Information asymmetry and price informativeness with short$horizon ...

Information asymmetry and price informativeness with
short-horizon investors
Qi Chen Zeqiong Huang Yun Zhang
April 21, 2011
Chen (qc2@duke.edu) and Huang (zh12@duke.edu) are from the Fuqua School of Business at Duke
University. Zhang (yunzhang@gwu.edu) is from George Washington University School of Business. We
bene…t from discussions with Hengjie Ai, Katherine Schipper, Vish Viswanathan, and workshop participants
at Duke.
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Information asymmetry and price informativeness with
short-horizon investors
Abstract
The paper analyzes the e¤ect of information asymmetry among investors on price informativeness
in a two-period trading model of perfect competition with short-horizon investors,
where price informativeness refers to the amount of private information revealed by price
in equilibrium. The main …nding is that information asymmetry exacerbates the loss of
price informativeness due to short-horizon. Several important implications of this …nding
are analyzed. We show that both short-horizon and information asymmetry are necessary
conditions for public information to a¤ect price informativeness. Public information improves
price informativeness only when it is of high quality (precision). When the quality
of public information is low, multiple equilibria can arise and increasing public information
could reduce price informativeness.

1 Introduction
The main purpose of this paper is to identify a mechanism under which information asymmetry
a¤ects price informativeness in stock markets with perfect competition. In doing so,
we seek to understand whether and how public information a¤ects the overall informational
e¢ ciency of stock prices via its e¤ect on information asymmetry. A unique aspect of stock
price is its ability to aggregate and reveal private information in the stock market (Grossman
(1976)). It does so via the trading process when traders condition their trades on the information
available to them (e.g., Grossman and Stiglitz (1980), Hellwig (1980), Glosten and
Milgrom (1985), Kyle (1985)). Price informativeness, which refers to the amount of private
information revealed by price that is otherwise not available to the general public, directly
contributes to the overall stock market e¢ ciency. 1
More price informativeness reduces investors’uncertainty
about …rms’fundamental and the risk premium they require to hold the
stocks, which can in turn lower …rms’costs of capital. 2
additional information to make better investment decisions. 3
It can provide corporate managers
That prices contain valuable
private information also underlies proposals for policy makers and regulators to base their
action on prices. 4
While it is without controversy that more private information increases price informativeness,
the e¤ect of distribution of private information among investors (i.e., information
1 The extent that price properly incorporates public information will also contribute to the overall market
e¢ ciency. We focus on price’s ability to reveal private information because such information would not
otherwise be available to the economy without price. In additional analysis, we also examine the overall
price e¢ ciency, i.e., the total amount of information (private and public) that price impounds.
2 For example, Easley and O’Hara (2004), and Lambert, Leuz and Verrecchia (2009).
3 See, e.g., Holmstrom and Tirole (1993), Dow and Gorton (1997), Subranmania and Titman (1999),
Goldstein and Gumbel (2008), Dow, Goldstein and Gumbel (2011), for theoretical models. See, e.g., Luo
(2005), Chen, Goldstein and Jiang (2007) for empirical evidence.
4 See Bond, Goldstein and Prescott (2010), and Bond and Goldstein (2011) for discussions on related
literature.
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asymmetry), and the related question of the role of public information, is less clearcut. On
the one hand, advocates for more public information disclosures argue that more public information
levels the playing …eld and improves price e¢ ciency. The implicit assumption is
that information asymmetry is deterimental. On the other hand, theoretical studies show
that in stock markets with perfect competition, price informativeness depends only on the
average precision of investors’private information: neither information asymmetry or public
information a¤ects the equilibrium price informativeness (see, e.g., Verrecchia (1982)). This
point was recently highlighted in the context of multiple assets by Easley and O’Hara (2004)
and Lambert, Leuz and Verrecchia (2009). Lambert, et al. (2009) further conjecture that
for information asymmetry to matter, researchers have to look into trading models of imperfect
competition (e.g., Kyle (1985), Glosten and Milgrom (1985), Kyle (1989), Diamond
and Verrecchia (1991)). 5
A key assumption for the "information asymmetry irrelevance" result is that investors’
investment horizon is the same as the …rm’s operating horizon, that is, investors hold stocks
until the …nal liquidation dates.
This assumption can be restrictive to the extent that
investors trade for various reasons and often sell stocks before the …nal payo¤ dates, either
because they are subject to exogenous liquidity needs, or they are simply outlived by the
…rm. In this paper we relax this assumption and study the impact of information asymmetry
when equilibrium prices are determined by investors with short investment horizons.
To capture the main gist of short horizon while maintaining tractability, we extend the
two-period version of Allen, Morris and Shin (2006) to allow information asymmetry among
investors. 6 In the model there are two overlapping generations of risk-averse investors. Each
5 The e¤ect of information asymmetry on price informativeness is also at the center of the debate over
insider trading. Proponents for insider trading argue that it allows price to be more informative; whereas
opponents argue that it reduces price informativeness by reducing the market liquidity. See Easterbrook
and Fischel (1991). Models of perfect competition (such as the one we study) assume that traders are price
takers and therefore liquidity is not a concern.
6 Similar models have been studied in Grundy and McNichols (1999) and Brown and Jennings (1991), and
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investor observes his own private information, but the quality (precision) of their private
information di¤ers. Investors from the …rst generation trade stocks of a risky security at the
beginning of the …rst period but liquidate their holdings for consumption goods in the second
period, before the stock’s fundamental value is realized. As a result, the consumption value of
their stock holdings depends on the stock price in the second period, which is determined by
the willingness of the second generation investors to pay for the stock. The second generation
investors live to receive the stock’s terminal value for consumption. As such, they behave
similar to investors with long investment horizon; whereas the …rst generation investors are
considered short-horizon investors.
Our primary interest is in how information asymmetry a¤ect the …rst period’s price informativeness
when investors have short horizons. Our main …nding is that information asymmetry
unambiguously lowers price informativeness; and may give rise to multiple equilibria
in situations where the equilibrium is otherwise unique without information asymmetry. The
intuition starts from the fact that short-horizon investors face the uncertainty of the next period
price, as opposed to the uncertainty of the fundamental value of the stock. This implies
that their trades are guided by their expectations of the future price, not of the fundamental.
Consequently, their tradings are less sensitive to their private information than long-horizon
investors. Lower sensitivity in turn reduces price informativeness, as shown in Allen, Morris
and Shin (2006) and Gao (2008). The new insight from our analysis is that while the sensitivity
is lowered uniformly when investors have identical private information precision, it is
no longer the case with information asymmetry. Whereas less informed investors (those with
less precise private information) do not reduce their sensitivities as much (because their sensitivities
are not high to begin with), the reduction is more pronounced among more informed
investors (those with more precise private information). Because price informativeness depends
on the average of individual sensitivities, it follows from Jensen’s inequality that the
recently in Gao (2008). None of these papers allow investors to have di¤erential precisions in their private
information.
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average of individual sensitivities is lower than the sensitivity of the average investor. This
leads to an aggregation loss in price informativeness, above and beyond the loss induced by
short horizon. Furthermore, more severe the information asymmetry is (as captured by a
