Discussion Papers In Economics
And Business

Moral Hazard and Target Budgets
Shingo Ishiguro, Yosuke Yasuda
Discussion Paper 18-03

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANMoral Hazard and Target Budgets
Shingo Ishiguro, Yosuke Yasuda
Discussion Paper 18-03

February 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANMoral Hazard and Target Budgets∗
Shingo Ishiguro†, Yosuke Yasuda‡
Graduate School of Economics, Osaka University
February 1, 2018

Abstract
In this paper we investigate a wide class of principal–agent problems with moral hazard and target budgets. The latter requires that the principal ﬁxes a total budget for
the wages paid to agents regardless of their outputs realized ex post. Target budgets are
relevant not just because they are exogenous institutional constraints in some cases, butthey can also endogenously arise in other cases, especially when agents’ performances
are not veriﬁable and thus the principal needs subjective evaluations. Although target
budgets impose an additional constraint, we show the irrelevance theorem that the principal is never worse oﬀ using target budgets when there are at least two risk-neutral
agents. Even when all agents are risk averse, we also show that the similar irrelevanceresult asymptotically holds if the number of agents is suﬃciently large. Furthermore,
we characterize optimal contracts when the target budget becomes a tight constraint sothat the irrelevance result cannot be applied.
Keywords: Moral Hazard, Multiple Agents, Subjective Evaluation, Target Budgets
JEL Classiﬁcation Numbers: D82, D86∗

We are grateful to Wing Suen and the seminar participants of the Contract Theory Workshop for theiruseful comments and discussions. All remaining errors are our own. Ishiguro appreciates the ﬁnancial
support from the Japanese Society for the Promotion of Science KAKENHI Grant Number 20245031. Yasuda
appreciates the ﬁnancial support from the Japanese Society for the Promotion of Science KAKENHI Grant
Number 17K13702.
†
Corresponding author: Shingo Ishiguro, 1-7 Machikaneyama, Toyonaka, Osaka 560 0043, Japan. Phone:+81 6 6850 5220, Fax: +81 6 6850 5256, E-mail: ishiguro@econ.osaka-u.ac.jp‡
E-mail: yosuke.yasuda@gmail.com

1

1

Introduction

1.1

Motivation and Results

Performance-based compensation is necessary to provide the right incentives to agents whose
actions are not publicly observable (Holmström (1979), Grossman and Hart (1983)). However, in many real-life situations, compensation is limited by predetermined budgets, as weshow in the several examples below. The presence of such budget constraints restricts the setof available contracts and this conﬂicts with the incentive provision considered in standard
models of moral hazard.
In this paper we investigate a wide class of principal–agent problems with moral hazard
and target budgets. A principal hires multiple agents who work on behalf of herself. Although the principal cannot directly observe the actions chosen by agents, she can observe
their performance signals, called outputs herein. In addition to the incentive compatibility(IC) and individual rationality (IR) constraints, we introduce the target budget constraintthat requires that once the principal determines a total budget for the sum of the wages paid
to all agents ex ante, she must follow the budget regardless of the agents’ outputs realized
ex post.
Target budgets are relevant for many organizational problems in real world. First andperhaps most importantly, they endogenously arise when the principal needs to evaluate
agents subjectively based on the privately observed outputs (we discuss more details about
the related literature on this issue later). Empirical evidence shows that workers are commonly evaluated by subjective rather than objective performance measures (see Murphy(1993), Bushman, Indjejikian, and Smith (1996), and Gibbs, Merchant, Van Der Stede, andVargus (2009) for related evidence). Under such subjective evaluations, the principal has anincentive to manipulate the outputs of agents so as to reduce the wage payments to them.
Such opportunistic behavior by the principal in turn undermines the ex ante incentives of
agents to work hard. Getting rid of the incentive for manipulation, the principal needs tocommit the total amount of wage payments whatever outputs are realized ex post.1 This isbecause if the total wages of agents vary with their outputs, the principal has an incentive
to misreport the outputs unless observed outputs require her to pay the minimum amount.
Second, target budgets are also relevant as exogenous institutional constraints even when
agents’ outputs are objective and veriﬁable. For example, according to their accounting andbudgeting policies, ﬁrms or divisions within a ﬁrm may meet the annual budget constraint for1

Although the principal may burn money to commit herself to paying a ﬁxed amount, money burning isnever optimal, as we show later.

2the wages paid to employees. Governments also set ﬁscal budgets for the payrolls of oﬃcials
working in the public sector. Similarly, some universities allocate budgets among diﬀerent academic units according to their performances. For example, over last three decades
several universities in U.S. have introduced a budget allocating rule, called Responsibility
Center Management (RCM), in order to induce diﬀerent academic units to compete for students (Whalen (1991)). Under RCM, academic units of an university are distributed tuition
revenues from the university’s budget based on veriﬁable outcomes such as the number of
attracted students (Wilson (2002)). Such a budget allocating rule creates the incentives
for academic units to improve educational qualities. As a related example, in several countries governments or independent institutes allocate ﬁxed budgets for research and teachingfunding among diﬀerent public schools and universities based on their research and teachingoutcomes. In U.K. an independent institute called Higher Education Funding Council for
England (HEFCE) receives a public funding from the U.K government and allocate it among
diﬀerent universities for research and teaching. In particular, the allocation of researchfunding is based on the research qualities of universities which are evaluated periodically by
Research Assessment Exercise (RAE).2
What are the optimal contracts when a target budget is imposed in addition to thestandard IC and IR constraints? We call a contract the third-best contract when it maximizes
the principal’s payoﬀ subject to all IC, IR and target budget constraints. If there areno target budget constraints, standard moral hazard models suggest that when agents aretechnologically independent of each other, the optimal contract should be separately andindependently designed for each agent. However, when the total amount of wages is ﬁxed at apredetermined constant, the optimal contracts for agents can no longer be independent, since
the wage of some agent must depend on outputs of others, even if they are technologicallyindependent of each other. When the wage of some agent, say agent i, increases as his output
rises, the wage of at least one other agent, say agent j ̸= i, must be reduced to make thetotal wages of all agents constant.
Such wage interdependency complicates the characterization of the third-best contract.
Our main objective in this paper is to address this problem and analyze how the presenceof target budgets aﬀects the characterization of third-best contracts.
Our ﬁndings are summarized as follows. First, we show the irrelevance theorem that the
principal is never worse oﬀ by target budgets as long as there are at least two risk-neutral
agents. Even in the presence of target budgets, the principal can achieve the second-best

2

For more details, see the web site of HEFCE: http://www.hefce.ac.uk/funding/annallocns/.

3

payoﬀ,3 which she would obtain if the target budget constraint were absent.
From this eﬃciency result, we can obtain a signiﬁcant implication about subjective evaluations. As already mentioned, our model is applied to the case of subjective evaluations
that agents’ outputs are not publicly revealed. In that case, the principal may manipulate
her reports about privately observed outputs. One traditional approach to address this issue
is to consider the long-term relationships between the principal and agents, which ensuresthat the principal will be punished in the future if she breaches the current promise (Levin(2002), Rayo (2007), Mukherjee and Vasconcelo (2011), and Ishihara (2017)). However,these previous studies have not fully investigated what the principal can attain in the static
benchmark. Indeed, they have simply stated that the principal can implement only the least
costly actions from agents in the static case. For example, Levin (2002) states as follows: “Inthis environment the static equilibrium is straightforward. Because the ﬁrm cannot commit
to reward performance, workers will do no more than the minimum, and the ﬁrm will dobest not to produce at all” (p. 1079)).
Although this statement is true when there is a single agent, the same conclusion may nothold when there are multiple agents. Indeed, the papers cited above all employ models withmultiple agents. Do static contracts always work worse than relational contracts do as these
papers have mentioned? Our eﬃciency result shows that this is not the case. In the static
environment in which agents’ outputs are not veriﬁable (more severely, they are privately
observed by the principal), the total wages of agents must be ﬁxed at a constant. However,the principal is never worse oﬀ from the second-best case with fully veriﬁable outputs if there
are at least two risk-neutral agents.4 Thus, there is no further room to improve eﬃciency byrelational contracting. We highlight this result in comparison with the existing literature on
relational contracts with multiple agents, which has emphasized the advantage of relational
contracting over static contracts (see the studies cited above).
The intuition behind our eﬃciency result is as follows. There must be at least two agents
whose wages are interdependent as already discussed. For the ease of exposition, supposethat there are two risk neutral agents while all others are risk averse.5 The principal canonly oﬀer risk neutral agents interdependent wage schemes, while oﬀering the second-best
independent contracts to risk-averse agents. Then, risk-averse agents face the same expected3

We use the terminology the second best because the principal is constrained by the IC constraints ofagents ensuring that any contract must induce agents to choose the right actions owing to their self-interest.
This is distinguished from the first best, which is the optimal outcome achieved when agents’ actions are
directly contractible.
4
Most studies of relational contracts with multiple agents have focused on the case of risk-neutral agents.
5
We can easily adapt our argument to the case of more than two risk neutral agents.

4

payoﬀs as those they would obtain in the second-best situation if no target budget constraint
were present. They thus choose the second-best actions and obtain the second-best payoﬀs.
In addition, the wage schemes of two risk-neutral agents are constructed such that they arepaid not only according to piece rates depending on their own individual outputs; they also
receive an equal share of the residual wage, which is deﬁned as the ﬁxed total payroll minus
the sum of the piece rate wages paid to all agents. Thus, risk-neutral agents serve to balance
total wage payments, thereby ensuring that the principal ends up paying an ex ante ﬁxed
amount regardless of the realized outputs. Since risk-neutral agents can absorb all the riskof the residual wage, no additional costs are associated with the risks caused under suchwage schemes. Hence, the principal can achieve the second-best payoﬀ even when she faces
a target budget constraint.
We can also connect our eﬃciency result to the literature on rank-order contracts (e.g.,
Lazear and Rosen (1981), Malcomson (1984)). Rank-order contracts motivate agents while
keeping the total prizes paid to them constant: the agent who performs the best obtains
the highest prize, the agent who performs the second highest obtains the second highest
prize, and so on. Since total prizes are ﬁxed, the principal has no incentive to misreport theoutputs of agents even when they are not veriﬁable. However, our eﬃciency result showsthat rank-order contracts never become optimal; rather, all agents except risk-neutral ones
should be oﬀered piece rate wages that depend only on their own outputs. Thus, it is not
optimal to use relative performance evaluations such as rank-order contracts for all agents
when the total budget for their wages is ﬁxed.
We next turn to the case of at most one risk-neutral agent so that the above eﬃciency
result no longer applies. Although it is complicated to fully characterize the third-best
contract in general settings, we provide the useful characterization result that the third-best
contract is given by a tractable linear formula, which we call the simple sharing rule: thewage of each agent is a linear combination of two parts. One is the piece rate wage which
depends only on his own output, and the other is the share of the residual wage deﬁned as thediﬀerence between a ﬁxed total wage and the sum of all agents’ piece rate wages. We showthat this simple sharing rule becomes optimal if and only if agents’ preferences over incomelotteries are represented by utility functions with constant absolute risk aversion (CARA).
The shares of the residual wage are determined by the relative magnitudes of agents’ risk
aversion. More risk-averse agents are rewarded by more piece rate wages but a lower shareof the residual wage.
Beyond the speciﬁc forms of utility functions, it is diﬃcult to characterize the third-best
contract in general. However, we show that such a contract has simple and important features5

in the asymptotic case as the number of agents becomes suﬃciently large. In particular, weshow that under the third-best contract, most agents are compensated according to almost
their own individual outputs but not those of others when the number of agents is suﬃciently
large. While the wage of each agent may depend on the outputs of others, such eﬀects
become negligible when the number of agents is suﬃciently large. In this way, most agentsare rewarded according to almost their piece rates or individual performance evaluation(hereafter, IPE). Hence, relative performance evaluations such as rank-order contracts tend
to have no value asymptotically in large organizations.
We show this asymptotic result as follows. Contrary to the above claim, if a large
proportion of agents is oﬀered wage schemes that are not IPE under the third-best contract,
the principal can then increase her payoﬀ by enlarging the set of agents oﬀered IPE wage
schemes. To this end, the principal divides the set of all agents into two subsets. The agents
belonging to one set are oﬀered IPE wage schemes, which induce them to choose the sameactions as those under the original third-best contract. All the other agents belonging to
the remaining subset either randomly face an IPE wage scheme or equally share the residual
wage deﬁned as the diﬀerence between the ﬁxed total payroll and the wages paid to theagents who work under IPE schemes. The agents paid residual wages serve to balance thetotal wage payments of all agents regardless of the outputs realized. This random wage
scheme imposes a risk on some agents because, with a positive probability, they must share
the residual wage depending on the outputs of others. However, each of them needs tobear only a small share of such risk as the number of agents is large because of the Law
of Large Numbers. In this way, by enlarging the set of agents oﬀered IPE wage schemes,
the principal can improve eﬃciency by reducing the risk that agents would otherwise incur
under the original third-best contract.
As a corollary of the above result, we also show that the principal can approximate thesecond-best payoﬀ on average as the number of agents becomes suﬃciently large. When
there are many agents, those who equally share the residual wage incur only low risk. Thus,
agents face virtually the same wage schemes as the second-best ones when the number of
agents is suﬃciently large. Then, the principal succeeds in achieving the second-best payoﬀ
in the limit as the number of agents tends to inﬁnity. We show this asymptotic eﬃciency
result by using the simple sharing rule mentioned above. Therefore, this rule is shown to
be asymptotically optimal in large organizations, even when no particular restrictions are
made on the utility functions of agents and their production technologies.

6

1.2

Related LiteratureThree strands of the literature are related to our study. As already discussed, our model
is connected to the literature on subjective evaluations. First, several recent studies haveconsidered models of relational contracts with multiple agents in which their outputs are
not veriﬁable and hence not contractible (see Malcomson (2013) for a recent survey on thedevelopments in relational contracts). Mukherjee and Vasconcelo (2011) and Ishihara (2017)
investigate the job design problem of how agents are responsible for diﬀerent tasks and how
they are formed as teams. Levin (2002) ﬁnds the value of making relational contracts multilateral, which beneﬁts the principal more than making relational contracts with separate
agents independently. Rayo (2007) focuses on the relational incentives in a team where
agents contribute to the team’s outputs repeatedly. Kvaløy and Olsen (2006) also consider
team-based incentives in the dynamic relationships among the principal and multiple agents.
Our main insight is that even in the static environment, the principal can recover the secondbest payoﬀ that would be attained if formal contracts contingent on outputs were enforced.
This is in sharp contrast to the studies above that implicitly assume that static contracts
cannot induce agents to work eﬃciently when their outputs are not veriﬁable.
Second, some studies have focused on target budgets as a solution to the problem ofsubjective evaluations in static settings. One approach to this issue is to use money burning(MacLeod (2003), Kambe (2006)).6 The principal commits herself to pay a ﬁxed amountwhatever outputs are realized. However, such ﬁxed expenditure is not always equal to what
the agent receives. When an agent’s output is low, the principal pays a low bonus from aﬁxed budget and discard the remaining amount of the budget. Since the principal always
spends the same expenditure regardless of the realized output of the agent, she has no
incentive to lie about agent’s outputs. In addition, the agent is given the right incentive
to work hard because his wage can vary with his output. However, money burning may be
an unrealistic solution to the problem of subjective evaluations because it is ineﬃcient forcontracting parties to discard useful resources or pay third parties ex post. In this paper
we show that, even when we allow money burning so that total wages of agents can be lessthan what the principal pays, it is never used at the optimal contract which endogenously
satisﬁes the budget-balancing condition that the principal pays the same amount as what
agents totally receive. This can thus avoid wasteful resource destruction.
Third, rank-order contracts are also connected to the role of target budgets in static
settings (Lazear and Rosen (1981), Malcomson (1984)). Under rank-order contracts, agentsare paid according to the ranking determined by their relative performance and the total
6

See also Fuchs (2007) for a dynamic extension of these static models.

7

prizes paid to all agents are ﬁxed. It is known that rank-order contracts work well tomotivate agents even when total budgets for prizes are ﬁxed if two important restrictions are
imposed: (i) agents are risk neutral and (ii) environments are symmetric in the sense thatagents are homogeneous and play a symmetric equilibrium strategy. Instead of imposing
these strong restrictions, we consider more general principal–agent problems with target
budget constraints in which agents are allowed to be heterogeneous. Then, we show in our
Propositions 1, 2, and 3 that the optimal contract has a diﬀerent feature from rank-ordercontracts. Under rank-order contracts, the wage schedule of an agent has a (discontinuously)larger slope with respect to his or her outputs when his or her peers perform worse. However,
we show that the third-best contract that solves the moral hazard problem with the target
budget constraint has no such property; rather, how the wage of each agent is sensitive tohis output is independent of those of other agents.
Furthermore, except for the case that all agents are risk-neutral, rank-order contracts
cannot attain the second best when the outputs of agents are statistically independent of eachother. This is because the wage of each agent depends on the outputs of others under rankorder contracts and hence risk-averse agents incur higher risk under rank-order contracts
than under piece rate contracts, which depend only on their own individual performances.
We avoid this problem by ensuring that the risk-neutral agents share the risk of variationsin the residual wage equally, while keeping the total wage of all agents constant.
Our asymptotic eﬃciency result is also related to Green and Stokey (1983) and Malcomson (1986), who show that rank-order contracts asymptotically perform at least as wellas piece rate contracts do as the number of agents becomes suﬃciently large.7 However,these studies focus only on the symmetric case in which agents are identical and follow asymmetric equilibrium strategy. Rank-order contracts may not perform better than piece
rates do when agents are heterogeneous because in such a case, the former must manage
the diﬀerent eﬀort incentives of heterogeneous agents by using the unique prize structure
for all agents.8 On the contrary, our eﬃciency result is valid in more general environmentsthat allow heterogeneous agents and impose no symmetric restrictions on the equilibrium
behaviors of agents.
The remaining sections are organized as follows. In Section 2, we set up the basic model.
In Section 3, we show that the principal can fully implement the second-best payoﬀ as long
7

Green and Stokey (1983) also show that rank-order contracts sometimes perform better than piece rate

contracts do when agents can privately observe some common shock aﬀecting their individual performances.
8
For example, under the standard rank-order tournament, the winner’s prize does not depend on whoever
wins (i.e., the identity of the winner).

