The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Working Paper Series
Optimal Contracts with Reflection
WP 16-14
Borys Grochulski
Federal Reserve Bank of Richmond
Yuzhe Zhang
Texas A&M University
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Optimal Contracts with Reflection∗
Borys Grochulski†
Yuzhe Zhang‡
September 29, 2016
Working Paper No. 16-14
Abstract
In this paper, we show that whenever the agent’s outside option is nonzero, the optimal
contract in the continuous-time principal-agent model of Sannikov (2008) is reflective at the
lower bound. This means the agent is never terminated or retired after poor performance.
Instead, the agent is asked to put zero effort temporarily, which brings his continuation
value up. The agent is then asked to resume effort, and the contract continues. We show
that a nonzero agent’s outside option arises endogenously if the agent is allowed to quit
and find a new firm (after a random search time of finite expected duration). In addition,
we find new dynamics of the reflection at the lower bound. In the baseline model, the
dynamics of the reflection are slow, as in Zhu (2013), i.e., the zero-action is used often.
However, if the agent’s disutility from the first unit of effort is zero, which is a standard
Inada condition, or if his utility of consumption is unbounded below, the reflection becomes
fast, i.e., the zero-effort action is used seldom.
1
Introduction
In an important contribution to the literature on incentives, Sannikov (2008) develops and
applies methods for computing optimal contracts in class of continuous-time principal-agent
problems. Sannikov (2008) makes this class of contracting problems tractable by providing
a representation of the agent’s continuation value as a diffusion process and by identifying
contracts with solutions to the Hamilton-Jacobi-Bellman equation, an ODE. The solutions
obtained in Sannikov (2008) provide several new insights into the structure of optimal dynamic
∗
The authors would like to thank Huberto Ennis, Boyan Jovanovic, Chris Phelan, Yuliy Sannikov, and Russell
Wong for their helpful comments. The views expressed herein are those of the authors and not necessarily those
of the Federal Reserve Bank of Richmond or the Federal Reserve System.
†
Federal Reserve Bank of Richmond, borys.grochulski@rich.frb.org.
‡
Texas A&M University, zhangeager@tamu.edu.
1
incentive contracts. In particular, Sannikov (2008) shows that the optimal contract terminates
after sufficiently poor performance.
In this paper, we qualify this result. We show that termination after poor performance is
only optimal when the agent’s outside option is his minimax payoff of zero. If the agent’s
outside option is even sightly better than the absolute lowest payoff the agent could get, the
optimal contract reflects off of the lower bound and does not terminate. Termination after
poor performance, thus, occurs in the optimal contract only under the assumption that the
agent’s outside option value is in a corner.
The agent’s continuation utility Wt is the state variable in this contracting problem. When
Wt reaches its lower bound B > minmax{Wt } = 0, the agent’s compensation and effort supply
requirement are zero. This means that both the sensitivity of Wt to the agent’s performance
and the agent’s current utility flow are zero when Wt = B. With zero current utility flow, the
contract delivers the agent’s continuation utility Wt = B > 0 by promising him more in the
future, i.e., by increasing Wt at the agent’s rate of time preference r > 0. Thus, Wt moves up
away from Wt = B, which allows the contract to continue without termination.
Figure 1 replicates Figure 3 from Sannikov (2008). The faint vertical line indicates the exogenous lower bound on the agent’s continuation utility, which is taken to be 0.1 in this example.
The curve labeled “optimum without reflection” is derived in Sannikov (2008) under the assumption that the contract terminates whenever the agent is asked to provide zero effort, so,
in particular, the contract terminates at the lower bound. The curve labeled “optimum with
reflection” is the optimal profit for the firm when we relax this assumption. As we see, allowing for reflection leads to a better contract. The upper bound Fmax is the firm’s optimal
profit function when the agent’s value of the outside option is his minimax payoff of zero. In
this case, the optimal contract does not reflect even if allowed to. The lower bound F0 is the
(negative) profit the firm can make by never asking the agent to exert effort and just paying
him a constant compensation that delivers his promised utility W .
In Section 6, we embed this contracting problem in a simple model of the labor market similar
to Phelan (1995). In this model, there are a large number of firms having access to a common
production technology. At any point in time, an agent matched with a firm is allowed to quit
and rejoin the labor market, where he can rematch with a new firm after paying a search cost.
In this setting, the agent’s outside option is determined endogenously. Due to competition
between firms, unless the search cost is infinite, the agent’s value of the outside option is
strictly larger than his minimax payoff. Therefore, the contract with reflection at the lower
bound is used in equilibrium.
With reflection, the optimal contract has interesting dynamics around the lower bound. When
2
0.2
Optimum without re.ection
Optimum with re.ection
Lower bound F0
Upper bound Fmax
0
Pro-t
-0.2
-0.4
-0.6
-0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
W
Figure 1: The firm’s profit function with the agent’s exogenous lower bound B = 0.1.
we follow Sannikov (2008) in assuming that the agent’s marginal disutility of effort is strictly
positive even as effort goes to zero, the dynamics of the reflection are similar to that at the
suspension contract studied in Zhu (2013). The contract calls for zero effort at the lower bound
but positive effort when Wt is just above the lower bound B. Incentive compatibility, thus,
implies that the volatility of Wt is zero at B and positive above B. Moreover, optimal effort
is discontinuous at B, which means the volatility of Wt jumps at B as well. The contract,
thus, returns to B frequently, which Zhu (2013) dubs slow reflection. We show that the same
dynamics arise in the Sannikov (2008) model with nonzero lower bound.
However, if the marginal disutility of effort at zero is zero, which is a standard Inada condition,
the dynamics of the reflection change. As Wt moves toward the lower bound, effort decreases
toward zero continuously and thus the volatility of Wt approaches zero smoothly, while the
drift of Wt remains strictly positive. With volatility extinguished near B, the contract does
not return to B frequently. We dub these dynamics “fast reflection” and study them in detail
in Section 7.
In Section 8, we study two extensions of the model. The first one follows Phelan (1995) in
3
allowing the firm to break the contract (i.e., walk away without delivering to the agent his
promised continuation utility Wt ) upon incurring a deadweight expense of K > 0. We show
that if K is not too large, a reflective upper bound emerges for the agent’s continuation value
process in the optimal contract. When Wt becomes close to this bound, the firm’s continuation
profit becomes close to −K. At the upper bound, the firm is indifferent between continuing and
walking away. At this point, the optimal contract shows reflective dynamics that mirror those
at the lower bound: the agent is asked to provide zero effort, which is incentive compatible
because Wt becomes insensitive to realized output. Compensation paid to the agent, however,
is positive and set at a level high enough to guarantee a negative drift of Wt , which makes
Wt move down and away from the upper bound, allowing for resumption of effort and further
continuation of the contract. In this extension, thus, the optimal contract does not terminate.
Under the optimal contract in the model with positive agent outside option and a finite-cost
firm outside option, neither the low nor high agent retirement points characterized by Sannikov
(2008) exist.
The second extension relaxes the assumption that the agent’s utility from consumption is
bounded from below. We show that this specification also gives rise to fast reflection at the
lower bound. However, the dynamics of this reflection are different from those with the Inada
condition on the marginal disutility of effort: the drift of the agent’s continuation value explodes
to plus infinity as Wt approaches its lower bound.
Contribution relative to the existing literature Our paper extends the analysis of Sannikov (2008) building on Zhu (2013). We show that slow reflection discovered by Zhu (2013)
is not specific to the risk-neutral principal-agent model without risk-sharing or intertemporal
consumption smoothing, which Zhu (2013) considers, but also appears in principal-agent relationships with risk-aversion and identical discounting, where risk-sharing and consumption
smoothing are valuable. In particular, we show that termination of the contract following the
agent’s poor performance is never optimal, outside of the corner case in which the agent’s outside option is equal to his minimax payoff. Given that termination of output of the relationship
after poor performance does not appear in dynamic contracting models in discrete time, e.g.,
Atkeson and Lucas (1995) or Phelan (1995), the result of Sannikov (2008) showing that productivity of the relationship seizes for incentives reasons when the agent’s continuation value
hits its lower bound is surprising. By qualifying this result, our analysis brings the lessons
from dynamic optimal contracting models in continuous time closer to the lessons obtained
previously in discrete time.
Continuous-time methods applied to incentive problems by Sannikov (2008), however, allow
us to study the dynamics of the optimal contract in more detail than what has been done
in discrete time. We show that, in addition to slow reflection discovered by Zhu (2013), the
4
optimal contract can have fast-reflection dynamics of at least two kinds. One with volatility
vanishing, and one with drift exploding at the lower bound.
Organization Section 2 describes the contracting problem we study, which is exactly the same
as the problem studied as in Sannikov (2008). Section 3 recalls the optimal contracts without
reflection derived in Sannikov (2008). Section 4 provides an informal description of how fully
optimal contracts, i.e., those allowing for reflection, can be constructed by combining lowand high-action ODEs in a fashion similar to Zhu (2013). Section 5 contains formal analysis
proving these results. Section 6 endogenizes the lower bound on the agent’s continuation value
by embedding the contracting problem with one-sided commitment in a simple model of the
labor market similar to Phelan (1995). Section 7 allows for Inada conditions on the marginal
disutility of effort and studies fast dynamics of reflection. Section 8 studies two extensions that
allow for, respectively, two-sided limited commitment and utility of consumption unbounded
below. Section 9 concludes.
2
The principal-agent problem
The principal-agent contracting problem is the same as in Sannikov (2008). The cumulative
output Xt produced by the agent up to date t follows
dXt = At dt + σdZt ,
where At ∈ A is the agent’s action (effort), Zt is a standard Brownian motion on (Ω, F, P ),
and σ > 0 is a constant. We assume that the set of feasible actions A is a compact interval [0, Ā] for some Ā > 0.1 The contract is a pair of progressively measurable processes
{(Ct , At ), 0 ≤ t < ∞}, where At is the action recommended for the agent to take at t and Ct is
his compensation. The agent and the principal evaluate the contract according to, respectively,
Z ∞
−rt
E r
e (u(Ct ) − h(At )) dt ,
0
and
Z
E r
∞
−rt
e
(At − Ct )dt ,
0
where r > 0. The agent’s utility function u : R+ → R+ is C 2 with u0 > 0, u00 < 0, limc→0 u0 (c) =
0, and u(0) = 0. The function h : A → R+ representing the agent’s disutility from effort is
increasing and convex with h(0) = 0. In addition, in this section we follow Sannikov (2008) in
1
That A includes a neighborhood of 0 is important in Section 7. Other parts of our analysis can be extended
to the more general case studied in Sannikov (2008), where A is a compact subset of R with the smallest element
0.
5
assuming that there exists γ0 > 0 such that h(a) ≥ γ0 a for all a ∈ A. Note that this assumption
means that if h is differentiable, it does not satisfy the usual Inada condition lima→0 h0 (a) = 0.
Under a given contract (C, A), the agent’s continuation value process is
Z ∞
e−r(s−t) (u(Cs ) − h(As )) ds .
Wt := Et r
t
Sannikov (2008) shows that there exists a progressively measurable process {Yt , 0 ≤ t < ∞}
such that the agent’s continuation value from the contract satisfies
dWt = r(Wt − u(Ct ) + h(At ))dt + rYt (dXt − At dt),
(1)
where dXt −At dt is the agent’s performance relative to the benchmark At dt. Here, Yt represents
the sensitivity of the agent’s continuation value to his performance observed by the firm. The
contract is incentive compatible (IC) at t if
At ∈ arg max Yt ã − h(ã).
ã
(2)
In the recursive form, the firm’s problem is to maximize the profit F (W ) that it can attain in
the relationship with the agent when the agent is owed the continuation value W . The HJB
equation for this problem is
1
F (W ) = max a − c + F 0 (W )(W − u(c) + h(a)) + F 00 (W )rσ 2 Y 2 ,
c,a,Y
2
(3)
where controls a and Y must jointly satisfy the IC constraint (2), and the state variable Wt
must satisfy the lower bound constraint
Wt ≥ B
at all t ≥ 0. The lower bound B ≥ 0, which represents the agent’s outside option, is taken as
exogenous by the firm. We follow Sannikov (2008) in assuming that the firm has the option of
not employing the agent. Thus, we focus on contracts that give a non-negative profit to the
firm as of t = 0.
3
3.1
Optimal contracts without reflection
Optimal contract with B = 0
Theorem 1 in Sannikov (2008) solves the above contracting problem in the case of the lower
bound B = 0. The optimal contract is obtained from the unique solution Fmax to the HJB
6
equation (3) that satisfies the following conditions: Fmax ≥ F0 , Fmax (0) = 0, there exists Wgp
0
such that Fmax (Wgp ) = F0 (Wgp ) and Fmax
(Wgp ) = F00 (Wgp ), where F0 is the retirement profit
function F0 (W ) = −u−1 (W ). In Figure 1, Fmax and F0 are depicted as, respectively, the upper
and the lower bound functions.
Until time τ = inf{t : Wt = 0 or Wt = Wgp }, the optimal contract is constructed from the
policy functions c(·), a(·), and Y (·) that attain the solution Fmax in the HJB equation (3). In
particular, for any W0 ∈ (0, Wgp ), Ct = c(Wt ) and At = a(Wt ), where Wt solves
dWt = r(Wt − u(c(Wt )) + h(a(Wt )))dt + rY (Wt )(dXt − a(Wt )dt)
(4)
for 0 ≤ t < τ . Under this contract, the support of the state variable Wt is [0, Wgp ]. The
optimal effort a(W ) is strictly positive for all W in the interior of [0, Wgp ]. Correspondingly,
the volatility of the agent’s continuation value is strictly positive in (0, Wgp ). The end points
of the support interval, 0 and Wgp , are referred to as retirement points. At either of these
points, effort a is zero, and drift and volatility of Wt are zero as well. The dynamics of the
state variable are stopped, i.e., 0 and Wgp are absorbing states for the process Wt .2
More precisely, Sannikov (2008) solves the HJB equation (3) in two steps. In the first step,
an additional restriction is imposed in the HJB equation requiring that the controls a and Y
satisfy a > 0 and Y ≥ γ0 at all W such that F (W ) > F0 (W ). Building on the intuition of
Zhu (2013), we will refer to such a restricted HJB equation as the high-action ODE, because
the zero-effort action a = 0 is explicitly excluded unless F (W ) = F0 (W ). In the second step,
it is verified that the above restriction is without loss of efficiency, i.e., that the firm could
not make a profit higher than Fmax by using the low-action a = 0 at some W such that
Fmax (W ) > F0 (W ). The optimal contract is absorbed when Wt hits 0 because c = a = Y = 0
is the unique set of controls that are feasible at this point, given that Wt ≥ 0 at all t. The
00 (W ) < 0, a = Y = 0
optimal contract is also absorbed when Wt hits Wgp because, with Fmax
gp
and c = u−1 (Wgp ) is the efficient policy at Wgp .
