Optimal project termination with an informed agent
Erik Madsen∗
January 7, 2016
Abstract
Economic decision-makers commonly face the problem of determining when to terminate a project whose state, monitored imperfectly, shifts over time. I analyze the
optimal provision of dynamic incentives in an agent-assisted version of this problem.
A firm must decide when to halt a project which yields a stochastic profit flow in continuous time but eventually becomes unprofitable; the firm is aided by an agent who
possesses a superior private monitoring technology, limited liability, and incentives
for delayed project termination. I develop techniques for deriving the firm’s optimal
contract, which involves occasional inefficient early project termination and a golden
parachute that is sensitive to public news and declines with project length. Central
to my analysis is a duality result linking payment and termination variables in an optimal contract, closely analogous to the equivalence of price and quantity as control
variables in monopoly theory. I analyze both the “price” and “quantity” approaches
to optimization, which yield complementary insights into features of the optimal contract. In particular, the latter approach maps the economic problem with an agent
onto a single-person decision-making problem exhibiting an artificial wedge between
the signal-to-noise ratio of the public news process and the firm’s expected profit flow.
∗
Graduate School of Business, Stanford University. Email: emadsen@stanford.edu. I am deeply indebted
to my advisers Andrzej Skrzypacz, Robert Wilson, and Sebastian Di Tella for the essential guidance they
have contributed to the crafting of this paper. I am especially grateful to Andrzej Skrzypacz for the time and
tireless energy he has devoted at critical junctures. I also thank Mohammad Akbarpour, Doug Bernheim,
Jeremy Bulow, Gabriel Carroll, Glenn Ellison, Ben Golub, Brett Green, Johannes Hörner, Chad Jones,
Nicolas Lambert, Edward Lazear, Roger Myerson, Michael Ostrovsky, Takuo Sugaya, Juuso Toikka, and
Jeffrey Zwiebel for valuable discussions and feedback.
1
1
Introduction
Economic decision-makers commonly face the problem of determining when to terminate a
project whose state, which is monitored imperfectly, deteriorates over time. A classic example
is a firm operating a depreciating plant that must eventually be retired. The depreciation
can take various forms, such as a decline in output quality or an increase in breakdowns. A
key feature of this environment is that the firm cannot directly observe whether the plant
has depreciated, but instead sees only a noisy indicator of the current state. For instance, it
might observe the quality of individual units of output; the market price of its output when
demand fluctuates exogenously; or the incidence of individual breakdowns when malfunctions
arrive randomly over time. Another example is a board of directors deciding when to replace
a CEO if firm performance indicates she is no longer a good match for the firm. In all these
cases the decision-maker faces an optimal stopping problem of deciding when to terminate
the project based on the history of observed performance. An optimal policy must balance
the incentive to stop based on current expectations of the project state against the option
value of continuing operation and gathering more information.
The single-person problem with a binary state and a single state transition has been
well-studied in operations research as the problem of “quickest detection” or “change-point
detection”; see, e.g., Peskir and Shiryaev (2006) for a textbook treatment. However, in
practice many monitoring tasks in economics are accomplished with the assistance of an
agent who brings additional expertise to the problem. For instance, the plant owner discussed
earlier might hire a manager to oversee the plant’s operation, who by virtue of experience and
proximity maintains direct knowledge of the plant’s state. Similarly, the board of directors
may ask the CEO to help make the decision as to when she should be replaced.
In such cases the firm naturally wishes to incorporate the agent’s superior information in
deciding when to terminate the project. However, agents attached to lucrative projects often
have misaligned incentives and prefer to prolong their involvement as long as possible, for
instance due to empire-building concerns, on-the-job perquisites, or job search costs following
termination. The presence of such incentives substantially complicates optimal elicitation
of the agent’s private information. As the agent must be compensated for revealing news
which instigates project termination, the firm may instead simply choose to terminate the
project early to economize on payments. In this paper I study the optimal incorporation of
an agent’s information into the quickest detection problem when the project’s state is binary
and transitions exactly once from good to bad.
The form of an optimal contract is not a priori obvious because the firm must incentivize
2
truthful reporting by agents in each state, a problem which generally requires designing
the paths of two continuation utility state variables. On top of this, the agent’s incentive
constraint in the bad state is complex due to the possibility of double deviations. If the
agent decides to withhold his knowledge of the state change at time t, he need not report it
a moment later; instead he has many possible delayed reporting strategies, whose profitability
depend on non-local features of the contract. It is therefore not even clear that continuation
utilities alone constitute sufficient state variables for the problem.
Despite the complexity of the contracting problem, I find that an optimal termination
policy takes the simple form of a threshold rule in “virtual beliefs.” At each moment the
firm asks the agent to report on the current state of the project, but updates its beliefs
about whether the state has transitioned as if no agent were present. It then terminates the
project the first time either the agent reports that the state has transitioned or the firm’s
virtual beliefs drop below a (time- and history-independent) threshold. This threshold is
strictly lower than the optimal threshold in the firm’s problem without an agent. The firm
is therefore able to partially utilize the agent’s information, fully eliminating late terminations and partially mitigating early terminations that it would have incurred in the agent’s
absence. The optimal threshold also satisfies clean comparative statics: it is decreasing in
the informativeness of news about the state, increasing in the rate of state transitions, and
increasing in the severity of the agent’s incentive misalignment.
Optimal payments take a similarly simple form: all payments are deferred until project
termination, at which point the agent receives a lump sum “golden parachute.” If the agent
reports a state switch, the optimal golden parachute is exactly the expected discounted
stream of project rents he would have received by never reporting the switch. On the other
hand, if the project is terminated before a reported state switch he is paid nothing. The
dynamics of the optimal golden parachute may be characterized by a one-dimensional HJB
equation which determines how fast the parachute drifts down over time and how sensitively
the parachute reacts to news about output.
A crucial technical tool in my analysis is a duality result tightly linking payments and
termination times in an optimal contract. These two contractual elements function as “dual
variables”: either may be eliminated in favor of the other, reducing the analysis to design
of a single contractual element. As the two variables control the cost of implementing the
contract and the amount of output produced, respectively, this duality is closely analogous
to the equivalence between price and quantity as control variables in monopoly theory. I
pursue both the “price” and “quantity” approaches to deriving an optimal contract. The
price approach is equivalent to a recursive analysis using a particular continuation utility as
3
a state variable, and is therefore the familiar route for practitioners of dynamic contracting.
However, this characterization yields an HJB equation with a non-standard free boundary
condition at a singularity of the ODE, a technically challenging feature not present in many
conventional dynamic contracting problems. The quantity approach, by contrast, turns out
to be quite tractable and powerful in my setting. I therefore demonstrate the value of the
duality approach to contractual design in an economically interesting setting.
1.1
Related literature
My paper sits closest to the literatures on dynamic contracting under moral hazard and
dynamic mechanism design, sharing the continuous-time setting and many analytical tools
with the former and the basic information elicitation problem of the latter. I briefly survey
each literature and its connection to my model and results.
A large literature studies the optimal provision of dynamic incentives for effort under
repeated moral hazard. Spear and Srivastava (1987) formulate the standard infinite-horizon
recursive problem in discrete time, while Sannikov (2008) and Williams (2008) study the
problem in continuous time when output follows a Brownian motion with drift, affording
considerable tractability compared to the discrete-time case. The continuous-time model
has been extended in various ways. Williams (2008, 2015) permits general forms of hidden
savings which impact the agent’s marginal utility of consumption. Sannikov (2014) assumes
actions have long-run impacts on output. Prat and Jovanovic (2013) build in symmetric ex
ante uncertainty about the quality of the agent, as in the career concerns literature. Cvitanic
et al. (2013) suppose instead that the agent possesses ex ante private information about his
marginal cost of effort, a setting which may lead the principal to offer a menu of contracts
to separate or screen agents of different types as in mechanism design.
These papers solve the optimal contracting problem using recursive techniques with the
agent’s continuation utility as a state variable and eliminate effort from the problem using
a first-order condition. Their basic technique is therefore in the same spirit as the price
approach of this paper, although with important distinctions. First, in moral hazard models
the agent’s continuation utility process is a martingale on-path, which is not true of the
relevant continuation utility in my model; so additional representation results are required to
analyze my problem recursively. Also, while in moral hazard models the incentive constraint
for effort is formulated naturally in terms of the local dynamics of continuation utility, an
analogous result for the binding incentive constraint in my model cannot be formulated for
general contracts and takes care to establish even for a very restricted class of candidate
4
contracts. Finally, the papers cited focus their efficiency analysis on consumption smoothing
and the wedge between induced and first-best effort rather than the incidence of early project
termination, which is the key inefficiency in my model.
Another paper in this literature, DeMarzo and Sannikov (2006), simplifies the moral
hazard problem by replacing effort with the ability to steal output inefficiently. This is
equivalent to assuming marginal cost of effort is constant and less than 1, so efficient effort
is always implementable and optimal for the principal. They then tightly characterize the
form of an optimal contract in the presence of hidding savings. They show that an optimal
contract takes the form of a constant-limit credit line which the agent can draw down at
will, with the caveat that the project is terminated when the credit line is exhausted. This
dynamic has close similarities to the optimal payment process in my paper, which features
a golden parachute that can be taken by the agent at any time and which when exhausted
due to poor project returns leads to termination. Further, their contracting problem boils
down to the one-dimensional choice of an optimal credit limit, just as my problem can be
reduced to the choosing of a one-dimensional threshold in belief space.
Another literature studies dynamic mechanism design with transfers when agents have
private payoff-relevant types which fluctuate over time. Baron and Besanko (1984) and
Courty and Li (2000) consider two-period problems with a single agent, while Besanko (1985)
and Battaglini (2005) study infinite-horizon settings in discrete time with one agent and
special type processes. Pavan et al. (2014) extend these results to an infinite horizon discretetime setting with many agents and general type processes. Williams (2011) analyzes an
infinite-horizon problem in continuous time with one agent whose type evolves as an OrnsteinUhlenbeck process, yielding additional tractability compared to the general discrete-time
problem. In this set of papers payments are eliminated from the principal’s objective function
in favor of allocations using an envelope theorem or other argument, yielding a virtual profit
function which is maximized to obtain an optimal allocation. Their analysis is therefore
analogous to the quantity approach in this paper. However, in my model the agent’s type is
payoff-relevant only to the principal, in contrast to the mechanism design convention where
type captures payoff-relevant information for the agent. Also unlike the mechanism design
literature, I assume that the agent has limited liability, so it is not feasible to charge the
agent for the private benefits he receives or to “sell him the firm.” Moreover, in my paper
the firm observes exogenous signals and uses them to resolve the balance between inefficient
early termination and paying the agent information rents.
Recently, Garrett and Pavan (2012) have studied a contracting problem with both repeated moral hazard and time-varying private information. In their model output is a
5
function of both effort and match value, match value is privately observed by the agent and
changes over time, and the agent can be replaced. As in this paper, they are primarily
interested in characterizing the firm’s optimal termination policy. Also in common with
my model, their principal observes output which is correlated with the current state conditional on effort. Unlike my model, their agent does not intrinsically value association with
the project. Moreover, given his lack of limited liability he cannot extract rents from the
moral hazard problem. This crucial distinction leads to starkly different optimal termination dynamics, which in particular do not depend on the history of project output and can
eventually become unresponsive to bad news from the agent.
My paper also connects to recent work in the dynamic delegation literature. Guo (2015)
studies an environment with public experimentation by an agent who has private time-zero
beliefs about the true state and a bias toward experimentation, when no transfers are permitted. In her setting the interests of the firm and the agent are not completely misaligned,
because after enough bad news even the agent prefers to stop experimentation. Thus it is
possible to elicit information from the agent without transfers. In contrast, in my setup
the agent prefers to continue the project in all cases, so payments are necessary to obtain
any information from him. Despite the differences in our settings, she derives an optimal
policy strikingly similar to mine: the principal delegates the right to stop experimentation
to the agent until a virtual belief about the state, which is updated as if the agent weren’t
present, reaches a threshold, after which experimentation is cut off forever. Grenadier et
al. (2015) consider a principal who elicits an agent’s information about the optimal exercise
time of a real option, when the agent has a bias for late exercise exactly equivalent to the
flow of benefits specification of my model. Unlike my model, there are no transfers, the
agent knows the optimal exercise date at time zero, and the principal can commit only the
delegation of exercise authority. Still, they predict an outcome with distinct similarities to
the optimal policy in my model: the principal follows the agent’s exercise recommendation
until a threshold in the value of the underlying asset is reached, at which point the option
is exercised immediately.
2
2.1
The model
The environment
A firm possesses a project with an uncertain lifespan and must decide when to irreversibly
scrap the project. The project’s lifespan τ θ is unobserved by the firm, who has prior beliefs
6
that τ θ is exponentially distributed with rate parameter α. I let θt denote the state of the
project at time t, with θt = G if t < τ θ and θt = B otherwise. Then from the perspective of
the firm θ is a hidden Markov process with initial state θ0 = G and transition rate α from G
(the “good state”) to B (the “bad state”). The project generates average profits rθ per unit
time when the current state is θ, with rG > 0 > rB and ∆r ≡ rG − rB denoting the spread
in average profits between the two states. However, variability in instantaneous flow profits
prevent the firm from immediately detecting a state transition.
For most of this paper, I study a project which generates observable profits in Brownian
increments. Letting Yt be cumulative profits up to date t, I assume the stochastic process Y
has increments
dYt = rθt dt + σ dZt ,
where Z is a standard Brownian motion independent of τ θ . (It is also possible to analyze
variants of the model in which the state of the project impacts the arrival rate of Poisson
profit shocks rather than the drift rate of profits. The qualitative conclusions are very similar
to the Brownian case.)
Mathematically, I model this setting with a canonical probability space (Ω, Σ, P) sufficiently rich to admit Z and τ θ , with E the expectation operator under P. I let FY = {FtY }t≥0
denote the natural filtration generated by Y ; this filtration captures the information available
to the firm based on its observation of past project profits. I write EYt for the conditional
expectation under P wrt FtY . I also let F = {Ft }t≥0 denote the natural filtration generated
by both Y and θ. Finally, I define PG and PB to be the equivalent probability measures
under which ZtG ≡ σ1 (Yt − rG t) and ZtB ≡ σ1 (Yt − rB t), respectively, are standard Brownian
B
motions, and write EG
t and Et for the conditional expectations under these measures wrt
FtY .
The firm is a risk-neutral expected-profit maximizer with discount rate ρ. Supposing the
firm operates the project until some (FY -stopping) time τ Y , it receives expected profits
"Z
Π=E
0
#
τY
e−ρt dY
"Z
τY
#
e−ρt (πt rG + (1 − πt )rB ) dt ,
=E
(1)
0
where πt = PYt {θt = G} = P{θt = G | FtY } are the firm’s posterior beliefs at time t about
the state of the project.
The firm may hire an agent to oversee the project and monitor its state. The agent is
an expert who costlessly and privately observes the state process θ. (He cannot, however,
observe τ θ in advance.) That is, the filtration F captures the information available to the
7
agent at any time. The agent also faces an incentive problem: he receives flow rents b > 0
per unit time while the project operates, regardless of its state.1 I assume that rB + b < 0,
so that it is jointly unprofitable for the firm to run the project in the bad state.2 The agent
possesses limited liability and has no initial wealth, so cannot be sold the project. He is
risk-neutral and possesses the same discount rate ρ as the firm.
2.2
Contracts
The firm commits in advance to a contract eliciting reports from the agent over time and
specifying a termination policy τ and a cumulative payment function Φ as a function of
the public history of output and the agent’s reports. By the revelation principle, I restrict
attention to contracts which at each moment in time elicit a report θet ∈ {G, B} of the current
state from the agent. Equivalently, as the only information to communicate is the time of a
state switch, the firm asks the agent to make a single report at the time of a state switch.
This simplification leads to a formulation of the contractual setting which is formally very
similar to a static mechanism design problem.
Definition 1. A revelation contract C = (Φ, τ ) consists of a family of real-valued stochastic
processes Φt and stopping times τ t for each t ∈ R+ ∪ {∞} such that:
• Each Φt is FY -adapted, non-decreasing, right-continuous, and satisfies Φt0 = 0 and
Φts = Φtτ t for s > τ t ,
• Each τ t is an FY -stopping time,
0
0
• For every t and t0 > t, Φts = Φts and 1{τ t ≥ s} = 1{τ t ≥ s} for s < t.
Intuitively, a revelation contract specifies a payment Φt and an allocation τ t as a function
of the reported type t, as in static mechanism design. However, in this setting payments and
allocations are dynamic variables that can be conditioned on output history observed both
before and after the agent’s report. Definition 1 enforces an appropriate form of dynamic
consistency for the setting, by requiring that the contract not “look ahead” and anticipate
the timing of a future report when making current payment and termination decisions. The
1
None of the results of this paper would be changed by allowing the agent’s flow rents to vary in different
states, so long as they are always positive and weakly higher in the good state. In the more general case
only the rents in the bad state will enter into the optimal contract.
2
It turns out that if rB + b ≥ 0 the optimal contract under limited liability is trivial: the firm pays the
agent nothing and operates the project ignoring the agent’s information entirely. Intuitively, when rB +b > 0
there is scope for gains from trade by operating the project in the bad state, but because the agent can’t
pay the firm none of these gains are realized.
8
definition additionally requires that a revelation contract be fully “cashed out” immediately
upon termination. As both parties have linear utility with the same discount rate and no
information arrives after termination, this is without loss of generality. (Definition 1 does
not allow contracts to condition on any exogenous randomization. This restriction is without
loss of generality, as the firm’s value function turns out to be strictly concave in a natural
state variable summarizing contract continuations.)
Definition 2. A revelation contract (Φ, τ ) is incentive-compatible if
"Z
ττ
θ
e
E
−ρt
τθ
b dt + dΦt
#
Z
≥E
0
τσ
−ρt
e
(b dt +
dΦσt )
0
for all F-stopping times σ.
I refer to an incentive-compatible revelation contract as an IC contract. Under any IC
contract, the agent truthfully reports τ θ . Therefore when there is no confusion, I suppress
the superscripts on Φ and τ and consider an IC contract to be a pair C = (Φ, τ ) with Φ an
F-adapted process and τ an F-stopping time. The firm’s problem is then to maximize
Z
τ
−ρt
e
E
(dYt − dΦt )
0
over all IC contracts (Φ, τ ). I refer to any contract achieving this maximum as an optimal
contract. Implicit in this formulation of the problem is the assumption that the firm requires
the agent to operate the project in addition to monitoring it. Therefore the firm cannot
terminate the agent without also ceasing operation of the project. I explore alternative
assumptions later in the paper.
2.3
Baseline contracts
Before deriving an optimal contract, I describe a pair of contracts which establish a useful
baseline for understanding the contractual design problem.
Remark. Let Φt = ρb 1{t ≥ τ θ }. Then (Φ, τ θ ) is an IC contract which is profit-maximizing
among all IC contracts implementing efficient project termination.
This contract stops the project whenever the agent reports that the state has changed,
and pays the agent a lump-sum transfer of b/ρ upon termination. Intuitively, such a contract
is incentive compatible because at each moment the additional flow rents b dt from waiting
a moment longer to report a state switch and terminate the project are exactly offset by
9
the interest expense −ρ(b/ρ) dt of delaying receipt of the contract’s terminal payment. Thus
the agent is indifferent between all reporting policies regardless of the true state transition
time, and in particular is willing to report truthfully. And it is profit-maximizing in the class
of efficient contracts because any IC contract implementing τ θ must commit to operating
forever if the agent never reports a state switch. Therefore it must pay the agent at least as
much to stop as he’d collect by letting the project operate forever, which is precisely b/ρ.
Thus at one extreme, the firm can fully utilize the information available to the agent
provided it pays the agent enough. The following remarks considers the opposite extreme,
when the firm disregards the agent’s reports entirely.
Remark. Let τ ∗ be the FY -stopping time maximizing (1). Then (0, τ ∗ ) is an IC contract
which is profit-maximizing among all IC contracts utilizing only public information.
Evidently, if the firm does not condition payments or project termination on the agent’s
reports, incentive compatibility is trivially satisfied. The firm’s problem is then to improve
upon τ ∗ by extracting some of the agent’s state information, but to economize on transfers
to the agent and avoid paying the agent’s entire lifetime rental stream whenever the project
is halted. I will show that the firm can do better than either of the baseline contracts
described above by striking a balance between efficient termination and the costs of obtaining
information from the agent.
3
Dual approaches to contractual design
Over the next several sections I develop two distinct approaches to analyzing the firm’s
contracting problem. In this section I show that the firm’s transfer scheme and termination
policy are “dual variables”: either may be eliminated in favor of the other, reducing the
analysis to design of a single contractual element. This duality is closely analogous to
the equivalence between choosing prices or quantities in monopoly theory, a perspective
I return to later in the section. I go on to develop the “price approach” of optimizing
the transfer scheme fully in Section 4, and then pursue the dual “quantity approach” of
designing the termination policy in Section 5. The two approaches provide distinct and
complementary insights into properties of the optimal contract, in particular the relationship
of the contracting problem with an agent to the firm’s unassisted quickest detection problem.
10
3.1
Preliminaries
In general an IC contract must satisfy incentive constraints deterring the agent from falsely
reporting a state change either too early or too late at each moment in time. In particular,
when θt = G the agent’s expected utility from waiting until τ θ to report a state switch must
exceed his expected utility from reporting a state switch immediately. And conversely, when
θt = B the agent’s expected utility from reporting the switch immediately must exceed the
payoff of any delayed reporting strategy. It will be very helpful to characterize incentive
compatibility as the combination of these two constraints.
Definition 3. A revelation contract (Φ, τ ) satisfies IC-G (respectively, IC-B) if
"Z
ττ
θ
e
E
−ρt
τθ
b dt + dΦt
#
Z
≥E
0
τσ
−ρt
e
(b dt +
dΦσt )
0
for all F-stopping times σ ≤ τ θ (respectively, σ ≥ τ θ ).
Lemma 1. A revelation contract is incentive-compatible iff it satisfies both IC-G and IC-B.
Both the IC-G and IC-B constraints substantively restrict the space of IC contracts.
