[12:40 9/9/2015 rdv017.tex] RESTUD: The Review of Economic Studies Page: 1533 1533–1567
Review of Economic Studies (2015) 82, 1533–1567 doi:10.1093/restud/rdv017
© The Author 2015. Published by Oxford University Press on behalf of The Review of Economic Studies Limited.
Advance access publication 28 April 2015
Uncertainty, Information
Acquisition, and Price Swings
in Asset Markets
ANTONIO MELE
Swiss Finance Institute, USI Lugano and CEPR
and
FRANCESCO SANGIORGI
Stockholm School of Economics
First version received March 2011; final version accepted March 2015 (Eds.)
This article analyses costly information acquisitionin asset markets with Knightian uncertaintyabout
the asset fundamentals. In these markets, acquiring information not only reduces the expected variability
of the fundamentals for a given distribution (i.e. risk). It also mitigates the uncertainty about the true
distribution of the fundamentals.Agents who lack knowledge of this distribution cannot correctly interpret
the information other investors impound into the price. We show that, due to uncertainty aversion, the
incentives to reduce uncertainty by acquiring information increase as more investors acquire information.
When uncertainty is high enough, information acquisition decisions become strategic complements and
lead to multiple equilibria. Swift changes in information demand can drive large price swings even after
small changes in Knightian uncertainty.
Key words: Asset Markets with Knightian Uncertainty, Asymmetric Information, Value of Parameter
Uncertainty, Information Complementarities, Multiple Equilibria
JEL Codes: G12, G14, D82
1. INTRODUCTION
A basic tenet of financial economics holds that asset markets help summarize information
dispersed across individual investors. But what information do asset prices transmit?
Grossman and Stiglitz (1980) argue that as more agents acquire information, it becomes easier
to free-ride on the (costly) learning of others merely by observing the asset price. The value of
information diminishes with information acquisition.
The case for such a well-articulated role of the asset price relies on a number of assumptions
that have become standard. This article relaxes what is arguably the most standard of them:
that uncertainty can be quantified probabilistically. We consider a model in which the market
fundamentals are subject to ambiguity, or Knightian uncertainty (Keynes 1921; Knight 1921).
In this market, aversion to ambiguity provides incentives to acquire information, incentives that
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1534 REVIEW OF ECONOMIC STUDIES
increase with information acquisition. This property leads to a host of new conclusions about the
informational role of asset prices.
Our results rely on a framework in which uninformed investors face Knightian uncertainty
while attempting to glean information from the equilibrium price. We solve for the equilibrium
in this asset market with asymmetric information, and then analyse endogenous information
acquisition. The central element of our analysis is the agents’ attitude vis-à-vis the information
that prices reveal: how much of a price change can be attributed to new information, and how
much to a liquidity shock? When uninformed investors are uncertain about the true distribution of
the information held by the informed, this question cannot be given a precise probabilistic answer.
In this market, the value of acquiring information has two components. The first accounts for the
marginal value of reducing the riskiness of the fundamentals for any given prior distribution of
returns—the standard Grossman and Stiglitz component. The second relates to how valuable it
is for an ambiguity averse agent to resolve his ambiguity—the “value of parameter uncertainty”.
We show that the value of parameter uncertainty increases precisely as prices become more
informative. If uncertainty is high enough, it dominates the standard Grossman–Stiglitz free-
riding effect. Information complementarities result: the larger the mass of informed agents, the
higher the benefits of becoming informed.
Why is resolving ambiguity more valuable when there are more informed agents? As it turns
out, the value of parameter uncertainty lies in the benefit of forming portfolio decisions based on
the true distribution of the information revealed by the price. This benefit is high precisely when
prices incorporate more information on the fundamentals. Consider two polar cases:
In the first, no agent pays for information, and hence the equilibrium price at t =1, say, contains
no information and is unambiguous (albeit risky because of liquidity trading) from an ex ante
perspective (i.e. before trading and before the arrival of information, at t =0). However, returns
from trading are ambiguous from an ex ante perspective because they amount to the ambiguous
asset payoff less the unambiguous price. In this case, knowledge of the true return distribution at
t =1 provides an informational advantage only when the true expected returns differ from those
the uninformed investors impound into the equilibrium price. Ex ante, however, this informational
advantage cannot be quantified probabilistically because the true return distribution is ambiguous.
