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Complementarities, multiplicity, and supply information∗ Jayant V Ganguli†and Liyan Yang‡
Abstract If traders can obtain private information about the payoff and the supply of a stock then there can exist (i) complementarity in information acquisition and (ii) multiple equilibria in the financial and information markets. The additional dimension of supply information increases coordination possibilities in the financial market, leading to multiple equilibria. The existence of two information sources can lead to information acquisition being complementary. The multiplicity of equilibria is suggestive of excess volatility and crashes. The different financial market equilibria imply differing patterns of cost of capital and volume of trade.
1
Introduction
Traders in financial markets may seek information about factors that are not directly related to the payoffs from stocks but nonetheless are useful in refining the information that they obtain about payoffs directly. For instance, information about liquidity trades of any stock constitutes an additional dimension in traders’ knowledge about the stock when making trading decisions. Alternatively, a trader ∗ For
comments and advice we thank Prasun Agarwal, Gadi Barlevy, Markus Brunnermeier, Jordi Caballe, Yan Li, Maureen O’Hara, Karl Shell, Vikrant Tyagi, Laura Veldkamp, and audiences at Cornell and IESE and at the Cornell / Penn State Macroeconomics, the 2006 NBER General Equilibrium, the 2007 FMA, the Fall 2006 Midwest Economic Theory, and the 2007 RES meetings. The suggestions of Larry Blume, David Easley, Douglas Gale, Xavier Vives, and anonymous referees were particularly helpful in improving the paper. Many of the results were originally circulated in 2006 under the title ‘Supply signals, complementarities, and multiplicity in asset prices and information acquisition.’ All errors are our responsibility. Keywords CARA-normal, supply information, complementarities, multiplicity. JEL codes D82, D83, G14. † Faculty of Economics, University of Cambridge, Sidgwick Avenue, Cambridge CB3 9DD, UK, . ‡ [Corresponding author] Department of Economics, Cornell University, Ithaca, NY 14850, USA, .
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may face risk in the income derived from labor and revenue from assets other than stocks. Traders may use a stock to hedge such risk if the risk in the nonstock income is correlated with the stock’s performance. This correlation may also provide investors with information about the stock’s future performance. These examples suggest that the information available to traders in markets is multidimensional. As may be obvious from the preceding examples, we seek to study one aspect of multidimensional information on stocks: that pertaining to the (net) supply of stocks to be traded. We adapt the CARA-normal REE (rational expectations equilibrium) framework, where CARA denotes constant absolute risk aversion, (Grossman 1976, Grossman and Stiglitz 1980, Diamond and Verrecchia 1981, and Admati 1985) which is commonly used for studying differential information economies, by allowing traders to obtain information in the additional dimension of supply.Using this adapted framework in which the price plays an informational role and provides a channel for refining information, we seek to address the following questions. (i) Does the stock price always reveal more information about payoffs when more investors with private information trade? (ii) Is this price uniquely predicted? The answers to these questions in most existing analyses of CARA-normal REE models have been positive. In our extension of the standard CARA-normal model we find that the answers can, in fact, be negative. In particular, when the private payoff or supply information is noisy enough, the financial market has two (linear) partially revealing REE (PR-REE) price functions—with opposing properties regarding information content—due to the possibility of coordination among the traders’ self-fulfilling expectations. One of these equilibria shares the informational properties of the unique REE in the studies just cited, but the other does not. Our extension allows us to uncover an equilibrium map of PR-REE prices indicating that the uniqueness of equilibrium in previous analyses is not robust. We also find that strategic complementarity in (costly) information acquisition can exist and lead to multiple equilibria in the information market. An important aspect of our results is that the multiplicity in the financial market is not generated by multiplicity in the information market. Rather, the extra dimension of supply information increases the possibility of coordination among the traders’ self-fulfilling expectations in an REE, leading to multiplicity in the financial market. Independent of this multiplicity, the extra dimension of supply information may also make traders’ decisions to acquire information complementary. Another consequence of our extension is also that, as a particular case, we are able to study the CARA-normal REE model in terms of a general equilibrium model with a continuum of traders whose stock endowments make the REE price noisy, thereby avoiding the need for unmodeled noise traders (see Dow and Gorton 2008 for a recent discussion). This allows us to avoid two critiques that apply 2
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to some other mehods of doing away with noise traders: “schizophrenia” (Hellwig 1980, Laffont 1985), when there are only a finite number of traders (Diamond and Verrecchia 1981); and infinite variance of aggregate supply with an infinite number of traders (Grundy and McNichols 1989). A closely related model in Medrano and Vives (2007) may also be used to avoid unmodeled noise traders by obtaining noise–trader demand from standard expected utility–maximizing hedgers who are extremely risk averse or whose individual endowment of an untraded asset, with return correlated to that of the stock, is uncorrelated with the aggregate endowment of the asset (see Section 4.4, particularly 4.4.3, in Vives 2008 for a discussion of this structure and its solutions). The multiplicity in the information and financial markets means that there can be a large number of overall equilibria in the model, a possibility that suggests price crashes and excess price volatility (Section 4.1). We further explore the differing implications of each REE for the cost of capital to a firm and the volume of trade. The two REE prices feature differing costs of capital. Also, the volume of trade in the new REE price is driven by speculative motives, and it features more aggressive trading as the number of informed traders increases (Section 3.1). The rest of this paper is organized as follows. We first briefly discuss the information structure and related literature. We then proceed to Section 2, which presents the formal analysis by describing the simplest model that captures the changed information structure: an endowment economy. For this basic model we discuss the financial market equilibria in Section 3 and the information market equilibria in Section 4. The two main results for this structure are given in Propositions 1 and 3. In Section 5, we present an alternative setting where traders may purchase supply information and find more general results (Proposition 5). Section 6 concludes the paper, and all the results are proved in the Appendix. The Information Structure Allowing traders to have and use information about stock supply enables us to capture some interesting information structures in financial markets in a simple manner. As already mentioned, one instance is provided by liquidity trades.1 Information on supply may also be obtained from the NYSE Open Book or by sentiment-oriented technical traders and front-runners. The information maintained by investment banks on the float or that obtained by dealers or market makers with access to the order book are examples of similar information. Supply information that is costly to obtain, as in the preceding examples, is an additional dimension of information and may lead to complementarity and substitutability in information acquisition (Section 5). This multidimensional information structure could also arise when traders have hedging motives. In particular, the income from nontradable assets—for instance, 1 We
thank a referee for suggesting this line of interpretation.
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human capital—may be correlated with the performance of the stock market. Similar formulatons of hedging motives appear in Duffie and Rahi (1995, Sec. 3.1), Dow and Rahi (2003), Lo et al. (2004), Biais et al. (2008), Goldstein and Guembel (2008, Section 7.2), and Watanabe (2008). Therefore, traders have an incentive to use stocks to hedge their nontradable asset-income risk. How strong this hedging motive is depends on how closely their nontradable assets co-move with the stock market. There may exist both a common component and an idiosyncratic component in the correlation between investors’ nontradable assets and stocks. For example, the business cycle has a significant impact on the stock market; at the same time, it affects the labor income of workers in almost all sectors, but to differing extents. Our framework with endowments of stock and no noise traders permits analysis of financial markets with such common and idiosyncratic hedging motives.2 Related Literature The financial and information market equilibria in the CARA-normal framework have been widely applied—for example, to study price crashes, contagion, contrarian and trend-following investor behavior, uniqueness of equilibria in coordination games, media frenzies, and excess covariance of asset prices. The consequences of information about supply of stocks have also been examined in the study of price crashes and informational efficiency of imperfectly competitive markets. Brunnermeier (2001) and Vives (2008) contain excellent discussions of these and other applications of the CARA-normal REE framework. Watanabe (2008) and Biais et al. (2008) study information asymmetry in an overlapping generations framework with multiple risky assets that have normally distributed payoffs and CARA-utility investors. Multiplicity of financial market equilibria obtains in these models as a consequence of the self-fulfilling prophecies of overlapping generations. In Grossman and Stiglitz (1980), information acquisition exhibits strategic substitutability and so leads to a unique equilibrium in the information market. Veldkamp (2006a, 2006b) introduces an information production sector that supplies the payoff information at an endogenous price into the Grossman-Stiglitz model in a dynamic setting to generate complementarity. Other models also generate complementarity in information acquisition outside the CARA-normal framework (see e.g. Barlevy and Veronesi 2000, 2008; Chamley 2007). Our results suggest that the analytically tractable static CARA-normal REE models can be used to analyze situations in which (i) information acquisition is complementary across traders and (ii) multiplicity is potentially useful to understand economic data and puzzling economic phenomena. 2 The
supplementary appendix (Ganguli and Yang 2008) provides details.
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2
The Model
This section presents the simplest model that captures the changed information structure by studying an economy where traders are endowed with stochastic amounts of a risky asset (the stock) and where the aggregate supply of the stock acts as noise, which makes the REE price partially revealing. We discuss most of the results on multiplicity and complementarities in the context of this model and provide an extension of the model with costly supply signals in Section 5. There is a continuum of traders, indexed by [0, 1], who live for two periods. There are two assets—one riskless (a bond) and the stock—traded in a financial market that opens in period 2. The payoff to each unit invested in bond is normalized to 1 and, at the beginning of period 1, trader i is endowed with xi units of the stock and n¯ i units of bond, where n¯ i is constant: ¯ 1/ρ x ), ρ x > 0, ηi ∼ N (0, 1/ρη ), ρη > 0, xi = x + ηi , with x ∼ N ( x, and where the (ηi )i∈[0,1] are assumed independent across the traders. The payoff to each unit invested in the stock is denoted by v, and ¯ 1/ρv ) with ρv > 0. v ∼ N (v, The realization of xi is observed only by trader i. We adopt the convention that a law of large numbers holds for this continuum economy (see the technical appendix in Vives (2008), so the aggregate supply of stock is3 Z 1 0
xi di = x.
