GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.225

Fair Contracts

Shingo Ishiguro

November 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Fair Contracts
Shingo Ishiguro∗
Graduate School of Economics
Osaka University
November 17, 2011

Abstract
In this paper we present an axiomatic approach to characterize the optimal
contracts, which we call “fair contracts,” in the general moral hazard model.
The two main axioms we employ are incentive eﬃciency and no–envyness. The
incentive eﬃciency requires that agents of organization select the Pareto eﬃcient
contracts among all possible incentive compatible contracts. No–envyness isequity requirement to ensure that each agent does not envy contracts of othersin the same organization. We then show that, due to the tension between
incentive eﬃciency and no-envyness, fair contracts have the very simple featurethat risk averse agents are oﬀered the ﬁxed wage to choose only the least costly
action.
Keywords: Moral Hazard, Incentive Contracts, Fairness.
JEL Classiﬁcation Number: D82, D86∗
Correspondence: Shingo Ishiguro, 1-7 Machikaneyama, Toyonaka, Osaka 560 0043, Japan. email: ishiguro@econ.osaka-u.ac.jp1

1

Introduction

In this paper we investigate the moral hazard model in which agents choose unobservable actions and provide an axiomatic approach to characterize the optimal contracts, which we call “fair contracts,” in the view point of both eﬃciency and equity.
The standard approach to the moral hazard problem (for example see Grossman andHart (1983), Rogerson (1985), Jewitt (1988)) has focused only on the eﬃciency inthe sense that the principal chooses a contract to maximize her expected payoﬀ subject to the set of constraints as follows. (i) incentive compatibility (IC) constraint:
contract must give the agent the proper incentive to choose a right action because
his action is not observable. (ii) individual rationality (IR) constraint: contract
should give the agent at least the reservation utility. (iii) Budget balanced (BB):total incomes of both parties sum up to the realized total revenue. We then saythat a contract satisﬁes incentive eﬃciency (IE) if it satisﬁes (IC), (BB) and there
exist no other Pareto improving contracts which satisfy (IC), (BB). In the standard
principal-agent model a contract is selected in favor of the principal from all (IE)
contracts by satisfying (IR) of the agent.
We provide a slightly diﬀerent approach to address the optimal contract design
problem in the moral hazard models. We impose several axioms which must be taken
into account when designing contracts to agents in organization and then characterize the contracts satisfying these axioms. The axioms we will employ include not
only the standard eﬃciency criterion such as (IE) mentioned above but also a sortof equity concern among agents, which we will call no-envyness (NE). Here (NE)requires that each agent does not envy any contracts oﬀered to others within thesame organization. This is analogous to the concept of “no-envyness” used in theliterature of social choice theory (see Thosmson (2010) for a comprehensive survey
on this topic): a resource allocation is said be no-envy if every individual does notenvy the consumption bundles allocated to others.1 We will modify this idea in thecontext of the principal-agent model and utilize it as an equity condition in orderto characterize the optimal contracts.
In addition to (IE), (IR) and (NE), we also consider the possibility that the wage
scheme of each agent is non-decreasing in his own veriﬁable performance (output).
We call this axiom monotonicity (MON). This is also a reasonable requirement
because most wage schemes used in practice satisfy this condition. We can also givetheoretical justiﬁcation for this condition by considering ex post moral hazard ofagents as follows. Each agent can secretly dispose his own output ex post after itis realized. Then, if his wage is decreasing in some parts of his outputs, the agent
has the incentive to discard some amount of the realized output and obtain higher
wages. Thus (MON) must be satisﬁed for ruling out such ex post moral hazard (seeInnes (1990) and Matthews (2001) for a similar argument).
We call a contract proﬁle of agents in organization fair contract when it satisﬁes
1
See also Foley (1967), Pazner and Schmeidler (1974, 1978) and Varian (1974) for the classic
works on equity in economics.

2

all these axioms (IE), (NE), (MON) and (IR).2 Then we show that fair contracts
have the very simple feature as follows: every risk averse agent is oﬀered a ﬁxed wage
so that he chooses the least costly action. On the other hand, risk neutral agents are
motivated to choose the actions which maximize the surplus of organization, given allrisk averse agents choose the least costly action. Hence the optimal contracts (“faircontracts” in our notion) become drastically simple once we take into account not
only the incentive eﬃciency but also non-envyness for contracting in organizations.
The basic intuition behind our characterization result is the following: since risk
averse agents are assumed to be heterogeneous with respect to their abilities for
performing tasks, they must be oﬀered diﬀerent wage schemes which diﬀer in the
degrees of how much their wages reﬂect their veriﬁable performances. If a wage
scheme responds more highly to the changes of the performances than another, it
is said to be higher-powered incentive scheme than the other. Put diﬀerently, ahigher-powered incentive scheme has “steeper slope” of reward with respect to theperformances than a lower-powered scheme. Of course, higher-powered incentive
scheme involves more risk on risk averse agent than lower-powered one. Then consider a fair contract which induces risk averse agents with diﬀerent abilities to work
hard. High ability agents can improve the probability distributions of their performances at lower marginal action costs than the low abilities. Thus, less risk isneeded for the high ability agents to work as hard as the low abilities. Then, inthe view point of incentive eﬃciency the high ability agents should be oﬀered lower
powered incentive scheme than the low abilities because the former can work as hardas the latter at smaller risk. However, then the high ability agents envy the higherpowered incentive scheme oﬀered to the low ability agents because the former can
exploit their (high) ability under the wage scheme which more highly responds tothe veriﬁable performances. When facing such conﬂict between incentive eﬃciency
and no-envyness, the organization ﬁnds it impossible to have risk averse agents with
diﬀerent abilities work hard together. As a consequence, any fair contract must
involve a ﬁxed wage for diﬀerent risk averse agents who are induced to choose only
the least costly action.
Our model has the relative advantage to the standard moral hazard models inthat we can provide a simple characterization result even in general production
environment. In the standard approach to the moral hazard problem, it is quite
diﬃcult to give sharp predictions about exact features of optimal contracts: how do
optimal wages vary with veriﬁable performances? What shapes does the optimal
wage schedule display? Is it linear or non-linear? In contrast our result shows thatthe optimal contracts become drastically simple once we take into account not only
eﬃciency but also some sort of equity among agents in organization.
Beside the theoretical interest, it is also important to extend the standard
2

This is also analogous to the concept of “fair allocation” used in the literature of social choicetheory. An allocation is said to be fair if it satisﬁes Pareto eﬃciency and no-envyness (Thomson(2010)). In our context the Pareto eﬃciency is replaced by incentive eﬃciency (IE) due to thepresence of the moral hazard problem.

3

principal-agent models to encompass some notions of equity or fairness. This isbecause in real world there are organizations which may seek diﬀerent objectives
from maximizing the principal’s proﬁt, e.g., non-proﬁt organizations, governments,and state–owned enterprises. These organizations may take into account the welfareof employees, citizens and so on, and may pursue not only eﬃciency but also somekinds of fairness, e.g., governments would try to design “fair” tax systems. Even in
proﬁt organizations equity is often one of the major concerns as well as eﬃciency
and proﬁtability. For example, top management of corporations might try to avoid
conﬂict among employees who are concerned about their relative wages in the same
workplaces.
Related Literature
There are two strands of the literature which are related to our paper. First, several
recent papers incorporate the aspects of fairness into the standard principal-agent
models.3 However, in these papers equity concern and other-regarding preferences
are directly introduced into the payoﬀ functions of agents in some speciﬁc ways: forexample, the payoﬀ function of an agent is assumed to depend on relative incomes
between him and his colleagues. Our approach is diﬀerent from these papers because
we do not impose direct speciﬁcations of fairness concern on the payoﬀ functions ofagents but rather we use an axiomatic approach by using (NE) as a natural axiom.
Thus our model can avoid the problem regarding what speciﬁcation of the payoﬀ
function is the most plausible to describe the preference over equity and fairness.
Second, some papers show the optimality of simple contracts in the moral hazard
environments. Holmström and Milgrom (1987) show that the optimal reward toagent becomes linear with respect to the eventual outcomes in the dynamic modelin which agent controls the drift rate of the stochastic process of outputs (see also
Hellwig and Schmidt (2002) for further elaboration of the model). Our paper isdiﬀerent from theirs because we focus on the standard static moral hazard environment only except for incorporating the concept of no-envyness as an equity concern.
Holmström and Milgrom (1994) also show that the optimal contract to agent becomes ﬁxed wage in the multi-task context in which agent performs multiple tasks
at a time. Although our model also shows the optimality of the ﬁxed wage as intheirs, our approach is diﬀerent from theirs again because we investigate the tension
between incentive eﬃciency and no-envyness, which results in the lower-powered
incentive schemes.
The remaining sections are organized as follows: in Section 2 we set up the basicmodel of moral hazard in an organization with multiple agents. In Section 3 weprovide a characterization result that the optimal contracts which satisfy incentive
eﬃciency and no–envyness become the ﬁxed wage scheme for all risk averse agents3

See for example Bartling (2010), Englmaier and Wamback (2010), Fehr and Schmidt (2000,
2004), Itoh (2004), Neilson and Stone (2010) and Rey-Biel (2008) for recent development of agency
models incorporating inequity aversion.

4

when we impose some technical conditions on the probability distributions of theveriﬁable performances. In Section 4 we will show more general characterization
result without imposing the speciﬁc conditions on the information structure, given
the heterogeneity among risk averse agents is suﬃciently small. In Section 5 wewill proceed to show the existence proof of fair contract. Our proof is constructive.
We ﬁnd a fair contract which satisﬁes all the axioms we have mentioned, (IE),(NE), (MON) and (IR), under certain conditions. In Section 6 we will discuss therobustness and extensions of our results: ﬁrst, we show that our characterization
results are not substantially changed by dropping the axiom of monotonicity (MON).
Second, we will discuss the implications about the priority of eﬃciency and equity
for designing incentive contracts. Our concept of fair contracts is interpreted as theone we are capturing the situation in which agents of organization ﬁrst seek the
incentive eﬃciency and then, as the second step, they select the contracts which
satisfy no-envyness as well as other axioms. We will call such contracts eﬃciency–
ﬁrst–optimal contracts. One might however think that in some organizations equity
may be taken as the ﬁrst priority before considering the incentive eﬃciency. Thismight be suitable in non–proﬁt organizations and public sectors. In such situations
agents of organization ﬁrst identify the set of no–envy contracts and then, as thesecond step, they select the Pareto eﬃcient contracts among all possible no-envy
contracts together with other axioms. We will call such contracts equity-ﬁrst-optimal
contracts. We then show that the set of equity–ﬁrst-optimal contracts is not smaller
than the set of eﬃciency–ﬁrst–optimal contracts. This result implies that there may
exist the possibilities that high–powered incentive contracts are oﬀered to risk averse
agents even when heterogeneity among agents is so small that all eﬃciency–ﬁrst–
optimal contracts must have the ﬁxed wage schemes of risk averse agents. Thus
incentive schemes oﬀered in organizations which take equity as the ﬁrst priority are
more likely to be higher powered than those in organizations which take eﬃciency
as the ﬁrst priority.

2

Model

We consider an organization (or a team) which consists of N agents. Let I be the setof all agents where #I = N . Let Ir be the set of risk averse agents where Nr = #Ir
and In the set of risk neutral agents where Nn = #In . Thus I = Ir ∪ In . We
assume that there exist at least two risk averse agents and one risk neutral agent inorganization:
Assumption 1. Nr ≥ 2 and Nn ≥ 1.
Agent i chooses an unobservable action ai ∈ A where A is ﬁnite and #A =M + 1 ≥ 2. We denote by an ∈ A a generic element of A where a0 ≡ 0 ≤ a1 ≤ a2 ≤· · · ≤ aM . Let a ≡ (a1 , a2 , ..., aN ) be a vector of actions taken by all agents.

5

Agent i generates a veriﬁable performance y i ∈ Y . We can interpret y i as output
of agent i which is used to yield the revenue of organization as we will explain more
below. We assume that Y is a ﬁnite set and #Y = L ≥ 2. Let yni be a genericelement of Y . We denote by Pni (a) ∈ (0, 1) the probability of the performance yni ofagent i being realized conditional on his action a ∈ A. Let y ≡ (y 1 , y 2 , ..., y N ) be avector of the realized performances of all agents. We assume that the performances
y are independently distributed among agents.
Agents are heterogeneous in terms of their abilities for performing the tasks assigned to them. Speciﬁcally, agents diﬀer from each other in terms of the probability
distributions of their performances (Pni (ai ))L
n=1 and their action costs Gi (a). Agenti has the following standard utility function which is separable between utility on
income w and disutility on action a ∈ A:
ui (w) − Gi (a)(1)

where Gi is increasing and we normalize Gi (0) = 0. Here we assume that the utility
functions on incomes are identical across all risk averse agents. Thus we set ui = u
for all i ∈ Ir . Let φ ≡ u−1 be the inverse function of u. We also suppose that theutility function on income of risk neutral agent is given by ui (w) = w for each i ∈ In .
Remark. We can allow all A, Y and u to depend on identities of agents. However,
our main results are not substantially changed as long as the diﬀerences in thesecharacteristics are small.
The organization can generate the total revenue R(y) which depends on a proﬁleof performances (outputs) of agents y. R is increasing in each argument. In what
follows we will use notations as follows: let Ey [·|a] be expectation over the performances of all agents y ∈ Y N conditional on an action proﬁle of them a. Let Eyi [·|ai ]be expectation over the veriﬁable performance y i of agent i conditional on his action
ai . Let also Ey−i [·|a−i ] denote expectation over the veriﬁable performances of allother agents than i conditional on their actions a−i = (aj )j̸=i .
Since the performances (or outputs) of agent y are veriﬁable but their actions a
are not, a wage scheme of agent i should be contingent only on the realizations of y.
A contract of agent i is deﬁned by C i = {wi (y), âi } which speciﬁes wi (y) the wage
scheme depending on realization of veriﬁable performances y and an action âi ∈ A
to be taken. Let C = (C i )i∈I be a proﬁle of contracts. We call a wage scheme wi (y)ﬁxed wage whenwi (y) = w
for some constant w for all y ∈ Y N .
We suppose that the wage schemes {wi (y)}i∈I must be budget balanced ex post,
i.e. the total wages of all agents should be equal to the organization revenue, as
follows:∑wi (y) = R(y) ∀ y ∈ Y N .
(BB)
i∈I

6

We denote by C the set of all contracts satisfying (BB). We will restrict our attention
to the class of the wage schemes in C in what follows.
Next we will consider several axioms to be satisﬁed in organization.
First, we deﬁne the eﬃciency concept in the presence of the moral hazard problem. One reasonable notion of eﬃciency is incentive eﬃciency (IE) which is deﬁned
as the Pareto eﬃcient contracts satisfying incentive compatibility (IC) of agents.
(IC) requires the following: since actions of agents are not observable, a contract
proﬁle C ∈ C should give them the proper incentives to choose speciﬁed actionsâ = (âi )i∈I . In other words a contract proﬁle C ∈ C should require that every agent
follows the instruction to choose speciﬁed action âi given all others doing so:
Ey [ui (wi (y))|âi , â−i ] − Gi (âi ) ≥ Ey [ui (wi (y))|a′ , â−i ] − Gi (a′ ) ∀ a′ ̸= ai .

