Simple Optimal Contracts with a Risk-Taking
Agent⇤
Daniel Barron, George Georgiadis, and Jeroen Swinkels†
April 12, 2017
Abstract
Consider an agent who can costlessly add mean-preserving noise to
his output. Then, the principal can do no better than offer weakly
concave incentives to deter risk-taking. If the agent is risk-neutral and
protected by limited liability, optimal incentives are strikingly simple:
linear contracts maximize profit. If the agent is risk averse, we characterize the unique profit-maximizing contract and provide conditions
under which it takes an intuitive form. We extend our model to analyze costly risk-taking, and we show that the model can be reinterpreted
as a dynamic setting in which the agent can manipulate the timing of
output.
We are grateful to Simon Board, Ralph Boleslavsky, Luis Cabral, Gabriel Carroll, Hugo
Hopenhayn, Johannes Hörner, Eddie Lazear, Erik Madsen, Niko Matouschek, Meg Meyer,
Konstantin Milbradt, Dimitri Papanikolaou, Andrea Prat, Lars Stole, Doron Ravid, Luis
Rayo, Balazs Szentes, Dimitri Vayanos, Glen Weyl, and seminar participants at Cambridge
University, New York University, Northwestern University, University College London, the
University of Miami, the University of Exeter, the University of Rochester, the University
of Warwick, CRETE 2016, the 2016 ASU Theory Conference, and the 2016 Transatlantic
Theory Conference for helpful comments and discussions.
†
Kellogg School of Management, Northwestern University, Evanston, IL 60208, U.S.A.
D. Barron: [email protected];
G. Georgiadis: [email protected];
J. Swinkels: [email protected]
⇤
1
1
Introduction
Organizations offer incentive schemes to motivate their employees, suppliers,
and partners to exert effort. However, in many settings, an agent can game
these contracts by taking on risks that increase his expected compensation
but do not benefit the principal. For example, portfolio managers can choose
riskier investments as well as influence their average returns; executives and
entrepreneurs control both the expected profitability of their projects and the
distribution over possible outcomes; and salespeople can both invest to increase
demand and adjust the timing of the resulting sales.1 If the principal fails to
take these opportunities to game the system into account, then she might offer
a contract that encourages risk-taking rather than productive effort.
This paper explores how a principal can effectively motivate an agent who
can engage in risk-taking to game his performance contract. In our setting, a
principal offers a contract to a potentially liquidity-constrained agent. If the
agent accepts this contract, then he exerts costly effort that produces a noncontractible intermediate output. The agent privately observes this output
and can then manipulate it by costlessly adding mean-preserving noise to it,
which in turn determines the final, contractible output.
Echoing Jensen and Meckling (1976) and others, we argue that the agent’s
ability to engage in risk-taking fundamentally constrains the principal’s ability
to offer effective incentives. In Section 3, we show that the agent optimally
takes on additional risk whenever the intermediate output lies in a region where
his utility is convex in output, effectively concavifying his utility. If the principal and agent are both weakly risk-averse, then the principal finds it optimal to
deter such gaming entirely by offering an incentive scheme that directly makes
the agent’s utility concave in output. Motivated by this logic, we consider how
the principal can optimally motivate the agent if she is constrained to offer
concave incentives.
1
See Brown, Harlow and Starks (1996), Chevalier and Ellison (1997), and de Figueiredo,
Rawley and Shelef (2014) on portfolio managers; Matta and Beamish (2008) and Repenning and Henderson (2010) on executives; Vereshchagina and Hopenhayn (2009) on entrepreneurs; and Oyer (1998) and Larkin (2014) on sales.
2
Suppose that the agent is risk-neutral and that higher intermediate output
indicates higher effort in the sense of an increasing likelihood ratio. Section
4 proves that the optimal contract is linear. Relative to a strictly concave
contract, a linear contract motivates the agent at lower cost because it concentrates pay on higher outcomes, which are more indicative of effort. We
show that a linear contract remains optimal regardless of the principal’s attitude towards risk, and is uniquely optimal if the principal is risk averse.
This linear optimal contract is tractable and highlights new and interesting
economic forces. In particular, we show that as the worst output over which
the agent can gamble becomes arbitrarily bad, the optimal contract for any
given positive effort gives the agent an arbitrarily large rent and so becomes
arbitrarily costly for the principal. Consequently, optimal effort tends to zero.
Section 5 considers optimal incentives if the agent is risk-averse, restricting
attention for simplicity to a risk-neutral principal. Our central result characterizes the unique profit-maximizing contract, and shows how it reflects the
fundamental trade-off between insurance and incentives. As in Mirrlees (1976)
and Holmström (1979), this contract balances the marginal cost to the principal of giving the agent higher utility at a given output to the marginal benefit
of relaxing the participation and incentive constraints. Absent the concavity
constraint, the optimal contract would set the marginal cost and benefit equals
at every output, leading to the classic Holmström-Mirrlees characterization.
However, this contract might not be concave.
We give a simple set of perturbations that respect concavity and can be
used to fully characterize the profit-maximizing contract. If this contract is locally strictly concave at a point, then we can freely perturb it at nearby points
without violating concavity. Therefore, the contract must set marginal benefits
and costs equal at such points and hence has exactly the Holmström-Mirrlees
form. However, local perturbations might violate concavity on a linear segment of the contract. In principle, there are many perturbations that respect
concavity on linear segments. Perhaps surprisingly, however, we show that it
is both necessary and sufficient to consider the two perturbations that change
either the level or the slope of the agent’s utility over that linear segment.
3
The expected marginal benefit of each of these perturbations must equal the
expected marginal cost. Hence, a natural generalization of the fundamental
economics identified by Holmström and Mirrlees also characterizes the contract
in our setting.
This characterization puts considerable structure on optimal incentives.
For example, if the participation constraint does not bind, then the optimal
contract is linear wherever the likelihood ratio is negative. We also show that
the contract will have at most one linear segment (with this segment beginning
at the lowest possible output) under mild conditions on the curvature of the
likelihood ratio and utility functions. And we find sufficient conditions under
which the optimal contract makes the agent’s utility linear for all output,
as well as conditions under which it coincides exactly with the HolmstromMirrlees contract.
Finally, Section 6 considers three extensions, all of which assume that both
principal and agent are risk-neutral. First, we analyze a variant of our model
in which the agent must choose his risk-taking distribution before he observes
the intermediate output. For example, an entrepreneur might be able to adjust
the riskiness of a project only before she learns whether or not it will bear fruit.
We extend our tools to this setting and give mild conditions under which linear
contracts remain optimal. Second, we consider optimal contracts if the agent
incurs a cost to engage in risk-taking that is increasing in the variance of that
risk. We extend our basic intuition to this case and show that the unique
optimal contract is convex and converges to a linear contract as the cost of
taking on risk converges to zero.
Our third extension reinterprets our model as a dynamic setting in which
the principal offers a stationary contract that the agent can game by shifting
output over time. Oyer (1998) and Larkin (2014) empirically document how
convex incentive schemes and long sales cycles can encourage such gaming.
We assume that the agent’s effort generates a stochastic output, but that he
can costlessly manipulate when that output is realized over an interval of time.
We show that this model is equivalent to our baseline setting; in particular, a
linear contract is optimal, since a convex contract would induce the agent to
4
bunch sales over a short time interval, while a strictly concave contract would
provide subpar effort incentives.
By identifying how optimal contracts deter the agent from gaming, our
analysis sheds light on the kinds of risk-taking that an improperly designed
incentive system can encourage. For example, in discussing the 2007-2008 financial crisis, Federal Reserve Chairman at the time Ben Bernanke argued that
“compensation practices at some banking organizations have led to misaligned
incentives and excessive risk-taking, contributing to bank losses and financial
instability” (Federal Reserve Press Release (10/22/2009)). See also Rajan
(2011) for an overview, and Garicano and Rayo (2016) for a case study of the
American International Group (AIG). In our model, if compensation packages
do not deter these kinds of gaming, then managers or employees exert too
little effort and take on too much risk.
Our analysis is inspired in part by Diamond (1998), which uses several
examples to argue that linear contracts are approximately optimal if the agent
has sufficient control over the distribution of output, and in particular are
exactly optimal if the agent can choose any distribution over output such
that expected output equals effort. Relative to that paper, our model puts
different constraints on the agent’s risk-taking by assuming that he can add
mean-preserving noise to an exogenous distribution, but cannot eliminate all
randomness in output. We cleanly characterize how risk-taking constrains
incentives in a flexible but tractable framework, which allows us to analyze
optimal contracts for more general risk preferences, output distributions, and
effort functions. Garicano and Rayo (2016) similarly builds on Diamond
(1998) but fixes an exogenous (convex) contract to focus on the social costs of
agent risk-taking. In a contemporaneous paper, Garicano, Matouschek, and
Rayo (2016) focuses on the social (rather than incentive) costs of risk-taking
in the context of a regulator who attempts to deter a firm from taking socially
destructive risks.
Palomino and Prat (2003) considers a delegated portfolio management
problem in which the agent chooses both expected output and the riskiness
of the portfolio (in the sense of second-order stochastic dominance) from a
5
parametric family of distributions. The resulting optimal contract consists of
a base salary and a fixed bonus that is paid whenever output exceeds a threshold. The threshold for the bonus is determined by features of the parametric
family. In contrast, our agent can choose any mean-preserving spread, which
means that our optimal contract must deter a more flexible form of gaming.
Hébert (2015) finds conditions under which, if an agent can manipulate the
output distribution at a cost that depends on the difference between that distribution and some exogenous baseline, then the optimal contract resembles a
debt contract. Demarzo, Livdan and Tchistyi (2014) characterize the optimal
contract if the agent can take on socially inefficient risk in a dynamic setting.
They argue that backloading can mitigate, but not necessarily eliminate, the
agent’s incentive to take such risks. Makarov and Plantin (2015) considers a
model of career concerns in which the agent can take excessive risk to temporarily manipulate the principal’s beliefs about her ability, and characterizes
a backloaded incentive scheme that eliminates these incentives. Attar, Mariotti, and Salanie (2011) argues that linear contracts are optimal in a market
with adverse selection if the seller can privately sign contracts for additional
products.
More broadly, our work is related to a long-standing literature which argues
that real-world contracts are simple in order to deter gaming. Holmström and
Milgrom (1987) displays a dynamic environment in which linear contracts are
optimal but notes that the point that linear contracts are robust to gaming
“is not made as effectively as we would like by our model; we suspect that it
cannot be made effectively in any traditional Bayesian model.” Recent papers,
including Chassang (2013), Carroll (2015), and Antic (2016), take up this
argument by departing from a Bayesian framework and proving that simple
contracts perform well under min-max or other non-Bayesian preferences. In
contrast, our paper justifies simple contracts in a setting that lies firmly within
the Bayesian tradition.
While the solution concept is quite different, Carroll’s intuition is related
to ours. In that paper, Nature selects a set of actions available to the agent in
order to minimize the principal’s expected payoffs. The key difference is in the
6
types of gambles available to the agent. In Carroll’s paper, the agent might
take on additional risk in order to game a convex incentive scheme, in which
case risk-taking behavior is similar to our paper. However, if the principal
offers a concave incentive scheme, then Nature might allow the agent to choose
a distribution with less risk. This second type of gaming is fundamentally
different from our setting, which assumes that the agent can only add risk. This
difference is most striking if the agent is risk-averse, in which case Carroll’s
optimal contract makes the agent’s utility linear in output, while ours might
make utility strictly concave.
2
Model
We consider a static game between a principal (P, “she”) and an agent (A,
“he”). The agent has limited liability, so he cannot pay more than M 2 R
to the principal. Let [y, ȳ] ⌘ Y ✓ R be the set of contractible outputs. The
timing is as follows:
1. The principal offers an upper semicontinuous contract s(y) : Y ! [ M, 1).2
2. The agent accepts or rejects the contract. If he rejects, the game ends,
he receives u0 , and the principal receives 0.
3. If the agent accepts, he chooses effort a
0.
4. Intermediate output x is realized according to F (·|a) 2 (Y), where (i)
F is infinitely differentiable with density f , (ii) F has full support for
(·|a)
all a 0, (iii) f is strictly MLRP-increasing in a with ffa(·|a)
uniformly
bounded for all a, and (iv) EF (·|a) [x] = a.
5. The agent chooses a distribution Gx 2
EGx [y] = x.
2
(Y) subject to the constraint
One can show that the restriction to upper semicontinuous contracts is without loss:
if the agent has an optimal action given a contract s(·), then there exists an upper semicontinuous contract that induces the same equilibrium payoffs and distribution over final
output.
7
6. Final output y is realized according to Gx , and the agent is paid s(y).
The principal’s and agent’s payoffs are equal to ⇡ (y s(y)) and u (s (y)) c(a),
respectively. We assume that ⇡(·) and u(·) are strictly increasing and weakly
concave, with u(·) onto. We also assume c(·) is infinitely differentiable, strictly
increasing, and strictly convex.
Let G = {G : Y ! (Y) | EGx [y] = x for all x 2 Y}. We can treat the
agent as choosing a and G 2 G simultaneously, since the agent chooses each
Gx to maximize his ex-ante expected payoff.
After the agent observes x but before y is realized, this is a problem with
both a hidden type and a hidden action. In such problems, it is often useful
to ask the agent to report his type, in this case x. By punishing differences
between this report and y, the principal may be able to dissuade some or all
gambling. We restrict attention to situations where such intermediate reports
are not useful. The simplest way to do so is to assume that the timing of x is
random, and gambling is instantaneous.3
3
Risk-taking and optimal incentives
This section explores how the agent’s ability to engage in risk-taking constrains
the contracts offered by the principal.
We find it convenient to rewrite the principal’s problem in terms of the
utility v(y) ⌘ u(s(y)) that the agent receives for each final output y. If we
define u ⌘ u( M ), then an optimal contract solves the following constrained
3
With a risk-neutral principal and agent, and a non-binding participation constraint,
Appendix E shows that a linear contract remains optimal even if the principal can demand
a report after x but before y is realized.
8
maximization problem:
max
a, G2G,v(·)
⇥
⇥
EF (·|a) EGx ⇡ y
u
1
(v (y))
⇤⇤
⇥
⇤
s.t. a, G 2 arg max EF (·|ã) EG̃x [v (y)]
(ObjF )
c(ã)
ã, G̃2G
EF (·|a) [EGx [v (y)]]
v (y)
c(a)
u0
u for all y.
(ICF )
(IRF )
(LLF )
The main result of this section is Lemma 1, which characterizes how the
threat of gaming affects the incentive schemes v(·) that the principal can offer.
The principal optimally offers a contract that deters risk-taking entirely, but
doing so constrains her to incentive schemes that are weakly concave.
Lemma 1. Suppose v(·) satisfies (ICF )-(LLF ) for some a
0 and G 2 G.
Then there exists a weakly concave v̂(·) that satisfies (ICF )-(LLF ) for that a
and degenerate G and gives the principal a weakly higher expected payoff.
The proof is in Appendix A. The intuition is simple. For an arbitrary incentive
scheme v(·), define v c (·) : Y ! R as its concave closure,
v c (x) =
sup
w,z2Y,p2[0,1] s.t. (1 p)w+pz=x
{(1
p)v(w) + pv(z)} .
(1)
At any outcome x such that the agent does not earn v c (x), he can engage in
risk-taking to earn that amount in expectation but no more. But then the
principal can do at least as well by directly offering a concave contract. Note
that if either the agent or the principal is strictly risk-averse, then offering a
concave contract is strictly more efficient than inducing risk-taking.
