Information Acquisition in Financial Markets: a Correction

Information Acquisition in Financial Markets: a Correction Gadi Barlevy Federal Reserve Bank of Chicago 30 South LaSalle Chicago, IL 60604 Pietro Veronesi Graduate School of Business University of Chicago Chicago, IL 60637 January 17, 008 Abstract This note provides a proper example for the channel of strategic complementarities e proposed in our 000 paper "Information Acquisition in Financial Markets". As pointed out in Chamley (007), our earlier example contained an error, hich, once corrected, reveals that our example actually exhibits strategic substitutability in information acquisition. JELCodes:G14,D8,D84 In our 000 paper Information Acquisition in Financial Markets e argued that contrary to the conventional isdom set forth in Grossman and Stiglitz (1980), it as theoretically possible that as more traders in financial markets acquire information, equilibrium prices ould change in such a ay that it became more difficult for remaining agents to infer the fundamentals from prices. We presented an example e thought demonstrated this claim. Hoever, as as subsequently pointed out to us by Christophe Chamley, the expression e used for the value of information in that paper (expression 3.5) as incorrect. As demonstrated by Chamley (007), using the correct expression for the value of learning reveals that learning is in fact a strategic substitute in our example. This leaves open the question of hether there is an example consistent ith our original conjecture. This note provides a proper example of the mechanism e previously attempted to model. 1 The example involves changing the ay fundamentals and noise are jointly distributed, in contrast ith our previous paper in hich e tried to change the functional form for the distribution of each variable. While this amounts to a different technical assumption, alloing fundamentals and noise to be correlated captures precisely hat e attempted to model in our earlier paper, a stochastic environment in hich more agents learning can make prices harder, not easier, to read. We are grateful to Christophe Chamley for discovering our error, and to Christian Hellig for his comments. 1 Since our paper as published, there have been other papers that demonstrated information acquisition in financial markets can exhibit strategic complementarity, e.g. Veldkamp (006), Chamley (006), and Ganguli and Yang (006). Hoever, these papers rely on different mechanisms than the one e conjectured in our 000 paper. 1

The intuition for our result is as follos. Consider an environment in hich as fundamentals change, additional factors that do not impact on fundamentals tend to change as ell, in a ay that has an offsetting effect on the price of the asset. For example, firm managers might take advantage of periods in hich fundamentals are high to issue more shares, perhaps because prices tend to be higher on average at these times or because good fundamentals tend to be associated ith ne investment opportunities that necessitate ne funds. This increase in supply may counteract the effect of more favorable fundamentals on the price of the asset. As another example, if good fundamentals are correlated ith higher future income for some agents, they ill liquidate some of their assets to consume more today. The resulting sales pressure ill counteract the positive nes about fundamentals. In such situations, as more agents acquire information, prices ill be pushed up (don) hen fundamentals are high (lo). Hoever, this change may no longer serve to make prices more informative. Due to the offsetting factors, such a shift ill cause the distribution of prices for different fundamentals to converge toards the same values, making it harder to infer fundamentals from prices. This is precisely hat e sought to demonstrate in our previous paper. We begin our discussion ith a simple example based on the setup in our 000 paper. This example illustrates hy alloing noise to be correlated ith fundamentals can produce strategic complementarities in information acquisition. We then argue that correlation can be a source of complementarities quite generally, including in the original specification of Grossman and Stiglitz (1980). 