Investment Busts, Reputation, and the
Temptation to Blend in with the Crowd∗
Steven R. Grenadier
Andrey Malenko
Ilya A. Strebulaev
Stanford GSB
MIT Sloan
Stanford GSB
September 2011
PRELIMINARY AND INCOMPLETE
COMMENTS WELCOME
We thank seminar participants at Stanford University for their helpful comments. We are
responsible for all remaining errors. Address for correspondence: Graduate School of Business,
Stanford University, 518 Memorial Way, Stanford, CA, 94305. Grenadier: sgren@stanford.edu;
Malenko: amalenko@mit.edu; Strebulaev: istrebulaev@stanford.edu.
∗
1
Abstract
Real investment decisions can be substantially affected by the temptation
of managers to blend in with the crowd and delay liquidation of unsuccessful projects. As we show, this could have a far-reaching repercussions for the
dynamics of whole industries and conceivably for the economy at large. The
underlying mechanism is the link between managerial reputation based on the
available information about the current project outcome and managerial payoffs from attracting future funding recourses and future project cash flows. If
liquidation of a current project reveals negative information about ability, the
manager will optimally delay abandonment decisions to make it more difficult
for the market to judge managerial ability correctly. In a dynamic model, we
show how this simple intuition can lead to amplification of modest negative
shocks and explain such phenomenon as industry-specific investment busts.
2
1
Introduction
None love the messenger who brings bad news. Love can be even harder to find if the
bad news speaks ill of the messenger’s abilities. When corporate managers face the
unappealing task of revealing unsuccessful outcomes that impact their reputations,
delay may be their first instinct.1 Delay becomes even more enticing if they can
wait for industry-wide downturns in order to hide individual failings and instead
“blend in with the crowd”. In this paper we argue that real investment decisions
can be substantially affected by the temptation to blend in with the crowd. More
importantly, this could have a far-reaching repercussions for the dynamics of whole
industries and conceivably for the economy at large.
To appreciate the simplicity and generality of our intuition, consider an example
from the venture-capital industry. Entrepreneurs care not just about the payoffs
on current projects, but also about their reputations, which is critical in attracting
future funding sources. Thus, perception of one’s ability by outsiders links the payoff
on a current project with future payoffs. Now suppose an entrepreneur realizes that a
project is likely to never pay off, and thus should be scrapped. However, if liquidation
reveals negative information about her ability, she will reconsider the decision to
scrap the project. First, she may delay her liquidation decision simply hoping for
some potential resurrection of the project’s fortunes. Second, and more importantly,
by delaying she may hope to time her decision so that it would be more difficult
for the market to judge her low ability correctly. She may strategically attempt to
blend in with the crowd at a time when higher-ability managers have to liquidate
their projects as well, either during a recession or a negative industry-wide shock.
In such a case she can successfully hide the negative information, avoiding truthful
revelation altogether. Such behavior leads to inefficiently low liquidation and the
continued survival of projects which, on the basis of economics alone, should long
ago have been liquidated.
This intuition is not constrained to entrepreneurial firms. General partners of
private equity funds get up to 50% of their compensation from future funds they
1
See e.g. Miller (2002) and Kothari, Shu, and Wysocki (2009).
3
will run (Chung, Sensoy, Stern, and Weisbach (2010)). If they realize that their
current fund is not doing well, they may want to delay liquidating their unsuccessful
investments until the time when other funds have to liquidate as well. Bank loan
officers may have substantial private information about the performance of the loans
they authorized in the first place and they may delay foreclosing on a bad loan for
fear of losing their reputation and job until the time recession hits and loan’s poor
performance could be attributed to systematic factors rather than their screening
and monitoring abilities.
Strategic delay decisions of individual managers, who attempt to blend in with
the crowd, may have far-reaching consequences for their industries. In particular,
even relatively small negative market-wide shocks could launch an avalanche of liquidations and lead to investment busts. While one may view a sudden and severe
economic downturn as causing massive failures, it may in fact be mostly revealing
past failures that were waiting for an opportune time to be revealed. Thus, this
economic mechanism can explain the magnitude of cyclicality of investment activity
which we observe in many industries, such as VC, banking, and real estate.
In order to explore this economic mechanism in detail, we build a continuous-time
dynamic signaling model that formalizes the blending in with the crowd intuition and
investigate whether relatively small shocks can lead to disproportionately large liquidations. Because our goal is to understand the efficiency of investment liquidation,
our starting point is the cross-section of projects that have already been funded
and running. A project’s expected payoff consists of an idiosyncratic project-specific
component (some projects are inherently less likely to be successful) as well as a
component that depends on the ability of managers who run them. We assume that
projects are either successful (with a random arrival of payoff) or unsuccessful (payoff never arrives). Neither the manager nor outside investors know the type of the
project, and gradually adjust their beliefs about the project’s type in a Bayesian
manner. Only managers, however, know their abilities. Outsiders are initially aware
only about the distribution of ability in the total pool of projects and subsequently
either observe managerial ability at the time of the project’s successful payoff or
learn about it by observing managerial liquidation decisions (or lack of them).
4
By operating projects managers incur running costs, which proxy not only for the
direct financing costs but also for indirect costs, such as lost opportunities elsewhere.
If these costs are the same across projects, then we should expect low ability managers
to liquidate their projects first.
Consider the trade-off that a low-ability manager faces. By liquidating the project
now she saves on the running costs of the project but at the same time she reveals
her low quality which adversely impacts her reputation and thus future returns.
This trade-off naturally leads to a single-jamming equilibrium, in which managers
strategically delay liquidation, but are in fact not able to hide their true ability from
the market upon liquidation (Grenadier and Malenko, 2011).
Now consider what happens in the presence of negative market-wide shocks which
cause a liquidation of a fraction of all projects, irrespective of their characteristics. A
low ability manager has an additional incentive to delay because now she has an opportunity to hide her true quality by liquidating at the same time as other managers
and blaming the market-wide shock. We find that the presence of systematic shocks
has two effects on industry dynamics. First, strategic delays caused by the temptation to blend in may cause a substantial increase in inefficient liquidation. Second,
at the time of the market-wide shock, the liquidation rate can be disproportionately
large relative to the size of the shock.
This mechanism can thus explain commonly observed dynamics in such different
industries as venture capital, real estate, and banking. Over a long period of time,
the incidence of negative outcomes and “news” can be unusually low. This period
is then followed by an investment bust with a very high rate of mortality among
projects. A typical explanation of this phenomena involves either markets’ unreasonable optimism (say, about the prospects of a particular industry or technology),
followed by the participants “waking up” to more sorrow prospects, or the availability of “cheap” credit which feeds bad projects until a large macroeconomic downturn
limits the availability of capital dramatically. Our simple and intuitively clear mechanism can explain such a phenomenon when the markets are rational and when the
market-wide shocks are relatively small.
An important implication of our results is that the temptation of individual man5
agers to blend in can serve as a multiplier effect in industry- and economy-wide
movements. For practical purposes, it could therefore be important to consider what
contractual and institutional mechanisms could limit the opportunity of managers
to blend in.
Our paper relates to several strands in the literature. Psychological research has
demonstrated that people tend to overinvest suboptimally in loss-making situations,
a trait called escalation of commitment or sunk cost fallacy (Arkes and Blumer
(1985), Staw and Hoang (1995)). Escalation of commitment has been shown to be
particularly pervasive is scenarios when additional investment can lead to the return
of the original investment (known as “gamble for resurrection”). Our mechanism
provides a rational interpretation of the escalation of commitment when managers
find it optimal to overinvest by attempting to blend in with the crowd later on.
Rajan (1994) builds a 2-period banking model with an inefficient delay of loan
liquidations in normal times. In his model, in the adverse economy all loans are the
same (bad) quality and the private information of banks drives loans liquidations. In
our mechanism, differential ability of managers and loans and the belief of the market
play the crucial role. A recent literature on dynamic lemons markets (e.g. Kurlat
(2010)) consider the case of buyers and sellers who choose to trade in the market
when there are many liquidity traders and the adverse selection problem is not as
severe. In this literature, high liquidity is good in the sense that entrepreneurs
can sell their projects to raise capital for the new ones. Thus, high liquidity is
associated with economic expansions and low liquidity is associated with economic
contractions. In our paper, negative economic shocks are associated with forced
liquidations. Acharya, DeMarzo, and Kremer (forthcoming) consider a model of
dynamic disclosure. In their model, a public shock reduces the value of the option to
delay disclosure and thereby leads to a wave in disclosures of bad information. There
is no blending in with the crowd effect because disclosure is assumed to be truthful
(as in the standard disclosure models).