mean-preserving spread of the distribution of the precision levels across investors), the larger
the aggregation loss, and the lower the equilibrium price informativeness.
The intuition for multiple equilibria is subtle and relates directly to the aggregation
loss mechanism discussed above. Since price is publicly observable, traders always use the
information they inferred from price in determining their trades. Speci…cally, if investors
conjecture equilibrium price to be more informative, they would rely less on their private
information and more on price. In standard models with no information asymmetry, this
would not constitute an equilibrium as lower sensitivity to private information would result in
lower equilibrium price informativeness, contradicting investors’initial conjecture. However,
in our model, more reliance on price would reduce the aggregation loss due to information
asymmetry. This e¤ect may be strong enough to result in higher price informativeness
in equilibrium, despite the fact that each individual trader has relied less on her private
information.
With respect to public information, we …nd that both information asymmetry and short
horizon are necessary conditions for public information to a¤ect price informativeness. 7 Public
information a¤ects price informativeness via the aggregation loss channel uniquely identi…ed
when both information asymmetry and short horizon are present. We show that
whenever the equilibrium is unique, more public information can improve price informativeness
by reducing the aggregation loss. The e¤ect is ambiguous when multiple equilibria
exist. Depending on the starting equilibrium, more public information can either increase
7 With only short-horizon but no information asymmetry, public information does not a¤ect price informativeness
(see, Allen, Morris and Shin (2006) and Gao (2008)). Neither does it a¤ect price informativeness
with only information asymmetry but no short-horizon (see, Verrecchia (1982), Easley and O’Hara (2004),
Lambert, et al. (2009)).
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or decrease price informativeness.
A su¢ cient condition for public information to improve price informativeness is that the
quality (precision) of public information is high enough. To see the intuition, note that
when public information is very precise, there is little uncertainty about the fundamental,
and the future price does not deviate much from the fundamental. Consequently, shorthorizon
investors’trading behavior is similar to that of the long-horizon investors, that is,
more public information has the similar e¤ect as lengthening investors’investment horizon.
Since price informativeness is higher when long-horizon investors than when short-horizon
investors, more public information increases price informativeness in this case.
To further relate to prior literature, we analyze the relation between information asymmetry
and the price discount in the …rst period, where the discount is the di¤erence between
the equilibrium price and the expected value of the fundamental. In the context of multiple
assets, Easley and O’Hara (2004) and Lambert, et al. (2009) interpret the …rm-speci…c discount
as the …rm’s cost of capital. To the extent that the risky asset in our model represents
the market portfolio, or a single …rm whose risk cannot be diversi…ed away, the discount
can be viewed as the expected return for the risky asset relative to the risk-free bond. We
…nd that holding the average information precision in the economy constant, the discount
is higher with information asymmetry than without. Further, the marginal (bene…cial) effect
of public information in reducing the discount is higher when the degree of information
asymmetry is higher.
Our paper contributes to the literature by identifying a mechanism in which information
asymmetry diminishes price’s e¤ectiveness in aggregating and revealing private information.
In addition to providing a theoretical support for the claim that information asymmetry is
deterimental to market e¢ ciency, even when the market is perfectly liquid, it also o¤ers a
channel for public information to a¤ect market e¢ ciency that has not been studied in the
prior literature. In most prior literature, public information improves market e¢ ciency (how
close price is to fundamental) by providing more information about the fundamentals. In
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our setting, public information can also move price closer to the fundamental by improving
price’s e¢ ciency in aggregating and revealing private information. It does so by a¤ecting
the degree of the aggregation loss discussed earlier. As such, our paper o¤ers a theoretical
framework for the claim that more public disclosure helps improve market e¢ ciency by
leveling the playing …eld. 8 9
Our paper relates to the accounting literature on the role of public information and of
information asymmetry. These issues are central to accounting research because accounting
information is a major source of public information and plays an important role in reducing
information asymmetry among investors. Our paper is closely related and indeed extends
Gao (2008) who analyzes a similar model with short horizon investors but without heterogeneous
information quality among investors. Gao (2008) analyzes the role of public information
on price e¢ ciency, which captures the amount of all information (public and private)
that price impounds. Gao (2008) quanti…es price e¢ ciency by how close price is to the fundamental.
Since price informativeness measures how much private information impounded
in price, ceteris paribus, more price informativeness will increase price e¢ ciency, although
the reverse is not true. Indeed in Gao (2008) public information improves price e¢ ciency
without improving the informativeness of price. We extend Gao (2008) to allow information
asymmetry and show that public information also plays a role in price informativeness.
Our study also sheds light on the recent debate on the e¤ect of information asymmetry on
…rms’costs of capital (e.g., Easley and O’Hara (2004), Lambert, et al. (2007, 2009)). These
8 Easley and O’hara (2004) and Lambert et al. (2007, 2009) show that more public information can decrease
…rms’costs of capital, by providing more information to investors. In these models, public information
helps price become closer to the fundamental but not more informative.
9 Our model is a version of the dynamic noisy rational expectations model. A large literature has used
this model to analyze various issues, including price volatility, trading volume, and technical analysis, among
others. We refer the readers to Brunnermeier (2001) for an excellent and comprehensive survey. To our
best knowledge, our paper is the …rst to analyze the impact of heterogenous information precision among
investors on price informativeness.
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studies analyze a static trading model, where both public and private information lower
…rms’costs of capital by increasing the average precision of all information in the economy.
Their analyses suggest that the distribution of information (between public and private, or
among private) does not matter as long as the average precision of the economy is controlled
for. In contrast, our model shows that the distributions matter, and that public information
can take on the addition role by reducing the aggregation loss due to information asymmetry
among investors. 10
In what follows, we …rst set up the model in Section 2 and then provide the model solution
in Section 3 before Section 4 concludes.
2 Model set-up
We study a two-period noisy rational expectation trading model with short-horizon investors
and independent supply shocks (e.g., Allen, Morris and Shin (2006) and Gao (2008)). In
each period (denoted by t = 1 or 2), a continuum of investors with a unit measure (indexed
by i 2 [0; 1]) choose their investments through trading between a risky asset and a riskless
asset (cash) to maximize their expected utility. The rate of return for cash is normalized
to 1, without loss of generality. The liquidation value of the risky asset, , is random,
has a di¤use prior, and will be realized at the end of the second period. Investors do not
observe the average per capita supply of the risky asset (denoted as s t ) but understand that
s t N
s; 1 and is assumed to be independent across periods, with s > 0:
t
At the beginning of period 1 before trading occurs, a public information signal z is
10 The model in the paper is based on the dynamic noisy rational expectations model. A large literature
has used this type of model to analyze various issues, including price volatility, trading volume, and technical
analysis, among others. We refer the readers to Brunnermeier (2001) for an excellent and comprehensive
survey. To our best knowledge, our paper is the …rst to analyze the impact of heterogenous information
precision among investors on price informativeness.
7