We can interpret x as a common shock due to the fluctuation of the entire economy and ηi as an idiosyncratic shock due to variation in individual ability.4 In the studies by Grossman and Stiglitz (1980), Diamond and Verrecchia (1981), and Grundy and McNichols (1989), where traders have random endowments, there is no common component across traders. These random endowments mean that there is also a hedging motive for trade. As mentioned previously, a similar structure on trader characteristics is commonly adopted in the literature. The supplementary appendix (Ganguli and Yang 2008) provides details on how a hedging motive driven by nonstock income risk, which is correlated with stock income, is represented by the endowment structure described here. 3 Doob
(1937, Thm. 2.2, p. 113) and later Judd (1985) pointed out the problem with an exact law of large numbers for a continuum of independently and identically distributed (i.i.d.) random variables; see also Sun (2006). 4 The existence of multiple equilibria in the financial market does not require a common shock to the endowments of all agents. A common shock to a positive fraction of the agents is enough to generate multiple equilibria. A proof of this assertion is available from the authors on request.
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At the beginning of period 1, the information market opens. At cost κ > 0, trader i can independently purchase a private information payoff signal yi , where yi = v + ε i , with ε i ∼ N (0, 1/ρε ), ρε > 0. We assume that (v, x, (ε i , ηi )i∈[0,1] ) are mutually independent. Trader i is called informed if she purchases yi and uninformed otherwise. Once the traders have made their acquisition decision, period 1 ends and the financial market opens in period 2. We relabel the traders so that the set of informed traders is [0, λ]; that is, the fraction of informed traders is λ. Each trader cares only about wealth W at the end of period 2 and has the (von Neumann–Morgenstern) utility function u with CARA parameter γ > 0, u(W ) = −e−γW . We normalize the price of bond to 1 and denote the price of the stock by P. We denote trader i’s demand for the stock by Di and for the bond by Bi . Then trader i0 s wealth at the end of period 2 is W2i = Bi + vDi . The price P of the stock is a function of diverse private information. As is common in much of this literature, we focus on REE in which the price aggregates all the diverse information and is a function only of the aggregates v and x (i.e., P = i ( P, x )) the demand function of an P(v, x )). If we denote by DiI ( P, xi , yi ) (resp. DU i informed (resp. uninformed) trader i in the financial market, an overall equilibrium in the model is a tuple i )i∈(λ∗ ,1] λ∗ , P, ( DiI )i∈[0,λ∗ ] , ( DU
i ( P, x )) such that ( P, ( DiI ( P, xi , yi ))i∈[0,λ∗ ] , ( DU i i ∈(λ∗ ,1] ) is an REE in the financial mar∗ ket and λ is the equilibrium fraction of informed traders given that the traders i ( P )) coordinate on ( P, ( DiI ( P))i∈[0,λ∗ ] , ( DU i ∈(λ∗ ,1] ) in the financial market. This is formalized in Definitions 1 and 2, where E[VIi (λ)| xi ] denotes the expected (indirect) utility of a trader i if she is informed and E[VUi (λ)| xi ] denotes the expected (indirect) utility if she is uninformed, for any λ ∈ [0, 1] , and Ri (λ) = E[VIi (λ)| xi ]/E[VUi (λ)| xi ] .5 The absence of wealth effects in CARA utility means that the portfolio choices do not depend on ( xi )i∈[0,1] and that it is possible to construct λ and Ri (λ) independently of ( xi )i∈[0,1] . For general utility functions, λ and Ri (λ) will depend on xi . This would introduce non linearities, making the financial market equilibria difficult to construct, so we do not pursue it here. 5 Throughout, we use
A|{ B1 , . . . , Bn } to mean the random variable A conditioned on the random
variables { B1 , . . . , Bn }.
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Definition 1 (Financial market equilibrium) Given a fraction (λ) of informed traders, i ( P, x )) a price function P( x, v) and demand functions ( DiI ( P, xi , yi ))i∈[0,λ] and ( DU i i ∈(λ,1] i ( P, x )) maximizes the expected utility of constitute an REE if (i) DiI ( P, xi , yi ) (resp. DU i informed (resp. uninformed) trader i conditional on her information, including that provided by the prices, given the price P and (ii) the markets for the stock and the bond clear for almost every realization of (v, x ). Noting that we have defined utility to be negative, an equilibrium λ∗ ∈ [0, 1] is given as follows. Definition 2 (Information market equilibrium) λ∗ = 0 if Ri (0) > 1, λ∗ = 1 if Ri (1) < 1, or λ∗ ∈ [0, 1] if Ri (λ∗ ) = 1 for all i ∈ [0, 1]. Hence, if a trader does not benefit from becoming informed when no other trader is informed (i.e., if Ri (0) > 1) then it is an equilibrium in the information market for no one to buy the information (λ∗ = 0) . On the other hand, if a trader is strictly better off from being informed when all other traders are also informed (i.e., if Ri (1) < 1) then in equilibrium all traders in the market will be informed (λ∗ = 1). In general, for a given fraction of informed traders λ∗ , if every trader is indifferent between becoming informed and staying uninformed (i.e., if Ri (λ∗ ) = 1) then that fraction λ∗ is an information market equilibrium.
3
Equilibria in the Financial Market
We now establish the existence of two REE in the financial market, focusing on REE with P(v, x ) as a linear function. Suppose the traders conjecture the price function as P = a + bv − cx, with b ≥ 0 and c ≥ 0. (1) Note that this price is always measurable with respect to the information of both types of traders because it is a function of the prior information of the traders. Then the information contained in P and xi can be expressed by the private signal zi for each trader i, zi = v − (c/b)( x − E[ x | xi ]) = v − (c/b)(ρ x + ρη )−1 [ρ x ( x − x¯ ) − ρη ηi ],
(2)
when b > 0 and c ≥ 0. Let θi = −(c/b)( x − E[ x | xi ]). Conditional on xi , we have θi ∼ N (0, ρθ ) with ρθ = (b/c)2 (ρ x + ρη ). (3) In the absence of xi , for trader i, the informativeness of the price about the payoff v is given by its precision in the estimation of v, that is (Var[v| P])−1 = (b/c)2 ρ x . 7
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Equation (3) captures the benefit to trader i from the supply signal xi in the form of the additional precision (b/c)2 ρη . This benefit depends on (b/c), which in turn will be determined by the value of λ in the (overall) equilibrium. Note that (b/c) measures the informativeness of the price about the payoff v. In particular, a price P with a larger value of (b/c) is, ceteris paribus, a “more sufficient” Blackwell experiment for v than a price with a lower value of (b/c) (see Kihlstrom 1984 or the technical appendix in Vives 2008). This is consistent with the measure used by Grossman and Stiglitz (1980, pp.396-397). Henceforth, when we refer to the informativeness of price P, it is with respect to the payoff v unless specified otherwise. Informed Traders An informed trader i has information { xi , yi , P} and uses { xi , yi , zi } to update her beliefs: v|{ xi , yi , zi } ∼ N (µiI , ρ I ), with 1 µiI = ρ− I ( ρv v¯ + ρε yi + ρθ zi ), ρ I = ρv + ρε + ρθ .
(4)
Given her posterior beliefs (4), the demand function of informed trader i is DiI ( P, xi , yi ) = γ−1 ρ I (µiI − P).
(5)
Note that each informed trader has two possibly profitable sources of information: the information about payoff v, which constitutes direct speculation; and the information about aggregate supply x. Uninformed Traders An uninformed trader i only has information { xi , P} (or, equivalently, { xi , zi }) and i , ρ ) with uses it to update her beliefs: v|{ xi , zi } ∼ N (µU U −1 i = ρU (ρv v¯ + ρθ zi ), ρU = ρv + ρθ . µU
(6)
Given her posterior beliefs (6), uninformed trader i’s demand function is i i DU ( P, xi ) = γ−1 ρU (µU − P ).
(7)
In equilibrium, the stock market clears: Z λ 0
DiI ( P, xi , yi )di +
Z 1 λ
i DU ( P, xi ) di = x.
(8)
We find the REE by solving equation (8) for P and then verifying that P is of the form conjectured in (1). Proposition 1 characterizes the REE. As we elaborate in what follows, one of the REE in our model exhibits the same properties as 8
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the REE in Grossman and Stiglitz (1980) and, when traders coordinate on it, information acquisition in the first period is a strategic substitute. So, we label the corresponding values of the variables for this REE by SUB. The other REE differs from that of previous studies and information acquisition is strategically complementary when traders coordinate on it. We label the corresponding values of the variables by COM. Proposition 1 If γ2 > 4λρε ρη > 0 then there exist two PR-REE in which P = a + bv − cx, where
a = K −1 (ρv v¯ + βρ x x¯ ), b = K −1 (λρε + ρθ ), c = K −1 ( βρ x + γ)
and where β = (b/c) takes one of two values, q SUB −1 2 β = (2ρη ) γ − γ − 4λρε ρη , q COM −1 β = (2ρη ) γ + γ2 − 4λρε ρη , with K = ρv + λρε + ρθ and ρθ = β2 (ρ x + ρη ).6 This result suggests that, for a linear PR-REE price function to exist, the signals cannot be very sharp, that is, we need low ρε or ρη .7 This dependence on the precisions of the signals is similar to that found in Bhattacharya and Spiegel (1991) and Medrano and Vives (2004), where a linear REE price exists when the adverse selection problem due to the presence of informed traders is moderate. In contrast, the existence results in Biais et al. (2008) and Watanabe (2008) require restrictions on the prior precisions of the payoff and supply (ρv and ρ x in our notation). In the model of Grossman and Stiglitz (1980), β increases as the fraction of informed traders increases, that is, (∂β/∂λ) > 0. This is true of the equilibrium with SUB variables in our model. In contrast, in the equilibrium with COM variables, β decreases as λ increases. However, for any given value of λ, the COM– REE price is always more informative about v than the SUB–REE price because β COM ≥ β SUB . The SUB–REE can be seen as a generalization of the equilibrium in Grossman (1976), Grossman and Stiglitz (1980), and Diamond and Verrecchia (1981) in our setup, whereas the COM–REE is a new (partially revealing) REE that does not exist in the previous studies. 6 Although all the variables here depend on λ, we suppress this dependence to simplify notation throughout. 7 Fully revealing REE prices exist irrespective of the sign of ( γ2 − 4λρ ρ ). However, we focus ε η on the more interesting PR-REE prices. Also, our result suggests nothing about the existence of a possibly nonlinear PR-REE price when γ2 < 4λρε ρη beyond indicating that a linear price function does not exist.