(IC)

Here agents simultaneously choose their actions and hence each of them takes theactions of others as given when he chooses his action because, by assumption, agentscannot observe their actions each other. Thus an action proﬁle â should be a Nashequilibrium in the game in which agents simultaneously choose their actions.
We call a contract proﬁle C ∈ C incentive feasible if it satisﬁes (IC). We denote
by C F ⊂ C the set of all incentive feasible contracts.
Although there are several ways to deﬁne the eﬃciency in the presence of private
information, we follow the argument by Holmström and Myerson (1983) and deﬁne
the eﬃciency on the ground of contract satisfying (IC). More formally, take an
incentive feasible contract C ∈ C F and deﬁne the resulting expected utility of agenti as∑
U i (C i ) ≡Ey [ui (wi (y))|â] − Gi (ai ).
(2)nThen, our ﬁrst axiom is incentive eﬃciency (IE) as follows:
Incentive Eﬃciency (IE): An incentive feasible contract C ∈ C F is incentive
eﬃcient if there exist no other incentive feasible contract C ′ ∈ C F with C ′ ̸= C′
such that U i (C i ) ≥ U i (C i ) for all i ∈ I with strict inequality being held for at least
one agent.
In the standard principal-agent model it has been assumed that one party, say
the principal, chooses her optimal contract among all (IE) contracts given the agent
receives at least the reservation utility. In other words the optimal contract isselected in favor of the principal on the Pareto frontier constrained by (IC). In thispaper we do not make any restrictions on bargaining powers of agents in organization
but simply we require that a contract must be incentive eﬃcient.
In addition to (IE), we impose an axiom representing a sort of equity requirement,
which we call no–envyness (NE), as follows:
Ey [ui (wi (y))|âi , â−i ] − Gi (âi ) ≥ max
Ey [ui (wj (y))|a′ , â−i ] − Gi (a′ ) ∀ j ̸= i, a′ ∈ A.
′
a ∈A

(NE)
7

(NE) states that each agent i does not envy the wage scheme of others wj (y) compared to his own scheme wi (y). If agent i were oﬀered the wage scheme of agent
j, he imagines that he could receive the expected utility corresponding to the righthand side of (NE) given he will choose his action a′ ∈ A appropriately and all otherschoose the speciﬁed actions â−i . (NE) then requires that agent i does not envy thecontract oﬀered to others.
Note that (NE) is analogous to the concept of no-envyness used in the literatureof social choice theory (Thomson (2010)). In our model each agent imagines what
would happen if he were oﬀered a contract of other agent before he chooses his
action. In this sense our notion of no-envyness is imposed at the ex ante stage
before agents choose actions. However, an alternative condition for no-envyness isalso possible. For example, some agent may envy others after they chose actions
and the ﬁnal performances were realized. We will discuss this ex post case in the
concluding remarks later.
We also impose individual rationality (IR) as an axiom to be satisﬁed. We denoteby U i the reservation utility of agent i who would receive outside the organization
if he rejected the contract C i and left the organization. Then a contract C i shouldigive agent i at least the reservation utility U , i.e. it should be individually rational
(IR):
i
(IR)
Ey [ui (wi (y))|â] − Gi (âi ) ≥ U ∀ i ∈ I.
Finally, we impose monotonicity (MON) that the wage scheme of each agent is
non-decreasing in his own veriﬁable performance (output). This is stated formallyas follows. For all i ∈ I,
′′

′

′′

′

wi (y i , y −i ) ≥ wi (y i , y −i ) ∀ y i > y i , ∀ y −i(MON)Many contracts used in practice satisfy such monotonicity and hence this is a reasonable condition. As in Innes (1990) and Matthews (2001), we can also give atheoretical justiﬁcation for this as well: each agent can secretly dispose his output
y i before he provides it to the organization.4 Then, if wi (y ′′ , y −i ) < wi (y ′ , y −i ) forsome y ′′ > y ′ , agent i will dispose y ′′ − y ′ > 0 and obtain higher wage wi (y ′ , y −i )even when y ′′ is realized. Thus wi must be non-decreasing in y i .
Remark. We will discuss below that we can dispense with (MON) (see Section 6).

We call a contract proﬁle C = (C i )i∈I ∈ C fair contract if it satisﬁes all the above
axioms (IE), (NE), (IR) and (MON). Let C f denote the set of all fair contracts. It
may be the case that C f = ∅.
4

We assume here that the supplied output after the decision to dispose was made is veriﬁable
but it is not veriﬁable what outputs each agent actually produced before the decision to dispose is
made.

8

3

Characterization of Fair Contracts

In this section we will characterize the fair contracts. To obtain sharper characterization result, we modify the notion of incentive eﬃciency slightly as follows:
Modiﬁed Incentive Eﬃciency (IE∗ ): Let C˜F ⊂ C be the set of all contracts
satisfying (IC) and (MON). Then a contract proﬁle C ∈ C˜F is said to be modiﬁed
incentive eﬃcient, denoted by (IE∗ ), if there exist no other contarcts C ′ ∈ C˜F with′C ′ ̸= C such that U i (C i ) ≥ U i (C i ) for all i ∈ I with strict inequality being held forat least one agent.
Let C∗f denote the set of all contracts satisfying (IE∗ ), (NE) and (IR).
Then we can verify that C f ⊆ C∗f .
Lemma 1. C f ⊆ C∗f .
Proof. Take C ∈ C f but suppose that C ∈/ C∗f . Then, since C satisﬁes (NE) and(IR) but C ∈/ C∗f , C must not satisfy (IE∗ ). Since C satisﬁes (IC) and (MON),
we have C ∈ C˜F . Then, since C is not modiﬁed incentive eﬃcient, there must
exist a diﬀerent contract C ′ ̸= C where C ′ ∈ C˜F such that C ′ Pareto dominates C.
However, since C ′ ∈ C˜F implies that C ′ satisﬁes (IC), we must have C ′ ∈ C F . Then,C ′ is incentive feasible and Pareto dominates C but this contradicts the fact that C
is incentive eﬃcient. Q.E.D.
In this section we will focus on the set C∗f which is larger than C f . Then we
will give a characterization result on C∗f which is sharper than the characterization
result on the set C f . With a slight abuse of deﬁnition, we call C ∈ C˜F fair contract
in the same way as deﬁned in the previous section. Then we will show that all
fair contracts C ∈ C˜F must involve ﬁxed wages oﬀered to risk averse agents undercertain conditions. Then Lemma 1 implies that all contracts in C f must also haveﬁxed wages oﬀered to risk averse agents.
We take any fair contract Ĉ ∈ C∗f where Ĉ i = {wi (y), âi } for each i ∈ I and ﬁx
it in this section.
We ﬁrst show the following result:Lemma 2. Any fair contract Ĉ i oﬀered to risk averse agent i ∈ Ir who is induced tochoose a higher action âi > 0 than the least costly one (zero) must be individualistic
in the sense that ŵi (y) = ŵi (y i ) for all i ∈ Ir .
Proof. Appendix.
Since the performance signals are statistically independent among agents, it iseﬃcient for wage scheme of every risk averse agent to depend solely on his own
9performances. If this is not the case for some risk averse agent, it is possible todesign the new wage scheme of that agent such that it induces the same expectedutility and the same action as under the original scheme while reducing the risk
imposed on him by eliminating the dependence of all other agents’ performancesfrom his wage scheme. Then some risk neutral agents are better oﬀ by reducing therisk on the risk averse agents while unchanging their incentives. Thus (IE∗ ) requiresthat the wage scheme of every risk averse agent must be individualistic.
Due to Lemma 2 we will denote by wi (y i ) the wage scheme oﬀered to risk averse
agent i, which solely depends on his own performance y i = yn ∈ Y .
Next we consider the following implementation problem where the expected wageof risk averse agent j ∈ Ir is minimized when implementing an action a ∈ A fromhim:
Problem (CMj − a)∑Pnj (a)w(yn )minw(yn )

nsubject tow(y ′′ ) ≥ w(y ′ ) ∀ y ′′ > y ′∑∑Pnj (a)u(w(yn )) − Gj (a) ≥Pnj (a′ )u(w(yn )) − Gj (a′ ), ∀ a′ ̸= a
n

∑

(MON)
(IC)

n

Pnj (a)u(w(yn ) − Gj (a) ≥ Û j

(IR∗ )nwhereÛ j ≡∑Pnj (a)u(ŵj (yn )) − Gj (âj )ndenotes the expected payoﬀ of agent i under the supposed fair contract Ĉ j .
Let ŵj (y|a) be the wage scheme which solves the above problem (CMj − a) toimplement a ∈ A from risk averse agent j. Let a = âj in Problem (CMj − a). Thenthe optimal wage scheme ŵj (y|a) which solves Problem (CMj − âj ) exists because
the fair contract Ĉ j satisﬁes all (MON), (IC) and (IR∗ ) by deﬁnition and hence theconstraint set is non-empty.5 Also the optimal scheme ŵj (y|a) is unique.6
Since a contract Ĉ i must be incentive eﬃcient, it must solve Problem (CMi −iâ ) for each risk averse agent i ∈ Ir . Otherwise there exists a diﬀerent contract
5
By a standard argument (Grossman and Hart (1983)), we can verify that the constraint set iscompact as well. Then the existence of the optimal wage scheme directly follows from the continuityof the objective function.
6
To see the uniqueness, deﬁne wn ≡ φ(un ) for each yn ∈ Y and note that (IC) and (IR∗ ) areboth linear constraints with respect to (un ). Thus the constraint set, denoted by Γ, is convex: takeany (u′n )n and (u′′n )n in Γ, and λu′n + (1 − λ)u′′n for λ ∈ (0, 1). Then we can verify that φ(λu′n + (1 −λ)u′′n ) − φ(λu′n−1 + (1 − λ)u′′n−1 ) ≥ φ′ (λu′n−1 + (1 − λ)u′′n−1 )(λ(u′n − u′n−1 ) + (1 − λ)(u′′n − u′′n−1 )) ≥ 0
due to convexity of φ and (u′n ), (u′′n ) ∈ Γ so that they satisfy (MON). Thus λu′n + (1 − λ)u′′n ∈ Γand hence Γ is convex.

10

C i ̸= Ĉ i which satisﬁes (MON), (IC) and (IR∗ ) for agent i but yields a smaller
expected wage than the minimum attained in Problem (CMi − âi ). By replacing Ĉ i
by such alternative contract C i , some risk neutral agent can be better oﬀ because theorganization can save the wage payment to the risk averse agent and gives some risk
neutral agent such saved amount while still implementing the same action proﬁle.
This is formally shown as in the following lemma:
Lemma 3. Suppose that Ĉ i = {ŵi (y i ), âi } is a fair contract of risk averse agenti ∈ Ir . Then Ĉ i should solve Problem (CMi − âi ) to implement âi from agent i, i.e.,ŵi (y) = ŵi (y|âi ) for each y ∈ Y and for each risk averse agent i ∈ Ir .
Proof. Appendix.
From now on, we assume that risk averse agents are heterogeneous in decreasingorder in terms of their abilities. To this end, we introduce some notation as follows:let ∆Gi (â, a) ≡ Gi (â) − Gi (a) > 0 be the marginal action cost of agent i for â > a.
Let Fni (a) be the distribution functionof agent i’s performance conditional on his∑iaction a ∈ A, deﬁned by Fn (a) ≡ l≤n Pli (a).
Then we impose the following weak condition:
Assumption 2. (i) For any i ∈ Ir and any â > a, we have Fni (a) > Fni (â) for any
n ≤ L − 1. (ii) For any i ̸= j, i, j ∈ Ir , and any a ∈ A, Fni (a) > Fnj (a) holds for allyn ∈ Y when i > j.
Assumption 2 (i) simply states that higher action can improve the probability
distribution of performances for any risk averse agent in the sense of the ﬁrst orderstochastic dominance. Assumption 2 (ii) says that the risk averse agent indexed bylower number j can improve the probability distribution of the performances in thesense of the ﬁrst order stochastic dominance than the risk averse agent indexed byhigher number i > j.
In addition to Assumption 2 we impose further restrictions on the ability ranking
among diﬀerent risk averse agents. We deﬁne the following: for any two risk averse
agents k = i, j ∈ Ir , i ̸= j, and any two actions â ̸= a,
λkn (â, a|i, j) ≡andFni (a) − Fnj (â)
Fnk (a) − Fnk (â)

k

λ (â, a|i, j) ≡ max λkn (â, a|i, j),nλ (â, a|i, j) ≡ min λkn (â, a|i, j).
knHere note that λkn (â, a|i, j) is well-deﬁned due to Assumption 2.