1 The Case To construct a relatively simple example of complementarities, e turn to the frameork of our 000 paper. Briefly, agents must choose to allocate their ealth beteen money and an asset that pays a random amount e per share. There is a unit mass of risk-neutral agents ho can observe e if they pay a cost c>0. The demand of informed traders is denoted x I ( e, P), and of uninformed is denoted x U (P ). Traders can spend at most their initial endoment, equal to one unit of money, and cannot sell assets short. Demand for the asset by noise traders is P ex for some >0, hereex is random. In our 000 paper, e assumed ex had a continuous distribution. Hoever, the intuition is more transparent if both e and ex take on only to values: e, ª here >,andx {x 0,x 1 } here x 1 >x 0. We return to the case here ex has a continuous distribution in the next section. We refer to the four states of the orld as ω 1 through ω 4, ith probabilities π 1 through π 4,asfollos: x 0 x 1 ω 1 ω ω 3 ω 4 x 0 x 1 π 1 π π 3 π 4 Let z denote the fraction of traders ho acquire information. Consider the case here z =0, so all traders are uninformed. An equilibrium price function assigns a price to each of the four states. Since all traders are uninformed, it seems natural to restrict attention to equilibria in hich P (ω 1 )=P(ω 3 ) and P (ω )=P(ω 4 ), i.e. here the price for a realization of ex is not measurable ith respect to e.wethusruleoutequilibriainhichagentscoordinatetobuytheassetbasedon unobservable realizations of e. One can The existence of such an equilibrium follos by construction. Define P 0 P (ω 1 )=P (ω 3 ) as the price of the asset hen ex = x 0 and P 1 P (ω )=P (ω 4 ) as the

price of the asset hen ex = x 1. Since e ish to demonstrate the possibility that prices become less informative once some agents become informed, e need prices to contain some information hen z =0. Thus, e restrict attention to parameters that ensure P 0 6= P 1. Belo e derive sufficient conditions for this assumption. The information set of an uninformed agent given this equilibrium price corresponds to the partition Ω 0 = {{ω 1,ω 3 }, {ω,ω 4 }}. (1) Next, suppose z>0. First, note that in general, z>0canonlybeanequilibriumifp (ω )=P (ω 3 ). This is because state ω 1 (respectively, ω 4 ) involves lo (high) supply and high (lo) fundamentals, implying P (ω 1 ) ill be strictly higher than the price in any other state hile P (ω 4 ) ill be strictly loer. It follos that if P (ω ) 6= P (ω 3 ), the price ould fully reveal ω. But then no agent ould agree to pay to learn ω, soz =0. Hence, the only candidate equilibrium in hich z>0is one in hich P (ω )=P (ω 3 ), and the information of an uninformed agent corresponds to the partition Ω z = {{ω 1 }, {ω,ω 3 }, {ω 4 }}. () Comparing Ω z to Ω 0 reveals that neither partition refines the other. Hence, there is no sense in hich uninformed agents kno more about the fundamentals hen some traders are informed than hen none are. Equilibrium prices do not inherently become more informative as more traders acquire information they simply convey different information. This contrasts ith hat Grossman and Stiglitz (1980) find in their model, here prices become more informative in a ell-defined sense as more agents become informed (specifically, prices are more informative in the Blackell sense). Since the fraction of informed agents z changes the informational content of prices, e ould expect complementarities to occur if the information conveyed by prices hen z > 0 is less helpful for making investment decisions than the information conveyed by prices hen z =0. Anextreme example of this occurs hen π 1 = π 4 =0, i.e. ex and e are perfectly correlated. In this case, the true state of the orld ω must be either ω or ω 3.SinceΩ z does not distinguish beteen these to states, equilibrium prices hen z>0convey no additional information beyond hat uninformed traders already kno. By contrast, the equilibrium price hen z =0does distinguish the to states so long as P 0 6= P 1. The agent ould have no need for additional information hen z =0, but ould value information on e hen z>0. Learning is thus a strategic complement. The same logic applies if x and are imperfectly but positively correlated. Intuitively, the fact that agents can perfectly deduce ex hen z =0implies that prices convey some information about e given the to are correlated. But since ex and e are positively correlated, a high (lo) realization of the supply shock ex is likely to be accompanied by high (lo) demand from informed agents. Having more informed agents respond to e thus jams the signal about ex that ould otherise be conveyed by the price of the asset. In sum, if the information content of prices is given by (1) and () and ex and e are positively correlated, the value of learning ill be higher for some z>0than for z =0. We no derive sufficient conditions for there to exist an equilibrium price function consistent ith (1) and (), and then sho that these conditions are compatible ith ex and e being positively correlated. Prices ill be extreme in states ω 1 and ω 4 if agents are not indifferent beteen the asset and money in these states. This can be ensured by choosing parameters appropriately. 3