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Figure 1
Payoffinformativeness of linear REE (1 − e−β )
Payoff-fully-revealing REE
COM-REE
SUB-REE
γ 2 / (4λρ ε )
0
ρη
Figure 1: Map of linear REE prices and the precision of supply information. The SUB–REE is the unique linear PR-REE in Diamond and Verrecchia (1981) as a limiting case when ρη → 0, whereas the COM–REE price approaches an REE price that fully reveals v. Our results show that this uniqueness result is not robust to small perturbations in the value of ρη and that multiple PR-REE prices exist for small positive values of ρη . Even when ρη = (γ2 /4λρε ) > 0, there is a unique PR-REE price, that, again, is not robust to small perturbations in the value of ρη .8 Figure 1 illustrates the map of linear PR-REE prices as ρη varies, using a renormalization f ( β) = 1 − e− β of β on the vertical axis that takes the value 1 when the REE price reveals v fully ( β = ∞) and the value 0 when the REE price reveals nothing about v ( β = 0). The map also contains an REE price that reveals v fully. This price always exists as an equilibrium price in the financial market owing to the self-fulfilling nature of an REE. However, as is well-known (see e.g. Chapter 4 and Section 3.1 in Vives 2008) this fully-revealing REE is not implementable as the equilibrium of a welldefined trading game even though each trader is “informationally small” given the assumption of a continuum of traders. The possibility of two REE stems from the coordination of the traders’ selffulfilling expectations given their stock endowment. The information contained in the endowment signals (xi ) is useful for traders to make their investment decisions, and the usefulness of xi depends on the informativeness of the price system. To see this, note that xi works through the signal zi whose precision (as given by equation 8 If
ρη = (γ2 /4λρε ) then the quadratic equation determining the coefficient (b/c) of the price function has a unique root. See the proof of Proposition 1 in the Appendix.
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(3)) ρθ = β2 (ρ x + ρη ) and is positively related to the informativeness measure β. Formally, the aggregate demand function shifts by ( βρη /γ) x, in response to the variation in xi . The aggregate demand function is given by D ( P; v, x ) =
)v + ( βρη γ−1 ) x − (γb)−1 (λρε ) P. {z } | {z }
CONSTANT + (λρε γ
|
−1
Effect of yi
(9)
Effect of xi
Therefore, how the price reacts to x depends in turn on the traders’ belief about β. The term ( βρη /γ) x in the aggregate demand function captures the coordination in the traders’ self-fulfilling expectations. Suppose traders conjecture the price function reveals more information (a high β > 0) about v. They respond to this conjecture by shifting their demand function. For example, at a given price, when x rises, the traders demand more of the stock. This increased demand means the equilibrium price falls less than it would if the demand shifted less (a low β) or did not shift (β = 0). As a result, the responsiveness of price to a given change in x decreases, which in turn self-fulfills the initial belief. This equilibrium is the COM– REE. Note that this coordination disappears once the individual endowments contain no information of the aggregate stock supply (ρη = 0). Similar reasoning applies to the SUB–REE (with a low β). See also Vives (2008, Chap. 4, pp.29–32), which discusses the role of self-fulfilling expectations in generating multiplicity of REE. The change in β as λ changes in the two REE is similarly due to the self-fulfilling expectations of the traders. Suppose the traders coordinate on the COM–REE and conjecture that an increase in λ leads to a decrease in β COM . Given the marketclearing condition D ( P; v, x ) = x, where the aggregate demand is given by equation (9), the implied value of β COM as a function of the conjectured value of β COM is given by (λρε /γ)/(1 − ( β COM ρη /γ)). Clearly, an increase in λ will lead to an increase in the numerator of the implied value. However, the increase in the denominator will be greater as the value of β COM is high (relative to that of β SUB ) in the equilibrium before the change in λ. This in turn will lead to a lower implied value of β COM in the new equilibrium after the change in λ, thereby self-fulfilling the beliefs of the traders. In order to ensure that the multiplicity of REE is not an artifact of the continuum of traders, we also considered an economy identical to the one just described but with a finite number of traders. If ρη > 0 and there is a large but finite number of traders in the economy, then there may exist three PR-REE in that economy, (for details see Ganguli and Yang 2008). As the number of traders converges to infinity, the three REE of the finite economy converge to the SUB–REE, the COM–REE, and an REE which fully reveals v. This suggests that the multiplicity of equilibria is not an artifact of the continuum economy but rather is generated by the changed information structure. Indeed, the fact that REE models with normally distributed random variables (and mean-variance utility) may have multiple equilibria due to nonlinearities was also noted by McCafferty and Driskill (1980) in an adaptation 11
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of Muth’s (1961) inventory model. The introduction of supply information in our model suggests that non-linearities and hence, multiplicity may be natural even in the case of portfolio choice, unlike the linear case described by Grossman and Stiglitz (1980) and Diamond and Verrecchia (1981).9 As mentioned earlier, multiple equilibria in the overlapping-generations structures of Watanabe (2008) and Biais et al. (2008) are generated by the self-fulfilling prophecies of the overlapping generations. Our results clarify that, in fact, the changed information structure from allowing traders to use supply signals is an independent new source that yields multiplicity even in the “static” CARA-normal setting without overlapping generations. Implications of Multiplicity in the Financial Market The multiplicity of market-clearing prices in our model is suggestive of price crashes and the possibility that stock prices can exhibit excess volatility. Also, the (unique) REE in the previous studies and its properties have been extensively used in the finance literature. However, the two REE in this model behave very differently in many aspects, and their differing implications in terms of observable variables like volume serve to suggest how one may identify equilibria based on data. Details of subsequently omitted derivations and illustrations are provided in the supplementary appendix (Ganguli and Yang 2008). Cost of Capital The information structure surrounding a company’s stock affects the cost of capital faced by that company (Easley and O’Hara 2004). The cost of capital (CC ) to a firm issuing the stock is measured by the equilibrium required return, that is, CC = E(v − P j ), j ∈ { SUB, COM }. By Proposition 1, ¯ j ∈ { SUB, COM }. E( P j ) = v¯ − (K j )−1 γ x,
(10)
Since K COM ≥ K SUB , on average the SUB–REE has a higher cost of capital than the COM–REE. In the SUB–REE an increase in the dispersion of private information (λ), or in the precision of the private signal (ρε ), leads to a decrease in the cost of capital through the same forces as in Easley and O’Hara (2004). However, in the COM–REE, an increase in λ or ρε raises the cost of capital because the information revealed by the price falls, making the stock riskier and thus raising the risk premium.10 These results are straightforward consequences of Proposition 1. So, although the information structure is important, its effects on the cost of capital depend strongly on the coordination decisions or self-fulfilling beliefs of the traders or other extrinsic factors. R1 Volume of Trade The volume of trade is Q = (1/2) 0 | Di − xi |di (Blume et al. 1994 Wang 1994, Watanabe 2008). Given the demand functions (5) and (7), we can derive E[ Q] analytically (see Ganguli and Yang 2008). 9 The
McCafferty and Driskill (1980) analysis was brought to our attention by Chamley (2008). we assume that traders commit to one REE for all values of the parameters.
10 Here
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Figure 2
(a)
(b)
0.45
0.45
0.4
0.4
SUB-REE
0.35
0.35
0.25
0.2
COM-REE
0.25
0.15
0.1
0.1
0.05
0.05
0
0.5
1
1.5
2
2.5
3
3.5
Precision of Payoff Signal, ρε
0
4
COM-REE
0.2
0.15
0
SUB-REE
0.3
E(Q)
E(Q)
0.3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fraction of Informed Traders, λ
1
Figure 2: The volume of trade in the two REE. Figure 2 shows comparative statics for the expected volume in the two REE with γ = 2, ρv = ρ x = ρη = 1, and x¯ = 0. In panel (a), λ = 1/4; in panel (b), ρε = 1. The hump-shape of the volume in the SUB–REE is similar to that of the equilibria in Watanabe (2008), but not the pattern of volume in the COM–REE. As in Watanabe (2008), the hump-shape can be understood as follows. When ρε is low, the informed traders have little information advantage and so trade mainly for hedging purposes. Similarly, when ρε is high, the SUB–REE price reveals much of their private information. This reduces their information advantage and hence reduces informational trading, leaving only the hedge-motivated trading. For intermediate values of ρε , informational trading will be large. Together with the hedge-motivated trading, which always exists, this generates large trading volume. The monotonic nature of volume in the COM–REE reflects more aggressive trading by informed traders as their signals become more precise. A similar effect is noted in Wang (1994, p.145). Note that ρε and λ affect β in the same way (Proposition 1), so the analysis is similar for λ. Comparative statics results for the volatility of prices are provided in the supplementary appendix (Ganguli and Yang 2008).
