J. Service Scie nce & Management, 2009, 3: 149-160
doi:10.4236/jssm.2009.23018 Published Online September 2009 (www.SciRP.org/journal/jssm)
149
Joint Contract under Inequity Aversion
Guangxing WEI, Yanhong QIN
School of Management, Chongqing Jiaotong University, Chongqing, China.
Received March 5th, 2009; revised May 7th, 2009; accepted June 12th, 2009.
ABSTRACT
The standard contract theory adopts the traditional hypothesis of pure self-interest. However, a series of game experi-
ments have proven that people are not any self-interest but also inequity averse. Then, how will the inequity aversion
influence the optimal contract for multiple agents? This paper attempts to obtain new theoretical insights by incorpo-
rating inequity aversion into the standard frame of optimal contract design. The optimal contract under relatively weak
inequity aversion is found to be the relative joint contract, by which pa yment to each independen t agent increases with
his own output and others’ and agents with higher output will be pa id more, while what under strong, even very strong,
inequity aversion is the egalitarian joint contract, by which payment to each independent agent is always equal and
hence agents with lower output will not be paid less. Moreo ver, it is shown that the inequity aversion results in incen-
tive efficiency losses as agents with inequity a version will suffer d isutility in fa ce of unfair allo catio n. Consequently, the
principal has to pay additiona l inequity rent and risk compensation for inequity aversion to th e agents, which both are
the incentive efficiency losses resulted from inequity aversion and have never been explored by the standard contract
theory, besides info rmation rent and risk comp ensation for asymmetric information, which both have been probed in the
standard contract theory deeply. In this way, this paper designs the optimal contracts for multiple agents with more
realistic assumpt ion s an d hence can explain real ec on o mic beh avio rs mo r e pro perly .
Keywords: inequity aversion, contract design, optimal contract, principal-agent, contract theory
1. Introduction
The hypothesis of pure self-interest has been proved cor-
rect in many situations and has been highly useful in
designing optimal incen tive co n tracts. It is normal for the
standard contract theory to assume that the agent maxi-
mizes his profit to the extent of his receipts minus the
cost of private effort while the principal maximizes his
wealth to the extent of profits generated by agents minus
payments made to the agents. However, in recent years, a
series of game experiments such as ultimatum game and
dictator game [1], trust game [2], gift exchange game [3]
and public good g ame [4], have been proving that not all
people are motivated exclusively by pure self-interest
considerations, which are reviewed in [5] and [6]. Actu-
ally, many people are inequity averse and prefer more
fair allocation. Both inequity aversion and self-interest
preference affect behaviors, but sometimes their effects
are not consistent. For example, people pursuing inequity
aversion will sacrifice some of their own material pay-
offs in order to realize a more fair allocation while peo-
ple pursuing pure self-interest won’t. When the related
material payoffs are relatively large, the inequity aver-
sion is the dominating factor in deciding individual be-
havior, while in case of relatively small, the self-interest
is the dominating. Some new theoretical models, such as
[7,8,9] and [10], have been developed to explain various
experimental results by incorporating inequity aversion
into the framework of utility maximization.
Then, how will the inequity aversion influ ence the op-
timal contract for multiple agents? This paper tries to
design the optimal contract for multiple agents with in-
equity aversion and further examines the influence of
inequity aversion on incentive efficiency in the approach
of FS Model developed by [7] with the improvement of
replacing the assumption of risk neutrality with risk
aversion. The optimal contract under inequity aversion is
found to be the joint contract by which payment to each
independent agent must depend on both his own output
and others’, instead of the independent contract which is
the optimal contract given by standard contract theory
and pays each independent agent according to his own
output. To be concrete, the optimal contract under rela-
tively weak inequ ity aversion is the relative joint contract,