3.2
Optimal contract with B > 0 without reflection
Theorem 3 in Sannikov (2008) solves for an optimal contract with B > 0 under the assumption
that, as in the case of B = 0, the zero-effort action a = 0 is not used unless F (W ) = F0 (W ),
at which point the agent is retired or fired, so, effectively, the zero-effort action a = 0 is not
used prior to contract termination. In particular, this solution maintains that the agent is
fired when Wt = B, from which the firm makes the profit of zero. Under this assumption,
2
In other words, at 0 and Wgp the contract is constructed from the policy functions that attain the retirement
profit F0 .
7
the optimal contract is obtained from the solution F̃ of the HJB equation that satisfies the
following conditions: F̃ ≥ F0 , F̃ (B) = 0, there exists Wgp such that F̃ (Wgp ) = F0 (Wgp ) and
F̃ 0 (Wgp ) = F00 (Wgp ). This solution curve is depicted in Figure 1 as the optimum without
reflection.
4
Optimal contracts allowing for reflection: intuition
In this paper, we relax the assumption that the zero-effort action is not used prior to contract
termination. In particular, we allow for the zero-effort action to be applied at the lower bound
B without terminating the contract. We show that this possibility is in fact optimal when
B > 0. The dynamics of the optimal contract are reflective at the lower bound and thus the
contract does not terminate at B > 0. The process Wt moves up from B and the contract
continues.
We proceed in a way similar to Zhu (2013). We split the HJB equation into two ODEs: the
high-action ODE with the restriction that a > 0, which is the case studied in detail in Sannikov
(2008), and a low-action ODE, where it is assumed that the recommended action is a = 0.
4.1
The low-action ODE
If a = 0 is the recommend action, the minimal volatility of Wt that makes this action incentive
compatible is Y = 0. With these two controls fixed, the HJB equation (3) reduces to
F (W ) = max −c + F 0 (W )(W − u(c)).
c≥0
(5)
The optimal c is zero if F 0 ≥ 0 and solves −1 − F 0 u0 (c) = 0 if F 0 < 0. Consider the following
cases.
Case 1a. Suppose F 0 > 0. Then c = 0 and the ODE reduces to F = F 0 (W − u(0)) = F 0 W .
dW
C1
Solving by separation of variables we have dF
F = W or d ln(F ) = d ln(W ), so F = W e .
Because eC1 > 0, we have F 0 = eC1 > 0 consistent with the assumption. Thus, solutions with
F 0 > 0 are straight lines F = αW for α > 0. These lines go though the origin of the phase
plain (W, F ). Since we do not consider negative W , our solutions are rays out of the origin
with a strictly positive slope.
Case 1b. If F 0 = 0, then optimal c = 0, and the ODE is F = F 0 (W − u(0)) = 0 (W − 0) = 0.
This solution coincides with the horizontal axis, i.e., it takes the same form as the solutions in
Case 1a.
8
Case 2 Suppose F 0 < 0. The optimal c > 0 solves −1 − F 0 u0 (c) = 0. The ODE is F =
−c + F 0 (W − u(c)), where c satisfies this FOC. It is not hard to show that solutions of this
form are a) the retirement profit function F0 and b) all tangent lines to F0 . These solutions
are of secondary interest to us for now, as they will not be part of the optimal contract until
we consider two-sided limited commitment in Section 8.1.
Let us now consider the dynamics of Wt along any Case-1 solution. From (1) we get
dWt = rWt dt,
i.e., Wt grows deterministically at the rate r.
4.2
Combining high- and low-action solutions
Let us now discuss informally how the solutions to the two ODEs can be combined to improve
the contract relative to the optimal contract without reflection. Figure 2 replicates again the
optimum without reflection, F̃ , for the example with B = 0.1 presented in Figure 3 (left panel)
of Sannikov (2008). As we see, there exists a unique Case-1 solution to the low-action ODE
tangent to F̃ at some W s > B. Let us denote this solution by L(W ) and note that the slope
of L is F̃ (W s )/W s .
Consider a new contract (C, A) defined by using the optimal c from the solution L to the lowaction ODE (with a = 0) at all W ∈ [B, W s ] and the optimal controls (c, a, Y ) from the solution
F̃ to the high-action ODE at all W ∈ (W s , Wgp ]. Because the two solutions satisfy at W s the
value-matching and smooth-pasting conditions, L(W s ) = F̃ (W s ) and L0 (W s ) = F̃ 0 (W s ), this
contract delivers to the firm profit L(W ) if W ∈ [B, W s ] and F̃ (W ) if W ∈ (W s , Wgp ]. Because
L(W ) > F̃ (W ) for all W ∈ [B, W s ), the new contract constitutes a Pareto improvement over
the optimal contract without reflection.
Note that because the optimal controls obtained from the solution L to the low-action ODE
are (c, a, Y ) = (0, 0, 0), the process Wt is deterministic in the interval [B, W s ]. The agent’s
continuation value Wt grows exponentially and moves out of [B, W s ). Once Wt hits W s , it
never returns into [B, W s ), i.e., Wt reflects off of W s and stays in [W s , Wgp ] thereafter.
Note also in Figure 2 that the second derivatives of L and F̃ are not equal at W s . Because
W s > B is an interior point in the feasible support for the continuation value process, this
means that the contract obtained by splicing L and F̃ at W s is not an optimal contract. In
fact, better combinations of low- and high-action ODE solutions exist. One such example is
provided in Figure 3. In that example, the splicing point W s is closer to the lower bound B,
the low-action ODE has a higher slope, while the high-action ODE is everywhere above F̃ . As
9
0.2
Optimum without re.ection
Lower bound F0
Upper bound Fmax
Tangent low-action ODE
Pro-t
0
-0.2
-0.4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
W
Figure 2: Combining the high- and low-action solutions. The splicing point is W s = 0.17 >
0.1 = B. Positive effort is exerted to the right of W s . Zero effort is exerted at and to the left
of W s .
before, the two ODEs are spliced at a point where the smooth pasting conditions are satisfied,
thus resulting in a consistent contract over the whole domain [B, Wgp ].3
The intuition for why this contract is better has to do with the “endogenous” support [W s , Wgp ].
The larger this support, the better insurance the firm can provide to the agent without shutting
down the agent’s effort, i.e., a higher solution curve F of the high-action ODE can be attained.
As we show formally next, the optimal contract obtains when the splicing point W s coincides
with the exogenous lower bound B, which case is depicted in Figure 1. In this case, the splicing
point cannot be moved further to the left, i.e., the “endogenous” support [W s , Wgp ] cannot be
further enlarged.
3
Wgp is not the same in these two examples. It is higher in the second case.
10
0.2
Optimum without re.ection
Lower bound F0
Upper bound Fmax
Tangent low-action ODE
Tangent high-action ODE
Pro-t
0
-0.2
-0.4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
W
Figure 3: Combining the high- and low-action solutions at the splicing point W s = 0.13 leads
to a better contract than F̃ at all W .
5
Solution of the contracting problem
In this section, we provide a formal solution verifying the intuition given in the previous section.
∗ the first-best effort shut-down threshold, i.e.,
As in Sannikov (2008), let us denote by Wgp
∗ = u(c), where c solves u0 (c) = h0 (0).
Wgp
∗ ] and for two numbers y and y 0 , we will denote by F
For B ∈ [0, Wgp
(B,y,y 0 ) the solution to
the high-action ODE that starts at W = B and satisfies the boundary condition F (B) = y,
F 0 (B) = y 0 .4
We start out by examining solutions F(B,0,0) , i.e., the high-action ODE solutions that start at
∗ ]. We
the horizontal axis with the initial slope of zero, where the starting W is B ∈ [0, Wgp
are interested in these solutions because they can be pasted smoothly with the lowest of the
4
Lemma 1 in Sannikov (2008) shows existence, uniqueness, concavity, and continuity in initial conditions of
solutions to the high-action ODE.
11
low-action ODE solutions, i.e., the solution that follows the horizontal axis (solution Case 1b).
∗ ] for which the solution
Lemma 1 (Largest lower bound) There exists a unique B̄ ∈ [0, Wgp
∗ ], and b) there exists W
F(B̄,0,0) satisfies a) F(B̄,0,0) (W ) ≥ F0 (W ) for all W ∈ [B̄, Wgp
gp ∈
∗
0
0
[B̄, Wgp ] such that F(B̄,0,0) (Wgp ) = F0 (Wgp ) and F(B̄,0,0) (Wgp ) = F0 (Wgp ).
Conditions a) and b) in the above lemma are analogous to the conditions in Lemma 3 of
Sannikov (2008). The above lemma identifies the largest lower bound, B̄, at which the firm can
find a contract that never violates this bound but also lets the firm break even in expectation
as of t = 0. As we will see in Theorem 1 below, the optimal contract subject to the agent’s
quitting constraint at the lower bound B̄ will be constructed by splicing the lowest of the
positively sloped low-action solutions, i.e., the ray that follows the horizontal axis, with the
high-action ODE solution F(B̄,0,0) .
For B > B̄, the solution curve F(B,0,0) stays above F0 for all W ≥ B, i.e., it fails to satisfy
condition b) of Lemma 1. This means that with B > B̄ there is no contract such that Wt ≥ B
at all t and F (W0 ) ≥ 0.
For 0 ≤ B < B̄, the solution curve F(B,0,0) crosses the retirement profit curve F0 at some
W > B. By pasting F(B,0,0) with the horizontal low-action ODE solution at W = B, it is
possible to obtain a feasible contract under which the agent’s quitting constraint is satisfied
and which the firm breaks even. But because F(B,0,0) crosses F0 , this contract would not
be optimal. A better contract can be obtained if F(B,0,0) is replaced with F(B,y,y0 ) such that
F(B,y,y0 ) > F(B,0,0) .
The next lemma describes how such a solution is obtained. For any B ∈ (0, B̄], for some y > 0
and y 0 > 0, a high-action ODE solution F(B,y,y0 ) satisfying conditions a) and b) can be pasted
at W = B with a strictly positively sloped low-action ODE solution (i.e., a solution of Case
1a). The requirement of smooth pasting between F(B,y,y0 ) and a straight ray coming out of the
origin pins down the initial slope y 0 for given B and y. We must have y 0 = By .
Lemma 2 (Largest initial
0 ≤ y < Fmax (B) such that
∗ ], and b) there
W ∈ [B, Wgp
0
0
F(B,y,
y (Wgp ) = F0 (Wgp ). If
)
level and slope) For each B ∈ (0, B̄], there exists a unique
the solution F(B,y, y ) satisfies a) F(B,y, y ) (W ) ≥ F0 (W ) for all
B
B
∗ ] such that F
y (Wgp ) = F0 (Wgp ) and
exists Wgp ∈ [B, Wgp
(B,y, B )
B = B̄, then y = 0. For B ∈ (0, B̄), y > 0.
B
Let us denote the unique y pinned down in this lemma by y ∗ (B). Also, we will denote the
point Wgp pinned down by the smooth pasting condition between F(B,y∗ (B), y∗ (B) ) and F0 by
B
Wgp (B).
12
5.1
Optimal contract
Following Zhu (2013), let us define a function V : [B, Wgp (B)] → R by splicing the high-action
ODE solution F(B,y∗ (B), y∗ (B) ) (W ) with, respectively, the low-action ODE solution L(W ) =
y ∗ (B)
B W
B
at B, and with F0 at
V (W ) =
Wgp (B). That is, let
L(W )
for W = B,
∗
F(B,y∗ (B), y (B) ) (W ) for W ∈ (B, Wgp (B)),
B
F0 (W )
for W = Wgp (B).
Theorem 1 For each B ∈ (0, B̄], the function V is the firm’s value function in the contracting
problem with the agent’s lower bound B. The optimal controls c, A, Y attaining V define an
optimal contract Ct = c(Wt ), At = a(Wt ), where {Wt , 0 ≤ t < ∞} is a solution to (4). In
particular, c(B) = a(B) = Y (B) = 0, with dWt = rBdt > 0dt when Wt = B; a(W ) > 0 and
Y (W ) > 0 for all W ∈ (B, Wgp (B)); and c(Wgp (B)) > 0, a(Wgp (B)) = Y (Wgp (B)) = 0, with
dWt = 0 when Wt = Wgp (B).
The proof follows Sannikov (2008) very closely with two exceptions. The technical argument
for the existence of a solution to (4) is modified to account for volatility of Wt vanishing at B,
and the step verifying the optimality of the contract is modified to account for the reflection
of the process Wt at B.
The dynamics of the optimal contract are shown in Figure 4. Because the super-contact
condition is not satisfied at B or Wgp , the optimal controls Y and a jump at these points. In
particular, volatility Y is extinguished at both boundaries. Since a jumps at the boundaries,
drift of Wt is also discontinuous at these points. But despite this discontinuity, drift of Wt
remains strictly positive at B, which generates reflection of the process Wt off of B. The
reflection is slow, as in Zhu (2013). That is, the process Wt spends a positive expected amount
of time at the lower bound.
5.2
Comparative statics
Next, we examine how the optimal contract depends on the value of the agent’s outside
option, B. In particular, we study how the agent’s value from the contract, W0 (B) :=
argmaxW ≥B V (W ), and the firm’s value, V (W0 (B)), depend on B.
Proposition 1 For all B ∈ [0, B̄], W0 (B) is strictly increasing in B but W0 (B) − B is strictly
decreasing in B. V (W0 (B)) is strictly decreasing in B.
We can interpret W0 (B) − B as the agent’s surplus from the contracting relationship, with
the firm’s surplus being V (W0 (B)). When the agent’s outside option improves, the value he
13
0.1
drift
volatility
Drift and volatility of Wt
0.08
0.06
0.04
0.02
0
-0.02
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
W
Figure 4: The dynamics of Wt . The two vertical lines mark B and Wgp . Volatility of Wt is
discontinuous jumping down to zero at both boundaries. Drift jumps as well but not all the
way to zero at B.
obtains increases in absolute terms. But the contracting relationship becomes less profitable,
as the firm has less room to provide incentives. As a result, both the firm’s surplus and
the worker’s surplus relative to his outside option are reduced. Figure 5 shows the inverses
relationship between W0 and V (W0 (B)) graphically.