Given Lemma 1, we henceforth restrict attention to contracts that make the
agent’s utility concave in output, with the caveat that our resulting solution
is one of many if (but only if) both the principal and agent are risk-neutral.
If v(·) is constrained to be concave, then for every x, the agent will optimally
choose a Gx that is degenerate at x, so y = x. Therefore, an optimal incentive
9
scheme solves the simplified problem:
⇥
max EF (·|a) ⇡ y
a,v(·)
u 1 (v(y))
⇤
s.t. a 2 arg max EF (·|ã) [v(y)]
ã
EF (·|a) [v(y)]
v(y)
c(a)
u for all y 2 Y
v(·) weakly concave.
u0
(Obj)
c(ã)
(IC)
(IR)
(LL)
(Conc)
For a fixed effort a 0, we say that v(·) implements a if it satisfies (IC)-(Conc)
for a, and does so at maximum profit if it also maximizes (Obj). An optimal
v(·) implements the optimal effort level a⇤ 0 at maximum profit.
Mathematically, the set of concave contracts is well-behaved. Consequently,
we can show that for any a 0, a contract that implements a at maximum
profit exists, and is unique if either ⇡(·) or u(·) is strictly concave.
Lemma 2. Fix a 0 and suppose that u > 1. Then there exists a contract
that implements a at maximum profit, and does so uniquely if either ⇡(·) or
u(·) is strictly concave.
This result, which follows from the Theorem of the Maximum, is an implication of Proposition 5 in Appendix D.4 The profit-maximizing contract is
unique if at least one player is strictly risk-averse because Jensen’s Inequality
implies that a convex combination of two different contracts that implement
a also implements a and gives the principal a strictly higher payoff.
Roadmap of our Main Results
The next two sections build on Lemma 1 to characterize optimal incentive
schemes. These sections are designed to be modular so that the interested
reader can focus on the results that most interest them. We offer here a
roadmap that describes how these analyses connect to one another.
4
Appendix D may be found online at https://sites.google.com/site/danielbarronecon/
10
Figure 1: Roadmap
Consider Figure 1. Both principal and agent are risk-neutral at the origin. Moving right makes the agent more risk averse, while moving upwards or
downwards makes the principal more risk-averse or more risk-seeking, respectively.
Section 4 explores contracting with a risk-neutral agent, which corresponds
to the vertical axis on Figure 1. Regardless of the principal’s risk preferences, a
linear contract is optimal in this setting, and it is uniquely so if the principal is
strictly risk-averse (above the origin). Section 5 considers a risk-neutral principal and a risk-averse agent (the horizontal axis in Figure 1). Here, we develop
necessary and sufficient conditions that characterize the unique contract that
implements a > 0 at maximum profit. In some situations, this contract makes
the agent’s utility either partially or fully linear in output; for instance, if (IR)
is slack (so that (LL) binds), then we show that the agent’s payoff under the
profit-maximizing contract is linear at any y such that F (y|a) has a negative
likelihood ratio.
So long as at least one player is strictly risk-averse (the upper quadrant
excluding the origin in Figure 1), the Theorem of the Maximum implies that
11
the (unique) contract that implements a at maximum profit is continuous in
the topology of almost everywhere pointwise convergence. See Proposition
5 in Appendix D for a proof. Therefore, the characterizations in Sections
4 and 5 shed light on profit-maximizing incentives in the rest of this quadrant. In particular, if the principal is risk-averse, the agent is approximately
risk-neutral, and effort is fixed at the optimal effort level for a risk-neutral
agent, then the profit-maximizing contract is approximately linear. Similarly,
if the agent is risk-averse and the principal is approximately risk-neutral, then
profit-maximizing incentives approximate the characterization in Section 5.
Continuity does not extend to the lower quadrant, so we should be cautious
about applying our intuition if the principal is risk-seeking and the agent is
risk-averse.
4
Optimal Contracts for a Risk-Neutral Agent
Suppose the agent is risk-neutral, so u(y) = y, v(·) = s(·), and u =
any effort level a, define
sLa (y) = c0 (a)(y
y)
M . For
w,
where w = min M, c0 (a)(a y) c(a) u0 . Intuitively, sLa (y) is the least
costly linear contract that implements a. Note that sLa (·) depends only on
the marginal cost of effort, the limited liability constraint, and the agent’s
outside option.5 Define the first-best effort aF B 2 R+ as the unique effort that
maximizes y c(y) and so satisfies c0 aF B = 1.
The following proposition proves that a linear contract is optimal if the
agent is risk-neutral.
Proposition 1. Let u(s) ⌘ s. If a⇤ is optimal, then a⇤  aF B and sLa⇤ (·) is
optimal.
The proofs for all results in this section can be found in Appendix A.
5
Without the assumption EF (·|a) [x] = a, sL
a (·) would depend on EF (·|a) [x] as well. However, it would not depend on any other features of F .
12
For the moment, suppose the principal is risk neutral, so that ⇡(y) ⌘ y,
and assume u0 + c(a) < M so (IR) never binds. The distribution over
intermediate output F (·|a) satisfies MLRP, so if we ignored the requirement
that s(·) be concave, then an approximately optimal incentive scheme would
pay large rewards for outcomes near ȳ and M otherwise.6 However, such a
contract is convex and hence susceptible to gaming.
Suppose a⇤ is the optimal effort level, and let s̃(·) be a strictly concave
contract that implements a⇤ . Consider the unique linear contract sL (·) that
starts at s̃(y) and gives the agent the same expected payoff as s̃(·). Then sL (·)
must cross s̃(·) exactly once from below. If the slope of sL (·) is larger than
1, then the proof of Proposition 1 shows that the principal strictly prefers to
induce first-best effort with the linear contract sLaF B (·) rather than offer s̃(·).
If sL (·) has a slope less than 1, then it assigns larger rewards to high output
and smaller rewards to low output than s̃(·), and hence motivates the agent to
exert strictly more effort than a⇤ . Consequently, we prove that sL (·) leads to
a higher payoff for the principal. So a linear contract is at least as profitable
as s̃(·) and hence sLa⇤ (·) is optimal.
If ⇡(·) is strictly concave, then the principal’s marginal utility of money
is strictly decreasing in y s(y). Therefore, so long as the slope of sL (·) is
no larger than 1, the principal strictly prefers it to s̃(·) holding effort fixed,
since y sL (y) is greater than y s̃(y) where the principal’s marginal utility
of money is low, and smaller than y s̃(y) where her marginal utility is high.
So the argument above holds a fortiori and a linear contract continues to be
optimal.
Note that Lemma 2 implies that sLa⇤ (·) is uniquely optimal if the principal
is risk-averse. If she is risk-neutral, then sLa⇤ (·) is optimal but not uniquely so;
in particular, any contract with a concave closure equal to sLa⇤ (·) would result
in identical expected payoffs.
In some applied settings, the principal might have risk-seeking preferences
over output, for instance because she also faces convex incentives. For instance,
6
Without (Conc), no optimal contract would exist, since the principal would want to
make the payments associated with high outputs arbitrarily large.
13
Rajan (2011) notes that investment funds and banks were motivated to take
on significant amounts of risk in the years leading up to the 2007 financial
crisis, while Chevalier and Ellison (1997) argue that mutual funds earn disproportionate profits from outperforming their competitors, which encourages
laggards to take on excess risk.
We can model such settings by allowing ⇡(·) to be any strictly increasing
and continuous function. Lemma 1 does not directly apply in this case because the principal might strictly prefer the agent to take on additional risk
following some realizations of x. Nevertheless, we can modify the argument
from Proposition 1 to show that a linear contract is optimal.
Remark 1. Let u(s) ⌘ s and ⇡(·) be an arbitrary continuous and strictly
increasing function. If a⇤ is optimal, then a⇤  aF B and sLa⇤ (·) is optimal.
As in (1), define ⇡ c (·) as the concave closure of ⇡(·). To see the proof of Remark
1, note that the principal’s expected payoff cannot exceed ⇡ c (·) for reasons similar to Lemma 1. Therefore, the contract that maximizes EF (·|a) [⇡ c (x s(x))]
subject to (IC)-(Conc) provides an upper bound on the principal’s payoff. But
Proposition 1 asserts that sLa⇤ (·) is optimal in this problem because ⇡ c (·) is concave. Given sLa⇤ (·), the agent is indifferent among distributions G 2 G, so he
is willing to choose G such that the principal’s expected payoff equals ⇡ c (·).
Our final result in this section considers how a⇤ changes with the lower
bound y on output. A decrease in y implies that the agent can take on more
severe left-tail risk by gambling over worse outcomes. We prove that a lower
y makes it costlier for the principal to induce any non-zero effort level. As
y approaches 1, inducing any positive effort becomes arbitrarily expensive
and so the agent exerts no effort in the optimal contract.
Corollary 1. Let a⇤ be the optimal effort level. Then limy!
c000 (a) 0 for all a and ⇡(y) ⌘ y, then a⇤ is increasing in y.
1
a⇤ = 0. If
To see the argument, observe that Proposition 1 implies that the principal’s
expected payment from inducing a⇤ 0 equals EF (·|a⇤ ) [⇡(y c0 (a⇤ )(y y)+w)].
As y ! 1, the limited liability constraint eventually binds at sLa⇤ (·) and so
14
w = M . But then implementing an a⇤ > 0 becomes arbitrarily costly as
y ! 1, in which case the principal is better off not motivating the agent
at all. If the principal is risk-neutral, then her payoff from inducing effort
⇥
⇤
a⇤ equals a⇤ EF (·|a⇤ ) sLa⇤ (y) ; if c000 (a)
0, then we can manipulate this
expression to perform comparative statics on a⇤ .
5
Optimal contracts if the agent is risk averse
This section characterizes the unique contract that implements a given a > 0
at maximum profit in a setting with a risk-averse agent and a risk-neutral
principal.
Let ⇡(y) = y, and define w as the infimum of the domain of u(·). We
assume that u : (w, 1) ! R satisfies limw#w u0 (w) = 1 and limw"1 u0 (w) = 0.
Our first step is to replace (IC) with the weaker condition that local incentives
are slack,
d
EF (·|a) [v (y)] c(a)
0.
(IC-FOC)
da
Given that v(·) is constrained to be concave, replacing (IC) with (IC-FOC)
entails no loss if F (·|·) satisfies weak regularity conditions.7,8 For a fixed effort
a
0, define the principal’s problem (P) as maximizing (Obj) subject to
(IC-FOC), (IR), (LL), and (Conc).
If u > 1, then Lemma 2 implies that a unique solution to (P) exists.
If u = 1, then Appendix D shows that a unique solution exists so long as
u0 (·) is not excessively convex. The characterization in this section applies to
either setting.
Define ⇢(·) as the function that maps u01(·) into u(·); that is, for every
⇣
⌘
1
w 2 (w, 1), ⇢ u0 (w) = u(w). This function is well-defined because u0 (·) and
Rz
A sufficient condition is that y Faa (y|a)dy 0 for all z 2 Y and a 0. In particular,
R
EF (·|a) [y] = a implies that Faa (y|a)dy = 0, so this condition holds if Faa (·|a) never changes
sign from negative to positive. See Jewitt (1988) and Chade and Swinkels (2016).
8
For expositional convenience, we use an indefinite integral to denote an integral from y
to y.
7
15
u(·) are strictly monotonic. Note that ⇢ 1 (v(y)) equals the marginal cost to
the principal of giving the agent extra utility at y. For a 0 and y 2 Y, define
the likelihood function
fa (y|a)
l(y|a) =
.
f (y|a)
Given the program (P), let and µ be the shadow values on (IR) and
(IC-FOC), respectively. For a fixed a 0 and an incentive scheme v(·) that
implements a, define the net cost of increasing v(·) at y as
n(y) ⌘ ⇢ 1 (v(y))
µl(y|a).
(2)
Intuitively, n(y) represents the marginal cost of increasing v(y) at y, taking into
account how that increase affects (IR) and (IC-FOC). In particular, consider
increasing v(y). Doing so increases the principal’s cost at rate ⇢ 1 (v(y))f (y|a).
It relaxes (IR) at rate f (y|a), which has implicit value , and similarly relaxes
(IC-FOC) at rate fa (y|a), which has implicit value µ. Taking the difference
between these costs and benefits and dividing by f (y|a) yields n(y).
If the principal could offer non-concave contracts, then the optimal contract
would set n(y) = 0 for all y. Indeed, this incentive scheme (with appropriate
and µ) is the Holmström-Mirrlees contract characterized in Holmström (1979).
However, this contract is not necessarily concave. Instead, we characterize the
profit-maximizing contract by identifying a set of perturbations that respect
concavity.
Given v(·), say that an interval [yL , yH ] is a linear segment if v(·) is linear
on [yL , yH ], but not on any strictly larger interval. Say that y is free if it is not
on the interior of any linear segment of v(·). Intuitively, if y is free, then v(·)
is strictly concave on at least one side of y. We can make small perturbations
to this strictly concave side without violating concavity. The points y and ȳ
are always free.
Consider the following two perturbations, formally defined in Appendix
B. Raise increases v(·) by a constant over an interval, while tilt increases the
slope of v(·) by a constant over an interval. Raising an interval typically
introduces non-concavities into v(·) at both endpoints of that interval. Tilting
16
Figure 2: Raise and tilt. These perturbations require care around yL and yH to ensure
that concavity is preserved. For this reason, we need both yL and yH to be free for raise.
For tilt up, we need yL to be free, while yH must be free for tilt down.
it a positive amount introduces a non-concavity at the lower end of the interval,
and tilting it a negative amount introduces a non-concavity at the upper end
of the interval. Appendix B shows that for small perturbations, these nonconcavities can be repaired on an arbitrarily small interval so long as the
relevant endpoints are free. Figure 2 presents simplified representations of
these perturbations.
Each of raise and tilt simultaneously affects both (IR) and (IC-FOC).
Appendix B shows that these effects are linearly independent, so we can construct a combination of the two perturbations to separately affect the agent’s
expected utility and effort incentives in any desired manner. Therefore, so
long as there exists at least one free point ŷ < ȳ such that v(ŷ) > u, we can
use these perturbations on [ŷ, ȳ] to establish candidate shadow values and µ
for (IR) and (IC-FOC).9
Consider raising v(·) on an interval between two free points yL < yH .
As we argued above, we can undo the effects of this perturbation using a
combination of raise and tilt on [ŷ, ȳ]. For v(·) to be optimal, the net cost of
9
If no such point exists, then v(·) is linear and v(y) = u.
17
this perturbation must be positive, or
Z
yH
n(y)f (y|a)dy
(3)
0.
yL
If v(yL ) > u, then we can similarly perturb v(·) on [yL , yH ] by raising it a
negative amount, which implies (3) must hold with equality.
We can make a similar argument using tilt. Suppose yL < yH with yL free.
Then v(·) is optimal only if it cannot be improved by applying positive tilt:
Z
yH
n (y) (y
yL )f (y|a) dy + (yH
yL )
yL
Z
ȳ
n(y)f (y|a)
0,
(4)
yH
where the first term represents the fact that tilt increases the slope of v(·) from
yL to yH and the second represents the resulting higher level of v(·) from yH
to ȳ. If yH is free, then applying negative tilt yields the reverse inequality:
Z
yH
n (y) (y
yL )f (y|a) dy + (yH
yL )
yL
Z
ȳ
yH
n(y)f (y|a)  0.
(5)
Our characterization combines these perturbations with the usual complementary slackness condition that = 0 if (IR) is slack (in which case v(y) = u).