GUANGXING WEI, YANHONG QIN 150
by which payment to each independent agent increases
with his own output and others’ and agents with higher
output will be paid more, while that that under strong,
even very strong, inequity aversion is the egalitarian
joint contract, by which payment to each independent
agent is always equal and hence agents with lower output
will not be paid less. Moreover, it is shown that the in eq-
uity aversion adds a new incentive constrain and surely
results in incentive efficiency losses because agents with
inequity aversion will suffer disutility in face of unfair
allocation. As a result, the princip al has to pay additional
inequity rent and risk co mpensation for in equ ity aversion
to the agents, which both are the incentive efficiency
losses resulted from inequity aversion and have never
been explored by the standard contract theory, besides
information rent and risk compensation for asymmetric
information, which both have been investigated deeply in
the standard contract theory.
There are some existing related literatures. Two brief
but good recent surveys are [11] and [12]. The optimal
contract for multiple agents showing jealousy, which is
one side of inequity aversion, is analyzed in [13] and
[14], while [15,16,17,18,19] and [20] studied the incen-
tive contract when agents exhibit inequity aversion or
some other forms of social preferences by restricting the
class of contracts to linear schemes. Literature [15, 16]
focused on the case of a single agent, while [17,18,19]
and [20] examined that of multiple agents. Tournament
amongst agents with inequity aversion is considered in
[21] and [22], while [23,24] and [25] probed the influ-
ence of inequity aversion on team incentives. Further-
more, [26,27] and [28] studied peer pressure, a special
form of social preference similar to inequity aversion. As
the most relevant literature, [29] established the optimal
contract for multiple agents showing inequity aversion
with assumption of risk neutrality and limited liability
constraints, while this paper assumes that agents with
inequity aversion are also risk averse and there is no lim-
ited liability constraint. Thus, some forms of contracts
outlined by [29], are unrealistic, while contracts offered
in this paper are all practicable. Furthermore, [29] didn’t
investigate the influence of inequity aversion on incen-
tive efficiency, while this paper analyzes it and con-
cludes that inequity aversion results in incentive effi-
ciency losses, including inequity rent and risk compensa-
tion for inequity aversion.
The remainder of this paper is organized as follows.
Section 2 briefly summarizes the theory of inequity
aversion. Section 3 presents the basic model and Section
4 provides the solution. Section 5 analyzes the optimal
joint contract. Section 6 studies th e influence of inequity
aversion on incentive efficiency. A numerical example is
given in Sectio n 7. Fi nal ly, Section 8 dra ws c onclusion.
2. Theory of Inequity Aversion
A lot of game experiments and literatures argue that
people often care about material payoffs of others, dis-
like unfair allocation, and reciprocate kind or unkind
behaviors of others. Generally speaking, these behaviors
are motivated by social preferences. There are two ap-
proaches to describe the social preferences: the distribu-
tional approach and the motivational approach, which are
reviewed in [30]. The distributional approach, such as
that of [7] and [8], is concerned only with effects of ac-
tions on final allocations. It neglects intentions behind
behaviors and solely focuses on fina l allocations but still
fares well in explaining observed experimental results
while remaining quite simple and tractable. The motiva-
tional approach, such as that of [9,10,31] and [32], pays
attention to the intentions behind behaviors and tries to
actually model reciprocity, by which any friendly action
will be returned and an y spiteful action will be retaliated .
It is certainly closer to a realistic modeling of human
behavior, but is not analytical ly tractable. As one kind of
the distributional approach, the theory of inequity aver-
sion emphasizes preference for fair allocation. When his
material payoff is below others’, the agent su ffers j ealous
disutility, while when his material payoff is above oth-
ers’, he suffers compassion disutility. The sum of dis-
utility arising from jealousy and compassion is defined
as inequity aversion disutility. The theory of inequity
aversion, especially the FS Model proposed by [7], is
reasonable enough, simple and quite tractable. Therefore,
it is widely accepted and applied. In this paper, we also
adopt the FS Model, expressed by
max( ,0)
1
i
ii ji
ji
ux xx
n


 

max( ,0)
1
iij
ji
xx
n



 (1)
where ui denotes the utility of i under the assumption of
risk neutrality, xi represents the material pa yoff of i while
xj illustrates that of j, n denotes the number of agents in
the reference group,
max( ,0)
1
iji
ji
xx
n



and
max( ,0)
1
iij
ji
xx
n



illustrate the jealous disutility and compassion disutility
respectively, where i

and i

denote the degree of
jealousy and that of compassion respectively with the
constraint ii


 and 01 
i

. Here, ii


 im-

GUANGXING WEI, YANHONG QIN 151
plies that the jealous disutility resulted from disadvanta-
geous inequity is greater than the compassion disutility
originated from advantageous inequity, while 01  i

implies that although the agent with inequity aversion
suffers compassion disutility, he still prefers more. Par-
ticularly, 0ii