6
2
Model Setup
In this section, we discuss the main ingredients of the model and the economic rationale behind our setup. Our goal is to explore the economic mechanism of “blending
in” by building a simple model that intuitively relates to a number of features observed in the real world.
Consider an entrepreneur (or manager) operating an investment project. The
manager is characterized by her ability, represented by θ, which is private information
of the manager. The manager’s investment project can be successful (with probability
p0 ) or unsuccessful (with probability 1 − p0 ). A successful project provides a payoff
of θ > 0 at a Poisson event which arrives with intensity λ > 0. Outsiders do not
know θ except for its prior distribution, which is common knowledge. It is given by
p.d.f. f (θ) and c.d.f. F (θ) with full support on θ, θ̄ . By contrast, an unsuccessful
project never pays off. Waiting for a project to pay off requires a cash outflow of c
per unit time. The manager learns about the quality of her investment project over
time and decides on the optimal timing of liquidation. The discount rate is equal to
r ∈ (0, λ).
The model of an individual entrepreneurial project thus contains four main ingredients: ability level θ, success probability p0 , success intensity λ, and waiting cost
c. All these ingredients have close equivalents in the business world. As an example,
consider financing entrepreneurial start-ups. As entrepreneurs typically know more
about their abilities and project’s potential is driven in large part by the quality
of start-up teams,2 we assume that θ is both the ability level and the payoff upon
successful realization and that it is private to the founders. At the same time, most
start–ups fail, often for reasons unrelated to entrepreneurial ability. The likelihood
of success or failure is becoming apparent over time to both managers and financiers.
For example, both the managers and investors of a solar energy firm observe the
2
There is substantial literature on venture capital and entrepreneurial firms showing that quality
of management teams determines in large part the outcome of the project and that VCs and
angels pay a great deal of attention to the characteristics of the founders. E.g., see Busenitz and
Barney (1997), Maxwell, Jeffrey, and Levesque (2011), Petty and Gruber (2011), Sudek (2006), and
Zacharakis and Meyer (2000).
7
trends in the industry, but the managers are better positioned to know how large
the payoff will be in the case of a particular technology or wider industry success.
Start-ups also incur (frequently substantial) costs while waiting for the resolution of
uncertainty. Such costs are not necessarily exclusively financial and often include the
opportunity cost of the manager.
The project may be liquidated at any time for two reasons. First, the project
may be liquidated because the manager chooses to do so. Liquidating the project
allows the manager to stop paying out the flow of c at the cost of foregoing any
potential payoff of θ. Second, the project may be liquidated exogenously because
it gets exposed to a market-wide (or an industry-wide) shock that impacts many
firms at once. In such an event, the manager has no choice but to liquidate. The
market-wide shock arrives as a Poisson event with intensity µ > 0 and is publicly
observable. A given project is exposed to the market-wide shock with probability q.
Whether the project is in fact exposed to the shock or not is learned by the manager,
but not by the market, at the time of arrival of the shock.
To illustrate an industry-wide shock, consider again the example of the solar
energy industry. Although the physical outcome, electricity generation, is common
across myriad of solar energy projects, technologies differ widely, and often subtle
changes in technologies can produce large variation in payoff structure. A substantial
reduction in the production cost of solar panels – a negative industry-wide shock
– pushes some producers out of business. The propensity to fail often depends
on subtle variation in technology that is difficult for outsiders to decipher. The
managers are thus better equipped to learn whether such a shock negatively affects
their technology.
Assume that the entrepreneur cares about her reputation in the eyes of the outsiders. When a successful project pays off, all parties observe θ. For simplicity, we
assume that the reputation payoff upon the project payoff is γθ, where γ > 0 is
the parameter measuring the degree with which reputation-building incentives are
important. However, in the event of liquidation, outsiders do not directly observe θ.
Instead, at any time t prior to the payoff on an investment or a liquidation, outsiders
hold the conditional expectation E [θ|It ], where It is the past history that outsiders
8
know at time t. The reputation payoff upon liquidation at time t is therefore γE [θ|It ].
The reputation payoff is the reduced-form specification of the manager’s interactions with outsiders after the given project. For example, entrepreneurs interact
repeatedly with investors and, once their low ability is established, are less likely to
get new funding or have access to top VCs. Of course, more sophisticated specifications of reputation-building incentives are also possible but the simple specification
we employ captures succinctly the main idea behind reputation importance.
Before proceeding to find optimal decisions and equilibrium solution of our model,
let us identify the main economic mechanism we want to capture. Consider a manager
with a relatively low θ and suppose that the market-wide shock has not arrived yet.
She thinks about the following trade-off. If she exercises the liquidation option
immediately, she will save on cash outflows. However, the exercise decision will
reveal θ to the market which will lead to a drop in the reputation component of the
manager’s utility function. If there were no market-wide shock, this would simply
lead to a separating equilibrium with the single-jamming property (as in Grenadier
and Malenko (2011): every type delays the liquidation decision in the hope of fooling
the market into believing that her type is higher, but in equilibrium the outsiders
learn θ with certainty. However, with the market-wide shock the equilibrium will be
more interesting. Specifically, the entrepreneur may choose to delay optimally the
exercise decision until the market-wide shock arrives and liquidate at the arrival of
the shock. By doing this, she pools (blends in) with all types that liquidate at the
arrival of the market-wide shock including types with high θ. We later explore how
this blending in can yield a number of interesting stylized facts such as frequently
observed industry-wide investment busts.
2.1
Learning Process
As the success of the project is uncertain, an important state variable is the belief that
the project is successful. Let p (t) denote the posterior probability at time t that the
project is successful, given that the project has not paid off until time t. Over a short
period between t and t + dt, the project pays off with probability λdt, conditional
9
on being successful. Hence, the posterior probability p (t + dt) conditional on not
getting a payoff during [t, t + dt] is equal to:
p (t + dt) =
(1 − λdt) p (t)
.
(1 − λdt) p (t) + 1 − p (t)
(1)
Rewriting (1) and taking the limit dt → 0, we obtain the following differential equation:
dp (t) = −λp (t) (1 − p (t)) dt,
(2)
with the boundary condition
p (0) = p0 .
(3)
p0
.
(1 − p0 ) eλt + p0
(4)
The solution to (2) – (3) is:
p (t) =
The dynamics of the belief process is intuitive. As long as the project does not
pay off, the belief that the project is successful decreases continuously over time.
The speed of adjustment is proportional to λ, the intensity with which the successful
project pays off. Once the project pays off, the belief that the project is successful
jumps up to one. Note that because the private information of the manager does
not enter the belief evolution process (4), the manager and outsiders always share
the same belief about the success of the project.
2.2
Benchmark Case
As a benchmark, consider the case of symmetric information in which outsiders also
know θ. We break up the value function into two regions depending upon whether
or not the market shock has yet occured. Let Vb∗ (p) and Va∗ (p) denote the value
to the manager before and after the arrival of the market-wide shock, respectively,
where p is the current belief that the project is successful. Working backwards in
time, consider first the liquidation problem after the market-wide shock has occured.
By Itô’s lemma for jump processes (see, e.g., Shreve (2004b), in the range in which
10
liquidation is not optimal the evolution of Va∗ (p) is given by
dVa∗ (p) = −cdt − λp (1 − p) Va∗′ (p) dt + (θ + γθ − Va∗ (p)) dN (t) ,
(5)
where N (t) is a point process describing the arrival of the project payoff. In (5), the
first component reflects the cash outflow due to keeping the project afloat, the second
component reflects the change in expected value due to learning about potential
success of the project, and the last component reflects the change in expected value
if the project pays off (the manager receives the project payoff θ, reputation payoff
γθ, and foregoes the expected value of the project). Dividing (5) by dt, taking
the expectation, and equating it with rVa∗ (p), we obtain the following differential
equation:
(r + λp) Va∗ (p) = −c − λp (1 − p) Va∗′ (p) + λp (1 + γ) θ.