Definition 1. A contract v(·) is Generalized Holmström-Mirrlees (GHM) if
(IC-FOC) holds with equality, (IR)-(Conc) are satisfied, and there exist
0
and µ > 0 such that
✓Z
v(y)f (y|a) dy
c(a)
u0
◆
= 0,
and for any yL < yH ,
1. if yL and yH are free, then (3) holds, and holds with equality if v (yL ) > u;
2. if yL is free, then (4) holds;
3. if yH is free, then (5) holds.
18
Our main result in this section characterizes the incentive scheme that
implements any a > 0 at maximum profit.
Proposition 2. Suppose u(·) is strictly concave and ⇡(y) ⌘ y. Then for any
a > 0, v(·) implements a at maximum profit if and only if it is GHM.
The proofs for all results in this section are in Appendix B. The necessity
of GHM follows from the arguments above. To establish sufficiency, we first
show that if any ṽ(·) implements a at higher profit than v(·), then there exists
a local perturbation that improves v(·). Then we show that among local perturbations, it suffices to consider tilt and raise on valid intervals. This perhaps
surprising result follows because any perturbation that respects concavity can
be approximated arbitrarily closely by a combination of valid tilts and raises.
Therefore, if any perturbation improves the principal’s profitability, then so
must some individual tilt or raise.
Proposition 2 implies that optimal incentive schemes must satisfy several
intuitive properties. For any free y 2 (y, ȳ), say y is a kink point of v(·) if two
linear segments meet at y, and a point of normal concavity otherwise.
Corollary 2. Suppose u(·) is strictly concave and ⇡(y) ⌘ y. For any a > 0,
let v(·) solve (P) and suppose y is free. Then n(y)  0, and n(y) = 0 if y is a
point of normal concavity.
The optimal contract must set the net cost equal to 0 in expectation between any two free points yL < yH with v(yL ) > u, since (3) holds with equality
between these points. If y is a point of normal concavity, then there exist two
free points that are arbitrarily close to y. Since (3) holds with equality between
these points, taking a limit as these points approach y proves that n(y) = 0.
Moreover, Lemma 3 in Appendix B proves that absent (Conc), the principal
would want to increase payments near the ends of a linear interval and decrease them somewhere in the middle of that interval. Therefore, n(y)  0 at
the right endpoint of any linear segment, which includes any kink point.
Suppose (IR) is slack, which implies that (LL) binds. Then Corollary 2
implies that the optimal contract is linear everywhere that l(·|a) is negative.
19
Corollary 3. Suppose u(·) is strictly concave and ⇡(y) ⌘ y. For any a > 0,
let v(·) solve (P), and assume that (IR) is slack, so that (LL) binds. Define
y0 by l (y0 |a) = 0. Then v is linear on [y, y0 ].
Intuitively, the agent’s incentive to exert effort is decreasing in his payment
following y < y0 , because fa (y|a) < 0. If v(·) is strictly concave for y < y0 ,
then making it “flatter” on [y, y0 ] by taking a convex combination of it with
the linear segment that connects v(y) and v(y0 ) both improves the agent’s
incentives and decreases the principal’s expected payment. This perturbation
is feasible so long as (IR) does not bind.
Finally, we identify sufficient conditions under which the optimal incentive scheme is linear, as well as conditions under which it coincides with the
Holmström-Mirrlees contract.
Corollary 4. Suppose u(·) is strictly concave and ⇡(y) ⌘ y. Fix a
0, let
v(·) solve (P), and let and µ be the corresponding shadow values on (IR)
and (IC-FOC).
1. If ⇢ ( + µl(·|a)) is convex, then v(·) is linear.
2. If ⇢ ( + µl(·|a)) is concave and (LL) does not bind, then n(·) ⌘ 0. If
(LL) binds, then v(·) has at most one linear segment on which n(·) is
not identically 0, and this segment begins at y.
Sufficient (but far from necessary) conditions for ⇢ ( + µl(·|a)) to be convex
(concave) are that ⇢(·) and l(·|a) are convex (concave).
To see this result, note that if ⇢( + µl(·|a)) is concave and the agent’s
liability constraint does not bind, then the Holmström-Mirrlees contract is
concave and hence profit-maximizing in our setting as well. If ⇢( + µl(·|a)) is
convex, then n(ŷ)  0 at any free ŷ by Corollary 2, so v(ŷ)  ⇢ ( + µl(ŷ|a)).
But ⇢( + µl(·|a)) is convex and v(·) is concave, so either v(y) < ⇢( + µl(y|a))
for all y > ŷ, or v(y) > ⇢( + µl(y|a)) for all y < ŷ. Since y, ŷ, and ȳ are all
free, either violates (3). Hence, v(·) has no interior free points and so must be
linear.
20
These corollaries show that the profit-maximizing contract is particularly
simple if (IR) does not bind, or if ⇢( + µl(·|a)) is either always convex or
always concave. However, Proposition 2 imposes substantial structure on the
optimal contract even if these conditions do not hold.
To illustrate this point, we establish mild conditions under which the profitmaximizing v(·) has no more than one linear segment, which must begin at y.
For v(·) to have two linear segments, Corollary 2 requires that ⇢( + µl(·|a))
must be at least as steep as the first linear segment but no steeper than the
second linear segment. Hence, ⇢( + µl(·|a)) must be first strictly concave
and then weakly convex on some region, a case which can be ruled out with
reasonably weak conditions. In particular, for any interval X ✓ R and any
twice continuously differentiable function h : X ! R+ , define the concavity of
00
0
h(·), con(h) = inf X 1 (hh )/(h )2 , as the largest value t for which ht /t is
concave, so that h(·) is concave if con(h) 1 and is log-concave if con(h) 0.10
Corollary 5. Assume that con (⇢0 ) + con (ly ) > 1 and that u(·) and F (·)
are analytic. Then for any a 0, the contract v(·) that solves (P) has at most
one linear segment, which must begin at y.
The condition con(⇢0 ) + con(ly ) > 1 is satisfied by many natural cases,
including if either u(w) = log w and log(ly ) is strictly concave or u(w) is HARA
and l12 is strictly concave.11
y
6
Extensions and Reinterpretations
This section considers three extensions of the baseline model, all of which
assume that both the principal and the agent are risk-neutral. Section 6.1
alters the timing so that the agent gambles before observing intermediate
output. Section 6.2 changes the agent’s utility so that he must incur a cost
that is increasing in the amount of gambling that he chooses to do. Section
6.3 reinterprets the baseline model as a dynamic setting in which, rather than
10
11
See Prekopa (1973) and Borell (1975) for details.
0
p
A utility function u(w) is HARA if uu00(w)
w.
(w) is linear, as, for example, if u(w) =
21
gambling, the agent can choose when output is realized in order to game a
stationary contract. Proofs for these extensions may be found in Appendix
C.12
6.1
Risk-Taking Before Intermediate Output is Realized
This section argues that linear contracts may be optimal even if the agent
cannot condition his risk-taking on intermediate output.
Consider the following timing:
1. The principal offers a contract s(y) : Y ! [ M, 1).
2. The agent accepts or rejects the contract. If he rejects, the game ends,
he receives u0 , and the principal receives 0.
3. The agent chooses an effort a
0 and a distribution G(·|a) 2 (Y)
subject to the constraint EG [x|a] = (a). We assume that (·) is a
smooth, increasing function such that c ( 1 (·)) is strictly convex.
4. The outcome of the gamble x ⇠ G(·) is realized, and final output is
realized according to y ⇠ F (·|x). We assume that F (·|x) has density
f (·|x) with full support, satisfies strict MLRP in x, and EF (·|x) [y] = x.
The principal and agent earn y s (y) and s (y) c(a), respectively. The
distribution G(·|a) can be interpreted as the agent mixing among distributions
F (·|x) over output, where a increases the expected value of that mixture. As an
example, suppose the agent is an entrepreneur who chooses a product to bring
to market. Effort a improves the quality of whichever product she chooses,
while G(·|a) captures the entrepreneur’s choice between a product that would
be modestly profitable regardless of economic conditions, and one that would
have more variable profitability. The distribution F (·|x) represents residual
demand uncertainty that depends on the economic conditions. Note that if
Rz
Fxx (y|x)dy 0 for all z 2 Y and x, then a riskier G(·|a) leads to a riskier
y
12
Available online at https://sites.google.com/site/danielbarronecon/
22
distribution over output: H(·|a) is increasing in G(·|a) in the sense of secondorder stochastic dominance.
Given s(·) and x, the agent’s expected payoff equals
Vs (x) ⌘
Z
(6)
s(y)f (y|x)dy.
Define Vsc (·) as the concave closure of Vs (·) as in (1). Analogous to Lemma 1,
the agent will optimally choose G such that EG(·|a) [Vs (x)] = Vsc ( (a)). Since
⇥
⇤
EG(·|a) EF (·|x) [y] = (a) for any G(·|a), the principal’s problem is
max
a ,Vs (·)
(a)
(7)
Vs ( (a))
s.t. a 2 arg max {Vs ( (ã))
ã
Vs ( (a))
c(a)
c (ã)}
u0
Vs (·) weakly concave,
with the additional restriction that Vs (·) must be the concave closure of (6)
for some s(·)
M.
We prove that a linear contract is optimal in this problem.
Proposition 3. For optimal effort a⇤
n
0 ⇤)
where w = min M, c 0(a
(a⇤ ) y
(a⇤ )
s⇤ (·) is optimal.
0
⇤
)
0, define s⇤ (y) = c 0(a
y
w,
⇤) y
(a
o
c(a⇤ ) u0 . Then a⇤  aF B and
To see the argument, relax this problem by ignoring the constraint that
Vs (·) must be the concave closure of (6). Redefine = (a). Then this relaxed
problem is identical to (Obj)-(Conc), except that Vs (·) is a function of effort
rather than intermediate output. Nevertheless, we can modify the proof of
Proposition 1 to show that a linear Vs (·) is optimal. But a linear Vs (·) can
be implemented by a linear s(·) because EF (·|x) [y] = x, so a linear contract is
optimal. Note that since both parties are risk-neutral, any contract such that
0 ⇤)
Vsc (y) = c 0(a
(y y) w is also optimal.
(a⇤ )
23
6.2
Costly Risk-Taking
Consider the model from Section 2, and suppose that the agent must pay
a private cost EGx [d(y)] d(x) to implement distribution Gx following the
realization of x, where d(·) is smooth, strictly increasing, and strictly convex,
with d(y) = 0. For example, this cost function equals the variance of Gx if
d(y) = y 2 . More generally, d(·) captures the idea that the agent must incur a
higher cost to take on more dispersed risk. The principal’s and agent’s payoffs
are y s(y) and s(y) c(a) EGx [d(y)] + d(x), respectively.13
For any contract s(·), define the agent’s “modified payment” and “modified
cost” as
ṽ(y) ⌘ s(y) d(y) and c̃(a) ⌘ c(a) EF (·|a) [d(x)],
respectively. Then the agent’s payoff equals ṽ(y) c̃(a). The principal’s payoff
equals ⇡
˜ (y) ṽ(y), where ⇡
˜ (y) ⌘ y d(y) is strictly concave. As in Section 3,
the agent optimally chooses Gx so that his expected payoff equals ṽ c (x). Since
⇡
˜ (·) is strictly concave, the principal prefers to deter risk-taking by offering
a contract that makes the agent’s payoff ṽ(·) concave. Consequently, we can
modify the proof of Proposition 1 to show that the principal’s optimal contract
makes ṽ(·) linear. Therefore, the optimal s(·) is convex and equals the sum of
a linear component and d(·).
Proposition 4. Assume c̃(·) is strictly increasing and strictly convex. For
optimal effort a⇤
0, define s̃⇤ (y) = c̃0 (a)(y y) + d(y) w̃, where w̃ =
min M, c̃0 (a)(a y) c̃(a) u0 . Then s̃⇤ (·) is optimal.
As in Proposition 1, a contract that makes ṽ(·) linear implements higher
effort than any strictly concave ṽ(·). Therefore, ṽ(·) is optimally linear, which
implies that s(·) has the desired shape. Intuitively, the profit-maximizing
contract is the most convex contract that does not induce the agent to gamble.
Importantly, the slope of ṽ(·), c̃0 (a), does not equal the slope c0 (a) of the
linear contract from Proposition 1. In fact, one can show that the principal
pays strictly less than in the optimal contract from Proposition 1, which is
13
We are very grateful to Doron Ravid for suggesting this formulation of the cost function.
24
intuitive: if the agent finds risk-taking costly, then convex contracts do not
necessarily induce gaming.
6.3
Manipulating the Timing of Output14
This section proposes a model in which the principal offers a stationary contract that the agent can game by shifting output across time, rather than by
engaging in risk-taking. We show that this setting is isomorphic to the model
in Section 4.
Consider a continuous-time game between an agent and a principal on the
time interval [0, 1]. Both parties are risk-neutral and do not discount time. At
t = 0:
1. The principal offers a stationary contract s(y) : Y ! [ M, 1).
2. The agent accepts or rejects. If he rejects, he earns u0 and the principal
earns 0.
3. The agent chooses an effort a
0.
4. Total output x is realized according to F (·|a) 2
(Y).
5. The agent chooses a mapping from time t to output at time t, yx :
R1
[0, 1] ! Y, subject to 0 yx (t)dt = x.
R1
6. The agent is paid 0 s(yx (t))dt.
R1
R1
The principal’s and agent’s payoffs are 0 [yt s(yt )] dt and 0 s(yt )dt c(a),
respectively. Let F (·|·) and c(·) satisfy the conditions from Section 2.
Crucially, this model constrains the principal to offer a stationary contract
s(·). Without this assumption, the principal could eliminate gaming incentives
entirely by paying only for output realized at time t = 1. While stationarity
is a significant restriction, we believe it is realistic: as documented by Oyer
(1998) and Larkin (2014), contracts tend to be stationary over some period
of time (such as a quarter or a year).
14
We are very grateful to Lars Stole for suggesting this interpretation of the model.
25
This problem is equivalent to one in which, rather than choosing the realized output yx (t) at each time t, the agent instead decides what fraction of
time in t 2 [0, 1] to spend producing each possible output y 2 Y. In particular,
define Gx (y) as the fraction of time for which yx (t)  y.15 Then Gx (·) is a distribution: it is increasing, with Gx (y) = 0 and Gx (ȳ) = 1. The amount of time
the agent spends producing exactly y equals dGx (y), so the agent’s and prinR
cipal’s payoffs are s(y)dGx (y) c(a) = EGx [s(y)] c(a) and EGx [y s(y)],
respectively, where Gx (·) must satisfy EGx [y] = x. Therefore, for each x, both
players’ expected payoffs are as in (ObjF )-(LLF ); consequently, the results
from Sections 3 and 4 apply.
Remark 2. The optimal contracting problem in this setting coincides with
(ObjF )-(LLF ) with u(y) ⌘ y and ⇡(y) ⌘ y. Hence, if a⇤ 0 is optimal, then
a⇤  aF B and sLa⇤ (·) is optimal.
Intuitively, the agent will adjust his realized output so that his total payoff
equals the concave closure of s(·). He does so by smoothing his output over
time if s(·) is concave, and bunching it in a short interval if s(·) is convex.
This behavior is consistent with Oyer (1998) and Larkin (2014), who find
that salespeople facing convex incentives concentrate their sales. Conversely,
Brav et al. (2005) find that CEOs and CFOs pursue smooth earnings to avoid
the severe penalties that come from falling short of market expectations.