However, I conjecture that the sole binding constraint under an optimal contract is the ICB constraint. Intuitively, given the agent’s preference to delay project termination absent
contractual incentives, the IC-B constraint must bind at least some of the time. But as
provisioning incentives is costly for the firm, it should impose them as lightly as possible; in
particular, it would be surprising if the firm provisioned such strong incentives for reporting a
switch that IC-G were violated and the agent wished to report a state switch prematurely. I
therefore solve the relaxed problem of characterizing optimal IC-B contracts, and then verify
at the end that such a contract is incentive-compatible and therefore an optimal contract.
A general IC-B contract could specify a complex series of payments to the agent over the
course of his employment, as well as a varied set of termination policies depending on the
project’s output history and the agent’s past reports. The following pair of lemmas prune
away much of this complexity by establishing the optimality of a subset of IC-B contracts
obeying parsimonious transfer and termination rules.
Lemma 2 (No late termination). Given any IC-B contract C = (Φ, τ ), there exists another
IC-B contract C 0 = (Φ0 , τ 0 ) satisfying τ 0 = τ θ ∧ τ . Further, the firm’s expected profits under
C 0 are weakly higher than under C, and strictly higher if τ 0 < τ with positive probability.
Finally, τ 0 = τ θ ∧ τ Y for some FY -stopping time τ Y .
11
Lemma 3 (Backloading). Let C = (Φ, τ ) be any IC-B contract such that τ = τ θ ∧ τ Y for
some FY -stopping time τ Y . Then there exists another IC-B contract C 0 = (Φ0 , τ ) satisfying
Φ0t = Fτ 1{t ≥ τ } for some FY -adapted process F ≥ 0. Further, C 0 yields the same expected
profits as C.
The first lemma establishes that optimal IC-B contracts never terminate inefficiently
late - that is, after the agent reports the state has switched. In principle late termination
could be desirable as a way to compensate the agent with rents. However, the assumption
that the project is jointly unprofitable in the bad state means that the firm can always
more cheaply compensate the agent with cash at the time of the state switch. Therefore an
optimal termination policy always consists of a rule of the form “Terminate the project as
soon as the agent reports a state switch or a publicly observable threshold has been reached,
whichever comes first”.
Remark. It is not clear a priori that an arbitrary IC contract can be improved upon by
terminating as soon as a state switch is reported without violating incentive-compatibility.
For the firm might enforce IC-G in a particular contract by monitoring output following
a report to ensure that the state has really switched. If such monitoring is removed by
truncating the project lifetime, an otherwise IC contract may create incentives to prematurely
report a state switch. Lemma 2 therefore illustrates the tractability brought by passing to
the relaxed problem, as well as the importance of verifying at the end that an optimal IC-B
contract does not violate IC-G.
The second lemma demonstrates that all payments can be backloaded to a single golden
parachute, denoted F, payable upon termination. This is because both parties are riskneutral and discount the future equally; thus expected profits and incentive-compatibility
are unchanged by deferring all promised payments until termination and accruing interest
on them at rate ρ. And because termination occurs no later than the date of a reported state
change, the size of the parachute need not be conditioned on the date of a past report, hence
is FY -adapted. It is then without loss of generality to study only transfer policies of the form
“Pay nothing until project termination, at which point grant the agent a single lump sum
golden parachute”.
I therefore restrict attention to backloaded contracts with no late termination. I summarize such contracts by pairs (F, τ Y ), with F a golden parachute and τ Y a (public) termination
policy. Under such a contract, an agent’s reporting rule can be thought of as a restricted
project termination decision: he can terminate the project immediately at any point by
reporting a switch, and can extend the project until τ Y by staying silent. Under such a
contract, the IC-B constraint can be characterized in a convenient recursive form.
12
Lemma 4. Fix a contract C = (F, τ Y ). Suppose that for every t and every FY -stopping time
τ 0 ≥ t,
Ft ≥ EB
t
"Z
#
τ Y ∧τ 0
e−ρ(s−t) b ds + e
−ρ(τ Y
∧τ 0 −t)
Fτ Y ∧τ 0
(2)
t
whenever τ Y > t. Then C satisfies IC-B.
Conversely, suppose C satisfies IC-B. Then there exists another IC-B contract C 0 =
(F 0 , τ Y ) such that F 0 satisfies (2) for every t and τ 0 ≥ t whenever τ Y > t, and further
such that Fτ0 Y ∧τ θ = Fτ Y ∧τ θ a.s. Hence C 0 yields the same expected profits as C.
This lemma formalizes the idea that at each moment in time, an agent in the bad state
must prefer collecting the golden parachute now rather than delaying termination and collecting additional flow rents from operating the project. As the agent’s delayed report can
be conditioned on the path of output going forward, a general delayed reporting policy is
an FY -stopping time τ 0 . Lemma 4 rules out the profitability of any such τ 0 at any time as
essentially a necessary and sufficient condition for satisfying IC-B. Without loss of generality,
I restrict attention to IC-B contracts satisfying (2) going forward and treat the condition as
necessary as well as sufficient without comment.
3.2
Designing the golden parachute
One approach to the optimal contracting problem is to design the golden parachute F,
treating the termination policy τ Y as an auxiliary variable and eliminating it. To do so, one
must determine the optimal termination policy which incentivizes truthful reporting under
a given golden parachute F.
It turns out that early termination serves a very limited role in an optimal contract. To
incentivize truthful reporting, the firm must commit to limit the agent’s continuation utility
(flow rents plus golden parachute) from delaying his report of a state switch. In particular,
if the golden parachute ever falls to zero, then any delayed report must also yield at most
zero expected utility. Given the agent’s limited liability, immediate project termination is
of course the only way to enforce such a promise. Is early termination optimal in any other
circumstance? The following lemma answers this question in the negative.
Lemma 5. Suppose C = (F, τ Y ) is an IC-B contract. Then τ Y ≤ inf{t : Ft = 0}, and there
exists another IC-B contract Ce = (Fe, τeY ) such that Fet = min{Ft , b/ρ} whenever τ Y ≥ t and
τeY = inf{t : Fet = 0}. Further, τeY ≥ τ Y and Ce yields expected profits at least as high as C.
13
The first claim of the contract is just a restatement of the conclusion that no contract
can operate past the point where the golden parachute hits zero. The lemma then constructs
a new contract illustrating two design principles: first, the golden parachute can be capped
at b/ρ; and second, if a contract (F, τ Y ) is ever terminated before F hits zero, it can be
extended in an incentive-compatible way without decreasing profits. The first principle is
closely related to the fact that (b/ρ, ∞) is an IC contract, as I demonstrated in Section 2.3.
Essentially, whenever Ft hits b/ρ, delayed reporting can be deterred indefinitely simply by
holding the fee at that level; in particular, it can be delayed until the next time Ft falls below
b/ρ. Thus min{Ft , b/ρ} is also incentive compatible. The second principle can be proven by
explicitly constructing an IC continuation contract. In particular, set
n
o
Y
Fet = max b/ρ − (b/ρ − min{Fτ Y , b/ρ})eρ(t−τ ) , 0
for t ≥ τ Y . Note that so long as Fet > 0, dtd Fet = −b + ρFet , which exactly offsets the flow rents
and interest expense of delaying termination by a moment. Hence incentive compatibility is
ensured for t ≥ τ Y . As for times before τ Y , Fet provides no more expected utility to the agent
than a contract that terminates at τ Y and pays Feτ Y . So it provides no additional incentives
to delay reporting prior to τ Y . Finally, because this extension of the contract pays no more
than Feτ Y to the agent while generating positive revenue as long as the state is good, it must
improve profitability relative to termination at τ Y .
Lemma 5 solves the design problem for τ Y given an incentive-compatible golden parachute
F. The firm’s problem therefore reduces to the design of the dynamics of the golden parachute,
subject to the requirement that the project terminate as soon as the parachute reaches zero.
I derive the optimal golden parachute in Section 4.
3.3
Designing the termination policy
The second approach to deriving an optimal contract is to design the termination policy τ Y
while treating the golden parachute F as an auxiliary variable. To proceed this way, one
must determine the optimal golden parachute which incentivizes truthful reporting under a
given termination policy τ Y .
Consider that one reporting strategy an agent may follow is to never report a state switch,
collecting rents until the firm halts the project according to τ Y . The value of any IC-B golden
14
parachute implementing τ Y must therefore exceed this amount, implying the inequality
Ft ≥ EB
t
"Z
#
τY
−ρ(τ Y
e−ρ(s−t) b ds + e
−t)
Fτ Y
t
after all histories. Setting Fτ Y = 0 and Ft to saturate this inequality minimizes the golden
parachute subject to deterring a deviation to never reporting a state change. The following
lemma establishes that such a payment scheme deters all other delayed reporting strategies
as well and is therefore optimal within the class of IC-B golden parachutes implementing τ Y .
Lemma 6. Every FY -stopping time τ Y is supportable as part of an IC-B contract, and the
profit-maximizing IC-B golden parachute implementing τ Y satisfies
Ft = EB
t
"Z
#
τY
e−ρ(s−t) b ds
t
for all t and all histories such that τ Y ≥ t.
The surprising conclusion of Lemma 6 is that the firm faces no real double deviation
problem when designing an optimal golden parachute. The parachute in the lemma is derived
by pretending that the agent can do just one of two things at the time of a state change report the change now, or stay silent forever. In general, the agent’s ability to deviate again
by reporting a state change at some later date implies additional constraints that might
bind at the optimum. In my model, however, delaying a report of a state change for any
interval of time leaves the agent in an “on-path” setting identical to a counterfactual agent
with the same output history but a later state switch. And under the contract of Lemma 6,
that agent’s golden parachute is exactly his expected rents from never reporting. Therefore
the true agent’s utility from delaying a report by that interval is identical to his utility from
never reporting. This argument establishes the lemma’s claim that no strategy of delayed
reporting yields a profitable deviation.
Lemma 6 allows F to be eliminated from the firm’s objective function, reducing the
contracting problem to the design of τ Y alone. I derive the optimal termination policy in
Section 5.
3.4
Dual variables and monopoly theory
Lemmas 5 and 6 establish the duality of F and τ Y as contractual design variables. Note that
F controls the cost of implementing the contract, while τ Y regulates the amount of output
15
produced; in addition, the longest IC-B τ Y is increasing in F. It is therefore tempting to
draw an analogy to the price and quantity variables of standard monopoly theory. Indeed,
price and quantity are also dual variables in that problem, so this analogy provides useful
intuition for understanding the duality result in my model.
Consider a standard monopsony problem, in which a firm faces an upward-sloping supply
function for procurement of a good. The firm can be considered to control either price P or
quantity purchased Q, with the remaining variable eliminated from the firm’s optimization
problem. One approach is to optimize profits over P , determining Q via the supply curve
along the way. The dual approach is to control Q, with P eliminated from the problem via
the inverse supply curve. In my problem, one can analogously think of Lemma 5 as giving
a generalized supply curve, with Lemma 6 computing the generalized inverse supply curve.
The firm then optimizes profits Π(F, τ Y ) either by using the generalized supply curve to
eliminate τ Y (the quantity variable), and then optimizing Π(F ) over F (the price variable);
or by controlling τ Y and optimizing Π(τ Y ), using the generalized inverse supply curve to
eliminate F .
The analogy may be pushed even further by examining the standard logic of equating
gains on marginal units and losses on inframarginal units. In the monopsonist’s problem,
the firm finds the point (in either P or Q space) where profits on a marginal unit purchased
equal losses due to the increased cost of inframarginal units already purchased. Analogously
in my problem, the firm determines when the increased profit from operating the project
when τ θ ∈ (τ Y , τ Y + dt) is balanced by the increased cost of golden parachutes paid to the
agent when τ θ ≤ τ Y . This is essentially the calculation carried out in Sections 4 and 5 using
F and τ Y , respectively, as control variables.
4
The price approach
In this section I pursue the approach suggested by Lemma 5 and eliminate τ Y from the
optimal contracting problem, reducing the contracting problem to the design of F alone.
I then use techniques of recursive dynamic contracting to maximize the firm’s profits wrt
incentive-compatible F and derive an optimal golden parachute. The main theorem of this
section derives a differential equation, the HJB equation, whose solution characterizes the
optimal incentive-compatible golden parachute. The associated optimal termination policy
is left implicit, but could in principle be calculated by Lemma 5. Throughout this section,
I will often summarize a contract by the golden parachute process F alone; whenever τ Y is
not explicitly stated as an element of the contract, it is understood to be the first time the
16
golden parachute reaches zero.
4.1
The golden parachute as state variable
In this subsection I show that the current value of the golden parachute constitutes a sufficient
statistic for the state of the contract. In the context of recursive contractual design, this
assertion amounts to two statements. First, the firm’s profit function can be written as a
discounted sum of flow payments involving only the current value of F at each moment of
time. Second, the agent’s IC constraints can be reduced to a set of local restrictions on the
dynamics of F. With these results in hand, standard techniques of continuous-time dynamic
contracting (see, e.g., Sannikov (2008)) can be used to derive the firm’s value function using
F as a state variable.
Recall that the firm’s ex post profits from an IC-B contract F consist of flow payments
rG up to the stopping time τ = τ θ ∧ τ Y , followed by a lump-sum payment of Fτ once τ
is reached. From an ex ante perspective, this payment flow can be analyzed as follows: at
each time t, there is a probability e−αt that the project has not yet terminated. In this case
the firm receives a discounted flow payment of e−ρt rG dt, and with probability α dt the state
changes in the next instant and the firm must pay out e−ρt Ft . The net expected contribution
of time t’s flows to time-zero profits are therefore e−(ρ+α)t (rG − αFt ). These flows are then
summed up to the public stopping time τ Y and appropriately averaged to obtain the firm’s
total expected profits from F. This result is formalized in the following lemma.
Lemma 7. The firm’s expected profits under any contract F when the agent reports truthfully
are
#
"Z Y
τ
e−(ρ+α)t (rG − αFt ) dt .
EG
0
Lemma 7 establishes the first prong of the claim that Ft appropriately summarizes the
state of the contract at time t. Note that the expectation in the profit function of Lemma 7
is wrt to the measure PG under which Z G is a standard Brownian motion. In other words,
it assumes a world in which the project is always good. In this formulation, the distortion
caused by state switching is entirely captured by the higher discount rate ρ + α applied to
flow profits.
As for the second prong, recall that IC-B means stopping immediately following a state
switch must provide higher utility to the agent than any delayed reporting policy. In particular, it must pay more, on average, than waiting a moment and then reporting. This
latter constraint is purely local, restricting only the instantaneous dynamics of the golden
17
parachute given its current value. In fact, in this model these local constraints collectively
imply global IC-B. Intuitively, an agent who deviates from immediate reporting of a state
switch to delay momentarily finds himself in a situation identical to an agent with the same
public output history and a slightly later state switch who has not (yet) deviated. Therefore if agents have incentives to report a state switch at time t + dt, an appropriate local
constraint will ensure that agents at time t do as well.
This intuition is difficult to formalize for general golden parachute processes, as I do
not rule out the possibility that F evolves very irregularly. However, when F is sufficiently
well-behaved incentive compatibility can be characterized quite sharply.
Lemma 8. Fix a contract C = (F, τ Y ). Suppose there exist FY -adapted, right-continuous
processes γ and β such that for each t, F satisfies
Z
Ft = F0 +
t
Z
γs ds +
0
t
βs dZsG
0
a.s. Then (F, τ Y ) is an IC-B contract iff
b − ρFt + γt −
∆r
βt ≤ 0
σ
whenever t < τ Y .
Lemma 8 decomposes the increments dFt of the golden parachute into two components:
an instantaneous average change γt dt, and a “surprise” term βt dZtG linked to the arrival of
information on project output. In a world where the project’s state is known to be good, the
surprise term has mean zero and γt dt captures the expected change in the golden parachute
in the next instant. The derivation of the incentive constraint deterring late reporting arises
from the observation that after the state has switched, Z B rather than Z G has zero-mean
increments. Further, the increments of the two processes are related via dZtG = dZtB − ∆r
β dt.
σ t
It is then straightforward to compute the expected change in utility of an agent who
momentarily delays reporting a state change. After the delay, the agent collects additional
flow rents b dt; pays an interest cost −ρFt dt from collecting the golden parachute a bit
later; and receives the expected change in the value of the golden parachute, which from his
perspective is γt − ∆r
β dt. If the sum of these three factors is non-positive everywhere,
σ t
then the agent never benefits from delaying a report of a state change.
Unfortunately Lemma 8 does not fully characterize the set of IC-B contracts, as it applies
only to a particularly well-behaved class of stochastic processes. Indeed, the most involved
18
technical step in the proof of Theorem 1 is establishing the suboptimality of more general
golden parachutes.3 However, Lemma 8 captures the key intuition for the incentive constraint
appearing in Theorem 1.
Lemmas 7 and 8 together indicate that the current value of the golden parachute is an
appropriate state variable for a recursive formulation of the firm’s contracting problem. The
next subsection carries out the recursive derivation of an optimal contract.
Remark. The choice of Ft as a state variable for this problem contrasts with standard contracting environments, which generally use the agent’s continuation utility as a state variable. One distinctive feature of my model is the presence of asymmetric information, which
in general requires tracking a pair of continuation utilities for the agent in each state. Letting
(UtG , UtB ) be the agent’s continuation utilities in each state at time t, these variables can be
computed explicitly for any IC contract satisfying backloading and no late termination:
UtG = EG
t
"Z
#
τY
−(ρ+α)(τ Y
e−(ρ+α)(s−t) (b + αFs ) ds + e
−t)
Fτ Y ,
UtB = Ft .
t
The choice of Ft as a state variable is therefore equivalent, after appropriate reduction of
the contract space, to choosing UtB as a state variable. The key property of my model which
further reduces the state space is the fact that IC constraints in the good state don’t bind
at the optimum. The analysis therefore amounts to designing the dynamics of UtB , i.e. the
terminal utility of the contract.
4.2
A recursive formulation of the optimal contract
Consider the class of all IC-B contracts promising a golden parachute of F0 ≥ 0 if the agent
reports a state switch at time zero, and let V (F0 ) be the maximal profit achievable by the
firm among all such contracts. V is then the firm’s value function. To calculate V , I utilize
the accounting identity that V (F0 ) should be equal to the firm’s instantaneous flow profits
at time 0 plus continuation profits from following an optimal contract a moment later. Of
course, what state is reached a moment later depends on the contract increment dF chosen
at time 0; as V is the value function dF must of course be chosen to maximize lifetime
profits over all incentive-compatible contract increments. Heuristically, V must satisfy the
3
Examples of such processes include Ito processes with drift or diffusion terms that are not rightcontinuous; semimartingales whose BV components are not absolutely continuous or which contain jumps;
as well as even more irregular processes which are not semimartingales, such as fractional Brownian motions.
19
“Bellman equation”
V (F0 ) =
max
dF ∈IC(F0 )
EG (rG − αF0 ) dt + e−(ρ+α)dt V (F0 + dF ) .
Supposing the firm chooses dF = γ dt + β dZtG , its control variables are (γ, β), which by
Lemma 8 must satisfy b − ρF0 + γ − ∆r
β ≤ 0. Derivation of a proper HJB equation (the
σ
continuous-time analog of a Bellman equation) is then accomplished by taking dt → 0
appropriately.
Ito’s lemma provides the proper machinery to take this limit. It says that, to first order,
1
V (F0 + dF ) ' V (F0 ) + γV (F0 ) + β 2 V 00 (F0 )
2
0
dt + βV 0 (F0 ) dZtG .
This expression differs from a standard Taylor expansion due to the presence of the secondorder term 21 β 2 V 00 (F0 ). Intuitively, the extra term captures the impact of the curvature of V
on EG [V (F0 + dF )], which by Jensen’s inequality differs from V (F0 + EG [dF ]) in the same
direction as the sign of V 00 .
Inserting the Ito expansion of V (F0 + dF ) into the “Bellman equation”, the final term
βV 0 (F0 ) dZtG has mean zero and vanishes.4 Expanding e−(ρ+α)dt to first order as 1−(ρ+α)dt,
eliminating second-order terms, and re-arranging yields the HJB equation
1 2 00
0
(ρ + α)V (F0 ) = max
rG − αF0 + γV (F0 ) + β V (F0 ) .
(γ,β)∈IC(F0 )
2
This is a second-order differential equation which, combined with appropriate boundary
conditions, pins down the firm’s value function. One boundary condition is easy: when
F0 = 0 the unique IC contract entails immediate termination, so V (0) = 0. The second
condition is more subtle, and turns out to be a so-called “free boundary condition”. Note
that an upper bound on the value of any contract to the firm is rG /(ρ + α), as this is what
the firm would earn if it could directly observe the state of the project. Conversely, as
discussed earlier there is no benefit to offering a golden parachute larger than b/ρ. Therefore
V is bounded above, and it should reach a maximum for some finite value of the golden
parachute. The second boundary condition is then that V 0 (F ∗ ) = 0 for some undetermined
value F ∗ of the golden parachute.
The following theorem provides a formal statement of the conditions under which the
4
Strictly speaking, stochastic integrals wrt Z G are martingales only if they satisfy appropriate regularity
conditions. Checking that these conditions are met is a standard step in a rigorous verification proof. See
the proof of Theorem 1 for details.
20
heuristic approach just outlined produces a solution to the firm’s contracting problem. It
constitutes the main technical result of this section.
Theorem 1. Suppose there exists a constant F ∗ > 0 and a C 2 function V : [0, F ∗ ] → R
satisfying V (0) = 0, V 0 (F ∗ ) = 0, and the HJB equation
(ρ + α)V (F0 ) =
sup
(γ,β)∈IC(F0 )
1 2 00
rG − αF0 + γV (F0 ) + β V (F0 )
2
0
for every F0 ∈ [0, F ∗ ], where IC(F0 ) ≡ {(γ, β) : b − ρF0 + γ − β ∆r
≤ 0}. Then:
σ
• V (F ∗ ) is an upper bound on the expected profits of any IC-B contract;
• F ∗ < b/ρ, V is strictly increasing and strictly concave on [0, F ∗ ], and V (F ∗ ) =
rG −αF ∗
;
ρ+α
• There exist unique continuous functions γ ∗ , β ∗ : R → R such that (γ ∗ (F0 ), β ∗ (F0 ))
maximize the HJB equation on [0, F ∗ ] and γ ∗ , β ∗ are constant on R \ [0, F ∗ ];
• β ∗ ≥ 0, β ∗ (F ∗ ) = 0, and γ ∗ (F ∗ ) < 0;
• There exists a weak solution F to the stochastic differential equation (SDE)
∗
Z
t
∗
Z
γ (Fs ) ds +
Ft = F +
t
β ∗ (Fs ) dZsG
(3)
0
0
for all time;
• If limF0 →F V 0 (F0 )V 000 (F0 ) exists and is finite, then there exists a unique strong solution
F to (3) for all time;
• If F is a strong solution to (3), then max{F, 0} is an optimal contract whose expected
profits are V (F ∗ ).