Consider first the case here z =0. We ish to ensure there is a unique equilibrium price ith P 0 6= P 1.Wespecifically look for an equilibrium in hich uninformed agents prefer the asset hen supply is high, i.e. ex = x 1, and money hen supply is lo, i.e. ex = x 0. Market clearing implies x I ( e, P)+x U (P )+ P ex =1. (3) Since x I ( e, P)+x U (P ) 0, e can rearrange (3) to conclude that the price must be at least hen ex = x 0. Uninformed agents ould strictly prefer money to the asset upon learning ex = x 0 if the conditional expectation E[ e ex = x 0 ] ere less than the price hen ex = x 0. Hence, if E[ e ex = x 0 ]= π 1 + π 3 < π 1 + π 3, (4) uninformed traders ould prefer money if they learned ex = x 0. Inthatcase,(3)impliesthat P 0 =. Similarly, since xi ( e, P)+x U (P ) 1 P, the price hen ex = x 1 is at most +1. If the equilibrium price reveals ex = x 1, e can be assured that uninformed agents ould prefer the asset if E[ e ex = x 1 ]= π + π 4 π + π 4 > +1 (5) +1 It then follos from (3) that P 1 must equal in equilibrium. As long as +1 6= (6) e are assured that hen z =0, there is a unique equilibrium ith P 0 6= P 1. To ensure there is no other equilibrium in hich P 0 = P 1,efurtherrequire E[ e ]=(π 1 + π ) +(π 3 + π 4 ) > +1 (7) Under (7), the unconditional expectation of e exceeds the highest possible market clearing price. Hence, if there ere an equilibrium in hich P 0 = P 1, uninformed traders ould prefer to buy the asset in that equilibrium. But if this is their demand, market clearing ould imply P 0 6= P 1,a contradiction. In sum, conditions (4) through (7) together imply that hen z =0, there is a unique equilibrium price that conveys information according to (1). Next, e provide conditions hich imply that there exists some z>0such for hich P (ω )=P (ω 3 ) is an equilibrium. We look for an equilibrium in hich uninformed agents buy the asset at this common price. Suppose h i E e ω {ω,ω 3 } = π + π 3 > +1 (8) π + π 3 This condition says that the market clearing price if all agents ere to buy the asset in state ω 3 is less than the expected value of the asset. Hence, uninformed agents ould ish to buy the asset in this state. If e sharpen (6) to require that there exists a particular z (0, 1) such that +1 z = +1 e can ensure there exists an equilibrium in hich P (ω )=P (ω 3 ) hen z = z,iththecommon price given by the expression in (9). (9) 4

To verify that conditions (4) through (9) are compatible ith ex and e being positively correlated, consider the case here π 1 = π 4 =0. In this case, conditions (4) through (9) reduce to the folloing: (i) < (ii) > +1 = +1 z for some z (0, 1) It is easy to find parameters satisfying these conditions. As an example, let =10, x 0 =0.5 and x 1 =0.51, soz =0.1. These values imply +1 =7.7 and =6.67. No,set =5and =10 to satisfy (10). If π 1 = π 4 =0, the unique equilibrium hen z =0involves P (ω )= +1 =7.7 and P (ω 3 )= =6.67; uninformed traders hold the asset in state ω and money in state ω 3.Since agents are acting optimally, there is no value to learning e. The only other potential equilibrium is hen z = z,inhichcasep (ω )= +1 +1 z x 1+1 =7.7 = x 0+1 = P (ω 3 ); at this price, uninformed traders prefer the asset. Since uninformed tradersendupbuyinganovervaluedassetinstateω 3, they ould benefit from learning e. In particular, their expected gain ould equal (10) π 3 (1 /P (ω 3 )) (11) hich is strictly positive as long as π 3 > 0. Since z (0, 1), agents must be indifferent beteen learning e and not learning. Hence, for z to be an equilibrium, the cost of information c must exactly equal the value of information in (11). Under this assumption, the complementarity in learning allos multiple equilibria: either no agents are informed or a fraction z of agents are informed. Note that conditions (4) through (9) ould continue to hold for these parameters even if e slightly increased π 1 and π 4 and slightly decreased π and π 3, so the value of information can be higher hen z = z than hen z =0even hen ex and e are imperfectly positively correlated. Correlation as a Source of Complementarities The previous section derived sufficient conditions in the case for the value of information to be higher hen some fraction of agents are informed than hen none are. The advantage of the case is that it yields distinct information sets Ω z for z =0and z>0 (expressions (1) and (), respectively) that sho prices need not convey more information hen more traders are informed. Hoever, the discrete nature of this example can make the results seem special and knife-edge, especially since an equilibrium price only exists at to isolated values of z. We no argue that alloing noisy supply and fundamentals to be positively