(6)
Let p̂a denote the threshold at which type θ finds it optimal to liquidate the project.
Equation (6) is solved subject to the following boundary conditions:
Va∗ (p̂a ) = γθ,
(7)
Va∗′ (p̂a ) = 0.
(8)
The value-matching condition (7) says that at the liquidation trigger point, the
manager receives her reputation payoff. The boundary condition (8 is a standard
smooth-pasting condition. Evaluating (6) at the point p̂a gives us
p̂a =
c + rγθ
.
λθ
(9)
The comparative statics of the liquidation trigger is intuitive: the liquidation
occurs earlier when the waiting cost is higher, the success intensity of the project is
lower, the interest rate is higher (i.e., the opportunity cost of waiting is higher), and
the reputation is more important (i.e., a higher utility from liquidating sooner and
getting on with other opportunities).
Before discussing the comparative statics with respect to ability, consider first
11
the problem of optimal liquidation before the arrival of the market-wide shock. The
evolution of Vb∗ (p) is given by:
dVb∗ (p) = −cdt − λp (1 − p) Vb∗′ (p) dt + (θ + γθ − Vb∗ (p)) dN (t)
(10)
+ [q (γθ − Vb∗ (p)) + (1 − q) (Va∗ (p) − Vb∗ (p))] dM (t) ,
where M (t) is a point process describing the arrival of the market-wide shock. Compared to (5), equation (10) includes an additional term reflecting the change in expected value due to the potential arrival of the market-wide shock. Upon arrival of
the market-wide shock, the project is exposed to it with probability q, in which case
forced liquidation takes place, so the change in expected value is γθ − Vb∗ (p) . With
probability 1 − q, the project is not exposed to the shock, in which case the change
in expected value is Va∗ (p) − Vb∗ (p). Dividing (10) by dt, taking the expectation, and
equating it with rVb∗ (p), we obtain the following differential equation:
(r + λp + µ) Vb∗ (p) = −c−λp (1 − p) Vb∗′ (p)+λp (1 + γ) θ+µ (qγθ + (1 − q) Va∗ (p)) .
(11)
Let p̂b denote the threshold at which type θ finds it optimal to liquidate the project,
prior to the arrival of the market shock. By analogy with (7) – (8), equation (11)
is solved subject to boundary conditions Vb∗ (p̂b ) = γθ and Vb∗′ (p̂b ) = 0. Evaluating
(11) at point p̂b and rearranging the terms gives us:
rγθ + c = λθp̂b + µ (1 − q) (Va∗ (p̂b ) − γθ) .
(12)
It is easy to see that p̂b = p̂a satisfies (12). It is also the unique solution, because
the right-hand side of (12) is strictly increasing in p̂b . Thus, the liquidation trigger is
the same both before and after the arrival of the market-wide shock. We can denote
this full-information trigger as p∗ (θ) = p̂a = p̂b . Intuitively, without information
frictions between the manager and outsiders, blending in is impossible and the only
consequence of the market-wide shock is to introduce noise in the liquidation process
by forcing the manager to liquidate the project excessively early if her project gets
12
exposed to the shock.
Although the liquidation thresholds are the same before and after the arrival of
the shock, the value functions are not. Because the market-wide shock may trigger
liquidation at a sub-optimal time, Vb∗ (p) < Va∗ (p) for any p above the liquidation
threshold.
Assuming that the starting point p0 is such that no type wants to liquidate
immediately (p0 > p∗ (θ)), the optimal liquidation strategy in the benchmark case
can be summarized in Proposition 1:
Proposition 1. Suppose that θ is known both to the manager and outsiders.
Then, the optimal trategy of the manager is to liquidate the project when p (t) decreases to trigger p∗ (θ). When the market-wide shock arrives, the project is liquidated
if and only if it is exposed to the shock.
Let us see how p∗ (θ) depends on θ:
c
d ∗
p (θ) = − 2 < 0.
dθ
λθ
(13)
We can see that the liquidation decision under symmetric information satisfies three
properties. First, the equilibrium liquidation strategy is optimal in the sense that
there is no other liquidation strategy that yields a higher expected payoff to the
manager. Second, higher types liquidate the project later. Intuitively, if the potential
payoff from the project is higher, then the manager optimally waits for a longer period
before scrapping it. Finally, the market-wide shock affects liquidation if and only if
the project is exposed to the shock. If the project is not exposed to the shock, then
the manager never liquidates it at the time when the shock arrives.
3
Private Information Case
In this section we provide the solution of the main case, in which there is asymmetric
information between the manager and outsiders. Even though there is no revelation
13
of θ when the project is liquidated, outsiders will try to infer θ from observing
the timing of liquidation. In turn, the manager will attempt to manipulate her
liquidation decision in order to shape the belief of outsiders. Because we assume
only one market-wide shock, we solve the model by backward induction. First, we
consider the optimal timing of liquidation after the market-wide shock has already
arrived. In this case, no future market-wide shock is possible, so the solution is quite
standard and follows Grenadier and Malenko (2011). Second, we move a step back
and consider the problem of liquidating the project at the time of the market-wide
shock. It turns out that many types will choose to liquidate projects together with
the market-wide shock, even when their projects are not exposed to the shock. We
call this property “blending in.” Finally, we consider the problem of liquidating the
project prior to the arrival of the market-wide shock.
In the first and third stages of the problem, our focus is on the separating equilibrium. Note, however, that there can also be pooling or semi-pooling equilibria. In
these equilibria, several types liquidate the project at the same instant even without
the market-wide shock due to self-fulfilling beliefs. There are two reasons for our
focus on the separating equilibrium. First, because we are mostly interested in how
market-wide shocks trigger waves of strategic liquidations, it is natural to abstract
from possible waves of liquidations in other periods. Second, even though pooling
and semi-pooling equilibria exist, they typically do not survive the D1 equilibrium
refinement.3 In this respect, focusing on separating equilibria is without loss of
generality.
3.1
Liquidation Decision after the Shock
Consider the decision of type θ to liquidate the project after the market-wide shock
has already arrived at some point in the past. In this case, no more market-wide
shocks can arrive, so the problem is relatively simple. Conjecture that the distribution of types after the market-wide shock is truncated at type θ̂ from below. This
3
See Cho and Sobel (1990) and Romey (1996) for the proof of this result for signaling models
with discrete types and a continuum of types, respectively.
14
conjecture is verified
the posterior
h iniSection 3.2. Then, belief of outsiders is that θ
is distributed over θ̂, θ̄ with p.d.f. f (θ) / 1 − F θ̂ .
Following Grenadier and Malenko (2011) we first solve for the manager’s liquidation strategy for a given inference function of outsiders, and then apply the rational
expectations condition that the liquidation strategy and the inference function must
be consistent with each other. Specifically, suppose that outsiders believe that type
θ liquidates the project at threshold p̄a (θ), which is a monotonic and differentiable
function of θ. Thus, if the manager liquidates the project
at threshold p̂, outsiders
infer that the manager’s type is θ̃ = p̄−1
a (p̂). Let Va p, θ̃, θ
denote the value to
the manager for a fixed belief of outsiders about the manager’s type θ̃, where p is
probability that the project is successful and θ is the true type of the manager. By
the standard valuationarguments
(e.g., Dixit and Pindyck (1994) ), in the region
prior to liquidation Va p, θ̃, θ must satisfy:
(r + λp) Va = −c − λp (1 − p)
∂Va
+ λp (θ + γθ) .
∂p
(14)
Suppose that the manager decides to liquidate the project at trigger p̂. Upon liquidation, the payoff to the manager is γ θ̂, implying the boundary condition
Va p̂, θ̃, θ = γ θ̃.
(15)
Solving (14) subject to boundary condition (15) yields the value to the manager for
a given liquidation threshold and the belief of the outsiders:
λr +1 1
Va p, θ̃, θ, p̂ = p
−1
U θ̃, θ, p̂
p
i
c
λp h c
− +
+ θ + γθ ,
r r+λ r
where
− λr −1 1 1
c
λp̂ c
U θ̃, θ, p̂ =
−1
+ γ θ̃ −
+ θ + γθ .
p̂ p̂
r
r+λ r
(16)
(17)
Given the hypothesized inference function p̄a , we can substitute p̄−1
a (p̂) for θ̃ in (16)
15
– (17) and take the first-order condition with respect to p̂:
dp̄−1 (p̂)
c + γ θ̃ (p̂) r − λp̂ θ + γθ − γ θ̃ (p̂) + λ (1 − p̂) p̂γ a
= 0.
dp̂
(18)
Equation (18) reflects the costs and benefits of postponing the liquidation. The first
two terms reflect the costs of waiting an additional instant: paying a cost c and
delaying the reputation payoff. The third term reflects the benefits of waiting an
additional instant: possibility that the project will pay off. Finally, the last term
reflects the effect of waiting an instant on the outsiders’ inference of the manager’s
type.