denotes pure self-interest.
Although the FS Model solely focuses on the final al-
location and ignores the behavioral intentions, it can ex-
plain almost all the experimental results and capture
many reciprocal behaviors. Therefore, it is widely ap-
plied. However, it is based on the assumption of risk
neutrality and, thus, is not practicable to some extent. For
example, we usually feel more jealous when receiving a
payment of 600 dollars while our colleagues receive 700
than when receiving 6000 while our colleagues receive
6100. But the FS Model supposes that we feel the same
degree of jealousy and hereby suffer the same jealous
disutility in the above two cases, which is obviously not
in accordance with normal feelings. Consequently, in
this paper, we improve the FS Model and suppose that in
order to judge whether the allocation is fair or not, each
agent compares the utility derived from his payment with
others’ in the reference group one by one, instead of di-
rectly comparing the payment, which is the assumption
of the FS Model. In this way, the revised FS Model,
which is based on risk aversion, can be denoted as
()max[() ()
1
i
iiijj ii
Uuxux ux
n

 
,0]
j
max[ ( )(),0]
1
ji
iii j j
ji
ux ux
n





)( ii xu i
i
x)( jj xu x
(2)
where represents the utility of , derived from
his payment , and illustrates the utility of ,
derived from his payment .
j
It is clear that the revised FS model is more practica-
ble than the FS Model because the agent surely suffers
more jealous disutility when he receives a payment of
600 dollars while his colleagues receive 700 than when
he receives 6000 while his colleagues receive 6100,
which is in accordance with normal feelings. Further-
more, the revised model is more general than the FS
Model. In fact, the FS Model is a special case of the re-
vised model.
3. The Basic Model
The following models the interaction between a risk
neutral, pure self-interest, profit maximizing principal
and two symmetric risk averse, utility maximizing agents
A and B, who are inequity averse towards each other and
engage in Task 1 and Task 2 separately. The degrees of
jealousy and compassion are denoted as

and

respectively, subject to


 and 01 

. Each
agent can choose an effort from the set to
exert, where low effort costs 0 while high effort
costs . When effort is made, the task generates
high output with probability , and low output xh
with probability
}
He
,{ Leee
L
i
e
e
i
H
ch
xi
p
p

1, which follow the constraint
10



H
Lpp
w
. The reservation utility of each agent is
supposed to be zero. The timing of the game is as fol-
lows. First, the principal offers a contract to the agents.
Second, the agents decide whether to accept or reject the
contract. If the contract is rejected, the game ends and
each agent receives zero, the reservation utility. Third,
after accepting the contract, the agents exert efforts si-
multaneously. Finally, the outputs of the tasks are real-
ized and the transfers are made.
The contract is represented by,
in which), ll
w,, lhhlhh www(W
) l,hb,( a
ab

denotes the payment to

(i
and denotes the payment toby the con-
tract, when the output of iis and the output of
is . If alah
),BA
jx
ba
w
w
ij 
a
x
bw

, it is an independent contract, by
which the payment to each agent depends on only his
own output. On the contrary, if alah , it is a joint
contract, by which the payment to each agent depends on
both his own outp ut and oth er s ’ output.
w
,0]
( )e
w
0]
ab
uw

Therefore, by (2), the utility of each agent can be de-
noted as
()
max
ab
wm
(ab uw
 ax
)
[()
( ),
ba
ba
uw
uw
()
[ 
)(
Uu
c (3)
where

u meets the conditions u, 0''' 0

u and
0)0(

u)u , represents the utility of one agent (A
or B) derived from his payment, illustrates that
of the other agent, and
(ab w)),() ab
wu
(wu(wu ba
ba ]0max[

de-
notes the jealous disutility, baab ]0),()wu(umax[ w

is
the compassion disutility, is the cost of effort and
)(ec
0)(

L
ec c and ce
H

)(
2
.
If both A and B make high effort, the expected utility
of each agent, derived from the payment, is
2
(|)()
()](1
(1 )[(
)()
)
W
H
H
e
(|
HH
e
H
p
(1
(1







hh
lh
w p

)(
[ ()
)(
[ ()
HH
hl
HH
lh
pp
uw
pp
uw




H
H
p
)
(
)
(
uw
uw
H hl
ll
pu w
uw

)],
)],
lh hllh
hl hl lh
w w
w w


EU
F
EU
e
)e
u
uw (4)
And the expected inequity aversion disutility of each
agent can be represented by