6
Endogenous outside option
In this section, we endogenize the agent’s outside option B by embedding the contacting
problem in a simple model of the labor market similar to Phelan (1995). In a meeting between
the firm and the agent, the firm designs the contract (C, A) and offers it to the agent as a
take-it-or-leave proposition. If the agent rejects, he goes into the market, where he searches for
a match with a new firm, while the old firm exits with the payoff of zero. While searching, the
14
0.2
Contract curve (W0 (B); V (W0 (B)))
Pro-t function V (W ) for several values of B
B=0
0
Pro-t
7
B=B
-0.2
-0.4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
W
Figure 5: Comparative statics with respect to B.
agent consumes zero and exerts zero effort.5 The new match arrives with Poisson intensity λ.
The firm in the new match is identical to the firm in the previous match, i.e., it operates the
same production technology and faces the same contract-design problem. If the agent accepts
the contract, he is not committed to it, i.e., he can quit and go back to the market at any time.
Let W0 denote the initial value that a new contract (C, A) delivers to the agent. Since the
agent receives zero utility while searching, his post-separation utility comes exclusively from
future matches, whose arrival time has density λe−λt . The consistency condition between the
agent’s outside option B and the initial value W0 therefore is
Z ∞
λ
W0 .
(6)
B=
λe−λt e−rt W0 dt =
r+λ
0
The firm takes B as given and solves the contracting problem so as to maximize its profit.
Definition 1 Given λ ≥ 0, competitive equilibrium consists of the agent’s initial contract
value W0 ≥ 0 and outside option B ≥ 0 and of a function V such that 1) V is the firm’s value
5
This normalizing assumption can easily be relaxed.
15
0.3
Equilibrium consistency condition
Contract curve
0.25
W0
0.2
0.15
0.1
0.05
0
0
0.05
0.1
0.15
0.2
0.25
0.3
B
Figure 6: Equilibrium conditions.
function given the agent’s outside option B, and 2) the consistency condition (6) holds with
W0 ∈ argmaxW ≥B V (W ).
Proposition 2 For every λ ∈ [0, ∞], there exists a unique competitive equilibrium (W0 , B, V ).
The mapping from the expected search time 1/λ to the equilibrium values W0 , B, and V (W0 )
is monotonic, with W0 and B strictly decreasing and V (W0 ) strictly increasing in 1/λ. If the
expected search time is zero, then W0 = B = B̄ and V (W0 ) = 0. If the expected search time is
infinite, then B = 0, W0 = argmaxW ≥0 Fmax (W ) and V (W0 ) = max Fmax (W ).
Proof of Proposition 2 follows from Figure 6. In that figure, the circled curve represents the
relationship between the lower bound B and the agent’s value W0 delivered by the optimal
contract, i.e., the function W0 (B) := argmaxW ≥B V (W ) studied in Proposition 1. We have
W0 (0) > 0, W0 (B̄) = B̄, and the slope of W0 (B) is everywhere smaller than one because the
agent’s surplus W0 (B)−B is decreasing in B. The straight solid line represents the consistency
condition W0 = r+λ
λ B, the slope of which is larger than one. The unique intersection of these
two lines determines the unique equilibrium.
16
Figure 6 also shows how equilibrium depends on the new match arrival rate λ or on the expected
search time 1/λ. If λ = ∞, i.e., the agent can get a new match immediately after quitting,
the consistency condition coincides with the 45-degrees line and equilibrium lower bound B
is determined at the highest possible level B̄. When search cost 1/λ becomes positive, both
B and W0 (B) decline.6 The lower bound B reaches zero only when search cost 1/λ becomes
infinite. This means that the equilibrium contract features termination of output at the lower
bound only if the agent’s search cost is infinite. For all finite values of the search cost 1/λ, the
lower bound B determined in equilibrium is strictly positive, which implies that the process
Wt is not absorbed at B but rather reflects off of it in equilibrium.
7
Fast reflection
In this section, we assume that h has a bounded second derivative and that it satisfies the
standard Inada condition lima→0 h0 (a) = 0. We show that the dynamics of the reflection at
the lower bound change. In particular, they become fast, meaning the process Wt spends zero
time at B.
Assumption 1 h(·) ∈ C 3 , that is, its third derivative is continuous. h(0) = h0 (0) = 0 and
0 < h00 (0) < ∞.
7.1
Boundary conditions
As before, we will construct solutions to the HJB equation by splicing the low- and high-action
ODEs at an exogenous lower bound B for the agent’s continuation value process Wt . Let L be a
solution to the low-action ODE and F a solution to the high-action ODE. The value-matching
and smooth-pasting conditions, F (B) = L(B) and F 0 (B) = L0 (B), imply
0 = F (B) − L(B)
1 00
2 0
2
0
=
max a − c + L (B) (B − u(c) + h(a)) + F (B)rσ h (a)
c,a
2
− max −c + L0 (B) (B − u(c))
c≥0
1
= max a + L0 (B)h(a) + F 00 (B)rσ 2 h0 (a)2 .
a
2
If lima→0 h0 (a) = 0, this condition cannot be met with a finite F 00 (B). Indeed, if F 00 (B) > −∞,
the objective under maximization attains the value of 0 at a = 0, but a = 0 is not a maximizer,
which can be easily seen by differentiating the objective and evaluating the derivative at a = 0:
6
By Proposition 1, the firm’s value V (W0 (B)) increases.
17
1 + L0 h0 (a) + rσ 2 F 00 h0 (a)h00 (a)|a=0 = 1 > 0. In order to satisfy the above condition, therefore,
we must allow for F 00 (B) = −∞, in which case the optimal action in the high-action ODE is
a = 0 at W = B. 7 8
In sum, in order to satisfy the value-matching and smooth-pasting conditions at B, a solution F
to the HJB equation for W ≥ B must satisfy F (B) = L(B), F 0 (B) = L0 (B), and F 00 (B) = −∞,
i.e., F must be singular at B. In the Appendix, we present a change-of-variable technique that
allows for solving of the HJB equation forward from B despite this singularity.
Because the firm and the agent discount at the same rate, the optimal contract under full
information (observable a) is static: a and c are constant. The firm’s first-best profit function
is
Ffb (W ) := max{ a − c : u(c) − h(a) = W }.
c,a
∗ does not exist with the Inada condition
Let us note that the first-best shut-down threshold Wgp
on h, because c that solves u0 (c) = h0 (0) does not exist. That is, a is always strictly positive
at the first best, even with very high W .
Note that F0 < Ffb because
F0 (W ) = max{ a − c : u(c) − h(a) = W and a = 0}
c,a
and the restriction a = 0 binds at all W . We will look for solutions F of the HJB equation
that satisfy F0 (W ) < F (W ) < Ffb (W ) for all W ≥ B.
For B > 0 and for two numbers y and y 0 , denote by F(B,y,y0 ,−∞) the solution to the HJB that
satisfies boundary conditions F (B) = y, F 0 (B) = y 0 , and F 00 (B) = −∞.
The next two lemmas are analogs of Lemma 1 and Lemma 2.
Lemma 3 (Largest lower bound) There exists a unique B̄ > 0 for which the solution
F(B̄,0,0,−∞) satisfies
F0 (W ) < F(B̄,0,0,−∞) (W ) < Ffb (W ) for all W ≥ B̄.
Lemma 4 (Largest initial level and slope) For every B ∈ (0, B̄], there exists a unique y
such that the solution F(B,y, y ,−∞) satisfies
B
F0 (W ) < F(B,y, y ,−∞) (W ) < Ffb (W ) for all W ≥ B.
B
As before, we will denote the unique y pinned down in this lemma by y ∗ (B).
Note that this problem does not arise if h0 (a) ≥ γ0 > 0 for all a. Indeed, F 00 h0 (a)2 ≤ F 00 γ02 and maxa∈A a +
L h(a) ≤ M for some M imply a + L0 h(a) + 12 rσ 2 F 00 h0 (a)2 ≤ M + 12 rσ 2 F 00 γ02 for all a, so a finite number F 00 < 0
exists with which the value-matching and smooth-pasting conditions can be met.
8
The high-action ODE is in this case the unrestricted ODE, where a = 0 is allowed.
7
0
18
7.2
Optimal contract
For each B ∈ (0, B̄], let us define a function V : [B, ∞) → R by splicing at W = B the
high-action ODE solution F(B,y∗ (B), y∗ (B) ,−∞) (W ) with the low-action ODE solution L(W ) =
y ∗ (B)
B W.
B
That is, let
(
V (W ) =
L(W )
for W = B,
F(B,y∗ (B), y∗ (B) ,−∞) (W ) for W > B.
B
Theorem 2 For each B ∈ (0, B̄], the function V is the firm’s value function in the contracting
problem with the agent’s lower bound B. The optimal controls c, A, Y attaining V define an
optimal contract Ct = c(Wt ), At = a(Wt ), where {Wt , 0 ≤ t < ∞} is a solution to (4). In
particular, c(B) = a(B) = Y (B) = 0, with dWt = rBdt at Wt = B; and a(W ) > 0 and
Y (W ) > 0 for all W > B.
7.3
The dynamics of reflection
Under the optimal contract, the process Wt is not Sticky Brownian Motion studied in Zhu
(2013). See Figure 7. As in the non-Inada case shown earlier in Figure 4, drift of Wt is strictly
positive at the lower bound B > 0 and in a neighborhood of B, while volatility is zero at B
and strictly positive in a neighborhood of B. In the non-Inada case, the volatility of Wt was
discontinuous at B. In the Inada case, however, the volatility of Wt converges to zero as W
goes to the lower bound B, i.e., it is right-continuous at B.
This means that when Wt gets close to B, its drift remains bounded away from zero while its
volatility becomes extinguished. Intuitively, the closer Wt is to B, the more deterministic it
becomes in its movement up and away from B. As a result, the (Lebesgue) measure of time
that Wt spends at B is zero with probability one. Thus, the reflection of Wt off of B is faster
in the Inada case than in the non-Inada case.
Proposition 3 B is nonsingular and, for any W0 ≥ B, we have PW0 (Leb({t ≥ 0 : Wt =
B}) = 0) = 1.
7.4
Remark on immiserization
In the dynamic contracting literature, starting with Thomas and Worrall (1990), typically
F 0 (Wt ) is a martingale. This leads to the well-known result of immiserization, that is, the
agent’s continuation value converges to the lowest possible value almost surely. In our model,
as in Phelan (1995), the immiserization result fails because the agent’s continuation value is
19
0.06
drift
volatility
0.05
Drift and volatility of Wt
0.04
0.03
0.02
0.01
0
-0.01
0
0.2
0.4
0.6
0.8
1
1.2
1.4
W
Figure 7: Drift and volatility of Wt with h(a) = 21 a2 . The lower bound is the largest equilibrium
lower bound B̄. Upper bound does not exist. Both drift and volatility are continuous on [B̄, ∞).
Reflection is fast.
reflective at the lower bound. Yet, the drift of F 0 (Wt ) is zero whenever Wt > B, i.e., F 0 (Wt )
continues to be a martingale when effort recommendation is nonzero.9 The fact that F 0 (Wt )
is no longer a martingale when Wt = B is key to eliminating the immiserization result. Since
Wt has a drift of r(Wt − u(Ct )) > 0 and a volatility of zero at Wt = B, F 0 (Wt ) changes
deterministically and its drift is F 00 (Wt )r(Wt − u(Ct )) < 0. One might be surprised that
nonzero drift at just a single point, B, is strong enough to make the entire process fail to be a
martingale. The subtlety here lies in Sticky Brownian Motion: even though B is a singleton,
the process Wt spends a positive amount of time at B (see Section 5). In Section 7, where
The drift and the volatility of Wt are, respectively, r(Wt − u(Ct ) + h(At )) and h0 (At )rσ. By Ito’s lemma,
the drift of F 0 (Wt ) can be calculated as
9
F 00 (Wt )r(Wt − u(Ct ) + h(At )) +
F 000 (Wt ) 0
(h (At )rσ)2 .
2
Differentiating the HJB equation with respect to Wt , we can show that F 00 (Wt )r(Wt − u(Ct ) + h(At )) +
F 000 (Wt )
(h0 (At )rσ)2 = 0. Cf. equation (12) in Sannikov (2008).
2
20
0.2
Optimum with two-sided refection
Lower bound F0
Upper bound Fmax
Pro-t
0
-0.2
-0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
W
Figure 8: Optimal contract with two-sided limited commitment. The high-action ODE solution
curve goes from the vertical line W = B = 0.1 to the horizontal line F = −K = −0.2. At both
ends, it pastes smoothly with low-action ODE solutions. The higher splicing point D = 0.49
becomes a reflective upper bound for Wt .
the Inada condition for h holds, Wt spends zero amount of time at B, but the size of the drift
|F 00 (Wt )r(Wt − u(Ct ))| is infinity because F 00 (B) = −∞. An infinite drift over a small time
interval (of measure zero) has a nontrivial effect on the dynamics of the process.
8
Extensions
In this section, we extend our analysis to include two significant extensions. First, we allow
for limited commitment on the side of the firm. Then, we analyze the case with u unbounded
below.
21
8.1
Two-sided limited commitment
In this section, we follow Phelan (1995) in introducing limited commitment also on the side
of the firm. In particular, we allow the firm to get out of any contract upon incurring a
deadweight cost of K ≥ 0.10 This possibility adds another constraint to the contract design
problem:
F (Wt ) ≥ −K at all t.
(7)
That is, a firm cannot credibly promise a contract that allows for the firm’s continuation value
to drop below −K in some state. In this section, we describe the solution to this contractdesign problem. We treat B as exogenous, although it can be endogenized in the same way
as in the previous section. As well, we return to the assumption h0 (0) = γ0 > 0, which gives
us slow reflection, although it is also possible to study two-sided limited commitment with
h0 (0) = 0 and fast reflection.