Theorem 1 states that solving a particular boundary value problem yields the firm’s value
function as well as an optimal contract. Importantly, the value function is actually increasing
over a range of golden parachutes. This is because when the parachute reaches zero, the firm
is forced to terminate the project and loses all future output from its operation. Thus for
sufficiently small values of F0 , the revenue from a longer project lifespan outweighs the cost
of increasing expected payments to the agent, leading to an upward-sloping value function.
The critical point F ∗ determines when these two factors exactly balance. Because γ ∗ (F ∗ ) < 0
and β ∗ (F ∗ ) = 0, an optimal contract never promises a golden parachute larger than F ∗ no
21
matter how good the history of output. Instead, as Ft approaches F ∗ the volatility of the
parachute vanishes and the parachute drifts downward deterministically, holding its value
below F ∗ .
The HJB equation also allows one to explore the economics of the firm’s contractual
design problem. Given concavity of V as assured by the theorem, the optimal controls
(γ ∗ (F0 ), β ∗ (F0 )) are readily computed as the first-order conditions of the HJB equation given
a binding IC constraint:
β ∗ (F0 ) = −
∆r V 0 (F0 )
,
σ V 00 (F0 )
γ ∗ (F0 ) = −(b − ρF0 ) +
∆r ∗
β (F0 ).
σ
Intuitively, the tradeoff the firm faces when designing a golden parachute is whether to
provision incentives through downward drift or volatility of the parachute. Both are costly
for the firm, because each brings the contract closer to premature termination on average.
An optimal contract therefore provisions incentives as lightly as possible, leading to a binding
IC-B constraint at all points along the contract. In other words, (γ ∗ (F0 ), β ∗ (F0 )) satisfy
b − ρF0 + γ ∗ (F0 ) −
∆r ∗
β (F0 ) = 0.
σ
The optimal mix of incentives is then determined by the slope and curvature of the value
function: each unit of downward drift added to the contract incurs a constant marginal
cost V 0 (F0 ), while each unit of volatility added incurs a linearly increasing marginal cost
−βV 00 (F0 ). Equating the ratio of marginal costs to the marginal rate of substitution ∆r
σ
between volatility and drift in the incentive constraint determines the optimal mix of incentives.
The fact that an optimal contract provisions incentives as lightly as possible and increases
the golden parachute in response to good news verifies an earlier conjecture that the IC
constraint for the agent in the good state is slack under an optimal contract. To see this,
consider any contract evolving as
dFt = γt dt + βt dZtG ,
β = 0 and β ≥ 0. Reasoning as in the paragraphs following Lemma
where b − ρFt + γt − ∆r
σ t
8, one may compute the expected change in the agent’s utility when the state is good and
he considers momentarily delaying a (false) report of a state change, which turns out to be
b − ρFt + γt ≥ 0. Thus it is always at least weakly optimal for an agent in the good state to
delay (falsely) reporting a state change a moment. Since delay continues to be optimal as
22
long as the state as good, global incentive compatibility obtains.
Finally, note that Theorem 1 falls short of completely solving the optimal contracting
problem, as it does not ensure that a solution to the HJB equation actually exists. The
problem is essentially technical, as in principle the firm’s value function may not have the
required degree of smoothness to satisfy a standard HJB equation. I defer a thorough
examination of the technical issues to Section 4.4. In the next subsection, I take a different
approach and exhibit an explicit solution to the HJB equation under a particular parameter
restriction. For this class of models, Theorem 1 ensures that such a solution is in fact the
firm’s value function.
4.3
A complete solution to a special case
For a particular subspace of model parameters, the boundary value problem posed in Theorem 1 has an elegant closed-form solution. The following proposition states the parameter
restriction and describes the form of the value function and the associated optimal controls.
Proposition 1. Suppose ρ = α + ∆r
σ
that a1 > 0 > a2 and (a1 , a2 ) satisfy
2
. Then there exist unique constants a1 and a2 such

ρa + 2ba + α = 0,
1
2
2
 1 ∆r a2 + ba a − r a = 0.
1 2
G 2
1
4
σ
Further, V (F0 ) = a2 F02 + a1 F0 satisfies the HJB equation of Theorem 1 on [0, F ∗ ] and the
boundary conditions V (0) = 0, V 0 (F ∗ ) = 0, and V 00 (F ∗ ) < 0, where
F∗ ≡
∆r
σ
−2
b + rG
ρ
−
α
s
!
2
ρ
2
(b + rG )2 + rG
−1
∈ (0, b/ρ).
α
The corresponding continuous maximizers of the HJB equation are
β ∗ (F0 ) =
∆r ∗
(F − F0 )
σ
and
γ ∗ (F0 ) = −α(F ∗ − F0 ) − (b − ρF ∗ ).
Further, V 000 = 0, so there exists a unique strong solution to (3).
Proposition 1 solves the firm’s optimal contracting problem when ρ = α +
23
∆r 2
σ
. In
this case the firm’s value function is quadratic and the associated controls are linear in the
current value of the golden parachute. One implication of this linearity is that the optimal
contract’s sensitivity to news increases the closer the contract moves to termination, and
conversely vanishes as F0 → F ∗ . This dynamic conforms to a sensible intuition that news
should “matter more” for incentives as the contract progresses and it becomes ex ante more
likely that the state has transitioned. In Section 5 I explore the issue of optimal sensitivity
to news from another angle, when considering how news affects progress toward project
termination.
Proposition 1 characterizes all the essential features of the firm’s optimal contract but
stops short of deriving the contract explicitly. Given the linearity of the optimal controls,
the optimal contract satisfies the linear SDE
dFt = −(α(F ∗ − F ) + (b − ρF ∗ )) dt +
∆r ∗
(F − F ) dZtG .
σ
Such SDEs are solvable in closed form, yielding an explicit solution for the optimal contract:
Corollary. Suppose ρ = α +
contract is given by
∗
∗
Ft = F − (b − ρF ) exp
∆r 2
σ
. Then letting F ∗ be as in Proposition 1, an optimal
Z t
3
3
1
∆r G
1
∆r G
exp −
α− ρ t−
Z
α− ρ s+
Z
ds
2
2
σ t
2
2
σ s
0
for t ≤ inf{t : Ft = 0}.
An immediate implication of this corollary is the following: fix a time t and two states of
the world ω1 , ω2 ∈ Ω satisfying Ys (ω1 ) > Ys (ω2 ) for all s ∈ (0, t) and Yt (ω1 ) = Yt (ω2 ). Then
Ft (ω1 ) < Ft (ω2 ). In words, ω1 and ω2 encode two different paths of output that reach the
same cumulative output by time t. However, output under ω1 outperforms early and lags
late compared to ω2 ; consequently the contract is brought closer to termination by time t
under ω1 . Thus in a very strong sense the optimal contract weights news more heavily later
in the contract.
4.4
Existence of solutions to the HJB equation
This subsection is devoted to discussing technical existence issues of solutions to the HJB
equation derived in Theorem 1. It can be safely omitted by readers uninterested in this
detail.
24
Theorem 1 establishes a program for characterizing an optimal contract by solving a
particular boundary value problem. However, it does not ensure that this program will
succeed, as it could be that no solution to the boundary value problem exists. A standard
solution to this problem is to apply ODE existence theorems directly to the HJB equation
to ensure existence of solutions to a family of initial value problems, and then to argue that
there exist initial conditions whose solutions satisfy the right boundary conditions. The
following proposition demonstrates the difficulties of that approach in this context.
Proposition 2. Let V : [0, F ∗ ] → R be a C 2 function satisfying V (0) = 0 and V 0 (F ∗ ) = 0.
Then V satisfies the HJB equation on [0, F ∗ ] iff it is strictly increasing and strictly concave
on [0, F ∗ ) and satisfies
1
V (F0 ) = −
2
00
∆r
σ
2
V 0 (F0 )2
(ρ + α)V (F0 ) − (rG − αF0 ) + (b − ρF0 )V 0 (F0 )
for each F0 ∈ [0, F ∗ ).
Further, suppose V is such a function. Then V (F ∗ ) =
0.
rG −αF ∗
,
ρ+α
F ∗ < b/ρ, and V 00 (F ∗ ) <
Proposition 2 demonstrates that finding solutions to the HJB equation boils down to
solving a particular nonlinear second-order ODE of the form V 00 = G(F0 , V, V 0 ). Morever,
the solutions of interest to the optimal contracting problem are singular solutions to this
ODE. To see this, consider the numerator and denominator of G as F0 → F ∗ . The numerator
2 0 ∗ 2
V (F ) , which vanishes by definition of the free boundary condition. As
becomes − 21 ∆r
σ
for the denominator, (ρ+α)V (F ∗ )−(rG −αF ∗ )+(b−ρF0 )V 0 (F ∗ ) = 0 given the proposition’s
−αF ∗
. This condition follows from the observation that the
final result that V (F ∗ ) = rGρ+α
−αF ∗
∗
constant function V (F0 ) = V (F ) is a local solution to the ODE whenever V (F ∗ ) 6= rGρ+α
.
Given that G is continuously differentiable and thus uniformly Lipschitz on sufficiently small
neighborhoods around (F ∗ , V (F ∗ ), 0), standard local ODE uniqueness results then imply
that there can be no strictly increasing solutions to the ODE near F ∗ . Therefore candidate
−αF ∗
solutions of the ODE must satisfy V (F ∗ ) = rGρ+α
.
For this approach to succeed, then, it is necessary at minimum to show local existence
of a solution to the ODE V 00 = G(F0 , V, V 0 ) around the singularity (F0 , V (F0 ), V 0 (F0 )) =
(F ∗ , (rG − αF ∗ )/(ρ + α), 0) for some range of F ∗ > 0. As far as I am aware no general
existence results apply in this setting. I return to this question in Section 5, assisted by the
full characterization of an optimal contract obtainable by the quantity approach.
25
5
The quantity approach
In this section I pursue the dual approach to Section 4, eliminating F from the firm’s problem
and optimizing the resulting objective function to derive an optimal termination policy.
The main result of this section shows that the optimal incentive-compatible termination
policy solves a particular quickest detection problem, closely analogous to the firm’s problem
without an agent. I then develop an optimal stopping approach to explicitly solve the quickest
detection problem.
5.1
The firm’s objective function
Recall that by Lemma 6, any FY -stopping time τ Y is implementable by an IC-B contract,
with associated profit-maximizing golden parachute
Ft =
EB
t
"Z
#
τY
−ρ(s−t)
e
b ds .
t
The following proposition proves that when F is eliminated from the firm’s profit function,
the resulting optimization problem for τ Y can be stated elegantly in terms of maximizing an
expected discounted flow of virtual profits.
Proposition 3. Let τ Y be any FY -stopping time and Π(τ Y ) be the supremum of profits
achievable by IC-B contracts implementing τ Y . Then
Π(τ Y ) = E
"Z
#
τY
e−ρt (πt rG − (1 − πt )b) dt .
(4)
0
Though the formal derivation of (4) is somewhat involved, the intuition behind it is
very simple. Consider any contract with termination policy τ Y and golden parachute as
specified in Lemma 6. Upon a reported state change the firm pays the agent his expected
discounted flow of rents, conditional on the state being bad, from allowing the project to
operate until τ Y . The firm’s expected profits under such a contract are therefore identical
to a counterfactual setting in which its project pays expected returns −b rather than rB
when the state is bad. And conditional on the public information available at time t, the
probability of being in the good state at that time is πt , so instantaneous expected returns
at any time are πt rG − (1 − πt )b.
Proposition 3 reduces the contracting problem with an agent to solving a particular
virtual quickest detection problem, closely related to the firm’s problem without an agent
26
(cf. equation (1)). In this virtual problem the firm learns about the state of the project as
if no agent were available but incurs a flow cost of operating the project in the bad state
of −b rather than rB . Importantly, this reduction does not imply that the optimal contract
implements the same termination policy as in a quickest detection problem with no agent
and rB replaced by −b. First, the firm’s termination policy is τ = τ θ ∧ τ Y , so the presence of
an agent prevents operation of the project in the bad state; this is in stark contrast to the
problem without an agent, where both early and late terminations occur. Second, the firm
learns about the current state more quickly in the virtual problem, as its signal-to-noise ratio
with an agent, versus rGσ+b in the quickest detection problem with rB replaced
(SNR) is ∆r
σ
by b.
5.2
Threshold termination policies
Given the form of Π(τ Y ) in Proposition 3 and the fact that posterior beliefs drift downward
over time, a natural conjecture for an optimal termination policy is a threshold rule in the
firm’s posterior beliefs, i.e. τ π ≡ inf{t : πt ≤ π} is an optimal policy for some π ∈ [0, 1].
To verify this conjecture, it is necessary to establish that πt is a sufficient statistic for the
history of the project when projecting the future path of beliefs; in other words, one must
rule out the possibility that the time-t conditional distribution of future beliefs depends on
{Ys }s≤t in a more complicated way than through πt . The following lemma accomplishes this
task by fully characterizing the dynamics of π :
Lemma 9. π satisfies the SDE
dπt = −απt dt +
∆r
πt (1 − πt )dZ t
σ
with initial condition π0 = 1, where Z is a P-standard Brownian motion adapted to FY with
increments
1
dZ t = (dYt − (πt rG + (1 − πt )rB ) dt).
σ
The SDE describing the evolution of πt is derivable from Bayes’ rule, and consists of
two contributions. First, the known transition rate α from the good to the bad state leads
to a deterministic drop in beliefs over time at rate απt . Second, the firm learns about the
state via unexpected deviations in observed incremental output. The quadratic dependence
of this learning on π is related to the fact that beliefs are most dispersed at π = 0.5, and
hence most subject to revision as new information is received; more extreme beliefs require
stronger evidence to change.
27
The process Z is known as the innovation process, and tracks deviations of the project’s
output from time-t expectations. The fact that Z is a standard Brownian motion implies
that the distribution of the future path of beliefs conditional on FtY depends on the public
history only through πt . Thus πt is indeed a sufficient statistic for the history of the project.
This insight reduces the firm’s contractual design problem to the choice of a single number
π ∈ [0, 1]. A natural baseline contract is the break-even termination policy, under which π
satisfies πrG − (1 − π)b = 0. This termination policy would be optimal if, at the breakeven
point, the firm had to either end the project immediately or commit to continuing it forever.
However, the firm retains a real option to terminate the project in the future, which increases
the value of continuing past the break-even point in the hopes of receiving good news about
the project’s state which raises flow returns. An optimal threshold policy should therefore
satisfy π < b/(b + rG ). The next subsection determines the optimal threshold by calculating
the value of the firm’s real option over time.
5.3
Solving the quickest detection problem
Let
Rt ≡ sup EYt
τY
≥t
"Z
#
τY
e−ρ(s−t) (πs rG − (1 − πs )b) ds
t
be the value of the firm’s real option from continuing the project at time t. Given that πt
is a sufficient statistic for the project history, this option value can be written Rt = Ve (πt )
for some function Ve : [0, 1] → R+ , which I refer to as the firm’s virtual value function. It is
natural to expect that Ve is a weakly increasing function which vanishes for sufficiently small
posterior beliefs, at which point the firm’s real option is worthless and an optimal contract
is immediately terminated. An optimal threshold is then the largest π such that Ve (π) = 0.
The following theorem derives a differential equation characterizing Ve and establishes
that the procedure just outlined maximizes Π(τ Y ).
Theorem 2. Suppose there exists a π ∈ (0, 1) and a C 2 function Ve : [π, 1] → R satisfying
1
ρVe (π0 ) = π0 rG − (1 − π0 )b − απ0 Ve 0 (π0 ) +
2
∆r
σ
2
π02 (1 − π0 )2 Ve 00 (π0 )
on [π, 1] and boundary conditions Ve (π) = Ve 0 (π) = 0. Then π ≤ b/(b + rG ), Ve (π0 ) > 0
for π0 ∈ (π, 1], and Ve may be extended to a C 1 , piecewise C 2 function on [0, 1] by setting
Ve (π0 ) = 0 for π0 ∈ [0, π). On this extended domain, Ve is the firm’s virtual value function.
28
Further, let τ ∗ ≡ inf{t : πt ≤ π}. Then τ ∗ maximizes Π(τ Y ) among all FY -stopping
times τ Y , and Π(τ ∗ ) = Ve (1).
The differential equation stated in Theorem 2 may be derived heuristically in a fashion
analogous to the method outlined in Section 4.2 for obtaining the HJB equation of Theorem
1. The boundary condition Ve (π) = 0 reflects the fact that at π the project is terminated.
It turns out that, given any choice of π ∈ (0, 1), the boundary condition Ve (π) = 0 along
with the requirement that Ve be well-behaved (in particular, finite) at π0 = 1 select a unique
solution Ve to the ODE in Theorem 2.5 Further, Ve (πt ) gives the time−t continuation profits
for the termination policy τ π = inf{t : πt ≤ π}.
The final boundary condition, Ve 0 (π) = 0, is a so-called smooth pasting condition pinning
down the optimal π. If Ve 0 (π) < 0, then π is too low, because continuation profits eventually
become negative prior to termination. The firm could increase profits by terminating sooner
in the region of beliefs where continuation profits are strictly negative. On the other hand, if
Ve 0 (π) > 0, then the threshold is too high. The reasoning here is more subtle due to the kink
in the value function at π, but it can still be understood heuristically. Consider a firm at
the threshold belief who instead of terminating immediately continues an instant dt further
and then reverts to terminating the next time beliefs fall below π. The firm’s expected gains
from such a deviation are, to first order,
∆r
e
π(1 − π)dZ t .
∆Π ' (πrG − (1 − π)b) dt + E V π − απ dt +
σ
√
Now, assume as a simplification that dZ t is a binary random variable taking values ± dt
with equal probability (i.e. having the same mean and variance as a Brownian increment
over the same time interval). Then for sufficiently small dt
√
∆r
1
∆r
E Ve π − απ dt +
π(1 − π)dZ t
= Ve π − απ dt +
π(1 − π) dt ,
σ
2
σ
or to first order in
√
dt,
√
∆r
1
∆r
e
E V π − απ dt +
π(1 − π)dZ t
' Ve 0 (π + ) π(1 − π) dt.
σ
2
σ
5
Technically speaking, the requirement that Ve be C 2 at the right boundary is a transversality condition
closely analogous to the growth conditions common in macroeconomics and growth theory. It is imposed for
the same reason that the growth condition is required to rule out spurious explosive solutions and select a
unique growth path for the economy in those models.
29
Thus any first-order gains or losses from the flow of profits over the (sufficiently short)
interval dt are dwarfed by the lower-order expected gain in continuation profits. Ve 0 (π) = 0
is therefore a necessary condition for optimality of the threshold π. Theorem 2 ensures that
smooth pasting is also sufficient for optimality.
The next lemma establishes that the function Ve stipulated in Theorem 2 actually exists
by explicitly exhibiting one. The lemma makes use of Tricomi’s confluent hypergeometric
function U (m, n, z), which is a solution to Kummer’s differential equation defined and may
be written in closed form as
Z ∞
1
e−zt tm−1 (1 + t)n−m−1 dt
U (m, n, z) =
Γ(m) 0
when m, z > 0.
√
k2 +2α+
∆r
σ
Lemma 10. Let k ≡
and β ≡
C > 0 and π ∈ (0, 1) such that
(k2 +2α)2 +8k2 ρ
.
2k2
b
rG + b
π0 − + Cπ0β (1 − π0 )1−β U
Ve (π0 ) =
ρ+α
ρ
Then β > 1 and there exist constants
2α 2α π0
β − 1, 2β − 2 , 2
k k 1 − π0
is a C 2 function on [π, 1) satisfying
1
ρVe (π0 ) = π0 rG − (1 − π0 )b − απ0 Ve 0 (π0 ) +
2
∆r
σ
2
π02 (1 − π0 )2 Ve 00 (π0 )
and Ve (π) = Ve 0 (π) = 0. Further, Ve may be extended to a C 2 function on [π, 1] satisfying the
ODE on the extended domain.
This lemma relies on the fortuitous fact that the ODE of Theorem 2 is solvable in closed
form in terms of known special functions. As the ODE is second-order, the general solution
contains two linearly independent components; however, one of them diverges as π0 → 1,
and so must be discarded. Most of the work of the lemma then goes toward establishing
existence of constants C and π satisfying the remaining boundary conditions.
The final theorem of this section ties the previous results together and verifies that the
stopping time maximizing Π(τ Y ) is actually incentive-compatible.
Theorem 3. There exists a π ∈ (0, b/(b + rG )] such that τ ∗ ≡ inf{t : πt ≤ π} maximizes
"Z
E
#
τY
e−ρt (πt rG − (1 − πt )b) dt
0
30
among all FY -stopping times τ Y . Letting F ∗ be the associated golden parachute defined in
Lemma 6, (F ∗ , τ ∗ ) is an optimal contract.
To see why (F ∗ , τ ∗ ) ish incentive compatible,
note first that by the martingale represeni
R τ Y −ρ(s−t)
∗
B
tation theorem Ft = Et t e
b ds evolves as
dFt∗
=
(ρFt∗
− b) dt +
βt dZtB
∆r
∗
= ρFt − b +
βt dt + βt dZtG
σ
for some FY -adapted, progressively measurable process β. Further, given that τ ∗ is a threshold policy in πt , time to termination and the size of the golden parachute are increasing in πt .
The arrival of good news about the state must therefore increase F ∗ , in which case β ≥ 0.
This contract is then a member of a class shown to be incentive compatible in the discussion
following Theorem 1.
5.4
More on existence of solutions to the HJB equation
The results of Section 5.3 prove affirmatively that the firm’s virtual value function Ve is a C 2
function. It is natural to wonder whether this result can be leveraged to establish that the
firm’s value function V in the dual problem satisfies the HJB equation derived in Theorem 1.
This subsection explores that possibility. As with Section 4.4, this subsection can be safely
skipped by a reader uninterested in technical existence details.
Let (F ∗ , τ ∗ ) be defined as in Theorem 3. Because πt is a sufficient statistic for the state
of the project, there exists a function f : [π, 1] → R+ such that Ft∗ = f (πt ) for all time a.s.