11

(3)

To see what λkn (â, a|i, j) says about, suppose that risk averse agent i chooses
an action a ∈ A and, as a thought experiment, imagine that his type is changed
from i to j as well as his action is also changed from a to â. Then the distributionfunction of the performance y i is changed from Fni (a) to Fnj (â). Such total change
Fnj (â) − Fni (a) can be decomposed into either
Fni (a) − Fnj (â) = (Fni (a) − Fni (â)) + (Fni (â) − Fnj (â))(4)

Fni (a) − Fnj (â) = (Fni (a) − Fnj (a)) + (Fnj (a) − Fnj (â)).

(5)orSuch decomposition is made along with two diﬀerent roots: ﬁrst, type is changed
from i to j by ﬁxing an action. Second, action is changed from a to â by ﬁxing a
type. In this respect λkn (â, a|i, j) can be interpreted as the inverse measure about
how much the change in agent k’s action (a → â) contributes to the total change
of Fni (a) − Fnj (â) relative to the change of types (i → j). Then 1/λ k (â, a|i, j) (resp.
k
1/λ (â, a|i, j)) represents the maximum (resp. minimum) inﬂuence of such relative
contribution of agent k’s action to the total change Fni (a) − Fnj (â).
Then we make the following assumption:
Assumption 3. For any two risk averse agents i, j ∈ Ir and any action â > a, if
i > j, thenjλ (â, a|i, j)∆Gi (â, a)> i.
∆Gj (â, a)λ (â, a|i, j)

The left hand side of the inequality in Assumption 3 is the marginal action costof agent i relative to agent j. Its right hand side is the ratio between the maximal
inﬂuence of the relative contribution by agent i’s action and the minimal inﬂuenceof the relative contribution by agent j’s action as we have deﬁned above. Thus theright hand side represents the upper bound for the relative contribution of agent
i’s action as compared to agent j. Then Assumption 3 says that agent i incurs
larger marginal action cost than agent j even when he performs his task at the bestcompared to agent j. In this sense we can say that agent i is less eﬃcient than agentj.
Given all these assumptions, by recalling that #Ir = Nr and #A = M + 1, we
can show the following result.
Theorem 1. Suppose that Assumption 1-3 hold. Suppose also that M + 1 ≤ Nr .
Then any fair contract must have a ﬁxed wage for at least Nr −M risk averse agents.
Proof. Appendix.

12

Theorem 1 states that oﬀering a ﬁxed wage to risk averse agents becomes prevalent feature in large organizations in which the number of risk averse agents is solarge relative to the number of actions each agent can take.
The intuition behind Theorem 1 is as follows. Since the number of actions M + 1
is larger than the number of risk averse agents Nr (M + 1 ≥ Nr ), if only less thanNr − M risk averse agents choose the least costly action, there must be at least two
risk averse agents who choose the same action â > 0. Let these agents be i andj. Then, by Assumption 3 we can label them as i > j without loss of generality.
Here agent i is less eﬃcient than j in the sense of Assumption 3. Then we canshow that, when both i and j choose the same and higher action than the least
costly one, (IC) of more eﬃcient agent (agent j) must be slack: the wage schemeof less eﬃcient agent i must be higher-powered than that of more eﬃcient agent
j when implementing the same action â > 0 from both agents. However, agent j
then envies agent i since the former can exploit more gains of higher ability fromhigher-powered scheme than the latter. Thus no-envyness and incentive eﬃciency
cannot be compatible with each other. Therefore, fair contract must involve a ﬁxed
wage for at least Nr − M risk averse agents.
When the number of veriﬁable performances y i ∈ Y is two (#Y = 2), Assumption 3 is greatly simpliﬁed as follows. When #Y = 2, we have y = y1 (lowoutcome) or y = y2 (high outcome). Then Assumption 3 is satisﬁed as long as∆Gi (â, a)/∆Gj (â, a) > (P2i (â)−P2i (a))/(P2j (â)−P2j (a)) for all â > a because then wekhave λ k (â, a) = λ (â, a) = (F1i (a) − F1j (â))/(F1k (a) − F1k (â)) and F1k (a) = 1 − P2k (a).
Here P2k (â) − P2k (a) denotes the marginal increase in success probability of obtaining high outcome y2 from exerting higher action â > a. Thus this condition simply
states that the marginal action cost of agent i relative to that of agent j is larger
than the marginal increase in success probability of agent i relative to that of agentj.

4More General Characterization: Small Heterogeneity

We will next consider an alternative condition to dispense with both Nr ≥ M + 1
and Assumption 3. In particular, we will show that the same characterization result
as Theorem 1 holds even when we drop Nr ≥ M + 1 and Assumption 2 if, instead,
we assume that heterogeneity among risk averse agents is suﬃciently small.
We ﬁrst deﬁne the marginal expected revenue of organization with respect toaction of agent i, given action of agent j, aj , and the actions of all others, denotedby a−i−j ≡ (ak )k̸=i,j , as follows:
′′

′

′′

′

∆i (aj |ai , ai ) ≡ Ey [R(y)|ai , aj , a−i−j ] − Ey [R(y)|ai , aj , a−i−j ].
Since A and Ir are ﬁnite, there exists some scalar α > 0 such that|∆i (z|z ′′ , z ′ ) − ∆j (z|z ′′ , z ′ )| ≤ α
13for all z, z ′′ , z ∈ A and all i, j ∈ Ir .
Then we make the following assumption:
Assumption 4. For any two risk averse agents i, j ∈ Ir , i ̸= j, and α deﬁned
above, (i) there exists some β̃ > 0 such that∆i (z ′′ |z ′′ , z ′ ) − ∆i (z ′ |z ′′ , z ′ ) ≥ β̃for all z ′′ > z ′ and (ii) β ≡ β̃ − α > 0
The ﬁrst part of Assumption 4 says that the marginal expected revenue of organization with respect to action of each risk averse agent is increasing in actions ofothers. Put diﬀerently, actions of agents are complement (supermodular) with eachother. The second part of Assumption 4 then means that such complementarity
gain β̃ outweighs the asymmetry α among risk averse agents.
We also weaken Assumption 3 as follows:
Assumption 5. For any two risk averse agents i, j ∈ Ir , if i > j, then∆Gi (â, a) > ∆Gj (â, a)
for all â > a.
Assumption 5 simply states that risk averse agents are ranked in decreasing orderin terms of only their marginal action costs.
Since A and I are ﬁnite, there exists some (ε, δ) > 0 such that
¯
¯
¯
¯
(6)
δ ≥ ¯∆Gj (a′′ , a′ ) − ∆Gi (a′′ , a′ )¯for all i, j ∈ Ir and all a′′ , a′ ∈ A, and¯¯¯¯ε ≥ ¯Pni (a) − Pnj (a)¯(7)

for all i, j ∈ Ir , all yn ∈ Y and all a ∈ A.
Then we can show the following result:
Theorem 2. Suppose that Assumption 1, 4 and 5 hold. Then there exists some non–empty set D ⊂ ℜ2+ such that for all (ε, δ) ∈ D any fair contract Ĉ must have ﬁxed
wages for all risk averse agents who are induced to choose the least costly actions
ai = 0 for all i ∈ Ir .
Proof. Appendix.
Note that Theorem 2 states that all risk averse agents are oﬀered ﬁxed wages inany fair contract when heterogeneity among them are suﬃciently small in the sense14that ε and δ deﬁned in (6) and (7) are both small enough. Thus we can dispensewith Assumption 2 and 3 and M + 1 ≥ Nr which were used for proving Theorem 1.
The reason for this is as follows.
First, when heterogeneity among risk averse agents is small, any fair contract
must have the feature that the wage schemes of risk averse agents which implement
actions (âi )i∈Ir become close to each other. More formally, recall Lemma 3 thatthe fair contract ŵi (y) for risk averse agent i must solve Problem (CMi − âi ): letŵi (y i |âi ) be the optimal wage scheme which implements action âi from risk averse
agent i in Problem (CMi − âi ). Here we must have ŵi (y) = ŵi (y|âi ). Then weshow that ŵi (y|â) becomes close to ŵj (y|â) for any y ∈ Y when their heterogeneity
is small enough (both ε and δ are small), provided agent i and j choose the sameaction â ∈ A.
Second, we show that all risk averse agents are actually induced to choose thesame action at any fair contract if the heterogeneity is small enough. This follows
from (IE∗ ) and Assumption 4. By making use of the complementarity condition in
Assumption 4, we can obtain the result that the actions of risk averse agents which
satisfy (IE∗ ) must be symmetric. To see this, let the organizational residual surplus
deﬁne by the ∑expected total revenue Ey [R(y)|a] minus the total wages of all risk
averse agents i∈Ir Eyi [ŵi (y i )|ai ]:H(a) ≡ Ey [R(y)|a] −∑Eyi [ŵi (y i )|âi ].

i∈Ir

This should be shared among all risk neutral agents due to (BB). Then (IE∗ ) requiresthat the actions of risk averse agents (ai )i∈Ir must maximize H(a) because otherwise
other actions can be implemented from risk averse agents and hence risk neutral
agents can be better oﬀ, which contradicts (IE∗ ). We then show that, if Assumption
4 holds, such optimal action becomes symmetric among all risk averse agents in thesense that ai = aj for all i, j ∈ Ir .7 Thus we can ensure that all risk averse agentschoose the same action âi = âj in any fair contract when the heterogeneity among
them is suﬃciently small, given Assumption 4.
Third, we can show that Assumption 3 is satisﬁed when risk averse agents areheterogeneous but such heterogeneity is larger for the diﬀerences in their marginalaction costs than the diﬀerences in the impacts of their actions on the probability
distributions of performances. In other words both ε and δ are small enough but δis larger than ε.
By combining all these facts, we can ensure Assumption 3 as well as the condition
that all risk averse agents choose the same action âi = âj for all i, j ∈ Ir when theheterogeneity among risk averse agents lies in some small range. Then, by using
Without Assumption 4, the optimal actions which maximize H over (ai )i∈Ir may be asymmetriceven when the expected revenue function Ey [R(y)|a] is symmetric with respect to (ai )i∈Ir and ŵi (·|·)
is also symmetric, i.e. ŵi (·|·) = ŵ(·|·) for all i ∈ Ir . This is essentially because H may have multiple
maximizes over (ai )i∈Ir some of which are asymmetric.
7

15

Theorem 1, we can show that the fair contract must oﬀer a ﬁxed wage to each risk
averse agent.
Next we characterize the optimal contracts oﬀered to risk neutral agents. Noteﬁrst that the wage scheme of each risk averse agent in any fair contract is individualistic by Lemma 2. Recall that ŵi (y i ) is the wage scheme of risk averse agent i ina fair contract which implements action âi from him. Due to (BB), the risk neutral
agents then obtain the∑residual revenue R(y) after subtracting total wages paid to
all risk averse agents i∈Ir ŵi (y i ).
Then we deﬁne the actions of risk neutral agents, denoted by (ãi )i∈In , which
maximize the expected residual surplus of organization after ﬁxing the action proﬁlear ≡ (âi )i∈Ir of risk averse agents:∑∑âi ∈ arg max Ey [R(y)|ar , ai , â−i ] −Gi (ai ) −Eyi [ŵi (y i )|âi ].
ai ∈A

i∈In

i∈Ir

We can show the following result:
Theorem 3. In any fair contract the risk neutral agents must choose the actions(âi )i∈In so as to maximize the residual surplus of organization, given action proﬁleof risk averse agents ar = (âi )i∈Ir .
Proof. Appendix.

5

Existence of Fair Contract

In this section we show the conditions under which there exists a fair contract.
Instead of showing Cf∗ ̸= ∅, we will return back to the original set of fair contracts
C f and proceed to show the stronger result that C f ̸= ∅. Due to Lemma 1, we knowthat C f ⊆ C∗f . Thus, if we can show that Cf ̸= ∅, this implies C∗f ̸= ∅. Note herethat any fair contract in Cf is deﬁned in terms of incentive eﬃciency (IE) but not
its modiﬁed version (IE∗ ).
Our strategy for existence proof is as follows: ﬁrst, we set the wage scheme ofevery risk averse agent as a ﬁxed wage, i.e., wi (y i ) = w for all y i ∈ Y . Thus every
risk averse agent chooses the least costly action, ai = 0 for all i ∈ Ir .
We then show that oﬀering a ﬁxed wage to every risk averse agent becomes
incentive eﬃcient when we take the ﬁxed wage w to be large enough. To see this,note that, if the ﬁxed wage w is not incentive eﬃcient, there must exist a proﬁleof other contracts (C i )i∈Ir for risk averse agents such that it solves the following
problem:
Problem (OP):∑max E[R(y)|ar , an ] −Pni (ai )wni
(C i )i∈Ir

i∈Ir

16

subject to∑∑Pni (ai )u(wi (yn ))−Gi (ai ) ≥Pni (a)u(wi (yn ))−Gi (a) ∀ a ̸= ai , ∀ i ∈ Ir (IC)nn∑Pni (ai )u(wi (yn )) − Gi (ai ) ≥ u(w) ∀ i ∈ Ir

(IR∗ )nwhere ar ≡ (ai )i∈Ir and an ≡ (ai )i∈In denote vectors of actions taken by risk averse

and neutral agents respectively. Here (IC) denotes incentive compatibility constraint
for risk averse agent i and (IR∗ ) denotes the constraint such that risk averse agent
i is not worse oﬀ compared to the case that he receives
ﬁxed wage w. Note that∑ theithe expected wage paid to risk averse agent i, i.e., n Pn (ai )wni , is bounded below
from φ(Gi (ai ) + u(w)) due to (IR∗ ) and concavity of u. Then such lower bound forthe expected wage of risk averse agent becomes large when w is taken to be large
as well. This is more likely to be so when higher action ai is implemented because
then higher payment must be compensated to risk averse agent for agreeing on analternative wage scheme. Thus risk neutral agents cannot be better oﬀ by oﬀeringother wage schemes than the ﬁxed wage w to risk averse agent when the status quo
utility u(w) is large enough. Then the optimal solution to Problem (OP) involves
the implementation of only the least costly action from risk averse agents.
Second, given this result, we ﬁnd the wage schemes of risk neutral agents which
are induced to choose the actions maximizing the residual surplus of organization
after subtracting the ﬁxed wages of risk averse agents. More formally, we deﬁne theoptimal actions of risk neutral agents, denoted by ãn ∈ ANn , which maximize thetotal residual of surplus given all risk averse agents choosing the least costly action
ai = 0, as follows:∑ãn ∈ arg max RS(an ) ≡ Ey [R(y)|0, an ] −Gi (ai )
(8)
n
N
n
a ∈A
i∈I
n

where 0 denotes Nr –dimensional vector of zeros. Then we can show that there
exists a single optimal action ã ≡ ãi for any risk neutral agent i ∈ In when theheterogeneity among risk neutral agents is small enough. Given this, we will design
the wage scheme which implements the optimal single action ã from every risk
neutral agent.
The most diﬃcult part of the proof is that we need to show that the wage schemes
constructed in the above way must satisfy no–envyness condition (NE). To deal withthis subtle issue, we will design the wage schemes of agents by exploiting the risk
aversion of risk averse agents such that they would face risky lottery if they wereoﬀered the wage scheme of risk neutral agent, which prevents them from envying
risk neutral agents. On the other hand, the wage scheme of risk neutral agent isdesigned such that his expected payoﬀ is equal to the ﬁxed wage oﬀered to risk
averse agent. Thus risk neutral agent does not envy risk averse agents.