Thus, an ambiguity averse agent will give this advantage little weight as he formulates his choice
over whether to pay for information. The value of parameter uncertainty is small.
At the other extreme, if all agents pay for information, then the price encodes important
information about payoffs and is therefore ambiguous from an ex ante perspective. In contrast
to the previous case, the anticipated returns are less ambiguous ex ante: they are the difference
between the ambiguous payoff and the ambiguous price, and so the ambiguity “cancels out”
as the price fully incorporates information about the asset payoff. However, knowledge about
the probability distribution of the asset payoffs is valuable, because it provides the uninformed
investors with a “code” for correctly interpreting the information conveyed through the price.
This feature of our model is illustrated by the following example. The three-color Ellsberg urn
contains 30 red balls and a total of 60 black and yellow balls.
1
Informed decision makers know
the ratio of black to yellow balls, whereas the uninformed do not. At date-0, decision makers face
a bet that pays 0 if a red ball is drawn and $1 otherwise—a clearly unambiguous bet. The bet will
be resolved at date-2. However, there is an interim date-1 at which it will become known whether
the drawn ball is black or not. If the drawn ball is black, then the date-2 payoffs are known at
date-1. However, if the drawn ball is not black, then at date-1 the decision maker knows that the
drawn ball will be either red or yellow, but does not know which one it will be. Informed decision
1. Similar examples have been employed in the decision theory literature (e.g. Epstein and Schneider, 2003;
Hanany and Klibanoff, 2007, 2009; Siniscalchi, 2011).
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MELE & SANGIORGI UNCERTAINTYAND INFORMATION INASSETMARKETS 1535
makers can use Bayes’ law and determine the probability of red or yellow, but the uninformed
cannot, as the payoffs are no longer unambiguous. That is, learning about the ratio of black and
yellow balls is valuable for forecasting the outcome of the bet precisely because of the information
available at date-1.
In our model, the asset payoff is f =θ +, where θ is ambiguous (its mean is unknown), but
can be learnt at some cost, and where has a known distribution. When all agents are informed,
the price, p, moves one-to-one with the investors’ private information θ, that is p=θ −z, where
z is the random asset supply. Asset returns, R, are unambiguous as a result, R =f −p = +z.
Even if the returns are unambiguous (tantamount to the unambiguous bet that places $1 on “not-
red” in the urn example), p reveals information that is useful for forecasting R: R and p have z in
common. However, p contains ambiguous information, θ, and is not very useful unless one knows
the probability distribution of θ (just as the information “not-black” is not very useful unless one
knows the original composition of balls in the urn in Ellsberg’s example). For instance, a low
realization of p should not be interpreted as good news for future returns if the unknown mean
of θ is very low (just as the information “not-black” does not imply high chances of getting $1
if there are few yellow balls in the urn). An uninformed agent’s portfolio decision will reflect
his aversion to this uncertainty. However, whether the resulting decision correctly accounts for
the true meaning of the price realization ultimately depends on the true distribution of θ, which
is ambiguous ex ante. Hence, while assessing the implications of remaining uninformed, an
ambiguity averse investor fears making the wrong portfolio decision in light of price information.
The value of parameter uncertainty is higher when prices incorporate information than when they
do not.
Akey prediction of our model is that in markets with ambiguity aversion, the value of parameter
uncertainty increases when asset prices are sufficiently informative. Note that what is crucial is
not merely the ambiguity, but the aversion to it. Critically, we show that in the smooth ambiguity
extension of our baseline model (see below), higher-order uncertainty does not lead to information
complementarities, unless agents are ambiguity averse.
It is well known that information complementarities can lead to multiple equilibria (Section 5).