Clearly for all x ∈ [π, 1], f (x) satisfies
B
Z
f (x) = E
τx∗
−ρt
e
b dt ,
0
where τx∗ is an FY -stopping time defined by τ ∗ (x) ≡ inf{t : pxt ≤ π} for the stochastic
process px satisfying px0 = x and
dpxt = −αpxt dt +
∆r x
1
pt (1 − pxt ) (dYt − (pxt rG + (1 − pxt )rB ) dt).
σ
σ
Informally, τx∗ gives the total remaining project lifetime when the current posterior belief is
x ∈ [0, 1].
Lemma 11. f is a continuous, strictly increasing function on [π, 1] satisfying f (π) = 0 and
f (1) = F0∗ .
31
Similarly, there exists a function V : [0, F0∗ ] → R such that
V
(Ft∗ )
=
EG
t
Z
τ∗
−(ρ+α)s
e
(rG −
αFt∗ ) ds
t
for all time a.s. Using f it is straightforward to relate V , the firm’s value function when
designing F, and Ve , the firm’s virtual value function when designing τ Y :
Lemma 12.
Ve (x) = xV (f (x)) − (1 − x)f (x)
(5)
for all x ∈ [π, 1].
Essentially, when the firm’s posterior belief is x, then either the state is good; in which
case the continuation value of the contract is V (f (x)); or the state is bad, in which case the
continuation value of the contract is the termination fee paid to the agent at that point.
Given the tight linkage between V and Ve provided by Lemma 5 and the smoothness of
Ve , it seems reasonable that V should also be smooth enough to satisfy the HJB equation.
The main stumbling block to this argument is establishing that the function f mapping
beliefs onto golden parachutes is sufficiently smooth. The following proposition represents
a closest approach to showing existence of solutions to the HJB equation of Theorem 1 via
this method.
Proposition 4. Suppose there exists a C 2 function g : [π, 1] → R satisfying
ρg(x) = b −
αx +
∆r
σ
2
!
1
x (1 − x) g (x) +
2
0
2
∆r
σ
2
x2 (1 − x)2 g 00 (x)
on [π, 1] and the boundary condition g(π) = 0. Then g = f , and V is a C 2 function satisfying
the HJB equation on [0, F0∗ ] and the boundary conditions V (0) = 0 and V 0 (F0∗ ) = 0.
Standard ODE existence results guarantee a unique C 2 solution to the ODE of Proposition 4 on [π, 1) for any choice of g 0 (π). However, the ODE is singular at x = 1 due to
the vanishing coefficient on g 00 there, and so it is not guaranteed that any solution is wellbehaved, in particular finite, as x → 1. Thus despite the explicit characterization of an
optimal contract afforded by the quantity approach, some additional regularity assumptions
are needed to ensure that V satisfies the HJB equation.
32
6
Conclusion
I have studied the problem of a firm seeking to dynamically elicit the information of a rentseeking agent about when to terminate a project with a limited lifespan. I show that the
firm’s problem can be analyzed through the lens of classic monopoly theory, as the firm faces
a tight price-quantity tradeoff between payments to the agent and output from the project.
I find that the firm’s optimal contract can be derived by eliminating either price or quantity
from the objective function and optimizing with respect to the remaining variable alone.
While the price approach is equivalent to a standard recursive analysis using continuation
utility as a state variable, the quantity approach provides a much sharper characterization
of the firm’s optimal contract.
In an optimal contract, the firm tracks its posterior beliefs about the project’s state as
if no agent were present even though the agent truthfully reports whether the state has
switched. It optimally terminates the project the first time either the agent reports the state
has changed or its “virtual beliefs” fall below a threshold. This threshold solves a modified
version of the unassisted optimal stopping problem, from which I obtain simple and intuitive
comparative statics. The threshold is strictly lower than the optimal threshold in the firm’s
unassisted optimal stopping problem, and is decreasing in the informativeness of news about
the state, increasing in the rate of state transitions, and increasing in the severity of the
agent’s incentive misalignment. The benefits of hiring an agent can therefore be succinctly
summarized: the agent’s information completely eliminates late termination and partially
mitigates early termination universally across all projects and output histories. The optimal
payment rule can also be stated briefly: when the agent is terminated for reporting a state
change, he receives a lump sum golden parachute equal to his expected lifetime rents from
withholding his report forever.
While I have assumed that the firm’s project produces a Brownian output flow, the
major conclusions of the paper all hold under alternative Poisson technologies involving
either steady output flows and occasional costly “breakdowns” or steady expenditures and
occasional profitable “breakthroughs.” Another natural generalization of the model is to
agents with imperfect knowledge of the state. This change creates an additional incentive
for the agent to delay reporting bad news inducing project termination, as there is a chance
his beliefs about the state will improve in the future and he will not have to report the bad
news after all. In this setting the agent must be additionally compensated for the value of
his real option to wait and see.
Another generalization is to projects with many states. In this setting the agent expects
33
the project to continue deteriorating over time after a termination threshold is reached, so
that delaying a report produces a growing belief asymmetry which can be profitably exploited
by the agent. (By contrast, with only two states the belief asymmetry is constant over time,
and equal to its initial value at the time of the state switch.) The agent must therefore be
additionally compensated for the wedge between the marginal belief asymmetry at the time
of a state switch and his average belief asymmetry over the lifetime of the contract.
References
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35
Appendices
A
A.1
Proofs of theorems
Proof of Theorem 1
Fix F ∗ and V as described in the theorem statement. Lemma 23 in the technical appendix
establishes that V is a positive, strictly increasing, strictly concave function on [0, F ∗ ), and
that maximizers γ ∗ , β ∗ of the HJB equation exist on [0, F ∗ ] and are continuous except possibly
−αF ∗
and F ∗ < b/ρ. Finally, as this proposition
at F ∗ . And by Proposition 2, V (F ∗ ) = rGρ+α
also implies V 00 (F ∗ ) < 0, the maximizers (γ ∗ , β ∗ ) are continuous on [0, F ∗ ].
Extend V to R+ by setting V (F0 ) = V (F ∗ ) for F0 > F ∗ . Then V is a C 1 , piecewise C 2
function on this extended domain and satisfies
1 2 00
0
(ρ + α)V (F0 ) ≥
sup
rG − αF + γV (F0 ) + β V (F0 )
2
(γ,β)∈IC(F )
everywhere, with equality for F0 ∈ [0, F ∗ ].
I now carry out a verification argument for IC-B contracts (Fe, τe) such that Fe is cadlag
and bounded above by b/ρ and τe = inf{t : Fet = 0}. Lemma 5 justifies the boundedness
restriction and form of the termination policy, while Lemma 22 in the technical appendix
justifies the focus on cadlag golden parachutes. Verification within this class therefore suffices
to prove that V is an upper bound on the lifetime value of any IC-B contract.
Fix an IC-B contract (Fe, τe) satisfying the conditions just described. Wlog assume that
Fe = F τe, as the stopped version of any contract remains IC-B and yields the same profits for
the firm in all cases. Define the process M by
b
Mt = (1 − e−ρt ) + e−ρt Ft .
ρ
By Lemma 21 in the technical appendix, M is a PB -supermartingale, and as F is cadlag
and bounded above by b/ρ, so is M. M then satisfies the conditions of the Doob-Meyer
decomposition theorem (see Theorem 1.4.10 of Karatzas and Shreve (1991) for a statement),
hence
M = Fe0 + X − A
for some FY -adapted process A which is increasing, right-continuous, and starts at zero; and
some FY -adapted process X which is a cadlag PB -martingale starting at zero.
36
Further, by the martingale representationn theorem there exists
an FY -adapted, progreso
Rt
sively measurable process β M , satisfying PB 0 (β M )2s ds < ∞ = 1 for all t, such that
t
Z
βsM dZsB
Xt =
0
for all t. And as ZtB = ZtG +
∆r
t,
σ
X may be equivalently decomposed as
Z
Xt =
0
t
∆r M
β ds +
σ s
Z
t
βsM dZsG .
0
Now write Fet in terms of Mt as
b
Fet = eρt Mt − (eρt − 1),
ρ
and use Ito’s lemma to obtain
Z t
Z t
Z t
∆r
ρs
M
ρs
Fet = Fe0 +
e
βs
ρMs − b +
ds +
e dAs +
eρs βsM dZsG ,
σ
0
0
0
or equivalently,
Z t
Fet = Fe0 +
0
∆r e
βs
ρFet − b +
σ
Z
dt −
t
ρs
Z
e dAs +
0
t
βes dZsG ,
0
R
et ≡ t eρs dAs . Then A
e is a monotone increasing right-continuous
where βet ≡ eρt βtM . Let A
0
process started at zero. Thus Fe is a semimartingale to which Ito’s lemma, generalized to
et ≡ A
et − lims↑t A
et track the jumps of A,
e of
processes with jumps, can be applied. Let ∆A
P
et −
e
which there are at most countably many pathwise. I will let Dt ≡ A
0≤s≤t ∆At denote
the continuous part of A.
R
In general it is not assured that βe is sufficiently regular to guarantee that βe dZ G is a PG martingale. I therefore perform the verification procedure using a localized
Rversion ofthe pron
o
t
Y
l,N
cess. For each N = 1, 2, ..., define the F -stopping time τ
≡ inf t : 0 βes dZsG ≥ N .
hR l,N
i
R t∧τ l,N
t∧τ
For each N , the Ito isometry implies that EG 0
βes2 ds ≤ N 2 , and so 0
βes dZsG is
a PG -martingale.
37
Now fix a time t. Ito’s lemma says that
e−(ρ+α)(t∧eτ ∧τ
l,N )
X
V (Fet∧eτ ∧τ l,N ) = V (Fe0 ) +
e−(ρ+α)s ∆Vs
0≤s≤t∧e
τ ∧τ l,N
Z
t∧e
τ ∧τ l,N
+
0
Z
t∧e
τ ∧τ l,N
+
0
Z
+
1 e2 00 e
e
−(ρ + α)V (Fs ) + βs V (Fs ) ds
e
2
∆r e
−(ρ+α)s 0 e
e
e
V (Fs )
ρFt − b +
βs ds − dDs
σ
−(ρ+α)s
t∧e
τ ∧τ l,N
e−(ρ+α)s βes V 0 (Fes ) dZsG ,
0
where ∆Vt ≡ V (Fet ) − lims↑t V (Fes ). Note that ∆V ≤ 0 given that V is monotone increasing
and all jumps in Fe are downward. As V 0 is bounded, the final term is a martingale. Taking
expectations wrt PG leaves

h
i
l,N
V (Fe0 ) = EG e−(ρ+α)(t∧eτ ∧τ ) V (Fet∧eτ ∧τ l,N ) − EG 

X
e−(ρ+α)s ∆Vs 
0<s≤t∧e
τ ∧τ l,N
− EG
"Z
t∧e
τ ∧τ l,N
0
G
"Z
t∧e
τ ∧τ l,N
−E
0
#
1
e−(ρ+α)s −(ρ + α)V (Fes ) + βes2 V 00 (Fes ) ds
2
#
∆r
e−(ρ+α)s V 0 (Fes )
ρFet − b +
.
βes ds − dDs
σ
As ∆V ≤ 0, D is monotone increasing, and V 0 ≥ 0, this expression implies the inequality
h
i
l,N
V (Fe0 ) ≥ EG e−(ρ+α)(t∧eτ ∧τ ) V (Fet∧eτ ∧τ l,N )
"Z
#
t∧e
τ ∧τ l,N
1
− EG
e−(ρ+α)s −(ρ + α)V (Fes ) + βes2 V 00 (Fes ) ds
2
0
"Z
#
t∧e
τ ∧τ l,N
∆r
− EG
e−(ρ+α)s V 0 (Fes ) ρFet − b +
βes ds .
σ
0
Letting γ
e on [0, t] via γ
es =
∆r e
β
σ s
− (b − ρFes ), the previous inequality may be written
h
i
l,N
V (Fe0 ) ≥ EG e−(ρ+α)(t∧eτ ∧τ ) V (Fet∧eτ ∧τ l,N )
"Z
#
t∧e
τ ∧τ l,N
1
− EG
e−(ρ+α)s −(ρ + α)V (Fes ) + γ
es V 0 (Fes ) + βes2 V 00 (Fes ) ds .
2
0
38
Note that (e
γs , βes ) ∈ IC(Fes ) for all s ∈ [0, t], so the extension of the HJB equation stated at
the beginning of the proof implies
"
−(ρ+α)(t∧e
τ ∧τ l,N )
V (Fe0 ) ≥ EG e
Z
t∧e
τ ∧τ l,N
#
e−(ρ+α)s rG − αFes ds .
V (Fet∧eτ ∧τ l,N ) +
0
Finally, take N → ∞ and t → ∞. The interior of the expectation is uniformly bounded over
all t and N , so the bounded converge theorem yields
Z
−(ρ+α)e
τ
G
e
e
V (Fτe) +
V (F0 ) ≥ E e
τe
−(ρ+α)s
e
e
rG − αFs ds .
0
As Feτe = 0 and V (0) = 0, this reduces to
G
Z
V (Fe0 ) ≥ E
τe
−(ρ+α)s
e
rG − αFes ds .
0
By Lemma 7, the rhs is the expected profit of the contract Fe. Therefore V (Fe0 ) is an upper
bound on the expected profits of Fe. As F ∗ is a global maximum of V, it follows that V (F ∗ )
is as well.
The final step of verification is establishing that the optimal controls to the HJB equation
trace out a contract whose profits are equal to V (F ∗ ). Extend γ ∗ and β ∗ continuously to all F0
by setting γ ∗ (F0 ) = γ ∗ (0) and β ∗ (F0 ) = β ∗ (0) for all F0 < 0, and similarly γ ∗ (F0 ) = γ ∗ (F ∗ )
and β ∗ (F0 ) = 0 for all F0 > F ∗ .
Lemma 13. There exists a weak solution to (3) for all time. If limF0 →F ∗ V 0 (F0 )V 000 (F0 )
exists and is finite, then there exists a unique strong solution F for all time. Any strong
solution F satisfies F ≤ F ∗ a.s.
Proof. Because γ ∗ and β ∗ are continuous and bounded, Theorem 5.4.22 of Karatzas and
Shreve (1991) ensures existence of a weak solution. To show existence of a unique strong
solution, I establish under the additional condition in the lemma statement that γ ∗ and β ∗
are both globally Lipschitz continuous, from which Theorem 5.2.9 and 5.2.5 of Karatzas and
Shreve (1991) ensure strong existence and uniqueness, respectively.
It is sufficient to show that β ∗ is differentiable on [0, F ∗ ] with bounded derivative. For
then γ ∗ is differentiable with bounded derivative as well, and the mean value theorem implies
that both are Lipschitz continuous on [0, F ∗ ]. Lipschitz continuity on the extended domain
follows trivially given that γ ∗ and β ∗ are constant outside of [0, F ∗ ].
39
Recall from the proof of Proposition 2 that for F0 ∈ [0, F ∗ ), V satisfies
00
V (F0 ) = −
∆r
σ
2
V 0 (F0 )2
,
Γ(F0 )
where Γ(F0 ) ≡ (ρ + α)V (F0 ) − (rG − αF0 ) + (b − ρF0 )V 0 (F0 ) > 0. As both V 0 and Γ are
continuously differentiable, V 000 (F0 ) exists and is a continuous function on [0, F ∗ ). Then the
derivative of β ∗ exists, is continuous, and is given by
∆r
dβ ∗
(F0 ) =
dF0
σ
V 0 (F0 ) 000
−1 + 00
V (F0 )
V (F0 )2
for F0 < F ∗ . Recall that by Proposition V 00 (F ∗ ) < 0. Then under the assumption of the
lemma statement, the limit of this expression exists and is finite as F0 → F ∗ . Finally, the
dβ ∗
dβ ∗
(F ∗ ) = limF0 →F ∗ dF
(F0 ). Thus β ∗ is continuously
mean value theorem implies that dF
0
0
differentiable on the compact interval [0, F ∗ ], meaning it has a bounded derivative on that
domain.
Now, fix any strong solution F. Suppose that on some ω ∈ Ω, F satisfies Ft (ω) > F ∗ for
some t > 0. Let t0 = sup{s < t : Fs (ω) = F ∗ }. Then a.s. Ft (ω) satisfies the ODE
df
= γ ∗ (F ∗ )
ds
on [t0 , t] with boundary conditions f (t0 ) = F ∗ and f (t) = Ft (ω) > F ∗ . This is a contradiction
of the fact that this ODE has a unique global solution f satisfying f (s) > f (t) for all s < t
given γ ∗ (F ∗ ) < 0. Thus such paths can occur with at most probability zero. In other words,
F ≤ F ∗ a.s.
Assume a strong solution F to (3) exists for all time, and let τ Y = inf{t : Ft =
0}. I show that the expected profits of F are precisely V (F ∗ ). Because β ∗ is bounded,
Rt ∗
β (Fs )V 0 (Fs ) dZsG is a martingale for all time, so no regularization is needed for this step
0
of verification. Fix t and use Ito’s lemma to write
Y
e−(ρ+α)(t∧τ ) V (Ft∧τ Y )
Z t∧τ Y
1 ∗
∗
−(ρ+α)s
∗
0
2 00
= V (F ) +
e
−(ρ + α)V (Fs ) + γ (Fs )V (Fs ) + β (Fs ) V (Fs ) ds
2
0
Z t∧τ Y
+
e−(ρ+α)s β ∗ (Fs )V 0 (Fs ) dZsG .
0
40
Take expectations wrt PG to eliminate the martingale term, leaving
Y
EG [e−(ρ+α)(t∧τ ) V (Ft∧τ Y )]
"Z
Y
t∧τ
= V (F ∗ ) + EG
e−(ρ+α)s
0
#
1
−(ρ + α)V (Fs ) + γ ∗ (Fs )V 0 (Fs ) + β ∗ (Fs )2 V 00 (Fs ) ds .
2
Because Ft ∈ [0, F ∗ ] for t ≤ τ Y and (γ ∗ (F0 ), β ∗ (F0 ))
the rhs of the HJB
equahR maximize
i
Y
t∧τ
tion for F0 ∈ [0, F ∗ ], the final term is equal to −EG 0
e−(ρ+α)s (rG − αFs ) ds , or after
rearrangement
V (F ∗ ) = EG
"Z
#
t∧τ Y
Y
e−(ρ+α)s (rG − αFs ) ds + EG [e−(ρ+α)(t∧τ ) V (Ft∧τ Y )].
0
Now take t → ∞. The interior of each expectation is uniformly bounded for all time, so the
bounded convergence theorem allows limits and expectations to be swapped. As V (Fτ Y ) = 0,
this leaves
"Z
#
τY
∗
e−(ρ+α)s (rG − αFs ) ds .
G
V (F ) = E
0
By Lemma 7, the rhs are the expected profits of F, which was the desired result.
Finally, note that max{F, 0} is an Ito process for t ≤ τ Y with progressively measurable
β = 0. Then
increments γ, β, where γt = γ ∗ (Ft ) and βt = β ∗ (Ft ) ≥ 0 satisfy b − ρFt + γt − ∆r
σ t
Lemma 20 in the technical appendix implies that (max{F, 0}, τ Y ) is an IC contract.
A.2
Proof of Theorem 2
Let Ve be as stated in the theorem, with τ ∗ ≡ inf{t : πt ≤ π} and τ x ≡ inf{t : πt ≤ x} for
any x ≥ π. Then by Ito’s lemma, for all t
−ρ(τ ∗ ∧t−τ x )
e
Z
τ ∗ ∧t
Ve (πτ ∗ ∧t ) = Ve (x) +
−ρ(s−τ x )
e
τx
Z
τ ∗ ∧t
+
1
−ρVe (πs ) − απs Ve 0 (πs ) +
2
x
e−ρ(s−τ ) Ve 0 (πs ) dZ s .
τx
41
∆r
σ
2
!
Ve 00 (πs )
ds
Now, using the fact that Ve satisfies the ODE in the theorem statement on s ≤ τ ∗ , the
previous expression reduces to
e
Z τ ∗ ∧t
x
e
e−ρ(s−τ ) (πs rG − (1 − πs )b) ds
V (πτ ∗ ∧t ) = V (x) −
τx
Z τ ∗ ∧t
x
e−ρ(s−τ ) Ve 0 (πs ) dZ s .
+
−ρ(τ ∗ ∧t−τ x ) e
τx
Because Ve 0 is continuous on a compact domain and therefore bounded, the final term is a
martingale. Further, by the strong Markov property of stochastic integrals wrt Brownian
motions, its conditional expectation wrt Fτ x vanishes. Taking conditional expectations then
yields
Z
−ρ(τ ∗ ∧t−τ x ) e
τ ∗ ∧t
V (πτ ∗ ∧t )] = Ve (x) − Eτ x
Eτ x [e
−ρ(s−τ x )
e
(πs rG − (1 − πs )b) ds .
τx
Now take t → ∞. As the terms in both expectations are bounded, limits and expectations
may be swapped, yielding
Z
τ∗
Ve (x) = Eτ x
−ρ(s−τ x )
e
(πs rG − (1 − πs )b) ds
τx
given the boundary condition Ve (π) = 0. In particular, Ve (1) = Π(τ ∗ ).
Now, suppose π > b/(b + rG ). Then by the expression just derived Ve (x) > 0 for all x > π,
as the flow benefits of operation are always strictly positive on s ≤ τ x and τ x < τ ∗ . But on
the other hand, by the ODE in the theorem statement it follows that
1
2
∆r
σ
2
π 2 (1 − π)2 Ve 00 (π) = −(πrG − (1 − π)b) < 0,
meaning Ve 00 (π) < 0 and implying that for x sufficiently close to π, Ve (x) < 0. Thus π ≤
b/(b + rG ).
Next I claim that Ve is strictly increasing on [π, b/(b + rG )]. This claim is trivial if π =
b/(b + rG ), so assume π < b/(b + rG ) for the following argument. By a similar argument
to the previous paragraph, Ve 00 (π) > 0; then given the boundary conditions at π, Ve 0 (x) > 0
for x sufficiently close to π. Suppose Ve 0 vanishes somewhere on [π, b/(b + rG )), and let x∗ =
inf{x > π : Ve 0 (x) = 0}. Continuity ensures Ve 0 (x∗ ) = 0, and by assumption x∗ < b/(b + rG ).