Applying the rational expectations condition that the liquidation strategy of the
manager must be perfectly revealing, we get the equilibrium differential equation:
h i
λγ p̄a (1 − p̄a )
dp̄a
=−
, θ ∈ θ̂, θ̄ .
dθ
c + rγθ − λp̄a θ
(19)
This equation must be solved subject to the initial value condition that the lowest
type liquidates at the symmetric information threshold. Intuitively, if the lowest type
did not liquidate at the symmetric information threshold, she would find it optimal
to deviate to the symmetric information threshold. This deviation not only would
improve the direct payoff from exercise but also could improve the reputation payoff,
since the current belief is already as bad as possible. Depending on the time at which
the market-wide shock arrives, there can be one of two cases:
1. If the shock occurs early relative to the unconstrained optimal liquidation
threshold of type θ̂, then the initial value condition is
c + rγ θ̂
p̄a θ̂ = p∗ θ̂ =
.
λθ̂
(20)
This
is the case if the market-wide shock arrives at time t such that p (t) >
∗
p θ̂ .
2. If the shock occurs late relative to the unconstrained optimal liquidation thresh16
old of type θ̂, then the initial value condition is pa θ̂ = p (t). This is the case
if the market-wide shock arrives at time t such that p (t) ≤ p∗ θ̂ .
In the appendix we show that this argument indeed
yields a unique separating equilibrium threshold p̄a (θ) by verifying that U θ̃, θ, p satisfies regularity conditions in
Mailath (1987). This result is summarized in Proposition 2 below:
Proposition 2. Let pa (θ) be the decreasing function
that solves
n differential
o equation (19) subject to the initial value condition p̄a θ̂ = min p∗ θ̂ , p (t) . Then,
pa (θ) is the liquidation threshold of type θ in the unique separating equilibrium.
It immediately follows that the equilibrium liquidation threshold after the arrival of the shock depends on the lowest type in this region, θ̂, and the posterior
probability of the project being successful right after the shock,
p. Thus, we denote
the equilibrium liquidation threshold in this section by p̄a θ, θ̂, p . We denote the
equilibrium value function, Va p, θ, θ, p̄a θ, θ̂, p
, by V̄a p, θ, θ̂ .
3.2
Liquidation Decision Upon the Arrival of the Shock
Consider the decision of type θ to liquidate the project upon the arrival of the marketwide shock. Analogously to the previous section, conjecture that the manager’s
timing of liquidation before the arrival of the market-wide shock is given by threshold
p̄b (θ), which is decreasing in θ. This conjecture is verified in Section 3.3. Then, the
posterior belief of outsiders upon the arrival of the market-wide shock is that θ is
distributed over θ̌, θ̄ with p.d.f. f (θ) / 1 − F θ̌ . Let p ≤ p∗ θ̌ denote the
posterior probability of the project being successful upon the arrival of the shock.
The conjecture that at the time of the shock p never exceeds p∗ θ̌ is also verified
in Section 3.3.
When the market-wide shock arrives, the project may be exposed to the shock
(with probability q) or not exposed to the shock (with probability 1 − q). In the
former case the manager has no choice but to liquidate the project. In the latter
17
case the manager may decide to liquidate the project strategically. If the manager
chooses to liquidate the project, she gets a reputation payoff that is independent
of her true type. If the manager chooses to postpone the liquidation decision, her
expected payoff is increasing in her true type. Hence, there exists type θ̂ ∈ θ̌, θ̄
such that the non-exposed project is liquidated if θ < θ̂ and not liquidated if θ > θ̂.
Provided that θ̂ < θ̄, there can be two scenarios depending on when type that is
marginally higher than θ̂ liquidates her project:
Scenario 1: Type θ̂ + ε, where ε is an infinitesimal positive value, liquidates the
project immediately after the market-wide shock. This happens if and only if:
p ≤ p θ̂ ,
∗
´ θ̂
´ θ̄
θf (θ) dθ + q θ̂ θf (θ) dθ
θ̌
= θ̂.
F θ̂ − F θ̌ + q 1 − F θ̂
(21)
(22)
Condition (21) states that the optimal liquidation time of type θ̂ is at the market-wide
shock or immediately after. Condition (22) states that type θ̂ is exactly indifferent
between pooling and liquidating at the time of the market-wide shock (the left-hand
side of (22)) and separating and liquidating immediately after the market-wide shock.
Scenario 2: Type θ̂ + ε, where ε is an infinitesimal
positive value, liquidates at
∗
the symmetric information threshold p θ̂ + ε . This happens if and only if:
p > p θ̂ ,
∗
´ θ̂
´ θ̄
θf (θ) dθ + q θ̂ θf (θ) dθ
1 ∗ θ̌
=
V p, θ̂ ,
γ a
F θ̂ − F θ̌ + q 1 − F θ̂
(23)
(24)
where
p, θ̂ is the value of the project to type θ in the symmetric information
framework, when the belief that the project is successful is p. Condition (23) states
Va∗
18
that the optimal liquidation time of type θ̂ is given by threshold p θ̂ . Condition
∗
(24) states that type θ̂ is indifferent between liquidating at the time of the market-
wide shock and waiting to liquidate at
her optimal separating threshold. Note that
because Va∗ p, θ̂ = γ θ̂ for all p ≤ p∗ θ̂ , condition (22) is a special case of condition
(24).4
Scenarios 1 and 2 imply that the equilibrium can have one of two forms. In the
first case, liquidation happens continuously with positive probability, while in the
second case, a wave of liquidations triggered by the market-wide shock is followed
by a period of no liquidation. The next proposition shows that there exists a unique
equilibrium threshold θ̂, it can be easily computed, and it is always strictly between
θ̌ and θ̄:
Proposition 3. There exists a unique equilibrium threshold θ̂, and θ̂ ∈ θ̌, θ̄ .
It is computed
in the following way. Letθ̂m denote the unique solution to ( 22). If
∗
p ≤ p θ̂m , then θ̂ = θ̂m . If p > p∗ θ̂m , then θ is determined as the unique
solution to ( 24).
The results of this section have two interesting implications. First, because θ̂ > θ̌,
the market-wide shock leads to liquidation of more projects than are affected by the
shock directly. Intuitively, the manager of a bad project has incentives to time
the liquidation decision in order to pool with better types, who have to liquidate
because of exposure to the market-wide shock. In other words, the manager liquidates
strategically at times when the liquidation decision is not very informative of the
manager’s ability.
Second, depending on whether the equilibrium belongs to scenario 1 or scenario
2, there can be different time-series properties of liquidation decisions. In the former
case, the market-wide shock triggers a wave of liquidations, but after the shock is
over the liquidation rate does not change compared to normal times. By contrast,
in the latter case, the market-wide shock triggers a wave of liquidations, which is
4
Conditions (24) and (22) implicitly assume that θ̌ < θ̂ < θ̄. If θ̂ = θ̌ or θ̂ = θ̄, then (24) and (22)
may hold as strict inequalities. In the proof of Proposition 3, we establish that indeed θ̂ ∈ θ̌, θ̄ .
19
followed by a quiet period of no liquidations. We discuss these implications in more
detail in Section 4.
Let V̄s p, θ, θ̌ denote the value of type θ when the shock arrives and before the
manager learns if her project is exposed to the shock. Then,
n
o
V̄s p, θ, θ̌ = qγ θ̂ θ̌ + (1 − q) max γ θ̂ θ̌ , V̄a p, θ, θ̂ θ̌
.
3.3
(25)
Liquidation Decision Before the Arrival of the Shock
To complete the solution of the model, consider the decision of type θ to liquidate
the project before the arrival of the market-wide shock. Similar to Section 3.1, we
solve for the separating equilibrium in two steps. First, we solve for the manager’s
liquidation strategy for a given inference function of outsiders. Second, we apply
the rational expectations condition that the inference function of outsiders must be
consistent with the liquidation strategy of the manager.
