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Optimal Incentive Contracts with Job
Destruction Risk
WP 17-11
Borys Grochulski
Federal Reserve Bank of Richmond
Russell Wong
Federal Reserve Bank of Richmond
Yuzhe Zhang
Texas A&M University
Optimal incentive contracts with job destruction risk∗
Borys Grochulski
Russell Wong
Yuzhe Zhang
September 29, 2017
Working Paper No. 17-11
Abstract
We study the implications of job destruction risk for optimal incentives in a long-term
contract with moral hazard. We extend the dynamic principal-agent model of Sannikov
(2008) by adding an exogenous Poisson shock that makes the match between the firm and
the agent permanently unproductive. In modeling job destruction as an exogenous Poisson
shock, we follow the Diamond-Mortensen-Pissarides search-and-matching literature. The
optimal contract shows how job destruction risk is shared between the firm and the agent.
Arrival of the job-destruction shock is always bad news for the firm but can be good news
for the agent. In particular, under weak conditions, the optimal contract has exactly two
regions. If the agent’s continuation value is below a threshold, the agent’s continuation
value experiences a negative jump upon arrival of the job-destruction shock. If the agent’s
value is above this threshold, however, the jump in the agent’s continuation value is positive,
i.e., the agent gets rewarded when the match becomes unproductive. This pattern of
adjustment of the agent’s value at job destruction allows the firm to reduce the costs of
effort incentives while the match is productive. In particular, it allows the firm to adjust
the drift of the agent’s continuation value process so as to decrease the risk of reaching
either of the two inefficient agent retirement points. Further, we study the sensitivity of
the optimal contract to the arrival rate of job destruction.
Keywords: dynamic moral hazard, job destruction, jump risk
JEL codes: D86
1
Introduction
Understanding ex-post income heterogeneity of ex-ante identical workers is an important question in economics. Two extensively studied explanations for this heterogeneity are search
∗
FRB-Richmond and Texas A&M. 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.
DOI: https://doi.org/10.21144/wp17-11
1
frictions and information frictions. Although these two sources of heterogeneity were traditionally studied in separation from each other, the literature has been moving recently toward
studying search and information frictions jointly in order to obtain models that can better account for the data. This paper’s goal is to contribute to this effort by studying the implications
of the risk of job destruction, which is commonly used in the search and matching literature
to generate separations, on the optimal long-term contract in a dynamic private-action (i.e.,
moral hazard) environment.
To this end, we study the implications of job destruction risk in the dynamic moral hazard
model of Sannikov (2008). Like most papers in dynamic contracting literature, Sannikov
(2008) assumes that productivity of the match between the firm and the worker/agent is timeinvariant, i.e., although output is subject to transitory idiosyncratic shocks, the match remains
productive indefinitely into the future. This assumption makes it difficult to integrate dynamic
contracting models with the search and matching theory, e.g., Mortensen and Pissarides (1994),
where persistent match productivity shocks are a basic source of heterogeneity. We take a step
toward removing this limitation of the dynamic contracting theory by allowing for persistent
shocks to the productivity of the match between the firm and the agent. In particular, we study
the implications of a job-destruction shock that makes the match permanently unproductive.
We follow Pissarides (1985) and Mortensen and Pissarides (1994) in modeling job destruction
as an exogenous, observable Poisson shock that arrives with a known intensity λ. Prior to the
arrival of this shock, our model is identical to Sannikov (2008): The agent chooses privately a
costly action at at all t, where more-costly actions have a larger positive impact on the firm’s
expected flow of revenue. At a random time θ, the match becomes unproductive: no further
revenue will be generated regardless of what actions the agent takes.
The job-destruction shock can be naturally interpreted in two ways. First, it can be viewed as
a productivity shock that affects the quantity of output produced inside the match. Second,
as in Mortensen and Pissarides (1994), it can be interpreted as a shock to the market price
of the differentiated good produced by the firm. Complete job destruction at shock arrival
means that either the physical productivity of the match or the market price of output become
permanently zero at date θ.
Our main results show how job destruction risk impacts the optimal contract. The contract has
the same qualitative features as in Sannikov (2008). The agent’s promised utility Wt is a state
variable sufficient for recursive characterization of the solution. The agent’s effort is positive
everywhere except at two absorbing boundaries of the support of Wt : the low retirement point
at W = 0 and the high retirement point Wgp > 0. The firm’s optimal profit function F (W ) is
hump-shaped with a unique maximum at W ∗ ∈ (0, Wgp ). The agent receives no compensation
when Wt ≤ W ∗ and positive compensation when Wt > W ∗ . At θ, the contract becomes static:
2
the agent is asked for no effort and is provided a constant retirement/severance payment flow
c0 .
However, the agent is not fully insured against the job-destruction shock. At θ, both the
agent’s continuation utility and the firm’s continuation profit jump, i.e., job destruction risk
is shared between the firm and the agent. We show that the arrival of a job-destruction shock
is always bad news for the firm but can be good news for the agent. In particular, under weak
conditions, the optimal contract has exactly two regions. If the agent’s continuation value is
below a threshold, denoted by Wnj , the agent’s continuation value experiences a negative jump
at θ. If the agent’s value is above this threshold, however, the jump in the agent’s continuation
value is positive, i.e., the agent gets rewarded when the match becomes unproductive.
This pattern of adjustment of the agent’s value at job destruction is optimal because it allows
the firm to reduce the costs of effort incentives before the job-destruction shock arrives, i.e,
while the match is productive and the agent exerts effort. In particular, the level of continuation
utility promised to the agent conditional on job destruction is inversely related to the growth
(drift) of the agent’s promised utility conditional on no job destruction. If the firm promises
more after job destruction, drift of Wt is lower prior to that event and vice versa. In the
optimal contract, the promise of utility after job destruction is therefore used to manipulate
the dynamics of Wt so as to maximize the firm’s profit. There are two effects. First, there is the
wealth effect: the agent with higher W is more expensive to elicit effort from, which eventually
leads to retirement of the agent at Wgp . This effect calls for using the post-job-destruction
utility promise, W 0 , to lower the drift of Wt . Second, there is the “poverty trap” effect: if Wt
hits 0, incentives no longer can be provided to the agent, as he has no further “skin in the
game.” This effect calls for using W 0 to increase the drift of Wt . In the optimal contract, the
poverty trap effect dominates at low W and the wealth effect dominates at high W , with the
unique threshold being Wnj .
In this paper, we study job destruction risk in the contracting problem between a single firm
and a single agent/worker without considering equilibrium in a broader labor market. We
do not explicitly model separations and transitions of agents from one job to another after a
job-destruction shock. Similar to Sannikov (2008), we use the simplifying assumption that the
agent retires after job destruction. However, our analysis provides a building block for solving
labor market equilibrium models with long-term contracts under moral hazard and subject
to job-destruction shocks. Any such model, e.g., one that would integrate moral hazard and
an explicit search friction a la Diamond-Mortensen-Pissarides, will need to solve a contracting
problem similar to ours. Our contract characterization results will continue to apply as long as
the equilibrium post-separation continuation value function for the firm satisfies the sufficient
condition on the post-retirement profit function that we identify in this paper.
3
Related literature Several studies explore the impact of jump risk on the optimal provision
of incentives in risk-neutral environments without the consumption smoothing motive, e.g.,
Hoffmann and Pfeil (2010), Piskorski and Tchistyi (2010), DeMarzo et al. (2014). We share
with these studies the optimality condition that equates, whenever possible, the firm’s marginal
value before and after the jump shock. Our model provides additional implications for optimal
compensation, which, due to risk aversion, should remain continuous at job destruction.
Li (2017) allows risk aversion, a recurrent match-productivity shock, and provides a recursive procedure for computing the optimal contract numerically. We study a permanent jobdestruction shock and provide analytical characterization of the optimal contract. Our analytical results can be extended to allow temporary, recurrent spells of zero productivity.