The assumption that the agent can continuously adjust his realized output
implies that he has substantial freedom to game the contract. For instance,
suppose that the agent could adjust her output only once, at t = 12 . Then for
each x, the agent could choose at most two output levels yL , yH 2 Y, each of
which would be produced one-half of the time. Consequently, Gx (y) = 0 for
y < yL , Gx (y) = 21 for y 2 [yL , yH ), and Gx (y) = 1 for y yH . In contrast, if
the agent can adjust output continuously, then he can choose any Gx (·) that
satisfies EGx [y] = x.
15
Formally, Gx (y) = L({t|yx (t)  y}), where L(·) denotes the Lebesgue measure.
26
7
Concluding Remarks
While we have focused on the relationship between a single principal and agent,
incentive contracts are rarely offered in a competitive vacuum. For instance,
Chevalier and Ellison (1997) describe how tournament-like incentives drive
financial advisors to make risky investments. More generally, an individual’s
(or a firm’s) competitive context shapes the incentives they face, which in
turn determine the kinds of risks they would pursue in our model. A more
complete analysis of how competition interacts with risk-taking could shed
light on behavior in both financial and product markets. See Fang and Noe
(2015) for a step in this direction.
We show how risk-taking can blunt convex incentives. As emphasized in
Makarov and Plantin (2015), an agent might also engage in risk-taking to
try to manipulate his superiors’ beliefs about his ability. Further research
is required to understand how risk-taking simultaneously affects effort and
learning.
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A
A.1
Proofs for Sections 3 and 4
Proof of Lemma 1
Fix a 0, and let v(·) implement a at maximum profit. We first claim that
following each realization x, the agent’s payoff equals v c (x) and the principal’s
payoff is no larger than ⇡(x v̂ c (x)).
Fix x 2 Y. Since v is upper semicontinuous, there exists p 2 [0, 1] and
z1 , z2 2 Y such that pz1 + (1 p)z2 = x and pv(z1 ) + (1 p)v(z2 ) = v c (x).
Since the agent can choose G̃x to assign probability p to z1 and 1 p to
z2 , his expected equilibrium payoff satisfies EGx [v(y)]
v c (x). But v c is
concave and v c (y) v(y) for any y 2 Y, so by Jensen’s Inequality EGx [v(y)] 
EGx [v c (y)]  v c (EGx [y]) = v c (x). So EGx [v(y)] = v c (x), and hence the
contract v c (x) satisfies (ICF )-(LLF ) for effort a and the degenerate distribution
G.
Next, consider the principal’s expected payoff. Since ⇡(·) is concave, applying Jensen’s Inequality and the previous result yields
EF (·|a) [EGx [⇡(y
u 1 (v(y)))]]  EF (·|a) [⇡ (EGx [y
 EF (·|a) [⇡ (x u 1 (v c (x)))] ,
u 1 (v(y))])]
where the first inequality is strict if ⇡ is strictly concave and the second is
strict if u is strictly concave (so that u 1 is also strictly concave). Therefore,
the principal weakly prefers the contract v c (x), and strictly so if either ⇡(·) or
u(·) is strictly concave.
To prove uniqueness, suppose at least one of ⇡(·) or u(·) is strictly concave,
and suppose that two contracts v(·) and ṽ(·) both implement a 0 at maximum profit, with v(x) 6= ṽ(x) for some x 2 Y. Since v(·) and ṽ(·) are upper
semi-continuous and concave, they must differ on an interval of positive length.
But then the contract v ⇤ (·) ⌘ 12 (v(·) + ṽ(·)) clearly satisfies (ICF )-(LLF ) for
effort a, and the principal’s payoff under v ⇤ is
⇥
⇤
EF (·|a) [⇡(y u 1 (v ⇤ (y)))] EF (·|a) ⇡ y 12 (u 1 (v(y)) + u 1 (ṽ(y)))
1
E
[⇡(y u 1 (v(y)))] + 12 EF (·|a) [⇡(y u 1 (ṽ(y)))] ,
2 F (·|a)
31
by Jensen’s Inequality, where at least one of the inequalities is strict.⌅
A.2
Proof of Proposition 1
Consider some s(·) that implements a. Define sL (y) ⌘ s(y) + m(y y), where
m = a 1 y EF (·|a) [s(y)] s(y) .
Suppose m > 1. Recall that sLaF B (·) implements aF B . There are two cases
to consider. If sLaF B (y) = M , then since sLaF B has slope 1, sLaF B (y) < sL (y)
for all y > y. Therefore,
⇥
⇤
EF (·|a) [⇡(y s(y))]  ⇡ EF (·|a) [y s(y)] = ⇡ EF (·|a) y sL (y) <
⇥
⇤
⇥
⇤
⇡ EF (·|a) y sLaF B (y) = EF (·|aF B ) ⇡ y sLaF B (y) ,
where the first inequality follows from Jensen’s inequality, the first equality
follows because EF (·|a) [sL (y)] = EF (·|a) [s(y)] by choice of m, the strict inequality
follows because sL (y) < sLaF B (y) for all y > y, and the final equality follows
because y sLaF B (y) is constant in y.
If instead sLaF B (y) > M , then (IR) holds with equality under sLaF B (·). So
EF (·|a) [⇡(y s(y))]  ⇡ EF (·|a) [y s(y)]  ⇡ aF B c(aF B ) + u0 =
⇥
⇤
⇥
⇤
⇡ EF (·|aF B ) y sLaF B (y) = EF (·|aF B ) ⇡ y sLaF B (y) ,
where the first inequality follows from Jensen’s inequality, the second inequality holds because EF (·|a) [s(y)]
c(a) + u0 to satisfy (IR) and a c(a) 
aF B c(aF B ) and strictly so if a 6= aF B , the first equality follows because
⇥
⇤
EF (·|aF B ) sLaF B (y) = u0 + c(aF B ), and the final equality holds because y
sLaF B (y) is constant in y. Therefore, any s(·) corresponding to an sL (·) with
m > 1 is dominated by sLaF B (·), and strictly so if s(·) implements a 6= aF B .
Suppose instead that m  1. Because s(·) is concave with sL (y) = s(y) and
⇥
⇤
EF (·|a) s(y) sL (y) = 0, there must exist some ỹ > y such that sL (y)  s(y)
if and only if y  ỹ. For any y 2 Y,
⇡(y
L
s (y))
⇡(y
s(y)) =
32
Z
y sL (y)
y s(y)
⇡ 0 (x)dx.
If y  ỹ, then y sL (y) y s(y). Furthermore, y sL (y)  ỹ sL (ỹ) because
m  1. Therefore, ⇡ 0 (x) ⇡ 0 (ỹ sL (ỹ)) for x  ỹ sL (ỹ) and so
⇡(y
sL (y))
⇡(y
s(y))
If y > ỹ, then ỹ sL (ỹ)  y
x ỹ sL (ỹ), and again
⇡(y
sL (y))
⇡(y
⇡ 0 (ỹ
sL (y)  y
s(y))
⇡ 0 (ỹ
sL (ỹ))(s(y)
sL (y)).
sL (ỹ)) for
s(y), ⇡ 0 (x)  ⇡ 0 (ỹ
sL (ỹ))(s(y)
sL (y)).
Consequently,
⇥
EF (·|a) ⇡(y
sL (y))
⇡(y
s(y))
⇤
⇡ 0 (ỹ
⇥
sL (ỹ))EF (·|a) s(y)
⇤
sL (y) = 0.
Next, we claim sL (·) induces effort ã > a. Because sL (y) s(y) is negative
⇥
⇤
and then positive, EF (·|a) sL (y) s(y) = 0, and ffa is monotonically increasing
in y, Beesack’s Inequality16 implies that
Z
ȳ
(sL (y)
s(y))
y
fa (y|a)
f (y|a)dy
f (y|a)
0.
Hence, (IC) is slack at a under sL (·). So sL (·) implements ã > a. Since
1 m > 0, the principal prefers effort ã to effort a under contract sL (y).
Finally, note that sL (y) sLã (y) > 0 for all y 2 Y. Combining the previous
arguments yields
EF (·|a) [⇡(y
⇥
s(y))]  EF (·|a) ⇡(y
⇤
⇥
sL (y))  EF (·|ã) ⇡(y
⇤
⇥
sL (y))  EF (·|ã) ⇡(y
The preceding shows that any s(·) that implements a is dominated by sLã (·)
for some ã 2 [0, aF B ], and strictly so if a > aF B . So a⇤  aF B , and sLa⇤ (·)
implements a⇤ at maximum profit. ⌅
16
Beesack’s inequality asserts that if g(·)
R and h(·) are two integrable functions such
that
g(·)
is
first
negative
then
positive,
g(z)dz = 0, and h(·) is non-decreasing, then
R
g(z)h(z)dz 0. See Beesack (1957) for details.
33
⇤
sLã (y)) .
A.3
Proof of Remark 1
Fix a > 0 and consider the problem (ObjF )-(LLF ) with an arbitrary ⇡(·) and
u(s) ⌘ s. Define EGx [⇡(y)] = ⇡ c (x), where ⇡ c (·) denotes the concave closure
of ⇡(·).
Modify (Obj)-(Conc) so that the principal’s utility equals ⇡ c (·). Since
⇡ c (y) ⇡(y) for any y, so the principal’s payoff in this modified problem must
be weakly larger than under the original problem. But ⇡ c (·) is concave and
sLa (y) = M , so Proposition 1 implies that sLa (·) implements a at maximum
profit in this modified problem. So the principal’s expected payoff equals
⇥
⇤
EF (·|a) ⇡ c (x sLa (x)) in this modified problem.
Now, consider the contract sLa (x) in the original problem (Obj)-(Conc).
⇥
⇤
For any distribution Gx 2 (Y) such that EGx [y] = x, EGx y sLa (y) =
x sLa (x) because sLa is linear. Therefore, as in Lemma 1, there exists some
⇥
⇤
GPx such that EGPx ⇡(y sLa (y)) = ⇡ c (x sLa (x)). Furthermore, conditional
⇥
⇤
on x, the agent’s expected payoff satisfies EGx sLa (y) c(a) = sLa (x) c(a)
for any Gx with EGx [y] = x. So sLa (·) satisfies (ICF )-(LLF ) for a > 0 and
Gx = GPx for each x 2 Y. The principal’s expected payoff if she offers sLa
⇥
⇤
equals EF (·|a) ⇡ c (x sLa (x)) , her payoff from the modified problem. So sLa a
fortiori implements a at maximum profit for any a 0. ⌅
A.4
Proof of Corollary 1
Suppose a⇤ > 0 equals the optimal effort level. Then Proposition 1 implies
that the principal’s expected payoff equals
⇥
EF (·|a⇤ ) ⇡(y
⇤
⇥
sLa⇤ (y)) = EF (·|a⇤ ) ⇡ y
c0 (a)(y
y) + min M, c0 (a)(a
y)
0 +M
Note that c0 (a)(a y) c(a) u0 > M whenever y < a c(a)+u
. So for y
c0 (a)
sufficiently negative, the principal’s payoff is bounded above by
⇡ (1
c0 (a))a + c0 (a)y + M
34
c(a)
u0
⇤
.
because ⇡(·) is concave. For any a > 0, c0 (a) > 0 and hence (1 c0 (a))a +
c0 (a)y + M < u0 for y sufficiently negative. But then a > 0 is strictly
dominated by a = 0 and s(·) = u0 . The same argument holds for any a > 0,
so a⇤ must converge to 0 as y ! 1.
Suppose c000 (·) 0 and ⇡(y) ⌘ y. The principal solves
a⇤ 2 arg max a
a 0
c0 (a)(a
y) + w(a) ,
(8)
where w(a) = min M, c0 (a)(a y) c(a) u0 . Define â as the smallest
a 0 such that c0 (a)(a y) c(a) u0 M for all a â, and note that the
LHS is strictly increasing in a.
First, we claim that if w(a⇤ ) < M (i.e., if (LL) is slack), then a⇤ = aF B .
Towards a contradiction, suppose that w(a⇤ ) < M and a⇤ < aF B . Then
the principal’s expected profit is equal to a⇤ c(a⇤ ) + u0 . However, because
a⇤ < min{â, aF B } and a c(a) is increasing for all a  aF B , the principal can
obtain a strictly bigger expected profit by choosing min{â, aF B }.
Next, we restrict attention to the case in which w(a⇤ )
M . The above
claim implies that it suffices to consider effort levels a â. In this case, (8)
can be written as a⇤ 2 arg maxa â a c0 (a)(a y) + M . This problem is
strictly concave in a because c000 (·)
0, and a⇤ = max â , af oc , where af oc
solves the first-order condition 1 c0 (af oc ) c00 (af oc )(af oc y) = 0.
In summary, we have that
8
<max â , af oc
⇤
a =
: aF B
if aF B < â
otherwise.
Using the implicit function theorem and that c000 (·)
and af oc , and hence a⇤ are increasing in y. ⌅
35
0, it follows that both â
B
Proofs for Sections 5
Consider an interval [yL , yH ]. The initial impact of raising this interval on the
agent’s utility at a given output y is given by
ryL ,yH (y) =
(
1 y 2 [yL , yH ]
.
0
else
Similarly, tilting this interval has an initial impact on the agent’s utility at y
given by
8
>
0
y  yL
<
tyL ,yH (y) =
y yL y 2 (yL , yH ) .
>
:
yH yL
y yH
We will carefully define the perturbations raise and tilt and show that they
respect concavity in Section B.1.2.
Our first result proves an important properties of any contract that is
GHM.
Lemma 3. Let v be GHM, and let [yL , yH ] be a linear segment of v. Then, for
each ŷ 2 (yL , yH ), there is ỹ 2 (ŷ, yH ) such that
n (ỹ)  0.
If v (yL ) > u, then such an ỹ exists in (yL , ŷ) as well. But, somewhere on
(yL , yH ), n (y) 0.
Proof. Note that for y > yH , tŷ,yH (y) = yH ŷ = (yH ŷ) ryH ,ȳ (y) . Since v
satisfies IC, since a > 0, and since v is concave and weakly increasing, v must
be strictly increasing near y. Hence, since yH > y, v (yH ) > u. We thus have
R
n (y) ryH ,ȳ (y) f (y|a) dy = 0 by Definition 1.1. Hence, by Definition 1.3, we
36
have
Z
0
=
=
n (y) tŷ,yH (y) f (y|a) dy
Z
Z
n (y) tŷ,yH (y) f (y|a) dy
(yH
ŷ)
yH
Z
n (y) ryH ,ȳ (y) f (y|a) dy
n (y) tŷ,yH (y) f (y|a) dy,
ŷ
and so at some point ỹ 2 (ŷ, yH ) , the integrand is weakly negative. Since
tŷ,yH (ỹ) > 0, it follows that n (ỹ)  0.
R
Similarly, note that if v (yL ) > u, then n (y) ryL ,ȳ (y) f (y|a) dy = 0 by
Definition 1.1, and so by Definition 1.2,
0 
=
=
Z
Z
Z
n (y) tyL ,ŷ (y) f (y|a) dy
n (y) tyL ,ŷ (y) f (y|a) dy
(ŷ
ŷ
n (y) [tyL ,ŷ (y)
(ŷ
yL )
Z
n (y) ryL ,ȳ (y) f (y|a) dy
yL )] f (y|a) dy,
yL
where, since the bracketed term is strictly negative on (yL , ŷ) , it follows that
n (y) is somewhere weakly negative on (yL , ŷ) .
R
Finally, since n (y) ryL ,yH (y) f (y|a) dy 0, and since we have established
that n (y) is weakly negative somewhere on (yL , yH ) , we must also have n (y)
weakly positive somewhere on the same interval.