17

Now we will proceed the formal argument for the proof of existence of fair contract.
First, we make the following assumptions:
Assumption 6. For any action a ∈ A and any risk neutral agents i, j ∈ In ,
¯ 1
¯
¯
¯
¯Gi (a) − Gj (a)¯ < β2where β > 0 is given in Assumption 4.
Assumption 7. u′ (+∞) = 0.
Assumption 6 says that the diﬀerence between action costs of risk neutral agentsare not so large relative to the complementarity gains of actions β deﬁned in Assumption 4. Assumption 7 ensures that φ′ (+∞) = +∞.
Then we can show the following:
Lemma 4. Suppose that Assumption 4 and 6 are satisﬁed. Then there exists a
single optimal action ã ∈ A such that ãi = ã maximizes the residual surplus oforganization RS(an ) for all risk neutral agents i ∈ In .
Proof. See Appendix.
For such optimal symmetric action ã, we make the following condition:
Assumption 8. For some risk neutral agent l ∈∑In there exists no M + 1–
dimensional vector (λm ) ≥ 0, not all zero, such that m λm = 1,∑
λm Fnl (am ) ≤ Fnl (ã)
m

and

∑

λm Gl (am ) < Gl (ã) + δmwhere δ > 0 is given in (6).
Assumption 8 says that there exists some risk neutral agent, say l, who cannot
improve the probability distribution of his performances Fnl (a) in the sense of ﬁrst
+1
order stochastic dominance by using a mixes strategy λm over (am )M
m=1 ̸= ã at lower
expected action cost than the action ã. This condition is more likely to be satisﬁed
when heterogeneity among risk averse agents δ is small enough, as long as Fnl (a)satisﬁes Convexity of Distribution Function Condition (CDFC) and Gl is convex.88CDFC means that Fni (a) is convex function of a. Suppose that Fnl (a) satisﬁes CDFC and Gl

18

Assumption 8 then ensures that there exists a non-decreasing wage scheme which
implements the optimal action ã from every risk neutral agent. Thus (MON) is
satisﬁed.
Finally, we make the following assumption:
Assumption 9. Ey [R(y)|a] = Φ(a) + F
Assumption 9 says that the expected revenue of organization can be decomposed
into two parts: one is the part which varies with unobservable actions of agents andthe other is the part which is independent of these actions. Here F can be interpreted
as the base revenue which is independent of non–contractible action choices.9
Then we can show the following result:
Theorem 4. Suppose that Assumption 1 and 6-9 are satisﬁed. Then, if the baserevenue of organization F is suﬃciently large, there exists a fair contract Ĉ which
has the following features:• the risk averse agents are oﬀered the ﬁxed wage scheme which induces them tochoose the least costly action ai = 0, and• the risk neutral agents are oﬀered the wage scheme which induce them to choose
the optimal action ã each so as to maximize the residual surplus of organization∑
E[R(y)|0, an ] −Gj (aj ).
j∈Ingiven all risk averse agents choose the least costly action ar = 0.
Proof. See Appendix.
The requirement that the base revenue F is large can ensure that the ﬁxed wage
oﬀered to each risk averse agent can be large enough without aﬀecting the incentivesof risk neutral agents. Then, large enough ﬁxed wage makes the payoﬀ of risk averse
agents large as well so that the designed contract proﬁle becomes incentive eﬃcientbecause it solves Problem (OP).

P
P
P
is convex. Then Fnl ( nP
λa(m)) < m λm Fnl (a(m)) ≤ Fnl (ã) implies that a′ ≡P m λm a(m) > ã.
′
However, then we have m λm Gl (a(m)) > Gl (a ) > Gl (ã) which implies that m λm Gl (a(m)) >Gl (ã). The last inequality cannot hold when δ is small enough.
9This does not necessarily mean that F is independent of any kinds of agents’ actions. It could
depend on contractible actions taken by agents.

19

6Discussion and Robustness

6.1

Eﬃciency First or Equity First?

When we have so far characterized fair contracts in Section 3 and 4, we have deﬁned
the set of fair contracts as C∗f which are the contracts satisfying (IE∗ ), (NE) and(IR). In this deﬁnition (IE∗ ) was referred to as the set of incentive feasible (i.e.,budget balanced and incentive compatible) and monotonic contracts which are not
Pareto dominated by all other incentive feasible and monotonic contracts. Thus wehave imposed Pareto eﬃciency only to the set of incentive feasible and monotonic
contracts which are not necessarily no–envy. This deﬁnition of fair contracts canbe thus interpreted as follows: as the ﬁrst step, agents of organization seek theeﬃciency constrained by incentive compatibility and monotonicity by ruling out theconsideration of no-envyness. Then, as the second step, they select among all thesecontracts the one satisfying no–envyness (and (IR)). In other words, eﬃciency istaken as the ﬁrst priority to be satisﬁed in organization whereas equity concern isthe objective to be accomplished after eﬃciency is attained. In this section, by
taking into account the above interpretation, we will call fair contracts which wehave so far deﬁned eﬃciency–ﬁrst optimal contracts. We will denote by CE∗ the setof eﬃciency–ﬁrst optimal contracts. Of course, CE∗ coincides with the set of faircontracts Ĉ which we have characterized in Section 4.
However, in contrast to this case, one might think that equity can be consideredas the ﬁrst priority to be satisﬁed in organization whereas eﬃciency is the second
objective. For example, suppose that agents of organization ﬁrst identify the set of
all no–envy contracts. Next, the agents ﬁnd the Pareto eﬃcient contracts among all
possible no–envy, incentive feasible and monotonic contracts. We deﬁne by CE∗∗ theset of all the eﬃcient contracts deﬁned in this way and satisfying (IR). We will callthis equity–ﬁrst optimal contracts.
Then, we can show the following result:
Theorem 5. CE∗ ⊆ CE∗∗ .
Proof. Take any eﬃciency–ﬁrst optimal contract C ∈ CE∗ . Then we will showthat C ∈ CE∗∗ . To see this, suppose that C ∈/ CE∗∗ . Then, C ∈ CE∗ implies that C
satisﬁes all (IC), (NE), (IR), (BB) and (MON). Thus, the fact that C ∈/ CE∗∗ implies
that there must exist some contract C ′ ∈/ C such that C ′ satisﬁes (IC) and (NE),and it Pareto dominates C. In particular C ′ satisﬁes (IC) but Pareto dominates C.
However, since C ∈ CE∗ , C satisﬁes (IC), and there exist no other contracts which
satisfy (IC), and Pareto dominate C. C ′ contradicts to this fact. Q.E.D.
Thus, by Theorem 5, we may have CE∗∗ \ CE∗ ̸= ∅ so that there may exist someequity–ﬁrst optimal contract which does not have the property shown in Theorem
1-2 even when heterogeneity among agents is so small. We will show this by the20

following example:
Example. Suppose that Ir = {1, 2} and In = {p}. Here agent p is the principal.
Suppose also that A = {0, 1} and Y = {y2 , y1 } where y2 > y1 . We also simplify
the model by assuming that R(y 1 , y 2 , y p ) = R(y 1 , y 2 ). Thus the principal’s action
does not aﬀect the expected revenue of organization. We thus set ap = 0. Without
loss of generality we take the principal as the residual claimant who receives all theresidual surplus of organization after subtracting the wages to risk averse agents.
Thus (BB) is satisﬁed and we ignore this.
Now consider the following optimization problem which minimizes the expected
wage of the principal when implementing actions a = (a1 , a2 ) from risk averse agents
subject to (IC), (MON) and (NE), given each risk averse agent is guaranteed acertain utility level Û i :∑Ey [wi (y)|a]
min
i=1,2

subject toEy [u(wi (y))|a] − Gi (ai ) ≥ Ey [ui (wi (y))|a, aj ] − Gi (a), ∀ a ̸= ai(IC)

Ey [u(wi (y))|a] − Gi (ai ) ≥ Û i

(PE)

j

Ey [u(w (y))|a] − Gi (a ) ≥ Ey [u(w (y))|a, a ] − Gi (a), ∀ a(NE)w (y2 , y ) ≥ w (y1 , y ) ∀ i ̸= j

(MON)

i

i

i

j

j

i

j

Note that the contract which solves the above problem becomes an equity–ﬁrst
optimal contract because it is Pareto eﬃcient among all feasible contracts satisfying(IC), (NE) and (MON).
Let EW ∗∗ (a1 , a2 ) be the expected total wage in the solution to the above problem. Then we deﬁne the principal’s expected payoﬀ as follows:Πp (a1 , a2 ) ≡ Ey [R(y 1 , y 2 )|a1 , a2 ] − EW ∗∗ (a1 , a2 ).
Then, we can verify that implementing high actions (a1 , a2 ) = (1, 1) from both risk
averse agents maximize the principal’s expected payoﬀ Πp (a1 , a2 ) when ∆(1, 1) ≡Ey [R(y)|1, 1] − maxa̸=1 Ey [R(y)|a] is suﬃciently large. This shows that the contract
which solves the above problem gives risk averse agents high–powered incentive tochoose high action instead of low action. Thus there may exist some equity–ﬁrst
optimal contract having the feature that high–powered incentive schemes are oﬀered
to risk averse agents when ∆(1, 1) is suﬃciently large.
The above example is in contrast to Theorem 1–2 that all eﬃciency–ﬁrst optimal
contracts must have ﬁxed wage schemes oﬀered to risk averse agents. This gives aninteresting testable implication as follows: rewards of agents are more sensitive totheir objective performances in organizations in which equity among them is taken
as the ﬁrst priority to be satisﬁed than in organizations in which eﬃciency is theﬁrst priority to be satisﬁed.
21

6.2

Robustness of the Results

We have so far paid our attention to the class of monotonic wage schemes (MON).
Although (MON) is a reasonable and realistic restriction, one might think that, once
we remove (MON) from the consideration of possible axioms, the set of contracts
is expanded so that all axioms may be satisﬁed at non-ﬁxed wages for risk averse
agents.
In this subsection we will discuss the robustness of our results when we remove
the axiom of monotonicity (MON).
First, we can show that our characterization results, Theorem 1 and Theorem 2,
still remain true even if we drop (MON) from the axioms to be considered when we
impose the monotone likelihood ratio property (MLRP) and convexity of distributionfunction condition (CDFC) on the probability distribution of each risk averse agent’s
performances Pni (a). MLRP means that the likelihood ratio Pni (a′ )/Pni (a) of agent
i’s performance is increasing in yn conditional on a′ > a. As we∑have alreadyimentioned, CDFC means that the distribution function Fni (a) ≡l≤n Pn (a) isconvex with respect to action levels a ∈ A. Then, we modify Problem (CMj − âj ) by
dropping (MON) from the constraint set. Then, by a similar argument to Lemma 3,
we can show that any fair contract Ĉ j of risk averse agent j must solve this modiﬁedproblem. Otherwise, it is not incentive eﬃcient. Then we can easily verify thatthe optimal wage scheme which solves this modiﬁed problem becomes monotone
increasing by the standard argument when MLRP and CDFC are satisﬁed (seeGrossman and Hart (1983)). Thus, as long as MLRP and CDFC are met, we canreach the same characterization results as Theorem 1-3.
Second, we will investigate the implications when we drop (MON) from theaxioms and still do not impose the restrictions on the probability distributions such
as MLRP and CDFC. We will then show that, even when we drop (MON), any
fair contract still has the ﬁxed wage oﬀered to every risk averse agent, provided
the heterogeneity among them is suﬃciently small and the wage of each agent isbounded below from some amount. We impose the limited liability (LL) instead of(MON): for some w > −∞,wi (y) ≥ w, ∀ i ∈ I, ∀ y ∈ Y N(LL)

Then we call a contract proﬁle C incentive feasible if it satisﬁes (BB), (IC) and(LL). Let C L be the set of all incentive feasible contracts. We focus on this set C L .
Similarly, we deﬁne the incentive eﬃcient contract as follows: with a slight abuse of
deﬁnition, a contract C ∈ C L is said to be incentive eﬃcient, denoted by (IE∗∗ ), if
no other contract C ∈ C L exists such that it Pareto improve the expected utilities
of agents. A fair contract C is deﬁned analogously: C is said to be fair if it satisﬁes(IE∗∗ ), (NE) and (IR).
Then we can show the following result:
Theorem 5. Suppose that Assumption 1, 4 and 5 hold. Suppose also that (MON)22is replaced by (LL). Then there exists some non-empty set D ⊂ ℜ2+ such that for all(ε, δ) ∈ D any fair contract Ĉ involves a ﬁxed wage oﬀered to all risk averse agents.
Proof. See Appendix.
By (LL), we can ensure that the utility payment for every risk averse agentu(w(yn )) is bounded below from u ≡ u(w). Also (BB) requires that the utility
payment u(w(yn )) is bounded above by u(R − (N − 1)w) where R ≡ maxy R(y).
Thus the utility payment un ≡ u(w(yn )) is bounded for each yn ∈ Y . Then we canshow that, when heterogeneity among risk averse agents is small enough so that εis suﬃciently small relative to δ, Assumption 3 is satisﬁed and hence we can reachthe same characterization result as Theorem 1 and 2.