available to all investors, including period 2 investors after they are born:
z = + " z where " z N 0; 1
.
Each investor also observes a private signal x ti prior to trading:
1
x ti = + " it where " ti N 0; :
it
We assume the p.d.f.
of i across investors in each period is g t () (with the associated
c.d.f. G t ()). We further assume " it is i.i.d. across periods, implying that we can drop the
subscript of t from the g (and G) function from now on. This is without of loss of generality:
as will be shown shortly, the distribution of i among investors in the second period does
not matter for our main results.
Each investor only lives for one period and has a constant absolute risk aversion (CARA)
c
utility function of U (c) = exp
t
where t is the risk tolerance parameter for period
t investors. The …rst generation of investors transfer their risky asset holdings to the next
generation through trading at the beginning of period 2. With CARA utility functions and
normal distributions, the standard result shows that the optimal demand for the risky asset
by a …rst generation investor i is
Similarly, the demand by a second generation investor is
D 1i = 1
E 1i (p 2 ) p 1
V ar 1i (p 2 ) : (1)
D 2i = 2
E 2i () p 2
V ar 2i () : (2)
In both expressions, the subscripts denote that all expectations are taken with respect to the
information set ti of investor i in the t th generation. Speci…cally, 1i fz; p 1 ; x 1i g where p 1
is the equilibrium price of the risky asset from the …rst round of trading, and x 1i is investor
i’s private information signal. Similarly, 2i fz; p 1 ; p 2 ; x 2i g. Note that because investors
are short lived, x 1i is not an element of 2i . However, the second generation investors will
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glean some information about the x 1i s from p 1 . Short-horizion is also re‡ected in the fact
that the payo¤ for the second generation investors is the …rm’s liquidation value , whereas
the payo¤ for the …rst generation is the risky asset’s price from the second round of trading,
p 2 . The fact that prices appear in investors’information set re‡ects the notion that trading
is a information aggregation process and investors attempt to infer others’information from
prices (Grossman and Stiglitz, 1980). In summary,our setup is identical to those in Allen
et. al. (2006) and Gao (2008) except that we now allow the precisions of investors’private
information to di¤er. Figure 1 illustrates the timeline of our model.
3 Solution
3.1 Preliminaries: Price informativeness in the second period
The solution for the second period equlibrium price is fairly standard and the Appendix of
Allen et. al. (2006) provides a detailed account. Here, we only highlight the part that’s
relevant to our analysis. Start by conjecturing a linear equilibrium where period t price is
given by 11 p 1 = b 1 z + c 1 d 1 s 1 (3)
and p 2 = a 2 p 1 + b 2 z + c 2 d 2 s 2 : (4)
11 p 1 enters into p 2 expression because traders in period 2 will be able to learn useful information from p 1
about . This seems to contradict the empirical notion of a weak form market e¢ ciency where past prices
should contain no information about future fundamentals. However, in a standard REE model, the supply
noise prevents prices from fully revealing all private information. That is, past prices are not a su¢ cient
statistics for all private information. Because traders are short-lived, the second generation traders do not
observe the actual private signals of the …rst generation investors. They can, however, infer from p 1 these
private signals and use it to help them forecast the fundamentals.
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As in the literature, the informativeness of p t with respect to can be summarized by
2 ct
t
d t : (5)
t
To elaborate, de…ne
P1 p 1 b 1 z d 1
= s 1 ;
c 1 c 1
(6)
P2 p 2 (a 2 p 1 + b 2 z) d 2
= s 2 :
c 2
c 2
(7)
Since there is a one to one mapping between p t and P
t , the information content in p t is
equivalent to that in P
t . As such, since t measures the precision (inverse of the variance)
of the noise term in P
t , we interpret it as period t’s price informativeness. Note that our
de…nition of price informativess excludes the direct e¤ect of the public signal and is only
related to how the price aggregates investors’private information.
We note here that price informativeness is closely related to, but distinct from, the
concept of market e¢ ciency.
In empirical studies, market e¢ ciency takes various forms
depending on the degree to which stock prices re‡ect value relevant information.
While
there is no consensus on the precise de…nition of market e¢ ciency, in this paper we follow
the tradition in the analytical literature and use these terms interchangeably to mean the
amount of all value relevant information embedded in price, including both public and
private information. A typical measure for price e¢ ciency is the distance (mean squared
error) between the equilibrium prices and the fundamentals (e.g., Gao (2008)): more e¢ cient
prices are closer to the fundamentals on average. Everything else equal, more informative
price improves price e¢ ciency while the reverse is not true. This is because price can be close
to fundamental from incorporating public information alone without re‡ecting any private
information. In contrast, informative prices provide unique information to the public that
is not available from other sources. In other words, if price is not informative, the overall
amount of information available to investors would not be reduced without price.
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The equilibrium p t is set such that the aggregate demand for risky assets equals the
aggregate supply, i.e.,
With normal distributions, we obtain
Z
i
D ti dG ( i ) = s t :
E 2i () = z + 1P1 + 2 P2 + i x 2i
+ 1 + 2 + i
V ar 2i () =
1
:
+ 1 + 2 + i
Substituting the two expressions for E 2i () and V ar 2i () into (2) and imposing the market
clearing condition for period 2, we have
p 2 = k 2
1
c 1
b 1
2
c 2
b 2
z +
1
c 1
2
c 2
a 2
p 1 +
1
where k 2
1
+ 1 + 2
1
and R
+ i i dG ( i ). Clearly, here measures the average
c 2
precision of all investors.
Matching coe¢ cients in (8) with those in (4), It is immediate that
s 2
2
(8)
2 = 2 2 2 2 : (9)
(9) demonstrates that holding the average of private precisions constant, price informativeness
in period 2 no longer depends on the distribution of private precisions among
investors g 2 ( i ). The intuition is that although investors with less precise information trades
less, those with more precise information trades more and in equilibrium these two e¤ects
exactly o¤set each other. In addition, since (9) doesn’t depend on , public information does
not a¤ect price informativeness in the second period. That is, while more precise public information
can move the second period price closer to the fundamentals through the z term
in (8), it does not a¤ect the extent to which the price aggregates investors’private information.
This result may appear counterintuitive at the …rst glance, as one may expect that
since more precise public information induces investors to place less weight on their private
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information relative to the public signal, the ability of price aggregates private information
may decrease. However, this view fails to recognize the fact that with more precise public
information investors trade more aggressively. In equilibrium, these two e¤ects again exactly
o¤set each other. We note these results are the same as those from the prior literature with
one-period model (e.g., Verrecchia (1982), Lambert et. al. (2007)). Thus, our main focus,
as is that of Gao (2008)’s, is on properties of the …rst period equilibrium price, 1 , which we
derive next.
3.2 First period equilibrium price
3.2.1 Three benchmarks
Before we derive a solution to our model and study its imiplications, it is useful discuss three
benchmark scenarios in order to hightlight di¤erences between our setting and the prior
literature. In the …rst benchmark (short-horizon homogeneous investors), both generations of
investors have the same precision for their private signals (i.e. i = , 8i). This benchmark is
studied in Gao (2008). In the second benchmark (long-horizon heterogeneous investors with
early maturity), we assume the risky asset’s payo¤ is realized at the end of the …rst period.
Like Gao (2008), we interpret it as representing a long-horizon economy as investors lives
as long as the asset’s life span. Finally, in the third benchmark (long-horizon heterogeneous
investors with late maturity), we study a setting similar to Brown and Jenning (1989) where
investors with (potentially) heterogeneous private precisions live for two periods and trade
twice before the risky assets matures at the end of second period. Note that the second and
third benchmark cases also allow for homogeneous precision investors as a special case. We
summarize our …ndings in the following lemma.
Lemma 1 Let B 1
1 , B 2
1 , and B 3
1 denote the price informativeness of p 1 under Benchmark 1
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through 3, respectively. Then,
2
B 1
1 = 2 1 2 2
1
2 + < B 2
1 = B 3
1 = 2 1 2 1 ;
where 2 = 2 2 2 2 is the price informativeness of p 2 .
Proof of Lemma 1 The derivation of B 1
1 and B 2
1 are similar to that in Gao (2008), hence
omitted. See the appendix for a derivation of B 3
1 .
Lemma 1 shows that a shorter horizon lowers the price informativeness in the …rst period,
compared to the two long horizon benchmarks. The intuition lies in the so-called "Beauty
Contest" phenomena, studied in Allen et al. (2006). It is well known that in models with
perfect competition the expected equilibrium price equals to the average investors’expectations
of their …nal payo¤. As such, the period 2 price equals a weighted average of the public
signal z and in expectation (taken over the random supply s). To see this, for expositional
ease, suppose for a second generation investor 2i, his assessment of is
E 2i () = wz + (1 w) x 2i .
The average of E 2i () across all second generation investors is thus
E 2i () = wz + (1 w) :
Since with a short horizon the …rst generation of investors have to liquidate their positions
in period 2, they are concerned with the risky asset’s terminal payo¤ only to the extent it
is re‡ected in the period 2 price. Consequently, the period 1 price attaches a larger weight
to the public signal than the period 2 price, thus magnifying the impact of the public signal
on price and making it harder to infer from the price. Speci…cally, suppose for a …rst
generation investor 1i, his assessment of is
E 1i () = wz + (1 w) x 1i :
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And his assessment of the second period price is
E 1i
E2i () = wz + (1 w) [wz + (1 w) x 1i ]
= 1 (1 w) 2 z + (1 w) 2 x 1i :
The average expectation of the second period price is thus
E 1i
E2i () = 1 (1 w) 2 z + (1 w) 2 :
Stated di¤erently, the period 1 price will be less informative with short-horizon investors
because it aggregates less of investors’private information. Our next lemma sheds on light
how price informativeness changes with the precision of public information for the three
benchmark cases.
Lemma 2 Public information has no e¤ect on price informativeness in any of the three
benchmarks.
Proof of Lemma 2 Lemma 2 results from a straightforward inspection of B 1
1 , B 2
1 and
B 3
1 .
That the public information has no e¤ect on price informativeness follows the same intuition
as that for why the public information does not a¤ect the second period price informativeness
(i.e., 2 ) and is discussed earlier. Again, though an increase in public information
signal precision induces investors to impose more weight on the public information, an overall
reduction in uncertainty due to high quality public information increases the investors’
trading intensity.
Public information still increases the overall market e¢ ciency, de…ned in Gao (2008) as
the mean squared error of price in predicting the fundamentals. 12
This is intuitive because
12 We use the term "market e¢ ciency" the same way as Gao (2008) in order to stay consistent, even though
the term may mean di¤erent things for others.
14