8

as there are at least two risk-neutral agents even when she faces a target budget constraint.
In Section 4, we characterize the third-best contract when there is at most one risk-neutral
agent. Then, we show that the third-best contract becomes a tractable linear formula if andonly if agents’ preferences over income lotteries are represented by CARA utility functions.
In Section 5, we show the asymptotic result. First, we show that the third-best contract
must entail the property that most agents must be compensated according to almost theirindividual outputs when the number of agents is suﬃciently large. Second, we show thatthe principal can approximate the second-best payoﬀ by adopting a simple wage scheme as
the number of agents is suﬃciently large.

2

Model

2.1

Principal–Agent Environment with Moral Hazard

We consider a static moral hazard problem with one risk-neutral principal and N (risk-averse
or risk-neutral) agents. With slight notational abuse, we use the same symbol N to denotethe set of agents. In what follows, we also use a feminine pronoun for the principal anda masculine pronoun for agents. Agent i chooses action ai ∈ A ⊂ R, which stochasticallydetermines his performance signal, called output yi ∈ Y ⊂ R. Agent i’s action ai is observable only to himself (i.e., it is not contractible). On the contrary, agent i’s output yi
is observed by the principal. As discussed in the Introduction, the realization of agents’
outputs (y1 , ..., yN ) may be subjective evaluations by the principal. In this respect, they
may be privately observed only by the principal and hence not veriﬁable.
To save notation, we assume that the sets of actions A and outputs Y are the same for
all agents, although this is not essential for the following analysis. Further, although someof the results that follow hold both when the output of agent yi is discrete and when it iscontinuous, we ﬁx Y to be a ﬁnite set. In particular, we assume that Y has K distinct
elements (K ≥ 2) and denote by y k a generic element of Y , i.e., Y ≡ {y 1 , ..., y K }. Later,
we discuss how we can drop this ﬁniteness assumption for some results. Furthermore, ourresults can be extended to the case that an agent’s action set A and output set Y are
multidimensional, although we do not pursue such a case to avoid complicated notation.
The outputs of agents y1 , ..., yN are independently distributed. We denote by pi (yi |ai ) ∈(0, 1) the probability of agent i’s output being yi conditional on his action ai ∈ A. Here,∑∏Ny∈Y pi (y|a) = 1 for all a ∈ A. We denote by P (y|a) ≡i=1 pi (yi |ai ) the joint probabilityof an output proﬁle y = (y1 , ..., yN ) of N agents conditional on their action proﬁle a =9(a1 , ..., aN ).9 In what follows, we use the notation Ey [ · |a] to denote the expectation over
output proﬁle y ∈ Y N of N agents conditional on their action proﬁle a ∈ AN . Similarly,
by letting y−i ≡ (y1 , ..., yi−1 , yi+1 , ..., yN ) and a−i ≡ (a1 , ..., ai−1 , ai+1 , ..., aN ), we denote by
Ey−i [ · |a−i ] the expectation over the outputs of others than agent i, y−i ∈ Y N −1 , conditionalon an action proﬁle a−i of those agents. We also denote Eyi [·|ai ] the expectation over agent
i’s output yi conditional on his action ai .
As mentioned in the Introduction, we rule out money burning; hence, the sum of thetransfers made by the principal and agents must balance ex post regardless of the outputsy ∈ Y N realized. Let ti (y) ∈ R denote the net transfer received by agent i and tp (y) ∈R that by the principal when output proﬁle y ∈ Y N is realized. Then, we require that∑NN
i=1 ti (y)+tp (y) = 0 for any y ∈ Y . Thus, without loss of generality, we set ti (y) ≡ wi (y),∑Nwhich is interpreted as the wage agent i receives, and tp (y) = − N
i=1 wi (y) for any y ∈ Y .
Agent i has the utility function deﬁned on his wage income wi and action ai , denoted by
Ui : R × A → R, and his utility is given by
Ui (wi , ai )(1)which is assumed to be increasing and concave with his wage wi . Agent i obtains thereservation utility U i when he rejects the contract oﬀered by the principal.
The principal obtains her private beneﬁt or revenue from the outputs of agents y ∈ Y N ,denoted by R(y). We assume that the principal’s private beneﬁt R is observable only toherself and hence non-veriﬁable. Given the wage proﬁle w ≡ {wi }N
i=1 paid to agents, theprincipal obtains the following expected payoﬀ:
Ey [R(y)|a] −

N
∑

wi

(2)

i=1

2.2

Second-Best Contract

We begin with the standard moral hazard problem as the benchmark case: the principal
maximizes her expected payoﬀ (see (2)) subject to the IC and IR constraints (Grossman
and Hart (1983)). The principal makes contracts contingent on the realization of outputsy ∈ Y N . The wage scheme for agent i is deﬁned as a mapping wi : Y N → R, which speciﬁes
his wage wi contingent on the realization of the output proﬁles y ∈ Y N of agents. We denoteby {wi , ai } a contract for agent i, where wi is the wage scheme and ai is an action instructed
for agent i to choose.
9

Throughout the paper, we use a bold letter to denote a vector of the variables.

10

In this benchmark, the principal chooses contracts {wi , ai }N
i=1 to solve the following
second-best problem:
Problem SB
max Ey [R(y)|a] −
w,a

N
∑

Ey [wi (y)|a]i=1

subject toEy [Ui (wi (y), ai )|ai , a−i ] ≥ Ey [Ui (wi (y), a′′i )|a′′i , a−i ],for any a′′i ̸= ai(IC)

Ey [Ui (wi (y), ai )|ai , a−i ] ≥ U i(IR)Here, IC denotes the incentive compatibility constraint that agents choose the desired actions
a as a Nash equilibrium and IR is the individual rationality constraint ensuring that eachagent accepts the oﬀered contract. We call a contract that solves Problem SB the secondbest contract. We also call the principal’s payoﬀ attained under the second-best contract thesecond-best payoﬀ.
Since the outputs of agents are statistically independent of each other and there are
no technological externalities between their actions, one might think that the second-best
contract for agent i should be contingent only on his own output yi . We call wage scheme
wi independent performance evaluation (IPE) or piece rate when it depends solely on agent
i’s output, yi , that is, wi = wi (yi ).
We denote by {ŵi , âi }N
i=1 the second-best contract solving Problem SB and make thefollowing assumption.
Assumption 1. There exists an IPE contract {âi , ŵi }N
i=1 that solves Problem SB where
each agent i’s wage scheme ŵi depends only on his own output yi .
Assumption 1 holds when the agent’s utility function satisﬁes the so-called risk independence condition commonly assumed in standard principal–agent models.

This con-dition means that the agent’s preference over income lotteries is independent of his action. Such condition holds if and only if the utility function of agent i takes the form
Ui (wi , ai ) ≡ ui (wi )Hi (ai ) − Gi (ai ) for some functions ui , Hi , and Gi (Keeney (1973)).
Assumption 1 is weaker than the additively separable utility function used in the literature. Indeed, we can show that the second-best contract becomes IPE without loss ofgenerality when the risk independence condition is satisﬁed.

11

Lemma 1.

Suppose that the utility function of each agent i takes the form given by

Ui (wi , ai ) = ui (wi )Hi (ai ) − Gi (ai ). Then, the second-best contract ŵi for agent i, whichsolves Problem SB, depends only on his own output yi , that is, ŵi (yi ).
Proof. See the Appendix.
Under Assumption 1, we conﬁne our attention to the second-best contract, which is IPE:
wi = ŵi (yi ) for each agent i. With this in mind, we consider the cost minimization problem
for implementing action ai ∈ A from agent i in the second-best problem as follows:
Problem M-SB
min Eyi [wi (yi )|ai ]wisubject toEyi [Ui (wi (yi ), ai )|ai ] ≥ Eyi [Ui (wi (yi ), a′′i )|a′′i ], ∀ a′′i ̸= ai(IC)Eyi [Ui (wi (yi ), ai )|ai ] ≥ U i(IR)

We denote by ŵi (·; ai ) the optimal solution to the above problem for implementing actionai ∈ A from agent i, provided the constraint set is non-empty. In particular, with slightnotational abuse, by dropping argument âi , we denote by ŵi (·) ≡ ŵi (·; âi ) the second-bestwage scheme for implementing the second-best action âi .
The principal pays the following total expected wages of N agents under the second-best
contract {ŵi , âi }N
i=1 :
Ŵ ≡

N
∑

Eyi [ŵi (yi )|âi ]i=1

and obtains the following second-best payoﬀ:
Π̂ ≡ Ey [R(y)|â] −N∑Eyi [ŵi (yi )|âi ].

i=1

3

Target Budget and Eﬃciency Result

We now turn to the case that the principal faces a target budget constraint as well as IC
and IR constraints. The principal ﬁrst determines a target budget W for the sum of thewages paid to all agents. She cannot change this target budget regardless of the outputs ofagents realized ex post once it is predetermined ex ante. As discussed in the Introduction,
one relevant case for this is subjective evaluations under which the realization of agents’

12outputs y is observable only to the principal, meaning that the total wages of agents must
be constant regardless of their outputs realized.
Target budgets require that the total wages of agents must be ﬁxed for any realizationof their outputs as follows:N∑wi (y) = W for any y ∈ Y N .

(FTW)

i=1

Then, the principal solves the following problem, which we call the third-best problem:
Problem TB
max Ey [R(y)|a] − W

a,w,W

subject to FTW deﬁned above together with
Ey [Ui (wi (y), ai )|ai , a−i ] ≥ Ey [Ui (wi (y), a′′i )|a′′i , a−i ],for any ai ̸= a′′i .

(IC)

and
Ey [Ui (wi (y), ai )|a] ≥ U i(IR)

for all i ∈ N .
Since FTW is an additional constraint, the principal is never better oﬀ from the secondbest case. However, FTW may not constrain the principal at all. Indeed, we show that thesecond-best contract {ŵi , âi }N
i=1 solves the above Problem TB when there are at least two
risk-neutral agents. In other words, FTW does not cause any eﬃciency loss compared withthe second-best case.
We say that agent i is risk-neutral if his utility function Ui is represented by
Ui (wi , ai ) = Hi (ai )wi − Gi (ai )for some functions Hi and Gi . Thus, Eyi [Ui (wi (yi ), ai )|ai ] = Ui (Eyi [wi (yi )|ai ], ai ) for any
action ai ∈ A and any wage scheme wi when agent i is risk-neutral. We denote by Ir ⊆ N
the set of risk-neutral agents and by L ≡ #Ir the number of them.
Then, we show that the principal can fully attain the second-best payoﬀ even when
she faces the target budget constraint (FTW) as long as there are at least two risk neutral
agents. Before proceeding, we give an intuition behind this result by using a simple example.
Suppose that there are only two agents and they are all risk neutral (N = {1, 2} and L = 2).

13

Let Ui (wi , ai ) = wi −Gi (ai ) denote agent i’s payoﬀ. Recalling that ŵi denotes the second-bestwage scheme for agent i = 1, 2, we deﬁne the wage scheme for agent i as follows
wi (yi , yj ) = w̃i (yi ) + (1/2){W − w̃i (yi ) − w̃j (yj )}for i, j = 1, 2 and i ̸= j, where we setw̃i (yi ) ≡ 2ŵi (yi ) − Eyi [ŵi (yi )|âi ]for i = 1, 2. We also set W =∑i=1,2 E[ŵi (yi )|âi ] which is equal to the total expected

wage at the second-best optimum. Then, by its deﬁnition, w1 (y1 , y2 ) + w2 (y2 , y1 ) = W
holds for any (y1 , y2 ) ∈ Y 2 so that FTW is satisﬁed. Furthermore, we can readily seethat E[wi (yi , yj )|ai , âj ] = Eyi [ŵi (yi )|ai ] for any action ai ∈ A. Thus, agent i faces thesame expected wage as what he obtains under the second-best scheme whatever actions hechooses, given agent j choosing the second-best action âj . Then, both agents choose thesecond-best actions so as to maximize their expected payoﬀs, Eyi [ŵi (yi )|ai ] − Gi (ai ), andobtain the second-best payoﬀs.
The key of the above argument is that agent i is oﬀered a piece rate contract w̃i (yi ), which∑is contingent only on his own output yi , and equally shares the residual wage W − i w̃i (yi )deﬁned as the ﬁxed budget W minus total wages. Such a sharing scheme ensures that total
wages of agents become a constant at W irrespective of their outputs y ∈ Y 2 while providingthem the right incentives to choose the second-best actions.
When there are more than two risk neutral agents together with risk averse agents,
we can generalize the above argument by oﬀering the similar sharing schemes only to risk
neutral agents while oﬀering the second-best (and hence piece rate) wage schemes to risk
averse agents.
Thus, we show the following result.
Proposition 1. Suppose that L ≥ 2. Then, the principal can exactly attain the secondbest payoﬀ Π̂ even when the total wages of agents must be ﬁxed for any realization of their
outputs.
Proof. We set the wage scheme of risk-averse agent n by the second-best one ŵn :wn (yn ) ≡ ŵn (yn ),

n∈
/ Ir

Then, agent n ∈/ Ir chooses the second-best action ân and obtains the same expected payoﬀ
Eyn [Un (ŵn (yn ), ân )|ân ] as that under the second-best contract.
14

We now consider risk-neutral agent i ∈ Ir . We deﬁne the wage scheme for risk-neutral
agent i ∈ Ir as follows:
∑wi (yi , y−i ) ≡ w̃i (yi ) + (1/L) Ŵ −ŵn (yn ) −∑n∈N \Ir

j∈Ir ,j̸=iw̃j (yj ) − w̃i (yi )(3)where w̃i is deﬁned asw̃i (yi ) ≡L1ŵi (yi ) −Ey [ŵi (yi )|âi ].
L−1
L−1 i

(4)

Here,
Ŵ ≡

N
∑

Eyn [ŵn (yn )|ân ]n=1denotes the total expected wages of all agents under the second-best contract. Note that w̃iis well deﬁned because of L ≥ 2. Moreover, Eyi [w̃i (yi )|âi ] = Eyi [ŵi (yi )|âi ] for each i ∈ Ir . In
addition, the wage proﬁle (w1 , ..., wN ) deﬁned above satisﬁes FTW:N∑

wn (y) = Ŵ for any y ∈ Y N

n=1

Given this, risk-neutral agent i ∈ Ir obtains the following expected wage conditional onothers choosing the second-best actions â−i :Ey [wi (yi , y−i )|ai , â−i ]


∑
L−1
1
ŵn (yn ) −= Ey 
w̃i (yi ) + Ŵ −
L
L
n∈I
/ r

∑



w̃n (yn ) ai , â−i n∈Ir ,n̸=i= Eyi [ŵi (yi )|ai ] − (1/L)Eyi [ŵi (yi )|âi ] + (1/L) Ŵ −∑Eyn [ŵn (yn )|ân ]

n̸=i= Eyi [ŵi (yi )|ai ]for any ai ∈ A because

(5)

∑N

n=1 Eyn [ŵn (yn )|ân ] = Ŵ holds. Thus, risk-neutral agent i obtains

the expected payoﬀ as follows:
Ey [Ui (wi (y), ai )|ai , â−i ] = Ui (Ey [wi (y)|ai , â−i ], ai )= Ui (Eyi [ŵi (yi )|ai ], ai )= Eyi [Ui (ŵi , ai )|ai ](6)

for any action ai ∈ A. This payoﬀ is the same as that attained under the second-best
contract ŵi . Thus, risk-neutral agent i chooses the second-best action âi provided all others
15choose the second-best actions â−i . The risk-neutral agents also obtain the same expected
payoﬀs under the second-best contract. Thus, all agents obtain at least the reservation
utilities, meaning that IR is satisﬁed. In this way, the principal can implement the secondbest actions â at the same total wage cost as that under the second-best contract. Q.E.D.
We obtain Proposition 1 by generalizing the intuition aforementioned in the example oftwo risk neutral agents. To avoid a trivial case, suppose that the second-best action âi is not
the least costly one for each i ∈ N . Since the total wages of all agents must be ﬁxed (FTW),
whenever all agents are induced to choose diﬀerent actions from the least costly ones, at least
two agents must be oﬀered the wage schemes, which depend not only on their own outputs
but also on those of others. To see this, suppose that all agents but agent i are oﬀered wage
schemes that depend solely on their individual outputs wj (yj ) for any j ̸= i. Then, FTW∑
implies wi = W − j̸=i wj (yj ); hence, agent i’s wage depends only on the outputs of others.
However, agent i never works hard. Thus, there must be a diﬀerent agent j ̸= i whose wage
depends on the outputs of others y−j . In this way, to make total wages constant, at least
two agents must be oﬀered some interdependent wage schemes that vary with the outputsof others.
When there are at least two risk-neutral agents, the principal can have these agentsbear all the risk under interdependent wage schemes. On the contrary, risk-averse agentsare oﬀered the second-best wage schemes {ŵi }i∈I
/ r , which are independent of the outputs ofothers. Thus, they incur no additional risk and choose the second-best actions. Risk-neutral
agent i is oﬀered the wage scheme (given by (3)) consisting of an IPE wage w̃i (yi ), whichis slightly modiﬁed from the second-best one ŵi , and an equal share of the residual wage∑∑
(1/L){Ŵ − n∈L/ w̃n (yn ) −n∈L ŵn (yn )}. Then, risk-neutral agents absorb all the risk of
the residual wage and are induced to choose the second-best actions. By construction, theprincipal ends up paying the second-best cost Ŵ regardless of the outputs realized.
We derive several implications from Proposition 1 as follows. First and most importantly,
Proposition 1 implies that the principal incurs no eﬃciency loss from the target budget constraint. Hence, she can achieve the second-best payoﬀ even when she must ﬁx the total
wages of agents because of the problem of subjective evaluations. This has a notable implication for the role of relational contracting, which can alleviate the problem of subjective
evaluations. Studies of relational contracts with multiple agents have implicitly assumed
that the principal cannot motivate agents to work eﬃciently at the static benchmark when
their performance signals are not veriﬁable (e.g., Levin (2002), Rayo (2007) and Ishihara(2017)). Indeed, most papers have focused on the beneﬁts of repeated transactions between

16the principal and agents, which can prevent the former from reneging on promised payments
to the latter. However, in contrast to these studies, we show that the principal can attainthe second-best payoﬀ that she would obtain if the outputs of agents were veriﬁable even in
the static environment in which they are not veriﬁable (and even when they are privately
observed by the principal). Since subjective evaluations never cause eﬃciency loss even in
the static setting, there are no gains from relational contracting even when the principal andagents contract repeatedly over time. Proposition 1 therefore suggests that we reconsider
the roles of static contracts in dynamic settings as well as the value of relational contracting
when the principal hires multiple agents whose outputs are not veriﬁable.
Second, Proposition 1 implies that when all agents are risk-neutral, the principal can
implement the ﬁrst-best even if she faces a target budget constraint. The ﬁrst-best is deﬁned
as the outcome attained when agents’ actions are veriﬁable. Suppose that all agents are risk
neutral and their utility functions are given by Ui (wi , ai ) = wi − Gi (ai ) without loss ofgenerality.10 Then, the ﬁrst-best action proﬁle, denoted by af b = (af1 b , ..., afNb ), is deﬁned tomaximize the total expected surplus of the principal and N agents:
Πf b ≡ max Ey [R(y)|a] −
a∈AN

N
∑

Gi (ai )(7)

i=1

Even when agents’ actions are not veriﬁable, the ﬁrst best is still implemented by some
IPE wage schemes as long as their outputs are veriﬁable. For example, we can consider thefollowing wage scheme for agent i:
wif b (yi ) ≡ Ey−i [R(yi , y−i )|af−ib ] + Ti

(8)where Ti is set such that the IR of agent i holds as an equality:
Ey [R(y)|af b ] + Ti − Gi (afi b ) = U i(9)Then, agents choose the ﬁrst-best actions af b given the above wage schemes. Thus, thesecond-best contract that solves Problem SB coincides with the ﬁrst best. In other words,
one of the second-best contracts solving Problem SB becomes ŵi (yi ) = wif b (yi ) for each agenti ∈ N , as deﬁned in (8). Note that we can dispense with Assumption 1 when all agents arerisk-neutral because in that case the IPE wage scheme wif b deﬁned in (8) solves Problem SB.
Then, Proposition 1 implies that even if the principal faces a target budget constraint(FTW), the ﬁrst-best is implemented when all agents are risk-neutral.
10

More generally, we can consider the utility function Ui = Hi (ai )wi −Gi (ai ) for some function Hi , although

we set Hi (ai ) = 1 for simplicity.

