Multiple Equilibria in the Grossman-Stiglitz Model∗
Dömötör Pálvölgyi
Eötvös Loránd University
[email protected]
Gyuri Venter
Copenhagen Business School
[email protected]
April 1, 2014
Abstract
This paper studies equilibrium uniqueness in a standard noisy rational expectations economy with asymmetric information à la Grossman and Stiglitz
(1980). We show that the standard linear equilibrium of the model is the
unique equilibrium with a continuous price function. However, we also construct equilibria with non-continuous prices that have very different economic implications, including (i) jumps and crashes, (ii) significant revisions
in uninformed belief due to small changes in the market price, (iii) “upwardsloping” demand curves, and (iv) higher prices leading to higher expected
dollar return for uninformed agents.
A previous version of this paper was circulated under the title “On the uniqueness of equilibrium in
the Grossman-Stiglitz noisy REE model”. We thank Christian Hellwig, Aytek Malkhozov, Lasse Pedersen, Rohit Rahi, and especially Dimitri Vayanos for helpful comments. Venter gratefully acknowledges
financial support from the Center for Financial Frictions (FRIC), grant no. DNRF102.
∗
In their seminal paper, Grossman and Stiglitz (1980) (GS, henceforth) present a framework for a noisy rational expectations economy (REE), which has since become a
workhorse model studying asymmetric information in competitive financial markets.
In such environments, prices have a dual role: to clear the market and to transfer information from informed to uninformed investors. By introducing noise in the process, the
model resolves the paradox of fully revealing equilibria: Those who expend resources to
obtain information achieve better allocation.
Since the information transmission process is noisy, a central question in financial
economics is to what extent prices reflect fundamentals versus noise in equilibrium. To
answer this question, GS conjecture an equilibrium price function that is linear in the
state variables and show that, when random variables are jointly normally distributed
and investors have exponential (i.e., CARA) utilities, such an equilibrium exists and its
endogenous parameters are uniquely pinned down. However, the question whether there
exist other equilibria of the model, which are potentially less tractable but offer more
realistic predictions, remains. In this paper we seek to explore this question, and to
characterize the issue under certain generalizations of the original model assumptions.
The first contribution of the paper is to show that the well-known linear equilibrium
of the GS model is the unique equilibrium when allowing for any continuous equilibrium
price function, linear or not. Our solution method is different from the usual “conjecture
and verify” approach, in which conjecturing in a specific functional form naturally limits
economists to study existence and uniqueness in the linear class of functions. We also
show that our proof of uniqueness can be applied even if we relax the distributional
assumptions of the GS model. In particular, we show that as long as the payoff of
the asset conditional on the information of informed traders is still Gaussian and the
distributions of other state variables satisfy mild conditions, there exists at most one
continuous equilibrium of the model. Moreover, we provide necessary and sufficient
conditions for the existence of this equilibrium, even if it is not possible to characterize
it in closed form.
Our main contribution is to show that once we relax the assumption of a continuous
price function, there exist other equilibria. Our leading example of a non-continuous but
1
still tractable equilibrium has many features that the standard linear equilibrium cannot
produce: First, small changes in the asset payoff can lead to large price changes, i.e.,
jumps and crashes, and larger jumps happen with more extreme fundamentals. Second,
a small change in the market price can lead to significant revisions in uninformed beliefs
(i.e., mean, variance, skewness of the asset payoff conditional on the price). Third,
the demand curve of uninformed agents is locally downward-sloping but globally not,
that is, uninformed agents can demand more when the price is higher (in GS it is
always downward-sloping). Fourth, expected dollar return of uninformed agents can
both increase or decrease in the price (in GS it always decreases).
Although the theory of fully-revealing REE is largely complete, with many studies
on generic existence and uniqueness, and some non-generic examples of non-existence
(see, for example, Radner (1979), Jordan (1982), and Jordan (1983)), much less is known
about partially-revealing REEs. Previous studies were mainly concerned about the existence of a rational expectations equilibrium, see for example Grossman (1976), Grossman
and Stiglitz (1980), Hellwig (1980), Diamond and Verrecchia (1981). Ausubel (1990a)
and Ausubel (1990b) study existence and uniqueness of partially-revealing REE without
noise but in which uninformed traders face uncertainty about the preference of informed
traders. In contrast, we study uniqueness of the equilibrium in the classic model of Grossman and Stiglitz (1980). Our study is also related to DeMarzo and Skiadas (1998), who
show uniqueness of a perfectly revealing REE in the model of Grossman (1976), and
give examples of partially-revealing equilibria when payoffs are non-normal.1
The closest papers to ours are Breon-Drish (2010) and Breon-Drish (2012). BreonDrish (2010) studies a GS model with normal mixture distributions, finds a continuous
equilibrium numerically, and demonstrates that it has very distinct features compared
to the usual GS model. Breon-Drish (2012) studies existence and uniqueness among
continuous equilibria in GS- and Hellwig (1980)-type models with distributions from the
1
In all aforementioned studies of REE, agents are price-takers. Our analysis is also related to papers
studying existence and uniqueness in models with strategic agents. Rochet and Vila (1994) show that
the linear equilibrium of Kyle (1985) is unique if the single informed trader can observe noise trader
demand, and Back (1992) proves uniqueness in a continuous-time version of Kyle (1985). Du and Zhu
(2013) study existence of ex post equilibrium in static and dynamic models of double auctions with
multiple strategic bidders and dispersed information, and establish uniqueness in the static case.
2
exponential class. Besides establishing uniqueness of the standard linear GS equilibrium
among all continuous price functions, our paper complements these studies by constructing non-continuous price functions and showing that they produce very different features
from the standard GS predictions. Our contribution also lies in the fact that, in contrast
to both of these studies, our uniqueness proofs are significantly simpler.
There is a related literature that builds on GS and Hellwig (1980) to display realworld economic phenomena that the basic models cannot bring forth, such as market
crashes, crises, and contagion. These papers either rely on multiple assets (see, e.g., Admati (1985)), different distributions of random variables or utility functions (e.g., Barlevy
and Veronesi (2003)), feedback from prices to fundamentals (e.g., Subrahmanyam and
Titman (2001), Ozdenoren and Yuan (2008)), traders with hedging or portfolio rebalancing motives (e.g., Gennotte and Leland (1990)), trading constraints such as short-sale
or borrowing constraints (e.g., Yuan (2005), Yuan (2006)), or higher-order expectations
and coordination motives (e.g., Angeletos and Werning (2006)). In contrast, we show
that non-continuous equilibria of the basic GS model, without any changes in the assumptions and with no additional ingredients, can produce market crashes and crises.
The remainder of the paper is organized as follows. Section 1 presents the model.
Section 2 shows that the well-known linear equilibrium is the unique continuous equilibrium of the textbook GS model, and studies existence and uniqueness when we relax
the distributional assumptions. Section 3 provides non-continuous price functions from
a certain class and studies their properties. Finally, Section 4 concludes.
1
Model
This section introduces the baseline model that follows Grossman and Stiglitz (1980).
There are two periods, t = 0 and 1. Two securities, a riskless and a risky asset, are
traded in a competitive market in Period 0, and pay off in Period 1. The riskless asset
pays off one unit with certainty. The risky asset is assumed to be in random aggregate
supply of u shares, and pays off d units that consists of two random parts: d = s + n.
We assume that the random variables s, n, and u are pairwise independent and have
3
multivariate normal distribution with means normalized to zero and variances σs2 , σn2 ,
and σu2 , respectively. These distributions constitute a common prior for all agents. We
use the riskless asset as numeraire, and denote the price of the risky asset in Period 0
by p.
The asset market is populated by a continuum of agents in measure one. Agents have
exponential utility over wealth W in Period 1, U (W ) = − exp (−αW ), where α > 0 is
the coefficient of absolute risk aversion. If in Period 0 an agent buys x units of the risky
asset, her terminal wealth in Period 1 is the sum of the starting wealth in Period 0,
W0 , and the capital gains from trading the risky asset: W = W0 + (d − p) x. Without
loss of generality, we assume that all agents have the same risk aversion parameter,
and that none of them are endowed with any shares of the risky asset to begin trading
with. Moreover, the assumption of CARA-utility implies that asset demands will be
independent of the starting wealth W0 , so we also normalize it to zero. None of the
traders face any trading (e.g., short-sale or leverage) constraints.
Agents are heterogeneous with respect to their information; they can be either informed or uninformed. Informed traders, in measure 0 < ω < 1, observe the realization
of component s of the risky asset payoff, hence their demand will depend on s and the
price p. The rest of the agents, in measure 1 − ω, are uninformed, and neither observe
s, nor receive any additional information about s that would be known only to them.
Component n is unobservable to everyone.2 Besides potentially knowing s, all agents’
information sets are identical; we denote it by P and refer to it as the set of all publicly
available information. It contains knowledge about the setup of the economy (e.g., the
common prior and agents’ preferences) and, since we consider rational expectations equi2
The unlearnable part n can also be interpreted as the remaining error in informed traders’ forecast
about the payoff after receiving their signal s, and its presence ensures that informed agents submit
a downward-sloping demand curve for the risky asset. On the other hand, s (and its variance) can
be interpreted as the informational advantage of informed traders. An alternative
presentation of the
model, in line with this interpretation, includes a terminal payoff F ∼ N 0, σ 2 , and informed traders
observing a signal S = F + ε, where ε ∼ N 0, σs2 is independent of F . Because it is straightforward
to derive the relationship between the (exogenous and endogenous) parameters of the two models, our
results naturally apply to this specification, too. In Appendix C we argue that when we allow random
variables to have non-Gaussian distributions, the two models are not equivalent any more; in this case
our setting is more tractable than the alternative.
4
libria, everything they can infer from the market price p.3 We denote the expectation
and variance conditional on information set I by E [.|I] and V ar [.|I].