Tsuyuhara (2016) studies long-term contracts with moral hazard and job destruction embedded
in a labor market with directed search. That paper, however, does not allow for payments from
the firm to the agent after job destruction. In our paper, we allow for such payments (severance
or retirement benefits) and show that they are important for the agent’s incentives inside the
match. A similar model with long-term contracts, moral hazard, and job-destruction shocks
is solved in Lamadon (2016). There, however, output in the match does not depend on the
agent’s effort. The moral hazard problem applies to the probability of a job-destruction shock.
Similar to Tsuyuhara (2016), Lamadon (2016) does not allow for compensation conditional on
job destruction.
Organization The rest of this paper is organized as follows. Section 2 lays out the model
and conducts preliminary analysis of the HJB equation. Section 3 provides our main results
on the jumps in the firm’s and the agent’s continuation values at job destruction. Section 4
examines contract dynamics and exit probabilities. Sections 5 and 6 study the sensitivity of
various contract features to the severity of the risk of job destruction, as measured by the rate
of arrival λ. Section 7 concludes.
2
The principal-agent problem
The principal-agent contracting problem is the same as in Sannikov (2008) except that at a
Poisson time θ the productivity of the match ends, i.e., the job is destroyed. Before θ, 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 ).
After θ, the cumulative output follows
dXt = 0,
4
(1)
i.e., no further output is produced inside the match. Time θ arrives with Poison intensity λ
and is independent of Zt .
The set of feasible actions A ⊂ R, as in Sannikov, is compact with the smallest element 0. 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 (u(Ct ) − h(At )) dt ,
E r
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, we follow Sannikov (2008) in assuming that
there exists γ0 > 0 such that h(a) ≥ γ0 a for all a ∈ A.
Under a given contract (C, A), the agent’s continuation value process is
Z ∞
−r(s−t)
Wt := Et r
e
(u(Cs ) − h(As )) ds .
t
The Sannikov model is a special case of this specification with λ = 0, i.e., the match remains
productive indefinitely.
Clearly, after the match productivity termination shock hits, the optimal action is At = 0
forever and the profit function for the firm is
F0 (W 0 ) = −c0 such that u(c0 ) = W 0 ,
(2)
with the agent receiving a constant payment c0 forever after the arrival of the shock, and W 0
represents the agent’s continuation utility after the arrival of the shock, i.e., after any jumpat-arrival. Here, c0 can be interpreted as the flow of compensation to the agent in retirement
or as a severance benefit paid out upon termination of the job. The firm’s after-shock profit
function, F0 , is negative, strictly decreasing and, by strict concavity of u, strictly concave.
Before the shock, the dynamics of the continuation value are
dWt = r(Wt − u(Ct ) + h(At ))dt + rYt (dXt − At dt) + ∆t (dNt − λdt),
(3)
where dXt − At dt is the agent’s observed performance relative to the benchmark At dt, rYt represents the sensitivity of the agent’s continuation value to his performance, ∆t is the sensitivity
5
to the Poisson shock, and Nt is the counting process stopped at 1. As in Sannikov (2008), the
contract is incentive compatible (IC) at t if
At ∈ argmax Yt a − h(a).
(4)
a∈A
In equation (3), we can see how the risk of job destruction affects the drift of the agent’s
continuation value while the job is active, i.e., before the arrival of the job-destruction shock.
With dNt = 0, equation (3) reduces to
dWt = (r(Wt − u(Ct ) + h(At )) − ∆t λ) dt + rYt (dXt − At dt)dt.
As we see, larger ∆t implies, ceteris paribus, a smaller rate of increase in Wt that is needed to
deliver Wt to the agent over time as the job continues to survive.
The same observation can be made using the agent’s post-job-destruction promised continuation value. The sensitivity process ∆t shows by how much the agent’s continuation value
changes on arrival of the job-destruction shock in any date and state. Therefore, we can express
it as
∆t = Wt0 − Wt = u(c0t ) − Wt ,
(5)
where Wt0 is the agent’s continuation value at t in the event θ = t, c0t is the (constant) level of
the retirement (or severance) benefit grated to the agent in the same event. Using Wt0 or c0t
instead of ∆t , we can express the expected change in Wt conditional on job survival, i.e., the
drift term in (3) before job destruction, as
(r + λ)Wt − r(u(Ct ) − h(At )) − λWt0
= (r + λ)Wt − r(u(Ct ) − h(At )) − λu(c0t ).
Since Wt is replaced with Wt0 = u(c0t ) in the event of job destruction at t, a higher promise Wt0 ,
or, equivalently, a higher severance promise c0t , decreases the drift in Wt before job destruction.
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
λ
0
(r+λ)F (W ) = max ra−rc+F (W )r W − u(c) + h(a) − ∆ + F 00 (W )r2 σ 2 Y 2 +λF0 (W +∆),
c,a,Y,∆
r
2
(6)
where W and W + ∆ must remain nonnegative because 0 is the agent’s minimax payoff.
The first order (FO) condition over ∆ is
−F 0 (W )λ + λF00 (W + ∆) ≤ 0
6
with strict equality if W + ∆ > 0.
Because F00 (W ) ≤ 0 for all W , this FO condition leads to two cases. If F 0 (W ) > 0, then
W + ∆ = 0, so ∆(W ) = −W . If F 0 (W ) ≤ 0, then F00 (W + ∆) = F 0 (W ), so ∆(W ) =
−W + (F00 )−1 (F 0 (W )).
The interpretation of the optimal adjustment ∆ is as follows. By increasing the promise of the
continuation value to be delivered to the agent after job destruction, the firm gains additional
(negative) profit F00 (W + ∆), which it discounts at the rate of the shock arrival λ. That same
increase in the agent’s value post-arrival decreases the drift of the agent’s continuation value
conditional on the job’s survival, at the rate λ. This lower drift increases the firm’s profit at
the rate F 0 (W )λ. The optimal ∆ is set where the marginal cost is equal to its marginal benefit.
Remark In the first best, the agent would get constant consumption forever and work until
the arrival of the job-destruction shock. Thus, his continuation utility would jump upward at
the moment of shock arrival, i.e., ∆ > 0 in the first best. The first-best profit function is
Ffb (W ) := max{
c,a
r
r
a − c : u(c) −
h(a) = W }.
r+λ
r+λ
(7)
With moral hazard, we will have ∆ < 0 at least for small W . But consumption will be
continuous over the jump moment both with moral hazard and in first best.
2.1
Equivalent expressions of the HJB equation
Since the static contract is optimal after the shock, we can equivalently use the after-shock
agent’s continuation value W 0 = W + ∆, or his constant retirement/severance flow of compensation, c0 , where u(c0 ) = W 0 , as controls in the HJB equation, instead of ∆.
With W 0 = W + ∆, the HJB can be written as
λ
1
0
0
(r+λ)F (W ) = max0 ra−rc+F (W )r W − u(c) + h(a) − (W − W ) + F 00 (W )r2 σ 2 Y 2 +λF0 (W 0 ).
c,a,Y,W ≥0
r
2
The maximization of terms that involve W 0 , i.e., maxW 0 ≥0 {−F 0 (W )W 0 + F0 (W 0 )}, implies
that W 0 = 0 if F 0 (W ) ≥ 0, and W 0 solves
−F 0 (W ) + F00 (W 0 ) = 0
if F 0 (W ) < 0. This says that, when possible, the slope of the firm’s profit function should match
before and after the shock. This is the same as the earlier discussion of the FO condition with
respect to ∆.
Further, if we use retirement/severance flow c0 , we can write the HJB as
λ
1
0
0
(r+λ)F (W ) = max0 ra−rc+F (W )r W − u(c) + h(a) − (u(c ) − W ) + F 00 (W )r2 σ 2 Y 2 −λc0 .
c,a,Y,c ≥0
r
2
7
The FO condition with respect to c0 , F 0 (W )u0 (c0 ) = −1, is the same as the FO condition for
c, the agent’s consumption before the shock. This shows that at the time of shock arrival
consumption does not change, although the continuation value typically will. So, the arrival
of the shock “freezes” the current c and makes it fixed forever after.