7

Concluding Remarks

In this paper we have characterized fair contracts which are incentive eﬃcient andno–envy in general moral hazard environments. Then we have shown that faircontracts must have the feature that risk averse agents are oﬀered the ﬁxed wage
scheme. On the other hand, risk neutral agents are motivated to work in orderto maximize the residual surplus of organization after subtracting the ﬁxed wagesof risk averse agents. Then the tension between incentive eﬃciency and equity as
no–envy makes characterization of contracts drastically simple. We have also shown
the conditions under which a fair contract exists.
Our concept of no–envyness is based on ex ante view before agents choose theiractions and hence ﬁnal outcomes are realized. One might think that no–envy constraint is binding even ex post after the ﬁnal performances are realized. In such caseex post no–envy constraint becomes more stringent than the ex ante one: supposethat the performances of agents y ∈ Y N are realized. Then agent i does not envy
agent j ex post if and only if u(wi (y i )) ≥ u(wj (y i )), conditional on his own performance y i . Then, we must have wi (y) = wj (y) for any y ∈ Y and all i ̸= j. Hence
we can reach the same conclusions as Theorem 1 and 2.

8

Appendix

8.1

Proof of Lemma 2

Suppose that some fair contract Ĉ i is not individualistic for some risk averse agentk ∈ Ir , i.e., ŵk (y) depends on y −k . Let ûk (y) ≡ uk (ŵk (y)). Then we deﬁne the newcontract ũk (y k ) for that agent k asũk (y k ) ≡ Ey−k [ûk (y k , y −k )|â−k ]and consider an alternative contract C ̸= Ĉ as follows:23• For the risk averse agent k, the new contract C = {ũk , âk } is oﬀered.
• Pick one risk neutral agent, say agent n. Then, for all other agents i ̸= n, k,
we deﬁne a new wage scheme asw̃i (y i ) ≡ Ey−i [ŵi (y i , y −i )|â−i ] ∀ y i ∈ Y.
• For the risk neutral agent n, the following wage scheme w̃n (y) is oﬀered∑w̃n (y) ≡ R(y) −w̃i (y) − w̃k (y k )i∈I,i̸=kwhere w̃k (y k ) ≡ φ(ũk (y k )).
First, note that such alternative wage scheme (ũkn )yn ∈Y oﬀered to risk averse
agent k satisﬁes (MON) because the original scheme (ukn )yn ∈Y satisﬁes (MON) andy i and y −i are statistically independent. By the same reason, the new scheme w̃i (y i )oﬀered to agent i ̸= k, n is also non-decreasing and hence (MON) is satisﬁed. Sincew̃i (y i ), i ̸= k, n, is independent of the performance of the selected risk neutral agentn, w̃n (y) is also non-decreasing in y n because R is non-decreasing in y n . Thus
(MON) is satisﬁed for the risk neutral agent n as well.
Second, since (IC) is satisﬁed at Ĉ k , we have
Ey [ûk (y)|âk , â−k ] − Gk (âk ) ≥ Ey [ûk (y)|ak , â−k ] − Gk (ak ), ∀ ak ̸= âk ,which, by independence of y i and y −i , implies thatEyk [ũk (y k )|âk ] − Gk (âk ) ≥ Eyk [ũk (y k )|ak ] − Gk (ak ), ∀ ak ̸= âk .
Thus the new contract ũk satisﬁes (IC) for the risk averse agent k who chooses thesame action âk , given â−k . All other agents than k and n also choose the sameactions â−k−n as well. It is also true that all agents except n will obtain the same
expected payoﬀs under the new contract as under the original one.
Now consider the incentive of the risk neutral agent n. Agent n would receive
the following expected payoﬀ under the new contract w̃n (y):max Ey [w̃n (y)|an , â−n ] − Gn (an ) ≥ Ey [w̃n (y)|ân , â−n ] − Gn (ân )∑= Ey [R(y)|â] −Ey [ŵi (y)|âi ]an ∈Ai∈I,i̸=k,n−Eyk [w̃ (yk )|âk ] − Gn (ân )∑> Ey [R(y)|â] −Ey [ŵi (y)|â] − Ey [ŵk (y)|â] − Gn (ân ).
k= Ey [ŵ24ni∈I,i̸=k,n(y)|â] − Gn (ân )Here the last equality follows from (BB) and the strict inequality from the risk
aversion of agent k (concavity of u), the Jensen’s inequality and the dependence ofŵk (y k , y −k ) on y −k :Ey [ŵk (y k , y −k )|â]= Ey [φ(ûk (y k , y −k ))|âk , â−k ]= Eyk [Ey−k [φ(ûk (y k , y −k ))|â−k ]âk ]> Eyk [φ(Ey−k [ûk (y k , y −k )|â−k ])|âk ]= Eyk [φ(ũk (y k ))|âk ]= Eyk [w̃k (y k )|âk ]respectively. Thus the risk neutral agent n can be strictly better oﬀ under the newcontract w̃n (y), which contradicts the fact that Ĉ is incentive eﬃcient. Q.E.D.

8.2

Proof of Lemma 3

Consider Problem (CMj − âj ) for implementing the action a = âj speciﬁed in thefair contract Ĉ j from risk averse agent j. As we have argued in the main text, theconstraint set of such problem is non-empty, compact and convex. Thus the optimal
wage scheme which solves Problem (CMj − âj ) exists and is unique. We denote byŵj (y|âj ) the optimal wage scheme which solves Problem (CMj − âj ).
Suppose that for some risk averse agent j ∈ Ir the the wage scheme ŵj (y)speciﬁed in the fair contract Ĉ j = {ŵj (y), âj } diﬀers from the optimal wage schemeŵj (y|âj ): ŵj (y) ̸= ŵj (y|âj ) for some y ∈ Y . Then, note that ŵj (y) satisﬁes (IC) and(IR∗ ) in Problem (CMj − âj ) so that it is feasible in Problem (CMj − âj ) and thatŵj (y|âj ) is uniquely determined to solve the problem (CMj − âj ) given Û j . Thus
we must have∑∑
Eyj [ŵj (y j )|âj ] ≡Pnj (âj )ŵj (yn ) > Eyj [ŵj (y j |âj )|âj ] ≡Pnj (âj )ŵj (yn |âj ).
nnThen consider an alternative contract oﬀered to the above risk averse agent j as
C j = {ŵj (y j |âj ), âj }. Without loss of generality we can also assume that the wage
schemes of other risk neutral agents than n depend solely on their own performances
because (y i , y −i ) are statistically independent and hence their action incentives are
not changed by removing the dependence of the others’ performances on their wage
schemes. Thus all others than n choose the same actions âi as before.
But now consider an alternative contract oﬀered to the selected risk neutral agentn ∈ In as C n = {w̃n (y), ãn } where∑w̃n (y) = R(y) −ŵl (y l ) − ŵj (y j |âj )
l̸=j

25

and ãn is deﬁned as the action to maximize the expected utility of agent n:ãn ∈ arg max Ey [w̃n (y)|a, â−n ] − Gn (a).
a∈A

Then such risk neutral agent n can be strictly better oﬀ because
Ey [w̃n (y)|ãn , â−n ] − Gn (ãn )≥ Ey [w̃n (y)|ân , â−n ] − Gn (ân )∑= Ey [R(y)|â] −Eyl [ŵl (y l )|âl ] − Eyj [ŵj (y j |âj )|âj ] − Gn (ân )l̸=j> Ey [R(y)|â] −∑Ey [ŵi (y i )|âi ] − Gn (ân )i∈I= Ey [ŵ (y)|â ] − Gn (ân )nnwhere the ﬁrst inequality follows from the deﬁnition of ãn which maximizes the agent
n’s expected utility, the second inequality from Eyj [ŵj (y j )|âj ] > Eyj [ŵj (y j |âj )|âj ]and the ﬁnal equality from (BB) respectively. Q.E.D.

8.3

Proof of Theorem 1

Take any fair contract Ĉ. Suppose contrary to the claim that there are less thanNr − M risk averse agents who choose the least costly action (zero). Then, by our
supposition that #A = M + 1 ≤ #Ir = Nr , there must exist at least two risk
averse agents, say i and j, who choose the same action, say â > 0. Without loss ofgenerality we suppose that i > j in the sense of Assumption 3.
Let ûkn ≡ u(ŵk (yn )), k = i, j, for each yn ∈ Y . Then we obtain the following

26

series of expressions:∑
Pnj (â)ûjn − Gj (â)
n≥∑Pnj (â)ûin − Gj (â)

n

=

∑

{
Pni (â)ûin − Gi (â) +

n

≥

∑

Gi (â) − Gj (â) +Gi (â) − Gj (â) +

∑

n

=

∑

=

(Pnj (â) − Pni (â))ûinnn

Gj (a) − Gi (a) +∑}Pnj (a)ûjn − Gj (a){+}(Pnj (â) − Pni (â))ûinn{Pni (a)ûjn − Gi (a) +∑∑}(Pni (a) − Pnj (a))ûin{+

Gi (â) − Gj (â) +n∑}(Pnj (â) − Pni (â))ûjnnPnj (a)ûjn − Gj (a)n{+∆Gi (â, a) − ∆Gj (â, a) +∑(Pnj (â) − Pni (â))ûin −n∑}(Pnj (a) − Pni (a))ûjnnfor any a < â. Here the ﬁrst inequality follows from (NE) for agent j and thesecond inequality from (NE) of agent i respectively. Thus we can reach the following
inequality:∑∑
Pnj (â)ûjn − Gj (â) ≥Pnj (a)ûjn − Gj (a) + W (â, a|ûi , ûj )
(A1)
n

n

where ûk ≡ (ûkn )L
n=1 denotes the vector of the utility payments for k = i, j and{}∑∑W (â, a|ûi , ûj ) ≡ ∆Gi (â, a) − ∆Gj (â, a) +(Pnj (â) − Pni (â))ûin −(Pnj (a) − Pni (a))ûjn .
nnSince the above inequality (A1) holds for any action a < â, we take an actiona < â such that (IC) of agent j is binding:∑∑Pnj (â)ûjn − Gj (â) =Pnj (a)ûjn − Gj (a).
nnSuch action a must exist: otherwise the action â must be the smallest among all
relevant actions when solving Problem (CMj − â) because (IC) of agent j in thatproblem is not binding at any a < â. Thus the optimal wage which solves Problem(CMj − â), denoted ŵj (y|â), must be the ﬁxed wage but then this does not satisfy(IC). Thus (IC) of agent j must be binding at some a < â.
27

Now we show that W (â, a|ûi , ûj ) > 0 for such action a < â that (IC) of agent j
is binding. To see this, note that ûi = (ûin )L
n=1 must satisfy (IC) of agent i:∑∑
Pni (â)ûin − Gi (â) ≥Pni (a)ûin − Gi (a)nnwhich can be written by∑(Fni (a) − Fni (â))∆uin ≥ ∆Gi (â, a)nwhere ∆uin ≡ ûin − uin−1 for each i = 2, ..., L. By (MON) ∆uin ≥ 0 must hold foreach i = 2, ..., L.
Then, since Fni (a) > Fni (â) for each n when â > a, we can show that∑∑
(Pnj (â) − Pni (â))ûin =(Fni (â) − Fnj (â))∆uin
n

n

=

∑
n

=

∑
n

=

∑

(
(

Fni (â) − Fnj (â)
Fni (a) − Fni (â))(Fni (a) − Fni (â))∆uin)
Fni (a) − Fnj (â)
− 1 (Fni (a) − Fni (â))∆uin
Fni (a) − Fni (â)λin (â, a|i, j)(Fni (a) − Fni (â))∆uinn≥ (λi (â, a|i, j) − 1)∑

(Fni (a) − Fni (â))∆uinn≥ (λi (â, a|i, j) − 1)∆Gi (â, a)where the last inequality follows from (IC) of agent i and λi > 1 (due to Fnj (â) <Fni (a) for i > j by Assumption 2 (ii)).
Next, the binding (IC) of agent j can be written by∑(Fnj (a) − Fnj (â))∆ujn = ∆Gj (â, a).
n28

By using this and Fnj (a) > Fnj (â) for â > a, we can show that∑∑
(Pnj (a) − Pni (a))ûjn =(Fni (a) − Fnj (a))∆ujn
n

n

=

∑
n

=

∑
n

=

∑

(
(

Fni (a) − Fnj (a))

Fnj (a) − Fnj (â)
Fni (a) − Fnj (â)
Fnj (a) − Fnj (â)(Fnj (a) − Fnj (â))∆ujn)− 1 (Fnj (a) − Fnj (â))∆ujn

λjn (â, a|i, j)(Fnj (a) − Fnj (â))∆ujnnj≤ (λ (â, a|i, j) − 1)∑

(Fnj (a) − Fnj (â))∆ujnnj= (λ (â, a|i, j) − 1)∆Gj (â, a)
By combining the above inequalities, we obtain∑∑(Pnj (â) − Pni (â))ûin −(Pn j(a) − Pni (a))ûjn
n

=

∑

n

(Fni (â) − Fnj (â))∆uin −∑n(Fni (a) − Fnj (a))∆ujnnj≥ (λ (â, a|i, j) − 1)∆Gi (â, a) − (λ (â, a|i, j) − 1)∆Gj (â, a)i> −∆Gi (â, a) + ∆Gj (â, a)due to Assumption 3. Thus we have established that W (â, a|ûi , ûj ) > 0 for theaction a < â such that (IC) of agent j is binding.
Then inequality (A1) implies that∑∑
Pnj (â)ûjn − Gj (â) >Pnj (a)ûjn − Gj (a)nnwhich contradicts the supposition that (IC) of agent j is binding at a < â.
Thus our ﬁrst supposition that there are less than Nr − M risk averse agents
who choose the least costly action is false. This completes our proof. Q.E.D.