We define an equilibrium of the above economy the standard way:
Definition 1. A rational expectations equilibrium (REE) consists of a price function
P (s, u), and individual strategies for informed and uninformed traders, xI (s, p) and
xU (p), respectively, such that
1. demand is optimal for informed traders:
xI (s, p) ∈ arg max E [− exp {−α (d − p) x} |s, P] ;
x
(1)
2. demand is optimal for uninformed traders:
xU (p) ∈ arg max E [− exp {−α (d − p) x} |P] ;
x
(2)
3. the asset market clears at the equilibrium price:
ωxI (s, P (s, u)) + (1 − ω) xU (P (s, u)) = u.
(3)
One remark is in place regarding our setup and equilibrium definition. The standard
discussion of the public information set P is to establish that uninformed traders know
they are trading with informed agents and noise traders, and realize the equilibrium relationship between the asset payoff and the price. Therefore, their source of information
about the asset payoff is the market-clearing price; formally, they solve the optimization problem (2) with the expectation computed conditional on P (s, u) = p. Our main
3
As it is standard in models with informational asymmetry, the presence of random supply u ensures
that the price does not reveal informed traders’ knowledge perfectly, and hence the Grossman-Stiglitz
paradox does not apply. There exist alternative terminologies for u that essentially result in the same
analysis. First, it would be possible to assume a fixed supply of the risky asset and the presence
of noise traders, who trade for reasons exogenous to the model and submit a random price-inelastic
demand. Second, following Wang (1994), we could endogenize noise by endowing informed traders with
an investor-specific technology or liquidity shock that is correlated with the asset payoff. Differentiating
between the sources of noise is not relevant in the basic model we study here, because we do not impose
any constraints on traders; our (non-)uniqueness results trivially extend to these setups. See, e.g., Yuan
(2005), Yuan (2006), Bai, Chang, and Wang (2006) and Venter (2011) for the differences in the setups
when informed traders face short-sale or leverage constraints.
5
observation is that uninformed traders can also make use of the information that in
equilibrium the market must clear, i.e., P = {P (s, u) = p, (3)}. In fact, the number of
units of the risky asset uninformed agents hold in equilibrium in aggregate, which they
know, equals the net supply u − ωxI (s, p), and this can serve as an additional source of
information beyond the price realization p.
It turns out that in the CARA-normal setting P is informationally equivalent to
knowing P (s, u) = p. Our approach, however, has the advantage that the the marketclearing condition, with the only assumption that P is continuous in s and u, provides
enough information to uninformed agents that we can sidestep the usual “conjecture
and verify” solution method to pin down the equilibrium of the economy.
2
The unique continuous equilibrium
The standard solution applied by the literature is the so-called “conjecture and verify”
method; see, e.g., Grossman and Stiglitz (1980), Brunnermeier (2001), Vives (2010), and
Veldkamp (2011). According to this, solving for an equilibrium of the financial market
requires three fairly standard steps: First, we postulate a REE price function P . Second,
given the price, we derive the belief and optimal demand of uninformed traders. Finally,
we check under what conditions the market clears at the conjectured price. The problem
with this method is that guessing a particular form for the equilibrium price function
naturally limits the set of equilibria available for consideration. Instead, we approach the
problem by looking at the optimal informed demand and the market-clearing condition
first, and we determine how much uninformed agents can learn through this channel,
at least and at most. If, under some conditions, the two coincide, the information
uninformed investors obtain through trade is perfectly pinned down, and the equilibrium
is readily obtained.
Suppose an equilibrium exists, and fix the function P (s, u). First, we make the
observation that informed demand is independent of the equilibrium price function.
Since informed traders know s and the only uncertainty they face is the unobservable part
n that is independent of the particular form of P (s, u), the price provides no additional
6
information to them and their information set is II = {s}. As it is well-known, the
normality of n and the exponential utility together imply that (1) is equivalent to a
mean-variance problem, and optimal informed demand is simply
xI (s, p) =
s−p
.
ασn2
(4)
Second, we substitute the informed demand into the market-clearing condition (3)
that gives
ω
s−p
+ (1 − ω) xU (P (s, u) = p) = u.
ασn2
After rearranging, we obtain
where C =
2
ασn
ω
s − Cu = g (p) ,
(5)
g (p) = p − (1 − ω) CxU (P (s, u) = p) .
(6)
and
Notice that since uninformed agents know the equilibrium form of P and observe p, from
(6) they know g (p). Therefore, from (5), p also reveals the linear combination s − Cu.
We summarize and reinterpret graphically this result:
Lemma 1. Suppose an equilibrium exists. Fix a price function P (s, u), and take any
realization p. Then the set of all possible (s, u) pairs for which P (s, u) = p is a subset
of a single straight line on the (s, u) plane with slope 1/C.
As such a line can be defined by its intercept with the horizontal axis, we can refer
to it both as the line of points that satisfy s − Cu = lp for a given constant lp , or simply
denote it lp .
The main question is whether a realization p can tell more about s than just revealing
s − Cu. In what follows, we make some simple observations based on (5) to argue
that if P (s, u) is continuous in both arguments, it cannot. Hence, p and s − Cu are
observationally equivalent.
Suppose that the converse is true, and P (s, u) is a function of s not only through
s − Cu, i.e., it is not (s − Cu)-measurable. Put graphically, this is equivalent to saying
7
(s ,u )
*
*
γ
(s ,u )
2
u
2
(s*,u*)
(s ,u )
1
0
1
l
p
0
s
Figure 1. Proof of Lemma 2
If there exist (s1 , u1 ) and (s2 , u2 ) such that they satisfy s−Cu = lp for a fixed lp constant
but P (s1 , u1 ) 6= P (s2 , u2 ), there also exist two points, one on the lp line and one outside,
on an arbitrary γ curve, such that P (s∗ , u∗ ) = P (s∗ , u∗ ), which contradicts that they
should be on a single line with slope 1/C.
that the information revealed by the market price is a strict subset of a straight line,
instead of the whole line. If this is the case, there must be two price realizations p1 6= p2
such that g (p1 ) = g (p2 ) = lp for some lp . Equivalently, there are two pairs (s1 , u1 ) 6=
(s2 , u2 ) that correspond to the two different prices, P (s1 , u1 ) = p1 and P (s2 , u2 ) = p2
while s1 − Cu1 = lp = s2 − Cu2 ; see Figure 1.
As P (s, u) is a continuous function of the random variables s and u, the Intermediate
Value Theorem implies that if we connect (s1 , u1 ) and (s2 , u2 ) with any simple curve of
the plane, there must be at least one point (s, u) on this curve such that P (s, u) =
p1 +p2
.
2
We apply this theorem to two curves. The first is simply the segment connecting
(s1 , u1 ) and (s2 , u2 ), part of the line lp . Thus, there exists at least one point, denoted by
(s∗ , u∗ ), such that s∗ − Cu∗ = lp and P (s∗ , u∗ ) =
p1 +p2
.
2
The second will be any γ curve
whose intersection with the line is only (s1 , u1 ) and (s2 , u2 ). This gives at least one point
outside lp , denoted by (s∗ , u∗ ), such that P (s∗ , u∗ ) =
8
p1 +p2
.
2
Given that (s∗ , u∗ ) ∈
/ lp , it
must be that s∗ − Cu∗ 6= lp . Hence we found two points of the (s, u) plane that admit
2
2
= s∗ − Cu∗ 6= s∗ − Cu∗ =
, but g p1 +p
the same price, P (s∗ , u∗ ) = P (s∗ , u∗ ) = p1 +p
2
2
2
g p1 +p
. Graphically (see Figure 1), (s∗ , u∗ ) and (s∗ , u∗ ) should be on a straight line
2
with slope 1/C, but they are clearly not, and it contradicts Lemma 1. We summarize
the above result in the following lemma:
Lemma 2. Suppose an equilibrium exists. Fix a continuous equilibrium price function
P (s, u), and take any realization p of it. Then the set of all (s, u) pairs for which
P (s, u) = p is a whole straight line on the (s, u) plane with slope 1/C.
Therefore, it must be that p 7→ lp is a one-to-one mapping, i.e., a price realization p
is equivalent to observing the realization of s − Cu, lp . Note that while this pins down
the nature of the information uninformed traders infer, we have not determined the
exact relationship between p and s − Cu yet. Formally, Lemma 2 implies P is (s − Cu)measurable, i.e., P (s, u) is a function of s and u only through s − Cu, however, it is still
possible that multiple price functions exist in equilibrium.
To determine the functional form of P , the final step is to use the prior belief of
uninformed traders to derive their optimal demand. If the prior about s and u is jointly
normal, s − Cu is normally distributed, too, and Bayesian updating leads to a Gaussian
posterior. Combining it with the exponential utility, uninformed agents’ optimization
program is necessarily a CARA-normal one, and optimal uninformed demand is simply
xU (lp , p) =
E [s|s − Cu = lp ] − p
,
α (V ar [s|s − Cu = lp ] + σn2 )
(7)
where the expectation is linear in lp and the variance is constant:
E [s|s − Cu = lp ] =
σs2 C 2 σu2
σs2
l
and
V
ar
[s|s
−
Cu
=
l
]
=
.
p
p
σs2 + C 2 σu2
σs2 + C 2 σu2
(8)
Combining (7) and (8) with market clearing (3), and using lp = s − Cu, it is imminent
that p is linear in lp . After some algebra, we conclude with the following result:
9
Theorem 1. If we restrict the equilibrium price function to be continuous, there exists
a unique equilibrium of the asset market. It is linear in the state variables:
PGS (s, u) = B (s − Cu) ,
where
B=
ασn2
σs2 σn2 + ω (σs2 + σn2 ) C 2 σu2
>
0
and
C
=
> 0.