4
Equilibrium in the Information Market
We now consider the existence and properties of equilibria in the information market. Information acquisition can be strategically complementary if traders coordi13
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nate on the COM–REE, leading to multiple equilibria. The next result shows that (i) ex ante, the expected indirect utility of becoming informed is proportional to that of staying uninformed for any trader, irrespective of the REE and (ii) the proportions are identical across traders and realizations of xi . Proposition 2 Suppose that traders coordinate on the SUB–REE (or the COM–REE). Then, for any given λ, q E[VIi (λ)| xi ] γκ (ρU /ρ I ), R(λ) = =e E[VUi (λ)| xi ] where ρU and ρ I are given in equations (3), (4), and (6) and Proposition 1 corresponding to the SUB–REE (or the COM–REE, respectively). We now make precise the sense in which information acquisition is a strategic complement (substitute), using the result of Proposition 2. Definition 3 (Strategic complement/substitute) If R0 (λ) < 0 then information acquisition is a strategic complement, and if R0 (λ) > 0 then information acquisition is a strategic substitute. Hence, strategic complementarity (substitutability) will give the traders more (less) incentive to become informed as the fraction of informed traders is getting larger. This definition corresponds to those in Grossman and Stiglitz (1980) and Barlevy and Veronesi (2000). We now present our second result: whether information acquisition is a strategic substitute or a strategic complement depends (in the simple model) on the REE on which the traders coordinate. Complementarity and Substitutability Suppose the traders on one of the REE. Following Definition 2 and p coordinate γκ Proposition 2, if (ρ I /ρU ) > e then a trader would decide to become informed; here (ρ I /ρU ) measures the benefit from being informed and eγκ is directly related to the cost. From equations (3), (4), and (6) it follows that
(ρ I /ρU ) = (ρε /ρU ) + 1 = ρε (ρv + β2 (ρ x + ρη ))−1 + 1. Here (ρε /ρU ) is the benefit from the private payoff signal. In the SUB–REE, as λ increases, the price reveals more about v (i.e., β rises) making it easier for uninformed traders to free-ride on the the informed (i.e., ρU becomes larger). Thus the benefit from the payoff signal decreases, making information acquisition substitutable. In the COM–REE, on the other hand, as λ increases β decreases, making it harder for the uninformed to obtain information from the price and so the relative 14
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benefit of being uninformed (ρU ) decreases. Hence, acquiring information will be complementary. This is consistent with the idea in Barlevy and Veronesi (2000, p.88) that “the crucial component generating learning complementarities is that learning makes identification more complicated for uninformed agents.” Whereas Barlevy and Veronesi (2008) note that the complementarity for the framework in Barlevy and Veronesi (2000) holds only when the fundamentals and noise are positively correlated, no such assumption is needed here. This result on is information acquisition formalized in the following statement. Proposition 3 If the traders coordinate on the COM–REE (resp. SUB–REE) in the financial market, then information acquisition is a strategic complement (resp. substitute). Hence, if traders coordinate on the SUB–REE then there is a unique equilibrium in the information market, as in Grossman and Stiglitz (1980). However, in the COM–REE, multiple equilibria are possible. A general characterization of the information market equilibria when traders coordinate on the COM–REE is provided next and is a direct consequence of Propositions 1 and 2. Corollary 1 Suppose traders coordinate on COM–REE; then #−1 −1/2 " 2 q ρε . R(λ) = eγκ 1 + ρv + (2ρη )−2 γ + γ2 − 4λρε ρη (ρ x + ρη ) If R(0) ≤ 1 then there exists a unique equilibrium λ∗ = 1 and if R(1) ≥ 1 then there exists a unique equilibrium λ∗ = 0. If R(1) < 1 < R(0) then there exist three equilibria: λ∗ = 0, λ∗ = 1, and s 1 2 λ∗ = γ − 2ρη 4ρε ρη
−1
2
ρε (e2γκ − 1) − ρv − γ . ρ x + ρη
From this characterization we can infer the following properties of the interior under the COM–REE: when κ increases, λ∗ increases but when ρε increases, λ∗ decreases. Figure 3 provides an illustration of the information market equilibria under the SUB–REE and the COM–REE with γ = 2, ρε = ρη = ρv = ρ x = 1, and κ ∈ {1/20, 1/10}. In Figure 3(a), traders coordinate on the SUB–REE and so acquiring information is strategically substitutable; this leads to a unique equilibrium of λ∗ = 0.92 when κ = 1/10 and and of λ∗ = 1 when κ falls to 1/20. In Figure 3(b), traders coordinate on the COM–REE and so information acquisition is strategically complementary. There is a unique equilibrium λ∗ = 0 when the cost is high at κ = 1/10; but are three equilibria, λ∗ ∈ {0, 0.89, 1}, when κ falls to 1/20. At λ∗ = 0, the expected indirect utility of a trader from being informed is strictly less than that from being uninformed given that all the other traders are uninformed—that is, R(0) > 1—whereas at λ∗ = 1 we have R(1) < 1. λ∗
15
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Figure 3
(a) SUB-REE
(b) COM-REE
1.15
1.16
1.14
1.1
κ=1/10
1.05
κ=1/10
1.12
1.1 1
R(λ)
1.08
0.95
1.06
1.04
0.9
1.02
κ=1/20
0.85 1 0.8
0.75
κ=1/20 0
0.2
0.4
0.6
0.8
0.98
1
Fraction of Informed Traders,λ
0.96
0
0.2
0.4
0.6
0.8
1
Fraction of Informed Traders,λ
Figure 3: Information market equilibria.
4.1
Implications of Multiplicity in the Information Market
i ) There can be many overall equilibria of the form (λ∗ , P, ( DiI )i∈[0,λ∗ ] , ( DU i ∈(λ∗ ,1] ) in our model. For example, there exist two REE for each equilibrium λ∗ and there can exist up to three information market equilibria depending on the REE, there are also overall equilibria in which each equilibrium λ∗ corresponds to a different REE. Hence, a change in the coordination decision of traders in the financial market could lead to a change in the equilibrium λ∗ and vice versa. Our results provide two sources of excess price volatility: multiplicity in the REE and multiplicity in the information market. A change in the equilibrium λ∗ when the traders coordinate on the COM–REE can cause the stock-price volatility to increase. This can also happen with a change in the REE given an equilibrium λ∗ . Figure 4 illustrates some of these phenomena for the COM–REE, where γ = 2, ρ x = ρv = ρε = ρη = 1, and x¯ = v¯ = 1. The cost of information, κ, can be used as a measure of exogenous technological progress; for example, a proliferation of sources of information about financial markets leading to easier access can be thought of as a decrease in κ. The z-shaped curve in Figure 4(a) depicts the change in the equilibrium λ∗ as κ changes, while Figures 4(b) and 4(c) depict the changes in the average price E( P COM ) (equation (10)) and price volatility Var( P COM ), respectively. From Proposition 1, it follows that 1 ∗ COM 2 1 COM Var( P COM ) = ρ− ) (K COM )−2 + ρ− ρ x + γ)2 (K COM )−2 , v (λ ρε + ρθ x (β
where ρθCOM and β COM are as in the proposition and λ∗ is given by Corollary 1. 16
Pre-print of paper published in the Journal of the European Economic Association, Vol. 7, pp. 90-115, 2009 Figure 4
(a)
(b)
(c) 10
0.85 1 0.8
9 0.8
0.75 8
λ*
E(P)
0.6
0.4
Var(P)
0.7
0.65
0.6
7
6 0.2
0.55 5 0.5
0
0
0.05
0.1
Information Cost,κ
0.45
0
0.05
0.1
4
0
0.05
0.1
Information Cost, κ Information Cost, κ
Figure 4: Information market equilibria and price movements (COM-REE). For comparative statics analysis of E( P COM ) and Var( P COM ) as κ varies, we use Corollary 1 to obtain a range of values for κ where an interior equilibrium λ∗ ∈ (0, 1) exists. Then, restricting attention to the interior equilibrium λ∗ , we use the preceding expressions and the chain rule to obtain our results. In Figure 4, as κ falls by a small amount below 0.07, for example, the number of information market equilibria jumps from one (λ∗ = 0) to three (λ∗ = 0, λ∗ = 1, and λ∗ ∈ (0, 1)); hence there may be a sharp fall in the average price from 0.77 to 0.5. The volatility of price, on the other hand, could spike to 9.6 from 4.9.