8.4

Proof of Theorem 2First, note that Assumption 3 is satisﬁed when ε is small enough because thenkλ (â, a) → 1 and λk (â, a) → 1 for all i, j ∈ Ir and all â, a ∈ A. Then Assumption 5
implies Assumption 3 when we take ε to be small enough given δ.
By Lemma 3, the wage scheme ŵj (y j ) speciﬁed in fair contract Ĉ j must be equal
to ŵj (y j |âj ) for any risk averse agent j ∈ Ir which solves problem (CMj − âj ). Now
29

i − âj )replace agent j by agent i in Problem (CMj − âj ), denoted by Problem∑ i(CMjwhich minimizes the expected wage paid to risk averse agent i, i.e. n Pn (â )w(yn ),subject to (IC) and (IR∗ ) for agent i (but not agent j) who is induced to choose
the action âj speciﬁed in the fair contract oﬀered to agent j (but not agent i). In
general, such problem may have no optimal solutions because the constraint set
may be empty as opposed to Problem (CMj − âj ). However, we can show that whenheterogeneity among risk averse agents is suﬃciently small ((ε, δ) is small enough)the optimal wage scheme which solves the problem (CMi − âj ) exists and is very
close to ŵj (y|âj ) the optimal wage scheme which solves Problem (CMj − âj ).

Lemma A1. There exist some δ > 0 and ε > 0 such that
¯
¯
¯∑
¯ 1
∑
¯
¯
Pnj (âj )ŵj (yn |âj ) −Pni (âj )ŵi (yn |âj )¯ < β
¯
¯
¯ 2
n

n

for any risk averse agents i, j ∈ Ir , i ̸= j, where β is given in Assumption 4.
Proof. Take any fair contract Ĉ and the corresponding expected utility of agent i
as Û i .
Then we ﬁrst show the following:
Claim A1-1. |Û i −Û j | goes to zero as the diﬀerence between agents i and j (capturedby δ and ε) becomes very small.
Proof. Let ŵnk ≡ ŵk (yn ) and ∆uin ≡ ûkn − ûkn−1 for each yn ∈ Y .
Then we derive∑Û j ≡Pnj (âj )u(ŵnj ) − Gj (âj )n≥∑Pnj (âi )u(ŵni ) − Gj (âi ) (by (NE))

n

=

∑

Pnj (âi )u(ŵni ) − Gi (âi ) + {Gi (âi ) − Gj (âi )}

n

=

∑

(1 − Fnj (âi ))∆uin − Gi (âi ) + {Gi (âi ) − Gj (âi )}

n

=

∑

(1 − Fni (âi ))∆uin − Gi (âi ) + {Gi (âi ) − Gj (âi )} +n

= Û + {Gi (â ) − Gj (â )} +
i

i

i

∑

∑

(Fni (âi ) − Fnj (âi ))∆uinn(Fni (âi ) − Fnj (âi ))∆uinn≥ Û + {Gi (â ) − Gj (â )} + (λ i (âi , a) − 1)∆Gi (âi , a)iii

≥ Û i − ε̃ + {Gi (âi ) − Gj (âi )}where ε̃ > 0 is deﬁned asε̃ ≥ |(λi (â, a) − 1)∆Gi (â, a)|
30

(A2)

for all â, a ∈ A and all i ∈ Ir . Note that ε̃ can be close to zero as ε → 0 because
then λi (â, a) → 1.
Thus we obtainÛ j ≥ Û i − ε̃ + {Gi (âi ) − Gj (âi )}(A3)
Similarly, we haveÛ i ≥ Û j − ε̃ + {Gj (âj ) − Gi (âj )}.

(A4)

By adding (A3) and (A4), we derive{Gi (âj ) − Gj (âj )} + ε̃ + Û i
≥ Û j≥ Û i − ε̃ + {Gi (âi ) − Gj (âi )}≥ Û i − ε̃ + {Gi (âj ) − Gj (âj )} − δfrom which Û i − Û j goes to zero as δ and ε go to zero (hence ε̃ → 0). Also, whenε = 0 and δ = 0, we have Û i = Û j . Thus there exists some small ρ̃ > 0 such that|Û i − Û j | ≤ ρ̃ for any i, j ∈ Ir , when (ε, δ) → 0. Q.E.D.
Now, we pick any risk averse agents i, j ∈ Ir . Then we consider the modiﬁedproblem of Problem (CMj − a) which implements a ∈ A from risk averse agent j bykeeping (MON) while replacing (IR) and (IC∗ ) by∑Pnj (a)u(w(yn )) − Gj (a) ≥ Û j − ρ(IR∗ρ )nand∑nPnj (a)u(w(yn )) − Gj (a) ≥∑Pnj (a′ )u(w(yn )) − Gj (a′ ) − ρ, ∀ a′ ̸= a(ICρ )

nrespectively where ρ > 0 (we will give a precise deﬁnition of ρ later.)
We call such modiﬁed problem of (CMj − a) Problem (CMjρ − a). Then we canshow the following result:
Claim A1-2. The solution to Problem (CMjρ − a), denoted (ŵρj (yn |a))n , exists andis uniquely determined if the constraint set is non-empty.
Proof. To save notation, we drop superscript j to index agent j in the following
proof. Let w(yn ) ≡ wn for each yn ∈ Y . We change the variable as un ≡ u(wn )and hence wn = φ(un ) where φ = u−1 is inverse of u. Then (MON) is replaced byφ(un ) ≥ φ(un−1 ) because of yn > yn−1 .
Let Γ(a) be the constraint the set of u ≡ (un )n satisfying all (ICρ ), (IR∗ρ ) and(MON) in Problem (CMjρ − a). Then we show that Γ(a) is a convex set. Takeu, u′ ∈ Γ(a) and let u′′ ≡ λu + (1 − λ)u′ for a scalar λ ∈ (0, 1). Then we can see
31that u′′ satisﬁes (ICρ ) and (IR∗ρ ) because these constraints are linear with u. Thus
it suﬃces to show that u′′ satisﬁes (MON). To see this, note thatφ(u′′n ) − φ(u′′n−1 ) ≥ φ′ (u′′n−1 )(u′′n − u′′n−1 )= φ′ (u′′n−1 )[λ(un − un−1 ) + (1 − λ)(u′n − u′n−1 )]≥ 0
due to φ′ > 0 and u′n ≥ u′n−1 and un ≥ un−1 by (MON). Thus u′′ ∈ Γ(a) and henceΓ(a) is a convex set.
Note that Γ(a) is closed because φ is continuous. Also Γ(a) can be bounded (seeGrossmanand Hart (1983)). Thus Γ(a) is compact set. Since the objective function∑ jP(a)φ(un ) is continuous and convex with respect to un , a solution to Problem
n nj(CMρ − a) exists and is unique if the constraint set Γ(a) is non-empty. Q.E.D.
Thus, if the constraint set of Problem (CMjρ − a) is non-empty, by the Berge’s
Maximum Theorem, ŵρj (y|a) is continuous in ρ.
Recall that ŵj (y|âj ) is the optimal wage scheme which solves Problem (CMj −âj ).
We denote ûin ≡ u(ŵi (yn )) and ûi (yn |a) ≡ ui (ŵi (yn |a)). Then Lemma 3 shows thatthe fair contract ŵj (y) must be equal to ŵj (y|âj ). Thus we have ûin ≡ ûi (yn |âi ) for
each i ∈ Ir .
Then we show the following claim.
Claim A1-3. There exists some K < +∞ such that |Û i | ≤ K for all i ∈ Ir .
i

Proof. By (IR), we have Û i ≥ U . Thus Û i is bounded below.
∑
Second, by using Problem (CMi −âi ) and convexity of φ, we also obtain n Pni (âi )φ(ûin ) =(∑ i i i)∑ i i iii
i
i
in Pn (â )(û (yn |â )) ≥ φn Pn (â )û (yn |â ) ≥ φ(Gi (â ) + Û ). Thus, if Û →+∞ for some i ∈ Ir , the expected wage of such agent i goes to inﬁnity, i.e.,∑i iin Pn (â )φ(ûn ) → +∞, so that the payoﬀ of some risk neutral agent must be
negatively inﬁnite but this violates (IR) of that agent. Thus Û i must be bounded
above as well. Q.E.D.
As we have already mentioned, ŵi (y|âi ) which solves Problem (CMi − âi ) for
implementing âi exists and is unique. Thus ŵi (y|âi ) is well-deﬁned. Then ŵi (y|âi ) isibounded for all Û i ∈ [U , K], i.e., there exists some M < +∞ such that |u(ŵi (y|âi ))| ≤i
M for all y ∈ Y and all Û i ∈ [U , K].
Then we can show the following:
Claim A1-4. For all a ∈ A and all i ̸= j, i, j ∈ Ir , we have
¯
{
}¯
¯
¯∑
∑
¯
¯
Pni (a)ûi (yn |âi ) − Gi (a) −Pnj (a)ûi (yn |âi ) − Gj (a) ¯ → 0
¯
¯
¯ n
n

32

as (ε, δ) → 0
Proof. We obtain

¯
¯
¯∑
¯
¯
¯
i
j
i
i
(P
(a)
−
P
(a))û
(y
|â
)
+
(G
(a)
−
G
(a))
¯
¯
n
j
i
n
n
¯ n
¯
∑¯
¯
¯
¯
¯Pni (a) − Pnj (a)¯ûi (yn |âi ) + ¯Gj (a) − Gi (a)¯
≤
n

∑¯
¯
¯
¯
¯Pni (a) − Pnj (a)¯M + ¯Gj (a) − Gi (a)¯≤nwhich goes to zero as (ε, δ) → 0 for all a ∈ A (note that A is a ﬁnite set). Q.E.D.
By Claim A1-4, we can ﬁnd some small enough ρ̂ > 0 such that
¯
}¯
{
¯
¯∑
∑
¯
¯
j
i
i
i
i
i
Pn (a)û (yn |â ) − Gj (a) ¯≤ ρ̂
Pn (a)û (yn |â ) − Gi (a) −
¯
¯
¯
n

(A5)

n

when (ε, δ) is small enough.
In what follows we deﬁne ρ ≡ max{ρ̃ + ρ̂, 2ρ̂}.
Now we consider Problem (CMjρ − âi ) for the deﬁned value ρ > 0 and then showthat the wage scheme ŵi (y|âi ) is feasible in Problem (CMjρ − âi ) for implementing âi ,i.e., it satisﬁes (MON), (IR∗ρ ) and (ICρ ). It is clear that (MON) is satisﬁed becauseŵi (y|âi ) is monotonic by deﬁnition. To see that it also satisﬁes (IR∗ρ ) and (ICρ ),note ﬁrst that∑∑
Pnj (âi )u(ŵi (yn |âi )) − Gj (âi ) ≥Pni (âi )u(ŵi (yn |âi )) − Gi (âi ) − ρ̂nn

≥ Û i − ρ̂≥ Û j − (ρ̃ + ρ̂)≥ Û j − ρwhere the ﬁrst inequality follows from Claim 4-4, the second inequality from the factthat ŵi (y|âi ) satisﬁes (IR∗ ) of Problem (CMi − âi ) to implement âi from agent i,
the third inequality from |Û j − Û i | ≤ ρ̃ and the last inequality from the deﬁnitionof ρ respectively. This shows that ŵi (y|âi ) satisﬁes (IR∗ρ ) in Problem (CMj − âi ) for
implementation of âi .

33

Also we have∑∑
Pnj (âi )u(ŵi (yn |âi )) − Gj (âi ) ≥Pni (a)u(ŵi (yn |a)) − Gi (a) − ρ̂
n

n

≥

∑

Pni (a′ )u(ŵi (yn |a)) − Gi (a′ ) − ρ̂

n

≥

∑

Pnj (a′ )u(ŵi (yn |a)) − Gj (a′ ) − 2ρ̂

n

≥

∑

Pnj (a′ )u(ŵi (yn |a)) − Gj (a′ ) − ρnwhere the ﬁrst and third inequalities follow from the deﬁnition of ρ̂ (see (A5)), thesecond inequality from the fact that ŵi (y|âi ) satisﬁes (IC) of Problem (CMi − âi )to implement âi ∈ A from agent i and the last inequality from the deﬁnition of ρ
respectively.
The above argument shows that ŵi (y|âi ) satisﬁes (IR∗ρ ), (IC∗ρ ) and (MON) of
Problem (CMjρ − âi ). Thus ŵi (y|âi ) is feasible in Problem (CMjρ − âi ). This implies
that the constraint set of Problem (CMjρ −âi ) is non-empty for implementation of âi .
Thus the optimal wage scheme ŵρj (y|âi ) which solves such problem is well-deﬁned.
Also, by optimality of ŵρj (y|âi ), we have∑∑
Pnj (âi )ŵi (yn |âi ) ≥Pnj (âi )ŵρj (yn |âi ).
(A6)
n

n

By changing the role of i and j in the above argument, we can show that ŵj (y|âj )is feasible in the problem (CMiρ − âj ). Thus we have∑∑
Pni (âj )ŵj (yn |âj ) ≥Pni (âj )ŵρi (yn |âj ).
(A7)
n

n

Then we show Lemma A1 as follows: ﬁrst, note that ŵρj (y|âi ) is continuous in ρand that limρ→0 ŵρj (y|âi ) = ŵj (y|âi ) = ŵi (y|âi ) for each y ∈ Y because Û i = Û j ,
Pnj (a) = Pni (a) and Gi (a) = Gj (a) are all satisﬁed when ρ = 0 so that (ε, δ) = (0, 0),and hence the optimal wage schemes which solve Problem (CMk − âi ) for agentsk = i, j must be the same for implementation of a given action âi . Then, sinceŵi (y|âi ) is well-deﬁned, ŵj (y|âi ) does so when ρ = 0. Then,∑ by letting ε and δ
to be small enough (thus ρ → 0), we obtain from (A6) that n Pni (âi )ŵi (yn |âi ) ≅∑ j i i∑∑(yn |âi ) ≥ n Pnj (âi )ŵρj (yn |âi ) ≅ n Pnj (âi )ŵj (yn |âi ). Also, from (A7)n Pn (â )ŵ∑∑ i j j
∑ i j ij jjjjj
we obtainn Pn (â )ŵ (yn |â ) ≅n Pn (â )ŵ (yn |â ) ≥n Pn (â )ŵρ (yn |â ) ≅
∑
j
i
j
i
n Pn (â )ŵ (yn |â ). Thus, by taking small enough (ε, δ) so that ρ is close to zero,
we can ensure the inequalities in Lemma A1. Q.E.D.
In what follows we ﬁx such ε > 0 and δ > 0 to ensure Lemma A1.