As Ve 0 (x) > 0 for all x ∈ (π, x∗ ), and as Ve (π) = 0, it must be that Ve (x∗ ) > 0. But then by
42
the ODE Ve 00 (x∗ ) > 0, implying that Ve 0 (x) < 0 for x below and sufficiently close to x∗ . This
contradicts the definition of x∗ as the minimal point at which Ve 0 vanishes. Thus no such
point can exist, i.e. Ve 0 > 0 on [π, b/(b + rG )), proving the claim.
Now fix x > b/(b + rG ). Using the expression for Ve (x) derived earlier and the law of
iterated expectations, one can show that
"Z
#
τ b/(b+rG )
Ve (x) = Eτ x
e
−ρ(s−τ x )
h
i
b/(b+rG ) −τ x )
(πs rG − (1 − πs )b) ds +Eτ x e−ρ(τ
Ve (b/(b + rG )) .
τx
Then as the first term is strictly positive while the second is non-negative, I conclude that
Ve (x) > 0 for all x > b/(b + rG ), and hence Ve (x) > 0 for all x > π.
I have now established enough properties of Ve to verify the optimality of τ ∗ . Extend Ve
to [0, 1] by setting Ve (x) = 0 for x < π. Note that on this extended domain Ve is C 1 and
piecewise C 2 , so Ito’s lemma still applies. Further, given π ≤ b/(b + rG ), Ve satisfies
1
ρVe (π0 ) ≥ π0 rG − (1 − π0 )b − απ0 V (π0 ) +
2
e0
∆r
σ
2
π02 (1 − π0 )2 Ve 00 (π0 )
on [0, 1] \ {b/(b + rG )}, with equality iff π0 > π. (Ve 00 is potentially discontinuous at π0 = π, so
I do not specify an inequality holding at that point. This will not matter for what follows.)
Fix any FY -stopping time τ Y (not necessarily a threshold policy), and use Ito’s lemma as
before to write
−ρ(τ Y
e
Z
∧t) e
τ Y ∧t
e−ρs
V (πτ Y ∧t ) = Ve (1) +
0
Z
+
1
−ρVe (πs ) − απs V 0 (πs ) +
2
∆r
σ
2
!
πs2 (1 − πs )2 Ve 00 (πs )
τ Y ∧t
e−ρs Ve 0 (πs ) dZ s .
0
Given the inequality just derived, I obtain
e
Z
−ρ(τ Y ∧t) e
V (πτ Y ∧t ) ≤ Ve (1) −
τ Y ∧t
−ρs
e
Z
(πs rG − (1 − πs )b) ds +
0
τ Y ∧t
e−ρs Ve 0 (πs ) dZ s .
0
Take expectations to eliminate the final term, and then take t → ∞ and swap limits and
expectations. I am left with
"Z
Ve (1) ≥ E
τY
#
Y
Y
e−ρs (πs rG − (1 − πs )b) ds + E[e−ρτ Ve (πτ Y )] = Π(τ Y ) + E[e−ρτ Ve (πτ Y )].
0
43
ds
Given that Ve ≥ 0, the final term is non-negative and we’re left with Ve (1) ≥ Π(τ Y ). Thus
given that Π(τ ∗ ) = Ve (1), τ ∗ is an optimal termination policy.
Finally, I verify that Ve is the firm’s virtual value function, in the sense that Ve (πt ) = Rt
for all time. To verify this, fix any τ Y ≥ t and use a variant of the argument above to
conclude that
"Z
#
τY
e−ρ(s−t) (πs rG − (1 − πs )b) ds ,
Ve (πt ) ≥ Et
t
with equality when τ Y = inf{s ≥ t : πt ≤ π} given that the two inequalities in the
verification argument hold with equality for such a policy. Thus
"Z
e−ρ(s−t) (πs rG − (1 − πs )b) ds
Ve (πt ) = sup Et
τ Y ≥t
#
τY
t
for all time a.s., as desired.
A.3
Proof of Theorem 3
Theorem 2 in conjunction with Lemma 10 imply the existence of a threshold π ∈ (0, b/(rG +b)]
such that τ ∗ = inf{t : πt ≤ π} optimizes Π(τ Y ) among all FY -stopping times τ Y . Therefore
by Proposition 3 (F ∗ , τ ∗ ) is an optimal IC-B contract. It remains only to establish that
(F ∗ , τ ∗ ) is an IC contract.
Define a process X by
Z
τ∗
Xt =
e−ρs b ds
EB
t
0
∗
∗
for t ≤ τ , with Xt = Xτ ∗ for t > τ . Then X is a FY -adapted square-integrable (indeed
Y
bounded) PB -martingale, so by the martingale representation
hR theorem ithere exists an F t
adapted, progressively measurable process β X satisfying EB 0 (βsX )2 ds < ∞ for all t such
that
Z t
βsX dZsB
Xt =
0
for all time. Next note that
b
Xt = (1 − e−ρt ) + e−ρt Ft∗
ρ
for t ≤ τ Y , hence by Ito’s lemma
Ft∗
∗
Z
t
ρs
Z
e (ρXs − b) ds +
=F +
0
t
eρs βsX
dZsB
Z
44
Z
(ρFs − b) ds +
=
0
0
t
0
t
βs dZsB ,
where βt ≡ eρt βtX . Written in terms of Z G , this becomes
∗
Z
Ft = F +
t
Z
γs ds +
0
t
βs dZsG ,
0
where γt ≡ ρFt − b + ∆r
β . So F is an Ito process satisfying b − ρFt + γt − ∆r
β = 0 for all
σ t
σ t
Y
t ≤ τ . Additionally, a change of measure
combined
with the Cauchy-Schwartz inequality
i
hR
t
and the Ito isometry implies that EG 0 βs2 ds < ∞ for all t.
If in addition β ≥ 0, then Lemma 20 in the technical appendix implies that (F ∗ , τ ∗ ) is
an IC contract. This must be true, because Ft∗ is increasing in the current value of πt (see
Lemma 11), and πt increases in response to positive news shocks. If the loading on dZtG
were ever negative, it would be possible to find paths of output over which πt grows but Ft∗
shrinks, a contradiction.
45
B
B.1
Proofs of propositions
Proof of Proposition 1
Divide the first equation of the system in the proposition statement through by a1 and solve
for 1/a1 to obtain
1
1
a2
=−
ρ + 2b
.
a1
α
a1
Divide the second equation through by a1 a2 and insert the equality above to obtain
1
4
∆r
σ
2
a1 2brG a2
rG ρ
+
+b+
= 0.
a2
α a1
α
Multiply through by a1 /a2 to obtain a quadratic equation satisfied by a1 /a2 :
1
4
∆r
σ
2 a1
a2
2
ρrG a1 2brG
+
= 0.
+ b+
α a2
α
Letting F ∗ = −a1 /(2a2 ), F ∗ is a root of the quadratic polynomial
α
φ(F ) ≡
2
∆r
σ
2
F 2 − (αb + ρrG )F + brG .
The two roots of φ are
F±∗ =
Now, by assumption
∆r 2
σ
(αb + ρrG ) ±
∆r 2
σ
.
= ρ − α. Therefore
2
q
(αb + ρrG )2 − 2αbrG
2
α ∆r
σ
(αb + ρrG ) − 2αbrG
∆r
σ
2
> (αb + ρrG )2 − 2αbρrG = (αb)2 + (ρrG )2 > 0.
So the discriminant of the quadratic is strictly positive, and there exist two distinct positive
roots of the equation. Note that
α
φ(b/ρ) =
2
∆r
σ
2
b2
b
α
b2
αb2
α 2 b2
−
(αb
+
ρr
)
+
br
=
(ρ
−
α)
−
=
−
< 0,
G
G
ρ2
ρ
2
ρ2
ρ
2ρ2
46
hence F−∗ < b/ρ < F+∗ . Recall also that
a1 = −
α
.
ρ − b/F ∗
Hence there exists one solution to the system of equations, corresponding to F−∗ , with a1 > 0,
and another, corresponding to F+∗ , with a1 < 0. As F ∗ is positive for both solutions, there is
one solution satisfying a1 > 0 > a2 , and another satisfying a2 > 0 > a1 . The first solution is
the one claimed in the theorem statement, and I will restrict attention to it going forward.
Now, let V (F ) = a2 F 2 + a1 F . Note that V is strictly increasing and strictly concave on
[0, F ∗ ], and that V (0) = 0 while V 0 (F ∗ ) = 0 and V 00 (F ∗ ) = 2a2 < 0. Then inserting V into
the rhs of the HJB equation, there are unique maximizers
β ∗ (F ) = −
∆r ∗
∆r V 0 (F )
=
(F − F )
00
σ V (F )
σ
and
∆r ∗
β (F ) − (b − ρF )
σ
!
2
∆r
=
− ρ (F ∗ − F ) − (b − ρF ∗ )
σ
γ ∗ (F ) =
= −α(F ∗ − F ) − (b − ρF ∗ ).
for all F ∈ [0, F ∗ ). The continuous extension of γ ∗ and β ∗ to F = F ∗ constitute the unique
continuous maximizers of the HJB equation on the extended domain. The HJB equation
then becomes
∆r
(α+ρ)(a2 F +a1 F ) = rG −αF −(α(F − F ) + (b − ρF )) (2a2 F +a1 )+
σ
2
∗
∗
2
(F ∗ −F )2 a2 .
As this equation must hold for all F, the two sides of the equation must match term by term.
The zeroeth order term is
∗
0 = rG − a1 ((α − ρ)F + b) +
47
∆r
σ
2
F ∗ 2 a2 .
∆r 2
σ
Using the fact that F ∗ = −a1 /(2a2 ) and ρ − α =
1
0 = rG − ba1 −
4
∆r
σ
, this may equivalently be written
2
a21
,
a2
which is just a rearrangement of the second equation defining a1 and a2 . So the zeroeth order
terms match.
Meanwhile the first-order terms match if
∗
(α + ρ)a1 = −α − 2a2 ((α − ρ)F + b) + a1 α − 2a2 F
Again using ρ − α =
∆r 2
σ
∗
∆r
σ
2
.
, this reduces to
ρa1 = −α − 2a2 b,
which is just a rearrangement of the first equation defining a1 and a2 . So the first-order terms
match.
Finally, the second-order terms match if
(α + ρ)a2 = 2αa2 +
This is implied by ρ = α +
equation on [0, F ∗ ].
B.2
∆r 2
σ
∆r
σ
2
a2 .
, so second-order terms match. Thus V satisfies the HJB
Proof of Proposition 2
Suppose first that V satisfies the HJB equation. Then Lemma 23 in the technical appendix
establishes that V is strictly increasing and strictly concave on [0, F ∗ ), and that
β ∗ (F0 ) = −
∆r V 0 (F0 )
,
σ V 00 (F0 )
γ ∗ (F0 ) =
∆r ∗
β (F0 ) + ρF − b
σ
are maximizers of the HJB equation on [0, F ∗ ). Substituting these maximizers into the HJB
equation yields
1
(ρ + α)V (F0 ) = rG − αF0 + (b − ρF0 )V (F0 ) −
2
0
48
∆r
σ
2
V 0 (F0 )2
V 00 (F0 )
for F0 ∈ [0, F ∗ ). And because V 0 (F0 ) > 0 and V 00 (F0 ) < 0, it must be that (ρ + α)V (F0 ) −
(rG − αF0 ) − (b − ρF0 )V 0 (F0 ) > 0, so this equation may be solved for V 00 (F0 ) to obtain the
ODE in the proposition statement.
Conversely, suppose V is strictly increasing and strictly concave and satisfies the ODE
in the proposition statement. Rearranging it yields
(ρ + α)V (F0 ) = rG − αF0 + b − ρF0 −
∆r
σ
2
V 0 (F0 )
V 00 (F0 )
!
1
V (F0 ) +
2
0
∆r V 0 (F0 )2
σ V 00 (F0 )
2
V 00 (F0 ),
or
1
(ρ + α)V (F0 ) = rG − αF0 + γ ∗ (F0 )V 0 (F0 ) + β ∗ (F0 )2 V 00 (F0 ).
2
∗
∗
And the proof of Lemma 23 shows that (γ (F0 ), β (F0 )) maximizes the rhs subject to IC-B
given any strictly increasing, strictly concave V. So V satisfies the HJB equation.
Now, for V satisfying the conditions of the proposition, write the ODE satisfied by V on
[0, F ∗ ) as V 00 = G(f, V, V 0 ), where
1
G(f, V, V ) = −
2
0
∆r
σ
2
(V 0 )2
(ρ + α)V − (rG − αf ) + (b − ρf )V 0
is defined on the domain U = R3 \ {(f, V, V 0 ) : (ρ + α)V − (rG − αf ) + (b − ρf )V 0 = 0},
−αF ∗
. Then x0 = (F ∗ , V (F ∗ ), 0) ∈ U, and as V
with U an open set. Suppose V (F ∗ ) 6= rGρ+α
is C 2 by assumption, V satisfies V 00 = G(f, V, V 0 ) on the entire domain [0, F ∗ ]. Let . For
sufficiently small open sets U 0 containing x0 , U 0 ⊂ U and G is continuous in f and has
bounded first derivatives wrt V and V 0 on U 0 , hence is uniformly Lipschitz continuous wrt
(V, V 0 ) for fixed f in U. Then by the Picard-Lindelof theorem, there exists a unique solution
of V 00 = G(f, V, V 0 ) locally around x0 . Now observe that Ve (f ) = V (F ∗ ) is a solution of
the ODE sufficiently close to x0 , thus V (F0 ) = V (F ∗ ) for F0 sufficiently close to F ∗ . This
−αF ∗
.
contradicts the fact that V is strictly increasing on [0, F ∗ ). So V (F ∗ ) = rGρ+α
Finally, define Γ : [0, F ∗ ] → R by
Γ(F0 ) ≡ (b − ρF0 )V 0 (F0 ) + (ρ + α)V (F0 ) − (rG − αF0 ).
As V is a C 2 function, Γ is a C 1 function. Using Γ, the ODE satisfied by V on [0, F ∗ ) may
be written
2 0
1 ∆r
V (F0 )2
Γ(F0 ) = −
.
2 σ
V 00 (F0 )
Given V 0 (F0 ) > 0 and V 00 (F0 ) < 0 for F < F ∗ , it must be that Γ(F0 ) > 0 for F < F ∗ . And
49
as V (F ∗ ) =
rG −αF ∗
ρ+α
and V 0 (F ∗ ) = 0, Γ(F ∗ ) = 0. It must therefore be that Γ0 (F ∗ ) ≤ 0. As
Γ0 (F0 ) = (b − ρF0 )V 00 (F0 ) + α(1 + V 0 (F0 ))
and V 0 (F ∗ ) = 0, this implies
(b − ρF ∗ )V 00 (F ∗ ) ≤ −α.
As V 00 (F ∗ ) ≤ 0, if F ∗ ≥ b/ρ this inequality would be violated. So F ∗ < b/ρ. The inequality
would also be violated if V 00 (F ∗ ) = 0, so V 00 (F ∗ ) < 0.
B.3
Proof of Proposition 3
For each FY -stopping time τ Y , let F be the golden parachute defined by
"Z
Ft = EB
t
#
τY
e−ρ(s−t) b ds .
t
Lemma 6 says that
Π(τ Y ) = E
"Z
#
τ θ ∧τ Y
−ρ(τ θ ∧τ Y
e−ρt rG dt − e
)
Fτ θ ∧τ Y .
0
Lemma 14. e−ρ(τ
θ ∧τ Y
)
Fτ θ ∧τ Y = Eτ θ
hR
τY
τ Y ∧τ θ
i
e−ρt b dt a.s.
Proof. Note that 1{τ Y ≥ τ θ } is measurable wrt Fτ θ . Then as
Z
τY
−ρt
e
Y
Z
θ
τY
Eτ θ
e−ρt b dt
b dt = 1{τ ≥ τ }
τ Y ∧τ θ
it follows that
"Z
τY
τθ
#
e−ρt b dt = 1{τ Y ≥ τ θ }Eτ θ
"Z
τ Y ∧τ θ
#
τY
e−ρt b dt = 0 = e−ρ(τ
τθ
on {τ Y < τ θ }. On the other hand, on {τ Y ≥ τ θ }
"Z
#
τY
e
Eτ θ
−ρt
"Z
b dt = Eτ θ
τ Y ∧τ θ
−ρt
e
τθ
50
#
τY
b dt .
θ ∧τ Y
)
Fτ θ ∧τ Y
Further, on {τ θ = s}, a standard property of stopping time sigma algebras yields
"Z
#
τY
e−ρt b dt = Es
Eτ θ
"Z
τθ
#
τY
e−ρt b dt = Es
"Z
τθ
#
τY
e−ρt b dt .
s
And on {s ≥ τ θ }
"Z
#
τY
e−ρt b dt = EB
s
Es
"Z
s
#
τY
e−ρt b dt .
s
Thus on {τ Y ≥ τ θ } ∩ {τ θ = s},
"Z
#
τY
Eτ θ
τY
∧τ θ
e−ρt b dt = EB
s
"Z
#
τY
e−ρt b dt = e−ρs Fs = e−ρ(τ
θ ∧τ Y
)
Fτ θ ∧τ Y .
s
This relationship holds for all s, proving the result.
Therefore by the law of iterated expectations,
Π(τ Y ) = E
"Z
τ θ ∧τ Y
0
e−ρt rG dt −
Z
#
τY
e−ρt b dt = E
"Z
τ θ ∧τ Y
τ θ ∧τ Y
e−ρt (rG + b) dt −
Z
#
τY
e−ρt b dt .
0
0
Another application of the law of iterated expectations yields
"Z
E
τ θ ∧τ Y
#
e−ρt (rG + b) dt = E
"Z
0
#
τ θ ∧τ Y
e−ρt (rG + b) dt
0
∞
Z
θ
Y
−ρt
1{t ≤ τ }1{t ≤ τ }e
=E
(rG + b) dt
0
∞
Z
EYt [1{t
=E
0
"Z
=E
τY
θ
Y
≤ τ }]1{t ≤ τ }e
#
πt e−ρt (rG + b) dt .
0
Thus
Π(τ Y ) = E
"Z
#
τY
e−ρt (πt rG − (1 − πt )b) dt .
0
51
−ρt
(rG + b) dt
B.4
Proof of Proposition 4
Let g be as hypothesized. Fix x ∈ [π, 1] and let px be the unique strong solution to the SDE
t
Z
−αps −
p t = p0 +
0
∆r
σ
2
!
p2s (1
− ps )
Z
t
ds +
0
∆r
ps (1 − ps ) dZsB
σ
with initial condition p0 = x satisfying 0 < pxt < 1 for all t > 0 a.s. (See the proof of Lemma
11 for a proof of existence and uniqueness.) Define τxY ≡ inf{t : pxt ≤ π}. By Ito’s lemma
Y
g(x) = e−ρτx g(pxτxY )
Z τxY
e−ρt −ρg(pxt ) −
−
αpxt +
0
1
+
2
τxY
Z
∆r
σ
2
∆r
σ
!
(pxt )2 (1 − pxt )2 g 00 (pxt )
2
!
(pxt )2 (1 − pxt ) g 0 (pxt )
dt
∆r −ρt x
e pt (1 − pxt ) dZtB .
σ
−
0
As px is pathwise continuous a.s., g(pxτxY ) = g(π) = 0. Hence
Z
τxY
g(x) = −
e−ρt
∆r
αpxt +
σ
!
−ρg(pxt ) −
0
1
+
2
τxY
Z
−
2
∆r
σ
e−ρt
0
(pxt )2 (1 − pxt )2 g 00 (pxt )
2
!
(pxt )2 (1 − pxt ) g 0 (pxt )
dt
∆r x
p (1 − pxt ) dZtB .
σ t
Given the ODE satisfied by g, this reduces to
Z
g(x) =
τxY
e
−ρt
τxY
Z
b dt −
0
0
e−ρt
∆r x
p (1 − pxt ) dZtB .
σ t
Now take expectations wrt PB to eliminate the martingale term, yielding
g(x) = EB
"Z
τxY
#
e−ρt b dt = f (x).
0
For the remainder of this proof, assume f satisfies the ODE in the theorem statement. I
52
first establish that f 0 > 0. Given that f is strictly increasing by Lemma 11 and f (1) < b/ρ,
f < b/ρ. Then for any x < 1 such that f 0 (x) = 0, the ODE implies that f 00 (x) < 0. This
contradicts the fact that f is strictly increasing, hence f 0 (x) > 0 for all x < 1. And when
x = 1 the ODE implies f 0 (x) = (b − ρf (1))/α > 0. Thus f 0 > 0.
Then using the implicit function theorem, f −1 exists and is a C 2 function, implying V
is also a C 2 function given Lemma 11. Differentiating the expression in that lemma twice
yields
Ve 0 (x) = xV 0 (f (x))f 0 (x) + V (x) + f (x) − (1 − x)f 0 (x)
and
Ve 00 (x) = 2V 0 (f (x))f 0 (x) + xf 00 (x)V 0 (f (x)) + f 0 (x)2 V 00 (f (x)) + 2f 0 (x) − (1 − x)f 00 (x).
Substituting these expressions into the ODE of Theorem 2 and rearranging yields
(ρ + α)V (f (x))
1 2 2
2 00
0
2
2 0
= rG − αf (x) + −αxf (x) + k x(1 − x) f (x) + k x (1 − x) f (x) V 0 (f (x))
2
1 2 2
+ k x (1 − x)2 f 0 (x)2 V 00 (f (x))
2
1−x
1 2 2
2 00
2 2
0
+
ρf (x) − b + (αx + k x (1 − x))f (x) − k x (1 − x) f (x) ,
x
2
where k ≡ ∆r
. Now, given the ODE satisfied by f, the final term vanishes and the coefficient
σ
0
in front of V may be simplified, yielding
(ρ + α)V (f (x)) = rG − αf (x) + (ρf (x) − b + k 2 x(1 − x)f 0 (x))V 0 (f (x))
1
+ k 2 x2 (1 − x)2 f 0 (x)2 V 00 (f (x)).
2
Defining β(x) ≡ kx(1 − x)f 0 (x) and γ(x) ≡ ρf (x) − b + kβ(x), this equation may be written
1
(ρ + α)V (f (x)) = rG − αf (x) + γ(x)V 0 (f (x)) + β(x)2 V 00 (f (x)).