price always incorporates public information as seen from (??). This is also consistent with
the main message from Lambert et. al. (2007) that it is the average quality of all information
in the economy that matters (in terms of increasing market e¢ ciency), not the distribution
of the information. Thus, public information improves market e¢ ciency only to the extent
that it improves the average quality of all information available to the market. This implies
that as far as policy implications are concerned, there is no justi…cation for spending more
resources in improving public information versus improving private information, as long as
the average information quality across all investors is kept constant. 13
The previous section shows that information asymmetry doesn’t a¤ect the second period
price informativeness holding the average private information precision constant. Our next
lemma deals with the e¤ect of information asymmetry on the …rst period price informativeness
and price discount for benchmark 2 and 3, where the …rst period price discount is
de…ned as E (
p 1 j).
Lemma 3 i ’s distribution doesn’t a¤ect B 2
1 and B 3
1 ; given is held constant.
i ’s distribution doesn’t a¤ect the …rst period price discount under Benchmark 2
and 3, given is held constant.
Proof of Lemma 3 See appendix.
Lemma 3 shows that our "irrelevance" result for information asymmetry in the second
period carries over to the …rst period as long as investors have long horizons. This is consistent
with Lambert et al. (2007) assertion that information asymmetry does not matter for
13 Public information has a welfare role in that it may help reduce costly private information aquisition,
as shown in Diamond (1985). But it still does not increase the incremental price informativeness even in
Diamond (1985). This is because in Diamond, there is one common private signal realization. It is called
private because only a subset of investors observe it, if they decide to acquire it at a cost. Therefore, as long
as public information does not make all investors obstain from acquiring private information, it will have
no e¤ect on price informativeness. If it does prevent all investors from acquire private informatioin, it will
actually decrease price informativeness.
15

cost of capital in a model of perfect competition. Having established the three benchmark
cases, we now turn to our main setting where short horizon investors di¤er from each other
in their private information precisions.
3.3 Heterogenous precisions of private information with short horizon
investors
The …rst period equilibrium can be solved in a similar fashion to the second period equilibrium,
with the exception that the …rst period investors care about only to the extent that
it a¤ects the expected second period price. Speci…cally, with short horizons, each …rst period
investor forms his expectation of the second period price based on his own private information,
the public information, the …rst period price and i ’s distribution. In equilibrium, the
…rst period price is such that the aggregate demand exactly equates the aggregate supply of
shares. Proposition 1 characterizes the …rst period price informativess with heterogeneous
short horizon investors.
Proposition 1 The …rst period equilibrium price informativeness 1 is (implicitly)
determined by
1 = 1 2 2 1
2 2
2 +
R
i
r ( i ) dG ( i )
r ! 2
; (10)
where r ( i ) =
i
+ 1 + 2 + i
;
where B 1
= 1 2 2 2
2.
1 2 +
1 < B 1
;
More generally, if F ( i ) is a mean-preserving spread of G ( i ) and there exists a
unique linear equilibrium under G ( i ),
1 (F ) < 1 (G) :
16

Proof of Proposition 1 (See Appendix).
Lemma 1 shows that price informativeness is reduced with short horizon investors than
with long horizon investors.
Proposition 1 further shows that this decrease is more pronounced
when investors’ private information has di¤erential precisions. Furthermore, the
reduction in price informativeness is larger when the degree of information asymmetry is
higher, as measured by a mean-preserving spread of the distribution function for . The
intuition is the following. As before, price informativeness in our equilibrium depends on
the average sensitivity of individual investors’trades toward changes in the fundamentals.
When all investors have identical private information precisions, their individual sensitivity
(toward their private information) is also identical and thus equals the average (toward
fundamentals). In contrast, with heterogeneous investors, an individual trader’s sensitivity
is concave in his private information precision. As such, the average of sensitivities acoss
all investors is always lower than the sensitivity of the average by Jensen’s inequality. Consequently,
information asymmetry further reduces price informativeness, above and beyond
the reduction induced by short horizons.
Interestingly, the concavity property in the sensitivity of individual trades toward changes
in fundamentals doesn’t arise for long horizon investors, as implied by Lemma 3. Intuitively,
long horizon investors ultimately care only about the terminal payo¤ , while short horizon
investors are concerned with only to the extent that it a¤ects the second period expected
price. Mathematically, let’s take Benchmark 2 (long-horizon heterogeneous investors with
early maturity) as an example. A …rst period investor’s demand is
D 1i = 1
E 1i () p 1
V ar 1i () :
Note that since
E 1i () = V ar 1i () [z + 1 P 1 + i x 2i ]
17

and
P1 p 1 b 1 z
;
c 1
(1 c 1 ) p 1 b 1 z
D 1i = 1 z + 1 +
c i x 2i ;
1
which is linear in investor 1i’s private precision. This is because V ar 1i () in the numerator
is exactly cancelled out by the same term in the denominator, a direct consequence of long
horizons. Hence,
(1 c 1 ) p 1 b 1 z
D 1i dG ( i ) = 1 z + 1 + :
Z i
c 1
In contrast, with short horizon investors, a …rst period investor’s demand is
D 1i = 1
E 1i (p 2 ) p 1
V ar 1i (p 2 )
= 1
E 1i [a 2 p 1 + b 2 z + c 2 d 2 s 2 ] p 1
V ar 1i (p 2 )
= 1
a 2 p 1 + b 2 z + c 2 V ar 1i () [z + 1 P 1 + i x 2i ] p 1
V ar 1i (p 2 )
Thus, the coe¢ cient in front of x 2i is c 2V ar 1i () i
V ar 1i (p 2
. Note, because of short horizons, V ar
)
1i ()
can be cancelled out by V ar 1i (p 2 ), giving rise to the aforementioned concavity property.
Having established information asymmetry reduces the …rst period price informativeness,
we now look at whether the …rst period investors demand a higher discount in the …rst period
price due to information asymmetry. De…ne the …rst period price discount as E ( p 1 j) :
We have the following corollary.
Corollary 1 Assume i 2 ( min ; max ), where max > min > 0. Holding average private
information precision constant, the …rst period price discount is higher under information
asymmetry than under information symmetry, if 1 2 1 is su¢ cient big. More
generally, if F ( i ) is a mean-preserving spread of G ( i ) and there exists a unique
linear equilibrium under G ( i ), then the price discount under F is higher than under
G when 1 2 1 is su¢ cient big.
18