34

Lemma A2. âi = âj must hold for all risk averse agents i, j ∈ Ir in any faircontract Ĉ.
Proof. Take any fair contract Ĉ. Then ﬁrst note that the action proﬁle âr ∈ ANrof risk averse agents must maximize the expected residual surplus which the risk
neutral agents totally receive after paying the total expected wages to all risk averse
agents:∑Ey [R(y)|ar , an ] −Eyi [ŵi (y i )|ai ].
i∈Ir

Otherwise, Ĉ is not incentive eﬃcient because some risk neutral agents can be strictly
better oﬀ by implementing the action proﬁle of risk averse agents ar which maximizes
the above expected residual surplus. In this proof we will omit the vector of the risk
neutral agents’ actions an from the argument of the above function because it does
not play any role in the proof.
Here recall that, since the fair contract must solve Problem (CMi − âi ) for eachrisk averse agent i ∈ Ir , we have ŵi (y) = ŵi (y|âi ) for each i ∈ Ir .
Now suppose that âj ≡ a′′ > âi ≡ a′ for some risk averse agents i, j ∈ Ir . Then
we haveEy [R(y)|âj , âi , â−i−j ] − Eyj [ŵj (y j |âj )|âj ]≥ Ey [R(y)|âi , âi , â−i−j ] − Eyj [ŵj (y j |âi )|âi ]and
Ey [R(y)|âi , âj , â−i−j ] − Eyi [ŵi (y i |âi )|âi ]≥ Ey [R(y)|âj , âj , â−i−j ] − Eyi [ŵi (y i |âj )|âj ].
From these inequalities, we derive
Eyi [ŵi (y i |a′′ )|a′′ ] − Eyi [ŵi (y i |a′ )|a′ ]≥ ∆i (a′′ |a′′ , a′ )≥ ∆j (a′′ |a′′ , a′ ) − α≥ ∆j (a′ |a′′ , a′ ) + β̃ − α (by Assumption 4)≥ Eyj [ŵj (y j |a′′ )|a′′ ] − Eyj [ŵj (y j |a′ )|a′ ] + β
and hence by Lemma A1
1
β > Eyi [ŵi (y i |a′′ )|a′′ ] − Eyj [ŵj (y j |a′′ )|a′′ ]2≥ Eyi [ŵi (y i |a′ )|a′ ] − Eyj [ŵj (y j |a′ )|a′ ] + β
1> − β+β
2
1
=
β,
2
35

a contradiction. Thus we must have âj = âi . Q.E.D.
By Lemma A2, we must have âj = âi for any i, j ∈ Ir , i ̸= j, in any fair contract
Ĉ. Thus we will denote âi = â for all i ∈ Ir .
Now we pick any two risk averse agents i, j ∈ Ir where i > j in the sense ofAssumption 5. Suppose then that â ≡ âi = âj > 0. Then, since Assumption 3 issatisﬁed under Assumption 5 when ε is small enough given δ, we know from theproof of Theorem 1 that (IC) of Problem (CMj − â) must be slack for more eﬃcient
risk averse agent j at any action a < âj = â under the fair contract ŵj (y). However,this contradicts to the fact that the optimal wage scheme ŵj (y) = ŵj (y|â) whichsolves Problem (CMj − â) must have the property that (IC) is binding at some a < â(otherwise, the contract which solves Problem (CMj − â) must be the ﬁxed wage
contract but then it cannot satisfy (IC) for agent j to choose â > 0). Thus the
supposition that â > 0 is false, so we must have â = 0. Since âl = âk for all l, k ∈ Ir
by Lemma A2, we then have âi = 0 for all i ∈ Ir . Then (IE∗ ) implies that ŵi (y)must be the ﬁxed wage contract, i.e., ŵi (y) does not depend on y ∈ Y for all i ∈ Ir(otherwise, âj = 0 but ŵj (y) is not ﬁxed wage for some j ∈ Ir . However this is
not incentive eﬃcient because oﬀering a ﬁxed wage contract to such agent j ∈ Ir
improves the welfare of some risk neutral agents while keeping all the actions andcontracts of others unchanged.)
We deﬁne the set of the values ε and δ representing the heterogeneity of agents
as follows:D ≡ {(ε, δ) ∈ ℜ+ | (ε, δ) satisﬁes (6), (7) and Lemma A1.}.
Then we obtain Theorem 2 for all (ε, δ) ∈ D. Q.E.D.

8.5

Proof of Theorem 3

Given an action proﬁle of risk averse agents in a fair contract ar = (âi )i∈Ir , themaximum residual surplus which all risk neutral agents can share is∑∑
E[R(y)|ar , an ] −Gi (ai ) −Eyi [ŵi (y i )|âi ]
i∈In

i∈Ir

where an = (ai )i∈In denotes the vector of actions of risk neutral agents.
By deﬁnition, (ãi )i∈In maximizes the above surplus. Let deﬁne the best responseof action ai for each risk neutral agent i ∈ In , given a−i ∈ ANn −1 :∑BRi (a−i ) ≡ arg max Ey [R(y)|ar , (ai , a−i )] −Gi (ai ).
ai ∈A

i∈In

Then, the deﬁnition of ãi means that ãi ∈ BRi (ã−i ) for each i ∈ In .
36

Now suppose contrary to the claim that the wage schemes of risk neutral agentsin some fair contract ŵi (y) do not implement the optimal actions (ãi )i∈In deﬁned
above. Let (âi )i∈In be the implemented action proﬁle of risk neutral agents under{ŵi (y)}i∈In . Since (âi ) does not maximize the residual surplus, there must exist
some risk neutral agent k such thatâk ∈/ BRk (â−k ).
Then we deﬁne the alternative wage schemes as follows:w̃i (y i ) ≡ Ey−i [ŵi (y i , y −i )|â−i ], ∀ i ̸= k ∀ i ∈ In , ,w̃i (y i ) = ŵi (y i ) ∀ i ∈ Ir ,andw̃k (y) = R(y) −∑w̃i (y i )i̸=k,i∈I

Note that the wage scheme of risk neutral agent k depends on y −k only throughR(y) so that its expected wage depends on a−k only through E[R(y)|ak , a−k ].
These new schemes satisfy (BB) by construction. Also, since {ŵi (y)}i∈In implements the action proﬁle (âi )i∈In and (y i )i∈In are statistically independent, the newscheme w̃i (y i ) still implements action âi from agent i ̸= k. Thus all agents but kchooses the same actions âi as before. Finally, risk neutral agent k has the strict incentive to choose some action in BRk (â−k ) rather than âk because of âk ∈/ BRk (â−k ).
Then we have∑max Ey [w̃k (y)|ar , (a, â−k )] − Gk (a) −Eyi [w̃i (y i )|âi ]
a∈A

i̸=k
−k

r

= max Ey [R(y)|a , (a, â
a∈A

r

−k

k

> Ey [R(y)|a , â , â)] − Gk (a) −] − Gk (a) −∑∑Ey [ŵi (y)|â]i̸=kEyi [ŵi (y i )|â]

i̸=k
k

r

k

−k

= Ey [ŵ (y)|a , (â , â)] − Gk (â )kwhere thefollows from âk ∈/ BRk (â−k ) and the last equality from (BB):∑ inequalityiR(y) = i∈I ŵ (y).
However, then this contradicts the fact that Ĉ i = {ŵi , âi } is incentive eﬃcient.
Q.E.D.

8.6

Proof of Lemma 4

Suppose that Assumption 4 and 6 hold. Suppose also contrary to the claim that
there exists some asymmetric optimal action proﬁle ãn which maximizes the residual
surplus RS(an ). Then there must exist some l, k ∈ In , l ̸= k such that ãl ̸= ãk .
37

Let a′′ ≡ ãl > a′ ≡ ãk without loss of generality. Then we have
Ey [R(y)|0, a′′ , a′ , ã−l−k ] − Gl (a′′ )≥ Ey [R(y)|0, a′ , a′ , ã−l−k ] − Gl (a′ )andEy [R(y)|0, a′ , a′′ , ã−l−k ] − Gk (a′ )≥ Ey [R(y)|0, a′′ , a′′ , ã−l−k ] − Gk (a′′ ).
By adding these inequalities, we have
Gk (a′′ ) − Gk (a′ ) ≥ ∆k (a′′ |a′′ , a′ )≥ ∆k (a′ |a′′ , a′ ) + β̃
≥ ∆l (a′ |a′′ , a′ ) − α + β̃
= ∆l (a′ |a′′ , a′ ) + β≥ Gl (a′′ ) − Gl (a′ ) + β
which implies that1β > Gl (a′ ) − Gk (a′ )2≥ Gl (a′′ ) − Gk (a′′ ) + β
1> − β+β21=
β,2a contradiction. Q.E.D.

8.7

Proof of Theorem 4

By Lemma 4, we can ensure the existence of a symmetric optimal action ã which
maximizes the residual surplus of organization RS(an ) over the actions of risk neutral
agents an after subtracting ﬁxed wages of risk averse agents. In what follows wetake such action ã and consider the implementation of ai = ã from all risk neutral
agents i ∈ In .
Now we will show that oﬀering a ﬁxed wage w to every risk averse agent i ∈ Ir
becomes incentive eﬃcient. To this end, it suﬃces to show that the ﬁxed wage
contract C ≡ {w, ai = 0} maximizes the residual surplus of organization which
belongs to all risk neutral agents, given any action proﬁle of risk neutral agentsan ∈ ANn . This is because then there exist no other feasible contracts which Pareto
dominate the ﬁxed wage contract.
38

Formally, we show the following lemma:
Lemma A3. When the ﬁxed wage w is large enough, it becomes incentive eﬃcient.
Proof. We consider the following maximization problem which states that the
expected residual surplus of risk neutral agents is maximized subject to (IC) andthe acceptance constraint ensuring that risk averse agent cannot be worse oﬀ byswitching to a new contract rather than sticking to the ﬁxed wage w:
Problem (OP)maxar ∈ANr ,wi (y)Ey [R(y)|ar , an ] −∑Eyi [wi (y i )|ai ]i∈Irsubject toEyi [u(wi (y i ))|ai , a−i ] − Gi (ai ) ≥ Eyi [u(wi (y i ))|a, a−i ] − Gi (a) ∀ a ̸= ai
Eyi [u(wi (y i ))|ai , a−i ] − Gi (ai ) ≥ u(w)

(IC)
(PE)

Here (PE) is the constraint to ensure that risk averse agent i ∈ Ir can be better oﬀ
by accepting a contract C i ̸= C.
Then we show that the contract to solve the above problem becomes the ﬁxed
wage contract C for any given action proﬁle of risk neutral agents an when w islarge enough. To see this, note that the objective function in the above problem isbounded above by∑
Ey [R(y)|ar , an ] −φ(Gi (ai ) + u(w))
i∈Irbecause, by concavity of u and (PE), we have
Ey [wi (y)|ai ] ≥ φ(Gi (ai ) + u(w)).
Now consider the following maximization problem:∑
φ(Gi (ai ) + u(w)).
max Ey [R(y)|ar , an ] −ar ∈ANri∈IrThen we can verify that ar = 0 solves this maximization problem whenφ(Gi (ai ) + u(w)) − φ(Gi (0) + u(w)) ≥ Ey [R(y)|0, an ] − Ey [R(y)|ar , an ].
In fact we can ﬁnd some w so that the above inequality holds because the right handside is ﬁnite (by ﬁniteness of A) but the left hand side goes to +∞ as w → +∞:φ(Gi (ai ) + u(w)) − φ(Gi (0) + u(w))> φ′ (Gi (0) + u(w))(Gi (ai ) − Gi (0))→ +∞ as w → +∞39

due to the convexity of φ and limw→+∞ φ′ (w) = +∞. Thus, when w is large enough,the optimal contract which solves Problem (OP) becomes the ﬁxed wage w for allrisk averse agents. Q.E.D.
Next we consider the incentives of risk neutral agents. We take the risk neutral
agent r ∈ In whose action cost is the largest among all risk neutral agents, i.e.,r ∈ arg max Gi (ã).
i∈In

We will deﬁne the wage scheme of the residual claimant later.
For the wage schemes of all other risk neutral agents than the residual claimantr, we will apply the following lemma.
Lemma A4. Suppose that Assumption 8 is satisﬁed. Then there exists a nondecreasing individualistic wage scheme w̃(y l ) for each risk neutral agent l ∈ In withl ̸= r such that it implements the optimal action ã:Eyl [w̃(y l )|ã] − Gl (ã) ≥ Eyl [w̃(y l )|a] − Gl (a) + δ
for all a ̸= ã where δ is given in (6) in the main text.
Proof. The implementation problem above is reduced to ﬁnd a L − 1 dimensional
vector (∆wn )L−1
n=1 such that∑∑
(1 − Fnl (ã))∆wn − Gl (ã) ≥(1 − Fn (a))∆wn − Gl (a) + δ, ∀ a ̸= ã.
n

n

and
∆wn ≥ 0, n = 1, 2, ..., L − 1.
The last condition is monotonicity requirement (MON) that the wage schedule wn
must be non-decreasing. By Avis and Kaluzny (Theorem 2, 2004), such non-negative
vector (∆wn ) exists under Assumption 8. Q.E.D.
By Lemma A4, we obtainEz [w̃(z)|ã] − Ez [w̃(z)|a] ≥ Gl (ã) − Gl (a) + δ≥ Gi (ã) − Gi (a).
for all i ∈ In with i ̸= n. Thus every risk neutral agent i ∈ In other than the residual
claimant r chooses the same optimal action ã under the same scheme w̃.
Deﬁne ŵ(y i ) ≡ w̃(y i ) + f for i ∈ In , i ̸= r, where f will be speciﬁed below, andoﬀer such scheme to all i ̸= r, i ∈ In . Then it is clear that every risk neutral agenti ̸= r, i ∈ In , does not envy other risk neutral agent j ̸= i, r, j ∈ In .