17

Corollary. Suppose that all agents are risk-neutral. Then, the principal can attain theﬁrst-best payoﬀ Πf b even when the total wages of agents must be ﬁxed for any realization of
their outputs.
Third, Proposition 1 is robust to the speciﬁcations of utility functions, action and output
spaces, and the probability distributions of outputs. Proposition 1 holds as long as there
exists an IPE contract solving Problem SB. This encompasses the standard additively separable utility function Ui (w, a) = ui (w) − Gi (a), ﬁnite and continuous actions, as well as ﬁnite
and continuous outputs. We also do not impose any technical restrictions on the probability
distributions of agents’ outputs such as the monotone likelihood ratio condition. Moreover,
we allow agents’ actions and outputs to be multidimensional.
Fourth, our eﬃciency result does not rely on money burning (MacLeod (2003), Kambe(2006)). When multiple agents exist, the principal can commit herself to paying a ﬁxed total
wage to all agents without burning money. This is a desirable feature of optimal contracts
because it is ineﬃcient for contracting parties to discard useful resources or pay third parties
ex post. Rank-order contracts also have a similar feature in that agents are paid diﬀerent
prizes depending on their relative performance ranking and these prizes sum to a constant(Lazear and Rosen (1981), Malcomson (1984, 1986)). However, such scheme cannot yield
the second-best payoﬀ to the principal in an environment in which some agents are riskaverse because such agents incur higher risk under rank-order contracts than under piece
rate contracts. The contract we constructed in the proof of Proposition 1 can avoid such
additional risk while implementing the second-best payoﬀ.11

4

Third-Best Contract

In this section, we turn to the case not covered by Proposition 1 by assuming that there
is at most one risk-neutral agent. We maintain this assumption throughout the remaining
sections. Then, we investigate the properties of the third-best contract for solving Problem
TB.
Here, we explicitly assume the risk independence condition that agents’ preferences overincome lotteries are independent of their actions. That is, we make the following assumption.1211
12

Only when all agents are risk-neutral do rank-order contracts work as well as piece rate contracts do.
Note that Assumption 2 implies Assumption 1 because of Lemma 1.

18

Assumption 2. (i) Ui (wi , ai ) ≡ ui (wi )−Gi (ai ) for each agent i ∈ N , where arg mina∈A Gi (a) =0 for each i ∈ N . (ii) There exists some w such that ui (w) − Gi (0) = U i for all i.
We normalize the least costly action that minimizes the action cost Gi (a) to zero forany agent. The assumption that the least costly action is the same for all agents is merely
made for simplicity. The following result also holds even when we consider a more generalutility function as ui (w)Hi (a) − Gi (a) for some function Hi , although we set Hi (a) ≡ 1 for
all a ∈ A to simplify the notation.
Given Assumption 2, we consider the problem of implementing action proﬁle a at theminimum total wage W subject to IC, IR, and FTW deﬁned in Section 3. We denoteby W (a) the optimal value of W in this minimization problem for implementing a.13 The
principal chooses action proﬁle a to maximize her expected payoﬀ E[R(y)|a]−W (a). We callthis action proﬁle the third-best action proﬁle, denoted by a∗ ∈ AN . To make the problem
non-trivial, we assume that the constraint set is non-empty for some a > 0 and that thethird-best actions a∗ satisfy a∗ > 0.
Under Assumption 2, we can write IC and IR in Problem TB as follows:∑

P (y|a∗i , a∗−i )ui (wi (y)) − Gi (a∗i ) ≥y∑P (y|a, a∗−i )ui (wi (y)) − Gi (a) for a ̸= a∗i , i ∈ N

y

∑

(IC)
P (y|a∗i , a∗−i )ui (wi (y)) − Gi (a∗i ) ≥ U iywhere P (y|ai , a∗−i ) ≡ pi (yi |ai )for i ∈ N

(IR)

∏

∗
j̸=i pj (yj |aj ) denotes the joint probability of output proﬁley conditional on action proﬁle (ai , a∗−i ).
Then, the third-best contract {wi (y)}N
i=1 should solve the following.

Problem TB
min W
subject to FTW, IC, and IR for implementing a = a∗ .
In what follows, we maintain the assumption that both the action and the output sets,
A and Y , are ﬁnite (we discuss some extensions to the continuous case in the Appendix).
If the constraint set satisfying IC, IR, and FTW is empty for a ∈ AN , we deﬁne W (a) = ∞. Note also∑that W (0) = N
i=1 (Gi (0) + U i ) holds when the principal implements the least costly action, zero, from all
13

agents. The wage schemes that achieve W (0) exist from Assumption 2 (ii). Thus, W (0) is well deﬁned.

19

We denote by M + 1 the number of all possible actions each agent can choose, #A = M + 1,
and then make the following assumption about the probability distributions of outputs andaction costs.
Assumption 3. There exist no non-negative M dimensional vectors (ρ(a))a∈A,a̸=a∗i of which∑
the elements are not all zero simultaneously and a̸=a∗ ρ(a) = 1 such thati∑ρ(a)pi (yi |a) = pi (yi |a∗i ),for each yi ∈ Y

a̸=a∗iand∑ρ(a)Gi (a) ≤ Gi (a∗i ).

a̸=a∗i

Assumption 3 states that by deviating from the targeted action a∗i and choosing any
mixed strategy ρ over his actions, agent i cannot induce the same probability distributions
over his output yi as when he follows the third-best action a∗i at a lower action cost. Similar conditions have been often imposed on principal–agent problems in the literature (e.g.,
Hermalin and Katz (1993)). Assumption 3 is fairly weak in the sense that it is genericallysatisﬁed when the number of possible outputs #Y = K is larger than that of possible actions#A = M + 1.
We now turn to characterize the third-best contract. The key factor that aﬀects thethird-best contract is the piece rate wage of each agent i, deﬁned as


)
(
∑
pi (yi |a) ξi (yi ) ≡ ln λi +µi (a) 1 −∗pi (yi |ai )
∗

(10)

a̸=ai

for some non-negative constants λi ≥ 0 and µi (a) ≥ 0 for each a ̸= a∗i , respectively.
To see what ξi means, it is useful to consider how the second-best contract ŵi is characterized. The second-best contract is the solution to Problem M-SB deﬁned in the previous
section. Under Assumption 2, the second-best wage scheme ŵi (·; a∗i ) for implementing thethird-best action a∗i from agent i is given by the familiar formula (Holmström (1979), Grossman and Hart (1983)):
1/u′i (ŵi (yi ; a∗i )) = λ∗i +()
pi (yi |a)µ∗i (a) 1 −pi (yi |a∗i )∗∑a̸=âifor non-negative Lagrange multipliers λ∗i and µ∗i (a) for each a ̸= a∗i associated with IR and
IC in Problem M-SB, respectively. By taking the logarithm of the right-hand side of this20

formula, we obtain essentially the same expression as ξi (yi ) deﬁned in (10), although theconstant terms λi and µi (a) may diﬀer from λ∗i and µ∗i (a).
On the other hand, in the third-best contracting problem, Problem TB, the principal
must take into account an additional constraint, FTW. To solve this problem, we replace∑′′
FTW by its weak inequality, that is, W ≥ N
i=1 wi (y), which we call FTW , and then consider the relaxed problem with IC, IR, and FTW′′ . When a certain constraint qualiﬁcation
is satisﬁed, as we see in the formal proof,14 the optimal solution to this relaxed problem
must satisfy the Kurash–Kuhn–Tucker (KKT) conditions as follows:


(
)

∑
pi (yi |a)µi (a) 1 −P (y|a∗ )u′i (wi (y)) λi += η(y)
pi (yi |a∗i ) 
∗

(11)

a̸=ai

where λi ≥ 0 denotes the Lagrange multiplier associated with the IR constraint for agenti, µi (a) ≥ 0 the Lagrange multiplier associated with the IC constraint for agent i’s actiona ̸= a∗i , and η(y) ≥ 0 the Lagrange multiplier associated with FTW′′ for output proﬁle∏∗y ∈ Y N . Here, P (y|a∗ ) = N
i=1 pi (yi |ai ) denotes the joint probability of N agents’ outputs
conditional on the third-best action proﬁle a∗ .
Then, we can show that η(y) > 0 for all y ∈ Y N and thus FTW′′ becomes binding for
any y ∈ Y N .15 This implies that the relaxed problem coincides with the original problem
TB. Thus, we can use the above KKT conditions for any pair of agents i ̸= j to obtain()∑
pj (yj |a)
λ
+
µ
(a)
1
−
′
∗
j
j
a̸
=
a
pj (yj |aj )jui (wi (y))() = eξj (yj )−ξi (yi ) .
=(12)∑
pi (yi |a)u′j (wj (y))λi + a̸=a∗ µi (a) 1 − pi (yi |a∗ )iiHence, in the third-best contract the ratio between the marginal income utilities of the two
agents u′i (wi )/u′j (wj ) must be equal to the relative magnitudes of the likelihood ratios of
their outputs measured by ξj (yj ) − ξi (yi ).
In this way, agent i is motivated via only ξi (yi ) in the second-best case because ξi (yi )varies with his output yi . On the contrary, in the third-best case, the diﬀerences between
the piece rate wages of agents ξi (yi ) − ξj (yj ) play an important role. Indeed, we show thefollowing lemma.
14
15

We check that the so-called Slater condition is satisﬁed in the Appendix.
The intuition behind this result is as follows. The principal can change the wage of any agent wi (yi , y−i )contingent on the outputs of others y−i to keep his IC and IR unchanged. That is, it is possible to have IC
′′
′
′′
′
and IR unaltered by changing wi (yi , y−i) and wi (yi , y−i) for y−i
̸= y−ifor a given yi . Such wage variations

lower total wages whenever FTW′′ is binding for some output proﬁle y′′ = (yi , y−i ) but not for another proﬁle′y′ = (yi , y−i). This can reduce total wages W further.

21

Lemma 2. Suppose that Assumptions 2 and 3 hold. Then, the third-best contract foragent i, denoted by wi (y), is given by a function of the diﬀerences in piece rate wages(ξi (yi ) − ξj (yj ))j̸=i , deﬁned as Qi : Y N → R and given bywi (y) = Qi (ξi (yi ) − ξ1 (y1 ), ..., ξi (yi ) − ξN (yN ))(13)

where ξi (yi ) is deﬁned in equation (10) and Qi is strictly increasing in ξi (yi ) and strictly
decreasing in ξj (yj ), j ̸= i.
Proof. See the Appendix.
Lemma 2 shows that under the third-best contract, the wage of each agent depends only
on the diﬀerences in piece rate wages between his own ξi (yi ) and those of others ξj (yj ), thatis, ξi (yi ) − ξj (yj ) for j ̸= i, and strictly increasing in his own piece rate ξi (yi ) but decreasingin the piece rates of others ξj (yj ). Thus, each agent is compensated more when his piece rate
wage ξi (yi ) increases but less as those of his peers ξj (yj ) increase. This is intuitive because
the total wage of all agents must be ﬁxed while they are motivated to work hard.

4.1

Third-Best Contract with CARA Utility Functions

Although deriving more detailed properties of the third-best contract Qi in general is challenging, we provide the conditions under which Qi is simpliﬁed. In particular, we provide
a tractable formula for the third-best contract when agents’ preferences over income lotteries are represented by utility functions with a constant degree of absolute risk aversion (orCARA). In that case, the third-best contract for agent i is given by the following simple
sharing rule:wi (y) = βi ξi (yi ) + αi

for some constants βi > 0 and αi such that

∑



W−

N
∑
j=1



βi ξ(yj )(14)i αi = 1, where ξi (yi ) represents the piece rate

wage of agent i deﬁned as in equation (10).
We suppose that risk-averse agents have CARA utility functions as follows:ui (w) =1(1 − e−ri w )
ri

(15)

where ri > 0 denotes the CARA of agent i. In the main text, we suppose that all agents arerisk-averse to avoid unnecessary complication. In the Appendix, we consider the case that
one agent is risk-neutral, while all the others are risk-averse. (Note that we are assumingthat there is at most one risk-neutral agent.)22

As we have shown in Lemma 2, the third-best contract for agent i depends on diﬀerences between his own piece rate wage and those of others, that is, ξi (yi ) − ξj (yj ). For
general income-utility functions ui , it is too complicated to see how the third-best contract
reﬂects these diﬀerences. However, under CARA utility functions, the ratio between themarginal income-utilities of agent i and j is simpliﬁed to u′i (wi )/u′j (wj ) = erj wj (y)−ri wi (y) .
By substituting this into the optimality condition (12), we obtainri wi (y) − rj wj (y) = ξi (yi ) − ξj (yj ).

(16)

We then sum this over all j ̸= i to show that that the third-contracts for agents are determined by a linear combination of piece rate wages ξk (yk ), k = 1, 2, ..., N , thus yielding theresult that the simple sharing rule (14) becomes optimal.
We thus show the following result.
Proposition 2. Suppose that Assumptions 2 and 3 hold and that agents’ preferences overincome lotteries are represented by CARA utility functions (see (15)). Then, the third-best
contract {wi (y)}N
i=1 that solves Problem TB is given by the simple sharing rule as follows:
1/ri

wi (y) = (1/ri )ξi (yi ) + ∑N

l=1 (1/rl )

{
W−

N
∑

}
(1/rl )ξl (yl )

l=1

where ξi (yi ) is deﬁned as in (10).
Proof. See the Appendix.
Here, the simple sharing rule shown in Proposition 2 corresponds to the one deﬁned in∑(14) by setting αi = (1/ri )/ N
j=1 (1/rj ) and βi = (1/ri ).
Proposition 2 provides a tractable characterization of the third-best contract. Each agent
obtains his piece rate wage (1/ri )ξi (yi ), which depends only on his output yi in addition to∑a constant share, (1/ri )/ N
l=1 (1/rl ), of the residual wage deﬁned as the ﬁxed total wage∑
minus the sum of the piece rate wages of all agents, that is, W − N
l=1 (1/rl )ξl (yl ). Theseshares of agents are determined by the relative magnitudes of their risk aversion. Morerisk-averse agents are rewarded with more piece rate wages based on their own outputs but
a lower share of the residual wage.
Three important remarks are in order. First, one might think that rank-order contracts
are an eﬀective way to motivate agents while keeping the total budget ﬁxed (Lazear and23

Rosen (1981), Malcomson (1984)). The principal commits herself to ﬁxed prizes, which arepaid to agents based on the relative ranking of their outputs. The rewards of agents are
given by step functions under rank-order contracts: if the output of an agent rises slightly
higher than those of his peers, his reward jumps to a higher prize, whereas it never changes
as it slightly increases more. Put diﬀerently, the diﬀerence in or “slope” of agent i’s wages
under rank-order contracts, wi (yi′′ , y−i ) − wi (yi′ , y−i ), with respect to his output yi′′ ̸= yi′
discontinuously changes with the outputs of others y−i . However, the third-best contract of
agent i that solves Problem TB has no such property; rather, the wage slope wi (yi′′ , y−i ) −∑∑wi (yi′ , y−i ) is given by ( j̸=i (1/rj )/ N
l=1 (1/rl ))(1/ri )ξi (yi ), which is independent of theoutputs of the other agents y−i .
Second, when we impose standard conditions such as the monotone likelihood ratio
property (MLRP) and convexity of distribution function condition (CDFC), we can showthat wi (y) is non-decreasing in i’s own output yi but non-increasing in the outputs of others
y−i . The MLRP is stated formally as follows.
(MLRP). pi (yi |a′ )/pi (yi |a′′ ) is non-increasing in yi given a′′ > a′ for each i ∈ N .
In addition, the CDFC is given as follows:(CDFC). Let Fi (z|a) ≡∑yi ≤z pi (yi |a) be the cumulative distribution function of pi (y|a).