Proof of Corollary 1 See appendix.
Lambert et al. (2007) shows that in a model of perfect competition where an individual
investor’s trading activities do not a¤ect the price, information asymmetry doesn’t matter
for cost of capital and price informativeness, as long as the average private information
precision is held constant.
And they conjecture that information asymmetry may have
an e¤ect in an imperfect competition setting such as Kyle (1985). In contrast, our paper
suggests that information asymmetry (in the sense of a mean-preserving spread) can play a
role even with perfect competitiion among investors. That is, the results so far suggest that,
in our two-period model of perfect competition, holding average private information precision
constant, information asymmetry strictly reduces the …rst period price informativeness,
when investors have short horizons. The next corollary deals with the impact of the public
information precision on …rst period equilibrium price informativeness.
Corollary 2 When 1 2 1 is su¢ ciently small and + 2 > 2 max , more precise public information
increases the …rst period price informativeness. That is, d 1
d > 0.
Proof of Corollary 2 See appendix.
As shown in Gao (2008), with homogeneous investors public information doesn’t impact
the …rst period price informativess. However, here we demonstrate that with heterogeneous
investors, more precise public information enables the …rst period price to better aggregate
investors’private information. As we have established earlier, introducing information asymmetry
reduces the …rst period price informativeness 1 because the concave term r ( i ) in
(10). That is
Note that
d
d 2 r
d 2 i
d 2 r
d 2 i
d
=
2 ( + 1 + 2 )
( + 1 + 2 + i ) 3 :
= 4 ( + 1 + 2 ) 2 i
( + 1 + 2 + i ) 4
> 0, when + 2 big.
19

This is to say, with more precise public information, the term r ( i ) becomes less concave
and hence more private information can be assimilated into the price. Our next result is
related to the existence of multiple equilibria.
3.4 Strategic uncertainties: existence of multiple linear equilibria
With short horizon homogeneous investors, Gao (2008) shows that there exists a unique
linear equilibrium. This can be immediately corroborated by inspecting the expression for
B 1
1 , which admits a unique value in equilibrium. However, with heterogeneous investors,
the …rst period price informativeness is implicitly determined by (10). A closer inspection
of (10) shows that 1 appears on both sides of the expression, thus raising the possibility
of multiple equilibrium 1 ’s.
multiple equilibria.
The following numerical example con…rms the existence of
Consider a binary distribution of i ’s, where i 2 f h ; l g with h = 100; l = 0;
Pr ( h ) = 0:01; 1 = 10; 2 = 1; 1 = 10; 2 = 2; and = 2: Under these parameter values,
it is straightforward to obain three equilibria.
Equilibrium 1:
p 1 = 0:5129z + 0:4871 1:1249s 1 ;
p 2 = 0:5600p 1 + 0:0037z + 0:4363 0:4363s 2 ;
1 = 1:8755; 2 = 2:
Equilibrium 2:
p 1 = 0:0648z + 0:9352 0:5507s 1 ;
p 2 = 0:9113p 1 + 0:0000z + 0:0887 0:0887s 2 ;
1 = 28:8346; 2 = 2:
20

Equilibrium 3:
p 1 = 0:0096z + 0:9904 0:2183s 1 ;
p 2 = 0:9858p 1 + 0:0000z + 0:0142 0:0142s 2 ;
1 = 205:7381; 2 = 2:
This example is illustrated in Figure 1, where the 45-degree straight line represents the left
hand side of (10), while the curvy line represents its right hand side. The two lines intersect
each other three times, thus yielding three solutions for 1 :
The existence of multiple linear equilibria may at the …rst glance appear qutie puzzling.
This is because, when the perceived …rst period price informativeness 1
increases, one
expects that a …rst period investor would increase the weight on the price and decrease the
weight on his private information, thus reducing the ability of price to incorporate private
information and leading to a smaller 1 . This seems to point to a unique solution. However,
this reasoning fails to recognize the subtle role played by information asymmetry to aggregate
private information into price. As discussed in the previous section, an individual investor’s
weight on his private information is c 2V ar 1i () i
V ar 1i (p 2
, which by the proof of Proposition 1 in the
)
2 E[r( i )]
2 +
i
+ 1 + 2 + i
.
appendix equals to
r() , where r ( i) =
Note that, Though r ( i ) is
concave in i , this concavity decreases as 1 increases. To see this, the second derivative of
r ( i ) with respect to i is
@ 2 r
@ 2 i
+
= 2
1 + 2
( + 1 + 2 + i ) 3 :
Note that
lim
@ 2 r
1 !1@ 2 i
= 0:
21

Furthermore, taking a partial derivative with respect to 1 on @2 r
,
@ 2 i
@
@ 2 r
@ 2 i
@ 1
= 2 2 ( + 1 + 2 ) i
( + 1 + 2 + i ) 4 > 0, when 1 big enough.
Let’s start with the equilibrium where 1 = 28:83. When investors’perceived 1 increases,
there are two opposing e¤ects at play. On the one hand, investors put a smaller weight on
their private information, thus leading to muted price informativenes. On the other hand,
the information aggregation loss due to concavity decreases, thus resulting in pronounced
price informativeness. When the second e¤ect dominates the …rst e¤ect, a second equilibrium
with a larger 1 may emerge. Conversely, When investors’perceived 1 decreases, there are
again two countervailing forces. On the one hand, investors put a larger weight on their
private information, thus leading to a higher price informativenes.
On the other hand,
the information aggregation loss due to concavity increases, thus resulting in lower price
informativeness. When the second e¤ect dominates the …rst e¤ect, a third equilibrium with
a smaller 1 may emerge.
4 Extension: three periods
Two key results have emerged from our analysis in the previous section. That is, introducing
information asymmetry to a two-period model populated by short horizon investors reduces
price informativeness and may lead to multiple equilibria. In this section, as a robustness
check, we extend the analysis to three periods. Speci…cally, let’s consider period t 2 f1; 2; 3g.
The risky asset’s terminal payo¤ is realized at the end of period 3. As before, period t
investors only live for one period, establish their holdings of the risky asset at the beginning
of period t, and have to liquidate these holdings at the end of period t. To simplify our
analysis, we set t (i.e. period t investors’risk tolerance) to be for all t. As before, we
22