correlated inherently captures the scenario e had attempted to model in our 000 paper, namely that hen some agents acquire information they can change prices in such a ay that makes it more difficult for remaining traders to infer the fundamentals from the price of the asset. We confirm this by demonstrating that correlation beteen noisy supply and fundamentals ill lead to complementarities hen ex has a continuous distribution, as ell as hen both ex and e are continuously distributed as in Grossman and Stiglitz (1980). In our original paper e questioned hether informed traders ould generally cause prices to be more informative as in the Grossman and Stiglitz (1980) model. A key feature of their model is that as more agents became informed, certain prices became more likely to be associated ith certain fundamentals, e.g. very high prices ould essentially emerge only hen fundamentals ere favorable. 5

We conjectured there might be circumstances in hich this as not the case. A positive correlation beteen noisy supply and fundamentals turns out to be just such a circumstance. When the to variables are positively correlated, ex and e are likely to either both be high or both be lo. When fe traders are informed, prices ill largely depend on, and generally vary ith, ex. As a result, different realizations of e are likely to be associated ith distinct prices, namely the prices associated ith the most common realizations of ex for a given e. By contrast, hen many traders are informed, prices ill depend on both e and ex. Since the to variables have offsetting effects on the price high values of e increase the price hile high values of ex decrease it different realizations of e ill tend to be associated ith similar prices. It is then harder to infer e from prices. Nothing in this intuition relies on the idiosyncratic features of the case. Before e confirm that our mechanism can occur in more general settings, it is orth commenting hether it is reasonable to allo noise to be correlated ith fundamentals. Grossman and Stiglitz originally assumed the to are independent, and their specification as subsequently adopted by others. Hoever, the motivation for this assumption as convenience rather than to capture some essential feature of the economic environment being modelled. In fact, there are various circumstances in hich the non-fundamental forces that noise trading is meant to capture ill naturally be correlated ith fundamental forces. For example, one justification for noise trading is liquidity: agents sell assets not in response to changes in the fundamentals but because they need immediate cash. Under this interpretation, one might expect that over the business cycle, fundamentals ill be lo hen liquidity demands are high. We even appealed to this scenario in our 000 paper to motivate our parameter choices. Ironically, the proper example for complementarities e provide here relies on the opposite correlation. But there are circumstances in hich a positive correlation is more appropriate than a negative one, e.g. if there is a strong hedging component to the demand for assets, or if financing frictions lead firms to issue more equity during favorable times. Indeed, one ay to motivate the correlation in the example (and the generalization e discuss belo) is to assume that a fraction of those ho initially on the asset have access to a private technology ith a rate of return R. If more private projects tend to be available in good times, a seemingly plausible assumption, then this fraction ill tend to be higher hen e is high, and as long as R is sufficiently large those traders ho have such opportunities ill sell their stock holdings to invest in the private technology. This implies the supply shock ex ill be positively correlated ith e. To demonstrate that the complementarity in the previous section does not hinge on the special features of the example, consider the case here ex has a continuous distribution. Since ex and e can be correlated, the distribution of ex ill generally vary ith the realization e. To ensure the to ill be positively correlated, e assume f(x e = ) is increasing in x and f(x e = ) is decreasing in x. Thus, a high value for e tends to be associated ith higher realizations of ex. In contrast to the case, the equilibrium price ill no be defined for any z [0, 1], and as e no sho it ill be possible for the value of information to increase in z over a range rather than at to distinct points. To simplify matters, suppose e equals and ith probability 1 each, so the unconditional expectation of e is. We further impose a symmetry assumption on the conditional density for x given e : Let x denote the value of ex for hich the market clearing price equals hen half of the ealth of non-noise traders is allocated to buying the asset, i.e. x solves +1/ x +1 = +. 6