Because c0 = c, it will be convenient for us to eliminate c0 and just use c as a single control
variable (representing consumption now and forever if the shock hits now). The HJB is
1
(r + λ)F (W ) = max ra − (r + λ)c + F 0 (W ) ((r + λ)(W − u(c)) + rh(a)) + F 00 (W )r2 σ 2 Y 2
c,a,Y
2
or
F (W ) = max
c,a,Y
r
r
r 1 00
a − c + F 0 (W )(W − u(c)) +
F 0 (W )h(a) +
F (W )rσ 2 Y 2 , (8)
r+λ
r+λ
r+λ2
where, up to the time of arrival of the job-destruction shock, Wt follows
dWt = ((r + λ)(Wt − u(c)) + rh(a))dt + rY σdZt .
(9)
Collecting terms, we can write the HJB in the following form
1 00
r+λ
0
0
0
2 2
F (W ) − F (W )W − max −c − F (W )u(c)
= max a + F (W )h(a) + F (W )rσ Y
.
c≥0
a,Y
r
2
Note that this equation reduces to the HJB equation of Sannikov (2008) when λ = 0.
2.2
Monotonicity of terms in the HJB equation
The left hand side of the HJB equation is monotone in λ. Whether the left hand side is increasing or decreasing in λ depends on the sign of F (W )−F 0 (W )W −maxc≥0 {−c − F 0 (W )u(c)}. But
this sign is always positive as long as we are solving for a curve F in the region F (W ) ≥ F0 (W )
at all W , and F is concave.
To see this, change c to W 0 again using u(c) = W 0 . We have
F (W ) − F 0 (W )W − max −c − F 0 (W )u(c) = F (W ) − F 0 (W )W − max
F0 (W 0 ) − F 0 (W )W 0
0
c≥0
W ≥0
0
= min
F (W ) − F (W )W − F0 (W 0 ) + F 0 (W )W 0
0
W ≥0
0
0
0
= min
F
(W
)
+
F
(W
)
W
−
W
−
F
(W
)
.
0
0
W ≥0
This quantity is the minimal vertical distance between the tangent to F at W and F0 . By
concavity of F , the tangent is always above F , i.e., F (W ) + F 0 (W ) (W 0 − W ) ≥ F (W 0 ) for all
W and W 0 . So
min
F (W ) + F 0 (W ) W 0 − W − F0 (W 0 ) ≥ min
F (W 0 ) − F0 (W 0 ) ≥ 0,
0
0
W ≥0
W ≥0
8
where the last inequality uses F ≥ F0 .
Thus, the left hand side of the HJB is increasing in λ as long as we are solving for a concave
F above F0 . The optimal solution will satisfy these conditions.
2.3
Option value of the agent’s effort
Let us denote this distance by S, i.e., let
0
0
0
S(W ) := F (W ) − max
F
(W
)(W
−
W
)
+
F
(W
)
0
W 0 ≥0
0
= F (W ) − max
F (W )(W − u(c0 )) − c0 .
0
c ≥0
Note that S is a function of F (W ), F 0 (W ), and W . With this notation, the HJB reads
1 00
λ
0
2 2
,
(1 + )S(W ) = max a + F (W )h(a) + F (W )rσ Y
a,Y
r
2
or, using the IC condition Y = h0 (a),
r
1 00
0
2 0
2
S(W ) =
max a + F (W )h(a) + F (W )rσ h (a) ,
r+λ a
2
which has an intuitive interpretation: S(W ) = F (W ) − maxW 0 ≥0 {F 0 (W )(W − W 0 ) + F0 (W 0 )}
is the firm’s surplus from being able to induce positive effort from the agent.1 The right hand
side of the HJB shows where this surplus is coming from. It is equal to expected output from
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 volatility Y = h0 (a) in the state variable (it is a cost because F 00 < 0).
r
represents the risk of productivity termination, i.e., job destruction.
The factor r+λ
Note that the same variable represents the surplus from the agent’s effort at the first best
contract. Indeed, the FOC in
0
Sfb (W ) := Ffb (W ) − max
Ffb (W )(W − u(c0 )) − c0
0
c ≥0
implies c0 = cfb (W ). Using this and (7), we have
Sfb (W ) =
=
=
=
r
0
afb − cfb − Ffb
(W )(W − u(cfb )) − cfb
r+λ
r
0
afb − Ffb
(W )(W − u(cfb ))
r+λ
r
r
0
afb + Ffb
(W )
h(afb )
r+λ
r+λ
r
0
(W )h(a) ,
max a + Ffb
r+λ a
If effort is no longer an option, then the firm’s profit is F0 (W 0 ) after incurring the cost F 0 (W )(W − W 0 ) of
optimally adjusting the agent’s value from W to W 0 .
1
9
where afb and cfb at evaluated at W .2 This shows that in the first best the calculation of the
surplus from the agent’s effort is the same except for the cost of volatility does not show up in
∗ and equals zero at W ∗ .
the formula. Also note that Sfb (W ) > 0 at all W < Wgp
gp
2.4
Existence, regularity, and computation
Solving the HJB for F 00 , equivalently, we have
F 00 (W ) = min
a≥0
(1 + λr )S(W ) − a − F 0 (W )h(a)
,
1
2 0
2
2 rσ (h (a))
(10)
which can be solved forward from W = 0 using boundary conditions (F (0), F 0 (0)) = (0, x) for
any x ≥ 0.
As in Sannikov (2008), the optimal profit curve is obtained by looking for the initial slope
F 0 (0) such that the solution curve stays above F0 and touches it at a point, which is denoted
by Wgp . Sannikov (2008) shows that the contract constructed from the policy functions that
attain this solution in the HJB equation is optimal. These results, proved in Lemmas 1, 2, 3,
Proposition 3, Lemma 4, and Proposition 4 of Sannikov (2008), also hold in our model. It is
easy to see that nothing changes in the proofs of these results when a constant factor (1 + λr )
multiplying terms F (W ) − F 0 (W )(W − u(c)) + c is inserted into the HJB equation.
Solving (10) for the optimal F results with a profit function qualitatively similar to the solution
in Sannikov (2008): F is strictly concave with a unique maximum at W ∗ := argmax F (W ),
where 0 < W ∗ < Wgp .
3
The jump at job destruction
In this section, we provide our main results that show how the optimal contract is affected by
the risk of job destruction and what happens at the arrival of a job-destruction shock.
3.1
The jump in the agent’s continuation value
Proposition 1 In the optimal contract, W 0 (W ) = 0 at all W ≤ W ∗ , and W 0 is strictly
increasing in W at all W ∗ < W < Wgp .
The first part follows from F 0 (W ) > 0 below W ∗ . The second part follows simply from strict
concavity of F0 , which is a direct implication of strict concavity of u.
2
Note that the third line uses the promise-keeping constraint in (7) and the last line uses the FOC for afb .
10
Figure 1 illustrates this proposition in a computed example. The jump in the agent’s continuation value at job destruction is represented by the vertical distance between the W 0 (W ) curve
and the 45 degrees line. For all W ≤ W ∗ , where F 0 (W ) ≥ 0 and c = 0, we have W 0 = 0, i.e.,
the agent loses all of his continuation value with c0 = 0. Above W ∗ , the retirement/severance
value W 0 is increasing in W , i.e., agents with larger continuation value at arrival receive a
higher retirement/severance value as well.
Further, Figure 1 suggests that the sign of ∆ = W 0 − W only changes once. That is, in the
region above W ∗ , agents with relatively small W are hurt by job destruction. However, agents
with high W gain from it. We prove this feature of the optimal contract under an additional
assumption.
Proposition 2 Assume F00 is weakly concave. Then there exists a unique Wnj such that
∆(W ) < 0 for all 0 < W < Wnj ; ∆(Wnj ) = 0; and ∆(W ) > 0 for all 0 < Wnj < Wgp . Also,
Wnj > W ∗ .