Suppose that outsiders believe that type θ liquidates the project at lower threshold p̄b (θ), which is a decreasing and differentiable function of θ. Hence, if the manager
liquidates the project at threshold p̂, outsiders
update their belief about the man−1
ager’s type to θ̃ (p̂) = p̄b (p̂). Let Vb p, θ̃, θ denote the value to the manager for
a fixed belief of outsiders θ, where p is the probability that the project is successful
and θ is the manager’s
true
type. Using the standard argument, in the region prior
to liquidation, Vb p, θ̃, θ satisfies the following differential equation:
∂Vb
(r + λp + µ) Vb = −c − λp (1 − p)
+ λp (θ + γθ)
∂p
+µV̄s p, θ, θ̃ (p) .
(26)
Suppose that the manager decides to liquidate the project at threshold p̂. Upon
liquidation, the payoff to the manager is γ θ̃ (p̂), implying the boundary condition
Vb p̂, θ, θ̃ = γ θ̃ (p̂) .
20
(27)
Equations (26) – (27) yield the value to the manager for a given liquidation threshold
p̂. The liquidation threshold that the manager chooses must satisfy the smoothpasting condition:
∂Vb
= γ θ̃′ (p̂) .
(28)
∂ p̂
Plugging (27) and (28) into (26) yields the following equation for the manager’s
liquidation threshold for a given outsiders’ inference function:
(r + λp̂ + µ) γ θ̃ (p̂) = −c − λγ p̂ (1 − p̂) θ̃′ (p̂) + µV̄s p̂, θ, θ̃ (p̂) .
(29)
Then, we can apply the rational expectations condition that the manager’s liquidation strategy must be consistent with outsiders’ inference function. Similar to Section 3.1, this condition implies θ̃ (p̂) = θ, and hence, p̂ = p̄b (θ) and θ̃′ (p̂) = 1/p̄b (θ).
Hence, the equilibrium liquidation threshold must satisfy
(r + λp̄b (θ) + µ) γθ = −c − λγ
p̄b (θ) (1 − p̄b (θ))
+ µV̄s (p̄b (θ) , θ, θ) .
p̄′b (θ)
(30)
Note that V̄s (p̄b (θ),θ,θ) is the value that type θ gets from the arrival of the market
wide shock, when she is the lowest type remaining. Because θ̂ θ̌ > θ̌, as established
in the previous section, when the market-wide shock arrives in this case, type θ will
always choose to liquidate the project, whether the project is exposed to the shock
or not. Therefore, V̄s (p̄b (θ),θ,θ) = γ θ̂ (θ), so we can write (30) as the following
equilibrium differential equation:
dp̄b
λγpb (1 − pb )
, θ ∈ θ, θ̄ .
=−
dθ
c + rγθ − λp̄ θ − µγ θ̂ (θ) − θ
(31)
b
Equation (31)
is similar
to Equation (19), but its denominator includes an additional
term −µγ θ̂ (θ) − θ . It reflects an additional incentive to delay liquidating due to
a potential arrival of the market-wide shock, which will allow the manager to blend
in with higher types.
Equation (31) is solved subject to the appropriate initial value condition. Anal21
ogous to Section 3.1, the appropriate initial value condition is that the lowest type
θ liquidates the project as if her liquidation decision did not reveal information to
outsiders:
c + rγθ − µγ θ̂ (θ) − θ
p̄b (θ) =
.
(32)
λθ
Note that the right-hand side of (32) is the most preferred threshold of type θ in
the symmetric information framework, taking payoff upon arrival of the market-wide
shock as given. The intuition behind (32) is simple. If the lowest type did not
liquidate at this threshold, she would find it optimal to deviate to it. Such deviation
not only would increase the direct payoff from liquidation but also could increase
the reputation payoff, because the current outsiders’ inference is already as low as
possible. Assuming that the prior on the project being successful p0 is such that
even the lowest type θ does not find it optimal to liquidate the project immediately,
we can summarize the results of this section
4. In the appendix we
in Proposition
prove the proposition by verifying that Ub θ̃, θ, p̂ satisfies regularity conditions in
Mailath (1987).
Proposition 4. Let pb (θ) be the decreasing function that solves differential equation (31) subject to the initial value condition (32). Then, pb (θ) is the liquidation
threshold of type θ in the unique separating equilibrium in the range before the arrival
of the market-wide shock.
Taken together, Propositions 2 – 4 imply that there exists a unique equilibrium, in
which different types separate via different liquidation strategies. At most times the
market-wide shock does not arrive, so different types liquidate projects at different
instances. However, on rare occasions when an external shock triggers liquidations
for an exogenous reason, a wave of liquidations occurs. Importantly, the wave exceeds
the fraction of firms affected by the shock: firms with bad enough projects choose
to blend in with the crowd and liquidate together with firms affected by the shock,
even though they are themselves not affected by the shock directly.
22
4
Model Implications
This section is to be completed.
In this section we analyze the economic implications of the model.
4.1
How Significant is Blending in with the Crowd?
The previous section implies that when liquidation decisions are informative and
the managers care about their reputation, there exists the blending with the crowd
effect: the managers have incentives to time their liquidation decisions to periods
when good projects are liquidated for an exogenous reason, such as the marketwide shock. It is interesting to see how significant the magnitude of this effect is.
Suppose that a fraction q of firms
is exposed
to
the
shock.Out
of firms that are not
exposed to the shock, fraction 1 − F θ̂ / F θ̂ − F θ̃
liquidates. To study
the importance of strategic liquidation, we introduce the magnification multiplier,
defined as the ratio of the probability that the project is liquidated at the time when
the market-wide shock arrives to the probability that the project is exposed to the
market-wide shock
M =
q + (1 − q) F
1−F (θ̂ )
(θ̂)−F (θ̃)
q
(33)
(1 − q) 1 − F θ̂
.
= 1+ q F θ̂ − F θ̃
If there is no blending with the crowd effect, as in the benchmark case in Section
2.2, then the magnification multiplier is equal to one. Otherwise, M > 1.
Figure 1 quantifies the magnification multiplier for different values of the fraction
of firms exposed to the shock, q, for the case of uniform distribution of types. It
appears that the magnification multiplier is very significant. For example, when the
project is exposed to the market-wide shock with probability 1%, the magnification
multiplier is around M = 10, meaning that for every exposed firm that liquidates
23
Figure 1: Plot M as a function of q
there are 9 firms that are not exposed to the shock but choose to liquidate for strategic
reasons. The magnification effect is even higher for smaller shocks. For example, if
the shock hits the firm with probability 0.1% (q = 0.001), then the magnification
multiplier is around M = 30. In general, the higher the size of the shock, the lower
the magnification effect.
Plot the magnification effect for scenario 1 vs. scenario 2. Intuition: in scenario
2 the firm has a valuable option to liquidate at the unconstrained optimal threshold,
which is likely to reduce the magnification effect. Thus, if the shock arrives early,
then the magnification effect is less significant.
4.2
Time-Series Patterns of Liquidations
Three distinct time-series effects that arise in our model:
Result 1: For any θ, p̄b (θ) < p̄a (θ, θ, p0 ). As a consequence, the probability of a
future market-wide shock incentivizes bad types to delay their liquidation decisions
(“the living dead”).
Result 2: However, it can be that pa (θ, θ, p) < pa θ, θ̂, p for some types. This is
always the case for types that are close to θ̂ under scenario 2 (when p > p∗ θ̂ , i.e.
if the market-wide shock arrives early). Intuitively, the worst type always liquidates
at the symmetric information threshold. Because the market-wide shock changes the
identity of the worst type, it increases the liquidation threshold (at least) for types
that are close to the new worst type. Thus, market-wide shocks have a positive effect:
without them, some types would delay their liquidation decisions even longer.
Result 3: in scenario 2 (when p > p∗ (θ)), some types speed up liquidation decisions compared to the case of no market-wide shock. Without pooling, they would
wait longer.