focus on linear pricing equations. That is,
p 1 = b 1 z + c 1 d 1 s 1 ;
p 2 = a 2 p 1 + b 2 z + c 2 d 2 s 2 ;
and p 3 = f 3 p 2 + a 3 p 1 + b 3 z + c 3 d 3 s 3 :
The following proposition derives expressions of price informativess.
Proposition 2
With three periods, price informativeness for each period is given by
3 = 3 2 2 ;
0
R 2 i
2 = 2 2 2 3
dG (
@ i + i + 1 + 2 + 3 i )
A
3 +
12
+ + 1 + 2 + 3
; (11)
2
32
1 = 1 2 2 4
2
2 + R 5 (12)
2 + i (+ 3 )
dG (
i + i + 1 + 2 + 3 i )
0R
2 0
i
R
2
+
dG (
@ i + i + 1 + 2 + 3 i )
i + 1 + 2
dG (
A @ i + i + 1 + 2 + 3 i )
A :
1
+ + 1 + 2 + 3
1
+ + 1 + 2
+ + 1 + 2 + 3
Proof of Proposition 2 Similar to the proof of Proposition 1, hence omitted.
R
i
Observe that (11) is quite similar to (10) in Proposition 1, except that the last term
i
+ i + 1 + 2 + 3
dG( i )
2
involves the …rst period price informativeness
1 . This is because
+ + 1 + 2 + 3
with three periods, the second period investors utilize information contained in the …rst
period price to form their portfolios. Proposition 3 sheds light on wether the two key insights
we have established before (i.e. that information asymmetry reduces the …rst period price
informativeness and can generate multiple equilibria) extends to three periods.
Proposition 3
There exists a ^ 1 > 0, such that 8 1 < ^ 1 , when investors have identical
private information precisions, there only exists one unqiue linear equilibrium.
23

Suppose parameter values are such that there is a unique equilibrium information
symmetry. When investors have di¤erring private information precisions,
where B 1
t
1 < B 1
1 and 2 < B 1
2 ;
is the period t price informativeness in the benchmark case 1 where
short horizon investors have identical private information precisions.
Suppose parameter values are such that there is a unique equilibrium with information
symmetry. When investors have di¤erring private information precisions,
there may exist multiple linear equilibria.
Proof of Proposition 3 See appendix.
Part 1 of Proposition 3 identi…es a su¢ cient condition (i.e. 1 < ^ 1 ) such that there
exists a unique linear equilibrium under information symmetry. Part 2 and 3 demonstrate
that, if one starts with a unique equilibrium under information symmetry, that introducing
information asymmetry reduces price informativenss and can lead to multiple lienar equilibria
continues to hold in a three period setup.
To give an example of multiple equilibria, let’s consider the following set of parameter
values ( = 1; 1 = 9; 2 = 1; 3 = 1 and = 1). Suppose the average private information
precision across all investors is 0:5. One can easily verify that there is only one equilibrium
under information symmetry (i.e. all investors have a identical private information precision
0:5). Now, let’s introduce information asymmetry while holding the average private
information precision constant at 0:5. Speci…cally, let’s assume a binary distribution of i ’s,
where i 2 f h ; l g with h = 50; l = 0; Pr ( h ) = 0:01: It is straightforward to obain the
following three equilibria.
Equilibrium 1:
24

p 1 = 0:2668z + 0:7332 1:4268s 1 ;
p 2 = 0:3815p 1 + 0:0159z + 0:6026 0:3210s 2 ;
p 3 = 0:0274p 1 + 0:1585p 2 + 0:0011z + 0:8130 0:1626s 3 ;
1 = 2:3770; 2 = 3:5244; 3 = 25:
Equilibrium 2:
p 1 = 0:0346z + 0:9654 0:5521s 1 ;
p 2 = 0:7959p 1 + 0:0004z + 0:2037 0:0830s 2 ;
p 3 = 0:0767p 1 + 0:4584p 2 + 0:0000z + 0:4648 0:0930s 3 ;
1 = 27:5160; 2 = 6:0271; 3 = 25:
Equilibrium 3:
p 1 = 0:0129z + 0:9871 0:3391s 1 ;
p 2 = 0:8862p 1 + 0:0001z + 0:1138 0:0379s 2 ;
p 3 = 0:0597p 1 + 0:6823p 2 + 0:0000z + 0:2580 0:0516s 3 ;
1 = 76:2460; 2 = 9:0241; 3 = 25:
Figure 2 illustrates the three solutions of 1 2 pairs. Curve A represents those 1
2 combinations such that the 2 expression in Proposition 2 is satis…ed, while Curve B
represents those 1 2 combinations such that the 1 expression in Proposition 2 is satis…ed.
The equilibria are represented by the three intersections of the two curves.
5 Conclusion
[To Be Finished.]
25

6 Appendix
Proof of Lemma 1 To be …nished.
Proof of Lemma 3 Part 1 results from a straightforward inspection of B 1
1 , B 2
1 and
B 3
1 .
[To be …nished].
Proof of Proposition 1
When traders have a short investment horizon, the informativeness
of p 1 will be di¤erent from that of p 2 . To see this, notice that the
…rst generation traders’terminal wealth depends on p 2 , instead of on the …rm’s
fundamental . As a result, the …rst generation traders’demand will be given by
D 1i = 1
E 1i (p 2 ) p 1
V ar 1i (p 2 )
Comparing D 1i with D 2i , we can see that it is the conditional expectation and
variance of p 2 (not ) that determines the …rst generation’s trade, which in turn,
will a¤ect the informativeness of p 1 . Substitute in the expressions for E 1i (p 2 ) and
V ar 1i (p 2 )
E 1i (p 2 ) = a 2 p 1 + b 2 z + c 2 E i ()
=
z +
a 2 p 1 + b 2 z + c 1 P1 + i x 1i
2
+ 1 + i
V ar 1i (p 2 ) = c 2 2V ar 1i () + d 2 2V ar 1i (s 2 )
= d2 2
(
2 V ar 1i () + 1)
2
= c2 2 2
+ 1 :
2 + 1 + i
into D 1i and apply the market clearing condition of R i
D 1i dG ( i ) = s 1 , we have
Z
E 1i (p 2 ) p 1
i
V ar 1i (p 2 ) dG ( i)
26
s 1
1
= 0

(a 2 1) p 1
Z
1
i
V ar 1i (p 2 ) dG ( i) + b 2 z
Z
E 1i () V ar 1i ()
+c 2
V ar 1i ()
i
Z
V ar 1i (p 2 ) dG ( i)
1
i
V ar 1i (p 2 ) dG ( i)
s 1
1
= 0
Since E 1i () = z+ 1 P 1 + i x 1i
1
and V ar
+ 1 + 1i () = , we have
i + 1 + i
Z
Z
1
(a 2 1) p 1
i
V ar 1i (p 2 ) dG ( 1
i) + b 2 z
i
V ar 1i (p 2 ) dG ( i)
+c 2
Z
Substitute in P 1 = p 1
(a 2 1) p 1
Z
+c 2 1
p 1 bz
c 1
i
1
i
i
(z + 1 P 1 + i x 1i ) V ar 1i ()
V ar 1i (p 2 ) dG ( i)
c 1
bz
and collect terms. We have
Z
V ar 1i (p 2 ) dG ( i) + b 2 z
Z
Z
V ar 1i ()
V ar 1i (p 2 ) dG ( i) + c 2
i
1
s 1
1
= 0
Z
V ar 1i (p 2 ) dG ( i) + c 2 z
i
V ar 1i ()
i
i
V ar 1i (p 2 ) dG ( i)
and therefore
2 R
1
b 2
p 1 = k 2
4
dG ( b
i V ar 1i (p 2 ) i) + c 2 (a 1 c 1
) R
i
R
V ar
+c 2 1i ()
dG ( i i V ar 1i (p 2 ) i)
where k 2 =
(1 a 2 ) R i
1
V ar 1i (p 2 ) dG ( i)
1
c 2 1
c 1
R
i
V ar 1i ()
V ar 1i (p 2 ) dG ( i)
s 1
1
= 0
V ar 1i ()
dG ( V ar 1i (p 2 ) i)
V ar 1i ()
dG ( V ar 1i (p 2 ) i) :
Thus, the informativeness of p 1 is given by
2 Z
c1
1 =
d 1 = 1 2 V ar 1i ()
1
c 2 i
1 i
V ar 1i (p 2 ) dG ( i)
Now, let’s solve for the expression of integrand:
Z
V ar 1i ()
c 2 i
i
V ar 1i (p 2 ) dG ( i)
Z
1
+
= c 2 1 +
i
i
dG (
c
2 2 i )
2
i + 1
2 + 1 + i
= 2
c 2
E [r ( i )] :
27
3
z
(13) 5
s 1
1
2