Proposition 2 shows that it is optimal to widen the spread of the agent’s value at the arrival of
the job-destruction shock: the agents with high W (higher than Wnj ) see their value increased,
∆(W ) > 0, while the agents with low W (lower than Wnj ) experience a drop in their continuation value at job destruction, ∆(W ) < 0. To see why doing so is profitable, recall that ∆(W )
has an inverse impact on the drift of Wt prior to job destruction: positive ∆ decreases the
drift of Wt and negative ∆ increases it. By suppressing the growth of Wt when Wt is high and
increasing it when Wt is low, the optimal policy ∆(W ) decreases the chance of hitting either
of the two inefficient agent retirement points, 0 and Wgp , while the match remains productive.
This decrease in the chance of early contract termination improves efficiency.
The assumption of weak concavity of F00 is a convenient sufficient condition that can be relaxed.
If u is trice differentiable, this assumption is equivalent to u000 (c)u0 (c) ≤ 3u00 (c)2 at all c, which
is very simple condition to verify.
Proposition 2 also implies that S(W ) is single peaked with a unique maximum at Wnj . Indeed,
differentiating S(W ) we have
dS
dW
= F 0 (W ) − F 00 (W )(W − W 0 (W )) − F 0 (W )
= F 00 (W ) W 0 (W ) − W
= F 00 (W )∆(W )
so ∆ and S 0 are of opposite sign. Proposition 2, by pinning down the sign of ∆(W ), implies
the following:
11
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Figure 1: Optimal jump of the agent’s continuation value at job destruction.
Corollary 1 S 0 (W ) > 0 at all W < Wnj , S 0 (Wnj ) = 0, and S 0 (W ) < 0 for all 0 < Wnj < Wgp .
Thus, Wnj is the unique peak point of S(W ).
This means that the jump in the agent’s continuation value at job destruction is zero only at
the single W at which the firm’s option value of the agent’s effort is maximal.3
3.2
The jump in the firm’s value
From the definition of S, we have S(W ) = F (W ) + F 0 (W )∆(W ) − F0 (W 0 (W )), which gives us
F (W ) − F0 (W 0 (W )) = S(W ) − F 0 (W )∆(W ).
This says that the firm’s loss of value at job destruction, F (W ) − F0 (W 0 (W )), equals the loss
of the option on the agent’s effort, S(W ), and the cost of the agent’s gain in utility, ∆(W ),
valued at the marginal price of utility, −F 0 (W ). The loss of productivity option value is always
3
Note that S(W ) does not have to be concave.
12
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 2: The firm’s loss of profit after the optimal jump in W at arrival of job destruction.
positive, as S ≥ 0. The second term may be positive or negative, as both ∆ and F 0 change
sings.
Proposition 3 Assume F00 is weakly concave. Then the firm always loses value at arrival of
the job-destruction shock: F (W ) ≥ F0 (W 0 (W )), strictly at all 0 < W < Wgp .
The above result is very intuitive at W equal 0 or Wgp . There, there is no loss of profits due
to job destruction because the job is dissolved endogenously at these two points anyway (i.e.,
both S and ∆ are zero there).
At all 0 < W < Wgp , the effort option value S(W ) is strictly positive. The firm’s value of the
agent’s utility jump depends on W . Above Wnj , ∆ > 0 and F 0 < 0, i.e., in addition to the
loss of productivity, the firm is hurt at job destruction by a positive jump in the value it owes
to the agent. Below W ∗ , the firm is again hurt by the adjustment to the agent’s value, but
for the opposite reason, as signs of both ∆ and F 0 are switched. With ∆ < 0 and F 0 > 0, the
agent’s loss of value actually hurts the firm’s value. In fact, the agent loses all promised value
as W 0 = 0 in this region, but this does not help the firm as all of the agent’s value in this
13
region comes from his future expected compensation. Finally, in the region between W ∗ and
Wnj we have ∆ < 0 and F 0 < 0, so the loss of the agent’s utility value does offset the firm’s
loss of productivity to an extent. However, it turns out that that this offset is insufficient.
Figure 2 provides a computed example. Note the small region in which F0 (W 0 (W )) > F0 (W ).
There, the negative jump in the agent’s continuation value at job destruction partially compensates the firm for the loss of the option on the agent’s effort.
4
Contract dynamics and exit probabilities
In this section, we discuss contract dynamics with particular attention to exits. We will
characterize these features of the optimal contract by finding an associated ODE and solving
it numerically using the policies from the optimal contract.
We will denote the drift of Wt under the optimal contract by µ(W ) and its volatility by ν(W ).
We have from (9) that
µ(W ) = (r + λ) W − u(c(W )) + rh(a(W )),
ν(W ) = rσY (W ),
where c(W ), a(W ), and Y (W ) are the policy functions from the optimal contract.
Figure 3 shows the drift and volatility functions in a computed example, where λ gives a
realistic contract duration. Following the labor literature, we target in our parametrization
an average duration of a job to be 10 calendar quarters. We approximate job duration here
as the expected time to arrival of a job-destruction shock, 1/λ. As we see, the contract has
interesting dynamics. At small W , the drift of W is positive and high with high volatility. The
contract likely moves out of this region toward the middle region of W . There, the contract
“slows down,” i.e., has drift close to zero and moderate volatility, which means the contract
will likely spend a lot of time in the middle region once it reaches it. Thus, the job-destruction
shock is likely to arrive while Wt is in that region.
4.1
Time remaining and exit probability
Let us denote by T (W ) the expected time until job end (including both exogenous job destruction and endogenous agent retirement). The job can “end” in three ways: the agent may be
retired at 0, retired at Wgp , or the job is ended by a job-destruction shock.
The probability of end/exit at 0 will be denoted by P0 (W ), and the probability of exit at Wgp
by Pgp (W ). The probability that the contract ends with the arrival of the job-destruction
shock will be denoted by PJD (W ). Clearly, P0 (W ) + Pgp (W ) + PJD (W ) = 1 for all W .
14
0.16
volatility
drift
0.14
0.12
0.1
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
W
Figure 3: Drift µ(W ) and volatility ν(W ) at the optimal contract under a parametrization with average
time till job destruction of 10 quarters.
We compute T and P by finding an ODE for each of them. These ODEs can then be easily
solved numerically using policy functions from the optimal contract.
Lemma 1 T and P satisfy the following ODEs
1
λT (W ) = 1 + T 0 (W )µ(W ) + T 00 (W )ν(W )2 ,
2
1
(r + λ)P (W ) = P 0 (W )µ(W ) + P 00 (W )ν(W )2 ,
2
with boundary conditions T (0) = T (Wgp ) = 0, P0 (0) = 1, P0 (Wgp ) = 0, Pgp (0) = 0,
Pgp (Wgp ) = 1, and PJD (0) = PJD (1) = 0.
Figure 4 shows T and the three P functions in our parametrized example. The probability of
exit at either end of the support of W drops very quickly in the distance between W and this
boundary. In the middle, the contract “slows down” very dramatically, as we saw in Figure 3.
It is therefore extremely unlikely that Wt reaches either retirement point before the arrival of
a job-destruction shock. Thus, in this region, T (W ) is very close to 1/λ and PJD (W ) is very
close to 1.
15
1.2
1.2
T
1/lambda
Pr0(W)
Prgp (W)
1
1
PrJD (W)
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
W
W
*
*
0
-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
W
0.4
0.5
0.6
0.7
W
Figure 4: T and P under parametrization with r/λ = 0.12, i.e., average job duration of 10 quarters.
4.2
Other contract features
This method, i.e., finding a function by numerically solving an ODE, can be used to compute
other features of the optimal contract and the value function. In this section, we illustrate this
point by computing a decomposition of F into its cost and revenue parts.
The expected, discounted remaining revenue will be denoted by R(W ), and the expected
discounted remaining wage bill by B(W ). That is:
Z ∞
Z ∞
−rs
−rs
R(Wt ) = Et r
e a(Wt+s )ds and B(Wt ) = Et r
e c(Wt+s )ds ,
0
0
so we have F (W ) = R(W ) − B(W ) for all W .