Then, Fi (z|a) is a convex function of action a ∈ A for each i ∈ N .
Under the MLRP, the piece rate wage deﬁned as ξi (yi ) is veriﬁed to be non-decreasingin yi as long as µi (a) > 0 holds only for less costly actions a ̸= a∗i such that Gi (a) < Gi (a∗ ).
This last condition is ensured because no agent has an incentive to choose more costly actions
than a∗i under the CDFC as is well known in standard moral hazard models (e.g., Grossman
and Hart (1983)). We provide a more detailed argument in the Appendix.
Third, when N is larger, the optimal contract wi (y) for agent i becomes asymptotically the same as the piece rate wage (1/ri )ξi (yi ). To see this, note that as N → ∞,∑the second term of the optimal wage scheme given in Proposition 2, (1/ri )/ l (1/rl ){W −∑l (1/rl )ξl (yl )}, converges to some constant in probability because of the Law of Large Numbers. We generalize this last point beyond the speciﬁc utility functions in Section 5.
Next, we show the converse of Proposition 2: the third-best contract is given by thesimple sharing rule (14) only if agents’ preferences over income lotteries are represented by24

CARA.
Recall that the third-best contract Qi is said to be the simple sharing rule when it isgiven by a linear combination of the diﬀerences in piece rate wages (ξi (yi ) − ξj (yj ))j̸=i in∑
a certain speciﬁc form deﬁned as in (14). In equation (14), we impose
i αi = 1 and∑∑∑αi l̸=i βl = βi l̸=i αl (or equivalently αi l̸=i βi = βi (1 − αi )). This last condition isrequired for the optimal contract to be consistent with Lemma 2; in other words, it should
depend only on the diﬀerences in piece rate wages (ξi (yi ) − ξj (yj ))j̸=i . Indeed, under thiscondition, we can rewrite equation (14) as follows:wi (y) = αi W +∑αi βj {ξ( yi ) − ξj (yj )}

(17)

j̸=i

so that the third-best contract depends only on diﬀerences in piece rate wages (ξi (yi ) −ξj (yj ))j̸=i in the linear fashion. Note also that since Qi is strictly increasing in ξi (yi ) as
shown in Lemma 2, we must have αi βj > 0 for each i and j, which implies that βj ̸= 0 for
any j.
The simple sharing rule which is a linear combination of the piece rate wages {ξi (yi )}N
i=1
is suﬃcient to manage agents’ incentives when agents’ preferences over income lotteries
are represented by CARA utility functions so that their risk attitudes do not change with
their incomes. One might however think that, when agents’ income-utility functions are not
CARA so that they exhibit income eﬀects, the linearity of the simple sharing rule becomesno longer optimal. In fact, we we show the converse of Proposition 2.
Proposition 3. Suppose that Assumptions 2 and 3 hold. Suppose also that N ≥ 3. Then, thesimple sharing rule given in (17) becomes the third-best contract only if agents’ preferences
over income lotteries exhibit CARA.
Proof. See the Appendix.
We need N ≥ 3 to show Proposition 3. In the case of two agents (N = 2), the principal
may be able to optimally manage agents’ incentives by using the simple sharing rule evenwhen they do not have CARA utility functions. This is the case when two agents are
symmetric. To see this, let (1/2)W + (1/2)β(ξ(yi ) − ξ(yj )) be the simple sharing rule foragent i. Then, the other agent j ̸= i faces (1/2)W +(1/2)β(ξ(yj )−ξ(yi )), which is symmetricto the wage scheme of agent i. Thus, only the single term β(ξ(yi ) − ξ(yj )) can perfectly work
to manage the eﬀort incentives of both agents at the same time by symmetry. A moredetailed example is given by the following.
25

Example: There are only two symmetric risk-averse agents, meaning that the third-best
contract is also symmetric and is given by w∗ (yi , yj ). Here, by symmetry, we have w∗ (y, y) =W /2 for any yi = yj = y. Suppose that Y = {y, y}. Then, given the original contractw∗ (yi , yj ), we ﬁnd β ̸= 0 such thatw∗ (y, y) = (1/2)βξ(y) + (1/2){W − βξ(y)}= (1/2)W + (1/2)β{ξ(y) − ξ(y)}where ξ(y) = ln(λ+∑∗a̸=a∗ µ(a)(1−p(y|a)/p(y|a ))) and we suppose that ξ(y) ̸= ξ(y). Then,

we deﬁne the simple sharing rule asw(yi , yj ) ≡ (1/2)W + (1/2)β{ξ(yi ) − ξ(yj )}.
Note that w(y, y) = W /2 for any yi = yj = y. We also havew(y, y) = (1/2)W + (1/2)β{ξ(y) − ξ(y)}= (1/2)W − (1/2){ξ(y) − ξ(y)}= W − w(y, y)= w(y, y)= w∗ (y, y)
by using W = w(y, y) + w(y, y). Thus, agents face the same wage scheme as the originalone. Hence, their incentives are unchanged and they obtain the same expected payoﬀs. The
principal can then achieve the same total wage W . This implies that the simple sharing rule
becomes the third-best contract even when agents’ preferences over income lotteries are notnecessarily represented by CARA utility functions.

5

Asymptotic Results

5.1

Optimality of Almost Piece Rate Contracts

Next, we turn to the case of more general utility functions beyond CARA. Although it isdiﬃcult to characterize third-best contracts in general environments, we show that the thirdbest contract becomes almost piece rates or individual performance evaluation (IPE) when
the number of agents is suﬃciently large.
In this subsection, we maintain Assumption 2 but drop Assumption 3. We also maintainthe basic assumption that output set Y is ﬁnite with K distinct elements, #Y = K, while
we allow an agent’s action to be discrete or continuous.
26

Moreover, we impose the following condition on the income utility function ui .
Assumption 4. (i) ui is continuously diﬀerentiable, increasing and concave with ui (∞) =u (u may be ∞) and limw→wmin ui (w) = −∞ for some wmin (wmin may be −∞). (ii)
limw→wmin u′i (w) = +∞ for risk averse agent i.
∗ ) satisﬁes IC and IR as follows:
Recall that the third-best contract w∗ = (w1∗ , w2∗ , .., wN

Ey [ui (wi∗ (yi , y−i ))|a∗i , a∗−i ] − Gi (a∗i ) ≥ Ey [ui (wi∗ (yi , y−i ))|ai , a∗−i ] − Gi (ai ),∀ ai ̸= a∗i (IC)Ey [ui (wi∗ (yi , y−i ))|a∗i , a∗−i ] − Gi (a∗i ) ≥ U i(IR)

The third-best contract for agent i, wi∗ (yi , y−i ), may depend on the outputs of others than i,y−i . We deﬁne a less risky wage scheme w̃i by eliminating the risk caused by the dependencyof wi∗ on y−i as follows:ui (w̃i (yi )) = Ey−i [ui (wi∗ (yi , y−i ))|a∗−i ](18)

for each yi ∈ Y conditional on the third-best action proﬁle of others than agent i, a∗−i . Sinceui is a concave function, we have w̃i (yi ) ≤ Ey−i [ui (wi∗ (yi , y−i ))|a∗−i ] for each yi ∈ Y and eachi ∈ N.
We say that for a given ε > 0, the wage scheme wi∗ satisﬁes ε-IPE whenε ≥ Ey−i [wi∗ (yi , y−i )|a∗−i ] − w̃i (yi ) (≥ 0)(19)

for all yi ∈ Y . When ε → 0, we say that a wage scheme wi∗ is “almost” IPE because theimpact of y−i on wi∗ becomes negligible. We also call a wage scheme wi∗ asymptotic non-IPE
or simply non-IPE when some ε′′ > 0 and yi ∈ Y exist such that for all large N ,Ey−i [wi∗ (yi , y−i )|a∗−i ] − w̃i (yi ) ≥ ε′′ .
By letting ⌊x⌋ denote the largest integer that does not exceed a given number x, we show
the following result.
Proposition 4. Suppose that Assumptions 2 and 4 hold. Then, for any α ∈ (0, 1) and any
ε > 0, there exists some N0 such that for all N ≥ N0 , at least ⌊αN ⌋ agents must be oﬀered
ε-IPE wage schemes at the third-best contract solving Problem TB.
Proof. See the Appendix.

27

The main insight of Proposition 4 is that most agents are compensated according to“almost” IPE wage schemes (ε → 0) when the number of them N is suﬃciently large.
Although the optimal wage scheme wi∗ of agent i may depend on the outputs of others y−i ,such an eﬀect asymptotically vanishes as N → ∞ (so that ε → 0). In other words, mostagents are virtually rewarded by piece rate contracts that depend on their own individual
performances. This result implies that the value of using rank-order contracts vanishes inlarge organizations. We discuss in more detail below how our result is related to existingstudies that show the asymptotic optimality of rank-order contracts (Green and Stokey(1983), Malcomson (1986)).
To understand the intuition behind Proposition 4, suppose, contrary to the claim, thatthe wages of only a limited proportion of agents are asymptotically IPE when N is suﬃciently
large. That is, most agents are paid according to non-IPE wage schemes. Then, we modify
the third-best contract w∗ in the following two respects, which can improve the principal’s
payoﬀ and hence contradicts the optimality of w∗ .
First, we expand the set of agents oﬀered IPE wage schemes while keeping their incentives
unchanged. More precisely, those oﬀered non-IPE wage schemes wi∗ under the originalcontract are now oﬀered the new schemes w̃i deﬁned by (18). While they incur lower risk
under the modiﬁed scheme than under the original one w∗ , they still choose the same thirdbest actions a∗i . Thus, such a modiﬁcation of wage schemes can reduce the total wage from
W as long as FTW remains satisﬁed. We denote by N ∗ the set of agents who are oﬀered
the new wage schemes w̃i deﬁned above.
To satisfy FTW, we divide the remaining set of agents N \ N ∗ into two disjoint subsets
L∗ and L∗∗ . The agents in N \ N ∗ are randomly selected into L∗ and L∗∗ by the principal
after she observes the outputs of agents. Then, the agents assigned to L∗ are paid according
to some IPE wage schemes wi (yi ), while those assigned to L∗∗ equally share the residual
wage deﬁned as the ﬁxed budget W minus total wages of all other agents than them:




∑
(1/#L∗∗ ) W −wj (yj )
(20)


∗
∗
j∈N ∪L

where #L∗∗ denotes the cardinality of set L∗ .
Thus, under the random wage scheme constructed above, agent i ∈ N \ N ∗ faces thefollowing wage:wi (yi , y−i ) ≡ wi (yi ) (1/#L∗∗ )

{

W−

}

∑

j∈N ∗ ∪L∗ wj (yj )28with probability β(N )with probability 1 − β(N )(21)where β(N ) ≡ #L∗ /#(N \ N ∗ ) is the equal probability for each agent in N \ N ∗ being
selected into the set of L∗ . The agents belonging to set L∗∗ serve to ﬁx total wages at W
regardless of their realized outputs, which ensures that FTW holds.
Second, the next step is to ﬁnd appropriate wage schemes {wi }i∈L∗ in the ﬁrst lineof (21) such that each agent in N \ N ∗ obtains the same expected payoﬀ as under theoriginal contract wi∗ given all the others choose the third-best actions a∗−i . This ensures thatthe agents in N \ N ∗ choose the same third-best actions a∗i as they do under the originalcontract wi∗ . We can show that such wage schemes actually exist (see the Appendix). Thenthe remaining problem is to check whether the agents oﬀered the above random wage incur
higher risk because their wages vary with the outputs of others y−i when they are paid theresidual wage (20). However, when the number of agents N is so large that #L∗∗ is so, theprobability that (20) aﬀects an agent’s payoﬀ tends to be suﬃciently small because of the
Law of Large Numbers. Thus, they equally share only negligible risk when N is suﬃciently
large, which causes no additional cost to the principal under the modiﬁed contract.
In this way, the principal can increase her payoﬀ by modifying the original contract w∗
if only a limited proportion of agents are oﬀered IPE wage schemes, even when the numberof agents N is large. This establishes Proposition 4.
The important lesson of Proposition 4 is that the principal has “most” agents working
under piece rate contracts contingent only on their individual outputs. In general, for FTW
to be satisﬁed, no agents can work under piece rate contracts because then their total wages
vary with their realized outputs. Hence, we use the random wage scheme deﬁned in (21) toensure that most agents are paid according to piece rates, while the remaining few agents are
randomly paid an equal share of the residual wage (20). The existence of the latter agentsalways ﬁxes total wages at a constant for any output realization. By taking the proportionof agents who face such a random scheme to be small enough as N rises, the principal can
improve her payoﬀ by having most agents work under piece rate contracts.

5.2

Asymptotic Eﬃciency

Next, we use Proposition 4 to show that the principal can approximate the second-best
payoﬀ, which is the outcome when FTW is dropped, when the number of agents N becomes
suﬃciently large. Thus, when many agents exist, the principal asymptotically incurs no loss
from the target budget constraint.
As the objective for the principal to achieve, we focus on the average payoﬀ per agent.
When we vary the number of agents N , a direct scale eﬀect arises such that N directly29increases or decreases the principal’s payoﬀ. To abstract the pure incentive eﬀect of largeorganizations from this direct scale eﬀect and make the comparison with the case of a single
agent transparent, we treat the average payoﬀ as the target of the principal. We deﬁne theaverage payoﬀ of the principal under the second-best contract {ŵi , âi }N
i=1 as follows:{π̂ ≡ (1/N ) Ey [R(y)|â] −N∑}Eyi [ŵi (yi )|âi ]i=1Then, we show the following result.
Proposition 5. Suppose that Assumptions 2 and 4 are satisﬁed. Then, for any ε > 0, there
exists some N such that for all N ≥ N the average payoﬀ of the principal is at least π̂ − ε;in other words, the principal can approximately attain the second-best payoﬀ π̂ on averageas N → ∞.
Proof. See the Appendix.
Proposition 5 is understood from the viewpoint of Proposition 4. In the proof of Proposition 4, we set the wage schemes for most agents as IPE, while we consider the random
scheme for the remaining agents as shown in (20). We modify this scheme by setting thewage schemes for most agents as the second-best ones ŵi and those for the remaining others
as that similar to the random wage (20). Because of the Law of Large Numbers, the agentsoﬀered the latter scheme face low income risk when the number of agents N is suﬃciently
large. In this way, most agents face almost the same wage schemes as the second-best onesŵi and hence choose the second-best actions âi . Thus, the principal can approximate thesecond-best payoﬀ per agent as the number of agents is suﬃciently large.
To show the above asymptotic eﬃciency result, we maintain the assumption that theutility function of each agent is additively separable between his income and action (Assumption 2). However, we can weaken this assumption when we focus on the case that
action set A is ﬁnite.
We now return to the general case considered in Section 3 that the utility function of
agent i is deﬁned as Ui : R × A → R, where agent i’s utility is given by U (wi , ai ) for his
income wi and action ai . We also maintain Assumption 1: some IPE contract {ŵi , âi } solves
Problem SB for each agent i, where ŵi depends only on his output yi . Such contract satisﬁes
both IC and IR in Problem SB for implementing the second-best action âi ∈ A from agent30

i:Eyi [Ui (wi (yi ), âi )|âi ] ≥ Eyi [Ui (wi (yi ), ai )|ai ], ∀ ai ̸= âi(IC)Eyi [Ui (wi (yi ), âi )|âi ] ≥ U i(IR)

We make the following assumption.
Assumption 5. For any δ > 0, there exists some contract {wiδ , âi } for each agent i suchthat |wiδ (yi ) − ŵi (yi )| < δ for all yi ∈ Y and both IC and IR above are satisﬁed with strict
inequalities.
Assumption 5 states that there exists a contract {wiδ , âi } in a neighborhood of the second–
best contract {ŵi , âi } such that agents have the strict incentive to choose the second-best
actions and strictly prefer accepting the second best-contract to rejecting it. We provide
the suﬃcient conditions for Assumption 5 below and discuss how this assumption is a weak
restriction.
We use a contract {wiδ , âi } deﬁned in Assumption 5 in order to approximate the second–
best contract while keeping both IC and IR as strict inequalities in the limit as the numberof agents N goes to inﬁnity. The reason that we need strict IC and IR is as follows. As
we have already discussed, in order to satisfy FTW, there must be some agents whose wage
schemes are interdependent in the sense that their wages vary with outputs of others. Thus,
these agents incur additional risks which may distort their action choices and the acceptancedecisions of contracts from the second-best optimum. Since such additional risks cannot be
exactly zero in the third-best case, we need to perturb wage schemes from the second-bestones in order to modify IC and IR corresponding to these additional risks. Strict IC and
IR ensure that we can incorporate such additional (but small when N is large) risks into IC
and IR without aﬀecting incentives of agents.
In fact we show the following result.
Proposition 6. Suppose that Assumptions 1 and 5 hold. Suppose also that agent i’s utility
is given by Ui (wi , ai ), where Ui is continuous and increasing in wi . Then, for any ε > 0,
there exists some N such that for all N ≥ N the principal can obtain at least π̂ − ε.
Proof. See the Appendix.