σs2 σn2 + (ωσs2 + σn2 ) C 2 σu2
ω
Notice that besides establishing the unique continuous equilibrium, Theorem 1 also
implies that our public information P is informationally equivalent to the standard conjectured linear equilibrium of GS. The distinguishing feature of our approach is that we
did not have to conjecture a price function and hence restrict ourselves to linear equilibria, but could derive the information uninformed traders have based on the marketclearing condition (see also the discussion in the paragraph following Definition 1).
While the main focus of the paper is on multiplicity in the standard GS model, there
is a straightforward extension of our result on uniqueness of the continuous equilibrium.
In Appendix C we show that under mild conditions there is still at most one continuous
equilibrium, and we provide sufficient conditions for the existence of this equilibrium.
3
Non-continuous price functions
In this section we show that if the price function P does not have to be continuous, only
(s, u)-measurable, there are more equilibria. We first argue that if an equilibrium exists,
it is perfectly pinned down by the information set of uninformed traders. Afterwards,
we determine certain properties of equilibrium sets, and use them to provide a tractable
non-continuous equilibrium that differs from the GS price almost everywhere, and is
non-continuous only on a zero-measure set.
Definition 2. Let P be an equilibrium price function. We call a region R of the (s, u)
plane p-homogeneous, if P (s, u) = p for all (s, u) ∈ R and P (s, u) 6= p for all (s, u) ∈
/ R.
If a region R is p-homogeneous for some p, we call it P -homogeneous.
10
Put differently, P -homogeneous regions are the set of (s, u) points that uninformed
investors cannot distinguish from each other in equilibrium, and hence are the graphic
representations of the building blocks of their information set. Moreover, we have the
following result on the relationship of P and the P -homogeneous regions:
Lemma 3. For any region region R and price function P for which R is P -homogeneous,
there is exactly one p such that R is p-homogeneous. Also, for any decomposition of the
(s, u) plane into a disjoint union of regions, there exists at most one P such that these
are the P -homogeneous regions.
From Definitions 1 and 2 it is clear that every region R can be p-homogeneous for
at most one p, regardless of how the price function P looks like on the rest of the plane.
This is because with the information (s, u) ∈ R and the priors on s and u, optimal
uninformed demand is resolved for any price level, which in turn determines the only p
that satisfies the market-clearing condition at (s, u), as both informed and uninformed
demand decrease in p. If these prices are the same for every (s, u) ∈ R, there exists
a unique p for which R is p-homogeneous.4 Moreover, if we have a partition of the
plane into regions, then from the above it follows that there can be at most one P for
which they are all P -homogeneous, as computing the only (if exists) possible price p for
each region determines what the function P can be. However, P must also satisfy a
consistency requirement: if we obtain the same p for two regions R1 and R2 , P is not
a valid equilibrium price function, because the p-homogeneous region should have been
R1 ∪ R2 rather than any of them alone.
With the help of Definition 2, we can rephrase the results of Section 2. Lemma 1
claims that for any equilibrium price function P , continuous or not, the P -homogeneous
regions are subsets of lines with slope 1/C. Lemma 2 states that the PGS -homogeneous
regions are exactly the lines with slope 1/C. Finally, Lemma 3 implies that if there is
a whole P -homogeneous line l for some P , then on this line P is the same as the GS
price: P (s, u) = PGS (s, u) for all (s, u) ∈ l.
4
These steps are illustrated in Appendix B for a special case, where we provide the constructive
proof for Theorem 2.
11
To understand further how another equilibrium P 6= PGS must behave, consider the
following. According to Lemma 3, P 6= PGS is equivalent to saying that they are different
on at least one line. That is, there exists at least one line l, with corresponding price
pGS under GS, that it is not P -homogeneous but is the disjoint union of at least two
P -homogeneous regions, denoted by li , i = 1, 2... Lemma 3 also implies that the price
realizations on these subsets are uniquely determined: P (s, u) = pi if (s, u) ∈ li , where
all pi s are different from each other. Thus, at least one of them is also different from
pGS ; call it p1 . If l1 denotes the (sole) line with slope 1/C on which PGS is p1 , we claim
that l1 cannot itself be a P -homogeneous region, only the disjoint union of at least two
P -homogeneous regions. This is because otherwise we would have P (s, u) = p1 for all
(s, u) ∈ l1 due to being P = PGS on this line, and thus the p1 -homogeneous region of
P would include points from two different lines with slope 1/C, contradicting Lemma
1. Therefore, P must be different from PGS on the line l1 , too. Following this line of
thought, it is imminent that a valid price function cannot be different from PGS on only
a finite number of lines, and take the same value as GS everywhere else.
We use the above observations to create a price different from PGS by constructing
a consistent “cut” of all PGS -homogeneous regions into two half-lines. Formally, we
look for a function s̄ : R → R that splits each line l into a left half-line, given by l− =
{(s, u) : s − Cu = l and s ≤ s̄ (l)}, and a right half-line l+ = {(s, u) : s − Cu = l and s > s̄ (l)},
and the P -homogeneous regions are these left and right half-lines. Our main result is
that it is possible to achieve such a P by a “linear cut” in which s̄ is linear in l.5
Theorem 2. There exist a continuum of non-continuous equilibria of the asset market,
created with a linear cut and given in closed form by

 P (s − Cu) = P (l)
+
+
PLC (s, u) =
 P (s − Cu) = P (l)
−
5
−
if s < s̄ (l) =
D−ωl
1−ω
(9)
if s ≥ s̄ (l)
Graphically, this cut can be thought of as the (s̄ (l) , ū (l)) points of the (s, u) plane, where ū (l) =
If s̄ is linear in l, ū is linear too, and the set {(s̄ (l) , ū (l)) | l ∈ R}, is a straight line on the plane.
See the bottom right panel of Figure 2 for illustration.
s̄(l)−l
C .
12
where D ∈ R is an arbitrary constant,
P+ (l) = Bl + ρ (Bl − D) − (1 − ω) ΣΨ−1
(λ
(Bl
−
D))
,
ρ
(10)
P− (l) = Bl + ρ (Bl − D) + (1 − ω) ΣΨ−1
ρ (−λ (Bl − D)) ,
(11)
and
and where Σ2 =
2
σs2 C 2 σu
2,
σs2 +C 2 σu
ρ=
C
,
αΣ2
and λ =
1+ρ
(1−ω)Σ
are all positive constants. Moreover,
if φ (.) denotes the pdf of the standard normal distribution and Φ (.) is the corresponding
cdf, then
Ψρ (x) = (1 + ρ) x +
φ (x)
Φ (x)
is an invertible function whose properties we collect in Appendix A.
Appendix B contains the details of the constructive proof. It includes four steps.
First we formally derive the belief of uninformed agents if the price reveals both the l
line, i.e., the linear combination of signal s and the noisy supply u, and that s is on
the right or left half of this line. Second, given the belief, we derive the inverse demand
function of uninformed agents. Third, we apply market clearing to obtain an implicit
equation that the equilibrium price must satisfy. Finally, we show that under this linear
cut the attained prices have all necessary properties to yield a valid price function.
To understand the intuition behind PLC , consider first its restriction to all left halflines, denoted by P− ; since P (s, u) is the same for all (s, u) ∈ l− this can be interpreted
as a function of which line we are on, i.e., P− (l) = PLC (s, u) for all (s, u) ∈ l− . Similarly
we define P+ (l) as the function telling us what the equilibrium price is on the right side
of the line l. If we choose s̄ (l) such that P− (l) and P+ (l) are both invertible functions
and their ranges are disjoint, then PLC is a valid equilibrium price function, because
every PLC (s, u) = p price realization reveals both the s − Cu = l line and on which
half of this line s is on. In Appendix B we show that P− (l) is increasing, infinitely
differentiable, and its range is (−∞, D), i.e., bounded from above. On the other hand,
P+ (l) is increasing, infinitely differentiable, and its range is (D, ∞), i.e., bounded from
below. Hence, PLC (l) takes all real values except for D.
13
GS price function
Linear cut price function
4
4
2
P
2
P
LC
GS
0
0
−2
−2
−4
−2
−4
−2
0
u
2
−4
0
4
2
0
−2
u
s
GS price as a function of l
2
2
P (s,u)
0
0
−2
−2
GS
−5
0
5
−4
−10
10
PGS−homogeneous regions
3
−5
0
5
10
PLC−homogeneous regions
2
1
1
u
0
0
−1
−1
−2
−2
−3
−4
s
l=s−Cu
2
u
4
2
LC
l=s−Cu
3
0
4
(s,u)
−4
−10
−2
−4
LC price as a function of l
4
P
2
−2
0
2
−3
−4
4
s
−2
0
2
4
s
Figure 2. The price function with linear cut, PLC
This figure illustrates the price function proposed in Theorem 2 and compares it to the
GS price of Theorem 1. The upper left panel shows PGS as a function of s and u, and
the middle left panel shows PGS as a function of l. The bottom left panel illustrates
the PGS -homogeneous regions for 3 different price realizations, with the actual prices on
these lines on the middle panel. The upper right panel shows the price function PLC
obtained by a linear cut as a function of s and u, and the middle right panel shows
PLC as a function of l. The price is not l-measurable any more, as it also depends on
whether s is on the right or left half-line of l; the upper (lower) solid blue curve shows
P+ (P− ). The bottom right panel illustrates the PLC -homogeneous regions for 4 different
price realizations, with the actual prices on the middle panel. The parameters are set
to σs = 0.6, σu = 0.4, σn = 0.5, α = 2, ω = 0.15, D = 0.