Lemma 2 R and B satisfy the following ODEs
1
(r + λ)R(W ) = ra + R0 (W )µ(W ) + R00 (W )ν(W )2 ,
2
1
(r + λ)B(W ) = (r + λ)c + B 0 (W )µ(W ) + B 00 (W )ν(W )2 ,
2
with boundary conditions R(0) = R(Wgp ) = 0, and B(0) = 0, B(Wgp ) = u−1 (Wgp ) =
−F0 (Wgp ).
Figure 5 shows the solutions of the ODEs for R and B, along with the profit function F ,
obtained with λ such that 1/λ is 10 quarters, as before. As we see, the wage bill by B(W )
accounts for most of the variation in F (W ) with expected revenue R(W ) being relatively flat.
However, it is the revenue part of F that gives it its hump shape, as B is monotone in W .
16
0.8
0.9
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
F(W)
R(W)
-B(W)
-0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
W
Figure 5: Profit, revenue and the wage bill.
5
Sensitivity of profit to job destruction risk
In this section, we study how the firm’s expected discounted profit depends on the expected
duration of the relationship, 1/λ, or, equivalently, on λ measuring job-destruction risk. For
˜ > λ ≥ 0 and denote the
comparisons with respect to the level of λ, let us always write λ
˜ as F˜ .
solution under λ
Lemma 3 Take two solutions F and F˜ of the HJB, where the first is with λ and the second
˜ > λ, such that F (W 0 ) ≤ F˜ (W 0 ) and F 0 (W 0 ) < F˜ 0 (W 0 ) at some W 0 ≥ 0. Then
is with λ
F 0 (W ) < F˜ 0 (W ) at all W > W 0 .
˜ respectively.
From here on, by F and F˜ we mean the optimal solution curves under λ and λ,
˜ gp , and Wgp > W
˜ gp .
Proposition 4 F (W ) > F˜ (W ) at all 0 < W ≤ W
Intuitively, the above result shows that principal-agent relationships with lower expected duration, i.e., faster rate of arrival of the job-destruction shock, are less profitable to the firm
for any fixed value W the firm might owe to the agent. Further, the upper agent retirement
17
point Wgp is always lower in the relationships with lower expected duration. That is, the firm
has a weaker incentive to invest in the agent’s incentives in relationships with a higher risk of
exogenous job destruction.
5.1
Application of the Feynman-Kac formula
Here, we use the approach of Lemma 4 of DeMarzo and Sannikov (2006) to derive the following
formula.
Lemma 4
∂F (W0 )
= −E
∂λ
Z
τ
e
−(r+λ)t
S(Wt )dt < 0,
(11)
0
where τ denotes the time of fist exit of Wt from (0, Wgp ).
Further, since S(0) = S(Wgp ) = 0 and the dynamics of Wt stop if either of these points is
reached, we can write the above equation as
Z ∞
∂F (W0 )
−(r+λ)t
= −E
e
S(Wt )dt .
∂λ
0
Proposition 4 was obtained by a different argument. The above formula provides additional
(W0 )
information of the magnitude of ∂F∂λ
by relating it to the firm’s option value on the agent’s
effort, S(W ). This expression will allow us to show, numerically, that the firm’s profit is less
sensitive to the job destruction risk with moral hazard than it would be without moral hazard
(i.e., in the first-best contract).
5.2
Profit sensitivity relative to the first best
Differentiating the first-best profit function Ffb given in (7) with respect to λ and using the
envelope condition, we obtain
∂Ffb (W0 )
∂λ
r
0
(afb + Ffb
(W0 )h(afb ))
(r + λ)2
1
Sfb (W0 )
= −
rZ + λ
= −
∞
= −
e−(r+λ)t Sfb (W0 )dt.
0
∗ . The above formula thus implies
In Section 2.3, we show that Sfb (W0 ) > 0 for any W0 < Wgp
(W0 )
that ∂Ffb∂λ
< 0, as in the case with moral hazard shown above.
18
0.08
S
S FB
0.07
0.06
0.05
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 6: The function S in the optimal contract with moral hazard and in the first best.
(W0 )
(W0 )
The comparison between ∂F∂λ
and ∂Ffb∂λ
is difficult to obtain analytically because the
first-best contract is static. This means that S takes into account two components of the costs
of higher effort: F 0 , and F 00 representing the cost of volatility. In contrast, Sfb only needs to
0 . However, F 0 will generally be less negative than F 0 , so it is unclear if
account for one cost, Ffb
fb
two smaller costs are larger than one bigger cost. For this reason, we compare these measures
of sensitivity numerically.
(W )
Figure 6 plots S and Sfb in our parametrized example. Since S < Sfb , we have ∂F∂λ
> ∂Ffb∂λ(W )
for all 0 ≤ W ≤ Wgp , regardless of the fact that the agent’s continuation value process has
complicated dynamics under moral hazard but is reduced to a constant under first best.
6
Sensitivity of compensation front-loading to job destruction
risk
In this section, we briefly discuss the impact of the risk of job destruction on the amount of
compensation front-loading in the optimal contract. We maintain the assumption that F00 is
weakly concave.
19
˜ gp ], c (W ) ≤ c˜ (W ) if
Proposition 5 There exists a unique W s such that for all W ∈ [0, W
and only if W ≤ W s . That is, the contract associated with higher λ involves more front-loaded
payments.
Following the terminology of Sannikov (2008), a contract involves more front-loaded payment
when it pays higher wage now rather than later. Proposition 5 concludes that a more likely
job-destruction shock (higher λ) induces front-loading of wages when the agent’s continuation
utility is low. When the continuation utility is high, the optimal contract front-loads less as
the job-destruction risk increases.
7
Conclusion
In this paper, we study the impact of exogenous job destruction risk on the optimal longterm contract in an dynamic moral hazard environment. We show that post-job-destitution
payments to the agent are an important incentive device. In particular, they help the firm
control the drift of the agent’s continuation value inside the contract. In the optimal contract,
these payments are used to keep the agent’s continuation value in the region where the firm’s
option value of using the agent’s effort is highest. The contract promises a positive jump in
the agent’s value at job destruction whenever the firm’s option value on the agent’s effort is
decreasing in the agent’s value. Likewise, a negative jump is promised whenever the firm’s
option value is increasing. These promised jumps help keep the state variable near the peak
of the firm’s option value.
In the model without job destruction risk, the optimal contract has two exit points: the low
and high agent retirement points 0 and Wgp , respectively. Job destruction adds the third
contract exit possibility. Further, our analysis suggests that with job destruction the high
retirement exit point becomes unreachable, as the contract dynamics become very slow before
the contract can get there. Only in relationships with very long expected duration, the high
retirement point may remain reachable. This conjecture can be further explored by studying
the limiting contract as the job destruction arrival rate goes to infinity.
Our model can be extended in several interesting directions. Job destruction can be endogenized in a way similar to Mortensen and Pissarides (1994): the job-destruction shock can be
partial, i.e., reducing the productivity of the match to a low but positive level. It is natural
to conjecture that the contract would terminate endogenously (retire the agent at 0 or Wgp )
shortly after the arrival of such a shock. The model can be embedded in a broader labor market
with search, as in Lamadon (2016) and Tsuyuhara (2016), to study the impact of severance
payments on the agent’s search behavior.
20
Appendix
Proof of Proposition 1
Above W ∗ , the FO condition for W 0 is F 0 (W ) = F00 (W 0 (W )). Differentiation yields
F 00 (W ) = F000 (W 0 )
dW 0
.
dW
That W 0 is strictly increasing in this region follows from strict concavity of F and F0 .
QED
Proof of Proposition 2
Step 1. Differentiating the HJB and canceling out like terms, we get
λ
1
00
0
0 = F r W − u + h − (W − W ) + F 000 r2 σ 2 Y 2
r
2
λ
1
= F 00 r h − (1 + )(W 0 − W ) + F 000 r2 σ 2 Y 2 .
r
2
From here we get that W 0 − W < 0 implies F 000 (W ) > 0 (i.e., F 0 must be strictly convex when
∆ < 0).