24
4.3
Welfare Effects of Market-Wide Shocks
Although it may seem that market-wide shocks are always detrimental because they
force liquidation of good projects and reduce information revelation due to the blending with the crowd effect, they actually have a positive effect of reducing the delay
in liquidation of some other projects. Without the economy-wide shock, costly liquidation delay could be even higher.
5
Conclusion
Real investment decisions can be substantially affected by the temptation of managers to blend in with the crowd and delay liquidation of unsuccessful projects. As we
show, this could have a far-reaching repercussions for the dynamics of whole industries and conceivably for the economy at large. The underlying mechanism is the link
between managerial reputation based on the available information about the current
project outcome and managerial payoffs from attracting future funding recourses and
future project cash flows. If liquidation of a current project reveals negative information about ability, the manager will optimally delay abandonment decisions to
make it more difficult for the market to judge managerial ability correctly. In a dynamic model, we show how this simple intuition can lead to amplification of modest
negative shocks and explain such phenomenon as industry-specific investment busts.
6
References
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Chung, Ji-Woong, Berk A. Sensoy, Lea H. Stern, and Michael S. Weisbach, 2010, Pay for Performance from Future Fund Flows: The Case of Private
Equity, Working Paper, Ohio State University.
25
Dixit, Avinash K., and Robert S. Pindyck, 1994, Investment under Uncertainty (Princeton University Press: Princeton, NJ).
Grenadier, Steven, and Andrey Malenko, 2011, Real Options Signaling
Games with Applications to Corporate Finance, Review of Financial Studies forthcoming.
Kothari, S., S. Shu, and P. Wysocki, 2009, Do managers withhold bad news?,
Journal of Accounting Research 47, 241–276.
Mailath, George J., 1987, Incentive compatibility in signaling games with a continuum of types, Econometrica 55, 1349–1365.
Miller, Gregory S., 2002, Earnings Performance and Discretionary Disclosure,
Journal of Accounting Research 40, 173–204.
Morellec, Erwan, and Norman Schurhoff, 2011, Corporate investment and
financing under asymmetric information, Journal of Financial Economics 99, 262–
288.
Ramey, Garey, 1996, D1 signaling equilibria with multiple signals and a continuum of types, Journal of Economic Theory 69, 508–531.
Shreve, Steven E., 2004, Stochastic Calculus for Finance II: Continuous-Time
Models (Springer-Verlag: New York, NY)
26
Appendix: Proofs
Proof of Proposition 2
We prove the proposition
by applying Theorems 1 - 3 from Mailath (1987). To do
this, we show that U θ̃, θ, p̂ satisfies regularity conditions from Mailath (1987).
h i2
2
Condition 1 (Smoothness): U θ̃, θ, p̂ is C on θ̂, θ̄ × (0, 1). It is straightforward to see that the condition is satisfied.
Condition 2 (Belief monotonicity): Uθ̃ θ̃, θ, p̂ never equals zero, and so is either
positive or negative. Differentiating (17) with respect to θ̃,
− λr −1
γ 1
−1
> 0.
Uθ̃ θ̃, θ, p̂ =
p̂ p̂
(34)
Hence, the belief monotonicity condition is
satisfied.
Condition 3 (Type monotonicity): Uθp̂ θ̃, θ, p̂ never equals zero, and so is either
positive or negative. Differentiating (17) with respect to θ,
− λr −1
1
λ (1 + γ)
Uθ θ̃, θ, p̂ = −
−1
.
p̂
r+λ
(35)
Differentiating (35) with respect to p̂,
Uθp̂
− λr −2
1+γ 1
θ̃, θ, p̂ = − 2
−1
< 0.
p̂
p̂
(36)
Hence, the type monotonicity condition is satisfied.
Condition 4 (“Strict” quasiconcavity): Up̂ θ̃, θ, p̂ = 0 has a unique solution in
p̂, which maximizes U θ̃, θ, p̂ , and Up̂p̂ θ̃, θ, p̂ < 0 at this solution. Consider the
27
derivative of (17) with respect to p̂ when θ̃ = θ
hc
i
r
λp̂
c
r r −1
−
−1
p̂ λ (1 − p̂) λ
+ γθ −
+ θ + γθ
Up̂ (θ, θ, p̂) =
λ
r
r+λ r
r
r
hc
i
r
λp̂
c
−
−2
+
+ 1 p̂ λ (1 − p̂) λ
+ γθ −
+ θ + γθ
(37)
λ
r
r+λ r
i
r
r
λ hc
−p̂ λ (1 − p̂)− λ −1
+ θ + γθ .
r+λ r
Equation Up̂ (θ, θ, p̂) = 0 has a unique solution in p̂, given by p∗ (θ), determined in
Section 2.2. The second derivative is
r
r
Up̂p̂ (θ, θ, p∗ (θ)) = −p∗ (θ) λ −1 (1 − p∗ (θ))− λ −2 θ < 0.
(38)
Hence, the “strict” quasiconcavity condition is satisfied.
Condition 5 (Boundedness): There exists δ > 0 such that for all (θ, p̂) ∈ θ, θ̄ ×
(0, 1) Up̂p̂ (θ, θ, p̂) ≥ 0 implies |Up̂ (θ, θ, p̂)| > δ. Differentiating Up̂ (θ, θ, p̂) with respect to p̂,
r
hr c
i
r
− 1 p̂ λ −2 (1 − p̂)− λ −2
+ γθ − p̂θ
λ
λh r
r
i
r
r
r c
−3
−
+
+ 2 p̂ λ −1 (1 − p̂) λ
+ γθ − p̂θ
(39)
λ
λ r
r
r
−p̂ λ −1 (1 − p̂)− λ −2 θ
r c
i
h r
r
r
− 1 + 3p̂
+ γθ − p̂θ − p̂ (1 − p̂) θ .
= p̂ λ −2 (1 − p̂)− λ −3
λ
λ r
Up̂p̂ (θ, θ, p̂) =
r
Condition Up̂p̂ (θ, θ, p̂) ≥ 0 is equivalent to
r
λ
− 1 + 3p̂ r c
+ γθ − p̂θ − p̂θ ≥ 0.
1 − p̂
λ r
(40)
If p̂ → 0, the left-hand side of (40) approaches λr − 1 λr rc + γθ < 0. If p̂ → 1,
the left-hand side of (40) approaches −∞ (this is because λr rc + γθ − θ < 0, since
p∗ (θ) < p0 < 1). Finally, if p̂ → p∗ (θ), the left-hand side of (40) approaches
28
−p∗ (θ) θ < 0. Therefore, (40) implies that
r
r
|Up̂ (θ, θ, p̂)| = p̂ λ −1 (1 − p̂)− λ −2 θ |p∗ (θ) − p̂|
(41)
>δ
for a low enough δ > 0. Thus, the boundedness condition is satisfied.
By Theorems 1 and 2 from Mailath (1987), any separating equilibrium liquidation
threshold p̄a (θ) is continuous, differentiable, satisfies Equation (19), and dp̄a (θ) /dθ
has the same sign as Uθp̂ . Because Uθp̂ < 0, the liquidation threshold p̄a (θ) is
decreasing in θ. To ensure that the decreasing solution to Equation (19) subject to
the initial value condition (20) is indeed the unique separating equilibrium, we check
the single-crossing condition:
Single-crossing condition: Up̂ θ̃, θ, p̂ /Uθ̃ θ̃, θ, p̂ is a strictly monotonic function of θ. The ratio of the two derivatives is equal to:
h i
r
r
−λ
−2 r
c
−1
λ
p̂
(1 − p̂)
+ γ θ̃ − p̂ θ − γ θ̃ − θ
Up̂ θ̃, θ, p̂
λ r
=
r
r
p̂ λ (1 − p̂)− λ −1 γ
Uθ̃ θ̃, θ, p̂
r
c
+
γ
θ̃
−
p̂
θ
−
γ
θ̃
−
θ
λ r
=
.
γ p̂ (1 − p̂)
(42)
Consider the derivative of Up̂ θ̃, θ, p̂ /Uθ̃ θ̃, θ, p̂ with respect to θ:
−
p̂ (1 + γ)
1+γ
=−
< 0.
γ p̂ (1 − p̂)
γ (1 − p̂)
(43)
Hence, the single-crossing condition is verified. By Theorem 3 in Mailath (1987), the
unique solution to (19) - (20) is indeed the separating equilibrium.