14
Figure 2 illustrates the price function proposed in Theorem 2 and compares it to
the GS price of Theorem 1. The upper left panel shows PGS as a function of s and u,
and the middle left panel shows PGS as a function of l = s − Cu. The price function is
represented by a plane and a line, respectively, because PGS is linear in l and hence in s
and u. The bottom left panel illustrates the PGS -homogeneous regions for three different
price realizations: if we fix p, the set of points that solve PGS (s, u) = p is a whole line
with slope 1/C. For instance, the price p = −1.8, illustrated by a triangle (△) in the
middle panel, reveals to uninformed investors that (s, u) is on the line s−Cu = l = −4.5,
also illustrated by triangles in the bottom left panel.
The upper right panel shows the price function PLC obtained by a linear cut as a
function of s and u, and the middle right panel shows PLC as a function of l = s − Cu.
As seen on the middle panel, the price is not l-measurable any more, as it also depends
on whether s is on the right or left half-line of l. The upper solid blue curve corresponds
to the function P+ , and the lower solid blue curve corresponds to the function P− .
For comparison, we illustrate PGS (l) by the dashed red line. The bottom right panel
illustrates the PLC -homogeneous regions for 4 different price realizations: if we fix p, the
set of points that satisfy PLC (s, u) = p is a half-line with slope 1/C. In this equilibrium
the level of p not only reveals the line but also which half it corresponds to: prices higher
than D indicate a right half-line, prices below D indicate a left half-line. For instance,
prices p◦ = −0.14 and p△ = 1.11, illustrated by the circle (◦) and the triangle (△) on
the middle right panel, reveal the same l = 2.75 line, but they also indicate whether
(s, u) is on the left side or the right side of this line, i.e., whether the payoff is high,
s ≥ s̄ (2.75) = −0.48, or low, s < −0.48. These two sets of points are illustrated by
triangles and by circles, respectively, on the bottom right panel. The dotted-dashed
green line indicates the cut s̄ (l).
The set of splitting points s̄ (l) has an interesting economic interpretation in this
setup. Suppose that a D-homogeneous region consists of only one point, i.e., is a singleton (s̄, ū) and this price is attained nowhere else on the (s, u) plane. Therefore, D
perfectly reveals s̄, the information of informed traders, and uninformed traders become
informed too. Since none of the agents know n, the equilibrium demand of all rational
15
agents is
s̄−D
2 .
ασn
Market clearing means this demand equals supply ū; rearranging, we
obtain s̄ − ωC ū = D, which is equivalent to the definition of s̄ (l) in (9). Therefore,
(s̄, ū) are the points that, one by one, would be D-homogeneous singletons. It follows
directly from this interpretation that the equilibrium price on all left half-lines must be
below D, because the asset payoff when uninformed agents learn s = s̄ as a random
variable first-order stochastically dominates the asset payoff when uninformed agents
learn s < s̄, hence investors demand more and push the equilibrium price higher.
The proposed price function has many interesting features that the standard linear
equilibrium cannot produce.
Proposition 1. Small changes in s and u can lead to large changes in the price, i.e.,
jumps and crashes, given by ∆P (l) = P+ (l) − P− (l). ∆P (l) is positive for all l, it
reaches its minimum at l = D/B and is symmetric around this point, and has limits
liml→±∞ ∆P (l) = ∞.
Proposition 1 describes price sensitivity to signal and supply shocks. We note that
jumps and crashes can happen independently of the asset’s value: since s̄ (l) is an infinite
line on the plane with finite slope, for all s ∈ R there exists a unique u ∈ R such that a
crash occurs at this (s, u). Crashes can be arbitrarily large, and large crashes happen to
more extreme fundamentals, i.e. when l is further away from D/B. Interestingly, jumps
and crashes occur to/from the intermediate region of prices close to D, as can be seen
on the middle right panel of Figure 2.
Proposition 2. First and second moments (i.e., mean and variance) of uninformed
belief about asset payoff and dollar return are non-monotonic in the price. Moreover, a
small change in the market price can lead to large revisions in the expected value of the
asset payoff conditional on the price.
The upper panels of Figure 3 illustrate the properties of the price function with linear
cut compared to the GS price. The upper left panel shows the expected payoff conditional
on the price realization, E [d|P = p], in the two equilibria. Notably, the conditional
expectation is non-monotonic in the LC equilibrium: a higher price realization does
16
Variance
Expected payoff
1.5
0.6
0.55
1
0.5
0.5
0.45
Var[d|P=p]
E[d|P=p]
0
0.4
0.35
−0.5
0.3
−1
−1.5
−3
0.25
−2
−1
0
1
2
0.2
−3
3
−2
−1
Expected dollar return
3
1
2
0.5
1
E[d−p|P=p]
p
0
−1
−1
−2
−1
0
2
3
0
−0.5
−2
1
Uninformed demand curve
1.5
−1.5
−3
0
p
p
1
2
−3
−3
3
−2
p
−1
0
1
2
3
xU(p)
Figure 3. Properties of PLC
This figure illustrates the properties of the price function with linear cut, PLC (solid
blue line on all panels), compared to the GS price (dashed red line). The upper left
panel shows that the expected payoff conditional on the price realization, E [d|P = p],
is non-monotonic under P . The upper right panel shows that the variance of the payoff
conditional on the price realization, V ar [d|P = p], is non-constant under PLC , and always lower than under PGS . The lower left panel shows that the expected dollar return
conditional on the price realization, E [d − p|P = p], can be non-decreasing. The lower
right panel shows that the inverse demand function of uninformed traders, xU (p), can
increase in the price. The parameters are set to σs = 0.6, σu = 0.4, σn = 0.5, α = 2,
ω = 0.15, D = 0.
not necessarily indicate a higher expected payoff. To understand the particular shape,
consider, for example, the case when the price decreases continuously from ∞ while
remaining always above D. The price in this case reveals both the linear combination
l and that s > s̄. The splitting point s̄ (l) is a decreasing function of l; when l is very
high, s̄ is very low, and uninformed agents’ belief about s is not very different from the
GS equilibrium, as the truncation at s̄ removes only the very extreme lower tail of the
distribution. When l decreases, the splitting point s̄ (l) increases and goes to infinity
17
when l → ∞. Therefore, even if uninformed agents learn that s would be low on average,
i.e. is from a line with low intercept, the truncation s > s̄ implies that expected payoff
goes to infinity:
lim E [d|P = p] = lim E [s|s − Cu = l, s > s̄ (l)] = ∞.
pցD
l→−∞
The upper right panel shows the variance of the payoff conditional on the price
realization, V ar [d|P = p], in the two equilibria. The conditional variance is non-constant
in the new equilibrium: it is always lower than in the GS equilibrium, because uninformed
investors do not only learn the linear combination l, as in GS, but also whether the payoff
is above or below a threshold. For example, when the price and thus l are very high, s̄
is very low, uninformed agents’ belief about s is similar to that in the GS equilibrium,
so the conditional variance of the payoff is close to that in the GS equilibrium. When
l decreases, s̄ increases, and the truncation reduces the uncertainty. Therefore, payoff
volatility is especially small when the price is close to the separating price level D.
Proposition 3. Non-monotonic belief moments about the payoff lead to “upward-sloping”
uninformed demand curve xU (p).
The lower right panel of Figure 3 shows the equilibrium inverse demand of uninformed
traders as a function of price in the two equilibria. Notably, the demand is locally
downward-sloping everywhere, but because it is discontinuous in the LC equilibrium, it
is globally non-monotonic: uninformed traders are willing to buy more when the price
increases from below D to above D. This is not surprising, since prices just below D
reveal a low payoff with little uncertainty, whereas prices above it reveal a high payoff
with little uncertainty (see the two upper panels).
Proposition 4. Expected dollar return, E [d − p|PLC = p], can be increasing in the price
level p, i.e., the equilibrium can exhibit price drift.
The lower left panel of Figure 3 shows the expected dollar return conditional on
the price realization in the two equilibria. Notably, the expected return of uninformed
traders is non-monotonic in the new equilibrium: a higher price realization can indicate
18
higher expected return, especially when the price is close to the separating price level
D, because it reveals a jump in the expected payoff. In the GS equilibrium this is
always downward-sloping, i.e., higher prices (higher contemporaneous returns) imply
lower subsequent returns, which Banerjee, Kaniel, and Kremer (2009) term reversal.
Since in the LC equilibrium higher prices can imply higher subsequent dollar returns,
the equilibrium displays price drift.
4
Conclusion
The standard method of conjecturing and then verifying a linear equilibrium price function has become widely used in models of asymmetric information. While papers following this technique show that the price function is unique in the linear class, they usually
claim we do not know anything outside the linear class. Hence it is important to study
whether there exist other equilibria of such a model, which are potentially less tractable
but more appealing in their predictions.
In this paper we explore this question. Our solution method is different from the
usual “conjecture and verify” approach in which guessing a specific functional form of
price naturally imposes limitations on what kind of equilibrium behaviour is observed.
Our contribution complements the usual techniques and leads to the conclusion that the
unique linear equilibrium of the Grossman and Stiglitz model is unique when allowing
for any continuous price function. We also construct non-continuous equilibria and show
they feature many realistic economic properties, including jumps and crashes, upward
sloping demand curves, and that expected dollar return of uninformed agents can both
increase or decrease in the price.
In this paper we restrict our attention to the assumptions of the original Grossman
and Stiglitz (1980) paper, and discuss only the generalization of our results by relaxing
certain distributional assumptions. There are many other directions we could look to
extend our results. First, it would be interesting to see whether we could provide similar
statements about the equilibrium in the same setting but with more general parameter
distributions and utility functions. Second, an important question is whether our result
19
would stay in a modification of the model with incorporating imperfect competition, as
in Kyle (1989) or the single-period version of Kyle (1985). Third, it would be interesting
to study whether other equilibria exist in settings with dispersed information as in, e.g.,
Hellwig (1980) or Diamond and Verrecchia (1981). These problems are left for future
research.
20
References
Admati, A. R. (1985): “A noisy rational expectations equilibrium for multi-asset securities
markets,” Econometrica, 53, 629–658.
