market efficiency arising from wealth redistribution, which will be a principal feature of the model I analyze below, is over and above the informational inefficiency that Grossman and Stiglitz discuss. The foregoing discussion has revolved around the question of how a competitive financial market processes information without considering specifically what "information" consists of. In the Grossman and Stiglitz framework, information is a datum which allows a trader to reduce his forecast variance of the next period price. If every trader possessed the information, they would have identical expectations about the future price. Clearly, this view of information is rather restrictive. We can easily think of things which not all market participants would consider even to be information at all. For example, the news that IBM had just completed a perfect "head and shoulders" top would be information to some, not to others. More generally, even if all traders accept a certain piece of informaThis content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:02:16 AM All use subject to JSTOR Terms and Conditions

MARKET“ EFFICIENC tion such as an earnings report as being important, they will not necessarily agree completely on its implications, It is not possible to separate the impact of elementary information such as news releases, crop reports, etc from the subjective evaluation of this information by the participants in the market. Thus, rather than dealing with differences in"information, it will be more convenient to work with differences in forecasting ability- aring in mind that access to elementary information is a major determi nant of forecasting ability. The operational definition of an efficient market, then, is one in which the market price at any time(plus normal profits)is the best, that is, minimum variance, estimate of the future price, given the individual forecasts of all the market participant In the next section I develop a model of a speculative market with two types of traders who have differing information. I derive the stochastic difference equation which describes the redistribution of wealth among the two groups and analyze the market's short-run behavior. In the following section, I approximate the difference equation by a discrete Marke model and analyze its long-run properties, Although it is not possible to derive the steady-state wealth distribution analytically, several illustrative examples show how the market behaves with different values of the under- lying parameters. This allows me to draw some tentative conclusions about how the informational efficiency of a decentralized market should be affected by heterogeneous information, differences in the quality of traders formation, risk aversion, and so on. The final section gives a summary and conclusion I. The model A competitive market weights a trader's information by the size of his investment, so a market's informational efficiency depends on the distrib tion of wealth among its participants. In this section I develop a model of a speculative market in which the interaction of information and wealth and the resulting effects on market efficiency can be analyzed The market is made up of equal numbers of two types of traders, a and b All a traders are alike, as are all b traders, but members of the two groups may differ in price expectations, risk aversion, predictive ability, and wealth. The assumption of just two types of traders is made purely fo expositional convenience, The model developed below can readily be extended to n traders with no change in the basic results. Although there are only two groups, we will assume that each trader himself rading in a perfectly competitive market. Otherwise we would have the problem that the a traders could solve back from the observed market price to obtain the b group's information and vice versa. In a market with more han two groups, it would not be possible to determine the information held by every other participant from the market price alone Sep201303:16AM I use subject to

MARKET "EFFICIENCY 585 tion such as an earnings report as being important, they will not necessarily agree completely on its implications. It is not possible to separate the impact of elementary information such as news releases, crop reports, etc., from the subjective evaluation of this information by the participants in the market. Thus, rather than dealing with differences in "information," it will be more convenient to work with differences in forecasting abilitybearing in mind that access to elementary information is a major determinant of forecasting ability. The operational definition of an efficient market, then, is one in which the market price at any time (plus normal profits) is the best, that is, minimum variance, estimate of the future price, given the individual forecasts of all the market participants. In the next section I develop a model of a speculative market with two types of traders who have differing information. I derive the stochastic difference equation which describes the redistribution of wealth among the two groups and analyze the market's short-run behavior. In the following section, I approximate the difference equation by a discrete Markov model and analyze its long-run properties. Although it is not possible to derive the steady-state wealth distribution analytically, several illustrative examples show how the market behaves with different values of the underlying parameters. This allows me to draw some tentative conclusions about how the informational efficiency of a decentralized market should be affected by heterogeneous information, differences in the quality of traders' information, risk aversion, and so on. The final section gives a summary and conclusion. I. The Model A competitive market weights a trader's information by the size of his investment, so a market's informational efficiency depends on the distribution of wealth among its participants. In this section I develop a model of a speculative market in which the interaction of information and wealth and the resulting effects on market efficiency can be analyzed. The market is made up of equal numbers of two types of traders, a and b. All a traders are alike, as are all b traders, but members of the two groups may differ in price expectations, risk aversion, predictive ability, and wealth. The assumption of just two types of traders is made purely for expositional convenience. The model developed below can readily be extended to n traders with no change in the basic results. Although there are only two groups, we will assume that each trader views himself as trading in a perfectly competitive market. Otherwise we would have the problem that the a traders could solve back from the observed market price to obtain the b group's information and vice versa. In a market with more than two groups, it would not be possible to determine the information held by every other participant from the market price alone. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:02:16 AM All use subject to JSTOR Terms and Conditions