Our model and its extensions do indeed predict multiple equilibria. Outcomes such as history-
dependent prices, market crashes, rebounds, and overshoots can result even from small changes
in the uncertainty about the fundamentals. These properties help isolate new testable predictions
regarding a largely unexplored issue: the market reaction to positive uncertainty shocks. Our
model predicts that the initial reaction to a series of uncertainty shocks will be a market drop,
led by reduced market participation, and followed by a sustained rally. The rally occurs because
the increased uncertainty induces the uninformed agents to learn about these shocks, which fuels
complementarities in information acquisition and price overshoots.
Our main model relies on a market in which agents have maxmin expected utility, as in
Gilboa and Schmeidler (1989). In this market, uninformed investors extract information from
the equilibrium price through full Bayesian learning, by updating each initial prior. Uninformed
investors are also sophisticated, in that they correctly anticipate their future choices (portfolio
policies) in light of new information (the equilibrium price), an assumption that has been known
as consistent planning since Strotz (1955–1956). However, our main conclusions are resilient to
a variety of model extensions and alternative treatments of ambiguity, including (i) maximum
likelihood updating, (ii) portfolio policies to which agents pre-commit before trading, and (iii)
smooth specifications of ambiguity aversion as in Klibanoff, et al. (2005), which allow us to
disentangle ambiguity from ambiguity aversion within the context of our study.
The article is organized as follows. The next section provides perspective on our contribution
in light of the existing literature. Section 3 sets the model assumptions and analyses the asset
market equilibrium. Section 4 characterizes the value of information in the asset market with
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1536 REVIEW OF ECONOMIC STUDIES
Knightian uncertainty. Section 5 analyses endogenous information acquisition and deals with
information complementarities, multiple equilibria, and the asset price swings that occur as a
result. Section 6 provides the extensions. Section 7 concludes. The appendices contain details
omitted from the main text. Proofs for Section 6 are in the Online Appendices D to F available
as Supplementary Material.
2. RELATED LITERATURE
This article contributes to two strands of literature. First, it analyses how Knightian uncertainty
about fundamentals affects assets and information markets in an otherwise standard noisy rational
expectations equilibrium (REE) model. Secondly, it provides insights into the economic incentives
for mitigating model uncertainty through costly information acquisition. To date, much of the
literature on Knightian uncertainty does rely on a representative agent framework;
2
on the other
hand, the REE literature typically ignores the distinction between risk and ambiguity. Some
exceptions are Caskey (2009) and Ozsoylev and Werner (2011), who rely on noisy supply for
partial revelation, as in our article, and Condie and Ganguli (2011, 2012) and Easley et al. (2011),
who do not. While these papers deal with informational properties of asset prices in markets with
ambiguity, our focus is on the value of fundamental information in these markets.
In the existing REE literature, information complementarities can occur because as more
agents acquire information, the price actually becomes less informative, making it more valuable
for uninformed agents to acquire private information. This property can arise due to different
mechanisms. In Barlevi and Veronesi (2008), it is a negative correlation between noisy supply
and fundamentals. In Chamley (2010), it is the possibility of independent jumps in noise trading
and fundamentals. In Breon-Drish (2010), it is departures from normality of noise trading and
fundamentals that lead to a failure of the monotone likelihood ratio property (MLRP) of the signal
conveyed by the price.
3
In Rahi and Zigrand (2014), the mechanism is the heterogeneity in agents’
private asset valuations. In Avdis’ (2011) dynamic model, more informed investors lead to prices
being more informative about dividends but less informative about the liquidity shocks that drive
short-term price movements. Finally, Ganguli and Yang (2009) and Manzano and Vives (2011)
show the existence of multiple linear equilibria in the price function when agents have private
information about both dividends and supply. In one of these equilibria, the price signal-to-noise
ratio decreases in the fraction of informed agents; when agents coordinate on this equilibrium,
there are complementarities in information acquisition.
In contrast to the previous papers, information complementarities arise in our model despite
the fact that more information acquisition leaves the uninformed agents with lower conditional
risk and lower conditional ambiguity.