Angeletos, G.-M., and I. Werning (2006): “Crises and prices: information aggregation,
multiplicity, and volatility,” American Economic Review, 96, 1720–1736.
Ausubel, L. M. (1990a): “Insider trading in a rational expectations economy,” American
Economic Review, 80, 1022–1041.
(1990b): “Partially-revealing rational expectations equilibrium in a competitive economy,” Journal of Economic Theory, 50, 93–126.
Back, K. (1992): “Insider trading in continuous time,” Review of Financial Studies, 5, 387–
409.
Bai, Y., E. C. Chang, and J. Wang (2006): “Asset prices under short-sale constraints,”
Working paper, MIT.
Banerjee, S., R. Kaniel, and I. Kremer (2009): “Price drift as an outcome of differences
in higher order beliefs,” Review of Financial Studies, 22, 3797–3734.
Barlevy, G., and P. Veronesi (2003): “Rational panics and stock market crashes,” Journal
of Economic Theory, 110, 234–263.
Breon-Drish, B. (2010): “Asymmetric information in financial markets: anything goes,”
Working paper, Stanford University.
(2012): “On existence and uniqueness of equilibrium in noisy rational expectations
economies,” Working paper, Stanford University.
Brunnermeier, M. K. (2001): Asset pricing under asymmetric information. Oxford Univeristy Press, New York, NY.
DeMarzo, P., and C. Skiadas (1998): “Aggregation, determinacy, and informational efficiency for a class of economies with asymmetric information,” Journal of Economic Theory,
80, 123–152.
Diamond, D. W., and R. E. Verrecchia (1981): “Information aggregation in a noisy
rational expectations economy,” Journal of Financial Economics, 9, 221–235.
Du, S., and H. Zhu (2013): “Dynamic ex post equilibrium, welfare, and optimal trading
frequency in double auctions,” Working paper, MIT.
Gennotte, G., and H. Leland (1990): “Market liquidity, hedging, and crashes,” American
Economic Review, 80, 999–1021.
Grossman, S. J. (1976): “On the efficiency of competitive stock markets where traders have
diverse information,” Journal of Finance, 31, 573–585.
Grossman, S. J., and J. E. Stiglitz (1980): “On the impossibility of informationally
efficient markets,” American Economic Review, 70, 393–408.
21
Hellwig, M. F. (1980): “On the aggregation of information in competitive markets,” Journal
of Economic Theory, 22, 477–498.
Jordan, J. (1982): “The generic existence of rational expectations equilibrium in the higher
dimensional case,” Journal of Economic Theory, 26, 224–243.
(1983): “On the efficient market hypothesis,” Econometrica, 51, 1325–1344.
Kyle, A. (1985): “Continuous auctions and insider trading,” Econometrica, 53, 1315–1336.
(1989): “Informed speculation with imperfect competition,” Review of Economic
Studies, 56, 317–355.
Ozdenoren, E., and K. Yuan (2008): “Feedback effects and asset prices,” Journal of
Finance, 63, 1939–1975.
Radner, R. (1979): “Rational expectations equilibrium: generic existence and the information revealed by prices,” Econometrica, 47, 655–678.
Rochet, J.-C., and J.-L. Vila (1994): “Insider trading without normality,” Review of
Economic Studies, 61, 131–152.
Subrahmanyam, A., and S. Titman (2001): “Feedback from stock prices to cash flows,”
Journal of Finance, 56, 2389–2413.
Veldkamp, L. (2011): Information choice in macroeconomics and finance. Princeton, Princeton, NJ.
Venter, G. (2011): “Short-sale constraints and credit runs,” Working paper, LSE.
Vives, X. (2010): Information and learning in markets: The impact of market microstructure.
Princeton, Princeton, NJ.
Wang, J. (1994): “A model of competitive stock trading volume,” Journal of Political Economy, 102, 127–167.
Yuan, K. (2005): “Asymmetric price movements and borrowing constraints: a REE model of
crises, contagion, and confusion,” Journal of Finance, 60, 247–292.
(2006): “The price impact of borrowing and short-sale constraints,” Working paper,
University of Michigan.
22
Appendix A Preliminary results
In this appendix we collect a few properties of two functions that will be interesting for us
later. First we focus on
φ (x)
,
Ψ−1 (x) =
Φ (x)
where φ (.) denotes the pdf of the standard normal distribution, and Φ (.) is the corresponding
cdf.
Lemma 4. The function Ψ−1 (x) has the following two sets of properties:
(i) Ψ−1 (x) > 0 ∀x ∈ R, and its limits are limx→−∞ Ψ−1 (x) = ∞ and limx→∞ Ψ−1 (x) = 0.
(ii) Ψ−1 (x) is decreasing, its slope satisfies −1 < Ψ′−1 (x) < 0 for all x ∈ R, and its limits
are limx→−∞ Ψ′−1 (x) = −1 and limx→∞ Ψ′−1 (x) = 0.
Proof. Throughout the proof we make repeated use of the fact that the first derivatives of
φ (x) and φ2 (x) are given by
′
φ′ (x) = −xφ (x) and φ2 (x) = −2xφ2 (x) .
(i) The first part of the statement is straightforward because φ (x) , Φ (x) > 0 for all x ∈ R.
The third part of the statement is also obvious, since limx→∞ φ (x) = 0 while limx→∞ Φ (x) = 1.
Regarding the second part of the statement, limx→−∞ φ (x) = limx→−∞ Φ (x) = 0, hence we
use l’Hôpital’s rule to obtain
φ (x)
φ′ (x)
−xφ (x)
= lim
= lim
= lim (−x) = ∞.
x→−∞ Φ (x)
x→−∞ Φ′ (x)
x→−∞ φ (x)
x→−∞
lim
(ii) First, differentiating Ψ−1 we have
Ψ′−1 (x) = −
φ (x) [φ (x) + xΦ (x)]
.
Φ2 (x)
(A-1)
It is easy to show that
lim [φ (x) + xΦ (x)] = 0 and lim [φ (x) + xΦ (x)] = ∞,
x→∞
x→−∞
moreover
d
[φ (x) + xΦ (x)] = Φ (x) > 0,
dx
hence φ (x) + xΦ (x) is positive and strictly increasing, and Ψ′−1 < 0 follows from here.
√
To show Ψ′−1 (x) > −1, we start from x x2 + 4+x2 +2 > 0, which trivially holds for x ≥ 0,
and easy to confirm for all x < 0. We perform the following series of equivalent arrangements:
p
p
p
1
2
1
−x + x2 + 4 .
< x+ x2 + 4 ⇐⇒ −x+ √
<
x2 + 4+x2 +2 > 0 ⇐⇒ √
2
x2 + 4
x2 + 4
√
Multiplying both sides by 21 φ (x) x + x2 + 4 > 0 we obtain
x
p
p
p
1
1
1
2
2
2
√
< φ (x) x + x + 4 −x + x + 4 = φ (x)
φ (x) x + x + 4 −x +
2
4
x2 + 4
23
√
d
1
2+4
for all x. It is easy to show that the expression on the LHS is dx
x
, so
φ
(x)
x
+
2
integrating the inequality between −∞ and x and rearranging yields
r
x2
1
φ (x) 1 +
< Φ (x) − xφ (x) .
4
2
Since the LHS is non-negative, we can take squares to get
2
1
x2
1
φ (x) 1 +
< Φ (x) − xφ (x) = Φ2 (x) − xφ (x) Φ (x) + x2 φ2 (x) ,
4
2
4
2
that is,
φ2 (x) + xφ (x) Φ (x) < Φ2 (x) .
(A-2)
According to (A-1), Ψ′−1 (x) > −1 follows straight from (A-2).
The limit part of the statement is straightforward for x → ∞ from limx→∞ Φ (x) = 1,
limx→∞ φ (x) = 0, and limx→∞ xφ (x) = 0. Finally, both the numerator and the denominator
of (A-1) converge to zero when x → −∞, hence with the repeated use of l’Hôpital’s rule we
obtain
−x [φ (x) + xΦ (x)] + Φ (x)
−xΦ (x)
φ (x) [φ (x) + xΦ (x)]
= lim
= lim
2
x→−∞
x→−∞ φ (x)
x→−∞
Φ (x)
2Φ (x)
2
x − 2 φ (x)
−xφ (x) − Φ (x)
x2 − 2
= lim
= lim
=
lim
= 1,
x→−∞
x→−∞ (x2 − 1) φ (x)
x→−∞ x2 − 1
−xφ (x)
lim
which concludes the proof of the lemma.
With the help of Lemma 4 we derive the following results:
Lemma 5. Suppose ρ ≥ 0 constant. The function
Ψρ (x) = (1 + ρ) x +
φ (x)
Φ (x)
has the following properties:
(i) If ρ = 0, Ψ0 (x) > 0 for all x ∈ R, and its limits are limx→−∞ Ψ0 (x) = 0 and
limx→∞ Ψ0 (x) = ∞. If ρ > 0, Ψρ (x) takes every real value, and its limits are limx→−∞ Ψρ (x) =
−∞ and limx→∞ Ψρ (x) = ∞.
(ii) Ψρ (x) is increasing, its slope satisfies ρ < Ψ′ρ (x) < 1 + ρ for all x ∈ R, and the limits
of the slope are limx→−∞ Ψ′ρ (x) = ρ and limx→∞ Ψ′ρ (x) = 1 + ρ.
Proof. We first prove everything for the case ρ = 0.
(i) The first and last parts of the statement follow from the fact that Ψ0 (x) = φ(x)+xΦ(x)
,
Φ(x)
and the properties of φ (x) + xΦ (x) are discussed in the proof of statement (ii) of Lemma 4.