29
Proof of Proposition 3
h i
Let φ θ̂ denote the left-hand side of (22) and (24). Note that φ θ̂ , θ̃, θ̄ is
h i
a continuous function that takes values in θ̃, θ̄ . Hence, by Brower fixed point
theorem, it has a fixed point. Taking the derivative of f θ̂ ,
´
´
(1 − q) f θ̂ θ̂ q + (1 − q) F θ̂ − F θ̃ − θ̃θ̂ θf (θ) dθ − q θ̂θ̄ θf (θ) dθ
.
φ′ θ̂ =
2
q + (1 − q) F θ̂ − F θ̃
(44)
Hence, φ′ θ̂ has the same sign as θ̂ − φ θ̂ . Thus, at any fixed point φ′ θ̂ = 0.
′
The fixed point is unique, because any point at which φ θ̂ = 0 must be a lo cal minimum. To see this, note that according to (44) we can write φ′ θ̂ =
ψ θ̂ θ̂ − f θ̂ for some strictly positive and differentiable function ψ θ̂ . Hence,
φ′′ θ̂ = ψ ′ θ̂ θ̂ − φ θ̂ + ψ θ̂ 1 − φ′ θ̂ .
′′
′
Around point θ̂ : φ θ̂ = 0, φ θ̂ = ψ θ̂ > 0. Hence, any point φ θ̂ = 0 must
′
be a local minimum. Therefore, the fixed point is unique. In particular, this result
implies
that equation (21) has a unique solution in θ̂, which is also a point at which
φ θ̂ reaches its minimum.
1
Next, consider any p. It must be the case that γ Va p, θ̂ ≥ θ̂ with the strict
equality if and only if p ≤ p∗ θ̂ . To the left of θ̂m : φ′ θ̂m = 0, φ θ̂ is
a monotonically decreasing function of θ̂. Because Va p, θ̂ is a strictly increasing
function for any p taking values from V p, θ̃ to V p, θ̂m ≥ θ̂m , equation (24) has a
h
i
unique solution on θ̂ ∈ θ̃, θ̂m , provided that Va p, θ̃ ≤ Eθ̃ [θ]. If p > p∗ θ̂m , this
∗
solution is θ̂ < θ̂m . If p ≤ p θ̂m , this solution is θ̂ = θ̂m . The case Va p, θ̃ > Eθ̃ [θ]
corresponds to the case when no type wants to blend in with the crowd, because the
option to wait is too valuable to liquidate immediately even for the most optimistic
30
belief possible.
i
Finally, it remains to show that equation (24) has no root in the range θ̂ ∈ θ̂m , θ̄ .
Note that because φ′ θ̂ > 0 for any θ̂ > θ̂m , θ̂ > φ θ̂ for any θ̂ > θ̂m . However,
1
V
p, θ̂ ≥ θ̂ by definition of the option. Hence, equation (24) has no root in the
a
γ
i
range θ̂ ∈ θ̂m , θ̄ .
Proof of Proposition 4
We check that the value function in the pre-shock region satisfies regularity conditions
1 - 5 in Mailath (1987). Then, we apply Theorems 1 and 2 from Mailath (1987).
The general solution to (26) is
c
λp
c
+
+ (1 + γ) θ
Vb (p) = C (1 − p)
p
−
r+µ r+λ+µ r+µ
−1
ˆ p x r+µ
λ
V̄s x, θ, θ̃ (x)
r+µ+λ
r+µ
µ
+ (1 − p) λ p− λ
dx.
(45)
r+µ
λ
0
(1 − x) λ +2
r+µ+λ
λ
− r+µ
λ
To pin down constant C, we use boundary condition (27). This gives us the following
value to the manager for a given liquidation threshold and inference function:
Vb p, θ̃, θ, p̂
r+µ
+1
λ
1
= p
−1
Ub θ̃, θ, p̂
p
c
λp
c
+
+ (1 + γ) θ
−
r+µ r+λ+µ r+µ
−1
ˆ x x r+µ
λ
V̄s x, θ, θ̃ (x)
r+µ+λ
r+µ
µ
+ (1 − p) λ p− λ
dx,
r+µ
λ
0
(1 − x) λ +2
31
(46)
where
− r+µ
ˆ
−1 1 1
λ
c
λp̂
c
µ p̂ x
Ub θ̃, θ, p̂ =
−1
+ γ θ̃ −
+ (1 + γ) θ −
p̂ p̂
r+µ
r+λ+µ r+µ
λ 0
(47)
Note that Ub θ̃, θ, p̂ is analogous to U θ̃, θ, p̂ except for two differences: (i) r + µ
instead of r; (ii) there is an additional term. The last term integrates over values
of θ̃ (p) , taking the functional form as given, while γ θ̃ denotes a specific value of
θ̃ ∈ θ, θ̄ .
Now we can check that Ub θ̃, θ, p̂ satisfies conditions from Mailath (1987):
h i2
2
Condition 1 (Smoothness): Ub θ̃, θ, p̂ is C on θ̂, θ̄ × (0, 1). The condition
is satisfied provided that the integral in the
is smooth.
last term
Condition 2 (Belief monotonicity): Ubθ̃ θ̃, θ, p̂ never equals zero, and so is either
positive or negative. Differentiating Ub θ̃, θ, p̂ with respect to θ̃:
− r+µ
−1
γ 1
λ
−1
> 0.
Ubθ̃ θ̃, θ, p̂ =
p̂ p̂
(48)
Hence, the belief monotonicity condition is satisfied.
Condition 3 (Type monotonicity): Ubθp̂ θ̃, θ, p̂ never equals zero, and so is either
positive or negative. Differentiating Ub θ̃, θ, p̂ with respect to θ,
− r+µ
−1
λ
λ (1 + γ)
1
−1
Ubθ θ̃, θ, p̂ = −
p̂
r+µ+λ
r+µ
ˆ p̂ x λ −1 V̄sθ x, θ, θ̃ (x)
µ
−
dx.
r+µ
λ 0
(1 − x) λ +2
(49)
Differentiating (49) with respect to p̂,
r+µ
µ
− r+µ
−2
1 + γ + V̄sθ p̂, θ, θ̃
Ubθp̂ θ̃, θ, p̂ = −p̂ λ (1 − p̂) λ
.
λp̂
32
(50)
r+µ
−1
λ
V̄s x, θ,
(1 − x)
r+µ
λ
Because V̄s p̂, θ, θ̃ is non-decreasing in θ, Ubθp̂ θ̃, θ, p̂ < 0. Therefore, the type
monotonicity condition is satisfied.
Condition 4 (“Strict” quasiconcavity): Ubp̂ (θ, θ, p̂) = 0 has a unique solution in
p̂, which maximizes
Ub(θ, θ, p̂), and Ubp̂p̂ (θ, θ, p̂) < 0 at this solution. Consider the
derivative of Ub θ̃, θ, p̂ with respect to p̂:
− r+µ
−1 λ
λp̂
c
1 1
c
Ubp̂ θ̃, θ, p̂ = − 2
−1
+ γ θ̃ −
+ (1 + γ) θ
p̂ p̂
r+µ
r+λ+µ r+µ
− r+µ
−2
λ
r+µ
1
1
c
λp̂
c
+
+1
−1
+ γ θ̃ −
+ (1 + γ) θ
λ
p̂
p̂3 r + µ
r+λ+µ r+µ
− r+µ
−1
λ
λ
1 1
c
−1
+ (1 + γ) θ
(51)
−
p̂ p̂
r+λ+µ r+µ
r+µ
−1 ¯
λ
Vs p̂, θ, θ̃
µ p̂
.
−
+2
λ (1 − p̂) r+µ
λ
We can rewrite this as
r+µ
− r+µ
−1
−1
λ
λ
Ubp̂ θ̃, θ, p̂ = −p̂
(1 − p̂)
λp̂
c
c
+ γ θ̃ −
+ (1 + γ) θ
r+µ
r+λ+µ r+µ
r + µ + λ r+µ −1
c
λp̂
c
−2
− r+µ
λ
+
p̂ λ
(1 − p̂)
+ γ θ̃ −
+ (1 + γ) θ
λ
r+µ
r+λ+µ r+µ
(52)
r+µ
r+µ
r+µ
λ
c
µ r+µ
+ (1 + γ) θ − p̂ λ −1 (1 − p̂)− λ −2 V̄s p̂, θ, θ̃ (p̂) .