2
Note that (γ(x), β(x)) ∈ IC(f (x)) for all x. Thus
1 2 00
0
(ρ + α)V (F0 ) ≤
sup
rG − αF0 + γV (F0 ) + β V (F0 )
2
(γ,β)∈IC(F0 )
53
for all F0 ∈ [0, F0∗ ].
To establish the opposite inequality, fix any F0 ∈ (0, F0∗ ) and (γ, β) ∈ IC(F0 ). Without
loss suppose that (γ, β) lies in the interior of IC(F0 ), i.e. b − ρF0 + γ − kβ < 0. For if
1
(ρ + α)V (F0 ) ≥ rG − αF0 + γV 0 (F0 ) + β 2 V 00 (F0 )
2
for all such (γ, β), then continuity of the rhs in (γ, β) shows that it holds everywhere on
IC(F0 ), and thus that
1 2 00
0
(ρ + α)V (F0 ) ≥
sup
rG − αF0 + γV (F0 ) + β V (F0 ) .
2
(γ,β)∈IC(F0 )
Define a fee process Fe by
Fet = F0 + γt + βZtG .
Also let φ : [0, F ∗ ] → R be the auxiliary function
1
φ(f ) ≡ −(ρ + α)V (f ) + rG − αf + γV 0 (f ) + β 2 V 00 (f ).
2
As V is a C 2 function, φ is continuous. It is sufficient to show that φ(F0 ) ≤ 0.
Suppose by way of contradiction that φ(F0 ) > 0. Define a stopping time τe by
τe ≡ inf{t : b − ρFet + γ − kβ = 0 or φ(Fet ) = 0 or Fet ∈ {0, F0∗ }}.
Note that τe > 0 a.s. given that φ is continuous and Fe is pathwise continuous a.s. Also, by
construction φ(Fet ) > 0 for all t < τe. By Ito’s lemma
e
−(ρ+α)e
τ
Z
V (Feτe) = V (F0 ) +
Z
+
τe
τe
−(ρ+α)s
e
0
1
2
00
0
−(ρ + α)V (Fes ) + γV (Fes ) + β V (Fes ) ds
2
βV 0 (Fes ) dZsG .
0
The final term is a martingale whose expectation wrt PG vanishes. Hence
G
E
h
−(ρ+α)(e
τ)
e
Z
i
G
V (Feτe) = V (F0 ) + E
τe
e
−(ρ+α)s
0
54
1
−(ρ + α)V (Fes ) + γV (Fes ) + β 2 V 00 (Fes )
2
0
ds .
By construction
G
Z
τe
e
E
0
−(ρ+α)s
1
−(ρ + α)V (Fes ) + γV 0 (Fes ) + β 2 V 00 (Fes )
2
so
G
τe
Z
e
V (F0 ) < E
−(ρ+α)s
0
G
Z
ds > −E
τe
e
−(ρ+α)s
e
(rG − αFs ) ds ,
0
−(ρ+α)(e
τ)
e
e
(rG − αFs ) ds + e
V (Fτe) .
But then the contract employing golden parachute Fe until τe and then following an optimal
parachute started at Feτe afterward satisfies IC-B and provides strictly higher expected profits
than using the optimal contract starting at F0 for all time. This is a contradiction, hence
φ(F0 ) ≤ 0. As discussed earlier, it follows that
1 2 00
0
(ρ + α)V (F0 ) ≥
sup
rG − αF0 + γV (F0 ) + β V (F0 )
2
(γ,β)∈IC(F0 )
for all F0 ∈ (0, F0∗ ). The fact that V is C 2 and IC(·) is a continuous correspondence implies
that the inequality holds also at the boundaries. Hence V satisfies the HJB equation.
As for the boundary conditions, the fact that V (0) = 0 is automatic from the fact that
any contract with initial golden parachute of 0 must terminate immediately. It remains only
to show that V 0 (F0∗ ) = 0. Surely V 0 (F0∗ ) ≥ 0, as otherwise at F ∗ the rhs of the HJB equation
would be infinite. So suppose by way of contradiction that V 0 (F0∗ ) > 0. Surely V 00 (F ∗ ) < 0,
or else again the rhs of the HJB equation would be infinite. Thus at F0∗ there exist unique
maximizers of the HJB equation
β ∗ (F0∗ ) = −k
V 0 (F0∗ )
,
V 00 (F0∗ )
γ ∗ (F0∗ ) = −b + ρF0∗ + kβ ∗ (F0∗ ).
In particular, β ∗ (F0∗ ) > 0. But we saw earlier that β(x) = kx(1 − x)f 0 (x) and γ(x) =
ρf (x) − b + kβ(x) saturate the rhs of the HJB equation for all x, hence given the uniqueness
of the maximizers β ∗ (F0∗ ) = 0. This is a contradiction, so V 0 (F0∗ ) = 0.
55
C
C.1
Proofs of lemmas
Proof of Lemma 1
Clearly IC-G and IC-B are implied by incentive-compatibility. For the converse result,
suppose a contract C = (Φ, τ ) satisfies IC-G and IC-B, and fix an arbitrary F-stopping time
σ. Let σ ≡ σ ∧ τ θ and σ ≡ σ ∨ τ θ . Then by IC-G,
"Z
ττ
θ
e−ρt b dt + dΦτt
E
θ
#
0
Z
≥E
τσ
−ρt
e
(b dt +
dΦσt )
0
"
= E 1{σ ≤ τ θ }
τσ
Z
e−ρt (b dt + dΦσt ) + 1{σ > τ θ }
ττ
Z
θ
τθ
−ρt
(b dt +
dΦσt )
(b dt +
dΦσt )
e−ρt b dt + dΦt
#
,
0
0
or after subtracting the final term from both sides,
"
Z
θ
ττ
θ
E 1{σ ≤ τ }
e
−ρt
τθ
b dt + dΦt
#
θ
Z
τσ
≥ E 1{σ ≤ τ }
e
.
0
0
A very similar argument using σ and the IC-B constraint yields
"
θ
Z
ττ
θ
E 1{σ > τ }
e
−ρt
τθ
b dt + dΦt
#
θ
Z
≥ E 1{σ > τ }
τσ
−ρt
e
0
0
Sum these two inequalities to obtain
"Z
ττ
θ
e
E
−ρt
τθ
b dt + dΦt
#
Z
≥E
0
τσ
−ρt
e
(b dt +
dΦσt )
.
0
As this inequality holds for arbitrary σ, C is an incentive-compatible contract.
56
.
C.2
Proof of Lemma 2
e τe) by τet = τ t ∧ t for
Suppose C = (Φ, τ ) is an IC-B contract. Define a new contract Ce = (Φ,
e ts = Φts for all t and s < τet , and
all t, Φ
e t = Φt t + EBt
Φ
s
τe
τe
"Z
τt
#
−ρ(s−e
τ t)
e
τet
(b ds + dΦts )
for all t and s ≥ τet .
e I will need to make a minor technical
To compute the agent’s expected utility under C,
e B be the conditional expectation under PB wrt Ft . (Recall that EB was defined
detour. Let E
t
t
to be the conditional expectation wrt
available to the agent at time t.)
FtY ,
which does not capture the full information
Lemma 15. For any F-stopping time σ and every time t,
e Bσ
E
τe
τσ
Z
e
−ρ(s−e
τσ)
τeσ
(b ds + dΦσs ) = EB
τet
"Z
τt
#
−ρ(s−e
τ t)
e
τet
(b ds + dΦts ) .
whenever σ = t.
Proof. Fix any F-stopping time σ. By a standard property of stopping time sigma algebras,
e Bσ
E
τe
τσ
Z
τeσ
σ
e Bt
e−ρ(s−eτ ) (b ds + dΦσs ) = E
τe
"Z
#
τt
τet
t
e−ρ(s−eτ ) (b ds + dΦts )
whenever σ = t. Now, the interior of the latter expectation is a function of the path of output
only; as the marginal distribution over future output under PB is the same conditional on
Ft and FtY , it is therefore the case that
e Bt
E
τe
"Z
τt
#
−ρ(s−e
τ t)
e
τet
(b ds + dΦts ) = EB
τet
"Z
#
τt
e
τet
−ρ(s−e
τ t)
(b ds + dΦts )
for all t.
The agent’s expected utility under Ce and an arbitrary reporting policy σ ≥ τ θ may
therefore be written
Z τeσ
Z τ σ
−ρ(s−e
τσ)
σ
σ
−ρt
σ
−ρe
τ σ eB
e
U =E
e (b dt + dΦt ) + e
Eτeσ
e
(b ds + dΦs ) ,
τeσ
0
57
Given σ ≥ τ θ , either τeσ ≥ τ θ or τeσ = τ σ . In either case
e Bσ
E
τe
Z
τσ
−ρ(s−e
τσ)
e
(b ds +
τeσ
eσ)
dΦ
s
τσ
Z
= Eτeσ
−ρ(s−e
τσ)
e
(b ds +
τeσ
eσ)
dΦ
s
,
as the conditional measure under P wrt the information in Ft is the same as under PB when
t ≥ τ θ . Then by the law of iterated expectations
σ
Z
τσ
−ρt
U =E
e
(b dt +
dΦσt )
.
0
The agent’s expected utility under a given reporting strategy is therefore the same under C
e so the fact that the former contract is IC-B implies the latter is as well.
and C,
e and τe to be F-adapted. The firm’s
As Ce is IC-B, I suppress superscripts and consider Φ
expected profits under Ce are
"Z
τ ∧τ θ
−ρt
e =E
Π
e
(rG dt − dΦt ) −
θ
eB θ
e−ρ(τ ∧τ ) E
τ ∧τ
Z
τ
−ρ(t−τ ∧τ θ )
e
#
(b dt + dΦt ) .
τ ∧τ θ
0
As before, it is true that
eB θ
E
τ ∧τ
Z
τ
−ρ(t−τ ∧τ θ )
e
Z
(b dt + dΦt ) = Eτ ∧τ θ
τ ∧τ θ
τ
e
−ρ(t−τ ∧τ θ )
(b dt + dΦt ) ,
τ ∧τ θ
and so by the law of iterated expectations
"Z
τ ∧τ θ
e−ρt rG dt +
e =E
Π
Z
τ
e−ρt (−b) dt −
Z
τ ∧τ θ
0
τ
#
e−ρt dΦt .
0
By comparison, the firm’s profits under C are
"Z
Π=E
0
τ ∧τ θ
e−ρt rG dt +
Z
τ
τ ∧τ θ
e−ρt rB dt −
Z
τ
#
e−ρt dΦt .
0
e ≥ Π, and this inequality is strict if
Recall that 0 > rB + b by assumption. Therefore Π
τ > τ ∧ τ θ = τe with positive probability.
θ
θ
θ
Finally, note that formally speaking τe should be written τeτ = τ τ ∧ τ θ , and τ τ ∧ τ θ =
θ
τ ∞ ∧ τ θ given that {τ ∞ ≤ t} = {τ τ ≤ t} whenever t ≤ τ θ . As τ ∞ is a FY -stopping time,
this proves the last claim of the lemma.
58
C.3
Proof of Lemma 3
Fix an IC-B contract C = (Φ, τ ) such that τ t = τ Y ∧ t for some FY -stopping time τ Y . Define
e τ ) as follows: when t < ∞, set Φ
e ts = 0 for s < τ t and
a new contract Ce = (Φ,
e t = eρτ t
Φ
s
Z
τt
e−ρu dΦtu
0
e t = 0. The
for s ≥ τ t . As τ t ≤ t this construction is well-defined. And for t = ∞ set Φ
agent’s expected utility under Ce and any F-stopping time σ is then just
σ
τσ
Z
U =E
e
−ρt
b dt + 1{σ < ∞}e
−ρτ σ
τσ
Z
σ
e
∆Φτ σ = E
0
−ρt
e
Z
τσ
b dt + 1{σ < ∞}
0
e
−ρt
σ
e
dΦt .
0
This is weakly lower than the agent’s expected utility from σ under C, and identical when
σ < ∞. As truthful reporting satifies this condition, the fact that C is IC-B means Ce is as
well. A nearly identical calculation shows that the firm’s expected profits under C and Ce are
the same.
e tt . Note that in general Φ
e ts = Φ
e t t 1{s ≥ τ t }
Now, define a FY -adapted process F by Ft = Φ
τ
e t = Fτ t 1{s ≥ τ t } whenever τ t = t. Consider the
for all t and s. Then in particular Φ
s
t
et t = Φ
e τ tt . For in this state of the
remaining case that τ < t. In this case it must be that Φ
τ
τ
world the project is immediately terminated regardless of whether the agent reports a state
switch at time t. If the payoffs at that time are not equal, then all agents will make the same
report regardless of their type, either all reporting a state switch at t or else refraining from
reporting depending on which yields a higher payoff. This violates IC, so the report at time
e t = Fτ t 1{s ≥ τ t } in all cases. This establishes the
t must not impact payments. Therefore Φ
s
representation in the theorem statement.
C.4
Proof of Lemma 4
Fix a contract (F, τ Y ). IC-B holds iff
"Z
#
τ Y ∧τ θ
e−ρt b dt + e
E
−ρ(τ Y
∧τ θ )
"Z
−ρ(τ Y
e−ρt b dt + e
Fτ Y ∧τ θ ≥ E
0
#
τ Y ∧σ
∧σ)
Fτ Y ∧σ
0
for every F-stopping time σ ≥ τ θ . This inequality may be rearranged to obtain
"
−ρτ θ
0 ≥ E 1{τ Y > τ θ }e
Z
τ Y ∧σ
!#
−ρ(t−τ θ )
e
τθ
59
−ρ(τ Y
b dt + e
∧σ−τ θ )
Fτ Y ∧σ − Fτ θ
.
Then by applying the law of iterated expectations, IC-B is equivalent to the condition that
"
0 ≥ E 1{τ Y > τ θ }e
"Z
−ρτ θ
τ Y ∧σ
#
−ρ(t−τ θ )
e
Eτ θ
b dt + e
−ρ(τ Y
∧σ−τ θ )
!#
Fτ Y ∧σ − Fτ θ
τθ
for every σ ≥ τ θ . I will rely on this alternate characterization of IC-B throughout this proof.
Now suppose that for each t and every FY -stopping time τ 0 ≥ t, equation (2) holds
whenever τ Y > t. Fix an F-stopping time σ ≥ τ θ and a time t, and define an FY -stopping
time τ 0 ≥ t by setting τ 0 = σ on {τ θ = t} and completing τ 0 uniquely to ensure its value is
independent of τ θ conditional on the history of output. Then by assumption
Ft ≥ EB
t
"Z
#
τ Y ∧τ 0
e−ρ(s−t) b ds + e
−ρ(τ Y
∧τ 0 −t)
Fτ Y ∧τ 0
t
on {τ Y > t}. In particular, on {τ Y > τ θ = t} this inequality can be written
"Z
τ Y ∧σ
Fτ θ ≥ Eτ θ
#
−ρ(s−τ θ )
e
b ds + e
−ρ(τ Y
∧σ−τ θ )
Fτ Y ∧σ .
τθ
(Recall that the conditional distribution on paths of output after t when τ θ ≤ t is PB
t .) This
argument holds for each t, therefore the inequality holds on {τ Y > τ θ }. In other words,
0 ≥ 1{τ Y > τ θ } Eτ θ
"Z
#
τ Y ∧σ
−ρ(t−τ θ )
e
b dt + e
−ρ(τ Y
∧σ−τ θ )
Fτ Y ∧σ − Fτ θ
!
.
τθ
Taking expectations establishes that F satisfies IC-B.
In the other direction, suppose (F, τ Y ) satisfies IC-B. Define a new golden parachute F 0
by setting Ft0 = Ft when t ≥ τ Y and
Ft0 = sup EB
t
τ 0 ≥t
"Z
#
τ Y ∧τ 0
e−ρ(s−t) b ds + e−ρ(τ
Y
∧τ 0 −t)
Fτ Y ∧τ 0
t
when τ Y > t, with the supremum ranging over all FY -stopping times. Clearly F 0 ≥ F by
taking τ 0 = t on the rhs.
I claim that for every t and FY -stopping time τ 0 ≥ t,
Ft0 = sup EB
t
τ 0 ≥t
"Z
#
τ Y ∧τ 0
−ρ(τ Y
e−ρ(s−t) b ds + e
t
60
∧τ 0 −t)
Fτ0 Y ∧τ 0
when τ Y > t, from which it follows that (F 0 , τ Y ) satisfies IC-B by the converse just proven.
Obviously the rhs is weakly greater than the lhs, so it is sufficient to establish that
Ft0 ≥ sup EB
t
"Z
τ 0 ≥t
#
τ Y ∧τ 0
−ρ(τ Y
e−ρ(s−t) b ds + e
∧τ 0 −t)
Fτ0 Y ∧τ 0 .
t
Fix a time t such that τ Y > t. For each ε > 0 there exists an FY -stopping time τ ε ≥ t such
that
"Z Y 0
#
τ ∧τ
e−ρ(s−t) b ds + e−ρ(τ
sup EB
t
τ 0 ≥t
"Z
≤ EB
t
Y
∧τ 0 −t)
Fτ0 Y ∧τ 0
t
#
τ Y ∧τ ε
e−ρ(s−t) b ds + e−ρ(τ
Y
∧τ ε −t)
t
ε
Fτ0 Y ∧τ ε + .
2
And by definition of F 0 , there exists a τeε ≥ τ Y ∧ τ ε such that
Fτ0 Y ∧τ ε ≤ EB
τ Y ∧τ ε
"Z
#
τ Y ∧e
τε
−ρ(s−τ Y
e
∧τ ε )
−ρ(τ Y
b ds + e
∧e
τ ε −τ Y
∧τ ε )
Fτ Y ∧eτ ε + eρ(τ
τ Y ∧τ ε
Y
∧τ ε −t) ε
2
.
(This stopping time may be constructed by using the definition of F 0 on each E s = {τ Y >
τ ε = s} for s ≥ t to choose τeε on E s . The inequality is trivially true when τ Y ≤ τ ε .) Thus
by the law of iterated expectations
sup EB
t
"Z
τ 0 ≥t
≤ EB
t
#
τ Y ∧τ 0
−ρ(τ Y
e−ρ(s−t) b ds + e
∧τ 0 −t)
Fτ0 Y ∧τ 0
t
"Z
τY
#
∧e
τε
e−ρ(s−t) b ds + e
−ρ(τ Y
∧e
τ ε −t)
Fτ Y ∧eτ ε + ε.
t
And as τeε ≥ t, the definition of F 0 implies
sup EB
t
τ 0 ≥t
"Z
#
τ Y ∧τ 0
e−ρ(s−t) b ds + e
−ρ(τ Y
∧τ 0 −t)
Fτ0 Y ∧τ 0 ≤ Ft0 + ε.
t
As this inequality holds for all ε > 0, the desired inequality must hold.
Finally, I show that Fτ0 θ ∧τ Y = Fτ θ ∧τ Y a.s. For each ε > 0, construct an F-stopping time
61
σ ε ≥ τ θ by setting σ ε = τ t,ε when τ θ = t, where τ t,ε ≥ t is an FY -stopping time satisfying
EB
t
"Z
#
τ Y ∧τ t,ε
−ρ(τ Y
e−ρ(s−t) b ds + e
∧τ t,ε −t)
Fτ0 Y ∧τ t,ε ≥ Ft0 − ε
t
whenever τ Y > t. (Such a τ t,ε must exist given earlier results about F 0 .) Then by construction
"Z
τ Y ∧σ
#
−ρ(t−τ θ )
e
Eτ θ
b dt + e
−ρ(τ Y
τθ
∧σ−τ θ )
Fτ Y ∧σ ≥ Fτ0 θ − ε
whenever τ Y > τ θ . Now as (F 0 , τ Y ) satisfies IC-B, the earlier characterization of this property
implies
h
i
θ
0 ≥ E 1{τ Y > τ θ }e−ρτ (Fτ0 θ − Fτ θ − ε) .
This inequality holds for all ε > 0, hence
h
i
θ
0 ≥ E 1{τ Y > τ θ }e−ρτ (Fτ0 θ − Fτ θ ) .
As F 0 ≥ F, it must be that Fτ0 θ = Fτ θ a.e. on {τ Y > τ θ }. And by construction Fτ0 Y = Fτ Y ,
hence Fτ0 Y ∧τ θ = Fτ Y ∧τ θ a.s.
C.5
Proof of Lemma 5
Fix an IC-B contract C = (F, τ Y ). I first claim that τ Y ≤ inf{t : Ft = 0}. For suppose not;
then there exists some time t and state of the world in which Ft = 0, τ Y > t, and θt = B.
In this case the agent’s utility from reporting a state change is 0, while his expected utility
from never reporting the change is strictly positive given that he collects strictly positive
expected flow rents and receives a non-negative golden parachute. Hence IC-B is violated, a
contradiction of our assumption.
I now show that the contract C 0 = (F 0 , τ Y ) with F 0 = min{F, b/ρ} satisfies IC-B. Consider
first the agent’s incentives under C 0 at time t in any state of the world such that θt = B and
Ft ≤ b/ρ. The agent’s utility from reporting the state change is the same under F and F 0 ,
while his expected utility from any delayed reporting strategy is weakly lower under F 0 than
under F. Thus given that C satisfies IC-B, so does C 0 .
Next consider the agent’s incentives at time t in any state of the world such that θt = B
and Ft > b/ρ. In this case his expected utility under F 0 from any delayed reporting strategy
is at most b/ρ, as F 0 is bounded above by b/ρ. Meanwhile his expected utility from reporting
62
the state change immediately is exactly b/ρ. Thus IC-B is satisfied for F 0 everywhere. Note
that as C 0 implements the same stopping strategy and pays the agent weakly less than C, it
must be weakly more profitable for the firm.
Now I construct a new contract Ce = (Fe, τeY ) as follows. Whenever t ≤ τ Y , set Fet = Ft0 .
For t > τ Y set
n
o
Y
Fet = max b/ρ − (b/ρ − Fτ0 Y )eρ(t−τ ) , 0 .
For the termination policy, let τeY = inf{t : Fet = 0}.
Suppose that in some state of the world τ Y ≥ τeY . This means that at FeτeY = Fτe0 Y , but
also by definition FeτeY = 0, meaning Fτe0 Y = 0 and thus τ Y ≤ τeY . Hence τ Y ≥ τeY implies
τ Y = τeY , meaning τeY ≥ τ Y in all cases.