Next, we prove some preliminary properties of optimal incentives schemes.
Lemma 1 shows that any optimal incentive scheme v(·) must be unique. We
prove that v(·) must be monotonically increasing and satisfy (IC-FOC) with
equality.
Suppose v(·) is concave and not everywhere increasing. Then, we can find
ỹ 2 Y such that if we replace v (y) by a constant v (ỹ) to the right of ỹ, the
resultant contract is concave, gives the same utility to the agent, is cheaper,
and, using MLRP and Beesack’s inequality makes (IC-FOC) slack. So any
37
optimal v(·) must be increasing.
Suppose v(·) does not satisfy (IC-FOC) with equality. Then, a convex
combination of v and the contract which gives utility constant and equal to
max {u, u0 + c (a)} 0 implements a, is strictly cheaper than v, and satisfies
(IC-FOC) with equality. So any optimal v(·) must satisfy (IC-FOC) with
equality.
B.1
B.1.1
Proof of Proposition 2
Preliminaries
Definition 1 and Proposition 2 are phrased in terms of free points. But, not
every free point is a convenient place to define a perturbation. Instead, for
any given v, let Cv be the set of points y at which there exists a supporting
plane L such that L (y 0 ) > v (y 0 ) for all y 0 6= y.
Clearly any kink point (see the discussion immediately before Corollary 2)
is an element of Cv . The next claim shows that for every other free point, there
is an arbitrarily close-by element of Cv .
Claim 1. Let ŷ be any point of normal concavity. Then, for each , there is a
point in (ŷ
, ŷ + ) \ŷ \ Cv . From this, it follows that for each " > 0, there
exists yL < yH such that yL , yH 2 Cv , and such that yL , yH 2 [ŷ ", ŷ + "] .
Proof of Claim. We will show first that for each , there is a point in (ŷ
, ŷ + ) \ŷ\
Cv . To see that this suffices to show the second part, apply the result first
to find a point y1 in Cv \ [ŷ ", ŷ + "] . Apply the result again to find y2 in
Cv \ [ŷ
, ŷ + ] , where = (1/2) |y1 ŷ| , and finally take yL and yH as the
smaller and larger of y1 and y2 .
So, fix > 0. Since ŷ is not on the interior of a linear segment and not a
kink point, there is at least one side of ŷ, without loss of generality the right
side, such that v (·) is not linear on (ŷ, ŷ + ). Let S (·) be the correspondence
which for each y assigns the set of slopes of supporting planes at y, and let
s (·) be any selection from S (·) . Note that since v is concave, for any y 00 > y 0 ,
max {S (y 00 )}  min {S (y 0 )} , and hence s is decreasing. Assume first that
38
there is a point ỹ 2 (ŷ, ŷ + ) where s (·) jumps discontinuously downward,
say from s00 to s0 < s00 . Then, the supporting plane at ỹ with slope (s0 + s00 )/2
qualifies. Assume instead that s (·) is continuous on (ŷ, ŷ + ). It cannot be
everywhere constant, since v (·) is not linear on (ŷ, ŷ + ). Hence, since s (·)
is continuous, there is a point ỹ at which it is strictly decreasing, so that
in specific, s (ỹ) < s (y) for all y < ỹ, and s (ỹ) > s (y) for all y > ỹ. The
supporting plane at ỹ with slope s (ỹ) then qualifies.
To see that why Claim 1 is helpful, assume that some part of Definition
1 is violated. For example, assume some optimal contract has a pair of free
R
points yL and yH such that n (y) ryL ,yH f (y) dy < 0. If either yL or yH is a
kink point, then it is also an element of Cv . If not, then we can apply Claim
1 to replace each relevant point by a sufficiently close-by element of Cv that
the strict inequality is maintained. Hence, it is enough to prove Proposition 2
when each restriction to a free point is tightened to a restriction to Cv .
B.1.2
Formal Definition and Properties of the Perturbations
This section defines raise and tilt, being careful in particular to maintain
concavity at the endpoints of the perturbed interval. We will need to consider
as many as three perturbations at once, where, given the previous discussion,
we will require the relevant points to be in Cv . First, we will have some small
amount "p of a perturbation p where p could be ryL ,yH or tyL ,yH in each case
with "p positive or negative. Second, for some ŷ 2 Cv , we will need to consider
some amount "t of tŷ,ȳ and "r of rŷ,ȳ . Intuitively, we will use tŷ,ȳ and rŷ,ȳ to
establish shadow values for (IC-FOC) and (IR), and then, for any particular
perturbation p, consider the three deviations together where one uses tŷ,ȳ and
rŷ,ȳ to undo the effect of p on (IC-FOC) and (IR).
Fix yL , yH , and ŷ. A priori, ŷ may have arbitrary position relative to yL
and yH , and moreover, in the case where p is tyL ,yH , one of yL or yH may not
be in Cv , depending on whether "p is negative or positive. Define y0 < y1 <
· · · < yK , K  4, as elements of the set y, yL , yH , ŷ, ȳ \ Cv . For any given
39
" = ("p , "t , "r ), let d (·; ") : [y, ȳ] ! R be given by
d (·; ") = "p p (·) + "t tŷ,ȳ (·) + "r rŷ,ȳ (·) .
If yL and yH are both elements of {y0 , · · · , yK } , as must be true if p is
ryL ,yH , then it follows that d is linear on each interval of the form (yk 1 , yk ) .
Assume that yH 2
/ {y0 , · · · , yK } . Then, it must be that p is tyL ,yH with "p 0.
In this case, if yH 2
/ (yk 1 , yk ) , then d (·; ") is linear on (yk 1 , yk ) , while if
yH 2 (yk 1 , yk ) , then, since "p 0, d (·; ") is concave with two linear segments
on (yk 1 , yk ) . Finally, assume yL 2
/ {y0 , · · · , yK } . Then, p is tyL ,yH with "p  0,
and once again, if yL 2
/ (yk 1 , yk ) , then d (·; ") is linear on (yk 1 , yk ) , while if
yL 2 (yk 1 , yk ) , then since "p  0, d (·; ") is once again concave with two linear
segments on (yk 1 , yk ) .
For each k, let Lk (·; ") be the line that coincides with the linear segment
of d (·; ") immediately to the right of yk 1 and let L+
k (·; ") be the line that
coincides with the linear segment immediately to the left of yk (these are the
same line if d is linear on (yk 1 , yk )), and let
8
>
y  yk 1
< Lk (y; ")
dk (y; ") =
d (y; ") y 2 (yk 1 , yk ) .
>
: +
Lk (y; ")
y yk
Note that dk is concave, and that as |"| = |"d | + |"t | + |"r | ! 0, dk converges
uniformly to the function that is constant at 0.
For each k, let Lk be a supporting line to v at yk , where since yk 2 Cv , we
can choose Lk such that Lk (y) > v (y) for all y 6= yk , and let
8
>
y  yk 1
< Lk 1 (y)
vk (y) =
v (y)
y 2 (yk 1 , yk ) ,
>
:
Lk (y)
y yk
40
so that vk (·) is concave. Define v̂ (·; ") by
v̂ (y; ") =
min
k2{1,··· ,K}
(vk (y) + dk (y; ")) .
As the minimum over concave functions, v̂ (·; ") is concave.
Fix k and consider any y 2 (yk 1 , yk ) . Since dk (y, 0) = 0, and by the fact
that for each k 0 , Lk0 (y) > v (y) for all y 6= yk0 , k is the unique minimizer of
vk (y) + dk (y; 0) . From this, it follows first that v̂ (y; 0) = vk (y) = v (y), and
second, that for all " in some neighborhood of 0 (where "p is restricted in sign
if p = tyL ,yH and if one of yL or yH is not in Cv ),
v̂"p (y; ") = d"p (y; ") = p (y) ,
v̂"t (y; ") = d"t (y; ") = tŷ,ȳ (y) , and
v̂"r (y; ") = d"r (y; ") = rŷ,ȳ (y) .
But then, except on the zero-measure set of points {y0 , · · · , yK },
v̂"p (·; 0) = p (·) ,
(9)
v̂"t (·; 0) = tŷ,ȳ (·) , and
v̂"r (·; 0) = rŷ,ȳ (·) .
B.1.3
Shadow Values
We need to establish that starting from " = 0 the effects of perturbation p can
be undone via tŷ,ȳ and rŷ,ȳ . To do so, let
" R
#
R
v̂"t (y, ") fa (y|a) dy
v̂"r (y, ") fa (y|a) dy
R
Q (") = R
.
v̂"t (y, ") f (y|a) dy
v̂"r (y, ") f (y|a) dy
The top row of Q tracks the rate at which "t and "r respectively affect (ICFOC), while the bottom row tracks the rate at which "t and "r respectively
41
affect (IR). Then, from (9),
and so
|Q (0)|
" R
#
R
tŷ,ȳ fa (y|a) dy
rŷ,ȳ fa (y|a) dy
R
R
Q (0) =
tŷ,ȳ f (y|a) dy
rŷ,ȳ f (y|a) dy
" R ȳ
#
R ȳ
(y
ŷ)
f
(y|a)
dy
f
(y|a)
dy
a
a
Rŷȳ
Rŷȳ
=
,
(y
ŷ)
f
(y|a)
dy
f
(y|a)
dy
ŷ
ŷ
=
Z
ȳ
(y
ŷ
R ȳ
(y
ŷ
= R ȳ
s
(y
ŷ
Z ȳ
=
ŷ
ŷ) fa (y|a) dy
ŷ) fa (y|a) dy
ŷ) f (y|a) dy
l (y|a) R ȳ
ŷ
(y
(y
Z
ȳ
f (y|a) dy
ŷ
R ȳ
ŷ
R ȳ
fa (y|a) dy
Z
ȳ
(y
ŷ) f (y|a) dy
ŷ
Z
ȳ
fa (y|a) dy
ŷ
f (y|a) dy
Z ȳ
ŷ) f (y|a)
f (y|a)
dy
l (y|a) R ȳ
dy,
ŷ) f (y|a) dy
f
(y|a)
dy
ŷ
ŷ
ŷ
where the symbol = means “has (strictly) the same sign as.”
s
Thus, |Q (0)| has the same sign as the difference between two expectations
of l (·|a) . Using that (y ŷ) is strictly increasing, the density in the first
integral strictly likelihood-ratio dominates the density in the second integral.
Since l (·|a) is strictly increasing, it follows that |Q (0)| is strictly positive (and
so remains so for all " in some ball around 0.) But then by the implicit function
theorem, for each p 2 {tyL ,yH , ryL ,yH } , we can on the appropriate neighborhood
implicitly define "r (·) and "t (·) by
Z
Z
v̂ (y; ", "t (") , "r (")) f (y|a) dy = c (a) + u0 , and
v̂ (y; ", "t (") , "r (")) fa (y|a) dy = c0 (a) ,
which is to say that starting from " = 0, if we make the small perturbation
"p to v, we can restore (IC-FOC) and (IR) by a suitable combination of small
applications "t and "r of tŷ,ȳ and rŷ,ȳ .
Let be the rate of change of costs as one increases the agent’s utility
42
using tŷ,ȳ and rŷ,ȳ . That is, if we let
!
qtIR
qrIR
then
=
Similarly, if
Z
⇢
1
= [Q (0)]
1
0
1
!
,
(v (y)) qtIR tŷ,ȳ (y) + qrIR rŷ,ȳ (y) f (y|a) dy.
!
qtIC
qrIC
= [Q (0)]
1
1
0
!
,
then the rate of change of costs as one increases the agent’s incentives using
tŷ,ȳ and rŷ,ȳ is
µ=
Z
⇢
1
(v (y)) qtIC tŷ,ȳ (y) + qrIC rŷ,ȳ (y) f (y|a) dy.
Given the shadow values
proof of Proposition 2. ⌅
B.1.4
and µ, the argument in Section 5 completes the
Proof of Sufficiency
Let v, with associated and µ, be GHM. Let us show that v is optimal. We
will argue by contradiction. Assume v is not optimal, and let v ⇤ be a lower cost
concave contract satisfying (IC)-(LL). As in the argument at the beginning of
Appendix B, v ⇤ can be taken to be increasing, satisfy (IC-FOC) exactly, and
as in the proof of Lemma 5 in Appendix D, v ⇤ (ȳ) and v ⇤ (y) can be taken to
be finite.
Enumerate the closed linear segments S1 , S2 , · · · , of v, and let S = [Si .
Let (y) = v ⇤ (y) v (y) , and let v̂ (y; ") = v (y) + " (y) , so that v̂ (·, 0) =
v (·) and v̂ (·, 1) = v ⇤ (·) . Then, for each ", v̂ (·; ") is a convex combination of
the concave contracts v and v ⇤ . Hence, v̂ (·; ") is concave and satisfies (IC)(LL). Since u 1 (·) is convex, and since v̂ (y; ") is linear in ", it follows that
43
R
u
1
(v̂ (y; ")) f (y|a) dy is convex in ". Thus, since
Z
u
1
Z
(v̂ (y; 0)) f (y|a) dx =
Z
>
Z
=
u
1
(v (y)) f (y|a) dy
u
1
(v ⇤ (y)) f (y|a) dy
u
1
(v̂ (y; 1)) f (y|a) dy,
it follows that
Z
d
0 >
u 1 (v̂ (y; 0)) f (y|a) dy
d"
Z
1
=
(y) f (y|a) dy
0
1
u (u (v̂ (y; 0)))
Z
=
⇢ 1 (v (y)) (y) f (y|a) dy
Z
Z
1
=
⇢ (v (y)) (y) f (y|a) dy +
⇢
1
(v (y)) (y) f (y|a) dy,
Y\S
S
and so, since every point in Y\S is a point of normal concavity (noting that
we took the sets Si to be closed, and so any kink point is in S), we have
Z
⇢
1
(v (y)) (y) f (y|a) dy <
S
=
=
Z
Z
⇢
1
(v (y)) (y) f (y|a) dy
Y\S
( + µl (y|a)) (y) f (y|a) dy
Z
Z
(y) f (y|a) dy µ
(y) fa (y|a) dy.
Y\S
Y\S
Y\S
where the first equality follows by Lemma 3.
R
Both v and v ⇤ satisfy (IC-FOC) with equality, and hence
(y) fa (y|a) dy =
0, from which
µ
Z
(y) fa (y|a) dy = µ
Y\S
Z
(y) fa (y|a) dy.
S
Similarly, either (IR) is binding at v, in which case
44
R
(y) f (y|a) dy
0, or
(IR) does not bind at v, in which case
Z
Y\S
= 0, and hence in either case
(y) f (y|a) dy 
Z
(y) f (y|a) dy.
S
Making these two substitutions thus yields
Z
⇢
1
(v (y)) (y) f (y|a) dy <
S
Z
(y) f (y|a) dy + µ
S
Z
(y) fa (y|a) dy.
S
Hence, since S = [Si , where the Si ’s are disjoint except possibly at their
zero-measure boundaries, there must be some i such that
Z
⇢
1
(v (y)) (y) f (y|a) dy <
Si
or equivalently,
Z
Z
(y) f (y|a) dy + µ
Si
Z
(y) fa (y|a) dy,
Si
n (y) (y) f (y|a) dy < 0.
Si
Fix such an i, and consider 1 , the restriction of to Si = [yL , yH ] . Since
v is linear on Si , and v ⇤ is concave, 1 is concave. For any given K, let
= (yH yL ) /K, and consider the function K on [yL , yH ] that agrees with
, · · · , yH } , and is linear in between these
1 on the set of points {yL , yL +
points. Note that K is concave and continuous on [yL , yH ] . Choose K large
enough that
Z
n (y)
K
(y) f (y|a) dy < 0.