Other models also predict complementarities despite the price signal-to-noise ratio increasing
with information acquisition. However, the mechanisms in these models hinge on different
channels. In Veldkamp (2006), the mechanism relies on a cost of information that decreases
with information demand. In García and Strobl (2011), the mechanism relies on consumption
complementarities that result from relative wealth concerns. In the sequential trade model of
Chamley (2007), more informed trading makes current prices more informative but future prices
more uncertain; the latter effect may increase the value of information for short-term investors.
2. Cao et al., (2005) and Easley and O’Hara (2009) contain early analyses of how ambiguity affects participation
in Walrasian markets with heterogeneous agents.
3. Breon-Drish (2010) also provides a numerical example of complementarities in a setup in which the MLRPholds
but uninformed agents’ demands are backward-bending over some range. This feature of backward-bending demands
further differentiates our article—the uninformed demand is downward sloping in all versions of our model.
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MELE & SANGIORGI UNCERTAINTYAND INFORMATION INASSETMARKETS 1537
Our channel relies on the incentives to form portfolio decisions based on the true distribution
of the fundamentals. Due to ambiguity aversion, these incentives increase when more investors
acquire information and impound it into the asset price.
3. THE MODEL
3.1. Agents and assets
We consider a market for a risky asset, with payoff equal to f =θ +, where θ ∼N
(
μ,ω
θ
)
and ∼
N
(
0,ω
)
. As in Grossman and Stiglitz (1980), the market is populated by a continuum of agents,
with a fraction λ of informed and a fraction 1−λ of uninformed agents. Informed agents observe
θ at cost c>0. We initially take λ as given and consider endogenous information acquisition in
the following sections. The asset supply is z ∼N
(
μ
z
,ω
z
)
. We assume that ω
θ
> 0 (asymmetric
information), ω
> 0 (partial resolution of risk by private information), ω
z
> 0 (partial information
revelation), and that all variables are independent. A riskless asset is also available for trading; it
is in perfectly elastic supply, and yields a rate of return equal to zero. All agents have negative
exponential utility u with constant absolute risk aversion τ , that is u
(
W
)
=−e
−τ W
. Initial wealth
is normalized to zero.
3.2. Ambiguity and ambiguity aversion
Our point of departure from Grossman and Stiglitz (1980) is the assumption that all agents are
ex ante uncertain about the expected value of the fundamentals f . Although they are unable
to assess what μ is, they believe it belongs to some interval, μ∈[μ
, ¯μ]. We assume that μ=
μ
0
−
1
2
μ and ¯μ =μ
0
+
1
2
μ, for some μ ≥0, and set μ
0
=0.
In this and the following two sections, we assume that agents display preferences in the form
of the maxmin expected utility model of Gilboa and Schmeidler (1989). This assumption does not
allow us to disentangle the notion of uncertainty from that of the attitude towards it. For example,
we cannot tell whether an increased length of interval, μ, reflects more uncertainty or more
uncertainty aversion. Unless otherwise stated, we shall favour a cognitive interpretation of μ.
In Section 6.3, we rely on the smooth ambiguity model of Klibanoff, et al. (2005), which allows
for a separation of tastes and beliefs. Our conclusions about complementarities and multiple
equilibria remain unaffected in this framework.
4
3.3. Informed agents
By observing the realization of θ, informed agents resolve their ambiguity straight away. They
choose portfolio holdings x
I
to maximize the expected utility of their final wealth W
I
=
(
f −p
)
x
I
−
c, where p is the observed asset price. Standard arguments yield
x
I
(
θ,p
)
=
E
(
f
|
θ,p
)
−p
τ Va r
(
f
|
θ,p
)
=
θ −p
τω
. (3.1)
Naturally, while informed agents are able to dissipate their ambiguity, they cannot eliminate
risk. Conditional upon θ, the fundamentals, f , are still normally distributed with expectation θ
and variance ω
.
4. Other non-smooth models include, among others, the Choquet expected utility model of Schmeidler (1989) and
the α-maxmin expected utility model of Ghirardato et al., (2004). The latter has the property of separating ambiguity and
ambiguity attitude. Gilboa and Marinacci (2013) provide a survey of the decision-theoretic literature on ambiguity.