Next I check incentive compatibility. First consider times t and states of the world in
which θt = B and τeY > t ≥ τ Y . In this case, note that Fet is pathwise differentiable, and
d e
F = ρFet − b. the change in the agent’s utility utility from waiting a moment dt to report
dt t
is
d
b dt − ρFet dt + Fet dt = 0.
dt
As this equality holds everywhere until τeY , the agent is indifferent between reporting now
and any delayed reporting strategy. Thus IC-B holds in these cases. Next consider states of
the world in which θt = B and τeY > t. In this case no delayed reporting strategy τ 0 such
that τ Y ≥ τ 0 is more profitable than immediate reporting given that C 0 is IC-B. And on the
other hand any delayed reporting strategy τ 0 yields exactly the same expected utility to the
agent as the reporting strategy τ 0 ∧ τ Y , as we’ve seen that the agent’s expected utility is
unchanged past that point. Thus IC-B holds in all cases.
Finally, I claim that Ce is at least as profitable for the firm as C 0 , and therefore as C.
Because Ce operates for at least as long as C 0 in the good state in all cases and pays the agent
no more than C 0 whenever τeY = τ Y , I need only check the cases in which τeY > τ Y . But in
this case the path of Fet is constructed to be (weakly) declining given b/ρ ≥ F 0 , and so the
firm pays the agent no more stopping at τeY than he would have stopping at τ Y ; in other
words, FeτeY ≤ Feτ Y = Fτ0 Y . Thus payments under Ce are weakly lower than under C 0 , proving
the claim.
C.6
Proof of Lemma 6
Fix an FY -stopping time τ Y . One feasible reporting policy for the agent at time t is to to
never report a state change, i.e. to implement the stopping time τ 0 = ∞. Then if F is any
63
IC-B golden parachute implementing τ Y , the IC-B constraint implies
Ft ≥ EB
t
"Z
#
τY
−ρ(τ Y
e−ρ(s−t) b ds + e
−t)
"Z
≥ EB
t
Fτ Y
t
#
τY
e−ρ(s−t) b ds .
t
It follows that if the golden parachute
Ft∗ = EB
t
"Z
#
τY
e−ρ(s−t) b ds
t
satisfies IC-B, then it minimizes fees paid to the agent state by state over all IC-B contracts
implementing τ Y , and is therefore profit-maximizing.
Fix a time t such that τ Y > t and a delayed reporting policy τ 0 ≥ t, and rewrite the
agent’s expected utility under τ 0 using the law of iterated expectations as
EB
t
"Z
#
τ 0 ∧τ Y
−ρ(τ 0 ∧τ Y
e−ρ(s−t) b ds + e
−t)
Fτ∗0 ∧τ Y
t
= EB
t
"Z
τ 0 ∧τ Y
−ρ(τ 0 ∧τ Y
e−ρ(s−t) b ds + e
−t)
EB
τ 0 ∧τ Y
t
=
EB
t
"Z
τ 0 ∧τ Y
−ρ(s−t)
e
b ds +
EB
τ 0 ∧τ Y
"Z
=
"Z
##
τY
−ρ(s−τ 0 ∧τ Y
e
)
b ds
τ 0 ∧τ Y
##
τY
−ρ(s−t)
e
b ds
τ 0 ∧τ Y
t
EB
t
"Z
#
τY
e
−ρ(s−t)
b ds = Ft∗ .
t
Hence every delayed reporting policy τ 0 yields the same expected utility under F ∗ as truthful
reporting, i.e. F ∗ is IC-B.
C.7
Proof of Lemma 7
Given truthful reporting, under any contract F the firm collects flow payments dY until
τ Y ∧ τ θ and pays out a lump sum of Fτ Y ∧τ θ at project termination. Its expected profits
under F are therefore
#
#
"Z Y θ
"Z Y θ
τ ∧τ
τ ∧τ
Y
θ
Y
θ
e−ρs rG ds − e−ρ(τ ∧τ ) Fτ Y ∧τ θ .
Π=E
e−ρs dYs − e−ρ(τ ∧τ ) Fτ Y ∧τ θ = E
0
0
64
Now I write out the expectation wrt uncertainty in the switching time explicitly, using the
fact that the conditional measure on output paths up to time t given τ θ ≥ t is equal to PG :
"Z
∞
Z
dt α e−αt EG
Π=
0
#
τ Y ∧t
−ρ(τ Y
e−ρs rG ds − e
∧t)
Feτ Y ∧t .
0
The expression in brackets is bounded, so I use Fubini’s theorem to exchange the order of
integration, yielding
Π = EG
"Z
∞
dt α e−αt
0
G
"Z
e−ρs rG ds − e
τY
t
Z
−αt
−ρs
dt α e
e
0
"Z
−ρ(τ Y
∧t)
Fτ Y ∧t
0
=E
G
!#
τ Y ∧t
Z
rG ds − e
−ρt
Fet + e
=E
−ρs
dt α e
e
0
rG ds + e
−ρτ Y
Fτ Y
0
t
Z
−αt
−ρs
e
0
τY
!#
τY
Z
−ατ Y
rG ds − e
−ρt
Fet + e
−ατ Y
Z
0
#
τY
−ρs
e
rG ds .
0
(In the last line I have used the fact that Feτ Y = 0.) Using integration by parts, the first
term can be evaluated as
Z
τY
−αt
Z
dt α e
0
t
e
−ρs
rG ds = −e
−ατ Y
0
Z
τY
−ρs
e
Z
rG ds +
0
Hence
Π = EG
"Z
τY
e−(ρ+α)t rG dt.
0
#
τY
e−(ρ+α)t (rG − αFet ) dt .
0
C.8
Proof of Lemma 8
I prove this lemma with the aid of Lemma 21 in the technical appendix, which establishes
Y
that a contract (F, τ Y ) satisfies IC-B iff M τ is a PB -supermartingale, where
b
Mt = (1 − e−ρt ) + e−ρt Ft .
ρ
Fix a contract (F, τ Y ) evolving as the Ito process
dFt = γt dt +
βt dZtG
∆r
= γt −
βt dt + βt dZtB
σ
for FY -adapted, progressively measurable processes γ and β. For every n = 1, 2, ..., define
65
o
n
Rt
Rt
τ n ≡ inf t : 0 |γs | ds + 0 βs2 ds ≥ n . And for every t and m = 1, 2, ..., define σ t,m ≡
inf{s ≥ t : Fs + |γs | + |βs | ≥ m}. (Right-continuity of FY under the usual conditions ensures
that each σ t,m is an FY -stopping time.)
Suppose first that F satisfies
b − ρFt + γt −
∆r
βt ≤ 0
σ
whenever t < τ Y . Fix times t and s > t, and let n be large enough that τ n > t. Ito’s lemma
implies that
τY
EB
t [Ms∧τ n ]
=
Y
Mtτ
+
EB
t
"Z
s∧τ Y ∧τ n
−ρu
e
t∧τ Y
#
Z s∧τ Y ∧τ n
∆r
−ρu
B
b − ρFu + γu −
βu du +
e βu dZu .
σ
t∧τ Y
Now, given the regularization imposed by τ n , the last term is a PB -martingale whose expectation vanishes. And the remaining term within the expectation is non-positive a.s. by
τY
τY
assumption. Hence EB
t [Ms∧τ n ] ≤ Mt . This holds for all n sufficiently large, so also
Y
Y
τ
τ
lim inf EB
t [Ms∧τ n ] ≤ Mt .
n→∞
Given that M is non-negative, Fatou’s lemma allows the limit and expectation on the lhs
τY
τY
τY
to be swapped, yielding EB
is a supermartingale, implying by the
t [Ms ] ≤ Mt . So M
previous theorem that F satisfies IC-B.
Conversely, suppose that F is an IC-B contract. For this direction, assume additionally
that γ and β are pathwise right-continuous. Fix a time t < τ Y and take m large enough
that σ t,m > t. (Right-continuity of F , γ, and β ensures that such an m exists.) Use Lemma
21 in the technical appendix to conclude that for all s > t,
EB
t
b
b
−ρ(s∧τ Y ∧σ t,m )
−ρ(s∧τ Y ∧σ t,m )
(1 − e
)+e
Fs∧τ Y ∧σt,m ≤ (1 − e−ρt ) + e−ρt Ft .
ρ
ρ
Using Ito’s lemma, this is equivalent to
EB
t
"Z
t
s∧τ Y ∧σ t,m
#
Z s∧τ Y ∧σt,m
∆r
e−ρu b − ρFu + γu −
βu du +
e−ρu βu dZuB ≤ 0.
σ
t
Due to the regularization imposed by σ t,m , the final term is a PB -martingale whose expec-
66
tation vanishes, yielding
EB
t
"Z
s∧τ Y ∧σ t,m
t
#
∆r
e−ρu b − ρFu + γu −
βu du ≤ 0.
σ
Rs
du
Now, given regularization under σ t,m , the integral t e−ρu 1{τ Y ∧σ t,m ≥ u} b + γu − ∆r
β
u
σ
is uniformly bounded state by state and thus in expectation, and so Fubini’s theorem permits
swapping the order of integration for this term. Meanwhile the non-negativity of F allows
the swapping of the order of integration in the remaining term by Tonelli’s theorem. This
procedure yields
Z
s
e
−ρu
EB
t
Y
1{τ ∧ σ
t,m
t
∆r
βu
du ≤ 0.
≥ u} b − ρFu + γu −
σ
Given τ Y ∧ σ t,m > t by assumption, 1{τ Y ∧ σ t,m ≥ u} must converge pathwise to 1 as
u ↓ t. Hence given pathwise right-continuity of γ and β, the expression
e
−ρu
Y
1{τ ∧ σ
t,m
∆r
≥ u} b − ρFu + γu −
βu
σ
is pathwise right-continuous at t. Further, 1{σ t,m ≥ u} b − ρFu + γu − ∆r
β is uniformly
σ u
bounded by b + 1 + ρ + ∆r
m for all u ≥ t. Therefore the dominated convergence theorem
σ
applies, and
∆r
−ρu B
Y
t,m
βu
e Et 1{τ ∧ σ ≥ u} b − ρFu + γu −
σ
is right-continuous at t.
Now I apply the fundamental theorem of calculus to conclude that
Z
s
−ρu
e
t
EB
t
Y
1{τ ∧ σ
t,m
∆r
≥ u} b − ρFu + γu −
βu
du
σ
is (right-)differentiable wrt s at t with derivative e−ρt b − ρFt + γt −
ε > 0,
∆r
β
σ t
. Hence for every
Z s
1
∆r
−ρu B
Y
t,m
βu
du
e Et 1{τ ∧ σ ≥ u} b − ρFu + γu −
s−t t
σ
∆r
−ρt
≥e
b − ρFt + γt −
βt − ε
σ
67
for s sufficiently close to t. As the lhs is bounded above by zero, this implies
e
−ρt
for all ε > 0, i.e. b − ρFt + γt −
C.9
∆r
b − ρFt + γt −
βt
σ
∆r
β
σ t
≤ε
≤ 0.
Proof of Lemma 9
See Propositions 8.10 and 8.11 of Harrison (2013).
C.10
Proof of Lemma 10
Throughout this proof I freely invoke well-known properties of U (m, n, z). In particular,
I will often invoke properties holding when m > 0 and n > m + 2, as is the case when
m = β − 1 and n = 2β − 2α
given β > 1 + 2α
.
k2
k2
I begin by deriving a general solution to the ODE
ρv = xrG − (1 − x)b − αxv 0 +
k2 2
x (1 − x)2 v 00 .
2
This is an inhomogeneous second-order linear ODE, whose solution can be found by conjecturing a particular solution and then solving the associated homogeneous equation. A
natural conjecture is linear in x; inserting v0 (x) = c1 x + c0 and matching coefficients reveals
that
rG + b
b
v0 (x) =
x−
ρ+α
ρ
is a particular solution to the ODE. The problem of solving the ODE then reduces to solving
the associated homogeneous equation
0
ρvH = −αxvH
+
Now I make the transformation z ≡
ρvH =
Next I guess that wH (z) ≡
x
,
1−x
k2 2
00
x (1 − x)2 vH
.
2
obtaining the transformed ODE
k2
−αz + k − α −
z+1
zβ
v (z)
1+z H
2
z
dvH k 2 2 d2 vH
+ z
.
dz
2
dz 2
satisfies a simpler ODE than vH itself for some positive
68
power of β. Inserting into the ODE yields
1 2
dwH k 2 2 d2 wH
2
α(β − 1)z + ρ + α + (α − k )(β − 1) − k (β − 1)(β − 2) wH = (k 2 β−α−αz)z
+ z
.
2
dz
2
dz
The right choice of β is therefore a solution to
1
ρ + α + (α − k 2 )(β − 1) − k 2 (β − 1)(β − 2) = 0,
2
which is the quadratic
2α
ρ
β − 1 + 2 β − 2 2 = 0.
k
k
2
It is straightforward to verify that a unique positive solution to this equation exists. Taking
this choice of β, the ODE for wH reduces to
α(β − 1)wH = (k 2 β − α − αz)
Finally, make the substitution t ≡
2α
z.
k2
dwH k 2 d2 wH
+ z
.
dz
2 dz 2
I arrive at the ODE
2α
dwH
d2 wH
(β − 1)wH = 2β − 2 − t
+t 2 .
k
dt
dt
This is Kummer’s differential equation, which has general solution
wH (t) = C1 U (m, n, t) + C2 M (m, n, t),
where U and M are Tricomi’s and Kummer’s confluent hypergeometric functions and m ≡
are both strictly positive. Transforming back to the original variables,
β − 1 and n ≡ 2β − 2α
k2
a general solution to the homogeneous equation is
vH (x) = x
m+1
−m
(1 − x)
2α x
2α x
CU m, n, 2
+ DM m, n, 2
.
k 1−x
k 1−x
Now, Kummer’s function M (m, n, t) is known to diverge in the limit as t → ∞. As the
leading term of vH also diverges in this limit, no solution with D 6= 0 can satisfy the desired
regularity conditions at x = 1. This leaves solutions to the ODE in the lemma statement of
the form
rG + b
b
2α x
m+1
−m
v(x) =
x − + Cx
(1 − x) U m, n, 2
ρ+α
ρ
k 1−x
69
as candidates for solving the desired boundary value problem.
Let
2α 2α x
β
1−β
VH (x) ≡ x (1 − x) U β − 1, 2β − 2 , 2
k k 1−x
for x ∈ (0, 1). U (m, n, ·) is known to be a strictly positive function on (0, ∞) when m > 0,
so the function Γ(x) ≡ log VH (x) is also well-defined on (0, 1). As U (m, n, ·) is an analytic
function on (0, ∞), VH and Γ are C 2 functions on (0, 1). En route to solving the boundary
value problem, I establish several basic facts about VH , Γ, and their derivatives in the limits
x → 0 and x → 1.
Lemma 16. The following limiting expressions hold as x → 1 :
β−1
k2
lim VH (x) =
,
x→1
2α
β−1
ρ k2
0
,
lim VH (x) = −
x→1
α 2α
ρ
lim Γ0 (x) = − ,
x→1
α
2 β+1
k
2α
2α
00
lim VH (x) =
β(β − 1) β − 2
β− 2 −1 .
x→1
2α
k
k
Proof. Let m ≡ β − 1, n ≡ 2β −
2α
,
k2
γ≡
k2
,
2α
x
and z ≡ γ −1 1−x
. Then I may write VH as
VH (x) = xγ m z m U (m, n, z).
Now I invoke the well-known asymptotic expansion of U as z → ∞ when z ∈ R. The
full expansion is U (m, n, z) ∼ z −m 2 F0 (m, m − n + 1; ; −1/z), where 2 F0 is the well-known
generalized hypergeometric series. To third order this expansion says that
U (m, n, z) = z
−m
1
−2
−1
a0 − a1 z + a2 z + Φ(z) ,
2
where a0 = 1, a1 = m(m − n + 1), a2 = m(m + 1)(m − n + 1)(m − n + 2), and Φ(z) ∼ O(z −3 ).
In other words, limz→∞ z N Φ(z) = 0 when N < 3. As U is analytic, so is Φ, and L’hopital’s
rule implies Φ0 (z) ∼ O(z −4 ) and Φ00 (z) ∼ O(z −5 ).
Now insert this expansion into VH to obtain
VH (x) = xγ
m
a0 − a1 z
−1
70
1
−2
+ a2 z + Φ(z) .
2
taking x → 1 implies limx→1 VH (x) = a0 γ m , which is the first formula in the lemma. Now
γ −1
dz
−1
= (1−x)
(1 + γz)2 . I obtain
differentiate wrt x, noting that dx
2 = γ
VH0 (x)
=γ
m
a0 − a1 z
−1
1
−2
+ a2 z + Φ(z) + xγ m−1 (1 + γz)2 a1 z −2 − a2 z −3 + Φ0 (z) .
2
Thus
lim
x→1
VH0 (x)
m
= γ a0 + γ
m+1
a1 =
k2
2α
m
+
k2
2α
m+1
m(m − n + 1).
Writing this explicitly in terms of model parameters, this is
lim
x→1
VH0 (x)
=
k2
2α
β−1 2 β−1 2 k2
2α
k
k
2α
k2 2
1+
(β − 1) −β + 2
=
1+ 2 β−
β .
2α
k
2α
2α
k
2α
Using the quadratic formula characterizing β, this simplifies to
lim
x→1
VH0 (x)
=−
k2
2α
β−1
ρ
,
α
which is the second formula in the lemma statement. The limiting expression for Γ0 (x) =
VH0 (x)/VH (x) may then be obtained by combining previous results.
Finally, the second derivative of VH is
VH00 (x) = (x + 1)γ m−1 (1 + γz)2 a1 z −2 − a2 z −3 + Φ0 (z)
+ 2xγ m−1 (1 + γz)3 a1 z −2 − a2 z −3 + Φ0 (z)
+ xγ m−2 (1 + γz)4 −2a1 z −3 + 3a2 z −4 + Φ00 (z) .
The first term converges to 2γ m+1 a1 as x → ∞. As for the rest, pull out the common factor
of xγ m−2 (1 + γz)3 and expand
2γ a1 z −2 − a2 z −3 + Φ0 (z) + (1 + γz) −2a1 z −3 + 3a2 z −4 + Φ00 (z) .
The result is
(−2a1 + γa2 )z −3 + 3a2 z −4 + 2γΦ0 (z) + (1 + γz)Φ00 (z).
All but the firm term are O(z −4 ), which when multiplied by (1 + γz)3 die out as z → ∞. I
conclude that
lim VH00 (x) = 2γ m+1 a1 + γ m+1 (−2a1 + γa2 ) = γ m+2 a2 ,
x→1
71
which is the final formula in the lemma statement.
Lemma 17. The following limiting expressions hold as x → 0 :
lim VH (x) = ∞,
x→0
lim inf Γ0 (x) = −∞.
x→0
Proof. Let m ≡ β − 1, n ≡ 2β −
2α
,
k2
γ≡
k2
,
2α
x
and z ≡ γ −1 1−x
. Then I may write VH as
VH (x) = (1 − x)γ m z m+1 U (m, n, z).
A standard property of U is limz→0 z m+1 U (m, n, z) = ∞ when m > 0 and n > m+2, proving
the result.
Given limx→0 VH (x) = ∞, clearly also limx→0 Γ(x) = ∞. Now suppose by way of contradiction that lim inf x→0 Γ0 (x) > −∞. In this case there exists an x > 0 and an M < ∞ such
that Γ0 (x) ≥ −M for all y ∈ (0, x]. The fundamental theorem of calculus then implies
Z
Γ(y) = Γ(x) −
x
Γ0 (t) dt ≤ Γ(x) + M (x − y) ≤ Γ(x) + M x.
y
Thus Γ(y) is bounded above on (0, x], contradicting limx→0 Γ(x) = ∞.
These lemmas provide the tools to demonstrate the existence of constants π ∈ (0, 1) and
C > 0 such that
b
rG + b
x − + CVH (x)
V (x; C) =
ρ+α
ρ
satisfies V (π; C) = V 0 (π; C) = 0. These boundary conditions are equivalent to CVH (π) =
G +b
G +b
x + ρb and CVH0 (π) = − rρ+α
. Dividing the second equation through by the first, exis− rρ+α
tence of an appropriate π is equivalent to existence of a solution to
Γ0 (x) = φ(x),
1
and x ≡ (rb(ρ+α)
> 0. If such a solution π exists, a corresponding C
where φ(x) ≡ − x−x
G +b)ρ
G +b
is C = − V 0 1(π) rρ+α
. And if Γ0 (π) < 0, then given strict positivity of VH it must be that
H
VH0 (π) < 0 and so C > 0.
Suppose first that x ≤ 1. Then limx↑x φ(x) = −∞ while φ(0) = −x−1 . Given the
continuity of φ, it must be bounded on [0, x0 ] for any x0 < x. Thus given lim inf x→0 Γ0 (x) =
−∞, there exists an x1 ∈ (0, x) such that Γ0 (x1 ) < φ(x1 ). And given that limx→1 Γ0 (x) is
72
finite and Γ0 is continuous on (0, 1), Γ0 must be bounded on the closed interval [x1 , x]. Thus
in particular there exists an x2 ∈ (x1 , x) such that Γ0 (x2 ) > φ(x2 ). Then as Γ0 (x) − φ(x)
is a continuous function on [x1 , x2 ], by the intermediate value theorem there exists an x∗ ∈
(x1 , x2 ) such that Γ0 (x∗ ) = φ(x∗ ). Further, Γ0 (x∗ ) < 0 given the negativity of φ on [0, x).
Now suppose that x > 1. In this case φ is continuous and decreasing on [0, 1], with
φ(1) = −1/(x − 1). The argument of the previous paragraph continues to establish existence
of an x1 ∈ (0, 1) such that Γ0 (x1 ) < φ(x1 ). Meanwhile limx→1 Γ0 (x) = −ρ/α, while
φ(1) = −
1
b(ρ+α)
(rG +b)ρ
1
ρ
< − ρ+α
=− .
α
−1
−1
ρ
Thus limx→1 Γ0 (x) > φ(1), and so there exists an x2 ∈ (x1 , 1) such that Γ0 (x2 ) > φ(x2 ). The
intermediate value theorem then ensures existence of a solution x∗ ∈ (x1 , x2 ) to Γ0 (x) = φ(x).
Given the negativity of φ, it must be that Γ0 (x∗ ) < 0.