Step 2. We know F 0 (0) > 0 = F00 (0). We also know that F approaches F0 from above as
W becomes close to Wgp , which means that for ε > 0 small enough, F (W ) > F0 (W ) for
all W ∈ (Wgp − ε, Wgp ) and F (Wgp ) = F0 (Wgp ). This implies that F 0 (W ) < F00 (W ) for all
W < Wgp close enough to Wgp . Therefore, the number of crossings between F 0 and F00 on
(0, Wgp ) is odd. We will show that this number is one. If this number is three or more, then
˜ somewhere between the second and the third crossing at which
there must exist a point W
0
F 0 is more concave than F0 . (If this was not true, the third crossing of F 0 and F00 would not
˜ ) < F 000 (W
˜ ) ≤ 0, where the weak inequality uses the assumption of weak
exist.) Thus, F 000 (W
0
0
˜ . Since W
˜ is between the second and the
concavity of F0 . I.e., F 0 is strictly concave at W
˜ ) > F 0 (W
˜ ), which implies that W 0 (W
˜)<W
˜ . We
third crossing of F 0 and F00 , we have F 0 (W
0
obtain a contradiction because we showed in Step 1 that F 0 must be convex when W 0 < W ,
˜)<W
˜ implies F 000 (W
˜ ) > 0.
i.e., W 0 (W
Step 3. Denote the unique crossing point between F 0 and F00 by Wnj . The sign of ∆(W ) is
the same as the sign of F00 − F 0 , which is negative for W < Wnj and positive W > Wnj by
the single crossing property shown in Step 2. Also, Wnj > W ∗ follows from F 0 (W ∗ ) = 0,
F 0 (Wnj ) = F00 (Wnj ) < 0, and F 0 continuous and decreasing.
QED
21
Proof of Proposition 3
If the contract reaches either of the two agent retirement points before the shock arrives, then
there is no jump.
For any fixed 0 < W < Wgp , we have
F (W ) > F0 (W ).
(12)
At Wnj , which satisfies 0 < Wnj < Wgp , we have W 0 (Wnj ) = Wnj , so if the shock arrives when
Wt = Wnj , the firm loses simply because of the loss of productivity, i.e., because of equation
(12). Indeed, we have F (Wnj ) > F0 (Wnj ) = F0 (W 0 (Wnj )).
At all Wnj < W < Wgp , the firm loses at shock arrival for two reasons. First, as in the previous
case, because of the productivity loss, i.e., equation (12). Second, the firm loses because the
agent gains and F is strictly decreasing in this region (recall W ∗ < Wnj ). Indeed W > Wnj
implies W 0 (W ) > W and F strictly decreasing implies F (W ) > F (W 0 (W )). Together, we have
F (W ) > F (W 0 (W )) > F0 (W 0 (W )).
At all 0 < W ≤ W ∗ , the adjustment from W to W 0 still hurts the firm, but its two components
switch signs. The agent gives up value, but this hurts the firm because F is increasing in this
region. In fact, W 0 (W ) = 0 < W for all 0 < W ≤ W ∗ , so we have F (W ) > 0 = F0 (0) =
F0 (W 0 (W )).
For W ∗ < W < Wnj , we have W 0 (W ) < W and F is decreasing, so the jump in W benefits the
firm. Thus, the loss of productivity effect and the jump in W effect work in opposite directions.
We will show that the productivity loss effect is stronger, i.e., the fact that the agent gives up
value does not make up for the loss of productivity.
Because F (W ∗ ) > 0 = F0 (W 0 (W ∗ )), showing that F (W ) − F0 (W 0 (W )) is increasing at all
W ∗ < W < Wnj will be sufficient for F (W ) > F0 (W 0 (W )) at all W ∗ < W < Wnj . We will
show that this is the case. Taking the derivative of F (W ) − F0 (W 0 (W )) and using the FO
condition F 0 (W ) = F00 (W 0 (W )) we have
0
dW 0
0
0
0
0 dW
= F (W ) 1 −
,
F (W ) − F0 (W )
dW
dW
0
∗
which means that it is sufficient to show that dW
dW > 1 at all W < W < Wnj . We know that
W 0 (W ∗ ) = 0 and W 0 (Wnj ) = Wnj , so W 0 starts below the 45 degrees line and catches up to it
over the interval W ∗ < W < Wnj , but we need to show that the catching up has no gaps.
Differentiating the FO condition F 0 (W ) = F00 (W 0 (W )) yields
F 00 (W ) = F000 (W 0 )
22
dW 0
.
dW
We have F 00 (W ) < F 00 (Wnj ) ≤ F000 (Wnj ) ≤ F000 (W 0 ). The first inequality follows from the strict
convexity of F 0 at all W < Wnj by Step 1 in the proof Proposition 2. The second inequality
follows from the fact that F 0 crosses F00 from above at Wnj . The third inequality follows from
the (weak) concavity of F00 and W 0 < W < Wnj . Finally, from F 00 (W ) < F000 (W 0 ) < 0 it follows
that F 00 (W )/F000 (W 0 ) > 1, which implies
dW 0
= F 00 (W )/F000 (W 0 ) > 1.
dW
QED
Proof of Lemma 1
R∞
For the expected time T , define H = 0 1s<θ 1s<τ ds, where θ is the arrival time of the jobdestruction shock, and τ is the time when Wt hits 0 or Wgp . Define a martingale Ht as
Rt
R∞
Rt
Ht = Et [H] = 0 Et [1s<θ ]1s<τ ds + Et [1t<θ ]E[ t 1s<θ 1s<τ ds|Ft , t < θ] = 0 e−λs 1s<τ ds +
e−λt 1t<τ T (Wt ). If t < τ , then its drift is
1 00
−λt
0
2
e
1 + T (W )((r + λ)(W − u) + rh) + T (W )(rσY ) − λT (W ) ,
2
which must be zero.
For the exit probability functions P , the argument is similar.
Proof of Lemma 2
Let Ft be the information set containing the sample path of the diffusion process of Wt , and
let Gt be the information set containing the realization of the job-destruction shock. Ft and
Gt are independent. We denote E[·|Ft ] as Et [·] to simplify notation.
R∞
For the revenue function R, define H = 0 1s<θ 1s<τ re−rs as ds, and a martingale Ht = Et [H] =
Rt
R
−rs a ds+E [1
−rt R(W ) = t re−(r+λ)s 1
−(r+λ)t 1
s
t t<θ ]1t<τ e
t
s<τ as ds+e
t<τ R(Wt ).
0 Et [1s<θ ]1s<τ re
0
If t < τ , then its drift is
1 00
2
−(r+λ)t
0
e
rat + R (W )((r + λ)(W − u) + rh) + R (W )(rσY ) − (r + λ)R(W ) ,
2
which must be zero because Ht is a martingale.
R∞
Rθ
For the wage bill B, define c˜t = ct∧τ and H = 0 re−rs (1s<θ c˜s + 1s≥θ c˜θ )ds = 0 re−rs c˜s ds +
23
e−rθ c˜θ . Define the martingale as
Ht = Et [H]
Z θ∧t
Z θ
re−rs c˜s ds + 1θ≥t
re−rs c˜s ds + Et [1θ<t e−rθ c˜θ + 1θ≥t e−rθ c˜θ ]
= Et
0
t
Z θ
Z θ∧t
re−rs c˜s ds + e−rθ c˜θ
re−rs c˜s ds + 1θ<t e−rθ c˜θ + Et 1θ≥t
= Et
t
0
Z θ
Z t
−rs
−rθ
−rs
−rθ
=
Et [1θ≥s ]re c˜s ds + Et [1θ<t e c˜θ ] + Et [1θ≥t ]E
re c˜s ds + e c˜θ |Ft , θ ≥ t
0
t
Z t
Z t
−(r+λ)s
λe−λs e−rs c˜s ds + e−λt e−rt B(Wt )
re
c˜s ds +
=
0
0
Z t
=
(r + λ)e−(r+λ)s c˜s ds + e−(r+λ)t B(Wt ).