5
An Alternative Setting: Costly Supply Signals
We now extend the simple model to one where traders can purchase supply information also. The economy is identical to that in Section 2 except for the following three changes. (i) Each trader is endowed with n¯ i units of the bond and no units of the stock at the beginning of period 1. (ii) The aggregate supply x of the stock is driven by noise traders (as in Grossman and Stiglitz 1980) and is normally distributed: ¯ 1/ρ x ) with ρ x > 0. x ∼ N ( x, (iii) In the information market, trader i can independently purchase at cost κ > 0 a two-dimensional private information signal Si = (yi , xi ), where yi and xi are 17
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mutually independent and ¯ 1/ρv ), ε i ∼ N (0, 1/ρε ), ρε > 0, yi = v + ε i , with v ∼ N (v, xi = x + ηi , with ηi ∼ N (0, 1/ρη ), ρη > 0. The noise (ε i )i∈[0,1] and (ηi )i∈[0,1] in the signals are i.i.d. across the traders (and we again adopt the “law of large numbers” convention). Trader i is informed if she purchases Si and is uninformed otherwise. Then the financial market opens in period 2. As before, we relabel the traders so that the set of traders who are informed is [0, λ]. Informed traders have the same information as in Section 3, but the uninformed traders do not possess information about x; their information is only that available from the prices. This structure captures the possibility that in some cases, obtaining information about the payoff and the supply (liquidity trades) may be tied together. i ) An equilibrium (λ∗ , P, ( DiI )i∈[0,λ∗ ] , ( DU i ∈(λ∗ ,1] ) is as described in Definitions 1 ˆ − cx ˆ is obtained in a and 2. The existence of two REE with linear price P = aˆ + bv manner similar to Proposition 1 and is stated next as Proposition 4. All proofs for this section are contained in the Appendix. Proposition 4 If γ2 > 4λ2 ρε ρη > 0 then there exist two PR-REE in which ˆ − cx, ˆ P = aˆ + bv where ˆ x x¯ ), bˆ = Kˆ −1 (λ(ρε + ρˆ θ ) + (1 − λ)ρˆ φ ), cˆ = Kˆ −1 ( βρ ˆ x + γ) aˆ = Kˆ −1 (ρv v¯ + βρ ˆ cˆ) takes one of two values, and where βˆ = (b/ q ˆβ GS = (2λρη )−1 γ − γ2 − 4λ2 ρε ρη , q ˆβ NGS = (2λρη )−1 γ + γ2 − 4λ2 ρε ρη , with Kˆ = λρˆ I + (1 − λ)ρˆU , ρˆ I = ρv + ρε + ρˆ θ , ρˆ θ = βˆ 2 (ρ x + ρη ), ρˆU = ρv + ρˆ φ , and ρˆ φ = βˆ 2 ρ x .11 The coefficients of the price function differ in Proposition 4 as compared to Proposition 1, but the informational properties of the two REE are similar. The GS-REE (Grossman-Stiglitz REE) corresponds to the SUB–REE because (∂ βˆ GS /∂λ) > 0 and the NGS-REE (non-Grossman-Stiglitz REE) corresponds to the COM–REE 11 As
before, we suppress the explicit dependence on λ throughout.
18
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because (∂ βˆ NGS /∂λ) < 0. However, the implications of the two equilibria for complementarity and substitutability need not be the same, so we use different labels for the REE. Information acquisition can be a strategic complement even when the traders coordinate on the GS-REE—unlike the preceding result for SUB–REE and results of previous studies with the exception of Veldkamp (2006a, 2006b), where information acquisition is a strategic substitute given the (unique) REE. Similarly, when the traders coordinate on the NGS-REE, it is also the case that information acquisition can be a strategic substitute. These results are presented in Proposition 5. Proposition 5 (i) Suppose the traders coordinate on the GS-REE. If (ρε /ρv ) < (ρη /ρ x ) (resp. (ρε /ρv ) > (ρη /ρ x )), then information acquisition is a strategic complement (resp. substitute). (ii) Suppose the traders coordinate on the NGS-REE. If (ρε /ρv ) < (ρη /ρ x ) (resp. (ρε /ρv ) > (ρη /ρ x )) , then information acquisition is a strategic substitute (resp. complement). As before, in order to establish strategic complementarity and substitutability p γκ ˆ ρˆU /ρˆ I , adapted to this setting, which is obtained we use the ratio R(λ) = e similarly to Proposition 2, and is noted in Lemma 1 of the Appendix. Much as in Section 4, the relative benefit to an informed trader is
(ρˆ I /ρˆU ) = (ρε /ρˆU ) + βˆ 2 (ρη /ρˆU ) + 1 where (ρε /ρˆU ) is contributed by the payoff signal and ( βˆ 2 ρη /ρˆU ) by the supply signal. These two terms are affected by βˆ in opposite ways. When βˆ rises, it is easier for an uninformed trader to learn from P, that is, ρˆU rises and so (ρε /ρˆU ) falls. However, as βˆ rises, an informed trader also gains more from using supply information xi to decrease the noise from x in P and make better inference about v. In other words, βˆ 2 ρη increases and so ( βˆ 2 ρη /ρˆU ) = ρη ((ρv / βˆ 2 ) + ρ x )−1 increases. In the GS-REE, βˆ GS increases as λ increases. Hence, (ρε /ρˆU ) decreases, so that acquiring the payoff signal yi is a strategically substitutable action, and ( βˆ 2 ρη /ρˆU ) increases, so that acquiring the supply signal is a strategically complementary action. Comparing the precision ratios of the two signals indicates which one contributes more to the relative benefit of an informed trader and thus indicates which effect dominates. This enables us to identify whether acquiring information Si = (yi , xi ) is strategically complementary or substitutable. A similar result using the comparative precision of two different signals is in Amador and Weill (2008, Lemma 2). When the traders coordinate on the NGS-REE, analogous arguments apply, keeping in mind that (∂ βˆ NGS /∂λ) < 0 . Chamley (2008) suggests that the multiplicity and complementarity results in Section 2 are not useful because the COM–REE is an “unstable” fixed point of a particular reaction function in that model, whereas the SUB–REE is “stable”.12 12 Under
Chamley’s (2008) analysis, the fully-revealing REE is a “stable” REE as well.
19
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However, the model in Section 2 (and in Chamley 2008) represents a static economy, and any useful notion of stability would require an explicit analysis of some dynamic process by which the participants may arrive at an equilibrium. Such an explicit analysis would require moving beyond the current static setup and thus is beyond the scope of this paper. Moreover, past analyses of stability and convergence to REE, (see e.g. Blume and Easley 1982, Bray 1982) lead to mixed results and suggest that the stability of REE depends heavily on the specifics of the learning process. Positive results may require “extensive knowledge about the structure and dynamics of the model while they learn, while less informationally demanding procedures need not lead to positive results, but are open to other criticisms” (Blume et al. 1982). Although stability of REE is certainly an interesting issue to be analyzed; as we point out in the alternative model of this Section 5, complementarity in information acquisition is driven by the multidimensionality of the information structure rather than by any specific REE. As Proposition 5 states, complementarity can exist even with the GS-REE, which is the analogue of the “stable” SUB–REE (and of the unique equilibrium in Grossman and Stiglitz 1980).
6
Conclusion
If traders use supply information in financial markets, then (i) acquiring information on payoffs may be complementary and (ii) this additional dimension of information may generate multiple equilibria in the financial market—a new REE with different informational properties may exist. Although other studies have explored the consequences of introducing information about supply within this framework, information remains a strategic substitute in those analyses. Our results mean that the analytically tractable CARA-normal framework can still be used to study financial markets when information complementarity seems a natural phenomenon. There are several possible extensions to the framework we present here, that could yield interesting insights. Multi-asset markets could shed light on co-movement of stock prices, and endogenizing the price of information would add an additional source of complementarity (Veldkamp 2006a, 2006b). Multiperiod trading would mean that prices in consecutive periods are correlated by the (random) asset supply, which may help inference from the price process and thus have an effect on the information acquisition. A multiperiod model would allow analyzing effects of higher-order beliefs on asset prices (Cespa and Vives, 2007). Possibly fruitful avenues of research include borrowing constraints, which may help explain asymmetric and correlated behavior of prices (Yuan, 2005), and allowing for consumption in the first period (Muendler 2007). Finally, we have competitive behavior on the part of the traders here and have not permitted strategic behavior. This is certainly an aspect that merits further 20
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research.
A A.1
Appendix Proofs of Results in Sections 3 and 4
Proposition 1. From equations (2) to (8), we solve for P and get
[λρ I + (1 − λ)ρU ] P = ρv v¯ + λρε v + ρθ
Z 1 0
zi di − γx
= [ρv v¯ + ρθ (c/b)(ρ x + ρη )−1 ρ x x¯ ] +(λρε + ρθ )v − (ρθ (c/b)(ρ x + ρη )−1 ρ x + γ) x. Comparing with equation (1) and recalling that ρθ = (b/c)2 (ρ x + ρη ), we have ρη (b/c)2 − γ(b/c) + λρε = 0. Then, after we define β = (b/c), the result follows directly. Proposition 2. The ex post indirect utility of informed trader i may be written as i
E[VIi | xi , yi , P] = E[−e−γW2I | xi , yi , P] = − exp −γ(n¯ i − κ + Pxi ) − ρ I ( E[v| xi , yi , P] − P)2 /2 . Conditioning on { xi , yi , P}is equivalent to conditioning on { xi , yi , zi }. Define h = Var( E[v| xi , yi , zi ]| xi , zi )and −1 1 use the conditional variance formula to get h = ρU − ρ− I . Defining Zi = ( E [ v | xi , yi , zi ] − P)h−1/2 we have E[VIi | xi , zi ] = eγκ u(n¯ i + Pxi ) E[exp(−(hρ I /2) Zi2 )| xi , zi ].Conditional on { xi , zi }, E[v| xi , yi , zi ]is normally distributed, and so, conditional on { xi , zi }, Zi2 has a noncentral chi-squared distribution. Then, for t > 0,the moment-generating function for Zi2 can be written as 2
E[e−tZi | xi , zi ] = (1 + 2t)−1/2 exp[−( E[ Zi | xi , zi ])2 t(1 + 2t)−1 ] with E[ Zi | xi , zi ] = ( E[v| xi , zi ] − P) h−1/2 . So, setting t = ( hρ I /2), we have
E[VIi | xi , zi ]
=e
γκ
q
(ρU /ρ I )u(n¯ i + Pxi ) exp[−( E[v| xi , zi ] − P)2 ρU /2].
(A.1)
The uninformed trader i has ex post indirect utility
E[VUi | xi , zi ] = u(n¯ i + Pxi ) exp[−( E[v| xi , zi ] − P)2 ρU /2].