Finally, I must establish that Ve can be extended to a C 2 function on [π, 1]. Once I have
done this, it is automatic that Ve satisfies the ODE on the extended domain simply by taking
limits of each side of the ODE as π0 → 1. I have already shown that limx→1 Ve (n) (x) all exist
and are finite for n = 0, 1, 2. Defining Ve (1) = limx→1 Ve (x) extends Ve continuously to [π, 1].
But I must check that the first two derivatives of the resulting function exist at 1 and are
continuous. To do this, I invoke the mean value theorem. For any x ∈ (π, 1) the MVT
ensures existence of a y(x) ∈ (x, 1) such that
Ve (1) − Ve (x)
= Ve 0 (y(x)).
1−x
Taking x → 1 implies y(x) → 1 by the squeeze theorem. Then as the limit of Ve 0 exists, so
does the derivative of Ve at x = 1, and Ve 0 (1) = limx→1 Ve 0 (x). Thus Ve is a C 1 function on
[π, 1]. Another application of the MVT in exactly the same fashion ensures that Ve is C 2 .
C.11
Proof of Lemma 11
Lemma 18. f is a continuous, strictly increasing function on [π, 1] satisfying f (π) = 0 and
f (1) = F0∗ .
By definition, each px solves the SDE
Z
t
−αps −
p t = p0 +
0
∆r
σ
2
!
p2s (1
− ps )
Z
ds +
0
73
t
∆r
ps (1 − ps ) dZsB
σ
with initial condition p0 = x. To avoid the possibility that this SDE does not have a unique
strong solution, I assume in particular that px solves the regularized SDE
Z
t
−αφ(ps ) −
pt = p 0 +
0
∆r
σ
2
!
2
φ(ps ) (1 − φ(ps ))
Z
ds +
0
t
∆r
φ(ps )(1 − φ(ps )) dZsB ,
σ
where φ(y) = min{max{y, 0}, 1}. (I will show in a moment that a solution to the regularized
SDE is also a solution to the original one.) The coefficients of this SDE are continuous,
bounded, continuously differentiable on [0, 1] (taking one-sided derivatives at the boundaries), and constant outside [0, 1]. Hence they are globally Lipschitz continuous and satisfy a
quadratic growth condition, so by Theorems 5.2.5 and 5.2.9 of Karatzas and Shreve (1991)
there exists a unique strong solution of the SDE for every initial condition x ∈ R for all time.
In particular, strong uniqueness ensures that for each x ∈ (0, 1], 0 < pxt < 1 for all t > 0
a.s. For note that p = 0 is a solution to the SDE as well, and so if pt = 0 then p0 = 0 < x.
And similarly pt = p0 − αt is a solution to the SDE for any p0 > 1 and t ≤ (p0 − 1)/α. Thus
if pt ≥ 1 for t > 0 then p0 = pt + αt > x. This verifies the earlier claim that a solution to
the regularized SDE is also a solution to the original one.
0
In addition, strong uniqueness ensures that for every x0 > x in (0, 1], pxt > pxt for all t
a.s. For suppose two solutions p and p0 to the SDE satisfy pt = p0t for some t. Then strong
0
uniqueness ensures that p = p0 a.s. In particular p0 = p00 a.s., establishing that pxt 6= pxt for
0
all time a.s. As solutions to the SDE are pathwise continuous and px0 > px0 , this proves the
claim. This result establishes that τxY0 > τxY a.s., and so f is strictly increasing in x.
As for continuity, use the fact that solutions to SDEs with globally Lipschitz coefficients
are continuous flows wrt time and initial data, i.e. the map (x, t) → ptx (ω) is continuous
in (x, t) for a.e. ω ∈ Ω. Fix a state ω ∈ Ω and for each x consider the path px (ω) with
associated stopping time τxY (ω). (Explicit references to ω will be suppressed to conserve on
notation.) By the reasoning in the previous paragraph, τxY is strictly increasing in x. So fix x
and consider first taking x0 ↑ x. Let limx0 ↑x τxY0 = L (guaranteed to exist given monotonicity
0
0
of τxY ); then limx0 ↑x pxτ Y = pxL . And as pxτ Y = π for every x0 , this means that pxL = π and
x0
x0
therefore that τxY ≤ L. So by monotonicity limx0 ↑x τxY0 = τxY .
Next take x0 ↓ x. Given pxτxY = π, the non-zero quadratic variation of px and the strong
Markov property imply that for every ε > 0 there exists a t0 ∈ (τxY , τxY + ε) such that pxt0 < π.
0
Now, pxt0 → pxt0 as x0 → x, so limx0 ↓x τxY0 < t0 < τxY + ε for every ε. Thus limx0 ↓x τxY0 = τxY
pointwise. This establishes that τxY is continuous in x a.s. The bounded convergence theorem
then implies continuity of f.
74
C.12
Proof of Lemma 12
By definition,
Ve (πt ) =
EYt
τ∗
Z
e
−ρ(s−t)
(πs rG − (1 − πs )b) ds
t
for all t ≤ τ ∗ , where τ ∗ = inf{t : πt ≤ π}. Given that πt = EYt [1{τ θ > t}], this may
equivalently be written
Ve (πt ) = EYt
"Z
t∨(τ ∗ ∧τ θ )
e−ρ(s−t) rG ds −
Z
#
τ∗
e−ρ(s−t) b ds .
t∨(τ ∗ ∧τ θ )
t
On {τ ∗ ≥ t} this latter expression may be partitioned as
EYt
"Z
t∨(τ ∗ ∧τ θ )
e−ρ(s−t) rG ds −
#
τ∗
Z
e−ρ(s−t) b ds
t∨(τ ∗ ∧τ θ )
t
"
= EYt 1{τ θ > t}
τ ∗ ∧τ θ
Z
e−ρ(s−t) rG ds −
Z
!#
τ∗
e−ρ(s−t) b ds
τ ∗ ∧τ θ
t
Z
Y
θ
− Et 1{t ≥ τ }
τ∗
e
−ρ(s−t)
b ds .
t
Now, Lemma 7 and the argument in the proof of Proposition 3 imply that V (Ft∗ ) may be
written
"Z ∗ θ
"Z ∗ θ
#
#
Z τ∗
τ ∧τ
τ ∧τ
∗
θ
e−ρ(s−t) rG ds − e−τ ∧τ Fτ∗∗ ∧τ θ = Et
e−ρ(s−t) rG ds −
e−ρ(s−t) b ds
V (Ft∗ ) = Et
t
τ ∗ ∧τ θ
t
on {τ ∗ ∧ τ θ > t}. And on {τ ∗ ≥ t ≥ τ θ },
Z
τ∗
e
Et
−ρ(s−t)
Z
B
b ds = Et
τ∗
e
−ρ(s−t)
b ds = Ft∗ .
t
t
Therefore by the law of iterated expectations
Ve (πt ) = EYt 1{τ θ > t}V (Ft∗ ) − EYt 1{t ≥ τ θ }Ft∗
= πt V (Ft∗ ) − (1 − πt )Ft∗
= πt V (f (πt )) − (1 − πt )f (πt )
on {τ ∗ ≥ t}. In particular, this holds whenever πt = x for each x ∈ [π, 1], proving the lemma.
75
D
Technical appendix
This section contains technical lemmas used in the proofs of results appearing in the main
text.
Lemma 19. Fix a contract C = (F, τ Y ). Suppose that for each t,
"Z
#
τ Y ∧τ θ
e−ρ(s−t) b ds + e
Et
−ρ(τ Y
∧τ θ −t)
Fτ Y ∧τ θ ≥ Ft
t
whenever τ Y ∧ τ θ > t. Then C satisfies IC-G.
Proof. Fix a contract C = (F, τ Y ). IC-G holds iff
"Z
E
#
τ Y ∧τ θ
e−ρt b dt + e
−ρ(τ Y
∧τ θ )
"Z
−ρ(τ Y
e−ρt b dt + e
Fτ Y ∧τ θ ≥ E
0
#
τ Y ∧σ
∧σ)
Fτ Y ∧σ
0
for every F-stopping time σ ≤ τ θ . This inequality may be rearranged to obtain
"
E 1{τ Y ∧ τ θ > σ}e−ρσ
!#
τ Y ∧τ θ
Z
−ρ(τ Y
e−ρ(t−σ) b dt + e
∧τ θ −σ)
≥ 0.
Fτ Y ∧τ θ − Fσ
σ
Then by applying the law of iterated expectations, IC-G is equivalent to the condition that
"
E 1{τ Y ∧ τ θ > σ}e−ρσ
"Z
#
τ Y ∧τ θ
−ρ(τ Y
e−ρ(t−σ) b dt + e
Eσ
∧τ θ −σ)
!#
Fτ Y ∧τ θ − Fσ
≥ 0.
σ
for every σ ≤ τ θ . Note that on for every time t, on the event {σ = t} the expression inside
the expectation is equal to
1{τ Y ∧ τ θ > t}e−ρt Et
"Z
#
τ Y ∧τ θ
e−ρ(s−t) b ds + e
−ρ(τ Y
∧τ θ −t)
!
Fτ Y ∧τ θ − Ft ,
t
which by assumption is non-negative. As this holds for every choice of t, it must be that
1{τ Y ∧ τ θ > σ}e−ρσ
"Z
Eσ
#
τ Y ∧τ θ
e−ρ(t−σ) b dt + e
σ
for every σ ≤ τ θ . Hence C satisfies IC-G.
76
−ρ(τ Y
∧τ θ −σ)
Fτ Y ∧τ θ − Fσ
!
≥0
Lemma 20. Fix a contract C = (F, τ Y ). Suppose there exist progressively measurable processes γ and β such that for all t, each of the following holds a.e. on {τ Y > t}:
• Ft = F0 +
Rt
0
γs ds +
Rt
0
βs dZsG ,
β = 0,
• b − ρFt + γt − ∆r
σ t
hR
i
t
• EG 0 βs2 ds < ∞,
• βt ≥ 0.
Then C is an IC contract.
Proof. The proof of Lemma 8 establishes that C is IC-B, as right-continuity of γ and β
are unnecessary for the sufficiency proof of that lemma. I check IC-G using the sufficiency
condition of Lemma 19 in the technical appendix. Define a process U by
"Z
#
τ θ ∧τ Y
e−ρ(s−t) b ds + e−ρ(τ
Ut = Et
θ ∧τ Y
−t)
Fτ θ ∧τ Y
t
when τ Y ∧ τ θ > t, and Ut = Uτ Y ∧τ θ otherwise. Then IC-G obtains if Ut ≥ Ft whenever
τ Y ∧ τ θ > t.
When τ Y ∧ τ θ > t, Ut may be written
Z
Ut =
∞
du αe−α(u−t) EG
t
"Z
t
#
u∧τ Y
−ρ(u∧τ Y
e−ρ(s−t) b ds + e
−t)
Fu∧τ Y .
t
Use Ito’s lemma to expand the final term as
e
−ρ(u∧τ Y −t)
u∧τ Y
Z
Fu∧τ Y = Ft +
e
−ρ(s−t)
Z
(γs − ρFs ) ds +
t
u∧τ Y
e−ρ(s−t) βs dZsG .
t
Due to the regularity condition imposed on β in the lemma statement, the final term is a
PG -martingale. Thus Ut reduces to
Z
Ut = Ft +
∞
du αe
−α(u−t)
EG
t
t
And for each s ∈ [t, τ Y ), γs + b − ρFs =
yielding Ut ≥ Ft .
"Z
#
u∧τ Y
−ρ(s−t)
e
(γs + b − ρFs ) ds .
t
∆r
β
σ s
77
≥ 0 a.s. So the final term is non-negative,
Lemma 21. Fix a contract C = (F, τ Y ) and define the FY -adapted process M by
b
Mt = (1 − e−ρt ) + e−ρt Ft .
ρ
Then C is an IC-B contract iff M τ
Y
is a PB -supermartingale.
Proof. Fix any t and FY -stopping time τ 0 ≥ t, and consider states of the world in which
τ Y > t. Then the definition of M implies
EB
t [Mτ Y ∧τ 0 ]
b
−ρ(τ Y ∧τ 0 )
−ρ(τ Y ∧τ 0 )
(1 − e
)+e
Fτ Y ∧τ 0
=
ρ
b
−ρt
−ρt B b
−ρ(τ Y ∧τ 0 −t)
−ρ(τ Y ∧τ 0 −t)
= (1 − e ) + e Et
(1 − e
)+e
Fτ Y ∧τ 0 .
ρ
ρ
EB
t
Now, suppose C is an IC-B contract. Then by Lemma 4 the final term is at most Ft , meaning
b
−ρt
) + e−ρt Ft = Mt ,
EB
t [Mτ Y ∧τ 0 ] ≤ (1 − e
ρ
Y
and in particular this holds when τ 0 = s > t. Thus M τ is a supermartingale.
Y
On the other hand, suppose M τ is a PB -supermartingale. Then EB
t [Mτ Y ∧τ 0 ] is at most
τY
Mt = Mt , implying
Ft ≥
EB
t
b
−ρ(τ Y ∧τ 0 −t)
−ρ(τ Y ∧τ 0 −t)
(1 − e
)+e
Fτ Y ∧τ 0 .
ρ
Thus by Lemma 4 C is an IC-B contract.
Lemma 22. Fix an IC-B contract C = (F, τ Y ) satisfying τ Y = inf{t : Ft = 0} and
F ≤ b/ρ. Then there exists another IC-B contract C 0 = (F 0 , τ 0 ) which is cadlag and satisfies
F 0 ≤ F and τ 0 = τ Y a.s. Thus C 0 yields weakly higher expected profits than C.
Proof. Fix any IC-B contract (F, τ Y ) satisfying F ≤ b/ρ. I first establish that F τ
PB -supermartingale. Let
b
Mt = (1 − e−ρt ) + e−ρt Ft .
ρ
Y
is a
Y
By Lemma 21, M τ is a PB -supermartingale. Thus for all t such that τ Y > t and all s > t,
τY
EB
t [Fs
]=
EB
t
ρ(s∧τ Y )
e
Y
Msτ
b ρ(s∧τ Y )
b
B
ρ(s∧τ Y )
τY
τY
− (e
− 1) = Et (e
− 1) Ms −
+ EB
t [Ms ].
ρ
ρ
78
As F ≤ b/ρ by assumption, M ≤ b/ρ as well. This establishes the bound
ρ(s∧τ Y )
(e
b
b
ρt
τY
τY
≤ (e − 1) Ms −
.
− 1) Ms −
ρ
ρ
Hence
b ρt
b ρt
Y
τY
τY
ρt B
ρt
τY
EB
= Ftτ .
t [Fs ] ≤ − (e − 1) + e Et [Ms ] ≤ − (e − 1) + e Mt
ρ
ρ
Y
Thus F τ is a PB -supermartingale.
Y
Now I assume wlog that F = F τ , if necessary passing to the stopped process, so that
F is a PB -supermartingale. Then by Proposition 1.3.14 of Karatzas and Shreve (1991), F
possesses pathwise right limits for all time almost surely, and the process F 0 constructed
by setting Ft0 (ω) = lims↓t Ft (ω) for each ω ∈ Ω is a PB −supermartingale wrt to the filtration {FtY+ }t≥0 . As FY is right-continuous under the usual conditions, this means F 0 is a
PB −supermartingale wrt FY . Further, the proposition says that Ft0 ≤ Ft for all time a.s.
Suppose additionally that τ Y = inf{t : Ft = 0}, and define τ 0 ≡ inf{t : Ft0 = 0}. I claim
that C 0 = (F 0 , τ 0 ) satisfies the conditions of the lemma statement. First note that as F is
non-negative, so is F 0 . So F 0 is a feasible payment process. Let τ 0 ≡ inf{t : Ft0 = 0}. I claim
that τ 0 = τ Y a.s. Given F 0 ≤ F, clearly τ Y ≤ τ 0 . In the other direction, suppose that for
some t and in some state of the world τ 0 = t, so that in particular lims↓t Fs = 0. Given that
F is a PB -supermartingale, the inequality Ft ≤ EB [Fs ] holds for all s > t. As F is uniformly
bounded, the bounded convergence theorem then implies that Ft ≤ EB [lims↓t Fs ] = 0. Thus
τ Y ≤ t, as desired.
From F 0 ≤ F and τ 0 = τ Y it follows that C 0 pays the agent weakly less than (F, τ Y ) while
operating the project for the same length of time in the good state in all cases. Thus C 0 is
weakly more profitable for the firm, as claimed. It remains only to show that C 0 is IC-B,
which I verify using the characterization of Lemma 4.
Whenever Ft0 = Ft IC-B is automatic, as the agent’s profits from any delayed reporting
strategy are weakly lower under F 0 . So consider any time t such that Ft > Ft0 . For every
s > t and FY -stopping time τ 00 ≥ s, incentive-compatibility of F implies that
Fs ≥
EB
s
b
−ρ((τ Y ∧τ 00 )−s)
−ρ((τ Y ∧τ 00 )−s)
(1 − e
)+e
Fτ Y ∧τ 00 .
ρ
−ρ(s−t)
Taking expectations wrt EB
, and taking the supremum over
t , multiplying through by e
79
all τ 00 ≥ s gives
−ρ(s−t)
EB
Fs ]
t [e
≥ sup
τ 00 ≥s
EB
t
b −ρ(s−t)
−ρ((τ Y ∧τ 00 )−t)
−ρ((τ Y ∧τ 00 )−t)
(e
−e
)+e
Fτ Y ∧τ 00 .
ρ
Rearranging yields
EB
t
b
−ρ(s−t)
−ρ(s−t)
B b
−ρ((τ Y ∧τ 00 )−t)
−ρ((τ Y ∧τ 00 )−t)
(1 − e
)+e
Fs ≥ sup Et
(1 − e
)+e
Fτ Y ∧τ 00 .
ρ
ρ
τ 00 ≥s
Now take s ↓ t. The rhs is increasing in s, so the limit of the rhs exists and is equal to
the supremum taken over all τ 0 > t. Meanwhile the interior of the expectation on the lhs
is uniformly bounded by b/ρ, so the bounded convergence theorem allows the limit and
expectation to be swapped. As lims↓t Fs = Ft0 by definition, the resulting inequality is
Ft0
≥ sup
τ 00 >t
EB
t
b
−ρ((τ Y ∧τ 00 )−t)
−ρ((τ Y ∧τ 00 )−t)
(1 − e
)+e
Fτ Y ∧τ 00 .
ρ
As F ≥ F 0 and τ Y = τ 0 a.s., the inequality is preserved when F is replaced by F 0 and τ Y is
replaced by τ 0 on the rhs, yielding
Ft0
≥ sup
τ 00 >t
EB
t
b
−ρ((τ 0 ∧τ 00 )−t)
−ρ((τ 0 ∧τ 00 )−t) 0
(1 − e
)+e
Fτ 0 ∧τ 00 .
ρ
Finally, the expected payoff of any stopping time τ 00 ≥ t is equal to a weighted average of Ft0
and the expected payoff from some τ 00 > t. As no stopping time τ 00 > t can improve on Ft0 ,
neither can a stopping time τ 00 ≥ t. It follows that C 0 satisfies IC-B.
Remark. The cadlag golden parachute F 0 is not necessarily a modification of F in the technical sense. That is, there may exist times t for which Ft 6= Ft0 with strictly positive probability.
However, Lemma 22 establishes that any such disagreement preserves incentive-compatibility
and weakly improves profitability of the contract.
Lemma 23. Suppose that for some F ∗ > 0, V : [0, F ∗ ] → R is a C 2 function satisfying
V 0 (0) = 0, V 0 (F ∗ ) = 0, and the HJB equation on [0, F ∗ ]. Then:
• V 0 (F0 ) > 0 and V 00 (F0 ) < 0 for all F0 ∈ [0, F ∗ ),
• The unique maximizers (γ ∗ (F0 ), β ∗ (F0 )) of the HJB equation for each F0 ∈ [0, F ∗ ) are
β ∗ (F0 ) = −
∆r V 0 (F0 )
,
σ V 00 (F0 )
γ ∗ (F0 ) = β ∗ (F0 )
80
∆r
+ ρF − b,
σ
• (γ ∗ (F ∗ ), β ∗ (F ∗ )) = (ρF ∗ − b, 0) are maximizers of the HJB equation for F0 = F ∗ .
Proof. Fix F0 ∈ [0, F ∗ ] and suppose V 0 (F0 ) < 0. Then γ may be taken arbitrarily large and
negative without violating IC, so that the rhs of the HJB equation equals ∞. However, the
lhs of the HJB equation is finite, a contradiction. So V 0 (F0 ) ≥ 0. An analogous argument
rules out V 00 (F0 ) > 0 by considering arbitrarily large and positive β. So both V 0 and −V 00
must be weakly positive. Next suppose that V 0 (F0 ) > 0 and V 00 (F0 ) = 0. Then again the
rhs of the HJB equation may be made arbitrarily large by taking both γ and β large and
positive. So either V 0 (F0 ) > 0 and V 00 (F0 ) < 0, or else V 0 (F0 ) = 0 and V 00 (F0 ) ≤ 0. In
particular, V is weakly increasing on [0, F ∗ ].
Suppose that V 0 (F0 ) = 0 and V 00 (F ) ≤ 0 for some F0 ∈ [0, F ∗ ). Then β ∗ (F0 )V 00 (F ) = 0
for any maximizer β ∗ (F0 ), in which case the HJB equation implies (ρ + α)V (F0 ) = rG − αF.
But at F ∗ the same argument applies, so that (ρ+α)V (F ∗ ) = rG −αF ∗ < rG −αF0 = ρV (F0 ).
This contradicts the fact that V is weakly increasing on [0, F ∗ ], so it must be that V 0 (F0 ) > 0
and V 00 (F0 ) < 0 for all F0 ∈ [0, F ∗ ).
The stated maximizers of the HJB equation for F0 ∈ [0, F ∗ ) follow from noting that
the IC constraint must bind, else increasing γ a sufficiently small amount would increase
the rhs of the HJB equation while preserving the IC constraint given V 0 (F0 ) > 0. Thus
β + ρF0 − b at the optimum, and substituting this expression into the rhs of the
γ = ∆r
σ
V 0 (F0 )
HJB equation yields a concave quadratic in β with unique maximum β = − ∆r
. And
σ V 00 (F0 )
00
∗
00
0
∗
∗
as V (F ) ≤ 0 by continuity of V while V (F ) = 0, the HJB equation at F is maximized
by (γ, β) = (ρF ∗ − b, 0) ∈ IC(F ∗ ).
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