Si
⇥ ⇤
Finally, define ˜ on y, ȳ by
˜ (y) =
8
>
<
0
y  yL
y 2 [yL , yH ] .
K (y)
>
:
y > yH
K (yH )
Note that yH and ȳ are free. Note also that as in the proof of Lemma 3,
v (yH ) > u. It thus follows from Definition 1.1 that since ˜ is constant on
45
[yH , ȳ] ,
and hence,
Z
ȳ
n (y) ˜ (y) f (y|a) dy = 0,
yH
Z
n (y) ˜ (y) f (y|a) dy < 0.
Let us next argue that ˜ can be expressed as a sum of raises and tilts. For
k 2 {0, . . . , K}, let yk = yL + k , and let sk be the slope of ˜ on (yk 1 , yk ) .
Then, we claim that for all y in [yL , yH ],
˜ (y) = (y0 ) ry ,ȳ (y) +
0
K
X1
(sk
sk+1 ) ty0 ,yk (y) + sK ty0 ,yK (y) .
(10)
k=1
To see (10) note first that for y < y0 = yL , both sides of the equation are 0. At
y0 , each side is (y0 ) , since ry0 ,ȳ (y0 ) = 1, and since ty0 ,· (y0 ) = 0. Thus, since
both sides are continuous and piecewise linear on [y0 , ȳ] , it is enough that the
two sides have that same derivative where defined. So, fix k̂ 2 {1, . . . , K} ,
and let y 2 yk̂ 1 , yk̂ . Note that for k < k̂, t0y0 ,yk (y) = 0, and for k
k̂,
0
ty0 ,yk (y) = 1. Hence, the derivative of the right-hand side is
K
X1
(sk
sk+1 ) + sK = sk̂ ,
k=k̂
as desired, and so, noting that ˜0 (y) = 0 for y > yK = yH , we have established
(10).
R
Since n (y) ˜ (y) f (y|a) dy < 0, we must thus have at least one of
R
1. (y0 ) n (y) ry0 ,ȳ (y) f (y|a) dy < 0,
R
2. for some k < K, (sk sk+1 ) n (y) ty0 ,yk (y) f (y|a) dy < 0, or
R
3. sK n (y) ty0 ,yK (y) f (y|a) dy < 0.
R
R ȳ
By Definition 1.1, and since y0 is free, n (y) ry0 ,ȳ (y) f (y|a) dy = y0 n (y) f (y|a) dy
0, and so 1. cannot hold. Since ˜ is concave on [yL , yH ] , it follows that
46
sk sk+1
0, and so, since y0 is free, it follows by Definition 1.2 that 2.
cannot hold either. Finally, since y0 and yK are both free, the integral in 3.
is in fact 0 by Definition 1.2 and Definition 1.3. We thus have the required
contradiction, and v is in fact optimal. ⌅
B.2
Proof of Corollary 2
If y is a kink point, then Lemma 3 applied to the left of y implies that n(y)  0.
If y is a point of normal concavity, then by Lemma 1 there exist sequences of
points {ykL }, {ykH } 2 Cv such that ykL < y < ykH for all k 2 N and limk ykL =
limk ykH = y. These points are free, so 3 holds with equality on each interval
[ykL , ykH ]. Hence, in the limit, n(y) = 0. ⌅
B.3
Proof of Corollary 3
Let v(·) be an optimal incentive scheme, and suppose (IR) does not bind.
Towards a contradiction, suppose that v(·) is strictly concave at some y < y0 .
Consider the alternative contract
8
h
i
<↵v(y) + (1 ↵) v(y) + (y y) v(y0 ) v(y)
y  y0
y0 y
ṽ(y) =
.
:v(y)
y>y
0
Note that ṽ(·) is concave, ṽ(y)  v(y) for all y 2 Y, ṽ(y) u, and there exists
an interval in [y, y0 ] such that ṽ(y) < v(y) on that interval. Therefore, ṽ(·)
is strictly less expensive than v(·) to the principal. Since (IR) does not bind,
there exists some ↵ 2 [0, 1) such that ṽ(·) satisfies (IR). Furthermore,
R
Ry
R ȳ
ṽ(y)fa (y|a)dy = y 0 ṽ(y)fa (y|a)dy + y0 v(y)fa (y|a)dy >
R y0
R ȳ
R
v(y)f
(y|a)dy
+
v(y)f
(y|a)dy
=
v(y)fa (y|a)dy,
a
a
y
y0
where the strict inequality follows because fa (y|a) is negative on y 2 [y, y0 ].
Hence, ṽ(·) satisfies (IC-FOC). So ṽ(·) implements a, contradicting that v(·)
is optimal. ⌅
47
B.4
Proof of Corollary 4
Suppose ⇢ ( + µl(·|a)) is convex. Let v(·) be optimal, and towards a contradiction assume that there exists a y 2 (y, ȳ) that is free. Then n(y)  0 by
Corollary 2. But v(·) is concave and ⇢( + µl(·|a)) is convex, so n(y)  0 for
all y 2 Y. But then (3) cannot hold between y and ȳ, so v(·) cannot be GHM,
which contradicts that v(·) is optimal. So v(·) cannot have any interior free
points, implying that v(·) is linear.
If ⇢ ( + µl(·|a)) is concave and (LL) does not bind, then it is GHM and
hence optimal by Proposition 2. If (LL) binds, suppose that v(·) has two
or more linear segments. Since v(·) is concave, Lemma 3 implies that ⇢( +
µl(y|a)) must be convex at some y; contradiction. So v(·) can have at most
one linear segment, on [yL , yH ]. Furthermore, yL = y, since otherwise Lemma
3 would imply that ⇢( + µl(y|a)) is convex somewhere on [yL , yH ]. ⌅
B.5
Proof of Corollary 5
Given the discussion immediately preceding the proposition, it is enough to
show that ⇢ ( + µl (·|a)) is never first strictly concave and then weakly convex.
For any analytic function q with domain a subset of the reals, let q (k) be
the k th derivative of q.
Lemma 4. Assume q > 0 is not everywhere a constant, is analytic, and has
con (q) = ! > 1. Assume also that for some ŷ on the interior of its domain,
q 0 (ŷ) = 0. Let k̂ = min k|q (k) (ŷ) 6= 0
2. Then, q (k̂) (ŷ) < 0.
Proof. Recall that q has concavity ! if q ! /! is concave, or, equivalently (cancelling the strictly positive term q ! 2 ), if for all y in the domain of q,
⇠ (y) ⌘ (!
2
1) (q 0 (y)) + q (y) q 00 (y)  0.
So, in particular, if k̂ = 2, then we must have q 00 (ŷ) < 0, since ⇠ (ŷ)  0. Note
that for k 2 {0, 1, 2, · · · }
⇠ (k) (ŷ) = d (ŷ) + q (ŷ) q (k+2) (ŷ) ,
48
where d is an expression involving derivatives of q of order less than k + 2.
(k̂ 2)
So, the first non-zero term of the Taylor expansion of ⇠ is ⇠ (k̂ 2)!(ŷ) (y ŷ)k̂ 2 ,
ŷ)k̂
where ⇠ (k̂ 2) (ŷ) = q (ŷ) q (k̂) (ŷ) . Hence, since (y
y > ŷ, q (k̂) (ŷ) must be strictly negative.
2
is strictly positive for
Using this lemma, we can prove the following claim, from which Corollary
5 is immediate.
Claim 2. Let g and h be strictly positive analytic functions with con (g 0 ) +
con (h0 ) > 1, and g 0 and h0 everywhere strictly positive. Then, (g (h (·))) is
never first strictly concave and then weakly convex.
Proof. Let
2
✓ (·) = (g (h (·)))00 = g 00 (h0 ) + g 0 h00 .
(11)
If both g and h are linear, then ✓ ⌘ 0, and we are done. Assume g and h are
not both linear, and consider any point ŷ at which ✓ = 0. We will show that
immediately to the right of ŷ, ✓ < 0. This rules out that ✓ is ever first strictly
negative and then weakly positive over any interval of non-zero length.
To see this, note that
3
✓0 = g 000 (h0 ) + 3g 00 h0 h00 + g 0 h000 .
(12)
Consider any point ŷ at which ✓ = 0. Consider first the case that g 00 (ŷ) h00 (ŷ) 6=
0. Then, since g 0 > 0, it follows by (11) that g 00 (ŷ) h00 (ŷ) h0 (ŷ) < 0, and so,
evaluated at ŷ,
✓0
g 000 (h0 )2
g 0 h000
3
s
g 00 h00
g 00 h00 h0
g 0 g 000
h0 h000
=
3
+
(g 00 )2
(h00 )2
 con (g 0 ) con (h0 ) 1
=
<0
49
where in the second line we substitute for (h0 )2 in the first term using (11) and
that ✓ (ŷ) = 0, and similarly for g 0 in the third term. Hence, ✓ is negative on
an interval to the right of ŷ.
Assume instead that g 00 (ŷ) h00 (ŷ) = 0, where, since ✓ (ŷ) = 0, it follows
that g 00 (ŷ) = h00 (ŷ) = 0. Thus, since con (g 0 ) > 1, it follows from Lemma
4 applied to q = g 0 that the first non-zero derivative of g 0 is strictly negative,
and similarly for h0 . But then, the first non-zero derivative of ✓ will be of the
form g (k) (h0 )k + g 0 h(k) with k 3, and at least one term strictly negative, and
so, taking a Taylor expansion, ✓ is strictly negative on an interval to the right
of ŷ, and we are done.
50
C
C.1
Proofs for Section 6
Proof of Proposition 3
Define
= (a) and ( ) = c( 1 ( )). Consider relaxing (7) so that the
principal can choose any Vs (·). Then the principal’s problem is equivalent to
choosing a value function Vs (·) that solves
max
,Vs (·)
s.t.
Vs ( )
⇣ ⌘
2 arg max Vs ˜
˜
n
u0
⇣ ⌘o
˜
Vs ( )
( )
Vs (y)
M for all y 2 Y
Vs (·) weakly concave.
This problem is identical to (Obj)-(Conc) with a degenerate distribution over
intermediate output. A simple modification of the proof of Proposition 1 implies that Vs (y) = 0 ( ) y y w, where w = min M, 0 ( )
y
( )
is a solution.
Next, let s (y) = 0 ( ) y y
w, and observe that EF (·|x) [s (y)] =
0
( )
y
w = Vs (x) for all x. Moreover, note that s (y)
M for all
0
0
y 2 Y, so s (·) implements a in the original problem. Given that ( ) = c 0(a)
,
(a)
we conclude that a profit-maximizing Vs (·) can be implemented by the linear
0
contract s⇤ (y) = c 0(a)
(y y) w. ⌅
(a)
51
u0
C.2
Proof of Proposition 4
Given the definition of ṽ(·), c̃, and ⇡
˜ , the optimal a and ṽ(·) solve
max EF (·|a) [EGx [˜
⇡ (y)
(13)
ṽ(y)]]
a,G2G,ṽ(·)
⇥
⇤
s.t. a, G 2 arg max EF (·|ã) EG̃x [ṽ(y)]
c̃(ã)
ã,G̃2G
EF (·|a) [EGx [ṽ (y)]]
ṽ(y)
c̃(a)
u0
d(y) 8y 2 Y.
M
As in Lemma 1, following any intermediate output x, the agent optimally
chooses Gx so that EGx [ṽ(x)] = ṽ c (x), where ṽ c (·) is the concave closure of
ṽ(·). Therefore, the principal’s payoff following x equals EGx [˜
⇡ (y) ṽ(y)] 
c
⇡
˜ (x) ṽ (x). Since ⇡
˜ (·) is strictly concave, this inequality holds with equality
only if Gx is degenerate. Consequently, we can restrict attention to contracts
in which ṽ(·) is concave, and hence for every x, the agent will optimally choose
Gx (y) = I{y x} .
Relax the limited liability constraint so that it must be satisfied only at
y = y. Then (13) can be written as
max EF (·|a) [˜
⇡ (y)
a,ṽ(·)
ṽ(y)]
s.t. a 2 arg max EF (·|ã) [ṽ(y)]
ã
EF (·|a) [ṽ (y)]
ṽ(y)
c̃(a)
c̃(ã)
u0
M
ṽ(·) is concave.
Fix any effort a 0 and any concave incentive scheme ṽ(·) that implements
a. As in the proof of Proposition 1, let ṽ L (·) be the unique linear incentive
⇥
⇤
scheme that satisfies ṽ L (y) = ṽ(y) and EF (·|a) ṽ L (y) = EF (·|a) [ṽ(y)] . Then
52
ṽ L (·)
ṽ(·) single-crosses 0 from below and hence Beesack’s inequality implies
Z
(ṽ L (y)
ṽ(y))
fa (y|a)
f (y|a)dx
f (y|a)
0
with strict inequality if ṽ L (y) 6= ṽ(y) for some y. Consequently, ṽ L (·) implements some ã a, with ã > a if ṽ L (y) 6= ṽ(y) for some y.
Define ṽ ⇤ (y) = c̃0 (a)(y y) w̃, where w̃ = min M, c̃0 (a)(a y) c̃(a) u0 ,
and suppose that ṽ ⇤ (y) = M . Then ṽ ⇤ (y)  ṽ L (y) for all y y and strictly
so if ã > a. Therefore, ṽ ⇤ (·) uniquely implements a 0 at maximum profit in
the relaxed problem. But ṽ ⇤ (y)
M
M d(y) for all y 2 Y, so ṽ ⇤ (·)
satisfies the limited liability constraint, and hence implements a in the original
problem.
Next, suppose that ṽ ⇤ (y) > M . Then by construction, EF (·|a) [ṽ ⇤ (y)] =
u0 +c̃(a)  EF (·|a) [ṽ(y)], which implies that EF (·|a) [˜
⇡ (y) ṽ ⇤ (y)] EF (·|a) [˜
⇡ (y) ṽ(y)];
⇤
i.e., ṽ (·) implements a at maximum profit.
Finally, note that the preceding holds for any a 0, proving that ṽ ⇤ (·), or
equivalently, s̃⇤ (y) = c̃0 (a)(y y) d(y) w̃ is optimal. ⌅
C.3
Proof of Remark 2
It suffices to prove that for any total output x,
max
yx :[0,1]![y,ȳ]
⇢Z
1
s(yx (t)) dt s.t.
0
Z
1
yx (t)dt = x
= sc (x) .
0
Consider the following yx : if s(x) = sc (x), then yx (t) = x for all t. If s(x) <
sc (x), then there exist w, z, and ↵ 2 [0, 1] such that ↵w + (1 ↵)z = x and
↵s(w) + (1 ↵)s(z) = sc (x). For t  ↵, yx (t) = w, with yx (t) = z for t > ↵.
This function yx guarantees that the agent earns sc (x).
R1
Now, s(yx (t))  sc (yx (t)) for all yx (t). Since sc is weakly concave and 0 yx (t)dt =
R1
R1
R1
x, we conclude that 0 s(yx (t))dt  0 sc (yx (t))dt  0 sc (x)dt = sc (x). So
the agent earns (and the principal pays) sc (x) following intermediate output
x, which proves the claim. ⌅
53
D
Online Appendix: Existence of Optimal Contracts
D.1
Proof of Existence and Uniqueness for u finite
Proposition 5. Let U and ⇧ be the set of increasing concave utility functions
for the agent and principal satisfying our assumptions and let V be the set of
concave (but not necessarily increasing) functions from [y, ȳ] to R, where each
of U, ⇧, and V has the topology of almost everywhere pointwise convergence.