(A.2)
Then, by (A.1) and (A.2),
E[VIi | xi , zi ] −
E[VUi | xi , zi ]
=
e
γκ
q
(ρU /ρ I ) − 1 E[VUi | xi , zi ].
Now taking expectations on both sides gives E[VIi | xi ] = eγκ proves the proposition.
21
p
(ρU /ρ I ) E[VUi | xi ], which
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A.2
Proofs of Results in Section 5
ˆ − cx ˆ can be expressed by the public signal The information from P = aˆ + bv ˆ bˆ )( x − x¯ ). s = bˆ −1 ( P − aˆ + cˆx¯ ) = v − (c/ ˆ bˆ )( x − x¯ ). Then φ∼ N (0, ρφ ) with Let φ = −(c/ ˆ cˆ)2 ρ x . ρˆ φ = (b/
(A.3)
Informed trader i uses the sufficient private signal
ˆ bˆ )( x − E[ x | xi ]) = v − (c/ ˆ bˆ )(ρ x + ρη )−1 [ρ x ( x − x¯ ) − ρη ηi ]. zˆi = v − (c/
(A.4)
ˆ bˆ )( x − E[ x | xi ]). Conditional on xi , we have θˆi ∼ N (0, ρˆ θ ) with Let θˆi = −(c/ ˆ cˆ)2 (ρ x + ρη ). ρˆ θ = (b/
(A.5)
An informed trader i has information { xi , yi , P} and uses { xi , yi , zi } to update her beliefs: v|{ xi , yi , zi } ∼ N (µˆ iI ,ρˆ I ), with 1 µˆ iI = ρˆ − I ( ρv v¯ + ρε yi + ρˆ θ zi ), ρˆ I = ρv + ρε + ρˆ θ .
(A.6)
Given her beliefs (A.6), the demand of informed trader i is
DiI ( P) = γ−1 ρˆ I (µˆ iI − P).
(A.7)
An uninformed trader has only information { P} (or, equivalently, {s}) to update her −1 beliefs: v|s ∼ N (µˆ U ,ρˆ U ), with −1 µˆ U = ρˆU (ρv v¯ + ρˆ φ s), ρˆU = ρv + ρˆ φ .
(A.8)
Given posterior beliefs (A.8) about the stock, each uninformed trader’s demand function is
DU ( P) = γ−1 ρˆU (µˆ U − P).
In equilibrium, the stock market clears; that is, Z λ 0
DiI ( P)di + (1 − λ) DU ( P) = x.
(A.9)
(A.10)
We now prove Proposition 4. Proposition 4. The proof is similar to that of Proposition 1 and uses equations (A.4)– (A.10) to solve for P and get
[λρˆ I + (1 − λ)ρˆU ] P ˆ bˆ )ρˆ θ ρ x (ρ x + ρη )−1 x¯ + (1 − λ)(c/ ˆ bˆ )ρˆ φ x¯ ] = [ρv v¯ + λ(c/ +[λ(ρε + ρˆ θ ) + (1 − λ)ρˆ φ ]v ˆ bˆ )ρˆ θ (ρ x + ρη )−1 ρ x + (1 − λ)(c/ ˆ bˆ )ρˆ φ + γ] x. −[λ(c/ 22
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ˆ cˆ)2 (ρ x + ρη ) and Comparing with the equation P = aˆ +bˆ v−cˆx and recalling that ρˆ θ = (b/ ˆ cˆ)2 ρ x , we have the polynomial λρ (b/ ˆ cˆ)2 −γ(b/ ˆ cˆ) + λρε = 0. Then define ρˆ φ = (b/ η ˆ cˆ and the result follows directly. βˆ = b/ The next lemma provides the expression for Rˆ (λ) noted in the text.
Lemma 1 Suppose traders coordinate on the GS-REE (or on the NGS-REE). Then, for any λ, p γκ ˆ ˆ ˆ R(λ) = e ρU / ρ I . Lemma 1. Again the proof is similar to that for Proposition 2. The ex post indirect utility of informed trader i is i
E[VIi | xi , yi , P] = E[−e−γW2I | xi , yi , P] = − exp{−γ(n¯ i − κ ) − (( E[v| xi , yi , P] − P)2 ρˆ I /2)}. Conditioning on { xi , yi , P} is equivalent to conditioning on { xi , yi , s}. Define hˆ = Var( E[v| xi , yi , P]|s) −1 1 ˆ −1/2 ˆ and then by the conditional variance formula, hˆ = ρˆ U −ρˆ − I . With Zi = ( E [ v | x i , yi , P ]− P ) h we have E[VIi |s] = eγκ u(W1i ) E[exp(−( hˆ ρˆ I /2) Zˆ i2 )| P]. Conditional on P or s, E[v| xi , yi , P] is normally distributed. Hence, conditional on P or s, Zˆ i2 has a noncentral chi-squared distribution. Then using the moment-generating function for Zˆ i2 yields
E[VIi |s]= eγκ
q
n o (ρˆU /ρˆ I )u(W1i ) exp −( E[v|s] − P)2 ρˆU /2 .
(A.11)
The uninformed traders have ex post indirect utility
n o E[VUi |s]= u(W1i ) exp −( E[v|s] − P)2 ρˆU /2] .
(A.12)
Then, by (A.11) and (A.12),
E[VIi |s]− E[VUi |s]=(eγκ
q
(ρˆU /ρˆ I ) − 1) E[VUi |s]. p Taking expectations on both sides then yields E[VIi ] = eγκ (ρˆ U /ρˆ I ) E[VUi ], which proves the result.
Proposition 5. We prove the result for the GS-REE and omit the similar proof for the NGS-REE. By Proposition 4,
Rˆ (λ)= eγκ
q
(ρˆU /ρˆ I )= eγκ
q
(ρv + βˆ 2 ρ x )(ρv + ρε + βˆ 2 (ρ x + ρη ))−1 .
ˆ 0 (λ) has the same sign as (∂(ρˆU /ρˆ I )/∂( βˆ 2 )). Since In the GS-REE, (∂ βˆ GS /∂λ) > 0. So R
(∂(ρˆU /ρˆ I )/∂( βˆ 2 ))=(ρ x ρε − ρη ρv )[ρv + ρε + βˆ 2 (ρ x + ρη )]−2 , ˆ 0 (λ) > 0 if and only if (ρε /ρv ) − (ρη /ρ x ) > 0. it follows that R 23
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References [1] Anat Admati (1985), A noisy rational expectations equilibrium for multi-asset securities markets, Econometrica, Vol. 53, Issue 3, pp. 629-658
[2] Manuel Amador and Pierre-Olivier Weill, Learning from prices: public communication and welfare, working paper, UCLA
[3] Gadi Barlevy and Pietro Veronesi (2000), Information acquisition in financial markets, Review of Economic Studies, Vol. 67, Issue 1, pp.79-90
[4] Gadi
Barlevy and Pietro Veronesi (2008), Information acquisition in financial markets: a correction, available online at hhttp://faculty.chicagogsb.edu/pietro.veronesi/research/BarlevyVeronesi Correction.pdfi, University of Chicago
[5] Utpal Bhattacharya and Matthew Spiegel (1991), Insiders, outsiders, and market breakdowns, Review of Financial Studies, Vol. 4, Issue 2, pp. 255-282
[6] Bruno Biais, Peter Bossaerts, and Chester Spatt (2008), Equilibrium asset pricing and portfolio choice under heterogeneous information, IDEI Working Paper number 474, forthcoming in the Review of Financial studies
[7] Lawrence Blume, Margaret Bray, and David Easley (1982), An introduction to the stability of rational expectations equilibrium, Journal of Economic Theory, Vol. 26, Issue 2, pp. 313-317
[8] Lawrence Blume and David Easley (1982), Learning to be rational, Journal of Economic Theory, Vol. 26, Issue 2, Pages 340-351
[9] Lawrence Blume, David Easley, and Maureen O’Hara (1994), Market statistics and technical analysis: the role of volume, Journal of Finance, Vol. 49 Issue 1, pp. 153-181
[10] Margaret Bray (1982), Learning, estimation, and the stability of rational expectations, Journal of Economic Theory, Vol. 26, Issue 2, pp. 318-339
[11] Markus Brunnermeier (2001), Asset pricing under asymmetric information, Oxford University Press
[12] Giovanni Cespa and Xavier Vives (2007), Dynamic trading and asset prices: Keynes vs. Hayek, WP716, SP-SP Center, IESE Business School
[13] Christophe Chamley (2007), Complementarities in information acquisition with short-term trades, Theoretical Economics, Vol. 2, pp.41-467
[14] Christophe Chamley (2008), Comment on equilibria on Ganguli and Yang (2006, 2007), mimeo, Boston University
24
Pre-print of paper published in the Journal of the European Economic Association, Vol. 7, pp. 90-115, 2009
[15] Douglas Diamond and Robert Verrecchia (1981), Information aggregation in a noisy rational expectations economy, Journal of Financial Economics, Vol. 9, Issue 3, pp. 221-235
[16] Joseph Doob (1937), Stochastic processes depending on a continuous parameter, Transactions of the American Mathematics Society, Vol. 42, pp.107–140
[17] James Dow and Gary Gorton (2008), Noise traders, The New Palgrave Dictionary of Economics, Second Edition, edited by S Durlauf and L Blume, Palgrave Macmillan
[18] James Dow and Rohit Rahi (2003), Informed trading, investment, and welfare, Journal of Business, Vol. 76, Issue 3, pp. 439-454
[19] Darrell Duffie and Rohit Rahi (1995), Financial market innovation and security design, Journal of Economic Theory, Vo. 65, Issue 1, pp. 1-42
[20] David Easley and Maureen O’Hara (2004), Information and the cost of capital, Journal of Finance, Vol. 59, Issue 4, pp. 1553-1584
[21] Jayant Ganguli and Liyan Yang (2008), Complementarities, multiplicity, and supply information: additional notes, mimeo, University of Cambridge
[22] Itay Goldstein and Alexander Guembel (2008), Manipulation and the allocational role of prices, Review of Economic Studies, Vol. 75, pp. 133-164
[23] Sanford Grossman (1976), On the efficiency of competitive stock markets where trades have diverse information, Journal of Finance, Vol. 31, Issue 2, pp. 573-585
[24] Sanford Grossman and Joseph Stiglitz (1980), On the impossibility of informationally efficient markets, American Economic Review, Vol. 70, Issue 3, pp. 393-408
[25] Bruce Grundy and Maureen McNichols (1989), Trade and the revelation of information through prices and direct disclosure, Review of Financial Studies, Vol. 2, Issue 4, pp. 495-526
[26] Martin Hellwig (1980), On the aggregation of information in competitive markets, Journal of Economic Theory, Vol. 22, Issue 3, pp. 477-498
[27] Kenneth Judd (1985), The law of large numbers with a continuum of i.i.d. random variables, Journal of Economic Theory, Vol. 35, Issue 1, pp. 19-25
[28] Richard Kihlstrom (1984), A “Bayesian” exposition of Blackwell’s theorem on the comparison of experiments, chapter 2 in Bayesian Models in Economic Theory, edited by M Boyer and R Kihlstrom, North Holland, 1984