where r ( i ) =
i
+ 1 + 2 + i
and the expectation operator is with respect to the
underlying distribution of i . Note r ( i ) is strictly increasing and concave in
(i.e., r 0 () > 0 and r 00 () < 0). Also note that c 2 can be expressed
Therefore, we have
c 2 =
2 +
+ 1 + 2 + = 2 +
r :
1 = 1 2 1
2
2 +
E [r ()]
r ! 2
:
Thus, the equilibrium informativenss of …rst period price would be lower under F
than under G. This proves our main result in Proposition 1. For completeness,
one can solve the pricing functions by setting up a system of equations where
coe¢ cient in (3) are equal to those in (13). These coe¢ cients are
a 2 =
b 2 =
1
c 1
+ 1 + + 2
b 1
c 1
1
+ 1 + + 2
+
c 2 =
2
+ 1 + + 2
d 2 = 1 + 2
+ 1 + + 2
2 = 2 2 2
b 1 =
c 1 =
1 =
d 1 =
+ 1 + 2 +
+ 2
+ 1 + 2 +
0
2 2 2
2
+ 1
@
2
+ 2
2
2 + 1 + 2 + 2
1 + R
dG (
i + 1 + 2 + i i )
R
1
2
dG (
i + 1 + 2 + i i )
R
i
dG (
i + 1 + 2 + i i ) + R
1
+
R
2
+ 1 + i
dG (
i + 1 + 2 + i i )
R
12
i
dG (
i + 1 + 2 + i i )
A
+ 1 + 2 +
R
i
i
dG (
+ 1 + 2 + i i ) + 1
R
+ 1 + i
dG (
i + 1 + 2 + i i ) R i
28
i
+ 1 + 2 + i +
+ 1 + 2 + i
dG ( i )
R
+ 1 + 2 + i +
+ 2 i
dG (
+ 1 + 2 + i i )
:
i
dG (
+ 1 + 2 + i i )

Clearly, for every solution of 1 , there is a unique set of equilibrium prices.
To prove part 2 of the proposition. Note that
1 = 1 2 1
! 2
2 E [r ()]
2 + r
2
2
;
< 1 1 2 1
2 +
where the inequality follows the Jensen’s inequality and the fact that r () is
strictly concave. It is easy to see that 1 is the price informativeness when the
precisions of investors’private information are identical. Thus, holding the average
constant, whenever i 6= j for any i 6= j, the informativeness of …rst period
price will be lower.
Given the concavity of r ( i ), it is immediate that if a distribution of i , say
F ( i ), is a mean preserving spread of G ( i ), then for every 1 the right hand
side of (10) is strictly smaller under F than under G. Furhtermore, if there exisits
a unique linear equilibrium under G, the slope of the right hand side of (10) at
the equilibrium has to be less than 1, otherwise there would be at least one more
equilibrium. Thus, it must be that any price informativess generated under F
must be strictly less than that under G. Q.E.D.
Proof of Corollary 1 Note that
E ( p 1 j) = E ( b 1 z c 1 + d 1 s 1 )
= d 1 s.
29

From the proof of Proposition 1,
2
R
+ 2
d 1 =
2 + 1 + 2 + 2
2
=
By (10), we have
Z
Thus,
+ 2
2
2 + 1 + 2 + 2
i
i
6
4
i
dG (
+ 1 + 2 + i i ) + 1 + 1 + 2 + i + dG (
i + 1 + 2 + i i )
R
+ 1 + i
dG (
i + 1 + 2 + i i ) R i
dG (
i + 1 + 2 + i i )
3
R
i
i
+ 1 + 2 + i
dG ( i ) =
+ 2
R
1
R
+
+ 1 + i
dG( i + 1 + 2 + i i )
1 1+ R
+ 2
dG( i + 1 + 2 + i i )
+ 1 + i dG( + 1 + 2 + i i ) R i
+ 1 + 2 +
i
+ 1 + 2 + i
dG( i )
p
1
7
5 .
+ 2
p :
2 1
d 1 =
2
+ 2
2 + 1 + 2 + 1
2
R
+ 1 + i
dG (
i + 1 + 2 + i i ) +
p
1 p 1
+ 2 1
+ 1 + 2 + 1 + R
dG ( + 2 i + 1 + 2 + i i )
R
+ 1 + i
dG (
+ 1 + 2 + i i )
i
Notice that
+ 1 + i
(+ 1 + 2 + i ) 1
is concave in i and
1 constant, d 1 is higher under F than under G. Also, note that
R i
is convex in
+ 1 + 2 + i i . Hence, holding
( + 2 ) 2
, and
2 (+ 1 + 2 + ) 2
dG (
+ 1 + 2 + i i ) are all decreasing in 1 , and that R + 1 + i
dG (
i + 1 + 2 + i i ) increasing
p 1 ( +
in 1 . Finally, simple algebra shows that
2 ) p 1
(+ 1 + 2 + ) is decreasing in 1 only when
1 > + + 2 . Consequently, everything else equal, d 1 decreases with 1 when
1 > + + 2 . Note
R
i
r ( i ) dG ( i )
r = Z
2
i
+ 1 + 2 + i
( + 1 + 2 + i ) dG ( i)
+ 1 + 2 + min
( + 1 + 2 + max ) ; + 1 + 2 + max
( + 1 + 2 + min )
where max and min are the highest and the lowest private precision, respectively. Since
min and max are strictly positive and …nite, the equilirium 1 implicitly determined
30
!
;

y (10) is positive and bounded from below and from above as a function of 1 holding
everything else constant. Speci…cally, 1 ’s lower bound can be obtained as
R
! 2
2
1 = 1 2 2 2 r (
i i ) dG ( i )
1
2 + r
" 2
> 1 2 2 2 + 1 + 2 + # 2
min
1
2 + ( + 1 + 2 + max )
" 2
> 1 2 2 2 + 2 + # 2
min
1
2 + ( + 2 + max ) > 0;
where the second inequality follows as (+ 1 + 2 + ) min
(+ 1 + 2 + max ) is increasing in 1. Thus, a
su¢ cient condition for 1 > + + 2 is 1 2 1 su¢ ciently big. The corollary is
immediate since a mean preserving spread entails a reduction in 1 as established in
Proposition 1. For completeness, 1 ’s upper bound can be established as
"R
# 2 2
1 = 1 2 2 2 r (
i i ) dG ( i )
1
2 + r
" 2
< 1 2 2 2 + 1 + 2 + max
1
2 + ( + 1 + 2 + min )
" 2
< 1 2 2 2 + 2 + # 2
max
1
2 + ( + 2 + min ) ;
# 2
where the second inequality follows as (+ 1 + 2 + ) max
(+ 1 + 2 + min )
is decreasing in i. Q.E.D.
[I noticed that when discussing the e¤ect of information asymmetry on cost of capital, we
didn’t control for the informativeness from price. When we increase the heterogeneity
in the precision of private information, the price becomes less informative; and the
average precision of the total information reduces. So, in order to isolate the e¤ect of
information asymmetry, we need to manipulate 1 such taht 1 is the same as when
there is no information asymmetry. In other words, we need to increase 1 such that
31