The solution to this contracting problem is constructed as follows. At the lower bound W =
B, we start solving the high-action ODE, in the direction of increasing W , with boundary
conditions F = y and F 0 = y/B for some y ≥ 0, as in Lemma 2. Recall that this choice of
boundary conditions guarantees that the resulting solution, F(B,y, y ) , pastes smoothly at B with
B
a positively sloped solution to the low-action ODE (Case 1a in section 4.1). We look for y such
that F(B,y, y ) stays between F0 and Fmax until it drops down to the level F(B,y, y ) (D) = −K
B
B
at some D > B.11 At W = D, we stop solving the high-action ODE and instead extend the
0
solution as a straight line with slope F(B,y,
y (D) < 0. This line can stay above F0 , cut through
)
B
F0 , or become tangent to F0 at some point D0 > D. In this last case, the straight line is a
type-2b solution to the low-action ODE (see section 4.1), and so D becomes a point of smooth
pasting between the high-action ODE solution F(B,y, y ) and a low-action ODE solution tangent
B
to F0 at D0 . Along this low-action solution, the agent’s continuation value process has zero
volatility and drift W − u(c0 ) + h(0), where u(c0 ) = D0 . For all W < D0 , this drift is negative.
In particular, it is negative at the splicing point D. With zero volatility and negative drift at
D, the agent’s continuation value process reflects downward off of D, i.e., D becomes an upper
bound for Wt .
Figure 8 provides one example of such a solution. Arguments similar to those in Theorem 1
can be used to verify that this solution indeed represents an optimal contract.
10
That is, if the firm pays K, it no longer has to meet its contractual obligation of delivering the continuation
value Wt to the agent.
11
If F(B,y, y ) becomes tangent to F0 before dropping down to −K, then the point of tangency becomes Wgp (B)
B
from Lemma 2 and the firm’s quitting constraint (7) does not bind. We will assume in this section that the
firm’s cost of breaking the contract, K, is low enough for this constraint to bind, i.e., K < −F0 (Wgp (B)). This
assumption would be vacuous in the fast-reflection case with the Inada condition on h, where Wgp (B) is infinite.
22
0.3
Optimum with refection
Lower bound F0
First Best
0.6
0.4
drift
volatility
0.25
0.2
Drift and volatility of Wt
0
Pro-t
-0.2
-0.4
-0.6
-0.8
0.2
0.15
0.1
-1
0.05
-1.2
-1.4
0
-3
-2.5
-2
-1.5
-1
-3
W
-2.5
-2
-1.5
-1
W
Figure 9: The case of unbounded u. Left panel: the firm’s value function. Right panel:
dynamics of the optimal contract.
8.2
Unbounded utility
Thus far, our analysis of the optimal contract requires that u be bounded below. In particular,
we have followed Sannikov (2008) in assuming that 0 is the lowest possible consumption level
for the agent and u(0) = 0. This specification disallows many utility functions that often are
used in applications, e.g., log or CRRA with the RRA coefficient larger than 1.
In this subsection, we allow for u unbounded below. In particular, we assume that limc→0 u(c) →
−∞. We show that the optimal contract is reflective at an exogenous lower bound B.12 Further, reflection dynamics are fast, even with h0 (0) = γ0 > 0, and the firm’s profit function is
decreasing.
With u(0) = −∞, by setting c = a = Y = 0, the firm can costlessly and instantaneously
shift the agent’s continuation utility upward. Indeed, with these controls, the expected flow
of payoff to the firm is zero and the drift of the agent’s utility is infinite. Since the firm can
always apply this instantaneous control at the peak of its profit function, the process Wt will
never go below the value at which the peak is attained, W ∗ , which restricts the support of Wt
to the values above W ∗ . By the same logic as in Section 4.2, it is optimal to locate the peak
of the profit function at the lower bound B, as this maximizes the support for Wt and allows
the firm to best insure the agent and hence achieve highest profit. With the peak of F at B,
F 0 (B) = 0 provides a boundary condition for the solution of the high-action ODE to the right
12
The lower bound can be endogenized as in Section 6 but now under the assumption that the agent receives
some positive flow of consumption while searching, which is necessary in order to obtain B > −∞.
23
of B.13
The analog of Lemma 2 is as follows. At B, we start solving the high-action ODE with
boundary conditions F (B) = y ≥ F0 (B) and F 0 (B) = 0.14 We search for an initial y such
that the solution curve F(B,y,0) remains above F0 touching it at some Wgp > B. Since F(B,y,0)
0
is strictly concave and F(B,y,0)
(B) = 0, the firm’s value function is strictly decreasing, i.e.,
compensation ct is strictly positive everywhere outside of the lower bound W = B. As Wt
becomes close the lower bound, drift of Wt becomes large. Therefore, reflection off of B is fast.
Figure 9 provides a computed example in which u is CRRA with relative risk aversion of 2.
9
Conclusion
We view this paper as making the following three contributions. First, we show that the
optimal contract in the dynamic principal-agent model of Sannikov (2008) is reflective at the
lower bound of the agent’s continuation value process, whenever this lower bound is greater
than the agent’s minimax payoff of zero. This means that the contract is never terminated
(the agent is not fired or retired) after poor performance. Rather, the agent is temporarily
asked to put zero effort, which brings his continuation value up, and effort is resumed.
Second, we endogenize the lower bound of the agent’s continuation value process by embedding
the contracting problem in a generalized version of the Phelan (1995) model of the labor market
with one-sided commitment. In this model, the agent can quit at any time, so his outside option
bounds his contract continuation value from below. We show that as long as the agent’s search
time for a new match is finite, this lower bound is strictly larger than the agent’s minimax
payoff, implying reflection at the lower bound in equilibrium.
Third, we find new dynamics of reflection of the agent’s continuation value off its lower bound.
We show that when the agent’s disutility of effort satisfies the usual Inada condition (marginal
disutility of the first unit of effort is zero), reflection off of the lower bound is fast, i.e., the
13
Informally, we can think of the low-action ODE (5) as being solved to the left of B, with the solution having
c = 0, infinite drift of Wt , and zero slope, so the low-action and high-action ODE solutions paste smoothly at
B. Having infinite drift at B, the state variable Wt receives at that point a positive instantaneous shift familiar
from the instantaneous control literature, see Stokey (2008). Since the firm’s cost of shifting Wt upward is zero,
F 0 (B) = 0 satisfies the smooth-pasting condition of the instantaneous-control problem, which requires that the
marginal cost of shifting the state be equal to its marginal value. Instantaneous shifts of the state variable also
occur in DeMarzo and Sannikov (2006) and Zhu (2013). There, the shifts are negative, carry a unit cost to the
firm, and occur at the upper bound of the state variable, where the firm’s marginal profit equals negative one.
Here, the shifts are positive, carry no cost to the firm, and occur at the lower bound of the state variable, where
the firm’s marginal profit is zero.
14
Despite the singularity at c = 0, this solution can be advanced because F 0 (W )u(c) remains bounded.
24
continuation value process spends zero total time at the lower bound. Another case of fastreflection dynamics obtains when the agent’s utility function is unbounded below.
Appendix
Similar to Sannikov (2008), we can express the high-action ODE as
F 00 (W ) = − max
a≥0
a + F 0 h(a) − F + maxc {F 0 (W − u(c)) − c}
.
rσ 2 (h0 (a))2 /2
|
{z
}
(8)
Ha (W,F,F 0 )
The following lemma, analogous to Lemma 2 in Sannikov (2008), orders the solutions to (8).
Lemma A.1 Consider two solutions F and F̃ to the high-action ODE that satisfy F (W ) ≤
F̃ (W ) and F 0 (W ) ≤ F̃ 0 (W ). If at least one of these inequalities is strict, then
F 0 (W 0 ) < F̃ 0 (W 0 ),
∀W 0 > W.
(9)
Proof This proof modifies the proof of Lemma 2 in Sannikov (2008). First, we show (9) in
a small neighborhood of W . This holds trivially if F 0 (W ) < F̃ 0 (W ). If F 0 (W ) = F̃ 0 (W ) and
F (W ) < F̃ (W ), then
F 00 (W ) ≤ − max Ha (W, F (W ), F 0 (W ) < − max Ha (W, F̃ (W ), F̃ 0 (W )) = F̃ 00 (W ).
a≥0
a≥0
It follows from F 0 (W ) = F̃ 0 (W ) and F 00 (W ) < F̃ 00 (W ) that (9) holds in a small neighborhood
of W .
Second, we show (9) for all W 0 > W by contradiction. Suppose (9) does not hold, then
there exists a smallest W ∗ > W at which F 0 (W ∗ ) = F̃ 0 (W ∗ ). Since F 0 (W 0 ) < F̃ 0 (W 0 ) for all
W 0 ∈ (W, W ∗ ), we have F (W ∗ ) < F̃ (W ∗ ) and again
F 00 (W ∗ ) ≤ − max Ha (W ∗ , F (W ∗ ), F 0 (W ∗ ) < − max Ha (W ∗ , F̃ (W ∗ ), F̃ 0 (W ∗ )) = F̃ 00 (W ∗ ).
a≥0
a≥0
It follows that F 0 (W ∗ − ) > F̃ 0 (W ∗ − ) for all sufficiently small > 0, a contradiction.
Proof of Lemma 1
Uniqueness: By contradiction, suppose for some B 0 < B, both F(B 0 ,0,0) and F(B,0,0) satisfy
0
0
conditions a) and b). Since F(B 0 ,0,0) (B) < 0 = F(B,0,0) (B) and F(B
0 ,0,0) (B) < 0 = F(B,0,0) (B),
it follows from Lemma A.1 that F(B 0 ,0,0) (W ) < F(B,0,0) (W ) for all W ≥ B. Since F(B 0 ,0,0) is
25
weakly above F0 , we have F0 (W ) ≤ F(B 0 ,0,0) (W ) < F(B,0,0) (W ) for all W ≥ B. This contradicts
condition b) for F(B,0,0) (Wgp ).
Existence: Define
∗
B̄ := inf{B ≥ 0 : F(B,0,0) (W ) ≥ F0 (W ), ∀W ∈ [B, Wgp
]}.
∗ ]} is nonempty because F
∗ ,0,0) is
The set {B ≥ 0 : F(B,0,0) (W ) ≥ F0 (W ), ∀W ∈ [B, Wgp
(Wgp
∗
∗
∗ ,0,0) is a singleton and F(W ∗ ,0,0) (Wgp ) = 0 > F0 (Wgp ).
above F0 . Indeed, the domain of F(Wgp
gp
Because F(B̄,0,0) satisfies condition a) of the lemma directly from the definition of the above
set, we only need to show that F(B̄,0,0) satisfies condition b) of the lemma.
If B̄ = 0, then we define Wgp = 0, as in Sannikov (2008). If B̄ > 0, then consider the
sequence {B̄ − n1 : n = 1, 2, ...}, which converges to B̄ from the left. Since B̄ − n1 ∈
/
∗ ]}, there exists some W ∈ [0, W ∗ ] such that
{B ≥ 0 : F(B,0,0) (W ) ≥ F0 (W ), ∀W ∈ [B, Wgp
n
gp
F(B̄− 1 ,0,0) (Wn ) < F0 (Wn ). Let Wgp be the limit of some subsequence {Wnk : k = 1, 2, ...}
n
of {Wn : n = 1, 2, ...}, then F(B̄,0,0) (Wgp ) = limk→∞ F(B̄− 1 ,0,0) (Wnk ) ≤ limk→∞ F0 (Wnk ) =
nk
F0 (Wgp ). This and F(B̄,0,0) (Wgp ) ≥ F0 (Wgp ) from condition a) imply F(B̄,0,0) (Wgp ) = F0 (Wgp ).
We have F(0B̄,0,0) (Wgp ) = F00 (Wgp ) because condition a) implies (F(B̄,0,0) )0+ (Wgp ) ≥ (F0 )0+ (Wgp )
and (F(B̄,0,0) )0− (Wgp ) ≤ (F0 )0− (Wgp ), where (F )0+ and (F )0− denote the right and left derivatives
of F . QED
Proof of Lemma 2
Uniqueness and existence arguments follow closely the steps in the proof of Lemma 1.
Uniqueness: By contradiction, suppose for some y1 < y2 , both F(B,y1 , y1 ) and F(B,y2 , y2 ) satisfy
B
B
0
0
conditions a) and b). Since F(B,y1 , y1 ) (B) < F(B,y2 , y2 ) (B) and F(B,y
y2 (B), it
y1 (B) < F
,
)
(B,y
,
2 B )
1 B
B
B
follows from Lemma A.1 that F(B,y1 , y1 ) (W ) < F(B,y2 , y2 ) (W ) for all W ≥ B. Since F(B,y1 , y1 )
B
B
B
is weakly above F0 , we have F0 (W ) ≤ F(B,y1 , y1 ) (W ) < F(B,y2 , y2 ) (W ) for all W ≥ B. This
B
B
contradicts condition b) for F(B,y2 , y2 ) (Wgp ).
B
Existence: Define
∗
y := inf{x ≥ 0 : F(B,x, Bx ) (W ) ≥ F0 (W ), ∀W ∈ [B, Wgp
]}.
The above set is nonempty because for each B ∈ (0, B̄] we have F(B,F
∗ ].
[B, Wgp
max (B),
Fmax (B)
)
B
F0 (W ), ∀W ∈
Indeed, strict concavity of Fmax means that Fmax (B) −
0
0
Fmax (0) = 0, so we have Fmax
(B)B < Fmax (B). This means Fmax
(B) < F 0
(B,Fmax (B),
Lemma A.1 now implies that Fmax (W ) < F(B,F
26
max (B),
Fmax (B)
)
B
(W ) >
0
Fmax
(B)B
Fmax (B)
)
B
>
(B).
(W ) for all W > B. Because
∗ , we have F
Fmax (W ) ≥ F0 (W ) for all W ≤ Wgp
F
(B) (W ) > F0 (W ) for all
(B,Fmax (B), max
)
B
∗
W ∈ [B, Wgp ], so the set is nonempty and y < Fmax (B).
As F(B,y, y ) satisfies condition a) of the lemma, we only need to show that it satisfies condition
B
b).
If y = 0, then the definition of B̄ and the fact that F(B,0,0) satisfies condition a) imply B ≥ B̄.