Fix a. Then, (i) for each z = (M, u0 , ⇡, u) , there exists an optimal contract
v that implements a given z and (ii) at any point z where at least one of ⇡
or u is strictly concave, the optimal contract implementing a is unique and
continuous in z.
Proof. The proof relies on Berge’s Theorem. Fix a. For any given z = (M, u0 , u, ⇡) ,
let v L (·|z) be given by v L (y|z) = c0 (a)(y y)+ , where = min u ( M ) , c (a) + u0
be the maximum-profit linear (in utils) contract that implements a. In particular v L (·|z) satisfies IC, since, under our assumptions, the agent’s utility
from income given v L (·|z) is linear in effort while c (·) is concave and so the
first order condition implies IC.
Let B : R ⇥ R ⇥ ⇧ ⇥ U !! V be the correspondence which for each
M 2 R, u0 2 R, ⇡ 2 ⇧, and u 2 U gives the set of contracts v such that
⇥
EF (·|a) ⇡ y
u
1
(v (y))
EF (·|a) [v (y)
⇤
⇥
EF (·|a) ⇡ y
u
1
v L (y|z)
a 2 arg max EF (·|ã) [v (y)
c (a)]
v(y)
ã
⇤
1, (14)
c (ã)] ,
u0 ,
u ( M ) , and
v 2 V,
where the second through fifth constraints are simply the translations of (IC)
(Conc) when z is a parameter, and the first constraint restricts attention
to contracts that come within 1 util of v L (·|z) . Since v L (·|z) 2 B (z) , this
constraint is innocuous, and it also follows that B is non-empty valued.
54
c0 (a) (a
y)
For any given v 2 V, define vmax = maxy2[y,ȳ] v (y) . We begin by proving
(*) For each compact subset Z ✓ R ⇥ R ⇥ ⇧ ⇥ U , there is ū such that
vmax  ū for all z 2 Z and v 2 B (z) .
To see (⇤) , begin by noting that v L (·|·) is continous on the compact set
⇤
y, ȳ ⇥ Z, and so 1 < m ⌘ miny2[y,ȳ]⇥Z ⇡ y u 1 v L (y|z)
. Using
that Z is compact, let u⇤ < 1 satisfy that for all z 2 Z, ⇡ (ȳ u 1 (u⇤ )) 
m 2, so that anytime the principal gives the agent utility u⇤ or above, the
principal is at least two utils worse off than under v L (·|z) .
Fix z 2 Z, and v 2 B (z) . Choose ymax so that v (ymax ) = vmax . Let
umin = minz2Z u ( M ) , and define v̂ as the function that equals umin at y and
ȳ, equals vmax at ymax , and is linear to the left and right of ymax . That is,
v̂ (ymax ) = ymax, and
⇥
v̂ (y) =
(
umin +
umin +
vmax umin
ymax y
vmax umin
ȳ ymax
y
y
y 2 [y, ymax )
y) y 2 (ymax , ȳ]
(ȳ
)
.
Note that
⇥
EF (·|a) ⇡ y
u
1
(v̂ (y))
⇤
⇥
EF (·|a) ⇡ y
u
1
v L (y|z)
⇤
1,
(15)
using that the concave function v is everywhere at or above v̂, and the first
constraint in (14).
We will show that (15) implies a uniform bound on vmax . Intuitively, when
vmax is large, the piece-wise linear function v̂ (y) is above u⇤ for nearly all of
[y, ȳ], implying losses compared to v L (·|z) that contradict (15).
A uniform bound on vmax is of course trivial for v such that vmax  u⇤ .
So, assume vmax > u⇤ . Let yL 2 [y, ymax ) solve v̂ (yL ) = u⇤ , where if ymax = y,
we let yL = y, and similarly, define yH 2 (ymax , ȳ] by v̂ (yH ) = u⇤ , where if
ymax = ȳ, yH = ȳ.
Since v̂ (·) is concave, v̂ (y) u⇤ for all y 2 [yL , yH ], and hence,
⇡ y
u
1
(v̂ (y))
⇡ y
55
u
1
v L (y|z)

2.
while for any y,
⇡ y
u
1
(v̂ (y))
⇡ y
u
1
L
v (y|z)
✓
 ⇡ ȳ + max M
z2Z
◆
m,
and so from (15) we must have
(F (yH |a)
F (yL |a)) ( 2)+(1
(F (yH |a)
◆
✓ ✓
F (yL |a))) ⇡ ȳ + max M
z2Z
m
or equivalently,
F (yH |a)
F (yL |a)  1
1
⇡ (ȳ + maxz2Z M )
m+2
,
(16)
where the RHS is strictly less than one because 1 > ⇡ (ȳ + maxz2Z M ) m >
0. But, if yL 6= y, then
u⇤ umin
ymax y
vmax umin
u⇤ umin
 y+
ȳ y ,
vmax umin
yL = y +
and so as vmax ! 1, yL ! y. Similarly, if yH 6= ȳ, then
yH = ȳ
ȳ
u⇤ umin
(ȳ
vmax umin
u⇤ umin
ȳ
vmax umin
ymax )
y ,
and so as vmax ! 1, yH ! ȳ. But then by (16) , vmax is bounded, establishing
(⇤) .
From (⇤) and the dominated convergence theorem, each expectation in (14)
is continuous in z, and hence, noting that each of IC and concavity can be
expressed as a collection of weak inequalities, B (·) is upper hemicontinuous.
Let us next show that B (·) is lower hemicontinuous. To see this, fix z,
let v 2 B (z) , and let zk ! z. For " 2 (0, 1) and > 0, define ṽ (·|", )
by ṽ (y|", ) = (1 ") v (y) + "v L (y|M
, u0 + , u, ⇡) . Consider the first
56
◆
1,
constraint. By Jensen’s inequality, for each y,
y u
1
(1
since
u
⇡ y
u
1
1
") v (y) + "v L (y|z)
(1
") y
u
1
(v (y)) +" y
u
1
v L (y|z)
,
is concave. Hence, since ⇡ is increasing and concave,
") v (y) + "v L (y|z)
(1
⇡ (1
(1
") y
") ⇡ y
1
u
u
1
(v (y)) + " y
u 1 v L (y|z)
(v (y)) + "⇡ y
u
1
and so the same is true in expectation. Since the first constraint in (14) holds
weakly for v (·) and strictly for v L (·|z) , we have that for each " 2 (0, 1) ,
⇥
EF (·|a) ⇡ y
u
1
(1
") v (y) + "v L (y|z)
⇤
⇥
> EF (·|a) ⇡ y
u
1
v L (y|z)
It follows from continuity that for each n 2 {1, 2, · · · } there exists n1 > n > 0
sufficiently small that the first constraint in (14) is slack for vn ⌘ ṽ ·| n1 , n .
It is immediate that the third and fourth constraints in (14) hold strictly at
vn , while the second and fifth constraints (which do not depend on z) continue
to hold, since vn is a convex combination of concave contracts satisfying IC.
Hence, for each n, there is Kn such that for all k Kn , vn 2 B (zk ) . Let kn =
max {n, maxn0 n Kn } . Then, for each n, vkn 2 B (zkn ) , and, since kn ! 1 and
n ! 0, vkn ! v. Hence, B (·) is lower hemicontinuous, and thus continuous.
Fix z, and let {vk } be a sequence in B (z) . Each vk , being concave, is
characterized by its values on the countable set Y containing y, ȳ, and the
rational points in [y, ȳ]. At each y 2 Y, vk (y) is an element of the compact
interval [u ( M ) , ū] for a suitably chosen ū. Hence, since Y is countable,
{vk } has a convergent subsequence. Thus, B is compact-valued, and from
Berge’s theorem the set of maximizers of EF (·|a) [⇡ (y u 1 (v (y)))] on B (·) is
non-empty valued and upper hemicontinous.
Finally, consider any z where at least one of ⇡ and u is strictly concave.
Then, if v1 ,v2 2 B (z), it is direct that (v1 + v2 ) /2 2 B (z) is strictly more
profitable that either v1 or v2 . Thus the maximum is unique, and hence continuous in z.
57
⇤
1.
v L (y|z)
.
D.2
Proof of Existence for u =
1
Suppose that u = 1, which corresponds to the agent having no liability
constraint. This section gives conditions underwhich a unique solution to (P)
exists and satisfies certain properties. Say that u(·) is regular if w = 1 and
u0 (w)u000 (w)
< 3 for all w 2 R. These conditions are quite mild; in particular,
u00 (w)
the second condition means that u0 (·) is not excessively convex, in the sense
that it has local concavity everywhere greater than 2. See Prekopa (1973)
and Borell (1975) for details.
Proposition 6. Suppose ⇡(y) ⌘ y, u(·) is strictly concave and regular, and
u = 1. Then for any a 0, there exists a unique contract v(·) that implements a at maximum profit. Furthermore, there exists ū < 1 independent of
u such that v(ȳ) < ū and v(y) > ū.
We begin this argument with an important lemma, which shows that if
vu (·) is an optimal contract for some limited liability constraint u and vu (y) >
u, then vu (·) remains optimal in the problem with any less binding limitedliability constraint u0 , including u0 = 1.
Lemma 5. Assume that for some u >
v u0 = v u .
1, vu (y) > u. Let u0 < u. Then,
Proof. Assume vu has vu (y) > u, but that when the limited liability constraint
is some u0 < u there exists a superior concave contract v̂ that implements a.
We will show that this leads to a contradiction.
Assume first that v̂(y) > 1 (as is automatic if u0 is finite). Then, for
small enough ", the contract (1 ") vu (·) + "v̂ (·) is both strictly cheaper than
vu (since u is strictly concave), and implements a subject to liability constraint
u, yielding the desired contradiction.
Assume instead that v̂(y) = 1. Begin by picking any point x0 > y where
x0 2 Cv̂ (since v̂(y) = 1, such points exist), and construct ṽ by applying a
sufficiently small positive amount of tx0 ,ȳ such that ṽ remains strictly cheaper
58
than vu . Since this adds a positive increasing function to v̂, both (IC-FOC)
and (IR) are strictly slack at ṽ.
For each y 2 [y, ȳ], let hy (·) be a supporting plane to ṽ at y. Let the concave contract vy (·) be given by vy (x) = ṽ (x) for x > y, and vy (x) = hy (x) for
x  y. For each x, vy (x) is weakly decreasing in y, with limy!y vy (x) = ṽ (x) .
R
Thus, by the monotone convergence theorem, as y ! y, vy (x) f (x|a) dx !
R
R
R
R
ṽ (x) f (x|a) dx, vy (x) fa (x|a) dx ! ṽ (x) fa (x|a) dx, and u 1 (vy (x))f (x|a) dx !
R 1
u (ṽ (x))f (x|a) dx. Hence, for y close enough to y, vy implements a and is
cheaper than vu . For any such y, vy (y) is finite, and we are back to the previous
case.
Proof. Given Lemma 5, it is enough to show that for some u, vu (y) > u.
Assume not, so that in particular, for all u, vu (y) = u. We will show that this
leads to a contradiction. We will henceforth restrict attention to u  0. For
u sufficiently negative, it cannot be the case that vu is linear. In particular, if
vu is linear, then since vu (ȳ) > u0 + c (a) , we have that
Z
vu (x) fa (x|a) dx =
Z
vu0 (x) ( Fa (x|a)) dx
u0 + c (a)
ȳ y
u
which diverges in u, contradicting that vu must satisfy (IC-FOC) with equality.
Hence, for each u, we can take a point xu 2 Cvu , and derive u and µu as in
the proof of Proposition 2.
Let zu (·) = ⇢ ( u + µu l (·|a)), where we follow the convention that ⇢ (s) = 1
for s  0. The contract vu will in general differ from zu since zu need be
neither concave nor satisfy the limited liability constraint. Note that nu (·) =
⇢ 1 (vu (·)) ( u + µu l (·|a)) =s vu (·) zu (·) .
Step 1 There is µ̄ < 1 such that µu  µ̄ for all u.
Proof of Step 1 Applying a small positive amount of txu ,ȳ adds cost at
R ȳ
R ȳ
rate at most ⇢ 1 (ū) xu (x xu )f (x|a) dx, adds incentives at rate xu (x
59
xu )fa (x|a) dx, and relaxes (IR). It follows that
1
µu  ⇢
R ȳ
(x
xu )f (x|a) dx
(x
xu )fa (x|a) dx
xu
(ū) R ȳ
xu
.
But, as in the proof that |Q (0)| > 0,
R ȳ
@ xu (x
R
@xu xȳ (x
u
R ȳ
(x
xu
= R ȳ
s
(x
xu
xu )f (x|a) dx
xu )fa (x|a) dx
xu )fa (x|a) dx
xu )f (x|a) dx
 0,
R ȳ
xu
fa (x|a) dx
xu
f (x|a) dx
+ R ȳ
and so we can take
µ̄ = ⇢
1
R
(x
(ū) R
(x
Step 2 There is µ > 0 and u⇤ >
Proof of Step 2 Choose
⇢
y)f (x|a) dx
< 1.
y)fa (x|a) dx
1 such that µu
µ for all u < u⇤ .
1 < u⇤  0 such that
1 1
⇢ (u0 + c (a)) , and
2
u0 + c (a) u⇤
c0 (a) <
,
ȳ y
1
(17)
(u⇤ ) <
where such a u⇤ exists since by assumption limw!
r⌘
⇢0 (⌧ ) .
sup
⌧ 2[ 12 ⇢
1 (u
0 +c(a)),1
60
1
1 u0 (w)
)
(18)
= 0. Let
Since ⇢ (1/u0 (w)) = u (w) , we have that
⇢
from which
⇢00
⇢0
⇣
⇣
1
u0 (w)
1
u0 (w)
0
✓
⌘
1
0
u (w)
◆
0
⌘ = u (w)
(u0 )3
=
(w) ,
u00
✓
u000 (w) u0 (w)
(u00 (w))2
◆
(19)
3 .
Since u is regular, it follows that ⇢00 < 0, and so r < 1. Let ¯lx = maxx lx (x|a) ,
and choose µ > 0 such that
1 ⇢ 1 (u0 + c (a))
, and
2 l (ȳ|a) l(y|a)
1 u0 + c (a)
µ < ¯
.
rlx ȳ y
(20)
µ <
(21)
Assume that for some u < u⇤ , µu < µ. We will show that this leads to a
contradiction, establishing the result.
Using Corollary 2 (which depends only on the necessity part of the proof
of Proposition 2, which is proven in Appendix B), and the fact that ȳ is free,
n (ȳ)  0, and so u + µu l (ȳ|a) ⇢ 1 (vu (ȳ)) ⇢ 1 (u0 + c (a)) . Thus,
u
+ µu l(y|a) =
⇢
=
+ µu l(ȳ|a)
u
1
1
⇢
2
µu l (ȳ|a)
1 ⇢ 1 (u0 + c (a))
l (ȳ|a)
2 l (ȳ|a) l(y|a)
(u0 + c (a))
1
(22)
l(y|a)
l(y|a)
(u0 + c (a)) ,
where the inequality follows from µu < µ and (20).
Since u < u⇤ , and by (17) , ⇢ 1 (vu (y)) = ⇢ 1 (u) < 12 ⇢ 1 (u0 + c (a)) . Thus,
using (22) , n(y) is strictly positive, and it follows by Corollary 2 that vu begins
with a linear segment, the slope of which (by concavity) is at least
u0 + c (a)
ȳ y
u
61
u0 + c (a)
.