For the second part of the statement, limx→−∞ Φ (x) = limx→−∞ [φ (x) + xΦ (x)] = 0, so using
l’Hoŝpital’s rule twice we obtain
φ (x) + xΦ (x)
Φ (x)
φ (x)
1
lim
= lim
= lim
= lim
−
= 0.
x→−∞
x→−∞ φ (x)
x→−∞ −xφ (x)
x→−∞
Φ (x)
x
24
(ii) Differentiating Ψ0 we have Ψ′0 (x) = 1 + Ψ′−1 (x), so the statements on the limits of Ψ′0
are straightforward from (ii) of Lemma 4.
Finally, the rest of the results for ρ > 0 follow from adding ρx to the function considered
so far; this changes the value of the function trivially, and increases its slope by ρ.
Lemma 6. The function
−1
d (x) = − Ψ−1
ρ (x) + Ψρ (−x)
is positive, convex and even, it reaches its minimum at zero, and diverges to ∞ for x → ±∞.
Proof. It is trivial from its definition that d (x) is an even function, i.e. d (x) = d (−x).
Moreover, Ψρ (x) is convex from Lemma 5, therefore Ψ−1
ρ is concave, and from here d is also
convex.
Since d is both symmetric with respect to the vertical axis and convex, it reaches its
minimum at zero. Let us denote the unique root of Ψρ by x̄, i.e. for which Ψρ (x̄) = 0;
it is easy to see from the definition of Ψρ that x̄ < 0. Therefore, the minimum of d is
d (0) = −2Ψ−1
ρ (0) = −2x̄ > 0.
Finally, symmetry and convexity, together with the fact that d’s domain is the whole R,
imply that d is increasing on R+ and hence trivially diverges: limx→∞ d (x) = ∞. Symmetry
then also implies limx→−∞ d (x) = ∞.
Appendix B Non-continuous price functions
This appendix provides a constructive proof that there exist non-continuous price functions
described in Section 3. In the first part we derive the relationship between price realizations
and the P -homogeneous regions for general price functions in three steps: As a start, we
formally derive the belief of uninformed agents if they learn something in addition to the usual
GS-information, i.e., if the price reveals both the linear combination of signal s and the noisy
supply u, and that s is inside (or outside) a set of intervals. Second, given the belief, we derive
the inverse demand function of uninformed agents. Third, we apply market clearing to obtain
an equation that the equilibrium price must satisfy. In the second part, we provide appropriate
’cuts’, i.e., choose the P -homogeneous regions to be half-lines in such a way that the attained
prices satisfy all the properties they must to produce a valid price function.
Appendix B.1 General derivation of non-continuous price functions
Posterior distributions. In the standard GS model (see also Section 2), uninformed agents
have prior s ∼ N 0, σs2 , and infer another piece of information from the price in the
form
2
of a linear combination of s with the noisy supply: l = s − Cu, where u ∼ N 0, σu where
C ∈ R+ is given in Theorem 1. Standard
Bayesian updating then implies that the posterior
distribution is s|s−Cu=l ∼ N L, Σ2 with mean and variance
L = E [s|s − Cu = l] =
σs2 C 2 σu2
σs2
2
l
and
Σ
=
V
ar
[s|s
−
Cu
=
l]
=
;
σs2 + C 2 σu2
σs2 + C 2 σu2
25
as in (8). Put it differently, the posterior pdf is
1
fs|l (s|s − Cu = l) = φ
Σ
s−L
Σ
.
Suppose now that besides the linear combination l, uninformed agents also learn that
s ∈ S ≡ ∪ni=1 [s̄2i−1 , s̄2i ] for a natural number n, s̄i ∈ R ∪ {−∞, ∞} for all i = 1, 2, ..., 2n and
s̄1 ≤ s̄2 ≤ ... ≤ s̄2n . Since the priors about s and u are normal, the posterior distribution
conditional on these two additional pieces of information is a generalization of the truncated
normal distribution, and it is easy to show that has pdf
φ s−L
1
Σ
i ,
fs|L (s|L) = 1s∈S P h (A-3)
Σ n Φ s̄2i −L − Φ s̄2i−1 −L
i=1
Σ
Σ
where we introduce the simplifying notation L for the information set {s − Cu = l, s ∈ S}.
Notice that if we fixl, pick n = 1, s̄1 = −∞ and s̄2 = ∞, uninformed agents’ belief becomes
s|s−Cu=l ∼ N L, Σ2 , i.e., the posterior belief in the standard GS equilibrium.
Naturally, the formulas above (and thus the results below) would still hold if we had open
intervals instead of closed ones, or if S consisted of a countable number of intervals. Also, from
(A-3) one could easily express the conditional distributions of d|s−Cu=l,s∈S , but they are not
necessary for the solution of the optimization programs, so we omit them here.
Uninformed optimization problems. Next, we solve the optimization problem
max E [− exp (−α (d − p) x) |L] .
x
(A-4)
Since d = s + n with n ∼ N 0, σn2 being independent of s, we have
1 2 2 2
E [− exp (−α (d − p) x) |L] = − exp αpx + α σn x E [exp (−αsx) |L] ;
2
moreover, from (A-3), we can show that
1 2 2 2
E [exp (−αsx) |L] = exp − αLx − α Σ x
2
s̄2i −(L−αΣ2 x)
s̄2i−1 −(L−αΣ2 x)
−Φ
i=1 Φ
Σ
Σ
i
h
,
Pn
s̄2i−1 −L
s̄2i −L
−
Φ
Φ
i=1
Σ
Σ
Pn
hence the optimization problem (A-4) is equivalent to
2
1
2
2
×
max − exp −α (L − p) x − α Σ + σn x
x
2
"
!#
!
n
X
s̄2i−1 − L − αΣ2 x
s̄2i − L − αΣ2 x
−Φ
,
Φ
×
Σ
Σ
i=1
where the first component is the usual term that illustrates the CARA-normal optimization
problem is equivalent to a mean-variance problem, and the second term adjusts the maximand
26
with the ’truncated’ belief. After deriving and rearranging the FOC, we obtain the inverse
demand function of uninformed agents:
Pn
s̄2i−1 −(L−αΣ2 x)
s̄2i −(L−αΣ2 x)
−φ
i=1 φ
Σ
Σ
2
2
i .
h
0 = L − p − α Σ + σn x − Σ P
(A-5)
2
s̄2i−1 −(L−αΣ2 x)
s̄2i −(L−αΣ x)
n
−
Φ
Φ
i=1
Σ
Σ
Market-clearing price. We rearrange the market-clearing condition
ω
s−p
+ (1 − ω) xU = u
ασn2
to get
xU =
p − (s − Cu)
p−l
=
.
(1 − ω) C
(1 − ω) C
(A-6)
With the help of (A-6), we replace the uninformed demand in (A-5). After rearranging and
using the definition of L, we obtain the following equation that the equilibrium price p corresponding to the information set L solves:
i
Pn h s̄2i +ϑp−ξl s̄2i−1 +ϑp−ξl
−
φ
φ
i=1
Σ
Σ
Σ
i ,
p = Bl − P h (A-7)
κ n Φ s̄2i +ϑp−ξl − Φ s̄2i−1 +ϑp−ξl
i=1
Σ
Σ
where B is the same as in Theorem 1, and
ϑ=
σ2
αΣ2
1
, ξ = ϑ + 2 s 2 2 , and κ = ϑ +
(1 − ω) C
σs + C σu
1−ω
satisfy 0 < ϑ < ξ < κB < κ.
Appendix B.2 The price function with linear cut and its properties
As a final step of the process described in Appendix B, we have to show that with an appropriately chosen ’cut’ of the plane, we obtain a valid price function, i.e., it is a bijection between
the price values and the P -homogeneous regions.
Proof of Theorem 2. We conjecture a simple linear cut of the plane into two halves by the
linear splitting rule s̄ (l) = Γl + ∆, where Γ, ∆ ∈ R are constants to be determined. Formally,
this is equivalent to either having n = 1, s̄1 = −∞ and s̄2 = s̄ (l), which implies an uninformed
information set L− (l) = {s − Cu = l, s < s̄ (l)}, or having n = 1, s̄1 = s̄ (l) and s̄2 = ∞, which
implies an uninformed information set L+ (l) = {s − Cu = l, s ≥ s̄ (l)}, for all l ∈ R. It is easy
to see from (A-7) that the corresponding equilibrium prices based on these beliefs, p+ and p− ,
solve
(ξ−Γ)l−ϑp+ −∆
1
φ
(ξl
−
ϑp
−
s̄)
φ
Σ
Σ
Σ
+
Σ
= Bl +
(A-8)
p+ = Bl +
κ Φ Σ1 (ξl − ϑp+ − s̄)
κ Φ (ξ−Γ)l−ϑp+ −∆
Σ
27
and
(ξ−Γ)l−ϑp− −∆
φ
−
(ξl − ϑp− − s̄)
Σ
Σ
Σ φ
,
= Bl −
p− = Bl −
1
κ 1 − Φ Σ (ξl − ϑp− − s̄)
κ Φ − (ξ−Γ)l−ϑp− −∆
Σ
1
Σ
(A-9)
where we omit the argument of functions p+ (l), p− (l) and s̄ (l) to simplify the expressions.
We start by analyzing p+ and derive the appropriate constants Γ, ∆ ∈ R from there; then
we look at p− and confirm it behaves as we want. For now we conjecture that p+ (l) and p− (l)
obtained from (A-8) and (A-9) are differentiable, and confirm it later.
Rearranging (A-8) we get
φ
(ξ−Γ)l−ϑp+ −∆
Σ
[(ξ − κB) − Γ] l + (κ − ϑ) p+ − ∆
(ξ − Γ) l − ϑp+ − ∆
=
+ + −∆
Σ
Σ
Φ (ξ−Γ)l−ϑp
Σ
(ξ − Γ) l − ϑp+ − ∆
= Ψ0
,
(A-10)
Σ
where in the last step we used the definition of Ψ0 . Note that p+ is a function of l, thus (A-10)
has to hold for all l as an identity. Therefore, the derivative of the two sides w.r.t. l must be
that same too, that is,
ξ−Γ
ϑ dp+
(ξ − Γ) l − ϑp+ − ∆
(ξ − κB) − Γ κ − ϑ dp+
′
+
=
−
Ψ0
.