We assume the support of x is restricted to [0, x ],andthat f(x e = ) =f(x x e = ) This assumption implies the folloing ill be an equilibrium. Uninformed agents hold money if the price of the asset exceeds thetoifthepriceequals clearing condition given this demand, and is given by P x, = if x +1 ³ if x +1 h +z if x +1, hold the asset if the price is belo, and are indifferent beteen. The price P (x, ) is the unique function consistent ith the market, +z +z, (+z) ³ (+z) if x +1 +1 if x +1 (+1) i, (+1) P (x, ) = if x +1 ³ if x +1, (+1 z) h (+1 z), +1 z ³ +1 z if x +1, +1 +1 z if x +1 +1 if x +1 +1 An implication of this price function is that for any p 6= beteen and, the equilibrium price is such that there exists a unique x 0 (p) such that P (x 0 (p),)=p and a unique x 1 (p) such that P x 1 (p), = p. Using the monotonicity of f(x e = ) and the direction of asymmetry beteen x 0 (p) x and x 1 (p) x for p> and p<, it follos that Pr = P (, ) =p R 1 for p Q This confirms that the demand schedule of uninformed is optimal given these prices. To generate complementarities, e need a sufficiently strong positive correlation beteen and x, hich amounts to a restriction on the rate at hich f(x e = ) decreases ith x. One example of a function that gives rise to complementarities is the truncated normal, i.e. 1 e x σ f(x e = ) = σ π Φ (x ) 1 for x [0, x ] here Φ ( ) denotes the CDF of a normal ith mean 0 and variance σ. The correct expression for the value of information, as described by Chamley (007), is given by g (z) = 1 Z +1 z 1 ³ 1 f x P (x, ) e = dx + 1 Z (+z) " # 1 ³ P x, 1 f x e = dx +z 1 (1) Figure 1 illustrates g (z) for the folloing parameter values: =1.0, =0.9, =1.1, andσ =0.. The value of information is increasing as z ranges from 0 to about 0.5 and is decreasing in z thereafter. This reflects the interaction of the to opposing forces. On the one hand, as z increases, the distribution of prices hen = concentrates mass on the same prices as the distribution of prices hen =. This force ould be present henever the conditional density function for high (lo) values of concentrated mass on high (lo) values of x, and ould tend to make information about fundamentals more valuable. At the same time, the value of using the information to trade optimally shrinks as prices are pushed closer to fundamentals and the payoff on the asset converges to that on money, and this force eventually causes the value of information to decrease ith z. Nothing in the above argument hinges on the distribution of e being discrete. Indeed, a recent short note by Ganguli and Yang (007) shos that if noise and fundamentals are positively correlated, i 7

0.011310 ghzl 0.0113118 0.0113116 0.0113114 0.1 0. 0.3 0.4 0.5 0.6 z Figure 1: The function g (z) information acquisition ill be a strategic complement in the original Grossman and Stiglitz frameork, here both ex and e have continuous distributions. Suppose agents have negative exponential utility U (c) = e γc rather than linear preferences as in our frameork. Suppose further that the dividend on the asset is e + eε here e and eε are i.i.d. normal ith mean zero and variances σ and σ ε respectively. The supply of the asset ex is equal to ρ e + ey here ey is independent of all other variables and is itself normally distributed ith mean zero and variance σ y. Ganguli and Yang sho that the value of becoming informed is increasing in z over the interval [0,ργσ ε) and decreasing otherise. This result reflects the same mechanism, namely that hen the to variables are correlated, informed traders can act to make the distribution of prices for different fundamentals more similar rather than more distinct, and this in turn can result in learning being a strategic complement even as prices are pushed closer to their fundamental values. 3 References Barlevy, Gadi and Pietro Veronesi, 000. Information Acquisition in Financial Markets Revie of Economic Studies, 67(1), p79-90. Chamley, Christophe, 006. Complementarities in Information Acquisition ith Short-Term Trades Mimeo (first version: 005) Chamley, Christophe, 007. Strategic Substitutability in Information Acquisition in Financial Markets forthcoming in Revie of Economic Studies. Ganguli, Jayant and Liyan Yang, 006. Supply Signals, Complementarities, and Multiplicity in Asset Prices and Information Acquisition Mimeo, Cornell University. Ganguli, Jayant and Liyan Yang, 007. Strategic Complementarities ith Correlated Noise and Fundamentals Mimeo, Cornell University. Veldkamp, Laura, 006. Media Frenzies in Markets for Financial Information American Economic Revie, 96(3), p577-601. 8