JOURNAL OF POLITICAL ECONOMY The market is for claims on an asset which pays a random return P at the end of each period. The net supply of claims is zero, so that the only way for one trader to buy a claim is for another to sell one short. All trad ing takes place between the two groups, since all members of a given group are identical. Two features of this setup lead to considerable simplification First, Pi is independent of the operation of the market, so we have avoided the"beauty contest" problem which Keynes talked about with respect to e stock market. Second, this is a zero-sum game, so that the"normal profits"to a trader are zero, and also the analysis is not complicated by hanges in the scale of the market over time. Both characteristics are fairly closely approximated by a typical futures market in which P, represents the spot price in the contract month At the outset of each period, the two groups receive information abe Pf. Next, the market opens and an equilibrium market price is achiev by a tatonnement process.(Expectations may be revised at any point up to equilibrium, so that the market clearing price P, is part of the information set upon which expectations are ultimately based. At the equilibrium the demands of the a traders, n ", are exac tly offset by the(algebraic)demands of the b traders, n Finally the market closes, Pi is revealed, and there is a net transfer of na(p* -Pi) from b to a traders. [Of course n (Pr-Pu) may be negative, so that b traders receive money from a traders. I There will only be a wealth transfer when Pt differs from Pr, tha hen the market price is an inaccurate forecast of the future price. But the market's forecast error is just a combination of the traders'forecast errors, so the wealth redistribution in period t is a function of the traders individual errors in forecasting PI. We will derive expressions for n, and P:-,)in terms of the traders characteristics such as forecasting ability and risk aversion and their random forecast errors. The latter drive the model, and their known distribution allows us to derive an equation for the stochastic process governing the redistribution of wealth within the We now consider the expectations formation of the two groups.(In what follows the subscript t will be dropped for simplicity when it is not essential There should be no confusion about what period the variables refer to No information about P* is available before the beginning of the period traders come into the d with nonin (flat)prior di ribu 2 arises when a price change is so large that one trader cannot cover hi losses. Treating the possibility of bankruptcy explicitly would greatly complicate th takes place through a well-capitalized clearing corporation which insures all trady A odel. Instead, we will assume that in th Thus traders can transact withe that their con ts will not be fulfilled. In a case, a trader's acceptable level of risk exposure de pends on his risk aversion. In this f traders are sufficiently risk averse the probabil

586 JOURNAL OF POLITICAL ECONOMY The market is for claims on an asset which pays a random return P* at the end of each period. The net supply of claims is zero, so that the only way for one trader to buy a claim is for another to sell one short. All trading takes place between the two groups, since all members of a given group are identical. Two features of this setup lead to considerable simplification. First, P,* is independent of the operation of the market, so we have avoided the "beauty contest" problem which Keynes talked about with respect to the stock market. Second, this is a zero-sum game, so that the "normal profits" to a trader are zero, and also the analysis is not complicated by changes in the scale of the market over time. Both characteristics are fairly closely approximated by a typical futures market in which P,* represents the spot price in the contract month. At the outset of each period, the two groups receive information about PJ*. Next, the market opens and an equilibrium market price is achieved by a t67tonnement process. (Expectations may be revised at any point up to equilibrium, so that the market clearing price Pt is part of the information set upon which expectations are ultimately based.) At the equilibrium the demands of the a traders, na, are exactly offset by the (algebraic) demands of the b traders, n'. Finally the market closes, P* is revealed, and there is a net transfer of n (Pt* - Pt) from b to a traders.2 [Of course na (p -Pt) may be negative, so that b traders receive money from a traders.] There will only be a wealth transfer when P,* differs from Pt, that is, when the market price is an inaccurate forecast of the future price. But the market's forecast error is just a combination of the traders' forecast errors, so the wealth redistribution in period I is a function of the traders' individual errors in forecasting P1*. We will derive expressions for n a and (Pt* - Pt) in terms of the traders' characteristics such as forecasting ability and risk aversion and their random forecast errors. The latter drive the model, and their known distribution allows us to derive an equation for the stochastic process governing the redistribution of wealth within the market. We now consider the expectations formation of the two groups. (In what follows the subscript t will be dropped for simplicity when it is not essential. There should be no confusion about what period the variables refer to.) No information about P* is available before the beginning of the period, so traders come into the period with noninformative (flat) prior distribu- 2 A question arises when a price change is so large that one trader cannot cover his losses. Treating the possibility of bankruptcy explicitly would greatly complicate the model. Instead, we will assume that in this market (as in many actual markets) trading takes place through a well-capitalized clearing corporation which insures all trades. Thus traders can transact without fear that their contracts will not be fulfilled. In any ease, a trader's acceptable level of risk exposure depends on his risk aversion. In this model, if traders are sufficiently risk averse the probability of a bankruptcy becomes arbitrarily small. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:02:16 AM All use subject to JSTOR Terms and Conditions