(A-11)
Σ
Σ dl
Σ
Σ dl
Σ
We want to choose Γ and ∆ such that liml→−∞ p+ is finite, which also means liml→−∞
Therefore, taking the limits of both sides of (A-11), we have
(ξ − Γ) l − ϑp+ − ∆
ξ−Γ
(ξ − κB) − Γ
′
=
lim Ψ
.
Σ
Σ l→−∞ 0
Σ
dp+
dl
= 0.
(A-12)
First, notice that from Lemma 5 Ψ′0 is bounded, and thus ξ = Γ would make the RHS zero
while the LHS is not. Therefore, ξ − Γ 6= 0, and rearranging (A-12) we obtain
κB
(ξ − Γ) l − ϑp+ − ∆
′
1−
= lim Ψ
.
(A-13)
ξ − Γ l→−∞ 0
Σ
Moreover, liml→−∞ p+ being finite and ξ − Γ 6= 0 together imply that
(ξ − Γ) l − ϑp+ − ∆
−∞ if Γ < ξ
lim
=
∞
if Γ > ξ,
l→−∞
Σ
hence from Lemma 5 we get
lim
l→−∞
Ψ′0
(ξ − Γ) l − ϑp+ − ∆
Σ
=
0
1
if Γ < ξ
if Γ > ξ.
κB
6= 0 means the LHS of (A-13) cannot be 1, and thus (A-13) can only hold if
However, ξ−Γ
Γ = ξ − κB < ξ. Substituting this into (A-10), we obtain
(κ − ϑ) p+ − ∆
κBl − ϑp+ − ∆
= Ψ0
.
(A-14)
Σ
Σ
28
Taking the limit of both sides when l → −∞ and using the above observations, we obtain
κ−ϑ
∆
κBl − ϑp+ − ∆
lim p+ −
= lim Ψ0
= 0,
Σ l→−∞
Σ l→−∞
Σ
thus introducing the notation D =
∆
κ−ϑ
we obtain liml→−∞ p+ = D; the limit is indeed finite.
Moreover, substituting Γ = ξ − κB and the definition of D into (A-10) and further rearranging, we obtain
κBl−ϑp+ −∆
φ
Σ
κρ
κBl − ϑp+ − (κ − ϑ) D
κBl − ϑp+ − ∆
= Ψρ
(Bl − D) = (1 + ρ)
+
Σ
Σ
Σ
Φ κBl−ϑp+ −∆
Σ
(A-15)
C
>
0.
Ψ
(.)
is
invertible
with the definition of the Ψ function and the notation ρ = κ−ϑ
=
ρ
2
ϑ
αΣ
for ρ ≥ 0, so after some algebra, (A-15) yields
Σ −1 κρ
Ψ
(Bl − D) ,
(A-16)
p+ = Bl + ρ Bl − D −
ϑρ ρ
Σ
which is identical to (10) if we note that ϑρ =
1
1−ω ,
κρ =
1+ρ
1−ω ,
and λ =
κρ
Σ.
Finally, to show that p+ is increasing in l, we differentiate (A-10) w.r.t l. Using Γ = ξ − κB
and rearranging:
ρ
κB κB
dp+
,
(A-17)
=
−
dl
ϑ
ϑ ρ + Ψ′ κBl−ϑp+ −∆
0
Σ
According to Lemma 5, Ψ0 is convex, i.e. its slope is increasing, and thus limx→−∞ Ψ′0 (x) =
0 < Ψ′0 (x) for all x ∈ R. Combining it with (A-17), we obtain 0 < dpdl+ , i.e., p+ is strictly
increasing for all l.
For p− , we rearrange (A-9) in two ways. First, it is possible to express it analogously to
(A-16): after some calculations, we obtain
Σ −1 κρ
Ψ
− (Bl − D) ,
p− = Bl + ρ Bl − D +
ϑρ ρ
Σ
which yields (11). Second, it can be expressed similarly to (A-14):
κBl − ϑp− − ∆
(κ − ϑ) p− − ∆
= −Ψ0 −
.
Σ
Σ
(A-18)
Taking the limit of both sides when l → ∞ and used the above observations, we obtain
∆
κBl − ϑp− − ∆
κ−ϑ
lim p− −
= − lim Ψ0 −
.
(A-19)
l→∞
Σ l→∞
Σ
Σ
Also, differentiating (A-18) and rearranging, we obtain
(κ − ϑ)
κB κB
dp−
.
=
−
dl
ϑ
ϑ (κ − ϑ) + ϑΨ′ − κBl−ϑp− −∆
0
Σ
29
(A-20)
Again, since Ψ0 is convex, its slope is increasing, i.e. Ψ′0 (x) > 0 for all x ∈ R, which means
that from (A-20), dpdl− > 0 for all l ∈ R.
Finally, as p− is increasing, its limit when l → ∞ can either be finite or ∞, and thus the
limit of the LHS of (A-19) is also either finite or ∞. On the RHS of this equation, however,
the limit of the term Ψ0 (.) be either zero, a finite value or ∞, depending on the argument of
the function; therefore the limit of the RHS is either finite or −∞. These together mean that
the only possible case is when liml→∞ p− is finite. Revisiting the RHS, it must be that the
argument of Ψ0 diverges to −∞, and thus
κBl − ϑp− − ∆
lim Ψ0 −
= 0.
l→∞
Σ
Plugging it back to (A-19) we obtain
lim p− =
l→∞
∆
= D = lim p+ .
l→−∞
κ−ϑ
Therefore, both p− and p+ are increasing, and there is no overlap in their ranges. Hence, PLC
is a valid price function.
Proof of Proposition 1. From (10) and (11), after some algebra, we obtain that the difference
between the price on the right and left side of s̄ (l) is given by
∆P (l) = P+ (l) − P− (l) = (1 − ω) ρΣd (λ (Bl − D)) ,
where d (.) is defined in (6). The properties of ∆P (.) are then straightforward from Lemma
6.
Proofs of Propositions 2-4. [TO BE ADDED]
30
Appendix C Continuous equilibria with non-normal distributions
In this appendix we extend the uniqueness result established in Section 2 by relaxing the
normality assumptions we made on s, u and n. We make use of the following lemma to express
optimal demand for general distributions:
Lemma 7. Suppose an agent has negative exponential utility with CARA-coefficient α, and her
belief about the Period-1 payoff of the risky asset is described by θ = θ + ε, where θ is the finite
expected value of θ, and ε is a zero-mean, non-degenerate random variable, whose cumulantgenerating function exists in an open interval around zero. The agent’s optimal demand is
given by
x (p) = h θ − p ,
(A-21)
where h = (H ′ )−1 and
H (x) =
1
log E [exp {−αxε}] .
α
(A-22)
The following proposition collects some properties of H and h that we employ later:
Proposition 5. The function H is infinitely differentiable, strictly convex, and H (0) = 0.
Thus, both H ′ and its inverse h are continuous and strictly increasing on their respective
supports. Moreover, H ′ (0) = 0, and hence h (0) = 0.
The properties mentioned in the first sentence hold because αH (x) is the cumulantgenerating function of −αε. Cumulant-generating functions, if exist, are infinitely differentiable, convex, and pass through the origin. The proof of Lemma 7 and the other statements in Proposition 5 are straightforward and hence omitted. Note that under normality
H (x) = 21 ασε2 x2 , where σε2 is the variance of ε, and hence (A-21) generalizes the linear CARAnormal demand to any distribution. Also note that Lemma 3 holds for arbitrary s, u, and n,
as the normality of the distributions is not needed for its proof.
Below we consider two main cases: (i) when the distribution of n remains normal, but the
distributions of s and u can be arbitrary (but independent), and (ii) when the distribution of
n is also arbitrary.
Appendix C.1 s and u have different distributions, but n is normal
We assume that the unlearnable part of the payoff, n, is still normally distributed, but s
and u, while still independent, can have arbitrary distribution functions fs : Ωs 7→ R and
fu : Ωu 7→ R with connected domains Ωs and Ωu —on R this means Ωs and Ωu are open,
closed, or open/closed intervals (including half-lines and the whole real line, too). It turns out
that our method from the standard setting can be extended to show that there exists at most
one continuous equilibrium. Moreover, we provide a sufficient condition for existence and thus
uniqueness.
Theorem 3. Suppose s and u have arbitrary distributions on the (potentially infinite) intervals
Ωs and Ωu such that the cumulant-generating function of s|s−Cu=l exists in an open interval
around zero. If E [s|s − Cu = l] is continuous in l, there exists a unique continuous equilibrium
of the asset market; if not, there exists no continuous equilibrium. If it exists, this price function
depends on s and u only through s − Cu. Finally, if E [s|s − Cu = l] is increasing in l, P is
also increasing in l.
31
Proof. Since n is still Gaussian, informed demand is still (4), and uninformed agents learn the
same linear combination l = s − Cu from the market-clearing condition as in (5). Due to Ωs
and Ωu being connected, the logic of Section 2 works for Lemmas 1 and 2, and P (s, u) depends
on s and u only through l.
What remains is to examine uninformed demand. If αHU denotes the cumulant-generating
function of the distribution of (−αs) |s−Cu=l and hU = (HU′ )−1 , Lemma 7 implies that uninformed investors demand
xU (l, p) = hU (E [s|s − Cu = l] − p) .