40

The wage scheme for residual claimant r is given by∑
wr (y r , y −r ) ≡ R(y r , y −r ) − Nr w −ŵ(y i ) + ŵ(y r ).
i∈In

Since ŵ(y i ) depends only on his performance y i for each risk neutral agent i except
the residual claimant r, the wage scheme of residual claimant wr (y r , y −r ) depends
on his performance y r only through R(y r , y −r ). Since R is monotone increasing in
y r , wr does so. Thus (MON) is satisﬁed for the residual claimant r as well.
We also deﬁne the ﬁxed wage which is oﬀered to risk averse agent:w=

1{E[R(y)|0, ã] − Nn Gr (ã)}.
N

Given such w, we set f to satisfy
E[w̃(z)|ã] + f = Gr (ã) + w.
Then, given such (w, f ), we can show thatEy [R(y)|0, ã] − Nr w − (Nn − 1)Ez [ŵ(z)|ã] − Gr (ã)= Ey [R(y)|0, ã] − Nr w − (Nn − 1)(Ez [w̃(z)|ã] + f ) − Gr (ã)= Ez [w̃(z)|ã] + f − Gr (ã)= w.
Then we show the following series of lemmas:
Lemma A5. The residual claimant r does not envy other agents.
Proof. By using the wage schemes deﬁned above, we can verify that the residual
claimant r does not envy other risk neutral agent i ̸= r, i ∈ In , because
max Ey [R(y)|0, (ar , ã−r )] − Nr w − (Nn − 1)E[ŵ(z)|ã] − Gr (ar )ar ∈A≥ Ey [R(y)|0, ã] − Nr w − (Nn − 1)Ez [ŵ(z)|ã] − Gr (ã)= Ez [ŵ(z)|ã] − Gr (ã)= Ez [w̃(z)|ã] + f − Gr (ã)≥ Ez [w̃(z)|a] − Gr (a) + f= Ez [ŵ(z)|a] − Gr (a) ∀ a ∈ A
Also the residual claimant r does not envy any risk averse agent j ∈ Ir because
max Ey [R(y)|0, ar , ã−r ] − Nr w − (Nn − 1)E[ŵ(z)|ã] − Gr (ar )ar ∈A≥ Ey [R(y)|0, ã] − Nr w − (Nn − 1)Ez [ŵ(z)|ã] − Gr (ã)= w
≥ w − Gr (a).
41

Q.E.D.
Lemma A6. Each risk neutral agent other than the residual claimant r does notenvy other agents.
Proof. First we show that any other risk neutral agent i ∈ In than the residual
claimant r does not envy any risk averse agent. This is because
Ez [ŵ(z)|ã] − Gi (ã) ≥ Ez [ŵ(z)|ã] − Gr (ã)= w
≥ w − Gi (a)for all a ̸= ã, where Gi (ã) ≤ Gr (ã) for all i ̸= r.
We also show that every risk neutral agent i ∈ In , i ̸= r, does not envy theresidual claimant r because the expected payoﬀ of agent i ̸= r would be the following
if he were oﬀered the wage scheme of agent r (i.e., he were residual claimant):∑
Ey [R(y)|0, ai , ã−i ] − Nr w −Eyj [ŵ(y j )|ã] − Gi (ai )j∈In ,j̸=iwhere agent i was supposed to choose ai . Since agent i obtains Eyi [ŵ(y i )|ã] − Gi (ã)
under his own contract, he prefers his own contract ŵ to the one oﬀered to theresidual claimant r if
Ez [ŵ(z)|ã] − Gi (ã)≥ Ey [R(y)|0, ai , ã−i ] − Nr w −∑Eyi [ŵ(y j )|ã] − Gi (ai )j∈In ,j̸=iwhere the right hand side cannot be greater thanEy [R(y)|0, ã, ã−i ] − Nr w − (Nn − 1)Ez [ŵ(z)|ã] − Gi (ã)
by deﬁnition of ã (which maximizes Ey [R(y)|0, ai , ã−i ] − Gi (a) over a ∈ A). Thus
it suﬃces to show thatEz [ŵ(z)|ã] − Gi (ã)≥ Ey [R(y)|0, ã] − Nr w − (Nn − 1)Ez [ŵ(z)|ã] − Gi (ã)
which is equivalent toEz [ŵ(z)|ã] ≥ Ey [R(y)|0, ã] − Nr w − (Nn − 1)Ez [ŵ(z)|ã].
This holds as equality due to the deﬁnitions of ŵ(z) = w̃(z) + f and w.
Finally, it is clear that any risk neutral agent j ̸= r does not envy other risk
neutral agent k ̸= r because they are oﬀered the same wage scheme. Q.E.D.
42

Lemma A7. Each risk averse agent i ∈ Ir does not envy other agents when δ issmall enough and F is large enough.
Proof. First we show that every risk averse agent does not envy any other risk
neutral agent j ∈ In than the residual claimant r when δ is small and w is large
enough. To see this, we deﬁne ηi for each i ∈ I, which satisﬁes
E[ŵ(z)|ã] − Gi (ã) − w = ηi .
Note here that ηr = 0 and ηi → 0 as δ → 0. Take any risk averse agent i ∈ Ir . Then
we can show that when δ > 0 is suﬃciently small and the ﬁxed wage w can be large
enough, we haveu(w) − u(Ez [ŵ(z)|a])≥ u′ (w){w − Ez [ŵ(z)|a]}= u′ (w){Ez [ŵ(z)|ã] − Gi (ã) − ηi − Ez [ŵ(z)|a]}≥ u′ (w){−Gi (a) − ηi }≥ −Gi (a) − ηiwhere the ﬁrst inequality follows from concavity of u, the second inequality fromthe fact that Ez [ŵ(z)|ã] − Gi (ã) ≥ Ez [ŵ(z)|a] − Gi (a) for all a ̸= ã, and the third
inequality from the fact that u′ (w) ≤ 1 for large enough w (by using u′ (+∞) = 0)
respectively. Thus we obtainu(w) ≥ u(E[z ŵ(z)|a]) − Gi (a) − ηi
> Ez [u(ŵ(z))|a] − Gi (a)for all a ∈ A when δ is so small that ηi can be close to zero if w becomes large
enough.
Next we show that every risk averse agent i ∈ Ir does not envy the residual
claimant r when ε is suﬃciently small. To see this, note that
u(w)

(

)
1
= u
{Ey [R(y)|0, ã] − Nn Gr (ã)}
N= u(Ey [R(y)|0, ã] − Nr w − (Nn − 1)Ez [ŵ(z)|ã] − Gr (ã)).

Then we want to show 
u(w) ≥ E u R(y) − Nr w −

∑
j∈Ir


¯
¯
¯
ŵ(y j ) + ŵ(y i ) ¯ai , â−i  − Gi (ai )
¯

43

(A8)

for all ai ∈ A. For this, by the Jensen’s inequality, it suﬃces to show thatu(Ey [R(y)|0, ã] − Nr w − (Nn − 1)Ez [ŵ(z)|ã] − Gr (ã))≥ u(Ey [R(y)|0, ai , ã−i ] − Nr w − Nn Ez [ŵ(z)|ã] + Ez [ŵ(z)|ai ]) − Gi (ai )for all ai ∈ A. To show this, note that by deﬁnition of w,
w = Ey [R(y)|0, ã] − Nr w − (Nn − 1)Ez [ŵ(z)|ã] − Gr (ã).
Let
B ≡ Ey [R(y)|0, ai , ã−i ] − Nr w − Nn Ez [ŵ(z)|ã] + Ez [ŵ(z)|ai ].
Then the above inequality (A8) can be written byu(w) ≥ u(B) − Gi (ai )which can be further written byu(w) − [u(B) − Gi (ai )]≥ u′ (w)(w − B) + Gi (ai )wherew − B = Ey [R(y)|0, ã] − Ey [R(y)|0, ai , ã−i ]+ Ez [ŵ(z)|ã] − Gr (ã) − Ez [ŵ(z)|ai ].
Under Assumption 9 Ey [R(y)|0, ã] − Ey [R(y)|0, ai , ã−i ] is independent of F . We
also know that Ez [ŵ(z)|ã] − Ez [ŵ(z)|a] = Ez [w̃(z)|ã] − Ez [w̃(z)|a] is independent ofF (because w̃(z) is independent of F ). Thus Ez [ŵ(z)|ã] − Ez [ŵ(z)|a] is independentof F as well. Also the optimal action of risk neutral agents ã does not vary with F .
Then the above term w − B is independent of F . We can also show that w increases
in F (because the optimal action ã is independent of F so that it does not changewith F ). Thus we can ensure that u′ (w) → 0 as F → +∞, and henceu(w) − [u(B) − Gi (ai )] ≥ u′ (w)(w − B) + Gi (ai )≅ Gi (ai )> 0for all ai > 0∑when F is suﬃciently large. Thus, since u is strictly concave andR(y)−Nr w− j̸=r ŵ(y j )+ ŵ(y r ) is random, we can verify that when F is suﬃciently
large,u(w)≥ u(B) − Gi (ai ) > Ey u R(y) − Nr w −

∑
j∈In


¯
¯
¯
ŵ(yj ) + ŵ(yi ) ¯0, ai , ã−i  − Gi (ai )¯44for all ai ∈ A. Thus agent i ∈ Ir does not envy the residual claimant r. Q.E.D.
From Lemma A5-A7, we have established the result that any agent does notenvy others under the constructed contract.
Finally, when F is large enough, we can take large f and w so thatEz [ŵ(z)|ã] − Gi (ã) ≥ U

ifor all i ∈ In but i ̸= r. Thus we can satisfy (IR) for all i ∈ In . Also, since theexpected payoﬀ of the residual claimant r is same as w, we can also ensure that theresidual claimant r and all risk averse agents i ∈ Ir obtain larger payoﬀs than theirireservation payoﬀs U as well when F is suﬃciently large (note that w can be large
when F is large).

8.8

Proof of Theorem 6

We will modify the proof of Theorem 2 by replacing (MON) by (LL). The main
modiﬁcation of the proof is that we show that Assumption 3 is satisﬁed when wedrop (MON) but impose (LL), if ε is suﬃciently small given δ.
Take two risk averse agents i and j. Here agent j is more eﬃcient than i in thesense that ∆Gi (â, a) > ∆Gj (â, a) for all â > a (Assumption 5). Recall that theseagents must choose the same action â due to Lemma 5, when their heterogeneity is
suﬃciently small.
Now suppose that â > 0 as in the proof of Theorem 2. Then recall that we havereached the following conclusion that (IC) of more eﬃcient risk averse agent j isslack at â > 0 if∑∑
∆Gi (â, a) − ∆Gj (â, a) +(Pnj (â) − Pni (â))ûin −(Pnj (a) − Pni (a))ûjn
(A9)
n

n

is strictly positive for all a < â.
From now on we will show that this is actually the case when ε is suﬃciently
small, given (LL). Since u ≥ ûin ≥ u for all yn ∈ Y due to (BB) and (LL), the second
term in (A9) is bounded below from∑∑Γij (â) ≡(Pnj (â) − Pni (â))u −(Pnj (â) − Pni (â))uyn ∈Y \Y (â)

yn ∈Y (â)

where Y (â) is deﬁned as Y (â) ≡ {yn ∈ Y | Pnj (â) ≥ Pni (â)}. Also, the third term in(A9) is bounded above by∑∑Ωij (a) ≡(Pnj (a) − Pni (a))u.
(Pnj (a) − Pni (a))u +yn ∈Y (a)yn ∈Y \Y (a)Thus (A9) is bounded below from
∆Gi (â, a) − ∆Gj (â, a) + Γij (â) − Ωij (a).
45Here, if we take ε to be suﬃciently small, both Γij (â) and Ωij (a) become smallenough. Then, since ∆Gi (â, a) > ∆Gj (â, a) for i > j by Assumption 5, we can showthat
∆Gi (â, a) − ∆Gj (â, a) + Γij (â) − Ωij (a) > 0
for all â > a when ε is suﬃciently small, given δ. Thus, when we take ε to be
suﬃciently small given δ, we can establish the result that (IC) of more eﬃcient risk
averse agent j is not binding at the action â > 0 at fair contract Ĉ j . However, thisis not incentive eﬃcient as we have argued in the proof of Theorem 2. Q.E.D.

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