MARKET“ EFFICIENCY3 tions on P*. Before the market opens and during the tatonnement process, they receive information sets and o(including P, the market clearing price), so that by the time the market has closed they have formed and traded on the basis of the following posterior distributions on P*:f(p*ay the a traders'posterior distribution, is normal with mean Pa and variance n2:f(P*Ia) is normal with mean Pb and variance n. We will assume that the posterior variances, which measure the quality of traders information and their ability to forecast from it, are constant over time and equal to the true forecasting variances. The(subjective) expected values of P* conditional on each group's information are P and pb. We assume that they are unbiased estimates of P* over all realizations fP*,Φ",andΦ, These distributions imply that we he traders' prediction errors as s h N(O, n2 to split each one into two independent parts, one of which is correlated with the other group's error and the other of which is not 6“~N(O,a2) 8~N(0,02); eb+68b~N(0,0) 8, 8, 8 are mutually independent a trader calculates his market demand by maximizing a two-parameter tility function defined on the expected value and variance of end of period wealth. Any portion of a trader's wealth not invested is held in a riskless asset earning zero return, and there are no limitations on borrowing A trader's expected wealth depends on the expected price change, which in turn depends on his expected value for P* The variance of wealth is determined by the variance of the change in the market price. We assume that traders estimate this variance by observ. ing the market's operation over time. In each period they treat the variance of (P*- P)as being a constant equal to the long-run average variance Notice that o2 is a characteristic of the market, not of the individual. The risk involved in taking a position in this market is the same for all traders even though they may have information of differing quality To derive his demand, n", an a trader solves the following maximization proble max UTE(wa1), var(wall Sep201303:16AM I use subject to

MARKET "EFFICIENCY 587 tions on P*. Before the market opens and during the t~tonnement process, they receive information sets V and Vb (including P, the market clearing price), so that by the time the market has closed they have formed and traded on the basis of the following posterior distributions on P* :fa(P* Iea), the a traders' posterior distribution, is normal with mean pa and variance 2; f (P * Ib) is normal with mean pb and variance tj2. We will assume that the posterior variances, which measure the quality of traders' information and their ability to forecast from it, are constant over time and equal to the true forecasting variances. The (subjective) expected values of P* conditional on each group's information are pa and pb. We assume that they are unbiased estimates of P* over all realizations of P*, (Da, and VDb. These posterior distributions imply that we can write the traders' prediction errors as pa _ p* - ,a (a N(O,1a2), pb p* = ,b ,b - N(O, 2) . Since ;a and ,b are not necessarily independent, it will be convenient to split each one into two independent parts, one of which is correlated with the other group's error and the other of which is not. pa _ p* =a + a Ea N(O, a2); 6 N(O, 02); (2) pb p* = 8b + $ ab .N(O, c2); ,6aI 8b, 6 are mutually independent. A trader calculates his market demand by maximizing a two-parameter utility function defined on the expected value and variance of end of period wealth. Any portion of a trader's wealth not invested is held in a riskless asset earning zero return, and there are no limitations on borrowing. A trader's expected wealth depends on the expected price change, which in turn depends on his expected value for P*. The variance of wealth is determined by the variance of the change in the market price. We assume that traders estimate this variance by observing the market's operation over time. In each period they treat the variance of (P* - P) as being a constant 2 equal to the long-run average variance. Notice that 02 is a characteristic of the market, not of the individual. The risk involved in taking a position in this market is the same for all traders, even though they may have information of differing quality. To derive his demand, na, an a trader solves the following maximization problem: max Ua[E( Wa+ 1), var ( Wta 1)] . (3) This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:02:16 AM All use subject to JSTOR Terms and Conditions

MARKET "EFFICIENCY 3 589 is a sufficient statistic. That is, given the market price, there should be no further advantage to be derived from having the specific information of the individual market participants. In this case, the efficient-market price, peff, would be the mean of the posterior distribution on P* of an omniscient observer who had access to both a and b traders' information.3 Jaffee and Winkler (1976, p. 57) derive the posterior distribution for this case and find it to be a normal distribution with V(V - p)Pa + (1 - pv)Pb mean: 2 v - 2pv + 1 v2(l _- 2q variance: a v - 2pv + 1 where qb 002 V 2' P= 'la 2la'2 If we substitute (02 + o2) and (02 + U2) for2 and ,2 and define k = u2/ab , these expressions can be simplified. We obtain E[p*jpa, pb] = pa I pb = pelf ] + k I + k (6) var LI*|pa' pb] = 1 k 2 + 02 = var [peff -P*] An efficient market should produce a market price which is a linear combination of the forecasts of the two types of traders where the weights vary inversely with the variance of the independent part of their forecasting errors.4 Notice from (5) that, while the actual market price is also a 3 We consider an omniscient observer who only deals with the posterior distributions of the traders. Since we have not ruled out the possibility of differences of opinion in evaluating a given set of elementary information, this tacitly involves one of two assumptions. Either the omniscient observer must always agree with a trader's evaluation of his information, or he must have access only to traders' posterior distributions and not their entire information sets. 4 Another way to derive the efficient-market price is to treat this as a forecasting problem in which what is wanted is the combination of the two estimates of P* which minimizes the squared forecast error. From normal theory we know the combination will be linear, and since both pa and pb are unbiased, their weights must sum to 1. min E{[Xpa + (1 - X)pb _ -*] 2} A = min E{1[Xa + (1 - ?)Fb + 8]2}, A - mmn ~[2ur2 + (1 - ?)2U2 + 02]. A This yields the first-order condition A = cr2/(ur2 + ut2), from which eq. (6) follows directly. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 03:02:16 AM All use subject to JSTOR Terms and Conditions