[29] Jean-Jacques Laffont (1985), On the welfare analysis of rational expectations equilibria with asymmetric information, Econometrica, Vol. 53, Issue 1, pp. 1-30
25
Pre-print of paper published in the Journal of the European Economic Association, Vol. 7, pp. 90-115, 2009
[30] Andrew Lo, Harry Mamaysky and Jiang Wang (2004), Asset prices and trading volume under fixed transactions costs, Journal of Political Economy, Vol. 112, No. 5, pp. 1054-1090
[31] Stephen McCafferty and R Driskill (1980), Problems of non-existence and uniqueness in nonlinear rational expectations models, Econometrica, Vol. 48, Issue 5, pp. 13131317
[32] Luis Medrano and Xavier Vives (2004), Regulating insider trading when investment matters?, Review of Finance, Vol. 8, pp. 199–277
[33] Luis Medrano and Xavier Vives (2007), Information, hedging and welfare, mimeo, IESE Business School
[34] Marc-Andreas Muendler (2007), On the possibility of an informationally efficient market, Journal of Economic Theory, 133(1), pp. 467–483
[35] John Muth (1961), Rational expectations and the theory of price movements, Econometrica, Vol 29, Issue 3, pp.315-335
[36] Yeneng Sun (2006), The exact law of large numbers via Fubini extension and characterization of insurable risks, Journal of Economic Theory, Vol. 126, Issue 1, pp. 31-69
[37] Laura Veldkamp (2006a), Media frenzies in markets for financial information, American Economic Review, Vol. 96, Issue 3, pp. 577-601
[38] Laura Veldkamp (2006b), Information markets and the co-movement of asset prices, Review of Economic Studies, Vol. 73, Issue 3, pp. 823-845
[39] Xavier Vives (2008), Information and learning in markets: the impact of market microstructure, Princeton University Press
[40] Jiang Wang (1994), A Model of Competitive Stock Trading Volume, Journal of Political Economy Vol. 102, pp. 127-167
[41] Masahiro Watanabe (2008), Price volatility and investor behavior in an overlappinggenerations model with information asymmetry, Journal of Finance, Vol. 63, p 229272
[42] Kathy Yuan (2005), Asymmetric price movements and borrowing constraints: a rational expectations equilibrium model of crises, contagion, and confusion, Journal of Finance, Vol. 60, Issue 1, pp. 379-411
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Complementarities, multiplicity, and supply information: Additional notes13 A Common Hedging Motive Here we show how the structure of the model in Section 2 arises when in a setting where x (resp. ηi ) represents a common (resp. an idiosyncratic) hedging motive of the traders by adapting the clear discussion in Biais et al. 2008). For simplicity, assume that traders receive zero units of stock endowment. Suppose each trader i has nonstock income—for instance, human capital whose payoff is labor income, denoted li . To analyze the distribution of the terminal wealth of the trader, we consider the projection of li onto the payoff from the stock v, li = x i v + ei . Then xi denotes the regression coefficient in this purely statistical relation, where ei denotes the regression residual. As noted in Biais et al. (2008, n.4), this projection can be performed by the traders because they know the parameters of the cash flows and the pricing equation. The economic interpretation of this is as follows. The coefficient xi captures the sensitivity of nonstock income to the stock payoff, while ei captures the part of nonstock income not spanned by the risky stock, that is ei is orthogonal to v. Here xi = x + ηi , where x denotes the sensitivity common to all traders and ηi denotes the sensitivity idiosyncratic to trader i. As an example, consider the following, adapted from Biais et al. (2008, p.11). There are two workers in Exxon— a laborer and a manager—each of whose income and wealth are exposed to the risk faced by the firm, and more generally, by the oil industry. How the workers’ income is affected by the risk from the firm—the sensitivity parameter xi —is determined by two components for each worker. One component is common to all workers, while the other is idiosyncratic (e.g., the worker’s position in the hierarchy and her individual labor contract with the firm). We refer the reader to the papers mentioned in the Introduction for more discussions and interpretation of this structure. Assume that (v, x, {ηi }i∈[0,1] , {ei }i∈[0,1] ) are mutually independent and that the ¯ 1/ρ x ), ηi ∼ “law of large numbers” convention holds. Then, assuming x ∼ N ( x, ¯ 1/ρv ), and ei ∼ N (0, 1/ρe ) leads us to the same structure N (0, 1/ρη ), v ∼ N (v, described in Section 2 as we show next. 13 Jayant
Vivek Ganguli, University of Cambridge, and Liyan Yang, Cornell University, . The figure, equation, page, and section numbers of this supplementary appendix are a continuation of those in the main paper to facilitate reading.
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Trader i’s information set is Fi = { xi , yi , P} if informed and { xi , P} if uninformed. She chooses stock holdings Zi to maximize E − exp{−γ[ Zi (v − P) + li ]}|Fi = E − exp{−γ[( xi + Z )v − ZP + ei ]} Fi . The optimal solution is Zi∗ =
E(v|Fi ) − P − xi . γ Var(v|Fi )
(A.13)
The market clearing condition is Z 1 0
Zi∗ di = 0.
(A.14)
Given the demand functions (A.13) and the market-clearing condition (A.14), we have Z 1 E(v|Fi ) − P di = x. 0 γ Var( v |Fi ) This coincides with the market-clearing condition in equation (8), which leads to Proposition 1.
The Finite Economy Here we provide details on our assertion that the multiplicity in the financial market is not an artifact of the continuum of traders. Consider an economy identical to that in Section 2 but with a finite number of traders. In this case, there may exist three partially revealing REE. Specifically, there are ( N + M ) < ∞ traders who live for 2 periods, trader n ∈ {1, . . . , N } is informed, and trader m ∈ { N + 1, . . . , N + M } is uninformed. In period 1, two assets—one riskless (a bond) and one risky (a stock)—are traded at a financial market. Normalize the bond price to 1, and denote the stock price by P. Trader i ∈ {1, . . . , N + M } is endowed with 0 units of the bond and xi units of the stock at the beginning of period 1, where xi = x + ηi and we assume x ∼ N (0, 1/ρ x ) with ρ x > 0 and ηi ∼ N (0, 1/ρη ) with ρη > 0. The realization of xi is observed only by trader i. The aggregate supply of stock is then ∑iN=+1 M xi . Each trader cares only about wealth W at the end of period 2 and has the (vonNeumann-Morgenstern) utility function u with CARA parameter γ > 0. In period 2, the payoffs to both assets are realized. The payoff to each unit invested in the bond is normalized to 1 and the payoff to each unit invested in the stock is denoted by v, v ∼ N (0, 1/ρv ) with ρv > 0. Before trading in the financial market, informed trader n ∈ {1, . . . , N } receives a private signal yn about v, yn = v + ε n , with ε n ∼ N (0, 1/ρε ), ρε > 0. 28
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We assume that all the underlying random variables v, x, {ε n }nN=1 , {ηi }iN=+1 M are mutually independent. This model is an extension of Diamond and Verrecchia (1981) in the sense that all traders behave competitively in the financial market. Actually, if M = 0 and ρη = 0, we return to the setup of Diamond and Verrecchia (1981). A rational expectations equilibrium (REE) is a price function P( x1 , . . . , x N + M , y1 , . . . , y N ) together N+M with demand functions { D nI ( P, xn , yn )}nn=1 and { D m I ( P, xm )}m= N +1 such that demand functions maximize traders’ conditional expected utilities and both markets clear almost surely. We are interested in REE prices that are linear in private signals and endowments. So, suppose traders conjecture the price function as
∑
N+M
N
N
P=F
yn − G I
n =1
∑
∑
xn − GU
n =1
xm ,
(A.15)
m = N +1
where F ≥ 0, G I ≥ 0, and GU ≥ 0 are nonrandom endogenous parameters. The magnitude of NF/( NG I + MGU ) measures the informativeness of the price about the payoff v as discussed in the model presented in Section 2. Informed Traders An informed trader n has information { xn , yn , P}. Let GI GU M P − Fyn + G I xn zn = + + E( x | xn ) F ( N − 1) F F N−1
=
N+M F ∑iN=1,i6=n yi − G I ∑iN=1,i6=n xi − GU ∑m = N +1 x m
F ( N − 1) GI GU M + + E( x | xn ) F F N−1 = v + θn , where θ n = ( N − 1 ) −1
N
∑
ε i − F −1 ( G I + GU M ( N − 1)−1 )[ x − E( x | xn )]
i =1,i 6=n
−( F ( N − 1))−1
GI
∑
!