0
If t < τ , then its drift is
1 00
−(r+λ)t
0
2
e
(r + λ)ct + B (W )((r + λ)(W − u) + rh) + B (W )(rσY ) − (r + λ)B(W ) = 0.
2
QED
Proof of Lemma 3
Let us use the following notation
Ha,Y,c;λ (W, F, F 0 ) :=
(1 + λr )(F − F 0 (W − u(c)) + c) − a − F 0 h(a)
.
1
2 2
2 rσ Y
By contradiction, let’s define W 1 as the smallest point at which F 0 (W 1 ) = F˜ 0 (W 1 ). Because
F (W 0 ) ≤ F˜ (W 0 ) and F 0 (W ) < F˜ 0 (W ) at all W ∈ [W 0 , W 1 ), we have F (W 1 ) < F˜ (W 1 ), and
F 00 (W 1 ) ≤ Ha,Y,c;λ (W 1 , F (W 1 ), F 0 (W 1 )) < Ha,Y,c;λ˜ (W 1 , F˜ (W 1 ), F˜ 0 (W 1 )) = F˜ 00 (W 1 ),
where (a, Y, c) are controls that attain F˜ 00 (W 1 ). The strict inequality is true because Ha,Y,c;λ (W, F, F 0 )
is strictly increasing in F and increasing in λ. This implies F 0 (W 1 − ε) > F˜ 0 (W 1 − ε) for a
sufficiently small ε > 0, which contradicts the definition of W 1 .
QED
Proof of Proposition 4
˜ > λ, F 0 (0) = F˜ 0 (0) implies F 0 (W ) <
1. Fix F and take a candidate solution for F˜ . Because λ
F˜ 0 (W ) for all W > 0, which shows that the solution F˜ never returns to F0 so it is not a feasible
24
candidate for an optimal contract. Lemma 3 now implies that F 0 (0) > F˜ 0 (0) at the optimal
solution F˜ . Thus, F (ε) > F˜ (ε) for all sufficiently small ε > 0.
2. We show that F and F˜ must meet on (0, Wgp ], i.e., it cannot be that F (W ) > F˜ (W ) for all
(0, Wgp ]. Indeed, we’d have F0 (Wgp ) = F (Wgp ) > F˜ (Wgp ), which contradicts F0 ≤ F˜ .
ˆ ≤ Wgp be the smallest W > 0 where F and F˜ have the same value. We show that
3. Let W
˜ gp ≤ W
ˆ . If not, then Lemma 3 implies that F˜ never returns to F0 , so W
˜ gp does not exist,
W
which is a contradiction.
˜ gp = W
ˆ = Wgp and W
˜ gp < W
ˆ < Wgp . We show that the first
4. There are two possibilities: W
˜ gp = Wgp the value matching and smooth pasting conditions
is not the case. Indeed, with W
F (Wgp ) = F˜ (Wgp ) = F0 (Wgp ),
F 0 (Wgp ) = F˜ 0 (Wgp ) = F00 (Wgp )
imply
˜ gp ) = F0 (Wgp ) − max F00 (Wgp )(Wgp − W 0 ) + F0 (W 0 ) = 0.
S(Wgp ) = S(W
0
W ≥0
˜ gp ) = 0 into the HJB equation and applying an Envelope Theorem,
Plugging S(Wgp ) = S(W
we obtain F 00 (Wgp ) = F˜ 00 (Wgp ) and hence a(Wgp ) = a
˜(Wgp ) and F 000 (Wgp ) = F˜ 000 (Wgp ). Repeating the same argument after differentiating the first-order condition for a at Wgp , we have
˜ 0 (Wgp ) /dW < 1. Thus there exists ε > 0
F (4) (Wgp ) < F˜ (4) (Wgp ), since dW 0 (Wgp ) /dW = dW
such that F 000 (W ) > F˜ 000 (W ), F 00 (W ) < F˜ 00 (W ) and F 0 (W ) > F˜ 0 (W ) for all W ∈ [Wgp −ε, Wgp ),
which contradicts the fact that F˜ 0 (W ) is cutting F 0 (W ) from above at W = Wgp . Thus, we
˜ gp < W
ˆ < Wgp .
must have W
QED
Proof of Lemma 4
Differentiating the HJB (8) wrt λ, we have
∂F (W )
∂λ
=
−r
1 00
r
∂F (W ) 0
0
2 2
h(a) +
a + F (W )h(a) + F (W )rσ Y
+
2
r+λ
∂λ
(r + λ)2
r 1 ∂F (W ) 00 2 2
∂F (W ) 0
rσ Y +
(W − u(c))
r+λ2
∂λ
∂λ
25
∂F (W )
∂λ
Denoting
G(W ) =
=
as G(W ), we have a second-order differential equation
r
1 00
r
−1
0
0
2 2
a + F (W )h(a) + F (W )rσ Y
+ G (W ) W − u(c) +
h(a) +
r+λr+λ
2
r+λ
r 1 00
G (W )rσ 2 Y 2
r+λ2
−1
r
r 1 00
0
S(W ) + G (W ) W − u(c) +
h(a) +
G (W )rσ 2 Y 2 ,
r+λ
r+λ
r+λ2
or
1
(r + λ)G(W ) = −S(W ) + G0 (W ) ((r + λ)(W − u(c)) + rh(a)) + G00 (W )r2 σ 2 Y 2 ,
2
with boundary conditions G(0) = G(Wgp ) = 0, where S(W ) =
is a known function, and where W follows
(13)
r
1 00
0
2 2
r+λ (a+F (W )h(a)+ 2 F (W )rσ Y )
dWt = ((r + λ)(Wt − u(c)) + rh(a))dt + rY σdZt .
As before, we denote drift of Wt by µ and its volatility by ν.
The derivation of the equality in (11) follows DeMarzo and Sannikov (2006). Let
Z t
Ht := −
e−(r+λ)s S(Ws )ds + e−(r+λ)t G(Wt ).
0
We have
dHt = −e−(r+λ)t S(Wt )dt − (r + λ)e−(r+λ)t G(Wt )dt + e−(r+λ)t dG(Wt ),
and thus, using Ito’s lemma,
e
(r+λ)t
1
2 00
= −S(Wt )dt − (r + λ)G(Wt )dt + G (Wt )µ(Wt ) + ν(Wt ) G (Wt ) dt +
2
0
G (Wt )ν(Wt )dZt .
dHt
0
The dt terms sum up to zero by (13), and E [Ht ] is bounded, i.e., Ht is a martingale. Thus,
Z τ
−(r+λ)t
−(r+λ)τ
G(W0 ) = H0 = E [Hτ ] = E −
e
S(Wt )dt + e
G(Wτ ) ,
0
which, with the boundary conditions G(Wτ ) =
∂F (W )
∂λ W =0
=
∂F (W )
∂λ W =W
= 0, which gives
gp
us the equality in (11).
∂F (W
)
gp
To verify the boundary conditions at exit time, consider first
, where Wgp also depends
∂λ
on λ. Differentiating the value-matching condition F (Wgp ) = F0 (Wgp ) totally with respect to
λ, we have
∂F (Wgp )
∂Wgp
∂Wgp
+ F 0 (Wgp )
= F00 (Wgp )
,
∂λ
∂λ
∂λ
26
so,
∂F (Wgp )
∂Wgp
= (−F 0 (Wgp ) + F00 (Wgp ))
= 0,
∂λ
∂λ
where the second inequality uses the smooth-pasting condition. The other boundary condition
is easy to verify because F (0) = 0 at all λ, so obviously ∂F∂λ(0) = 0.
The strict inequality in (11), i.e., G(W0 ) < 0, follows from S > 0 everywhere in (0, Wgp ).