The assumption B ≤ B̄ implies then that B = B̄. The conclusion follows because Lemma 1
has shown that F(B̄,0,0) satisfies condition b). If y > 0, then consider the sequence {y − n1 : n =
1, 2, ...}, which converges to y from below. Since y − n1 ∈
/ {x ≥ 0 : F(B,x, Bx ) (W ) ≥ F0 (W ), ∀W ∈
∗
∗ ] such that F
[B, Wgp ]}, then for each n there exists some Wn ∈ [B, Wgp
1 (Wn ) <
1 y− n
(B,y− n ,
B
)
F0 (Wn ). Let Wgp be the limit of some subsequence {Wnk : k = 1, 2, ...} of {Wn : n = 1, 2, ...},
then F(B,y, y ) (Wgp ) = limk→∞ F
This
y− n1 (Wnk ) ≤ limk→∞ F0 (Wnk ) = F0 (Wgp ).
B
(B,y− n1 ,
k
k
B
)
and F(B,y, y ) (Wgp ) ≥ F0 (Wgp ) from condition a) imply F(B,y, y ) (Wgp ) = F0 (Wgp ). We have
B
B
0
0
0
0
F(B,y,
y (Wgp ) = F0 (Wgp ) because condition a) implies (F(B,y, y ) )+ (Wgp ) ≥ (F0 )+ (Wgp ) and
)
B
B
(F(B,y, y ) )0− (Wgp ) ≤ (F0 )0− (Wgp ). QED
B
The next auxiliary lemma will be useful in verification proofs.
Lemma A.2 Take B ∈ [0, B̄,] and denote the solution F(B,y∗ (B), y∗ (B) ) simply by F . Then
B
min F (W ) + c + F 0 (W )(u(c) − W ) ≥ 0 at all W ≥ B.
c≥0
(10)
Proof We first show that all tangent lines to F stay above F0 , i.e., for all W ≥ B
F (W ) + F 0 (W )(W 0 − W ) ≥ F0 (W 0 ) ∀W 0 ≥ 0.
(11)
If W 0 ≥ B, then concavity of F implies F (W ) + F 0 (W )(W 0 − W ) ≥ F (W 0 ), which is above
F0 (W 0 ) because F(B,y∗ (B), y∗ (B) ) satisfies condition a) of Lemma 2. If W 0 < B, then concavity
B
of F implies F (W ) + F 0 (W )(W 0 − W ) = F (W ) + F 0 (W )(B − W ) + F 0 (W )(W 0 − B) ≥ F (B) +
W0
0
F 0 (W )(W 0 − B) ≥ F (B) + F 0 (B)(W 0 − B) = F (B) + F (B)
B (W − B) = B F (B) ≥ 0, which is
above F0 (W 0 ) because 0 ≥ F0 . Inequality (10) follows now from (11) by changing the variable
W 0 ∈ [0, ∞) to u(c) ∈ [0, ∞), where c attains F0 (W 0 ).
Proof of Theorem 1
First, we show that any incentive compatible contract (C, A) achieves profit at most F(B,y∗ (B), y∗ (B) ) (W0 (C, A)).
B
To simplify the notation, we will often drop the subscript in F(B,y∗ (B), y∗ (B) ) and refer to this
B
solution simply as F . Denote the agent’s continuation value by Wt = Wt (C, A), which follows (1). As in Sannikov (2008), it is without loss of generality to only consider contracts
27
such that u0 (Ct ) ≥ γ0 at all t, with which the restriction that (Ct , At ) belongs to the compact set [0, (u0 )−1 (γ0 )] × A at all t. By Lemma 4 in Sannikov (2008), the profit is at most
∗ . If W ∈ [B, W ∗ ], define
F0 (W0 ) ≤ F (W0 ) if W0 ≥ Wgp
0
gp
Z
Gt := r
t
e−rt (As − Cs )ds + e−rt F (Wt ).
(12)
0
By Ito’s lemma, the drift of Gt is
00
−rt
0
2 2 F (Wt )
.
re
At − Ct − F (Wt ) + F (Wt )(Wt − u(Ct ) + h(At )) + rσ Yt
2
Let us show the drift of Gt is always nonpositive. If At > 0, then incentive compatibility
requires Yt = h0 (At ). Then the fact that F solves the high-action ODE implies that the drift
of G is nonpositive. If At = 0, then (10) and F 00 < 0 imply that the drift of Gt is nonpositive.
It follows that Gt is a bounded supermartingale until the stopping time τ 0 (possibly ∞) defined
∗ . At time τ 0 , the principal’s future profit is less than or equal
as the time when Wt reaches Wgp
∗ ) ≤ F (W ∗ ). Therefore, the principal’s expected profit at time 0 is less than or equal
to F0 (Wgp
gp
to
"Z
#
τ0
E
0
∗
e−rt (At − Ct )dt + e−rτ F (Wgp
) = E[Gτ 0 ] ≤ G0 = F (W0 ).
0
Second, we show that the contract (C, A) described in the statement of the theorem achieves
profit F (W0 ) if W0 ∈ [B, Wgp ]. Existence of a weak solution to (4) follows from Engelbert and
Peskir (2014). Defining Gt as in (12), but now specifically for the stated contract, we have
from Ito’s lemma that the drift of Gt is
00
−rt
0
2 0
2 F (Wt )
if Wt > B,
re
At − Ct − F (Wt ) + F (Wt )(Wt − u(Ct ) + h(At )) + rσ h (At )
2
and
re−rt −Ct − F (Wt ) + F 0 (Wt )(Wt − u(Ct )) if Wt = B.
Given the construction of F(B,y∗ (B), y∗ (B) ) , the drift of Gt is zero in both cases. It follows that
B
Gt is a bounded martingale until the stopping time τ (possibly ∞) when Wt reaches Wgp . At
time τ , the principal’s future profit is equal to F0 (Wgp ) = F (Wgp ). Therefore, the principal’s
expected profit at time 0 is equal to
Z τ
−rt
−rτ
E
e (At − Ct )dt + e F (Wgp ) = E[Gτ ] = G0 = F (W0 ).
0
28
Proof of Proposition 1 (comparative statics)
Define X as the collection of points starting from which the solution F(W,V,0) stays above F0
and touches it. That is, X is the set of possible peak points of the firm’s value function in
equilibirum. Precisely:
X := (W, V ) : V ≥ 0, the solution F(W,V,0) satisfies two conditions:
∗
a) F(W,V,0) (W 0 ) ≥ F0 (W 0 ) for all W 0 ∈ [W, Wgp
],
0
(Wgp ) = F00 (Wgp )
b) F(W,V,0) (Wgp ) = F0 (Wgp ) and F(W,V,0)
∗
for some Wgp ∈ [W, Wgp
].
We show that X is a strictly decreasing curve.
Lemma A.3 Suppose (W, V ) ∈ X and (W̃ , Ṽ ) ∈ X . If W = W̃ , then V = Ṽ . If W < W̃ ,
then V > Ṽ .
Proof We prove it by contradiction. If W = W̃ , then suppose V 6= Ṽ . Assume without
loss of generality that V < Ṽ . Lemma A.1 implies that F(W̃ ,Ṽ ,0) (W 0 ) > F(W,V,0) (W 0 ) ≥
F0 (W 0 ), ∀W 0 ≥ W , hence violating condition b).
If W < W̃ , then suppose V ≤ Ṽ . Then F(0W̃ ,Ṽ ,0) (W̃ ) = 0 = F(W,V,0) (W ) > F(W,V,0) (W̃ ). Again
Lemma A.1 implies that F(W̃ ,Ṽ ,0) (W 0 ) > F(W,V,0) (W 0 ) ≥ F0 (W 0 ), ∀W 0 ≥ W̃ , hence violating
condition b).
Since for each W there is a unique V such that (W, V ) ∈ X , we will denote this unique V
as V (W ). Moreover, in this proof we will denote the solution curve F(W,V (W ),0) as FW to
simplify the notation. To prove the proposition, we will use the following result.
0 (W − x) > F 0 (W̃ − x) for any x > 0. F (W − x) −
Lemma A.4 If W > W̃ , then FW
W
W̃
0 (W − x)(W − x) is strictly decreasing in both W and x.
FW
00 (W ) <
Proof First, we show the inequality for small x > 0. It is sufficient to show that FW
00 (W̃ ), which follows from (14) and V (W ) < V (W̃ ).
FW̃
Second, we show the the inequality for any x > 0 by contradiction. Suppose the inequality
0 (W − x) = F 0 (W̃ − x). Since
does not hold, then there exists a smallest x at which FW
W̃
0 (W − x0 ) > F 0 (W̃ − x0 ) for all x0 ∈ (0, x), we have F (W − x) < F (W̃ − x). This and
FW
W
W̃
W̃
W > W̃ imply
0
FW (W − x) − FW
(W − x)(W − x)
0
< FW̃ (W̃ − x) − FW̃
(W̃ − x)(W̃ − x).
29
00 (W − x) < F 00 (W̃ − x) and therefore F 0 (W − x + ) <
It follows from (14) again that FW
W
W̃
0 (W̃ − x + ) for all sufficiently small > 0, a contradiction.
FW̃
0 (W − x)(W − x) in W . Because
Third, we show the monotonicity of FW (W − x) − FW
0
FW (W − x) − FW
(W − x)(W − x)
Z x
0
0
FW
(W − y)dy − FW
(W − x)(W − x)
= FW (W ) −
0
Z W
0
0
= V (W ) −
min(FW
(W − x), FW
(W − y))dy,
0
0 (W − x)(W − x) in W relies on the fact that V (W )
the monotonicity of FW (W − x) − FW
0 (W − x) and F 0 (W − y) are increasing in W . These properties
is decreasing in W , and FW
W
have been shown above.
0 (W − x)(W − x) is decreasing in x is because
Finally, FW (W − x) − FW
0 (W − x)(W − x))
∂(FW (W − x) − FW
00
= FW
(W − x)(W − x) < 0.
∂x
We can now prove Proposition 1.
Suppose B > B̃ and define x := W0 (B) − B, x̃ := W0 (B̃) − B̃. Because both profit functions,
FW0 (B) and FW0 (B̃) , satisfy the smooth pasting condition with a low-action ODE solution at
their respective lower bounds, B and B̃, we have
0
0 = FW0 (B) (W0 (B) − x) − FW
(W0 (B) − x)(W0 (B) − x)
0 (B)
0
= FW0 (B̃) (W0 (B̃) − x̃) − FW
0 (B̃)
(W0 (B̃) − x̃)(W0 (B̃) − x̃).
Lemma A.4 now implies that either W0 (B) > W0 (B̃) and x < x̃, or W0 (B) ≤ W0 (B̃) and
x ≥ x̃. The latter case cannot occur because W0 (B) − x = B > B̃ = W0 (B̃) − x̃. Thus, W0 (B)
is strictly increasing and W0 (B) − B is strictly decreasing in B. That V (W0 (B)) is strictly
decreasing in B follows from the fact that function V is strictly decreasing, which was shown
in Lemma A.3.
Proofs for Section 7
Writing the high-action ODE (8) as a system of first-order equations, we have
dF
dW
dF 0
dW
= F 0,
= − max
a≥0
(13)
a + F 0 h(a) − F + maxc {F 0 (W − u(c)) − c}
.
rσ 2 (h0 (a))2 /2
30
(14)
As we discuss in Section 7.1, F 00 must be −∞ and effort a must be 0 at the lower bound W = B.
The system (13)-(14) is therefore singlular at W = B. We now use a change-of-variable
technique to obtain an alternative system that is equivalent to (13)-(14) and well-behaved in
the neighborhood of W = B.
Change of variable
Dependent variable S. If we define
S := F (W ) − max{−c + F 0 (W ) (W − u(c))},
c≥0
then the HJB equation (3) can be written as
1
S = max{a + F 0 (W )h(a) + F 00 (W )rσ 2 h0 (a)2 }.
a
2
(15)
Economically, S represents the firm’s surplus from being able to induce positive effort from
the agent at W . To see this note that
F (W ) − max{−c + F 0 (W ) (W − u(c))} = F (W ) − max{F0 (U ) + F 0 (W ) (W − U )}.
U ≥0
c≥0
If positive effort could not be used at W , then the firm’s profit would be F0 (U ) minus the cost
−F 0 (W )(W − U ) of optimally adjusting the agent’s continuation value from W to U , where
−F 0 (W ) is the firm’s marginal cost of delivering utility to the agent in state W . Therefore,
the difference between F (W ) and maxU ≥0 {F0 (U ) + F 0 (W ) (W − U )} is the surplus the firm
can generate by inducing positive effort from the agent. In the HJB equation (15), we see
that this surplus comes from the expected output due to the agent’s effort, a, less the cost of
compensating the agent for his disutility of effort h(a), less the firm’s cost of having to induce
positive volatility Y = h0 (a) in the state variable. Because the firm always has the option to
ask for zero effort, S is always non-negative. Further, because
F (W ) − max{F0 (U ) + F 0 (W ) (W − U )} = F (W ) + min{−F0 (U ) − F 0 (W ) (W − U )}
U ≥0
U ≥0
= min{F (W ) + F 0 (W ) (U − W ) − F0 (U )},
U ≥0
the geometric interpretation of S is the minimum vertical distance between the line tangent to
the solution curve F at W and F0 . Because F is concave, the tangent line is always above F ,
which again shows that S is non-negative. For future reference, note that with S we can also
write the HJB equation as
a + F 0 h(a) − S
.
(16)
F 00 = − max 1 2 0
2
a≥0
2 rσ (h (a))
31
Inependent variable X. Now, instead of treating W as the independent variable and F
and F 0 as dependent variables, we change the independent variable to
X := −F 0
and treat W and S = F (W ) − maxU ≥0 {F0 (U ) − X (W − U )} as dependent variables.
The dynamics of W and S in terms of X are as follows. From dX/dW = −F 00 we have
dW
−1
= 00 .
dX
F
dF dX
dX dW
dF
00
dX (−F )
= FX00 . Further,
d
dF dW
dW
=
F + XW − max{F0 (U ) + XU } =
+W +X
− U (X)
U ≥0
dX
dW dX
dX
dW
dW
= −X
+W +X
− U (X) = W − U (X),
dX
dX
From −X =
dS
dX
dF
dW
(17)
=
=
we have
dF
dX
where U (X) = arg maxU ≥0 {F0 (U ) + XU }.15 Using the HJB equation (16) to eliminate F 00
from (17), we get the following system:
dS
dX
dW
dX
= W − U (X),
=
(18)
1
maxa≥0 a−Xh(a)−S
1
rσ 2 (h0 (a))2
2
.