31

To show this result, we use the following simple sharing rule:{}N∑wi (yi , y−i ) ≡ wiδ (yi ) + (1/N ) Ŵδ −wnδ (yn )

(22)

n=1

for the wage scheme {wiδ , âi } deﬁned in Assumption 5. Here, Ŵδ is the total expected wagesof all agents under the schemes {wnδ , ân }N
i=1 :
Ŵδ ≡

N
∑

Eyn [wnδ (yn )|ân ].

n=1

This contract is a special form of the simple sharing rule deﬁned in the previous section by
setting an equal share αi = 1/N for all i. Then, the agents face virtually the second-best wage
schemes {ŵi }N
i=1 when N is suﬃciently large and δ is taken to be small because, according toAssumption 5, wiδ can be suﬃciently close to the second-best wage scheme ŵi and the equal∑share of the residual wage (1/N ){Ŵδ − n wnδ } approaches zero in probability as N → ∞because of the Law of Large Numbers. Moreover, when δ is taken to be suﬃciently small,Ŵδ → Ŵ holds, where Ŵ is the total expected wage cost in the second best. Thus, theprincipal can approximately obtain the second-best payoﬀ π̂ on average when the numberof agents N is suﬃciently large. Green and Stokey (1983) and Malcomson (1986) show thatrank-order contracts perform as well as piece rate contracts do when there are many agents.
Green and Stokey (1983) assume that the set of agents is ﬁnite and obtain the asymptotic
eﬃciency result as their number tends to inﬁnity. Malcomson (1986) analyzes the case of acontinuum of agents. When there are many agents, the rank ordering of each agent’s relative
performance almost exactly reﬂects only his own performance. Thus, rank-order contracts
do not impose higher risk on agents than piece rate contracts do in the presence of inﬁnitely
many agents.
Our asymptotic result diﬀers from these studies in several important aspects. First, theabove studies restricted their analyses only to the case of identical agents and focused only
on the symmetric equilibrium in which agents follow an identical eﬀort strategy given thewage scheme. However, we allow agents to be heterogeneous and impose no such symmetricrestrictions on their equilibrium behaviors. Although rank-order contracts may perform wellin symmetric environments, they may be ineﬀective at mitigating the incentive problem inmore general cases when agents are heterogeneous. For example, under a standard rankorder tournament, agents are paid diﬀerent prizes based on the rankings of their relative
outputs and these prizes do not depend on their identities (i.e., the same prize is given to
the winner whoever he is). However, such prize structure may be an insuﬃcient instrument
to eﬃciently manage the diﬀerent incentives of heterogeneous agents who do not necessarily
32behave in a symmetric manner. By contrast, we show that the simple sharing rule given in(22) is suﬃcient to achieve the second-best eﬃciency regardless of agents’ heterogeneity as
the number of agents goes to inﬁnity.
Second, in ﬁnite environments in which the action set and output set are both ﬁnite,
we can allow more general utility functions of agents than the additively separable form
used in the studies cited above (see Proposition 6). Even when agents’ preferences exhibit
the income eﬀect, our approximate eﬃciency result remains valid provided a weak ﬁnitenesscondition, Assumption 5, is guaranteed.
Remark. Assumption 5 is satisﬁed under fairly weak restrictions. When we focus on a ﬁnite
environment with ﬁnite action set A and ﬁnite output set Y , we can show that Assumption
5 is weaker than the standard assumption that the utility functions of agents are additively
separable as long as a weak restriction is imposed on the probability distributions of outputs
as made in Assumption 3. In fact, Assumption 5 is satisﬁed under Assumptions 2 and3. Since Assumption 3 generically holds when the number of outputs is larger than thatof actions (#Y > #A), Assumption 5 is generically weaker than assuming the additive
separable utility functions.
From Assumption 3, we can ﬁnd a vector (v δ (y))y∈Y such that for a given δ > 0,∑(pi (y|âi ) − pi (y|a))viδ (y) ≥ δy∈Yholds for any a ̸= âi (Rockafellar (1970, p. 198, Proposition 22.1)).
Recall that ŵi is the second-best wage scheme implementing the second-best action âi .
Let ûi (y) ≡ ui (ŵi (y)) be the corresponding utility payments. Then, we deﬁne uδi (y) ≡ûi (y) + viδ (y) for each y ∈ Y , which ensures that∑∑∑(pi (y|âi ) − pi (y|a))viδ (y)(pi (y|âi ) − pi (y|a))ûi (y) +(pi (y|âi ) − pi (y|a))uδi (y) =yyy≥ Gi (âi ) − Gi (a) + δ> Gi (âi ) − Gi (a)for any a ̸= âi . By adding a constant vi to uδi (y) for each y ∈ Y , we can also ensure that∑pi (y|âi )(uδi (y) + vi ) − Gi (âi ) > U i .
yδ
By letting δ → 0, we can take viδ (y) → 0 and vi → 0. By deﬁning wiδ (yi ) ≡ u−1
i (ui (yi )) for

each yi ∈ Y , wage scheme {wiδ , âi } satisﬁes IC and IR with strict inequalities. As long as
action set A and output set Y are both ﬁnite with #Y > #A, Assumption 5 is genericallysatisﬁed under Assumptions 2.
33

6Concluding Remarks

In this paper we investigated a wide class of principal–agent problems with moral hazard
and target budgets. The latter requires that the principal ﬁxes a budget for the total wages
paid to all agents regardless of their outputs realized. We showed that the presence of suchtarget budgets cause no loss of eﬃciency when there are at least two risk-neutral agents.
We then discussed that our result contributes to the literature on relational contracts withmultiple agents, which has not thus far fully addressed what the principal can achieve in thestatic benchmark. In contrast to most studies in that strand of the literature, we showed that
subjective evaluations never constrain the principal, even in static environments with at least
two risk-neutral agents. We characterized the third-best contracts as when there is at mostone risk-neutral agent, which no longer guarantee that the second-best eﬃciency is achieved.
Then, we showed that the optimal contract is given by the presented simple sharing rule if
and only if agents have CARA preferences. In addition, beyond CARA utility functions,
we showed that under the third-best contract, most agents are compensated according toalmost their individual outputs when their number is suﬃciently large. We then showed thatthe principal can approximate the second-best payoﬀ on average as the number of agents is
suﬃciently large.
We conclude the paper by discussing some possible extensions of the model. First,
we ruled out money burning from the model. When this is allowed, the principal can∑weaken the target budget constraint as W ≥ N
i=1 wi (y). Thus, the principal can dispose∑Nof W − i=1 wi (y) > 0 if she wants. However, as we show in Lemma A2 in Appendix A,∑′′′′
this never happens under the third-best contract. If W > N
i=1 wi (y ) for some y , whereas∑N
W = i=1 wi (y′ ) for other y′ , then the principal can change the wages of some agent i for′′ and y ′ to make his IC and IR unchanged. Intuitively,
the diﬀerent outputs of others y−i−isince agent i’s action ai aﬀects only his own output yi , which is also not correlated withthe outputs of others y−i , it is still possible to keep agent i’s expected utility unchanged
by changing his wages for the diﬀerent outputs of others y−i . However, such variationalchanges in agent i’s wage can reduce total wages further from the optimal contract. Thus,∑
W = n wn (y) must be satisﬁed at the optimum.
Second, we focused on the situation in which the outputs of agents are technologicallyindependent of each other. Thus, we ruled out the case that the actions taken by one agent
directly and technologically aﬀect the outputs of others. However, considering how the
principal designs teamwork and job structure so that agents help and cooperate with eachother is an important issue (e.g., Itoh (1991), Ishihara (2017)). It would thus be interesting34

in future work to examine the extent to which the target budget constraint aﬀects teamwork
formation and job design in organizations.

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[15] Levin, J. (2002), “Multilateral Contracting and the Employment Relationships,” Quarterly Journal of Economics, Vol. 117, 1075-1103[16] MacLeod, B. W. (2003), “Optimal Contracting with Subjective Evaluation,” American
Economic Review, Vol. 93, 216-240.
[17] Malcomson, J. M. (2013), “Relational Incentive Contracts,” Chapter 25 in Handbook of
Organizational Economics edited by Gibbons, R. and J. Roberts, Princeton University
Press.
[18] Malcomson, J. M. (1984), “Work Incentives, Hierarchy, and Internal Labor Markets,”
Journal of Political Economy, Vol. 92, 486―507.
[19] Malcomson, J. M. (1986), “Rank-Order Contracts for a Principal with Many Agents,”
Review of Economic Studies, Vol. 53, 807-817.
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[21] Murphy, K. (1993), “Performance Measurement and Appraisal: Merck Tries to Motivate
Managers to Do It Right,” Employment Relations Today, Vol. 20, 47-60.
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[23] Rockafellar, R. T. (1970), Convex Analysis, Princeton University Press.
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36[26] Whalen, E. L. (1991), Responsibility Center Budgeting: An Approch to Decentralized
Management for Institutions of Higher Education. Indianapolis: Indiana University
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[27] Wilson, J. D. (2002), “Models of Centralized and Decentralized Budgeting within Universities,

Michigan State University.

37

7

Appendix A: Omitted Proofs

7.1

Proof of Lemma 1

Suppose that Ui (wi , ai ) = Hi (ai )ui (wi ) − Gi (ai ) for each i ∈ N . Suppose also that the wage
scheme of some agent i depends on the outputs of others y−i , that is, ŵi (yi , y−i ) under thesecond best contract. Then we deﬁne the new wage scheme w̃i as follows
ui (w̃i (yi )) = Ey−i [ui (ŵi (yi , y−i ))|â−i ]for each yi ∈ Y . Then we have
Ui (w̃i (yi ), ai ) − Ey−i [Ui (ŵi (yi , y−i ), ai )|â−i ] = Hi (ai ){ui (w̃i (yi )) − Ey−i [ui (ŵi (yi ))|â−i ]}= 0
for any yi ∈ Y and any ai ∈ A. Thus agent i obtains the expected payoﬀ under the above
new scheme w̃i as follows
Eyi [Ui (w̃i (yi ), ai )|ai ] = Ey [Ui (ŵi (yi , y−i ), ai )|ai , â−i ]which is maximized at the second-best action ai = âi . Then all agents choose the second-best
actions â.
The principal can however weakly reduce the wage cost by the new wage scheme becauseui (w̃i (yi )) = Ey−i [ui (ŵi (yi , y−i ))|â−i ] ≤ ui (Ey−i [ŵi (yi , y−i )|â−i ])which implies that w̃i (yi ) ≤ Ey−i [ŵi (yi , y−i )|â−i ] and hence
Eyi [w̃i (yi )|âi ] ≤ Ey [ŵi (y)|â]
for every agent i. Q.E.D.

7.2

Proof of Lemma 2

We proceed to prove Lemma 2 by several steps. First, we consider the relaxed problem of theoriginal problem, Problem TB, by replacing FTW by its weak inequality. Second, we showthat the so called Slater condition is satisﬁed in such relaxed problem, which ensures thatthe optimal solution to the relaxed problem must satisfy the Kurash-Kuhn-Tucker (KKT)
conditions. Third, we show that the optimal solution to the relaxed problem coincides withthe one of the original problem, Problem TB. Finally, by using the KKT conditions, we
prove Lemma 2.

38

By replacing FTW by the version of a weak inequality, we consider the relaxed problemof the original Problem TB as follows.
Problem RP
Min W
subject to IC, IR and
W ≥

N
∑

(FTW′′ )wi (y) ∀ y ∈ Y N

i=1

In what follows we denote by P (y−i |a−i ) ≡∏j̸=i pj (yj |aj ) the joint probability of outputs∏
Also we denote by P (y|a) = N
i=1 pi (yi |ai ) the jointof all other agents than agent i.

probability of all agents’ outputs y ∈ Y N conditional on an action proﬁle of them a ∈ AN .
We can use the change of variables by letting ui (y) = ui (wi (y)) and wi (y) = ϕi (ui (y))
for the inverse function ϕi of ui . Then, the above problem RP is equivalent to the following.
Problem RP∗
Min{ui (y)}i∈I ,W W
subject to∑y−i

P (y−i |a∗−i )∑pi (yi |âi )ui (y)−Gi (âi ) ≥yi∑

P (y−i |a∗−i )y−i∑

P (y−i |a∗−i )

∑

y−i

∑

pi (yi |a)ui (y)−Gi (a) ∀ a ̸= âi ,

yi

(IC)
pi (yi |a∗i )ui (y) − Gi (a∗i ) ≥ U i

(IR)

yi

W ≥

N
∑

ϕi (ui (y)) ∀ y ∈ Y N(FTW′′ )i=1

Then, since ϕi is a convex function, this can be a problem of convex programming. It is known
that, when the Slater condition is satisﬁed, the necessary condition for the optimal solution
to the above problem RP∗ satisﬁes the following KKT conditions (Takayama (Theorem
1.D.2, 1985, p.92): there exist some non-negative multipliers λi ≥ 0 for i ∈ N , µi (a) ≥ 0 fori ∈ N and a ̸= a∗i , and η(y) ≥ 0 for y ∈ Y N such that∑−1 +η(y) = 0,

(A1)

y∈Y N

P (y|a∗ )u′i (wi (y))





λi +


)
(
pi (yi |a)− η(y) = 0µi (a) 1 −pi (yi |a∗i ) 
∗

∑

a̸=ai

39

(A2)

We ﬁrst show the following.
Lemma A1. The Slater condition is satisﬁed in Problem RP∗ .
Proof. Since ϕi is a convex function as well as both IC and IR are linear functions of ui (y)in Problem RP∗ , it suﬃces to show that there exists an interior point (W , u1 (y), ..., uN (y))
in the constraint set. Under Assumption 3, by using the theorem on existence of solutions
to a system of inequalities (Rockafellar (Proposition 22.1, 1970, p.198)), we can ﬁnd a vector(ui (yi ))yi ∈Y such that∑∑pi (yi |a)ui (yi ) − Gi (a).
pi (yi |a∗i )ui (yi ) − Gi (a∗i ) >
yi ∈Y

yi ∈Y

Thus, IC of Problem RP∗ is satisﬁed as strict inequalities. By adding some constant vi tothese values of (ui (yi ))yi ∈Y , we can also ensure that IR of Problem RP∗ is satisﬁed as strict
inequalities. Finally, by taking a large enough W , FTW′′ is satisﬁed as strict inequalities
given the utility payments we have deﬁned above. Q.E.D.
Lemma A2. η(y) > 0 for all y ∈ Y N .
Proof. By (A1), η(y) > 0 must hold for some y ∈ Y N . Let y = (y1 , ..., yi , ..., yN ) for suchy.
Then, when we look at (A2) for agent i, we have


(
)

∑
pi (yi |a)µi (a) 1 −= η(y1 , ..., yi , ..., yN ) > 0
P (y|a∗ )u′i (wi (y)) λi +pi (yi |a∗i ) ∗a̸=aiso thatλi +()
pi (yi |a)µi (a) 1 −> 0.
pi (yi |a∗i )
∗

∑

(A3)

a̸=ai

′′ ) by ﬁxing y of agent i’s output but
Now we take any output proﬁle of y′′ = (y1′′ , ..., yi , ..., yNi′′ . Then (A2) and (A3) imply that
arbitrary outputs of others y−i)(∑pi (yi |a)′′= η(y1′′ , ..., yi , ..., yN)/P (y′′ |a∗ )u′i (wi (y′′ ))0 < λi +µi (a) 1 −
∗)
p
(y
|a
i i i
∗
a̸=ai

′′ ) > 0 for any y′′ ∈ Y N −1 . Then we take any agent j ̸= i andso that η(y1′′ , ..., yi , ..., yN−iobtain()′′ |a)


∑
p
(y
j j
′′
0 < η(y1′′ , ..., yi , ..., yj′′ , ..., yN) = P (y′′ |a∗ )u′j (wj (y′′ )) λj +µj (a) 1 −pj (yj′′ |a∗j ) 
∗
a̸=aj

40

so that
λj +

(

∑

pj (yj′′ |a)µj (a) 1 −pj (yj′′ |a∗j )
∗

)
>0

a̸=aj

for any yj′′ ∈ Y . Since this holds for any yj′′ ∈ Y and any agent j ̸= i, when we take any
arbitrary ỹ = (y1 , ..., yi , ..., yj′′ , ..., yN ) ∈ Y N , we obtain0 < λj +(∑pj (yj′′ |a)µj (a) 1 −pj (yj′′ |a∗j )∗)= η(y1 , ..., yj′′ , ..., yN )/P (ỹ|a∗ )u′j (wj (ỹ))

a̸=aj

so that η(y1 , ..., yj′′ , ..., yN ) > 0 for any ỹ = (yj′′ , y−j ) ∈ Y N . Q.E.D.
Lemma A2 implies that FTW′′ must be binding at the optimum of Problem RP∗ orequivalently Problem RP. Thus the optimal solution to Problem RP is same as the original
problem TB.
By Lemma A2, we have
λj +

(

∑

pj (yj |a)µj (a) 1 −pj (yj |a∗j )
∗

)
>0

a̸=aj

for all j ∈ N and all yj ∈ Y . Then we can rearrange (A2) as follows()∑
pj (yj |a)
µ
(a)
1
−
λ
+
j
a̸=a∗j jpj (yj |a∗j )u′i (wi (y))() = eξj (yj )−ξi (yi )=∑′pi (yi |a)uj (wj (y))λi + a̸=a∗ µj (a) 1 − pi (yi |a∗ )
i

(A4)

i

for any i ̸= j.
First, suppose that all agents are risk averse, that is, u′′i < 0 for all i. Then, we show
Lemma 2. From (A4), by letting gi be the inverse function of u′i , we obtain()
wj (y) = gj u′i (wi (y))eξi (yi )−ξj (yj )(A5)for any j ̸= i. Summing both sides of this over all j ̸= i, we obtain∑wj (y) =j̸=iwhich, by using W =

∑

∑

(
)
gj u′i (wi (y))eξi (yi )−ξj (yj )j̸=i

j̸=i wj (y) + wi (y) (FTW), can be written by

W = wi (y) +∑()
gj u′i (wi (y))eξi (yi )−ξj (yj ) .

j̸=i

41

(A6)

Here u′i is strictly decreasing and gj is strictly decreasing for all j ̸= i. Thus,∑′j̸=i gj (ui (·))

is strictly increasing. Then, the right hand side of (A6) is strictly increasing in wi (y). Also,
the right hand side goes to +∞ (−∞) as wi (y) → ∞ (−∞). Thus, there exists a uniquesolution of wi (y) to equation (A6). We denote it bywi (y) = Qi (ξi (yi ) − ξ1 (y1 ), ..., ξi (yi ) − ξN (yN ))which depends on diﬀerences in piece rate wages (ξi (yi ) − ξj (yj ))j̸=i between agent i andothers j ̸= i.
Furthermore, we can verify from (A6) that its right hand side is strictly decreasing in∑ξi because j̸=i gj (·) is strictly decreasing. Thus Qi is strictly increasing function of ξi .16
Furthermore, gj (·) is strictly decreasing for all j ̸= i. Thus, Qi is strictly deceasing in ξj forj ̸= i.
Next suppose that one agent is risk neutral bu all others are risk averse. Let agent
1 be risk neutral so that his utility is given by w1 − G1 (a1 ). Then, (A4) implies that
1/u′i (wi (y)) = eξi (yi )−ξ1 (y1 ) for any i ̸= 1 from which we have wi (y) = gi (eξ1 (y1 )−ξi (yi ) ) whichis strictly increasing in ξi but strictly decreasing in ξ1 . By summing this over all i ̸= 1 and∑using FTW, we obtain w1 (y) = W − i̸=1 gi (eξ1 (y1 )−ξi (yi ) ) which is also strictly increasingin ξ1 but decreasing in ξi . Q.E.D.

7.3

Proof of Proposition 2

Under CARA utility functions we have assumed in the main text, (A4) can be written byri wi (y) − rj wj (y) = ξi (yi ) − ξj (yj )where
ξi (yi ) ≡ ln λi +
()
pi (yi |a) µi (a) 1 −.
∗pi (yi |ai )
∗

∑

(A7)

a̸=ai
16

Suppose that some agents are induced to choose the least costly action, a∗i = 0. For such agent i, we

drop his IC from the Problem RP and obtain the KKT condition as follows:u′i (wi (y))=∑
u′j (wj (y))λj +λiµj (a)a̸=a∗j()
p (y |a)1 − pjj(yjj|a∗ )jfor any j ̸= i. Then, we can set ξi ≡ ln λi which is independent of agent i’s output yi and derive the optimal
contract for agent i as wi (y) = Qi (ξi − ξ1 (y1 ), ..., ξi − ξN (yN )). Since Qi does not change with yi , agent i isactually induced to choose the least costly action ai = 0, which trivially satisﬁes his IC. Thus, without lossof generality, we can set µi (a) = 0 for all a ̸= a∗i so that ξi = ln λi when agent i is induced to choose the least
costly action ai = 0.