QED
Proof of Proposition 5
It is equivalent to prove that there exists a unique W s ∈ [0, Wgp ] such that F 0 (W ) ≥ F˜ 0 (W ) if
and only if W ≤ W s .
˜ gp ≤ Wnj . We want to show that there does not exist W s < W
˜ gp that solves
0. Suppose W
˜ gp < Wnj implies F˜ 0 (W
˜ gp ) =
F 0 (W s ) = F˜ 0 (W s ). Since F 0 (0) > F˜ 0 (0), the fact that W
˜ gp ) < F 0 (W
˜ gp ) following Proposition 2. So, generically, either there exists an even number
F00 (W
of W s solving F 0 (W s ) = F˜ 0 (W s ), or the solution does not exist.
˜ gp ,
1. Suppose there are at least two solutions, denoted as W 1 and W 2 , where W 1 < W 2 < W
such that F 0 (W j ) = F˜ 0 (W j ) for j = 1, 2. The fact that the signs of second derivative alternate,
i.e., F 00 (W 1 ) < F˜ 00 (W 1 ) and F 00 (W 2 ) > F˜ 00 (W 2 ) implies
λ
)S(W 1 ) < (1 +
r
λ
(1 + )S(W 2 ) > (1 +
r
(1 +
˜
λ
˜ 1 ),
)S(W
r
˜
λ
˜ 2 ).
)S(W
r
Notice that by construction we have
F 0 (W ) < F˜ 0 (W ) for W ∈ W 1 , W 2 .
And there exist W 1.5 such that W 1 < W 1.5 < W 2 and
F 00 (W 1.5 ) = F˜ 00 (W 1.5 )
F 00 (W ) < F˜ 00 (W ) for W ∈ [W 1 , W 1.5 ),
F 00 (W ) > F˜ 00 (W ) for W ∈ W 1.5 , W 2 .
The goal is to contradict the equality.
27
(14)
2. For any W ∈ W 1.5 , W 2 , the fact that F 0 (W ) < F˜ 0 (W ) and W < Wgp < Wnj implies
˜ 0 (W ) < W 0 (W ) < W . The derivative of S (W ) is given by
W
S 0 (W ) = −F 00 (W )(W − W 0 (W )),
˜ −W
˜ 0 (W )),
< −F˜ 00 (W )(W
= S˜0 (W ) .
˜
Then the premise that (1 + λr )S W 2 > (1 + λr )S˜ W 2 implies
!
Z W2
˜
λ
λ
S 0 (W ) dW > (1 + )S˜ W 2 ,
⇒ (1 + ) S W 1.5 +
r
r
W 1.5
Z
2
W
˜
˜
λ
λ
λ
S˜0 (W ) dW > (1 + )S˜ W 2 ,
⇒ (1 + )S W 1.5 + (1 + )
r
r W 1.5
r
˜
λ
λ
⇒ (1 + )S W 1.5 > (1 + )S˜ W 1.5 ,
r
r
˜ gp .
which contradicts (14). In this case Proposition 5 holds by setting W s = W
˜ gp ∈ (Wnj , Wgp ). From
3. [Existence of W s and odd number of crossing points.] Suppose W
˜ gp ) = F˜0 (W
˜ gp ) > F 0 (W
˜ gp ),
Proposition 4, we have that F˜ 0 (0) < F 0 (0). We also have F˜ 0 (W
where the equality is the smooth-pasting condition and the inequality follows from Proposition
˜ gp ∈ (Wnj , Wgp ). Thus, F˜ 0 and F 0 cross on (0, W
˜ gp ), and the number of crossing
2 because W
˜ gp ), i.e., solutions to F˜ 0 (W s ) = F 0 (W s ), is odd.
points W s ∈ (0, W
4. Suppose there are at least three solutions, denoted as W 1 , W 2 , and W 3 , where W 1 < W 2 <
W 3 , such that F 0 (W j ) = F˜ 0 (W j ) for j = 1, 2, 3. The fact that the signs of second derivative
alternate, i.e. F 00 (W 1 ) < F˜ 00 (W 1 ), F 00 (W 2 ) > F˜ 00 (W 2 ) and F 00 (W 3 ) < F˜ 00 (W 3 ) implies
λ
)S W 1 < (1 +
r
λ
(1 + )S W 2 > (1 +
r
λ
(1 + )S W 3 < (1 +
r
(1 +
˜
λ
)S˜ W 1 ,
r
˜
λ
)S˜ W 2 ,
r
˜
λ
)S˜ W 3 .
r
Notice that by construction we have
F 0 (W ) < F˜ 0 (W ) for W ∈ W 1 , W 2 ,
F 0 (W ) > F˜ 0 (W ) for W ∈ W 2 , W 3 .
28
And there exist W a , W b and W c such that W 1 < W a < W 2 < W b < W c < W 3 and
F 00 (W a ) = F˜ 00 (W a )
00
b
˜ 00
(15)
b
F (W ) = F (W )
00
˜ 00
(16)
a
b
F (W ) > F (W ) for W ∈ (W , W ),
F 00 (W ) < F˜ 00 (W ) for W ∈ (W b , W c ).
The goal is to contradict the two equalities.
5. Consider two cases: W 2 < Wnj and W 2 ≥ Wnj . In the case of W 2 < Wnj , for any
˜ 0 (W ) < W 0 (W ); the fact that
W ∈ W 1 , W 2 , the fact that F 0 (W ) < F˜ 0 (W ) implies W
W a < W < W 2 < Wnj implies W 0 (W ) < W and F 00 (W ) > F˜ 00 (W ). The derivative of S (W )
is given by
0 < S 0 (W ) = −F 00 (W ) W − W 0 (W ) ,
< −F˜ 00 (W ) W − W 0 (W ) ,
˜ 0 (W ) ,
< −F˜ 00 (W ) W − W
= S˜0 (W ) .
˜
Then the premise that (1 + λr )S W 2 > (1 + λr )S˜ W 2 implies
!
Z W2
˜
λ
λ
S 0 (W ) dW > (1 + )S˜ W 2 ,
(1 + ) S (W a ) +
r
r
Wa
Z
2
W
˜
˜
λ
λ
λ
⇒ (1 + )S (W a ) + (1 + )
S˜0 (W ) dW > (1 + )S˜ (W2 ) ,
r
r Wa
r
˜
λ
λ
⇒ (1 + )S (W a ) > (1 + )S˜ (W a ) ,
r
r
˜
(1 + λr )S˜ (W a ) − a − F˜ 0 (W a )h(a)
(1 + λr )S (W a ) − a − F 0 (W a )h(a)
>
,
⇒
1
1
2 0
2
2 0
2
2 rσ (h (a))
2 rσ (h (a))
which contradicts (15).
6. In the second case W 2 ≥ Wnj , for any W ∈ W 2 , W 3 , the fact that F 0 (W ) > F˜ 0 (W ) implies
˜ 0 (W ) > W 0 (W ) > W . Then the premise that F 00 W b = F˜ 00 W b implies S 0 W b >
W
S˜0 W b , thus there exists ε > 0 such that W b + ε ≤ W c and 0 > S 0 (W ) > S˜0 (W ) for all
W ∈ [W b , W b + ε). The fact that F 00 W b = F˜ 00 W b implies
˜
(1 + λr )S W b − a − F 0 (W b )h(a)
(1 + λr )S˜ W b − a − F˜ 0 (W b )h(a)
=
,
1
1
2 0
2
2 0
2
2 rσ (h (a))
2 rσ (h (a))
˜
(1 + λr )S W b + ε − a − h(a)F 0 (W b + ε)
(1 + λr )S˜ W b + ε − a − h(a)F˜ 0 (W b + ε)
⇒
>
,
1
1
2 0
2
2 0
2
2 rσ (h (a))
2 rσ (h (a))
⇒ F 00 (W b + ε) > F˜ 00 (W b + ε),
29
which contradicts the premise that F 00 (W ) < F˜ 00 (W ) for W ∈ W b , W c .
QED
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