1 =
2
+ 2
2
2 2 1
R
i
i dG( + 1 + 2 + i i )
+ 1 + 2 +
2
is still satis…ed. Then we can see clearly
that cost of capital increases when there is more information asymmetry.
Proof of Corollary 2 Rewrite (10) as
" 2 Z
1 = 1 2 2 2 i + 1 + 2 + # 2
1
2 + i
( + 1 + 2 + i ) dG ( i) :
Take a total derivative with respect to , treating 1 as a function of .
2
0 1 = 2 1 2 2 2
1
2 +
" Z
i + 1 + 2 +
i
2
4
( + 1 + 2 + i ) dG ( i)
3
dG (
(+ 1 + 2 + i ) 2 i ) +
5 :
( i ) i
dG (
R i (+ 1 + 2 + i ) 2 i )
R i
( i ) i
0 1
#
Zeqiong]
Rearranging,
where
(1 K) 0 1 = K;
2
K 2 1 2 2 2
1
2 +
" Z
i + 1 + 2 +
i
( + 1 + 2 + i ) dG ( i)
" Z
i #
i
i
( + 1 + 2 + i ) 2 dG ( i)
#
If we can show K 2 (0; 1), then 0 1 > 0. Note that a su¢ cient condition for
to be convex in i is + 2 > 2 max and that
( i ) i
(+ 1 + 2 + i ) 2
( i ) i
(+ 1 + 2 + i ) 2
evaluated at i = is zero.
Thus, when + 2 > 2 max , the second square bracket in K is strictly positive, so is
32

K. Furthermore, holding other exogenous parameters constant, we have
i + 1 + 2 +
( + 1 + 2 + i )
min + 1 + 2 +
2
( + 1 + 2 + max ) ; max + 1 + 2 + !
( + 1 + 2 + min )
min + 2 +
( + 2 + max ) ; max + 2 + !
;
( + 2 + min )
and
Z
i i
i
( + 1 + 2 + i ) 2 dG ( i)
max !
max
2 0;
( + 1 + 2 + min ) 2
max !
max
0;
( + 2 + min ) 2 :
Notice all these bounds independent of 1 or 1 . Thus, when 1 2 1 is su¢ ciently small,
K will be less than 1. And 0 1 > 0 follows. Q.E.D.
Proof of Proposition 3
To prove the …rst part, note that when all investors have the
same private information precision, we have
3 = 3
2 ;
2
2 = 2 2 2 3
;
3 +
2
1 = 1 2 2 6 2
4
2 + 2 +(+ 3)
32
7
5
++ 1 + 2 + 3
: (14)
Observe that when 1 = 0 ( 1 = 1), the left hand side of (14) is smaller (larger)
then it right hand side. Thus, there exists at least one solution of 1 . Also note
33

that the right hand side is strictly increasing in 1
derivative with respect to 1 has the same sign as
and that its second order
2 + ( 3 2 ) 2 2 ( + 1 + 2 + 3 ) :
Thus, the right hand side is concave in 1 if and only if
1 2 + ( 3 2 )
2 2
( + 2 + 3 ) :
Clearly, a su¢ cient condition for a unique 1
satisfying (14) is that the …rst
order derivate of the right hand of (14) with respect to 1 , evaluated at 1 =
2 + ( 3 2 )
2 2
( + 2 + 3 ), is less than 1, the slope of the left hand side with
respect to 1 . Hence, simple algebra leads to the following condition:
i
27
h1 + 3 3 2 + 3 2 2 3 4 + 3 2 3 ( 2 + 3 ) 6
1 ^ 1
8 4 :
2 2 3 8
It is also straightforward to see that at the unique equilibrium price informativeness,
the slope of right hand side of (14) with respect to 1 is strictly less than
1. Otherwise, there would be at least one other equilibrium 1 satisfying (14), a
contradiction.
To prove the second part of Proposition 3, note that the last term on the right
hand side of (11),
0R
@
i
12
i
A
+ + 1 + 2 + 3
+ i + 1 + 2 + 3
dG ( i )
is strictly smaller than 1 when investors have di¤erent private precisions. This is
because
re-write (12) as
i
+ i + 1 + 2 + 3
is concave in i . Hence, 2 < B 1
2 . To establish 1 < B 1
1 ,
0R
@
i
1 = 1 2 2 M (15)
2 0
i
R
2
+
dG (
+ i + 1 + 2 + 3 i )
i + 1 + 2
dG (
A @ i + i + 1 + 2 + 3 i )
A ;
1
+ + 1 + 2 + 3
1
+ + 1 + 2
+ + 1 + 2 + 3
34

where,
2
6
M 4
1 + R i
+ i + 1 + 2 + 3
dG ( i ) + R i
1
i
+3
2
+ i + 1 + 2 + 3
dG ( i )
3
7
5
2
:
Note,
+3
i 2
dG (
Z i
+ i + 1 + 2 + i ) = Z
+ 3
3 2 i
=
+ 3
R i
i
+ i + 1 + 2 + 3
dG ( i )
3
+ 3
2
2 2 2
=
i
+3
2
+ i + 1 + 2 + 3
dG ( i ) increases when there is het-
Hence, holding 2 constant, R i
erogeneity among investors, since
+ 3
3
R
i
i
+ i + 1 + 2 + 3
dG ( i )
! 2
i dG( + i + 1 + 2 + 3 i )
++ 1 + 2 + 3
2
++ 1 + 2 + 3
2 3 2 2 2
R i
i
+ i + 1 + 2 + 3
dG ( i ) :
i
+ i + 1 + 2 + 3
is concave in i . Also,
+ i + 1 + 2 + 3
is convex in i and thus R i
+ i + 1 + 2 + 3
dG ( i ) increases with heterogeneity.
As a result, M decreases as investors have di¤erent private information precision.
What’s more, notice that M is increasing in 2 , everything else equal. As
such, information asymmetry a¤ects M via two channels: …rst, holding 2 constant,
information asymmetry reduces it through concavity/convexity; second,
information asymmetry decreases the equilibrium 2 which in turn reduces the
…rst bracket even further. Similarly, due to concavity, introducing information
asymmetry makes
R i
2
dG( i + i + 1 + 2 + 3 i )
and
+ + 1 + 2 + 3
R + i + 1 + 2
2
dG( i + i + 1 + 2 + 3 i )
+ + 1 + 2
in (15)
+ + 1 + 2 + 3
strictly less than 1, while with identical private information precisions these two
terms equal to 1. This implies that for every 1 , introducing information asymmetry
reduces the right hand side of (15). Hence, the solution to (15) must be
smaller than B 1
1 .
35

Part 3 is proved by the numerical example following the proposition in the main
text.
Q.E.D.
36