42

Note that, since λi and µi (a) are independent of outputs, ξi (yi ) depends only on agent i’s
output yi through the likelihood ratio pi (yi |a)/pi (yi |a∗i ).
We sum both sides of (A7) over all j ̸= i to obtain∑(1/rj )ri wi (y) −

j̸=i

∑

wj (y) =∑∑(1/rj )ξi (yi ) −(1/rj )ξj (yj )

j̸=i

j̸=i

j̸=i

which is further arranged as∑(1/rj )ri wi (y) − W + wi (y) =∑(1/rj )ξi (yi ) −

j̸=i

j̸=i

∑

(1/rj )ξj (yj )j̸=idue to FTW in Problem TB. Then we can derive


∑
∑
∑
 (1/rj )ri + 1 wi (y) = W +(1/rj )ξi (yi ) −(1/rj )ξj (yj )
j̸=i

j̸=i

j̸=i

which yields the following desired result
1/ri

wi (y) = (1/ri )ξi (yi ) + ∑N

l=1 (1/rl )

{
W−

N
∑

}
(1/rl )ξl (yl ) .

l=1

Q.E.D.

7.4

Proof of Proposition 3

From (A4), we obtainln u′j (wj (y)) − ln u′i (wi (y)) = ξi (yi ) − ξj (yj )(A8)for each i ̸= j.
We deﬁne the following function
hi (wi (y)) ≡ ln u′i (wi (y)) + γi wi (y)(A9)

for some positive constant γi > 0. Then, the above optimality condition (A8) is written by−γj wj (y) + γi wi (y) = ξi (yi ) − ξj (yj ) + hi (wi (y)) − hj (wj (y)).
By dividing both sides of (A9) by γj and summing them over all j ̸= i, we obtain
−

∑
j̸=i

wj (y) +∑∑(1/γj )γi wi (y) =(1/γj ) {ξi (yi ) − ξj (yj ) + hi (wi (y)) − hj (wj (y))} .
j̸=i

j̸=i

43

Since W =∑n wn (y), this can be written by




∑
∑
(1/γj ) wi (y) =(1/γj ) {ξi (yi ) − ξj (yj ) + hi (wi (y)) − hj (wj (y))} .
−W + γi 1 +
j̸=i

j̸=i

Thus we derive the third best contract for agent i in the following implicit form:wi (y) = Hi (ξi , ξ−i ) ≡ ∑∑1/γi(1/γj ) {ξi (yi ) − ξj (yj ) + hi (wi (y)) − hj (wj (y))} .
n (1/γn )
j̸=i

(A10)
Now suppose that the third best contract is given by the simple sharing rule as follows{}
N∑wi (y) = βi ξi (yi ) + αi W −βl ξl (yl )
l=1

where

∑

n αn = 1 and βi ̸= 0 for any i ∈ N . Note that βi ̸= 0 holds for all i due to Lemma

2 (Qi is strictly increasing in ξi ).
This is a linear function of {ξn (yn )}N
n=1 . Keeping this in mind, we diﬀerentiate Hi twicewith respect to ξi (yi ) in order to obtain∑∂ 2 Hi ∑2 2 ′′
(1/γ
)(1
−
α
)
β
h
(w
(y))
−
(1/γj )αj2 βi2 h′′j (wj (y))
=
j
i
i
i
i
∂ξi2
j̸=i

j̸=i

which must be zero because wi is linear with respect to ξi :∑∑(1/γj )(1 − αi )2 βi2 h′′i (wi (y)) −(1/γj )αj2 βi2 h′′j (wj (y)) = 0.
j̸=i

(A11)

j̸=i

From this, it must be that for any i ̸= m,∑

2 2 ′′(1/γj )(1 − αi )2 βi2 h′′i (wi (y)) − (1/γm )αm
βi hm (wm (y)) −

j̸=i

∑

(1/γj )αj2 βi2 h′′j (wj (y)) = 0.

j̸=i,m(A12)
By changing the roles of i and m, we obtain the similar equation:
∑

2 ′′
2 ′′
(1/γj )(1 − αm )2 βmhm (wm (y)) − (1/γi )αi2 βmhi (wi (y)) −

j̸=m

∑

2 ′′
(1/γj )αj2 βmhj (wj (y)) = 0.

j̸=i,m

(A13)
From (A12), (A13) and βn2 ̸= 0 for all n, we have




∑
∑
2  ′′
 (1/γj )(1 − αi )2 + (1/γi )αi2  h′′i (wi (y)) = (1/γj )(1 − αm )2 + (1/γm )αmhm (wm (y)).
j̸=i

j̸=m

(A14)
44

This implies that, if h′′i (wi (y)) > (<)0, then h′′m (wm (y)) > (<)0 must hold for any m ̸= i.
Also, we diﬀerentiate Hi twice with respect to ξj for any j ̸= i to obtain∑∂ 2 Hi ∑
2 ′′
2 ′′
2 ′′
hl (wl (y))
hm (wm (y))−(1/γl )αl2 βmhi (wi (y))−(1/γm )(1−αm )2 βm=(1/γj )αi2 βm2∂ξjl̸=i,mj̸=iwhich must be zero again:
∑

∑

2 ′′
2 ′′
(1/γj )αi2 βmhi (wi (y)) − (1/γm )(1 − αm )2 βmhm (wm (y)) −

j̸=i

2 ′′
(1/γl )αl2 βmhl (wl (y)) = 0.

l̸=i,mBy using (A12) and (A14), we obtain∑(1/γj )(2αi − 1)h′′i (wi (y)) = (1/γm )(1 − 2αm )h′′m (wm (y))

(A15)

j̸=i

for any i ̸= m.
Now suppose that h′′i (wi (y)) > 0 holds for some i. Then, we know that h′′m (wm (y)) > 0
for all m ̸= i. Then, from (A15) we have αi > 1/2 and αm < 1/2 for any m ̸= i. Takej ̸= m, i (recall that N ≥ 3). Then αj < 1/2. However, by replacing i by m and m by j ̸= iin the above equation (A15) respectively, we obtain αj > 1/2 which is a contradiction to
αj < 1/2 for j ̸= i. Similarly, if αi < 1/2, we have a contradiction as well.
Next suppose that h′′i (wi (y)) < 0. Then, h′′m (wm (y)) < 0 for any m ̸= i. By multiplying−1 both sides of (A15), we can use the same argument as above.
Finally, suppose that αi = 1/2. Then, αm = 1/2 must hold for any m ̸= i by (A16).
∑
However, this implies that 1 = n αn = N/2, contradicting to N ≥ 3.
Thus we have established the result thath′′i (wi (y)) = h′′i (zi )≡ h′′i (1 − αi )βi ξi + αi W −∑βj ξj j̸=i

= 0must hold for all (ξ1 , ..., ξN ) ∈ RN and all i ∈ N when the optimal contract becomes thesimple sharing rule (which is the linear function of (ξ1 , ..., ξN )). Since ξn varies over R foreach n = 1, 2, ..., N , zi does so. Then, since h′′i (zi ) = 0 for all zi ∈ R, it must be thath′′i (zi ) =( ′′)ui (zi ) ′=0u′i (zi )45

for all zi ∈ R. This implies that the degree of the absolute risk aversion of agent i,u′′ (z)− i′,ui (z)
must be constant for any i.17 Q.E.D.

7.5

Proof of Proposition 4

Suppose contrary to the claim that some α′′ ∈ (0, 1) and ε′′ > 0 exist such that for any N0
there exists N ≥ N0 which ensures that more than N − ⌊α′′ N ⌋ agents are oﬀered the wage
schemes wi∗ at the third-best contract, which solves Problem TB, as follows
Ey−i [wi∗ (yi , y−i )|a∗−i ] − w̃i (yi ) ≥ ε′′for some yi ∈ Y . Recall that we called the wage scheme satisfying the above inequality nonIPE wage scheme. Let Ñ denote the set of all risk averse agents who are oﬀered non-IPE
18
wage schemes under the original contract {wi∗ , a∗i }N
i=1 .

By deﬁnition of w̃i (yi ),ui (w̃i (yi )) = Ey−i [ui (wi∗ (yi , y−i ))|a∗−i ].
Then w̃i (yi ) ≤ Ey−i [wi∗ (yi , y−i )|a∗−i ] for each yi ∈ Y because ui is a concave function. For
any risk averse agent who is oﬀered the non-IPE wage scheme wi∗ (yi , y−i ) under the originalcontract w∗ , we must have thatw̃i (yi ) < Ey−i [wi∗ (yi , y−i )|a∗−i ]for some yi ∈ Y , which in turn implies thatEyi [w̃i (yi )|a∗i ] < Ey [wi∗ (yi , y−i )|a∗i , a∗−i ].
We deﬁneρi (N ) ≡ Ey [wi∗ (yi , y−i )|a∗i , a∗−i ] − Eyi [w̃i (yi )|a∗i ]for agent i ∈ Ñ who is oﬀered a non-IPE wage scheme under {wi∗ , a∗i }. Thus it must be that
ρi (N ) ≥ ε′′ > 0 for all large N for any i ∈ Ñ . Then, by our supposition it must be that for17
18

If there is (at most) one risk neutral agent i, u′′i = 0 holds so that (u′′i /u′i )′ = 0 as well.
Let N ′′ denote the set of all agents who are oﬀered non-IPE wage schemes under the original contract.

Let r denote a unique risk neutral agent if exists. We deﬁne Ñ ≡ N ′′ \ {r} if r ∈ N ′′ and Ñ = N ′′ if r ∈/ N ′′
respectively.

46

any i ∈ Ñ we have limN →∞ ρi (N ) ≥ ρ > 0 for some ρ > 0. Then we can take some ρ ∈ (0, ρ)such thatEyi [w̃i (yi )|a∗i ] + ρ < Ey [wi∗ (yi , y−i )|a∗i , a∗−i ](A16)

for any i ∈ Ñ .
Since w̃i yields the same expected payoﬀ to agent i as that under the original wage
scheme wn∗ , i.e.,
Eyi [ui (w̃i (yi ))|ai ] = Eyi ,y−i [ui (wi∗ (yi , y−i ))|ai , a∗−i ]for any action ai ∈ A, agent i still has the incentive to choose the third-best action a∗i andobtains the same equilibrium payoﬀ as that under the original third-best contract wi∗ . Thismeans that the modiﬁed contract w̃i satisﬁes IC and IR in Problem M-SB for implementing
the third-best action a∗i . Thus, w̃i is feasible in Problem M-SB for implementing the thirdbest action a∗i . Recall that ŵn (·; a∗n ) is the optimal wage scheme solving Problem M-SB for
implementing the third-best action a∗n (by replacing an by a∗n in Problem M-SB). In what
follows we will use the shorthand notation ŵn∗ (·) ≡ ŵi (·; a∗i ) by suppressing its dependency
on a∗n . By the deﬁnition of the second-best wage scheme ŵi∗ implementing the third-best
action a∗i in Problem M-SB, we must then have
Eyi [ŵi∗ (yi )|a∗i ] ≤ Eyi [w̃i (yi )|a∗i ]so that due to (A16)
Ey [wn∗ (y)|a∗ ] > Eyi [ŵn∗ (yn )|a∗n ] + ρ(A17)

for any i ∈ Ñ , that is, for any risk averse agent i who is oﬀered the non-IPE wage schemewi∗ (yi , y−i ) under the supposed third best contract.
Now we modify the original contract {wi∗ , a∗i }N
i=1 as follows:• Let N ∗ be the set of agents with #N ∗ = ⌊αN ⌋ for some α ∈ (α′′ , 1). Let N ∗ include
all the agents i ∈/ Ñ who are not oﬀered non-IPE wage schemes at the original contract{wi∗ , a∗i } and one risk neutral agent (if such agent exists). Thus all agents in N \ N ∗are risk averse and oﬀered non-IPE schemes which depend on the outputs of others
under the supposed wage scheme wi∗ (yi , y−i ). We take a large enough N to ensure that⌊αN ⌋ > ⌊α′′ N ⌋.
• For agent n ∈ N ∗ , the principal oﬀers the second best wage scheme ŵn which implements the third best action a∗n . Thus ŵn satisﬁes IC and IR in Problem M-SB for
implementing the action a∗n . As we have explained above, such ŵn exists because w̃n∗
is feasible in Problem M-SB.
47

• N \ N ∗ is divided into the two disjoint sets L∗ and L∗∗ where L∗ ∪ L∗∗ = N \ N ∗ andL∗ ∩ L∗∗ = ∅. Let #L∗ = ⌊β(N − #N ∗ )⌋ and #L∗∗ = N − #N ∗ − #L∗ for someβ ∈ (0, 1). Here note that #L∗∗ = (N − #N ∗ ) − ⌊β(N − #N ∗ )⌋ ≥ (1 − β)(N − #N ∗ ) =(1 − β)(N − ⌊αN ⌋) ≥ (1 − β)(N − αN ) = (1 − β)(1 − α)N . Thus #L∗∗ → ∞ whenN → ∞.
• Deﬁneβ(N ) ≡ #L∗ /#(N \ N ∗ )which is the equal probability for each agent in N \ N ∗ to be assigned to the set L∗ .
Then, for agent i ∈ N \ N ∗ , the new wage scheme is deﬁned as{wi (yi )w.p. β(N )w̃i (yi , y−i ) =}{∑∑(1/#L∗∗ ) W − n∈N ∗ ŵn∗ (yn ) − i∈L∗ wi (yi ) w.p. 1 − β(N )
where
W ≡

N
∑

(Eyn [ŵn∗ (yn )|a∗n ] + ρn ),

(A18)

n=1

and ρn ≡ ρ > 0 satisﬁes (A17) for any n ∈ L∗ , and ρn = 0 for any n ∈/ L∗ respectively.
Note that the total wages of all agents becomes constant at W for any realization of
the outputs y ∈ Y N .
The new contract consists of the following: After the principal observes the realized
output proﬁle y ∈ Y N , she divides the set of agents in N \ N ∗ randomly into the two
subsets L∗ and L∗∗ . Each agent in N \ N ∗ is equally likely to belong to either L∗ or
L∗∗ , although agents in N ∗ do not face such randomization.
Under the above random scheme, each agent i ∈ N \N ∗ is selected into L∗ and paid wi (yi )with probability β(N ) while he is selected into the remaining set L∗∗ and paid (1/#L∗∗ ){W −∑∑∗n∈N ∗ ŵn (yn )− j∈L∗ wj (yj )} with probability 1−β(N ) respectively. Also, by construction,
the above contract satisﬁes FTW: the principal always pays W whatever outputs are realized.
Moreover, the principal is willing to assign the agents in N \ N ∗ randomly into L∗ and L∗∗
because her total payment does not depend on whoever agents are selected from N \ N ∗ intoL∗ .
Step 1. Agent n in N ∗ faces the following expected payoﬀ:
Eyn [un (ŵn (yn ))|an ] − Gn (an ),which is maximized at an = a∗n because ŵn implements the action a∗n by its deﬁnition. Eachagent n ∈ N ∗ also obtains at least the reservation utility because ŵn∗ satisﬁes IR in Problem
48

M-SB with the implementation of ai = a∗i .
Step 2. Now consider any agent i ∈ N \ N ∗ . Note thatβ(N − #N ∗ )β(N − #N ∗ ) − 1
1
=
β
≥
β(N
)
≥
≥β−
∗
∗
N − #NN − #N(1 − α)Nso that β(N ) → β as N → ∞. When we ﬁx an output of agent i ∈ N \ N ∗ at each yi ∈ Y
and take expectation over y−i ∈ Y N −1 , agent i’s expected payoﬀ is given by



 
∑
∑
1 
wj (yj ) a∗−i −Gi (ai )ŵn (yn ) −W−β(N )ui (wi (yi ))+(1−β(N ))Ey−i ui 
#L∗∗
∗
∗
j∈L

n∈N

conditional on a∗−i . Recall that #Y = K and y k ∈ Y for k = 1, 2, ..., K.
Our next step is to ﬁnd a K × #L∗ dimensional vector w ≡ ((wi (y 1 ), ..., wi (y K ))i∈L∗ tosatisfyβ(N )ui (wi (y k )) + (1 − β(N ))Ey−i ui = ui (ŵi∗ (y k ))


1 
W−
#L∗∗

∑
n∈N ∗

ŵn (yn ) −

∑
j∈L∗



wj (yj ) a∗−i (A19)

for each k = 1, 2, ..., K and each i ∈ L∗ . We will show the existence of (wi (y k ))i∈L∗ ,yk ∈Ysolving (A19) by sequence of several steps below.
Step 2-1. We take β ∈ (0, 1) to be close to 1 and a large N so that β(N ) is close toβ. Then, since ui is increasing with ui (∞) = u and limw→wmin ui (w) = −∞ together withthe fact that the second best contract ŵi is bounded19 so that ui (ŵi∗ (y k )) < u for eachy k ∈ Y , there exists a unique wi (y k ) satisfying the above equation (A19) for each output
yi = y k (k = 1, 2, ..., K) for given {wj (y k )}j̸=i,j∈L∗ ,yk ∈Y . We denote by ϕki (w−i ) such solution
where w−i ≡ (wj (y 1 ), ..., wj (y K ))j̸=i,j∈L∗ . Note that ϕki is increasing in each argument andcontinuous.
Step 2-2. We ﬁnd lower bounds for ϕki , which we denote by wi (y k ), as follows. We set
ui (wi ) = u for i ∈ L∗ and wi (y k ) ≡ wi for all y k ∈ Y . Then we show that ϕki (w−i ) ≥ wi for
19

The constraint set ΓSB (a∗n ) in Problem M-SB to implement the action a∗n is non-empty because by our

supposition w̃n satisﬁes IC and IR in Problem M-SB for implementing a∗i . The compactness of ΓSB (a∗n ) was
shown by Grossman and Hart (1983). Then the second best contract ŵn exists (Grossman and Hart (1983)),
which implies that the set of the second best contracts is bounded.

49all w−i and all i ∈ L∗ , that is,








∑
∑
1
W −
β(N )ui (wi ) + (1 − β(N ))Ey−i ui 
ŵn∗ (yn ) −wj (yj ) a∗−i 
#L∗∗
∗
∗
n∈N

j∈L

≤ ui (ŵi∗ (y k )).
for all y k ∈ Y . Since the left hand side is decreasing in wj (yj ) for each j ∈ L∗ , the above
inequality holds for any wj (yj ) ≥ wj if

β(N )ui (wi ) + (1 − β(N ))Ey−i ui 
≤


1 
W−
#L∗∗

∑

ŵn∗ (yn ) −

n∈N ∗

∑
j∈L∗





wj  a∗−i 

ui (ŵi∗ (y k )).