(19)
In the new variables (X, W, S), the initial conditions at the lower bound B are as follows. We
F
B = 0 due to the smooth pasting with
have W0 = B > 0 and S0 = F (B) − F 0 (B)B = F − B
the low-action ODE at B. The starting X is unknown. It is some value X0 ≤ 0. When B is
the highest lower bound B̄, then X0 = 0. For B < B̄, X0 < 0.
√
Example For u(c) = c and h(a) = 12 a2 , which are the functions used in Figure 7, we have
u(c(X)) =
1 +
2X ,
so
dS
dX
= W − 12 X + and
dW
dX
= (maxa
maximization problem gives us a = 2S, so dW
dX =
around the initial condition point, where S = 0.
a−X 12 a2 −S −1
) .
1
rσ 2 a2
2
1
rσ 2 4S 2
2
2S−X 12 4S 2 −S
=
rσ 2 S
1
−XS
2
The FOC from this
, which is well-behaved
We will show that the system (18)-(19) is well-behaved in general. We start with Lipschitz
continuity of (18)-(19) around S = 0.
Define
(
ψ(X) := max a − Xh(a) =
a≥0
15
Ā − Xh(Ā), X ≤ 0
ā − Xh(ā), X > 0
U (X) has the interpretation of the continuation value to which the firm would adjust W if effort must be
zero and the marginal cost of utility is X. Clearly, U (X) = 0 for X ≤ 0, and U (X) solves F00 (U ) + X = 0 for
X > 0.
32
where ā satisfies 1 = Xh0 (ā). There are four possibilities:
1. If S ≤ 0, then maxa≥0
a−Xh(a)−S
1
rσ 2 (h0 (a))2
2
2. If S ∈ (0, ψ(X)), then maxa≥0
≥ maxa≥0
a−Xh(a)−S
1
rσ 2 (h0 (a))2
2
a−Xh(a)
1
rσ 2 (h0 (a))2
2
= ∞.
> 0 is positive and finite.
3. If S = ψ(X), then maxa≥0
a−Xh(a)−S
1
rσ 2 (h0 (a))2
2
= 0.
4. If S > ψ(X), then maxa≥0
a−Xh(a)−S
1
rσ 2 (h0 (a))2
2
is negative and finite.
Lemma A.5 Fix a point (X̂, Ŵ , Ŝ) in the domain of the ODE system (18)-(19).
1. If Ŝ ∈ [0, ψ(X̂)), then the ODE system in (18)-(19) satisfies the Lipschitz condition in a
neighborhood of (X̂, Ŵ , Ŝ). This system violates the Lipschitz condition if Ŝ = ψ(X̂).
2. If Ŝ = ψ(X̂), then the ODE system in (13)-(14) satisfies the Lipschitz condition in a
neighborhood of (Ŵ , F̂ , F̂ 0 ), where F̂ = Ŝ + maxU ≥0 {F0 (U ) − X̂(Ŵ − U )} and F̂ 0 = −X̂.
Proof
1. In this proof, we let M (X, S) denote maxa≥0
a−Xh(a)−S
(h0 (a))2
to simplify notation. Pick a small
> 0 such that S < ψ(X) for all X ∈ [X̂ − , X̂ + ] and S ∈ [Ŝ − , Ŝ + ]. Lipschitz
1
1
continuity of (19) in this neighborhood requires that | M (X,S
− M (X,S
| ≤ K(S1 − S2 )
2)
1)
for all 0 ≤ S1 < S2 in [Ŝ − , Ŝ + ] and for some K > 0. We do not consider negative
1
values of S because M (X,S)
= 0 is constant when S ≤ 0. The rest of this proof relies on
the following equation,
Z S2
1
1
1
1
−
=
dS,
(20)
2
0
∗
2
M (X, S2 ) M (X, S1 )
S1 M (X, S) (h (a (X, S)))
where a∗ (X, S) is the maximizer in maxa≥0
tically by the envelope theorem, because
∂
a−Xh(a)−S
.
(h0 (a))2
We can understand (20) heuris-
1
1
1
/∂S =
.
2
0
∗
M (X, S)
M (X, S) (h (a (X, S)))2
The technical issue here is that M (X, S) is nondifferentiable when the maximizer a∗ (X, S)
is not unique. To deal with this issue, we justify (20) at the end of this proof using an
generalized envelope theorem from Milgrom and Segal (2002). Here we shall proceed,
assuming (20) is correct.
To show Lipschitz continuity, it is sufficient to show that the integrand in (20) is bounded
when X ∈ [X̂ − , X̂ + ] and S ∈ [Ŝ − , Ŝ + ]. We show the boundedness as follows.
Because lima→0 h0 (a) = 0, we can pick a small δ > 0 such that for all a ∈ (0, δ) and
X ∈ [X̂ − , X̂ + ],
1
1 − Xh0 (a) > .
(21)
2
33
First, if a∗ (X, S) ≥ δ, then
1
1
2
0
∗
M (X, S) (h (a (X, S)))2
≤
≤
1
1
2
0
M (X, S) (h (δ))2
1
Ŝ+) 2
(maxa≥0 a−(X̂+)h(a)−(
)
(h0 (a))2
1
,
(h0 (δ))2
where the first inequality follows from a∗ (X, S) ≥ δ, the second inequality is because
maxa≥0 a−Xh(a)−S
is decreasing in both S and X.
(h0 (a))2
Second, suppose a∗ (X, S) ∈ (0, δ). The first-order condition for an interior a∗ is
(1 − Xh0 (a∗ ))h0 (a∗ )2 − (a∗ − Xh(a∗ ) − S)2h0 (a∗ )h00 (a∗ )
= 0,
h0 (a∗ )4
which implies
(a∗ − Xh(a∗ ) − S) =
(1 − Xh0 (a∗ ))h0 (a∗ )
h0 (a∗ )
>
,
2h00 (a∗ )
4h00 (a∗ )
where the inequality is from (21). Therefore, the integrand in (20) satisfies
1
(h0 (a∗ ))2
(h0 (a∗ ))2
1
=
≤
= 16(h00 (a∗ ))2 .
h0 (a∗ ) 2
M (X, S)2 (h0 (a∗ ))2
(a∗ − Xh(a∗ ) − S)2
( 4h00 (a∗ ) )
The last term 16(h00 (a∗ ))2 is bounded by maxa∈[0,δ] 16(h00 (a))2 .
Finally, we prove (20) accounting for the fact that a∗ (X, S) may not be unique and
M (X, S) is not differentiable everywhere (with respect to S). In this case a∗ (X, S) is
any selection from {a : a−Xh(a)−S
= M (X, S)}. First, we consider the case of S1 > 0.
(h0 (a))2
Because
a−Xh(a)−S
(h0 (a))2
is supermodular in (a, S), a∗ (X, S) is increasing in S, which implies
M (X, S) = max
a≥0
a − Xh(a) − S
a − Xh(a) − S
= max
,
(h0 (a))2
(h0 (a))2
a≥a∗ (X,S1 )
∀S ≥ S1 .
Because ∂ a−Xh(a)−S
/∂S = (h0−1
is bounded and continuous on the compact set [a∗ (X, S1 ), A],
(h0 (a))2
(a))2
Corollary 4 in Milgrom and Segal (2002) implies that M (X, S) is absolutely continuous
in S, and ∂M (X, S)/∂S exists and equals (h0 (a∗−1
almost everywhere. Because
(X,S)))2
M (X, S) is bounded away from zero when S ∈ [S1 , S2 ] (i.e., M (X, S) ≥ M (X, S2 ) > 0),
1
1
1
function M (X,S)
is also absolutely continuous in S and its derivative equals M (X,S)
2 (h0 (a∗ (X,S)))2
1
almost everywhere. Absolute continuity of M (X,S)
implies that
Z S2
1
1
1
1
−
=
dS.
(22)
2
0
∗
2
M (X, S2 ) M (X, S1 )
S1 M (X, S) (h (a (X, S)))
Second, we consider the case of S1 = 0. Taking limit S1 ↓ 0 in (22) yields
Z S2
1
1
1
1
−
=
dS.
2
0
∗
M (X, S2 ) M (X, 0)
M (X, S) (h (a (X, S)))2
0
34
2. To show that (14) satisfies the Lipschitz condition, we first show that the optimal a∗
in Ha (W, F, F 0 ) is uniformly bounded away from zero whenever (W, F, F 0 ) is in a neighborhood of (Ŵ , F̂ , F̂ 0 ). That is, maxa≥0 Ha (W, F, F 0 ) = maxa≥ Ha (W, F, F 0 ) for some
> 0. It follows from Ŝ = ψ(X̂) > 0 that â∗ + F̂ 0 h(â∗ ) − Ŝ = 0, where â∗ stands
for a∗ (Ŵ , F̂ , F̂ 0 ). Continuity and â∗ + F̂ 0 h(â∗ ) − Ŝ = 0 > 0 + F̂ 0 h(0) − Ŝ imply that,
there exists a small > 0 such that if W ∈ (Ŵ − , Ŵ + ), F ∈ (F̂ − , F̂ + ), and
F 0 ∈ (F̂ 0 − , F̂ 0 + ), then
â∗ + F 0 h(â∗ ) − S
a + F 0 h(a) − S
>
,
rσ 2 (h0 (â∗ ))2 /2
rσ 2 (h0 (a))2 /2
∀a ∈ [0, ).
This means a ∈ [0, ) cannot be the optimal effort at (W, F, F 0 ) as it is dominated by â∗ .
Therefore, maxa≥0 Ha (W, F, F 0 ) = maxa≥ Ha (W, F, F 0 ) whenever W ∈ (Ŵ − , Ŵ + ),
F ∈ (F̂ − , F̂ + ), and F 0 ∈ (F̂ 0 − , F̂ 0 + ).
Second, function Ha (W, F, F 0 ) is differentiable in (F, F 0 ), with
∂Ha (W, F, F 0 )
∂F
∂Ha (W, F, F 0 )
∂F 0 (W )
=
=
−1
rσ 2 (h0 (a))2 /2
,
W − u(c) + h(a)
.
rσ 2 (h0 (a)2 /2
These derivatives are uniformly bounded over a ∈ [, A], W ∈ (Ŵ − , Ŵ + ), and
F 0 ∈ (F̂ 0 − , F̂ 0 + ). As Sannikov (2008) argues in his Lemma 1, maxa≥ Ha (W, F, F 0 )
is Lipschitz continuous in (F, F 0 ).
If S = 0, we can solve (19) in its neighborhood; if S = ψ(X), we can solve (14); if S ∈ (0, ψ(X)),
we can solve either (14) or (19) because they are equivalent.
Lemma A.6 Suppose the initial conditions are S = 0 and X < 0, and suppose F (B) <
Ffb (B).
1. There is a > 0 such that the solution to (18)-(19) satisfies S > 0 and
X ∈ (X, X + ).
dW
dX
> 0 for
2. For X ≥ X + , the solution to (18)-(19) belongs to one of three cases: (1) S ∈ (0, ψ(X))
for all X ≥ X + , (2) S = ψ(X) at some finite X = X̄, and (3) S = 0 at some finite
X = X̄. In case (1), limX→∞ W (X) = ∞. In case (2), limX→X̄ W (X) = ∞. In case
(3), W (X̄) < U (X̄) < ∞. So in the first two cases, function F (·) implied by the solution
dS
to (18)-(19) is a global solution, but in case (3), it is not. In case (3), dX
|X=X̄ < 0,
which implies that we cannot extend the solution curve to X > X̄ because our model
requires S ≥ 0.
35
3. If S > 0 for all X > X, then F is a global solution and stays above F0 .
4. If there exists a smallest W ∗ ≥ B such that F (W ∗ ) = Ff b (W ∗ ), then F 0 (W ) > (Ff b )0 (W ∗ ), ∀W ≥
W ∗ . Moreover, case (2) in part 2 applies, which implies that S > 0, ∀W ≥ W ∗ .
Proof
dS
1. S > 0 because dX
|X=X = W −U (X) = B > 0. If S ∈ (0, ψ(X)), then maxa≥0
0 is positive and finite. Therefore,
dW
dX
=
1
maxa≥0 a−Xh(a)−S
r 2 0
σ (h (a))2
2
> 0.
a−Xh(a)−S
r 2 0
σ (h (a))2
2
>
(23)
2. In case (1), suppose by contradiction that limX→∞ W (X) < B for some finite B > 0.
dS
Then pick a sufficiently large X̂ such that U (X) > B + 1 for all X ≥ X̂. Then dX
=
W (X) − U (X) < B − (B + 1) = −1, ∀X ≥ X̂. This contradicts the assumption that S
is always positive.
In case (2), by contradiction, suppose W̄ := W (X̄) < ∞. Then (14) implies F 00 (W̄ ) = 0.
We can verify that the straight line F̃ (W ) := F (W̄ ) + F 0 (W̄ )(W − W̄ ) is a solution to
(13)-(14). On the other hand, the solution to (18)-(19) satisfies S < ψ(X) for X < X̄,
which implies that F 00 (W ) < 0 for W < W̄ . Since F 6= F̃ , we have two solutions to (13)(14) at (W̄ , F (W̄ ), F 0 (W̄ ), contradicting the result that (13)-(14) satisfies the Lipschitz
condition (part 2 of Lemma 5).
dS
|X=X̄ = W − U (X̄) < 0. First, we show X̄ > 0. If X̄ ≤ 0,
In case (3), we show that dX
then U (X) = 0, ∀X ∈ [X, X̄] and
Z X̄
S(X̄) = S(X) +
S 0 (X)dX
X
Z
X̄
(W (X) − U (X))dX
= S(X) +
X
Z
X̄
= 0+
W (X)dX > 0,
X
which contradicts the the assumption that S = 0 at X = X̄.
Second, we show that W − U (X̄) < 0. By contradiction, suppose W − U (X̄) ≥ 0. If
dS
W − U (X̄) > 0, then dX
|X=X̄ > 0, contradicting the fact that S > 0 for X slightly below
X̄. If W − U (X̄) = 0, then
d2 S
dW
dU
dU
|X=X̄ −
|X=X̄ = −
|
< 0.
|X=X̄ =
2
dX
dX
dX
dX X=X̄
This again contradicts the fact that S > 0 for X slightly below X̄.
36
3. F is a global solution because case (3) in part 2 does not apply. We have F0 (W ) < F (W )
because
0 < S(X) = F (W ) + XW − max (F0 (U ) + XU )
U
≤ F (W ) + XW − (F0 (W ) + XW ) = F (W ) − F0 (W ).