−p̂ λ (1 − p̂)− λ −1
r+λ+µ r+µ
λ
When θ̃ = θ,
Ubp̂ (θ, θ, p̂) = p̂
where
A=
r+µ
−1
λ
(1 − p̂)−
r+µ
−2
λ
× A,
c + rγθ
µ
− p̂θ −
V̄s (p̂, θ, θ) − γθ .
λ
λ
(53)
(54)
Note that V̄s (p̂, θ, θ) is weakly increasing in p̂. Therefore, the right-hand side of (54)
33
is strictly decreasing in p̂. Hence, equation Ubp̂ (θ, θ, p̂) = 0 has the unique solution
p∗2 (θ) =
c + rγθ − µγ θ̂ (θ) − θ
λθ
(55)
.
The second derivative of Ub (θ, θ, p̂)with respect to p̂ at p∗2 (θ) is equal to
Ubp̂p̂ (θ, θ, p∗2 (θ)) = −p∗2 (θ)
r+µ
−1
λ
(1 − p∗2 (θ))−
r+µ
−2
λ
µ
θ + V̄sp̂ (p∗2 (θ) , θ, θ) .
λ
(56)
Because θ̂ (θ) > θ, as established in Section 3.2, in the neighborhood of p∗2 (θ)
V̄s (p̂, θ, θ) is equal to θ̂ (θ). Therefore, V̄sp̂ (p∗2 (θ) , θ, θ) = 0. Hence,
Ubp̂p̂ (θ, θ, p∗2 (θ)) = −p∗2 (θ)
r+µ
−1
λ
(1 − p∗2 (θ))−
r+µ
−2
λ
(57)
θ < 0.
Therefore, the “strict” quasiconcavity condition is satisfied.
Condition 5 (Boundedness): There exists δ > 0 such that for all (θ, p̂) ∈ θ, θ̄ ×
(0, 1) Ubp̂p̂ (θ, θ, p̂) ≥ 0 implies |Ubp̂ (θ, θ, p̂)| > δ. Differentiating (53) with respect to
p̂,
Ubp̂p̂ (θ, θ, p̂) = p̂
r+µ
−1
λ
(1 − p̂)
−2
− r+µ
λ
r + µ − λ r + µ + 2λ
+
λp̂
λ (1 − p̂)
µ
× A − θ − V̄sp̂ (p̂, θ, θ) .
λ
(58)
Condition Up̂p̂ (θ, θ, p̂) ≥ 0 is equivalent to
r + µ − λ r + µ + 2λ
+
λp̂
λ (1 − p̂)
×A−θ−
µ
V̄sp̂ (p̂, θ, θ) ≥ 0.
λ
If p̂ → 1, the left-hand side of (59) approaches −∞. This is because
r
λ
(59)
c
r
+ γθ − θ <
0, since p∗ (θ) < p0 < 1. If p̂ → p∗2 (θ), the left-hand side of (59) approaches −θ < 0.
Finally, if p̂ → 0, the left-hand side of (59) approaches −∞, if r + µ > λ, ∞, if
r + µ < λ, and −θ − µγ
θ̂ (θ) < 0, if r + µ = λ.
λ
First, consider the case r +µ ≥ λ. By the argument above, there exists a constant
∆ > 0 such that any p̂ at which Ubp̂p̂ (θ, θ, p̂) ≥ 0 must satisfy p̂ > ∆, p̂ < 1 − ∆, and
34
|p̂ − p∗2 (θ)| > ∆. Therefore, for any p̂ at which Ubp̂p̂ (θ, θ, p̂) ≥ 0:
|Ubp̂ (θ, θ, p̂)| = p̂
r+µ
−1
λ
(1 − p̂)−
r+µ
−2
λ
θ |p∗2 (θ) − p̂|
(60)
>δ
for a low enough δ > 0. Therefore, the boundedness condition is satisfied in this
case. Second, consider the case r + µ < λ. Analogously to the case r + µ ≥ λ, any
p̂ bounded away from zero at which Ubp̂p̂ (θ, θ, p̂) ≥ 0 must satisfy |Ubp̂ (θ, θ, p̂)| > δ
for sufficiently low positive δ. Therefore, to verify the boundedness condition in this
case it is sufficient to check that |Ubp̂ (θ, θ, p̂)| > δ if p̂ is arbitrarily close to zero.
When p̂ → 0, |Ubp̂ (θ, θ, p̂)| converges to
lim
p̂→∞
θp∗2 (θ)
p̂
λ−r−µ
λ
= ∞.
(61)
Therefore, the boundedness condition is satisfied in the case r + µ < λ too.
By Theorems 1 and 2 from Mailath (1987), any separating equilibrium liquidation
threshold p̄b (θ) is continuous,
differentiable,
satisfies
Equation
(31), and dp̄b (θ) /dθ
has the same sign as Ubθp̂ θ̃, θ, p̂ . Because Ubθp̂ θ̃, θ, p̂ < 0, the liquidation threshold p̄b (θ) is decreasing in θ. To ensure that the decreasing solution to Equation (31)
subject to the initial value condition (32) is indeed the unique separating equilibrium,
we check the single-crossing condition:
Single-crossing condition: Ubp̂ θ̃, θ, p̂ /Ubθ̃ θ̃, θ, p̂ is a strictly monotonic function of θ. The ratio of the two derivatives is equal to:
Ubp̂ θ̃, θ, p̂
1
c
λp̂
c
=−
+ γ θ̃ −
+ (1 + γ) θ
γ p̂ r + µ
r+λ+µ r+µ
Ubθ̃ θ̃, θ, p̂
r+µ+λ
1
c
λp̂
c
+
+ γ θ̃ −
+ (1 + γ) θ
γλ
p̂ (1 − p̂) r + µ
r+λ+µ r+µ
(62)
λ
c
µ
1
−
+ (1 + γ) θ −
V̄s p̂, θ, θ̃ (p̂) .
γ (r + λ + µ) r + µ
γλ p̂ (1 − p̂)
35
Consider the derivative of Ubp̂ θ̃, θ, p̂ /Ubθ̃ θ̃, θ, p̂ with respect to θ:
1
−
γ (1 − p̂)
µ
1 + γ + V̄sθ p̂, θ, θ̃ (p̂)
< 0.
λp̂
(63)
Hence, the single-crossing condition is verified. By Theorem 3 in Mailath (1987) the
unique solution to (31) - (32) is indeed the separating equilibrium.
Proof of Results from Section 4.2
Proof of Result 1. Noting that µ > 0 and θ̂ (θ) > θ, initial value conditions (20)
and (32) imply that p̄b (θ) < p̄a (θ, θ, p0 ). Consider any point θ at which p̄b (θ) =
p̄a (θ, θ, p0 ). Equations (19) and (31) imply that p̄aθ (θ, θ, p0 ) > p̄′b (θ), that is, p̄b (θ)
decreases faster than p̄a (θ, θ, p0 ). Hence, p̄b (θ) < p̄a (θ, θ, p0 ).
n o
Proof of Result 2. Initial value condition (20) implies that p̄a θ̂, θ̂, p = min p∗ θ̂ , p .
∗
∗
In particular, if p ≥ p θ̂ , then p̄a θ̂, θ̂, p = p θ̂ . However, equation (19) im
plies that p̄a θ̂, θ, p < p∗ θ̂ for all θ̂ > θ: whenever p̄a (θ, θ, p) approaches p∗ (θ),
p̄aθ (θ, θ, p) approaches −∞. Therefore, if p ≥ p∗ θ̂ , there exists type θ̂′ > θ̂ , such
h
i
′
that p̄a θ̂, θ̂, p > p̄a θ̂, θ, p for all θ ∈ θ̂, θ̂ .
Proof of Result 3. Without the market-wide shock, the liquidation threshold
of
any type θ > θ, denoted p̄n (θ) would be below p∗ (θ). However, if p ≥ p∗ θ̂ , then
∗
type θ̂ (θ) liquidates at the symmetric information threshold: p̄a θ̂, θ̂, p = p θ̂ .
h
i
By continuity, there exists type θ̂′′ , such that p̄a θ, θ̂, p > p̄n (θ) for all θ ∈ θ̂, θ̂′ .
36