N+M
N
ηi + GU
i =1,i 6=n
29
∑
m = N +1
ηm
.
(A.16)
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Conditional on xn , θn ∼ N (0, ρθ I ), where 1 1 GI GU M 2 = + + Var( x | xn ) ρθ I ( N − 1) ρ ε F F N−1 # " GU 2 M 1 GI 2 1 + . + 2 F N−1 F ( N − 1) ρ η
(A.17)
Trader n’s information set is equivalent to { xn , yn , zn }, and v|{ xn , yn , zn } ∼ N (µnI , ρ I ) where ρε yn + ρθ I zn µnI = , ρ I = ρv + ρε + ρθ I . (A.18) ρv + ρε + ρθ I Given CARA utility and the normality assumption, the demand of informed trader n is µnI − P n . (A.19) D I ( P, xn , yn ) = 1 γρ− I Uninformed Traders An uninformed trader m has information set { xm , P}. Let GI GU M − 1 P + GU xm + + E( x | xm ) zm = FN F F N
=
F ∑nN=1 yn − G I ∑nN=1 xn − GU ∑iN=+NM +1,i 6=m xi
FN GI GU M − 1 + + E( x | xm ) F F N = v + θm ,
(A.20)
where θ m = N −1
N
∑ ε n − F−1 (GI + GU ( M − 1) N −1 )[x − E(x|xm )]
n =1
−( FN )−1
GI
N
N+M
n =1
i = N +1,i 6=m
∑ ηn + GU
∑
! ηi
.
Conditional on xm , θm ∼ N (0, ρθU ) where 1 GI GU M − 1 2 1 1 = + + ρθU Nρε F F N ρ x + ρη " # GI 2 1 GU 2 M − 1 1 − + . F N F ρη N2 30
(A.21)
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m ,ρ Trader m’s information set is equivalent to { xm , zm }, and v|{ xm , zm } ∼ N µU U where ρ zm m µU = θU , ρU = ρv + ρθU . (A.22) ρv + ρθU Trader m’s demand schedule is therefore m DU ( P, xm ) =
m−P µU
−1 γρU
.
(A.23)
In equilibrium, the stock market clears: N
∑
N+M
D nI ( P, xn , yn ) +
n =1
∑
m DU ( P, xm ) =
m = N +1
N+M
N
∑
xn +
n =1
∑
xm .
(A.24)
m = N +1
We find REE by first solving equation (A.24) for P in terms of private signals and stock endowments via (A.16)–(A.23) and then comparing coefficients with those in the conjectured price function (A.15). As a result, the coefficients ratios G I /F and GU /F are characterized as roots of the following system of polynomials, ρη GU M GI M GI GI γ + ρ + ρ − ρ + θU F N θI F θI F F N −1 ρ x + ρ η GI = , (A.25) F ρε + ρθ I + ρθU M N ρη GU N GU M −1 GU M −1 GI γ + ρ + ρ − ρ + θ I F N −1 θU F N θU F F N ρ x +ρη GU . (A.26) = F ρε + ρθ I + ρθU M N If such a root exists, then we can conclude that an REE might exist. In particular, if γ = 1, M = 0, and ρη = 0, then our model is identical to Diamond and Verrecchia (1981). In this case, for any finite N, equation (A.25) has 1 a unique root G I /F = ρ− ε ‘. It is easy to show that the existence results on REE in a finite economy coincide in the limit with those in a model with a continuum of traders. Formally, let ( M/N ) = (1 − λ)/λ and take the limit as N → ∞; then, by (A.17) and (A.21), 1 ρ− θI
=
1 ρ− θU
GI G = + U F F
1 −1 λ
2
1 . ρ x + ρη
Equations (A.25) and (A.26) become GI F
=
GU F
=
ρη GU 1−λ GI F + F λ ρ x +ρη , ρε + ρθ I + ρθU 1−λ λ ρ γ + ρθ I GFU + ρθU GFU 1−λ λ − ρθU GFI + GFU 1−λ λ ρx +ηρη ρε + ρθ I + ρθU 1−λ λ
γ + ρθU GFI 1−λ λ + ρθ I GFI − ρθ I
31
(A.27)
.
(A.28)
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Table 1: Financial market equilibria in the finite economy.
N = 100, M = 300
N = 1000, M = 3000
N = 10000, M = 30000
G I /F 1.87 0.15 0.02 1.87 0.14 0.001 1.87 0.13 0.0001
REE1 REE2 REE3 REE1 REE2 REE3 REE1 REE2 REE3
GU /F 1.87 0.12 0.01 1.87 0.13 0.001 1.87 0.13 0.0001
NF NG I + MGU
0.13 2.03 22.9 0.13 1.88 248.1 0.13 1.87 2498
1 −1 Combining equations (A.27) and (A.28) with ρ− θ I = ρθU , we have G I = GU ≡ G. Further simplifying equation (A.27) yields
λρε
G λF
3
−γ
G λF
2
+ ρη
G λF
= 0,
where G/λF is the reciprocal of price informativeness measure (i.e., 1/β in the main text). Hence in the limit, if γ2 > 4λρε ρη > 0 then three REE exist: a fully revealing REE and two partially revealing REE corresponding to the SUB–REE and the COM–REE, respectively. Now we give a numerical example to help illustrate the possibility of multiple REE in a finite economy (Table 1). Suppose γ = 2, λ = 1/4, and ρ x = ρε = ρη = 1. In the limit, β SUB = 0.134 and β COM = 1.866. We solve equations (A.27) and (A.28) for the cases N = 100, N = 1000, and N = 10000. In each case, there exist three partially revealing linear REE: REE1, REE2, and REE3.14 As we increase the number of traders, REE1 converges to the SUB–REE, REE2 converges to the COM–REE, and REE3 converges to a fully revealing REE. Although we have been unable to prove the general existence of multiple linear REE prices for the finite economy, the numerical results suggest that multiplicity does not depend on the assumption of a continuum of traders. The information properties of the finite-economy REE2 are the same as those of the continuumeconomy COM–REE, so our results on complementarity in the information market still go through. 14 If
we do not restict G I > 0 and GU > 0 in the price functions, we would have five partially revealing REE. The two not reported here converge to a fully revealing REE as N → ∞.
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Expected Volume Assuming x¯ = 0 for simplicity, we see that E[ Q] is given by
E[ Q] =
Z 1q 0
j
(2π )−1 Var( Di − xi )di = (2π )−1/2 [λE( QiI ) + (1 − λ) E( QU )], j
where QiI (resp., QU ) is the volume of trade by the informed trader i (resp., uniformed trader j) and q 1 2 −1 2 −1 E( QiI ) = γ−1 (ρε +ρθ −ρ I b)2 ρ− v + ρε + ( βρ x − ρ I c + γ ) ρ x + ( βρη − γ ) ρη , q j −1 1 2 −1 2 −1 E ( QU ) = γ ( ρ θ − ρU b )2 ρ − v +( βρ x − ρU c + γ ) ρ x +( βρη − γ ) ρη .
Price Volatility The volatility of price P is measured by Var( P). Figure 5 illustrates changes in the volatility of prices in the SUB-REE and the COM–REE with γ = 2, ρη = ρv = 1, and λ = 1/4. Two cases are presented with constant volatility of payoff—that is ρv = 1—and different values of aggregate supply volatility ρ x . This allows us to compare the changes in volatility of the REE prices as the volatility of x relative to v increases (moving across Figures 5(a)–5(b)). As the volatility of x increases (from Figure 5(a) to 5(b)), the volatility of the COM–REE price relative to that of the SUB– REE price decreases. This follows from because, ceteris paribus, greater weight is placed on v in the COM–REE price than in the SUB–REE price ( β COM ≥ β SUB ). Figure 5 also illustrates changes in Var( P) as the amount of private information (measured by ρε or λ) varies. Since λ and ρε affect Var( P) similarly, we present just the case of ρε . When x is less volatile, the price that tracks v more closely will be more volatile. Figure 5(a) illustrates this: as ρε increases, the SUB–REE price is closer to v and hence its volatility increases, whereas the COM–REE price’s volatility decreases as it is closer to x. Analogous reasoning applies to Figure 5(b), where x is relatively more volatile.
References [1] Bruno Biais, Peter Bossaerts, and Chester Spatt (2008), Equilibrium asset pricing and portofolio choice under heterogeneous information, IDEI Working Paper number 474, forthcoming in the Review of Financial studies
[2] Douglas Diamond and Robert Verrecchia (1981), Information aggregation in a noisy rational expectations economy, Journal of Financial Economics, Vol. 9, Issue 3, pp. 221235
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Pre-print of paper published in the Journal of the European Economic Association, Vol. 7, pp. 90-115, 2009
Figure 5
(a) ρ =5
(b) ρ =1
x
x
0.98
4
0.96 3.5 0.94 3
Var(P)
Var(P)
COM-REE
0.92
0.9
0.88
SUB-REE
SUB-REE
2.5
2
COM-REE
0.86 1.5 0.84 1 0.82
0.8
0
0.5
1
1.5
2
2.5
3
3.5
Precision of Payoff Signal, ρε
0.5
4
0
0.5
1
1.5
2
2.5
3
Figure 5: Price volatility in the two REE.
34
3.5
Precision of Payoff Signal, ρε
4