By the deﬁnition of wi , the above inequality is equivalent to
 



∑
∑
1 
 a∗−i Fi (u) ≡ β(N )u + (1 − β(N ))Ey−i ui W−ŵn∗ (yn ) −
u−1
j (u)
#L∗∗
∗
∗
n∈N

≤

j∈L

ui (ŵi∗ (y k ))for all y k ∈ Y .
We then show thatFi′ (u) = β(N ) −
×







1 − β(N )1 Ey−i u′i 
W−
∗∗
#L
#L∗∗

∑

∑

ŵn∗ (yn ) −

n∈N ∗

∑
j∈L∗





 a∗−i 
u−1
j (u)

1

u′ (u−1 (u))
j∈L∗ j j′so that, since u′i (∞) < ∞, u−1
j (−∞) = −∞ and uj (wmin ) = +∞, by letting u → −∞, weobtainFi′ (u)(1 − β(N ))#L∗
1
×
′
#L∗∗
minj uj (u−1
j (u))
[ (
(
))
]
N
−1
′
∗
∗
×Ey−i ui
min E[ŵn (yn )|an ] − max ŵn (yn ) − max uj (u)
a−i
n
n,yn
j
#L∗∗
→ β(N )≥

β(N ) −

where N/#L∗∗ → 1/(1 − α)(1 − β) and (1 − β(N ))#L∗ /#L∗∗ → β when N → ∞.
We can also show that Fi′′ (u) < 0 so that Fi is a concave function. This implies that
Fi (u) ≤ Fi (b) + Fi′ (b)(u − b)50

for any b ∈ (−∞, ∞). By the above result, taking a small enough b, we can ensure thatFi′ (b) ≥ β(N ) > 0. Then, given a suﬃciently small b such that Fi′ (b) ≥ β(N ) > 0, we showthat
Fi (u) ≤ Fi (b) + Fi′ (b)(u − b) → −∞when u → −∞. This is the desired result.
Step 2-3. We next ﬁnd an upper bound for ϕki , which we denote by wi (y k ), that is,ϕki (w−i ) ≤ wi (y k ) for each k = 1, 2, ..., K.
We setwi (y k ) ≡ ŵi∗ (y k ) + ρfor each y k ∈ Y and each i ∈ L∗ . Here recall the deﬁnition of ρ > 0 given by (A16).
We want to show thatβ(N )ui (wi (y k )) + (1 − β(N ))Ey−i ui 



∑

1 
W−
#L∗∗

ŵn∗ (yn ) −

n∈N ∗

∑
j∈L∗





wj (yj ) a∗−i > ui (ŵi∗ (y k ))for all wj (yj ) ≤ wj (yj ) for j ̸= i, j ∈ L∗ and all yj ∈ Y . Since the left hand side is decreasing
in wj (yj ) for j ∈ L∗ , the above inequality holds if
 



∑
∑
1 
W−
ŵn∗ (yn ) −wj (yj ) a∗−i β(N )ui (wi (y k )) + (1 − β(N ))Ey−i ui 
#L∗∗
∗
∗
n∈N

>

j∈L

ui (ŵi∗ (y k )).

Consider the second term in the above left hand side:
 



∑
∑
1 
Ey−i ui W−ŵn∗ (yn ) −wj (yj ) a∗−i  .
#L∗∗
∗
∗
n∈N



Let
Z(y) ≡

1 
W−
#L∗∗

j∈L

∑
n∈N ∗

ŵn∗ (yn ) −∑
wj (yj )

j∈L∗

for y = (yj )j∈N ∗ ∪L∗ . Here, since (A18)
W

=

N
∑

(Eyn [ŵn∗ (yn )|a∗n ] + ρn )

n=1

=

∑

n∈N ∗

Eyn [ŵn∗ (yn )|a∗n ] +∑(Eyn [ŵn (yn )|a∗n ] + ρ) +

n∈L∗

∑
n∈L∗∗

51

Eyn [ŵn∗ (yn )|a∗n ],and wj (yj ) ≡ ŵj∗ (yj ) + ρ for j ∈ L∗∗ , we obtain
Z(y) =

1
#L∗∗

{

∑

(Eyn [ŵn∗ (yn )|a∗n ] − ŵn (yn )) +n∈N ∗ ∪L∗∑}
Eyn [ŵn∗ (yn )|a∗n ]

n∈L∗∗

∑
1
≡ Z̃(y) +Eyn [ŵn∗ (yn )|a∗n ]
#L∗∗
∗∗
n∈L

where
#(N ∗ ∪ L∗ )Z̃(y) ≡
#L∗∗(

1#(N ∗ ∪ L∗ )

)(

∑

)
(Eyn [ŵn∗ (yn )|a∗n ] − ŵn∗ (yn )).

n∈N ∗ ∪L∗

Here note that#(N ∗ ∪ L∗ )
#L∗∗

=
=
≤
=
≤
=

We then obtain([

Ey−i [ui (Z(y))|a∗−i ] = Ey−iui#N ∗ + #L∗#L∗∗⌊αN ⌋ + ⌊β(N − ⌊αN ⌋)⌋N − ⌊αN ⌋ − ⌊β(N − ⌊αN ⌋)⌋⌊αN ⌋ + β(N − ⌊αN ⌋)N − ⌊αN ⌋ − β(N − ⌊αN ⌋)βN + (1 − β)⌊αN ⌋(1 − β)(N − ⌊αN ⌋)βN + (1 − β)αN
(1 − β)(N − αN )β + (1 − β)α.
(1 − α)(1 − β)

)
]
∑
1
∗
∗
∗
Z̃(y) +Eyn [ŵn (y)|an ] a−i .
#L∗∗
∗∗

(A20)

n∈L

Let
h≡

∑
1
Eyn [ŵn∗ (y)|a∗n ],
∗∗
#L
∗∗
n∈L

∗

h ≡ min Eyn [ŵn∗ (yn )|a∗n ]
n∈N

and
h∗∗ ≡

max ŵn∗ (yn )n∈N,y∈Y

respectively. Since the second best contract ŵn is bounded, h∗ and h∗∗ are bounded aswell.20 We denote by Pr(z; a∗−i ) the probability that an event z occurs conditional on the20

See footnote 19.

52

action proﬁle a∗−i of others than i. Then (A20) can be further written by
Ey [ui (Z(y))|a∗−i ]()= Pr |Z̃(y)| < ε; a∗−i Ey−i [ui (Z̃(y) + h)|a∗−i , |Z̃(y)| < ε]()+ Pr |Z̃(y)| ≥ ε; a∗−i Ey−i [ui (Z̃(y) + h)|a∗−i , |Z̃(y)| ≥ ε]()≥ Pr |Z̃(y)| < ε; a∗−i Ey−i [ui (Z̃(y) + h∗ )|a∗−i , |Z̃(y)| < ε]()+ Pr |Z̃(y)| ≥ ε; a∗−i Ey−i [ui (Z̃(y) + h∗ )|a∗−i , |Z̃(y)| ≥ ε]()≥ Pr |Z̃(y)| < ε; a∗−i ui (−ε + h∗ ))() ( α + (1 − α)β∗∗∗∗∗(h − h ) + h+ Pr |Z̃(y)| ≥ ε; a−i ui(1 − α)(1 − β)When N is suﬃciently large, #(N ∗ ∪ L∗ ) = ⌊αN ⌋ + ⌊β(N − ⌊αN ⌋)⌋ → ∞. By using#(N ∗ ∪ L∗ )/#L∗∗ ≤ (α + (1 − β)α)/(1 − α)(1 − β) for all N , the Law of Large Numbers
implies that(lim PrN →∞)Z̃(y) ≤ ε; a∗−i → 1

To see this, note that the variance of ŵi∗ (yi ) conditional on a∗i , denoted by Var(ŵi∗ (yi )|a∗i ),is ﬁnite because the optimal solution ŵi∗ to Problem (M-SB) is bounded as we have noted∑2∗∗
above. Thus limN →∞ N
n=1 (1/n )Var(ŵn (yn )|an ) < +∞ so that the Law of Large Numbers
is applied (see Sen and Singer (1993, p.67, Proposition 2.3.10)).
Thus we have
{ ()
Pr |Z̃(y)| < ε; a∗−i ui (−ε + h∗ )
N →∞
)}
(
) ( α + (1 − α)β∗∗∗∗∗(h − h ) + h+ Pr |Z̃(y)| ≥ ε; a−i ui(1 − α)(1 − β)= ui (−ε + h∗ ) > −∞.
lim

We take β ∈ (0, 1) and ﬁx it to satisfy
βui (ŵi∗ (y k ) + ρ) + (1 − β)ui (−ε + h∗ ) > ui (ŵi∗ (y k ))for all y k ∈ Y and all i ∈ L∗ . Such β ∈ (0, 1) exists because ρ > 0 and ui is strictlyincreasing.
For such a given β ∈ (0, 1), we can take a suﬃciently large N so that β(N ) → β and

 


∑
∑
1
W −
β(N )ui (wi (y k )) + (1 − β(N ))Ey−i ui 
ŵn∗ (yn ) −wj (yj ) a∗−i 
#L∗∗
∗
∗
n∈N

> ui (ŵi∗ (y k ))53

j∈Lfor each y k ∈ Y and each i ∈ L∗ . This completes Step 2-3.
Step 3. If necessary, we can take a small enough wi for i ∈ L∗ such that wi < wi (y k ) for eachy k and each i ∈ L∗ . We then consider wi (y k ) ∈ [wi , wi (y k )] for each y k ∈ Y (k = 1, 2, ..., K)
and each i ∈ L∗ .
By Step 2 above, we can ensure that the continuous mapping ϕki (w−i ) which we haveobtained in Step 1 lies in the compact set [wi , wi (y k )] for each k = 1, 2, ..., K and i ∈ L∗ .
We then deﬁne a continuous mappingϕ ≡ (ϕ1i , ..., ϕKi )i∈L∗from the compact set

∏

i∈L∗

∏K

k
k=1 [w i , w i (y )] to itself. Then there exists a ﬁxed point of ϕby the Brouwer’s ﬁxed point Theorem
w = ϕ(w)which can be written byβ(N )ui (wi (y k )) + (1 − β(N ))Ey−i ui 


1 
W−
#L∗∗

∑

ŵn∗ (yn ) −

n∈N ∗

∑
j∈L∗



wj (yj ) a∗−i = ui (ŵi∗ (y k ))for each y k ∈ Y (k = 1, 2, ..., K) and i ∈ L∗ . This is the desired result (A19). Each agenti ∈ L∗ thus obtains the same expected payoﬀ as that under the original contract wi∗ :β(N )Eyi [ui (wi (yi ))|ai ] + (1 − β(N ))Ey−i ui 

1 
W−
#L∗∗

=

∑

ŵn∗ (yn ) −

n∈N ∗

∑
j∈L∗





wj (yj ) |a∗−i  − Gi (ai )Eyi [ui (ŵi∗ (yi ))|ai , a∗−i ] − Gi (ai )for each action ai ∈ A. Thus agent i ∈ L∗ chooses the original third best action a∗i (becauseŵi∗ implements a∗i by its deﬁnition) and obtains the same expected payoﬀ as that attained
under the original third best contract {wi∗ , a∗i }i∈L∗ . Thus the principal can implement theoriginal actions a∗ ∈ AN from all agents.
Finally, by construction the total wage is given by (A18)∑∑W =Eyn [ŵn∗ (yn )|a∗n ] +(Eyn [ŵn∗ (yn )|a∗n ] + ρ)
n∈L∗

n∈L
/ ∗
∗

< W (a )∑∑=Ey [wn∗ (y)|a∗ ] +Ey [wn∗ (y)|a∗ ]
n∈L
/ ∗

n∈L∗

54



because Eyn [ŵn∗ (yn )|a∗n ] ≤ Ey [wn∗ (y)|a∗ ] for all n ∈ N and, in particular, Eyn [ŵn∗ (y)|a∗n ]+ρ <Ey [wn∗ (y)|a∗ ] holds for any n ∈ L∗ due to the deﬁnition of ρ > 0 (see (A17)). Thus
the principal can reduce the total wage payments, which shows that the original contract{wi∗ , a∗i }Ni=1 is not optimal to solve Problem TB, a contradiction. Q.E.D.

7.6

Proof of Proposition 5

Recall that â = (â1 , ..., âN ) denotes the second-best action proﬁle and ŵ = (ŵ1 , ..., ŵN ) the
second-best wage proﬁle, which solve Problem SB. As shown in Lemma 1, the second-bestwage scheme can be an IPE under the supposed utility functions of agents. Thus, without
loss of generality, we can suppose that ŵi depends only on agent i’s output yi , that is, ŵi (yi ).
We modify the proof of Proposition 4 by replacing the action a∗i by the second-best oneâi . We also replace the wage scheme wi∗ in Proposition 4 by the second-best one ŵn whichsolves Problem M-SB for implementing the second-best action âi from agent i. All agentsin N ∗ are oﬀered the second-best wage schemes ŵn implementing the second-best action ânwhile all others in N \ N ∗ are oﬀered the same random wage scheme as used in the proof of
Proposition 4:{
wi (yi , y−i ) ≡wi (yi )(1/#L∗∗ )w.p. β(N ){}∑∑W − n∈N ∗ ŵn (yn ) − n∈L∗ wn (yn ) w.p. 1 − β(N )Here we also modify the total wage W as follows
W ≡∑(Eyn [ŵn (yn )|ân ] + ρn )n∈Nwhere ρn ≡ ρ > 0 for n ∈ L∗ and ρn = 0 for n ∈/ L∗ respectively.
Then the same argument as the proof of Proposition 4 shows that the principal can attainthe total wage W for implementing the second-best action proﬁle â. To see this, we needto ﬁnd the wage schemes appeared in the ﬁrst line of the above expression {wi }i∈L∗ which
satisfyβ(N )ui (wi (y k )) + (1 − β(N ))Ey−i ui = ui (ŵi (y k ))


1 
W−
#L∗∗

∑
j∈L∗

wj (yj ) −

∑





ŵn (yn )

â−i 

n∈N ∗

(A21)

for each output y k ∈ Y (k = 1, 2, ..., K). The proof of Proposition 4 is modiﬁed as follows:
as in Step 2-1 we ﬁnd a continuous function ϕki (w−i ) which solves (A21) for wi (y k ), foreach output y k ∈ Y . Step 2-2 can be also adapted as well. To show Step 2-3 we set55

wi (y k ) ≡ ŵi (y k ) + ρ for agent i ∈ L∗ and each y k (k = 1, 2, ..., K) where ρ > 0 is a small
positive constant. Then, when N is large enough and β(N ) is taken to be close to β ≃ 1,
we can show that the left hand side of (A21) is larger than its right hand side evaluated atwi (y k ) = wi (y k ) + ρn for k = 1, 2, ..., K and i ∈ L∗ , completing Step 2-2. Finally, letting
ϕi ≡ (ϕki (w−i ))k=1,2,...,K , the continuous mapping ϕ = (ϕ1 (w−1 ), ..., ϕN (w−N )) deﬁned on
∏
∏
k
i∈L∗
k=1,2,...,K [w i , w i (y )] to itself has a ﬁxed point which corresponds to the desired
wage schemes. Here ρ > 0 can be small enough when N is suﬃciently large.
Then the average payoﬀ of the principal per agent is given by(1/N ){Ey [R(y)|â] − W } = (1/N ){Ey [R(y)|â] − Ŵ } − (1/N )

∑

ρ

n∈L∗

≥ (1/N ){Ey [R(y)|â] − Ŵ } − β(1 − α)ρ
≡ π̂ − εwhere ε ≡ β(1 − α)ρ > 0. We can take ε > 0 to be small enough when N → ∞ so that ρ > 0
is small enough. Q.E.D.

7.7

Proof of Proposition 6

We take the wage schemes {wiδ }N
i=1 which satisfy Assumption 5 and then deﬁne the total
wage asŴδ ≡N∑

Eyi [wiδ (yi )|âi ]i=1where Ŵδ → Ŵ as δ → 0.
Consider the following simple sharing rule for agent i: for each output proﬁle (yi , y−i ) ∈Y N,{w̃i (yi , y−i ) ≡ wiδ (yi ) + (1/N )

Ŵδ −

N
∑

}
wnδ (yn ).

n=1

Under the wage scheme w̃i deﬁned above, agent i obtains the following expected payoﬀ:
Ey [Ui (w̃i (yi , y−i ), ai )|ai , a−i ]where Ey [·|ai , a−i ] denotes the expectation over output proﬁles (yi , y−i ) ∈ Y N conditionalon an action proﬁle (ai , a−i ).
Our target is to implement the second best action proﬁle â = (â1 , ..., âN ) at the total
cost Ŵδ by using the wage scheme (w̃1 , ..., w̃N ) deﬁned above. Thus we need to show that
E[Ui (w̃i (yi , y−i ), âi )|âi , â−i ] ≥ E[Ui (w̃i (yi , y−i ), ai )|ai , â−i ]56(IC∗ )

for any ai ̸= âi , and(IR∗ )

E[Ui (w̃i (yi , y−i ), âi )|âi , â−i ] ≥ U i

When N is large enough, IC∗ is almost identical to the original IC in the second-best problem,
Problem SB, given in the main text. This is because the second term of w̃i has only negligible
impacts on the agent i’s expected payoﬀ when N → ∞ due to the Law of Large Numbers.
We also need to check that, given (ai , â−i ) for ai ̸= âi (only agent i deviates to choose
ai ̸= âi ), the same limit argument holds. Also, when N is large enough, agent i faces thealmost same expected payoﬀ as what he obtains under the second-best wage scheme so thatIR∗ is satisﬁed for a suﬃciently large N . We will show these results below.
Since |wiδ (yi ) − ŵi (yi )| < δ for all yi ∈ Y and the second best wage scheme ŵi is bounded,wiδ is bounded as well. Thus, its variance Var(wiδ (yi )|âi ) conditional on âi is ﬁnite. Thisimplies that
lim

N
∑

N →∞

because∑∞n=1(1/n2 )Var(wiδ (yi )|âi ) < +∞n=1

(1/n2 ) < 2. Then, the Law of Large Numbers shows that{
(1/N ) W δ −N∑}wiδ (yi )→0

n=1

in probability when N → ∞ (see Sen and Singer (1993, p.67, Proposition 2.3.10)).
Deﬁne{Z(y) ≡ (1/N ) W δ −N∑}wnδ (yn ).

n=1

By the above argument the probability that the event of |Z(y)| ≥ ε occurs conditional on
the second best action proﬁle (âi , â−i ) must be
Pr(|Z(y)| ≥ ε; âi , â−i ) → 0
as N → ∞, given ε > 0.
We also denote by
Ey [ · |ai , â−i , |Z(y)| < ε]the expectation over y ∈ Y N conditional on an action proﬁle (ai , â−i ) and the event that
|Z(y)| < ε occurs.