4. It follows from F (W ) ≤ Ff b (W ), ∀W ≤ W ∗ that F 0 (W ∗ ) ≥ (Ff b )0 (W ∗ ) ≡ −X. To show
F 0 (W ) > (Ff b )0 (W ∗ ), ∀W ≥ W ∗ , suppose by contradiction W ∈ [W ∗ , ∞) is the smallest
W such that F 0 (W ) = (Ff b )0 (W ∗ ) = −X. Since F is concave, F (W )+F 0 (W )(W ∗ −W ) ≥
F (W ∗ ) = Ff b (W ∗ ), which implies
F (W ) + XW ≥ Ff b (W ∗ ) + XW ∗ = F0 (U0 (X)) + XU0 (X) + ψ(X),
or S ≥ ψ(X). The fact that W is finite contradicts case (2) in part 2. Therefore,
F 0 (W ) > (Ff b )0 (W ∗ ), ∀W ≥ W ∗ and the solution curve does not belong to case (1) in
part 2.
To rule out case (3) in part 2, suppose by contradiction that S = 0 at some X̄ ≥ −F 0 (W ∗ ).
Proof of case (3) in part 2 shows that U (X̄) > W (X̄) ≥ W ∗ , which and the property
F 0 (W ) > (Ff b )0 (W ∗ ), ∀W ≥ W ∗ imply
F (W (X̄)) + F 0 (W (X̄))(U (X̄) − W (X̄))
≥ Ff b (W ∗ ) + (Ff b )0 (W ∗ )(W (X̄) − W ∗ ) + F 0 (W (X̄))(U (X̄) − W (X̄))
≥ Ff b (W ∗ ) + (Ff b )0 (W ∗ )(U (X̄) − W ∗ )
≥ Ff b (U (X̄)) > F0 (U (X̄)),
which contradicts F (W (X̄)) + X(W (X̄) − U (X̄)) − F0 (U (X̄)) = S = 0.
We now conclude that case (2) of part 2 holds, because cases (1) and (3) have been ruled
out.
Proof of Lemma 3
Let M > 0 be defined by Ffb (M ) = 0. Define U as the set of initial conditions under which
the solution crosses the upper bound Ffb .
U := {B ∈ (0, M ] : there exists a smallest W ∗ ≥ B such that F(B,0,0,−∞) (W ∗ ) = Ffb (W ∗ ).}
Define L as the set of initial conditions under which the solution returns to S = 0.
00
L := {B ∈ (0, M ] : there exists a W > B such that F(B,0,0,−∞)
(W ) = −∞, i.e., S = 0.}
37
It follows from part 4 in Lemma A.6 that U ∩ L = ∅. The proof consists of the following six
steps.
1. Both U and L are nonempty. If B = M , then F(B,0,0,−∞) (B) = 0 = Ffb (B). Therefore,
M ∈ U and U is nonempty.
To show L =
6 ∅, we show that B ∈ L when B is sufficiently small. If B = 0, then
dU (X)
dS
d2 S
dW
< 0. This implies that
dX |X=0 = W − U (X) = B = 0, and dX 2 |X=0 = dX − dX
∗
∗
S(X ) < 0 for some small X > 0. Because the ODE system satisfies the Lipschitz
condition, its solution is continuous in the initial condition B. That is, S(X ∗ ) < 0
whenever B > 0 is sufficiently small. The intermediate value theorem implies S(X) = 0
at some X > 0 whenever B > 0 is sufficiently small. Therefore, B ∈ L.
2. Both U and L are open subsets of (0, M ].
If B ∈ U, then F(B,0,0,−∞) (W ) > Ffb (W ) for some W > B. Suppose this happens at
W = W (X) in the ODE (14). Since the solution to (14) is continuous in its initial
conditions, (W (X, B), S(X, B)) are continuous in y. Moreover, Ffb (W ) is continuous in
W and F(B,0,0,−∞) (W ) = S(X, B) + F0 (U (X)) + X(U (X) − W ) is continuous in (W, X).
Therefore,
F(B,0,0,−∞) (W (X, B)) − Ffb (W (X, B))
= S(X, B) + F0 (U (X)) + X(U (X) − W (X, B)) − Ffb (W (X, B))
is continuous in y. There exists an > 0 such that for all B̃ ∈ (B − , B + ),
F(B̃,0,0,−∞) (W (X, B̃)) − Ffb (W (X, B̃)) > 0. Therefore, (B − , B + ) ⊂ U.
L is an open subset of (0, M ]. If B ∈ L, then S(X) = 0 for some X > 0. Part 3
of Lemma A.6 shows that S(X 0 ) < 0 if X 0 is slightly above X. Since the solution to
(14) is continuous in its initial conditions, S(X 0 , B) is continuous in B. There exists an
> 0 such that for all B̃ ∈ (B − , B + ), S(X 0 , B̃) < 0. For each B̃ ∈ (B − , B + ),
S(X 0 , B̃) < 0 and S(X̃, B̃) > 0 when X̃ is slightly above 0, therefore S(., B̃) reaches 0.
Therefore, (B − , B + ) ⊂ L.
3. Because of the above properties, U ∪ L =
6 (0, M ]. Therefore, there exists B̄ ∈ (0, M ] \
(U ∪ L). Obviously, F(B̄,0,0,−∞) is below Ffb . Part 3 in Lemma A.6 shows that F(B̄,0,0,−∞)
is above F0 .
4. B̄ is unique. By contradiction, suppose B̄ 1 < B̄ 2 both belong to (0, M ] \ (U ∪ L). Lemma
1 implies that F(B̄ 1 ,0,0,−∞) (W ) − F(B̄ 2 ,0,0,−∞) (W ) is increasing in W . In particular, if
38
B̄ 2 < W ,
F(B̄ 2 ,0,0,−∞) (B̄ 2 ) − F(B̄ 1 ,0,0,−∞) (B̄ 2 )
< F(B̄ 2 ,0,0,−∞) (W ) − F(B̄ 1 ,0,0,−∞) (W )
< Ffb (W ) − F0 (W ) ≤ a∗ (W ),
where a∗ (W ) is the optimal effort in Ffb (W ). Taking limit W → ∞, we have
−F(B̄ 1 ,0,0,−∞) (B̄ 2 ) ≤ lim a∗ (W ) = 0,
W →∞
which is a contradiction.
5. U = (B̄, M ]. By contradiction, suppose B ∈ (B̄, M ] is in L, then F(B,0,0,−∞) reaches
S = 0 at some W . Lemma A.1 implies that F(B̄,0,0,−∞) (W ) < F(B,0,0,−∞) (W ) and
0
F(0B̄,0,0,−∞) (W ) < F(B,0,0,−∞)
(W ) ≡ −X. Therefore,
F(B̄,0,0,−∞) (W ) + F(0B̄,0,0,−∞) (W )(U (X) − W )
0
< F(B,0,0,−∞) (W ) + F(B,0,0,−∞)
(W )(U (X) − W )
= F0 (U (X)),
where the inequality uses U (X) > W . This contradicts the fact that F(B̄,0,0,−∞) satisfies
S > 0 at all W .
6. L = (0, B̄). By contradiction, suppose y ∈ (0, B̄) is in U, then F(B,0,0,−∞) (W ) = Ffb (W )
for some W . Lemma A.1 implies that F(B̄,0,0,−∞) (W ) > F(B,0,0,−∞) (W ) = Ffb (W ), which
means that B̄ ∈ U, a contradiction.
Proof of Lemma 4
Similar to the proof of Lemma 3, this proof proceeds in six steps. Define U as the set of initial
conditions under which the solution crosses the upper bound Ffb .
U := {y ∈ [0, Ffb (B)] : there exists a smallest W ∗ ≥ B such that F(B,y, y ,−∞) (W ∗ ) = Ffb (W ∗ ).}
B
Define L as the set of initial conditions under which the solution reaches S = 0.
L := {y ∈ [0, Ffb (B)] : there exists a W > B such that F 00 (W ) = −∞, i.e., S = 0.}
It follows from part 4 in Lemma A.6 that U ∩ L = ∅.
1. Both U and L are nonempty. If y = Ffb (B), then F(B,y, y ,−∞) (B) = Ffb (B). Therefore,
B
Ffb (B) ∈ U. L is nonempty because y = 0 ∈ L. Because B < B̄, the proof of Lemma 3
implies F(B,0,0,−∞) reaches S = 0.
39
2. Both U and L are open subsets of [0, Ffb (B)].
If y ∈ U, then F(B,y, y ,−∞) (W ) > Ffb (W ) for some W > B. Suppose this happens at
B
W = W (X) in the ODE (14). Since the solution to (14) is continuous in its initial
conditions, (W (X, y), S(X, y)) are continuous in y. Moreover, Ffb (W ) is continuous in
W and F(B,y, y ,−∞) (W ) = S(X, y) + F0 (U (X)) + X(U (X) − W ) is continuous in (W, X).
B
Therefore,
F(B,y, y ,−∞) (W (X, y)) − Ffb (W (X, y))
B
= S(X, y) + F0 (U (X)) + X(U (X) − W (X, y)) − Ffb (W (X, y))
is continuous in y. There exists > 0 such that for all ỹ ∈ (y−, y+), F(B,ỹ, ỹ ,−∞) (W (X, ỹ))−
B
Ffb (W (X, ỹ)) > 0. Therefore, (y − , y + ) ⊂ U.
L is an open subset of [0, Ffb (B)]. If y ∈ L, then S(X) = 0 for some X > − By . Step 3 in
Lemma A.6 shows that S(X 0 ) < 0 if X 0 is slightly above X. Since the solution to (14)
is continuous in its initial conditions, S(X 0 , y) is continuous in y. There exists an > 0
such that for all ỹ ∈ (y − , y + ), S(X 0 , ỹ) < 0. For each ỹ ∈ (y − , y + ), S(X 0 , ỹ) < 0
and S(X̃, ỹ) > 0 when X̃ is slightly above − By , therefore S(·, ỹ) reaches 0. Therefore,
(y − , y + ) ⊂ L.
3. Because of the above properties, U ∪ L 6= [0, Ffb (B)]. Therefore, there exists y ∗ ∈
[0, Ffb (B)] \ (U ∪ L). Obviously, F(B,y∗ , y∗ ,−∞) is below Ffb . Part 3 in Lemma A.6 shows
B
that F(B,y∗ , y∗ ,−∞) is above F0 .
B
4.
y∗
is unique. By contradiction, suppose y ∗ < ỹ ∗ both belong to [0, Ffb (B)] \ (U ∪ L).
Lemma 1 implies that F(B,ỹ∗ , ỹ∗ ,−∞) (W ) − F(B,y∗ , y∗ ,−∞) (W ) is increasing in W . In parB
B
ticular,
ỹ ∗ − y ∗ < F(B,ỹ∗ , ỹ∗ ,−∞) (W ) − F(B,y∗ , y∗ ,−∞) (W ) < Ffb (W ) − F0 (W ) ≤ a∗ (W ).
B
where
a∗ (W )
B
is the optimal effort in Ffb (W ). Taking limit W → ∞, we have
ỹ ∗ − y ∗ ≤ lim a∗ (W ) = 0,
W →∞
which is a contradiction.
5. U = (y ∗ , Ffb (B)]. By contradiction, suppose y ∈ (y ∗ , Ffb (B)] is in L, then F(B,y, y ,−∞)
B
reaches S = 0 at some W . Lemma 1 implies that F(B,y∗ , y∗ ,−∞) (W ) < F(B,y, y ,−∞) (W )
B
B
0
and F 0 ∗ y∗
(W ) < F(B,y,
(W
)
≡
−X.
Therefore,
y
,−∞)
(B,y ,
B
,−∞)
B
0
F(B,y∗ , y∗ ,−∞) (W ) + F(B,y
(W )(U (X) − W )
∗
∗ , y ,−∞)
B
B
0
y
< F(B,y, y ,−∞) (W ) + F(B,y,
,−∞) (W )(U (X) − W )
B
B
= F0 (U (X)),
40
where the inequality uses U (X) > W . This contradicts the fact that F(B,y∗ , y∗ ,−∞)
B
satisfies S > 0 at all W .
6. L = [0, y ∗ ). By contradiction, suppose y ∈ [0, y ∗ ) is in U, then F(B,y, y ,−∞) (W ) = Ffb (W )
B
for some W . Lemma A.1 implies that F(B,y∗ , y∗ ,−∞) (W ) > F(B,y, y ,−∞) (W ) = Ffb (W ),
B
B
which means that y ∗ ∈ U, a contradiction.
Proof of Theorem 2 (verification in the Inada case)
First, we show that any incentive compatible contract (C, A) achieves profit at most F(B,y∗ (B), y∗ (B) ,−∞) (W0 (C, A)).
B
To simplify the notation, we will often drop the subscript in F(B,y∗ (B), y∗ (B) ,−∞) and refer to
B
this solution simply as F in this section. Let > 0 be a small number. Since F asymptotically
approaches Ffb as W → ∞, there is a large W̄ > W0 (C, A) such that F (W̄ ) + ≥ Ffb (W̄ ).
Define
Z
t
Gt := r
e−rt (As − Cs )ds + e−rt F (Wt ).
(24)
0
By Ito’s lemma, the drift of Gt is
00
−rt
0
2 2 F (Wt )
.
re
At − Ct − F (Wt ) + F (Wt )(Wt − u(Ct ) + h(At )) + rσ Yt
2
Let us show the drift of Gt is always nonpositive. If At > 0, then incentive compatibility
requires Yt = h0 (At ). Then the fact that F solves the high-action ODE implies that the drift
of G is nonpositive. If At = 0, the same argument as in Lemma A.2 implies that (10) holds
also for F = F(B,y∗ (B), y∗ (B) ,−∞) . Inequality (10) and F 00 < 0 imply that the drift of Gt is
B
nonpositive also when At = 0.
It follows that Gt is a supermartingale until the stopping time τ 0 (possibly ∞) defined as the
time when Wt reaches W̄ . For all t < ∞,
" Z 0
#
τ
E r
0
e−rt (At − Ct )dt + e−rτ F (W̄ )
0
"
= E Gt∧τ 0 + 1t≤τ 0
Z
τ0
r
!#
e−rs (As − Cs )ds + e
−rτ 0
F (W̄ ) − e−rt F (Wt )
t
"
= E [Gt∧τ 0 ] + e−rt E 1t≤τ 0
Z
r
τ0
!#
−r(τ 0 −t)
e−r(s−t) (As − Cs )ds + e
F (W̄ ) − F (Wt )
t
≤ G0 + e−rt Ā + 2 max F (W ) .