(A-23)
Substituting (A-23) into (3) and rearranging, we obtain
l − p + (1 − ω) ChU (E [s|s − Cu = l] − p) = 0,
(A-24)
where both E [s|s − Cu = l] and hU are uniquely determined by prior distributions, known
by uninformed agents, and do not depend on the actual price function P . Hence, there is a
simple constructive way of looking at existence and uniqueness: for every l ∈ R, uninformed
agents can manually compute E [s|s − Cu = l], substitute it into (A-24), and look for the price
p that satisfies the equation. Since the first term of the left-hand side, (l − p), is decreasing and
continuous in p, and takes every value on the real line, and the second term is also decreasing
and continuous in p due to Proposition 5, their sum is also decreasing, continuous, and takes
every value. Therefore, for all l there exists exactly one p that satisfies (A-24), and the pairing
l 7→ p gives us the only possible equilibrium price function. Proposition 5 implies that the
function hU is continuous, so this l 7→ p mapping is continuous if and only if E [s|s − Cu = l]
is a continuous function of l.
Suppose further that the prior distributions of s and u are such that E [s|s − Cu = l]
is increasing in l. Because hU is increasing, the left-hand side of (A-24) is increasing in l
and decreasing in p. Therefore, the implicit constructive price function determined above is
increasing in s − Cu. On the other hand, if E [s|s − Cu = l] is not an increasing function of l,
the price function can be non-monotonic. This completes the proof.
In textbook rational expectations economies (e.g., Grossman and Stiglitz (1980), Brunnermeier (2001), Vives (2010), and Veldkamp (2011)), uninformed agents’ information set is
defined as the equilibrium price function, because it contains all the relevant public information
available, just as we have shown in Section 2. When we relax the normality assumption of at
least one variable, this is not the case any more. In particular, Theorem 3 also implies the
following result:
Proposition 6. Assume in Definition 1 we replace (2) with the following condition:
2’. demand is optimal for uninformed traders, given the equilibrium price:
xU (p) ∈ arg max E [− exp {−αx (d − p)} |P (s, u) = p] .
x
Moreover, assume that E [s|s − Cu = l] is not an increasing function of l; therefore, the unique
(s − Cu)-measurable price function derived in Theorem 3 is non-monotonic. In this case there
exists no continuous equilibrium in the redefined economy.
Proof. Suppose the only possible equilibrium (s − Cu)-measurable price function of Theorem
3 is non-monotonic, i.e., there exist l1 6= l2 such that P (s − Cu = l1 ) = P (s − Cu = l2 ) = p
32
for some p. Then, in the redefined economy where uninformed agents can only condition on
the price, they cannot distinguish lines s − Cu = l1 and s − Cu = l2 , and submit the same
demand function xU (p). This violates Lemma 1; contradiction.
Proposition 6 illustrates the difference between the standard and our approach regarding
the information set of uninformed traders. It shows that with non-normal random variables
the textbook setting, in which uninformed traders can only condition on the price, under some
conditions there exists no continuous equilibrium. This is the case because the price function is
not invertible, and thus observing the net order flow of informed traders and the noisy supply
can provide information over and above the price alone.6
Appendix C.2 n has non-normal distribution [INCOMPLETE]
Informed agents know s and do not have to learn from the price, i.e., are not affected by the
distribution of u. Therefore, the only random variable whose prior affects informed demand is
n. Changing its distribution means that informed demand is not linear in s any more, and the
market-clearing condition does not reveal the linear combination l to uninformed agents.
Suppose an equilibrium exists, and fix the price function P . Informed traders’ expectation
of the asset payoff d is s + E [n] = s, and the residual random part is n, hence their optimal
demand, from Lemma 7, is
xI (s, p) = h (s − p) ,
(A-25)
where h = (H ′ )−1 and
H (x) =
1
1
log E [exp {−αx (d − s)} |s] = log E [exp {−αxn}] .
α
α
Importantly, since the distribution of n is common knowledge in our model, H and hence h
are known to uninformed agents.7 Moreover, the function h does not depend on p; the price
only shows up directly on the right-hand side of (A-25), as indicated.
Plugging (A-25) into the market-clearing condition (3) and rearranging, we obtain
ωh (s − p) − u = − (1 − ω) xU (p) = −c,
(A-26)
where we also introduce the simplifying notation c = (1 − ω) xU (p) for aggregate uninformed
demand. For the first part of our analysis when we analyze what uninformed agents learn from
this market-clearing condition only, we take c as a constant and abstract from the fact that it
depends on the price p. What is important is that since uninformed traders know the function
h, the price realization p, and their aggregate demand c, the only two unknown constituents
of (A-26) are s and u, just as in the benchmark GS model.
The steps of our analysis can be seen as generalizations of those in Section 2. Our first
result, similar to Lemma 1, is a straightforward graphic interpretation of (A-26):
6
This result is similar to Proposition 4.2 of Breon-Drish (2010) on the information content of adjusted
volume.
7
This is where our specification d = s+n becomes important. If instead we considered the alternative
model, with terminal payoff F and informed signal S = F +ε, relaxing the joint distribution assumption
would mean that both the expectation E [F |S] and the corresponding cumulant-generating function in
(A-21) would depend on S, possibly making the inference of uninformed traders intractable.
33
Lemma 8. Fix a pair (p, c) ∈ R2 . The set of points satisfying (A-26) is a strictly increasing
but not necessarily straight curve on the (s, u) plane that goes through (p, c).
Proof. Proposition 5 states that h (0) = 0, which implies that (s = p, u = c) indeed solves
(A-26). The same proposition also states that h is increasing, which implies that keeping s
fixed, there is at most one u that solves (A-26), and keeping u fixed, there is at most one s that
solves (A-26). Moreover, again from h increasing, for any two solutions (s1 , u1 ) and (s2 , u2 )
we have that s1 < s2 ⇔ u1 < u2
We denote such a curve by C (p, c). The next result is the equivalent of Lemma 2.
(s ,u )
1
2
δ
(s2,u2)
u
(s2,u1)
(s1,u1)
0
C(p ,c )
C(p,c)
2 2
0
s
Figure 4. Proof of Lemma 9
If P (s2 , u2 ) 6= p, P cannot take the value p anywhere on the rectangle δ besides (s1 , u1 ).
Since P is continuous, p must be an extreme value, and P = p + ǫ for some ǫ > 0 on
both the left and bottom side of δ, which is a contradiction.
Lemma 9. If the price function P is continuous, every p-homogeneous region R of the (s, u)
plane is a strictly monotone increasing infinite curve, i.e., R = C (p, c) for some c ∈ R.
Proof. If R is the p-homogeneous region of P , all (s, u) ∈ R satisfy (A-26) for some c, i.e., is
on C (p, c), and thus R ⊂ C (p, c). We need to show that it cannot be a strict subset. Suppose
it is, then there exists (s2 , u2 ) ∈ C (p, c) such that (s2 , u2 ) ∈
/ R.8
8
As there must be a price attained on the (s2 , u2 ) point, if this different price is p2 , we must also
have (s2 , u2 ) ∈ C (p2 , c2 ) for some c2 . Therefore, Lemma 9 also implies that two C curves belonging to
the same P with different ps cannot intersect.
34
Take an arbitrary point (s1 , u1 ) ∈ R, i.e. for which the equilibrium price is actually p;
without loss of generality we can assume s1 < s2 , u1 < u2 . Define δ as the boundary of the
rectangle whose bottom-left corner is (s1 , u1 ) and whose upper-right corner is (s2 , u2 ); see also
Figure 4.
We claim that the price function takes p on δ only at (s1 , u1 ). Indeed, because (s2 , u2 ) ∈
C (p, c), Lemma 8 implies that neither line s = s2 nor line u = u2 intersect C (p, c) anywhere
else besides (s2 , u2 ). Similarly, since (s1 , u1 ) ∈ C (p, c), neither line s = s1 nor line u = u1
intersect C (p, c) anywhere else besides (s1 , u1 ). By continuity, this means that p is an extremal
value of the price function on δ; without loss of generality, the minimum. But in this case for
a sufficiently small ǫ > 0 we have that p + ǫ is taken on both the left and the bottom side of
the rectangle, which contradicts Lemma 8 for the curve that corresponds to p + ǫ.
The final question is how p, or equivalently, Cp,c , affects uninformed demand. This obviously
depends on distributional assumption that we have not used yet. While we are unable to
characterize existence and uniqueness in the general case, we have the following result.
Theorem 4. If s is normally distributed, u has arbitrary distribution on the (potentially infinite) interval Ωu , and n has arbitrary distribution such that its cumulant-generating function
exists in an open interval around zero, there exists at most one continuous equilibrium of the
asset market.
Proof. [TO BE ADDED]
Appendix D Additional results on equilibrium properties [INCOMPLETE]
Here we collect a few results that are useful for the characterization of price functions in
general, be continuous or non-continuous. The first lemma describes when and how a candidate
equilibrium price function can be “fixed” if it is not invertible.
Lemma 10. Suppose that, as implied by Lemma 3, two disjoint regions R1 and R2 would
imply the same candidate market-clearing price p0 . If the equilibrium uninformed holding on
R1 is x1 and on R2 is x2 , and x1 = x2 , then R1 ∪ R2 is a p0 -homogeneous region. If x1 6= x2 ,
then R1 ∪ R2 cannot be p-homogeneous for any p.
Proof. [TO BE ADDED]
Lemma 10 can also be used to discover what the relationship is between the candidate
price on a P -homogeneous region that is the union of two P -homogeneous regions with the
same price p. Next we generalize this result for the union of two P -homogeneous regions with
different prices.
Lemma 11. If region R1 is p1 -homogeneous, region R2 (disjoint from R1 ) is p2 -homogeneous
for some p1 ≤ p2 and R = R1 ∪ R2 is p-homogeneous, then p1 ≤ p ≤ p2 and equality holds if
and only if p1 = p2 . As a corollary, similar results hold for the disjoint union of more regions.
Proof. [TO BE ADDED]
35