57Under the wage scheme deﬁned above, agent i obtains the following expected payoﬀ:
Ey [Ui (w̃i (yi , y−i ))|ai , â−i ][ ()]= Ey Ui wiδ (yi ) + Z(y), ai |ai , â−i
= Pr(|Z(y| ≥ ε; ai , â−i )Ey [Ui (wiδ (yi ) + Z(y), ai )|ai , â−i , |Z(y)| ≥ ε]
+ Pr(|Z(y| < ε; ai , â−i )Ey [Ui (wiδ (yi ) + Z(y), ai )|ai , â−i , |Z(y)| < ε]Then we can show that the above limit argument still holds even when we replace (âi , â−i )by (ai , â−i ) (thus only agent i’s action is changed from âi to any ai ∈ A). That is,lim Pr (|Z(y)| ≥ ε; ai , â−i ) → 0

N →∞for any ai ∈ A. To see this, deﬁne∆i (ai ) ≡ Eyi [wiδ (yi )|âi ] − Eyi [wiδ (yi )|ai ]for any ai ∈ A and let| max ∆i (ai )|/N < ε/2ai ∈A

by taking a large N . Then, for an arbitrary η > 0, we can ﬁnd a large enough N to ensurethat
Pr (|Z(y)| ≥ ε; ai , â−i )
(
)
N
N
∑
1 ∑
δ
δ
= Pr
Eyn [ŵn (yn )|ân ] −wn (yn ) ≥ ε; ai , â−i
N
n=1
n=1


N
N
∑
∑
1
= Pr 
wnδ (yn ) ≥ ε; ai , â−i 
Eyn [ŵnδ (yn )|ân ] + Eyi [wiδ (yi )|ai ] + ∆i (ai ) −
N
n=1
n̸=i


N
∑
∑
1
1
≤ Pr 
Eyn [wnδ (yn )|ân ] + Eyi [wiδ (yi )|ai ] −wnδ (yn ) +∆i (ai ) ≥ ε; ai , â−i 
N
N
n=1
n̸=i


N
∑
∑
1
≤ Pr 
Eyn [wnδ (yn )|ân ] + Eyi [wiδ (yi )|ai ] −wnδ (yn ) ≥ ε/2; ai , â−i 
N
n=1

n̸=i

≤ η
where the last inequality follows from the Law of Large Numbers.
Deﬁnew ≡ max wiδ (y), w ≡i∈N,y∈Y58

min wiδ (y).

i∈N,y∈Y{

Then we havew − w ≥ Z(y) ≡ (1/N ) Ŵδ −N∑}wnδ (yn )≥ w − w.

n=1

We take Γ > 0 and γ > 0 such thatmin Ui (2w − w, a) ≥ −γi∈N,a∈AandΓ > max Ui (2w − w, a)i∈N,a∈A

Then we can show that
Pr(|Z(y)| ≥ ε; ai , â−i )Ey [Ui (wiδ (yi ) + Z(y), ai )|ai , â−i , |Z(y)| ≥ ε]
≥ Pr(|Z(y)| ≥ ε; ai , â−i )Ui (w + (w − w), ai )≥ −ηγwhen N → ∞.
Thus, when N is large enough, agent i obtains at least the following expected payoﬀ bychoosing the second best action âi :Pr(|Z(y)| ≥ ε; âi , â−i )Ey [Ui (wiδ (yi ) + Z(y), âi )|âi , â−i , |Z(y)| ≥ ε]
+ Pr(|Z(y)| < ε; âi , â−i )Eyi [Ui (wiδ (yi ) − ε, ai )|âi ]≥ −ηγ + Pr(|Z(y)| < ε; âi , â−i )Eyi [Ui (wiδ (yi ) − ε, âi )|âi ]= Eyi [Ui (wiδ (yi ), âi )|âi ] + Ui∗
whereUi∗ ≡ −ηγ + Pr(|Z(y)| < ε; âi , â−i )Eyi [Ui (wiδ (yi ) − ε, âi )|âi ] − Eyi [Ui (wiδ (yi ), âi )|âi ]which converges to zero as N → ∞ so that η → 0 and ε → 0.
On the other hand, when agent i deviates to choose ai ̸= âi , he can obtain at mostPr(|Z(y)| ≥ ε; ai , â−i )Ey [Ui (wiδ (yi ) + Z(y), ai )|ai , â−i , |Z(y)| ≥ ε]
+ Pr(|Z(y)| < ε; ai , â−i )Eyi [Ui (wiδ (yi ) + Z(y), ai )|ai ]≤ Pr(|Z(y)| ≥ ε; ai , â−i )Ui (2w − w, ai )+ Pr(|Z(y)| < ε; ai , â−i )Eyi [Ui (wiδ (yi ) + ε, ai )|ai ]≤ Pr(|Z(y)| ≥ ε; ai , â−i )Γ + Pr(|Z(y)| < ε; ai , â−i )Eyi [Ui (wiδ (yi ) + ε, ai )|ai ]≤ ηΓ + Pr(|Z(y)| < ε; ai , â−i )Eyi [Ui (wiδ (yi ) + ε, ai )|ai ]≤ Eyi [Ui (wiδ (yi ), ai )|ai ] + Ui∗∗
59

where{}Ui∗∗ ≡ ηΓ + max Pr(|Z(y)| < ε; ai , â−i )Eyi [Ui (wiδ (yi ) + ε, ai )|ai ] − Eyi [Ui (wiδ (yi ), ai )|ai ]a∈Awhich converges to zero when N → ∞ because
lim Pr(|Z(y)| < ε; ai , â−i ) → 1

N →∞together with ε → 0 and η → 0.
Now we take δ̃ > 0 such thatEyi [Ui (ŵiδ (yi ), âi )|âi ] ≥ Eyi [Ui (wiδ (yi ), ai )|ai ] + δ̃,for any ai ̸= âiandEyi [Ui (ŵiδ (yi ), âi )|âi ] ≥ U i + δ̃Such δ̃ > 0 exists due to the strict inequalities of IC and IR under wiδ .
Then, when N is large enough so that Ui∗∗ − Ui∗ ≤ δ̃ for all i, we can ensure thatEyi [Ui (wiδ (yi ), âi )|âi ] ≥ Eyi [Ui (wiδ (yi ), ai )|ai ] + δ̃≥ Eyi [Ui (wiδ (yi ), ai )|ai ] + Ui∗∗ − Ui∗for all ai ̸= âi . Then IC∗ is satisﬁed.
Also agent i obtains at least the following expected payoﬀ by choosing âi :Eyi [Ui (wiδ (yi ), âi )|âi ] + Ui∗ .
When N is so large that |Ui∗ | ≤ δ̃ for all i, we have
Eyi [Ui (wiδ (yi ), âi )|âi ] + Ui∗ ≥ Eyi [Ui (wiδ (yi ), âi )|âi ] − δ̃≥ Ui
showing that the individual rationality constraint IR is satisﬁed as well.
Thus we have established the result that the principal can approximately attain thefollowing total wage for implementing the second best actions â from all agents:(1/N )Wδ

≡≤(1/N )(1/N )

N
∑
n=1
N
∑

Ey [wiδ (y)|ân ]Ey [ŵn (y)|ân ] + δ

n=1=(1/N )Ŵ + δ→ (1/N )Ŵ
where δ → 0 as N → ∞. Q.E.D.
60

8

Appendix B: Extensions

8.1

Monotonicity of the Third-Best Contract

First, we show that the third-best contract wi (y) shown in Lemma 2 is non-decreasing in yi
but non-increasing in y−i when MLRP and CDFC are satisﬁed.
Proposition B1. Suppose that MLRP and CDFC are satisﬁed. Then the third best contract
wi (y) = Qi (ξi (yi ) − ξ1 (y1 ), ..., ξi (yi ) − ξN (yN )) shown in Lemma 2 is non-decreasing in yi
and non-increasing in yj for j ̸= i.
Proof. It is readily veriﬁed that wi is non-decreasing in yi and non-increasing in y−i under
MLRP when µl (a) > 0 holds only for a < âl for all l ∈ N because ξi is non-decreasing in
yi under MLRP. It thus suﬃces to show that the third best contract solving Problem RP
without upward IC constraints actually satisﬁes upward IC constraints, that is, IC is neverbinding at any a > âi for each agent i.
We can re-write the expected payoﬀ of agent i as
EUi (ai ) ≡∑P (y−i |â−i )∆ui (y k+1 , y−i )(1 − Fi (y k |ai )) − Gi (ai )y−iwhere, since wi is non-decreasing in yi , we have∆ui (y k+1 , y−i ) ≡ ui (wi (y k+1 , y−i )) − ui (wi (y k , y−i )) > 0
for each output y k , k = 1, 2, ..., K. Since IC must be binding for some a′ < âi (otherwise,
wi is non-increasing in yi under MLRP so that agent i never chooses âi > 0), if IC is alsobinding at some a′′ > âi , then we have EUi (âi ) = EUi (a′′ ) = EUi (a′ ). However, since someα ∈ (0, 1) exists such that âi = αa′′ + (1 − α)a′ , CDFC and strict convexity of Gi imply that
EUi (âi ) > αEUi (a′′ ) + (1 − α)EUi (a′ ) = EUi (âi ), a contradiction. Q.E.D.

8.2

Third-Best Contract with CARA Utility Functions

In the main text we have assumed that all agents are risk averse and have the CARA utility
function as ui (w) = (1/ri )(1 − e−ri w ) for ri > 0. In this subsection we modify the third-best
contract when there is one risk neutral agent while all others are risk averse (recall that weare assuming that there is at most one risk neutral agent in Section 4).
Suppose that agent 1 is risk neutral and his utility function is given by w − G1 (a) while
any other j ̸= 1 has the CARA utility function. Then equation (A4) is modiﬁed for agent
61

1 and any other j ̸= 1 as −rj wj (y) = ξ1 (y1 ) − ξj (yj ). By dividing both sides of this by rj ,summing this over all j ̸= 1 and using FTW, we obtainw1 (y) = W +∑(1/rj )(ξ1 (y1 ) − ξj (yj )).
j̸=1

When N = 2 so that i = 1, 2, we have w2 (y) = W − w1 (y) = (1/r2 )(ξ2 (y2 ) − ξ1 (y1 )).
Now suppose that N ≥ 3. For any pair of risk averse agents i ̸= j where i, j ̸= 1, we still
have (A4) from which we can derivewi (y) = ∑




1/ri
W − w1 (y) +l̸=1 (1/rl ) ∑(ξi (yi ) − ξj (yj ))

j̸=i,1by summing (A4) over all j ̸= i, 1 and using FTW: W =∑l̸=1 wl (y) + w1 (y). Then, we cansubstitute the above w1 (y) into this expression to obtainwi (y) = (1/ri )(ξi (yi ) − ξ1 (y1 ))for each i ̸= 1. Thus, all but risk neutral agent should be compensated according to only
the diﬀerence between his own piece rate ξi (yi ) and the risk neutral agent’s one ξ1 (y1 ).
These optimal contracts can be also obtained by letting r1 → 0 while keeping ri > 0 for
any i ̸= 1 in the expression of the simple sharing rule wi (y) stated in Proposition 2 as well.
Next, we extend the result about the third-best contract with CARA utility functions(Proposition 2) to allow agents to choose continuous actions.
We suppose that agent’s action is given by A = [0, ∞) and maintain the assumption that
Y is ﬁnite. We slightly change notations which we have used in the main text by lettingpi (yi |ai ) denote the probability of agent i’s output being realized as yi . Here superscript i
denotes an index of agent whereas we have used subscript i to denote it in the main text. We
assume that the action cost Gi (a) and the probability of output pi (yi |a) are all continuously
diﬀerentiable functions of action ai ∈ A. In particular we assume that Gi is increasing andstrictly convex, G′ > 0 and G′′ > 0 for all a > 0. We also denote by pia (y|a) ≡ ∂pi (y|a)/∂athe partial derivative of pi with respect to agent i’s action ai .
We make the following conditions which are continuous versions of MLRP and CDFC.
Condition (MLRP∗ ). pia (y|i|ai )/pi (yi |ai ) is increasing in yi for each agent i and actionai ∈ A.

62

Condition (CDFC∗ ). Let F i (y|a) =∑iiz≤y p (z|a) denote the CDF of p (y|a).
i ≥ 0..
F i (y|a) is a convex function of a ∈ A, that is, Faa

Then

We then show the following result.
Proposition B2. Consider the continuous action model given above and suppose that
MLRP∗ and CDFC∗ are satisﬁed. Suppose also that agents’ preferences are represented
by CARA utility functions as in the main text. Then the third-best contract is given as
follows{

1/ri

wi (y) = (1/ri )ψi (yi ) + ∑N

l=1 (1/rl )

W−

N
∑

}
(1/rl )ψl (yl )

l=1

))
(
(
pi (yi |a∗i ).
ψi (yi ) ≡ ln λi + µi 1 − aip (yi |a∗i )whereProof. We replace the incentive compatibility constraint for implementing the third best
action a∗i∑∑∑∑
P (y−i |a∗−i )pi (yi |âi )ui (wi (y))−Gi (a∗i ) ≥P (y−i |a∗−i )pi (yi |ai )ui (wi (y))−Gi (ai )
y−i

yi

y−i

yi

(IC)
by the corresponding ﬁrst order condition:∑∑
P (y−i |a∗−i )pia (yi |a∗i )ui (wi (y)) − G′i (a∗i ) = 0
y−i

(FIC)

yi

whenever a∗i > 0. In what follows we assume that a∗i > 0 for all i and discuss the case ofa∗i = 0 later.
We further replace the above FIC by the weaker one:∑∑
P (y−i |a∗−i )pia (yi |a∗i )ui (wi (y)) − G′i (a∗i ) ≥ 0y−iyiThen we consider the following doubly relaxed problem:
Problem D-RP
min W
subject to FIC′′ , IR and FTW′′ :
W ≥

N
∑

wi (y) ∀ y ∈ Y N .

i=1

63

(FIC′′ )Here, the above problem is relaxed from the original third best problem, Problem TB, in two
respects: one is that IC is doubly relaxed by FIC′′ and the other is that FTW is replaced
by a weaker constraint FTW′′ .
Lemma B1. The Slater condition is satisﬁed in Problem D-RP.
Proof. We can change variables as ui (y) ≡ ui (wi (y)) for each y ∈ Y N and wi (y) = ϕi (ui (y))
for ϕi ≡ u−1
in the above problem, Problem D-RP. Since MLRP holds, there exists someiŷi such that pia (y|a∗i ) > (<)pia (y|a∗i ) for all y > (<)ŷi . Then we can consider an individual
utility scheme ui (yi ) such that ui (yi ) takes a large positive value for any yi > ŷi while ittakes a large negative value for any yi < ŷi . Then we can ensure that∑pia (yi |a∗i )ui (yi ) +yi >ŷi∑pia (yi |a∗i )ui (yi ) > G′i (a∗i ).

yi <ŷi

Thus FIC′′ is satisﬁed as a strict inequality. By adding a large constant v to ui (yi ), (ui (yi ) +v)yi ∈Y can satisfy IR as a strict inequality as well. Finally, by taking a large enough W ,FTW′′ is strictly satisﬁed given the above wage scheme. Q.E.D.
By Lemma B1, the necessary condition for the optimal solution to Problem D-RP mustsatisfy the following KKT conditions: there exist some non-negative λi ≥ 0, µi ≥ 0 andη(y) ≥ 0 such that
−1 +∗P (y|a )u′i (wi (y)){∑η(y) = 0,

(B1)

y∈Y N

)}
(
pia (yi |a∗i )− η(y) = 0.
λi + µi 1 − i
p (yi |a∗i )(B2)Then we can show
Lemma B2. η(y) > 0 for all y ∈ Y N .
Proof. This is shown by a similar argument to Lemma A2. Q.E.D.
By Lemma B2, we obtain the following formula for the optimal contract which solves
Problem D-RP:
1/riwi (y) = (1/ri )ψi (yi ) + ∑N

l=1 (1/rl )

64

{
W−

N
∑
l=1

}
(1/rl )ψl (yl )))((pia (yi |a∗i )ψi (yi ) ≡ ln λi + µi 1 − i.
p (yi |a∗i )whereHere, ψi is non-decreasing in yi under MLRP.
Finally, we can check that FIC′′ holds as equality and agent’s expected payoﬀ function
is a concave function of his action. Thus the relaxed problem D-RP is exactly same as theoriginal problem, Problem TB. Note ﬁrst that, if FIC′′ holds as a strict inequality, µi = 0
must be satisﬁed. This yields ψi (yi ) = ln λi so that wi (y) is independent of yi , that is,∑
wi (y) = wi (y−i ). Then, since yi ∈Y pia (y|a) = 0 holds, agent i would choose ai = 0 and
hence

∑

P (y−i |a∗−i )ui (wi (y−i ))y−i∑pia (yi |a∗i ) − G′i (a∗i ) = −G′i (a∗i ) ≤ 0,yicontradicting to our supposition that FIC′′ holds as a strict inequality. Second, note thatagent i’s expected payoﬀ is given by
∑
y−i

=

P (y−i |a∗−i )∑∑pi (yi |a)ui (wi (yi , y−i )) − Gi (a)yi

P (y−i |a∗−i )∆ui (y k , y−i )(1 − F i (y k |a)) − Gi (a)y−iwhere ∆ui (y k , y−i ) ≡ ui (wi (y k , y−i )) − ui (wi (y k−1 , y−i )) ≥ 0 for each k = 1, 2, ..., K because
wi and ψi are non-decreasing in yi . Since F i is convex with respect to action ai due to
CDFC∗ , the above expected payoﬀ is a concave function of ai .
Finally, we treat the case that a∗i = 0 for some agent i. This is stated as the following
IC constraint:∑

P (y−i |a∗−i )pia (yi |0)ui (wi (y)) − G′i (0) ≤ 0.

y−i

Letting µ0i ≥ 0 be the Lagrange multiplier for this constraint, we can modify the piece rate
wage appeared in the optimal contract for agent i as follows)(
i00 pa (yi |0)ψi (yi ) ≡ ln λi − µi i
p (yi |0)which is non-increasing in yi under MLRP. Thus, wi is non-increasing in yi so that agent i’s
expected payoﬀ is always non-increasing in his action ai . This implies that agent i optimallychooses the least costly action ai = 0. Q.E.D.

65