W ∈[B,W̄ ]
h R 0
i
0
τ
Taking t → ∞ yields E r 0 e−rt (At − Ct )dt + e−rτ F (W̄ ) ≤ G0 = F (W0 ). At time τ 0 , the
principal’s future profit is less than or equal to Ffb (W̄ ) ≤ F (W̄ ) + . Therefore, the principal’s
41
expected profit at time 0 is less than or equal to
" Z 0
τ
E r
e
−rt
(At − Ct )dt + e
−rτ 0
#
(F (W̄ ) + ) ≤ F (W0 ) + .
0
Since is arbitrary, we conclude that the principal’s expected profit at time 0 is less than or
equal to F (W0 ).
Second, we show that the contract (C, A) described in the statement of the theorem achieves
profit F (W0 ) if W0 ∈ [B, ∞). Defining Gt as in (24), but now specifically for the stated
contract, we have from Ito’s lemma that the drift of Gt is
00
0
2 0
2 F (Wt )
−rt
At − Ct − F (Wt ) + F (Wt )(Wt − u(Ct ) + h(At )) + rσ h (At )
if Wt > B,
re
2
and
re−rt −Ct − F (Wt ) + F 0 (Wt )(Wt − u(Ct )) if Wt = B.
Given the construction of F = F(B,y∗ (B), y∗ (B) ,−∞) , the drift of Gt is zero in both cases. It
B
follows that Gt is a martingale. For all t < ∞,
Z t
−rs
−rt
E r
e (As − Cs )ds + e F (Wt ) = G0 = F (W0 ),
0
which implies
Z t
−rs
E r
e (As − Cs )ds ≥ F (W0 ) − e−rt max F (W ).
W
0
(25)
R∞
Rt
Since r 0 e−rs (As − Cs )ds converges to r 0 e−rt (At − Ct )dt almost surely and is bounded
above by Ā, Fatou’s lemma and (25) imply
Z ∞
Z t
−rt
−rs
E r
e (At − Ct )dt ≥ lim E r
e (As − Cs )ds
t→∞
0
0
−rt
≥ lim F (W0 ) − e max F (W ) = F (W0 ).
t→∞
W
R∞
Therefore, the principal’s expected profit at time 0, E r 0 e−rt (At − Ct )dt , is at least F (W0 ).
QED
The next lemma is used in the proof of Proposition 3.
Lemma A.7 Under Assumption 1, limX→−F 0 (B)
h0 (a)2
(W −B)
=
4B
rσ 2
> 0.
Proof Using L-Hopital’s rule,
dh0 (a)
2h0 (a) dX
(h0 (a))2
lim
=
lim
.
dW
X→−F 0 (B) W − B
X→−F 0 (B)
dX
42
(26)
We need to evaluate
(3) is
dh0 (a)
dX
and
dW
dX .
The FOC with respect to a taken in the HJB equation
1 + F 0 (W )h0 (a) + F 00 (W )rσ 2 h0 (a)h00 (a) = 0.
(27)
The sufficiency of this first-order condition is shown in Lemma A.8. With effort a that satisfies
this FO condition, we can drop the max operator in (16) and write
F 00 (W ) = −
a + F 0 (W )h(a) − S
,
1
2 0
2
2 rσ (h (a))
(28)
which we use in (17) to obtain
1
rσ 2 (h0 (a))2
dW
−1
= 2
= 00
.
dX
F (W )
a − Xh(a) − S
(29)
Next, we substitute (28) into (27) to obtain
1 + F 0 (W )h0 (a) +
S − a − F 0 (W )h(a) 00
h (a) = 0,
1 0
2 h (a)
or
h0 (a) − X(h0 (a))2 + 2(S − a + Xh(a))h00 (a) = 0.
Using (19), we now totally differentiate this equation with respect to X
dh0 (a)
dh0 (a)
da
da
− (h0 (a))2 − X2h0 (a)
+ 2 W − u(c(X)) −
+ h(a) + Xh0 (a)
h00 (a)
dX
dX
dX
dX
da
+2(S − a + Xh(a))h000 (a)
= 0.
dX
Using
da
dX
=
1 dh0 dh00
h00 dX , dX
da
= h000 dX
=
h000 dh0
h00 dX ,
we solve for
dh0 (a)
dX
to obtain
dh0 (a)
2h(a)h00 (a) − (h0 (a))2 + 2(W − u(c(X)))h00 (a)
=
,
000
dX
1 − 2(S − a + xh(a)) h 00 (a)
h (a)
which converges to 2(B − u(0))h00 (0) = 2Bh00 (0) > 0 at the lower bound. Using this in (26),
we have
h0 (a)2
h0 (a)
.
lim
= 4Bh00 (0)
lim
X→−F 0 (B) W − B
X→−F 0 (B) dW
dX
0
da
Also, from dhdX(a) → 2Bh00 (0) follows that dX
=
substituting from (29) and simplifying, we obtain
1 dh0
h00 dX
→ 2B at the lower bound. Next,
8Bh00 (0)
h0 (a)2
a − Xh(a) − S
=
lim
.
2
0
0
rσ
h0 (a)
X→−F (B) W − B
X→−F (B)
lim
43
Using L-Hopital’s rule again,
h0 (a)2
X→−F 0 (B) (W − B)
lim
=
=
=
=
8Bh00 (0)
lim
rσ 2 X→−F 0 (B)
da
dX
da
− h(a) − Xh0 (a) dX
− (W − u(c(X)))
da
h00 (a) dX
8Bh00 (0) 2B − 0 + F 0 (B)02B − (B − u(0))
rσ 2
h00 (a)2B
8Bh00 (0)
B
rσ 2 h00 (0)2B
4B
.
rσ 2
Proof of Proposition 3 (fast reflection)
We consider the diffusion of Wt with no killing measure (i.e., k ≡ 0):
dWt = r(W − u(c(W )) + h(a(W )))dt + rh0 (a(W ))σdZt ,
where Zt is a standard Brownian motion. Let m be the speed measure of the diffusion process
Wt and s be the scale function. Using the formula on page 17 of Borodin and Salminen
(2002), we calculate the density function of the speed measure m as follows. Define B(x) :=
R x 2r(W −u(c)+h(a))
dW , where C ∈ (B, ∞) is a constant. Then
C
(rh0 (a)σ)2
2eB(x)
m(x) =
(rh0 (a(x))σ)2
Z
m((a, z)) =
,
∀x > B,
z
∀z > a > B.
m(x)dx,
a
The scale function s satisfies
Z
x
s(x) =
Z
0
x
s (y)dy =
C
e−B(y) dy.
C
First, we show that B is a nonsingular point, that is, B is both an exit and an entrance for
the diffusion Wt . To do so, we must estimate upper and lower bounds of the aforementioned
0 (a(W ))2
4B
functions. By Lemma A.7, limW ↓B h(W
−B) = rσ 2 > 0. Therefore,
2r(W −u(c)+h(a))
(rh0 (a)σ)2
lim
1
W ↓B
W −B
1
= .
2
That is, for any > 0, if W is sufficiently close to B,
1
1
2r(W − u(c) + h(a))
1
1
≤
( − )
≤ ( + )
2
W −B
(rh0 (a)σ)2
2
W −B
4B
4B
( 2 − )(W − B) ≤ h0 (a(W ))2 ≤ ( 2 + )(W − B).
rσ
rσ
44
Then, if C is sufficiently close to B and x < C,
1
1
( + ) (ln(x − B) − ln(C − B)) = ( + )
2
2
Z
x
1
dW
W −B
ZC x
2r(W − u(c) + h(a))
≤ B(x) :=
dW
(rh0 (a)σ)2
C
Z x
1
1
≤ ( − )
dW
2
W
−
B
C
1
= ( − ) (ln(x − B) − ln(C − B)) .
2
Then, limx↓B s(x) > −∞ because
Z x
Z
−B(y)
lim
e
dy ≥ lim
x↓B
x↓B
C
x
1
e−( 2 +)(ln(y−B)−ln(C−B)) dy
C
x
Z
= lim
x↓B
C
y−B
C −B
−( 1 +)
2
dy > −∞.
Moreover, if x < C,
( 1 +)
2
x−B
C−B
4B
r2 σ 2 ( rσ
2 + )(W −
2
( 1 +)
2
x−B
C−B
(rh0 (a(x))σ)2
2
≤
B)
2eB(x)
(rh0 (a(x))σ)2
( 1 −)
≤ m(x) :=
2
x−B
C−B
4B
r2 σ 2 ( rσ
2 − )(W
2
≤
− B)
.
Then, lima↓B m((a, z)) < ∞ because
Z
a
a↓B
a
B is an exit because
Z z
Z
m((a, z))s0 (a)da ≤ m((B, z))
B
z
Z
m(x)dx ≤ lim
lim
a↓B
z
z
( 1 −)
2
x−B
C−B
4B
r2 σ 2 ( rσ
2 − )(W −
2
B)
dx < ∞.
s0 (a)da = m((B, z))(s(z) − s(B)) < ∞.
B
B is an entrance because
Z z
Z
(s(z) − s(a))m(da) ≤ (s(z) − s(B))
B
z
m(da) = (s(z) − s(B))m((B, z)) < ∞.
B
Because the boundary point B is nonsingular, Wt reaches B with positive probability.
Second, we show that m({B}) = 0. Pick a bounded and smooth function f , and define
1
g := Gf = (rh0 (a(x))σ)2 f 00 (x) + r(x − u(c(x)) + h(a(x)))f 0 (x).
2
45
−( 1 −)
0 (x)
2
x−B
= ∞. Then
Also, define f + (x) := fs0 (x)
. Note that limx↓B s0 (x) ≥ limx↓B C−B
equation (c) in page 16 of Borodin and Salminen (2002) implies that
f 0 (B)
limx↓B f + (x)
limx↓B s0 (x)
m({B}) =
=
= 0.
g({B})
rBf 0 (B)
Finally, since m({B}) = 0, the diffusion process Wt is reflecting. A reflecting process has the
property that Px (Leb({t ≥ 0 : Wt = B}) = 0) = 1 for all x ≥ B.
QED
The last lemma justifies the first-order condition (27) used in Lemma A.7.
Lemma A.8 Pick δ > 0 such that
1
h0 (a)h000 (a)
< ,
00
2
(h (a))
2
∀a ∈ [0, δ].
There exists > 0, such that for all W ∈ [B, B + ], the maximum maxa a + F 0 (W )h(a) +
F 00 (W )rσ 2 (h0 (a))2 a) is not achieved on [δ, Ā], b) is strictly concave in [0, δ] and c) the maximizer is given by the first-order condition.
0
0
(W )h(a)
(W )
Proof Both mina∈[δ,Ā],W ∈[B,Wgp ] −a−F
and mina∈[0,δ],W ∈[B,Wgp ] −2F
are finite. It
rσ 2 (h0 (a))2
rσ 2 h00 (a)
00
follows from limW ↓B F (W ) = −∞ that, there is > 0 such that for all W ∈ [B, B + ],
−2F 0 (W )
−a − F 0 (W )h(a)
00
,
min
.
F (W ) < min
min
rσ 2 (h0 (a))2
a∈[0,δ],W ∈[B,Wgp ] rσ 2 h00 (a)
a∈[δ,Ā],W ∈[B,Wgp ]
Equivalently, if a ∈ [δ, Ā] and W ∈ [B, B + ], then
a + F 0 (W )h(a) + F 00 (W )rσ 2 (h0 (a))2 < 0,
(30)
and if a ∈ [0, δ] and W ∈ [B, B + ], then
1
F 0 (W ) + F 00 (W )rσ 2 h00 (a) < 0.
2
(31)
It follows from (30) that, if W ∈ [B, B + ], then a + F 0 (W )h(a) + F 00 (W )rσ 2 (h0 (a))2 cannot
achieve a maximum in a ∈ [δ, Ā].
To show that a + F 0 (W )h(a) + F 00 (W )rσ 2 (h0 (a))2 is strictly concave in [0, δ], we check its
second derivative. The second derivative is
F 0 (W )h00 (a) + F 00 (W )rσ 2 (h00 (a)h00 (a) + h0 (a)h000 (a))
1
< F 0 (W )h00 (a) + F 00 (W )rσ 2 h00 (a)h00 (a)
2
0
00
2 1 00
= (F (W ) + F (W )rσ h (a))h00 (a)
2
< 0,
46
where the second inequality follows from (31).
Finally, we verify that the first-order condition
1 + F 0 (W )h0 (a) + F 00 (W )rσ 2 h0 (a)h00 (a) = 0
pins down a unique a ∈ (0, δ). If a = 0, then 1 + F 0 (W )h0 (a) + F 00 (W )rσ 2 h0 (a)h00 (a) > 0,
violating the first-order condition.
References
Atkeson, A. and R. E. Lucas, Jr. (1995). Efficiency and equality in a simple model of efficient
unemployment insurance. Journal of Economic Theory 66 (1), 64–88.
Borodin, A. and P. Salminen (2002). Handbook of Brownian Motion - Facts and Formulae,
2nd ed. Birkhauser.
DeMarzo, P. and Y. Sannikov (2006). Optimal security design and dynamic capital structure
in a continuous-time agency model. The Journal of Finance 66 (6), 2681–2724.
Engelbert, H. and G. Peskir (2014). Stochastic differential equations for sticky brownian
motion. Stochastics 86 (6).
Milgrom, P. and I. Segal (2002, March). Envelope theorems for arbitrary choice sets. Econometrica 70 (2), 583–601.
Phelan, C. (1995). Repeated moral hazard and one-sided commitment. Journal of Economic
Theory 66 (2), 488–506.
Sannikov, Y. (2008). A continuous-time version of the principal-agent problem. Review of
Economic Studies 75 (3), 957–984.
Stokey, N. L. (2008). The Economics of Inaction: Stochastic Control Models with Fixed Costs.
Princeton University Press.
Thomas, J. and T. Worrall (1990). Income fluctuations and asymmetric information: An
example of a repeated principal-agent problem. Journal of Economic Theory 51 (2), 367–
390.
Zhu, J. Y. (2013). Optimal contracts with shirking. Review of Economic Studies 80 (2), 812–
839.
47
@FedFRASER