ȳ y
(23)
But, using (22) and the definition of r, we have that for all x,
zu0 (x) = ⇢0 (
u
+ µu l(x|a)) µu lx (x|a)
 rµu ¯lx
u0 + c (a)
<
,
ȳ y
where the strict inequality follows from (21) . Hence, the initial linear segment
of vu crosses zu at most once (from below). This implies that the entire contract
is in fact linear with slope at least (u0 + c (a) u) /(ȳ y). In particular, let
xH be the right end of the linear segment. If xH is at or before the crossing
point, then vu violates (3) and so cannot be optimal by part 1 of Proposition
2. If xH < ȳ is after the crossing, then we violate Corollary 2. It follows that
vu generates incentives at least
u0 + c (a)
ȳ y
u⇤
> c0 (a)
using (18) . But we have shown that (IC-FOC) binds at vu , leading to the
desired contradiction.
Step 3 There is u0 +c (a) > uc >
uc , then zu (·) is concave at x.
1 such that if u < u⇤ and ⇢ (
u
+ µu (l (x|a))) <
Proof of Step 3 Note first that ⇢ is trivially concave anywhere that it is
equal to 1, and that by assumption, lims!0 ⇢ (s) = 1. Hence, it is enough
to prove concavity where ⇢ ( u + µu (l (x|a))) is finite. But, it follows from (19)
and the fact that u is regular that limt#0 ⇢00 (t) /⇢0 (t) = 1, and so ⇢00 (t) /⇢0 (t)
62
is negative for t below some t0 . Assume
@2
⇢(
@x2
u
+ µu (l (x|a)))
u
+ µu (l (x|a)) < t0 . Then,
@ 0
(⇢ ( u + µu (l (x|a))) µu lx (x|a))
@x
= ⇢00 ( u + µu (l (x|a))) (µu lx (x|a))2 + ⇢0 (
=
⇢00
(
s ⇢0
⇢00
 0 (
⇢
=
u
+ µu (l (x|a))) µu +
u
+ µu (l (x|a))) µ +
u
+ µu (l (x|a))) µu lxx (x|a)
lxx
(x|a)
lx2
lxx
(x|a) .
lx2
The second term is bounded by assumption. The first term diverges to 1 as
0
u + µu (l (x|a)) ! 0. Hence, since ⇢ is monotone, and since limw! 1 u (w) =
1, the result follows.
Step 4 As in the derivation of r in Step 2, let r̂ be such that for all t
⇢ 1 (uc ) , ⇢0 (t)  r̂. Let 1 < û  u⇤ satisfy
ŝ ⌘
u0 + c (a)
ȳ y
û
max c0 (a) , µ̄¯lx r̂ ,
and assume that u < û. Then, zu (y)  u.
Proof of Step 4 Assume that zu (y) > u. Then, since vu (y) = u, vu begins
with a linear segment of positive length of slope at least ŝ, and so by Proposition 2 and Part 1 of Definition 1, crosses zu from below, and is strictly above zu
for an interval of positive length as well. Let xu,c be defined by zu (xu,c ) = uc .
If vu has its initial crossing of zu at or before xu,c , then since zu is concave until
xu,c , vu remains above zu until xu,c . But then, since for x > xu,c , ŝ zu0 , vu in
fact never recrosses zu . On the other hand, if the initial crossing of zu by vu is
after xu,c , then again, since vu has slope greater than zu0 for x > xu,c , vu never
recrosses zu . In either case, by Corollary 2, vu is thus linear on all of [y, ȳ], a
contradiction.
63
Step 5
Let uy0 = u0 + c (a)
c0 (a) (ȳ
y0 ) >
1. Then, vu (y0 )
u y0 .
Proof of Step 5 Since vu (ȳ)
u0 + c (a) , it follows that everywhere on
[y, y0 ), vu (·) is below the line L(·) that goes through (y0 , vu (y0 )) and (ȳ, u0 +
c (a)), and everywhere on (y0 , ȳ], vu (·) is above L(·). Hence, since fa < 0 on
[y, y0 ) and fa > 0 on (y0 , ȳ],
0
c (a) =
Z
Z
=
vu (x) fa (x|a) dx
L (x) fa (x|a) dx
u0 + c (a) vu (y0 )
.
ȳ y0
Rearranging yields the desired result.
Step 6 Choose 1 < us < min uy0 , uc , ⇢( µ̄l(y|a)), û small enough that
for all t  us ,
✓
◆
1
0
⇢
µ̄lx ŝ,
(24)
u0 (u 1 (t))
where lx = minx lx (x|a) > 0. Since ⇢0 (⌧ ) diverges to 1 as ⌧ # 0, and since
1/u0 (u 1 (t)) goes to 0 as t # 1, such a us is guaranteed to exist.
Step 7 Choose u < us . Let z̄u (·) = ⇢ ¯ u + µ̄l (·|a) , where ¯ u solves ⇢( ¯ u +
µ̄l(y|a)) = u. By Step 4, zu (y)  u, and so, since µu  µ̄, zu (·)  z̄u (·) . Let
xu,s be defined by z̄u (xu,s ) = us . Since ¯ u + µ̄l(y|a) = 1/u0 (u 1 (u)) > 0, it
follows that ¯ u + µ̄l(y0 |a)
µ̄l(y|a), and hence
⇢ ¯ u + µ̄l (y0 |a) > ⇢( µ̄l(y|a)) > us ,
where the last inequality is by definition of us in Step 6. Thus, xu,s < y0 .
64
Step 8
For all x < xu,s , vu (x)  z̄u (x) .
Proof of Step 8 Let xu,c be defined by zu (xu,c ) = uc . By construction, z̄u (·)
is concave where x  xu,c . Using (24), z̄u0 (·) > ŝ for x < xu,s , and z̄u0 (·) < ŝ
for x
xu,c . Assume that for some x̃ < xu,s , vu (x̃) > z̄u (x̃)
zu (x̃) . By
0
0
Corollary 2, vu is linear at x̃. If vu (x̃)  z̄u (x̃) , then, since z̄u is concave on
[y, xu,s ], and again using Corollary 2, vu is also above z̄u , and hence linear, for
all x in [y, x̃]. But then,
vu (y)
z̄u (y)
vu (x̃)
z̄u (x̃) > 0,
contradicting that vu (y) = u. Thus, vu0 (x̃) > z̄u0 (x̃) > ŝ. But then, vu remains
linear, and hence strictly above the concave function zu at least until xu,c . For
x xu,c , z̄u0 (x̃)  ŝ, and so as before v can never re-cross z̄u , and so a fortiori
can never re-cross zu . Hence vu is linear on [x̃, ȳ] , with slope at least ŝ. Let L
be the line that agrees with vu on [x̃, ȳ] . To the left of x̃, vu , being concave,
lies below L. But, x̃ < xu,s < y0 , and so, since fa (·|a) is negative on [y, x̃],
Z
vu (x) fa (x|a) dx
Z
L (x) fa (x|a) dx
ŝ > c0 (a) ,
again a contradiction.
R
Step 9 We show that limu! 1 vu (x) fa (x|a) dx = 1. For u sufficiently
negative, this provides the necessary contradiction to the original supposition
that vu (y) = u for all u, proving the result.
Proof of Step 9 By Step 8, for u sufficiently negative, vu (x)  z̄u (x) for all
x  xu,s . Let vuT truncate vu to never pay more than us . Since max (0, vu (x) us )
R
is an increasing function, max (0, vu (x) us ) fa (x|a) dx
0, and hence,
R
R T
vu (x) fa (x|a) dx
vu (x) fa (x|a) dx. Note also that since vu (y0 ) > us ,
T
vu (x) = us for all x
y0 . Let z̄uT similarly truncate z̄u to pay us to the
65
right of xu,s . Then, z̄uT is everywhere at least as large as vuT , but equal to vuT
everywhere to right of y0 . Hence, since fa is negative to the left of y0 , we have
Z
vu (x) fa (x|a) dx
Z
vuT
Z
(x) fa (x|a) dx
z̄uT (x) fa (x|a) dx.
R
To arrive at a contradiction, it would thus be enough to show that z̄uT (x) fa (x|a) dx
diverges as u ! 1. But, by Moroni and Swinkels (2014, Lemma 4), under
R
our regularity conditions, z̄u (x) fa (x|a) dx does diverge as u ! 1.
Let
u⇤⇤ = ⇢ 1 + µ̄ l (ȳ|a) l(y|a) < 1.
Then, for all u sufficiently negative that
Z
z̄u (x) fa (x|a) dx
Z
z̄uT
u0 (u
 1, z̄u (ȳ)  u⇤⇤ . Hence,
1
(x) fa (x|a) dx =

1 (u))
Z
Z
z̄u (x)
z̄uT (x) fa (x|a) dx.
ȳ
y0
 (u⇤⇤
< 1,
z̄u (x) z̄uT (x) fa (x|a) dx
Z ȳ
us )
fa (x|a) dx
y0
where the first inequality follows because z̄u (x) z̄uT is weakly positive, and
the second because it is bounded above by u⇤⇤ us .
E
Agent sends a contractible message
In this section, we consider an alternative mechanism for our baseline model
in which, after the intermediate output x has been observed by the agent but
before the final output y has been realized, the principal asks the agent to
voluntarily report x̃ and then rewards the agent as a function of x̃ and y.
Throughout this section, we assume that both parties are risk-neutral.
Because the agent is cash constrained, without loss, we can restrict attention to mechanisms in which, given a report x̃ and final output realization y,
66
the agent is paid s(y)I{y=x̃} M I{y6=x̃} for some upper-semicontinuous function
s(·). We show that as in the baseline model, this mechanism results in a linear
contract being optimal. The principal’s problem can be written as
⇥
max EF (·|a),G y
⇤
s(y)I{y=x̃} + M I{y6=x̃}
a, s(·)
n
h
s.t. a, G, x̃ 2 arg max EF (·|ã),G̃ s(y)I{y=x̃˜}
˜
ã,G̃2G,x̃
⇥
EF (·|a),G s(y)I{y=x̃}
s(·)
M I{y6=x̃}
M
⇤
c(a)
M I{y6=x̃˜}
u0
i
c(ã)
o
(25)
Note that x̃ is a function that maps the intermediate output x to a report
made to the principal. The first step towards an optimal contract involves
characterizing the agent’s optimal reporting and risk-taking strategy. For any
intermediate output x > y and contract s(·), define
s (x)
= max
:
(y
y)
M = s(y) for some y
x .
(26)
Intuitively, s (x) is the smallest slope such that s (x)(y y) M s(y) for
all y
x. The following lemma shows that given any realization x > y, the
agent will optimally choose the distribution Gx and his report such that his
expected payoff is equal to s (x)(x y) M .17
Lemma 6. For any contract s(·) and any intermediate output x 2 Y, the
principal’s expected payment to the agent equals:
s (x)
⇥
⌘ max EGx s(y)I{y=x̃}
Gx , x̃
M I{y6=x̃}
⇤
=
(
s(y)
s (x)(x
y)
M
if x = y
.
if x > y
(27)
Proof. Fix an arbitrary s(·) and x > y. First, we show that there exists
⇥
⇤
some Gx and x̃ such that EGx s(y)I{y=x̃} M I{y6=x̃} = s (x)(x y) M .
17
If x = y, then the agent is compelled to choose Gy (y) = 1, so his expected payoff is
equal to s(y).
67
Fix some arbitrary x, and let ŷ
x satisfy s (x)(ŷ y) M = s(ŷ), and
recall that by the definition of s (·), such ŷ exists. Next, let x̃ = ŷ and
x y
Gx (y) = (1 pŷ )+pŷ I{y ŷ} , where pŷ = ŷ y ; i.e., y = y with probability 1 pŷ ,
and y = ŷ with probability pŷ . Then the agent’s expected payoff is equal to
⇤ ŷ x
x y
x y ⇥
pŷ s (ŷ) (1 pŷ )M = ŷ y s (ŷ) ŷŷ xy M = ŷ y s (x)(ŷ y) M
M=
ŷ y
y) M .
s (x)(x
Next, we show that the agent cannot achieve a greater payoff by choosing a
different x̃ and Gx . Because the agent is paid s(y)I{y=x̃} , for any given report
x̃, he will choose a distribution Gx to maximize the probability that y = x̃
subject to the constraint that EGx [y] = x. This is accomplished by choosing
Gx (·) such that y = x̃ with some probability px̃ and y = y with probability
1 px̃ , where px̃ x̃ + (1 px̃ )y = x. It follows from algebraic manipulations
x y
that Gx (y) = (1 px̃ ) + px̃ I{y x̃} , where px̃ = x̃ y .
Suppose that there exists some x̃ such that px̃ s(x̃) (1 px̃ )M > pŷ s (ŷ)
(1 pŷ )M = s (x)(x y) M . Then there must exist some ˜ > s (x) such
that ˜ (x̃ y) M = s(x̃). However, this is not possible by the definition of
s (x). Therefore, for all x, the maximal achievable payoff is equal to s (x)(x
y) M .
Because the agent receives a positive pay only if y = x̃, he will optimally either report x̃ = x and choose Gx (y) = I{y x} , or report some x̃ > x and choose
x y
Gx (y) = (1 p) + pI{y x̃} , where p = x̃ y ; i.e., a distribution that puts mass p
on x̃ and mass 1 p on y. The set
: (y y) M = s(y) for some y x
characterizes the set of feasible gambles, and (x y) M is the agent’s payoff
corresponding to the gamble associated with when the intermediate output
realization is equal to x. Notice that s (x) is the steepest slope in that set,
and hence it represents the gamble that maximizes the agent’s payoff at x.
68
Using Lemma 6, we can re-write the principal’s problem as
max EF (·|a) [x
s (x)]
a, s(·)
s.t. a 2 arg max EF (·|ã) [ s (x)]
ã
EF (·|a) [ s (x)]
s(·)
c(a)
c(ã)
u0
M
where for any contract s(·), s (·) is given by (27).
Let sLa (y) = c0 (a)(y y) w, where w = min M, c0 (a)(a y) c(a) u0 .
The following result shows that this linear contract, which was shown to be
optimal in Section 4, implements any effort level a such that the agent’s limited
liability constraint binds at maximum profit.
Proposition 7. Fix any effort a
implements a at maximum profit.
0. If sLa (·) =
M , then the contract sLa (·)
Proof. First note that for any contract s(·), s (·) is weakly decreasing. If s(·)
is affine and satisfies s(y) = M , then s (·) is constant. Suppose that ŝ(·)
implements a at maximum profit and ŝ (x) is strictly decreasing for at least
some x.
Next, let sL (y) = (y y) M , where is chosen such that it satisfies
⇥
⇤
EF (·|a) sL (y)
y) = 0. By the intermediate value theorem, such
ŝ (y)(y
exists, and there exists an interior y ⇤ such that ŝ (y)
if and only if
R
⇤
y  y . Because
[
ŝ (y) is first negative and then positive,
ŝ (y)] (y
fa (·|a)
y)f (y|a)dy = 0 by construction, and f (·|a) is strictly increasing, Beesack’s
inequality implies that
Z
[
ŝ (y)] (y
y)fa (y|a)dy > 0 .
Therefore, sL (·) implements some effort level a0 > a, which implies that
c0 (a).
69
>
Observe that sLa (y) < sL (y) for all y > y, because sLa (y) = M by assumption and c0 (a) < . Moreover, sLa (·) implements effort level a and satisfies
both the individual rationality and limited liability constraints. Therefore,
sLa (·) implements effort a at strictly higher profit than ŝ(·), which completes
the proof.
70