(5)
So, the expected utility of each agent can be denoted
as

GUANGXING WEI, YANHONG QIN 152
(|)(|) (|)
WF
HHHH HH
EUe eEUe eEUeec (6)
where )|(
H
H
WeeEU illustrates the expected utility
derived from the payment, )|(
H
H
FeeEU c is the ex-
pected inequity aversion disutility and is the cost of
high effort . H
e
On the other hand, if
)
,
(
B
A
i
H
e makes low effort
while makes high effort , the expected utility
of , derived from the payment, is illustrated as
L
e
ij 
i
(| )()
(1 ) ()
WLHLH hh
(1 )( )
(1)(1)()
LH hl
LH lh
L
Hll
pp
uw
ppuw

 
EUeeppu w
ppuw


(1)[ ()()]
(1)[ ()
()],
LHhllh
LH hl
ppuw uw
pp uw
uw ww








(7)
And the expected inequity aversion disutility of
can be represented by i
(| )(1)[ ()()]
(1)[ ()
()],
lh hl lh FLH LHlhhl
LH lh
hl hl lh
EUeeppuwuw
pp uw
uw w w








L
eH e
(8)
So, the expected utility of the agent who makes low
effort while the other makes high effort can be
denoted as
(| )(| )(| )
WF
L
HLHL
EUe eEUeeEUe e
H
(9)
where )|( H
L
WeeEU is the expected utility derived
from the payment and )|(
H
L
FeeEU is the expected
inequity aversion disutility.
The principal designs the optimal contract subject to
the participation constraint and the incentive compatibil-
ity constraint to maximize the expected profit, which is
illustrated as
[P1]
2
,,,
min(1 )()(1 )
hh hl lh ll
2
H
hhHH hllhHll
wwww
pwpp wwpw
()(|)0PC EU ee
.() (|)(|)
HH
HH LH
st
I
CEUeeEUe e




2()(1 )[()()]
Hhh HHhllh
puwpp uwuw
c

 
where PC denotes the participation constraint and IC
illustrates the incentive compatibility constraint. The
next section solves [P1], and hereby derives the optimal
contract for multiple agents with inequity aversion.
4. Solutions to the Model
In order to solve [P1], and obtain the optimal contract,
the following lemma is necessary, where a contract is
feasible if it satisfies the participation constraint and the
incentive compatibility constraint.
LEMMA: For every feasible contract with
, there exist a feasible contract W with
where the expected payment to each agent by
contract equals that by contract .

W
 lhhl ww
lhhl ww 
W W
Proof of Lemma : Contract is feasible and here-
by satisfies both the participation constraint and the in-
centive compatibility constraint. From (4), (5), (6), PC
and , the participation constraint for contract
can be denoted as

W
 lhhl ww

W
2
(1 ) ()
(1)()[ ()()]
Hl l
HHlh hl
pu w
pp uwuw 






hl
w

(10)
From (4), (5), (6), (7), (8), (9), IC and , the
incentive compatibility constraint for contract can
be represented by

lh w

W
[()( )]
(1)[()) ()]
[(1)][()(
Hhh lh
Hhlll
HHlh
HL
puw uw
puw uw
ppuwuw
c
pp





 


)]
hl
(11)
The expected payment to each agent by contract
is

W
2
2
(1 )()
(1 )
H
hhHH hllh
Hll
EWp wppww
pw



 


c
(12)
Contract is defined by ,,
and . Obviously, it satisfies,
and the expected payment to each agent by contract
equals that by contract . Therefore, if contract
is feasible, the lemma is proved. From (4), (5), (6),
PC and , the participation constraint for con-
tract is
W
lh
w
w

hhhh ww  llll ww
lhhl ww 

lhhl ww
W
Ww
W

hl w
lh

W
hl
2
2
()(1 )[()()]
(1)()
(1)()[ ()()]
Hhh HHhllh
Hll
HHhllh
puwpp uwuw
puw
pp uwuw 
 


 
(13)
According to the definition of contract and (10),
it is clear that (13) holds. Thus, contract satisfies the
participation constraint. And from (4), (5), (6), (7), (8),
(9), IC and , the incentive compatibility con-
straint for contract can be denoted as
W
W
lhhl ww 
W

GUANGXING WEI, YANHONG QIN 153
[()( )]
(1)[ ()()]
[(1)][()(
Hhh lh
Hhlll
HHhl
HL
puw uw
puw uw
ppuwuw
c
pp


 


)]
lh
(14) (14)
which by the definition of contract is equivalent to which by the definition of contract is equivalent to
WW
[()( )]
(1)[ ()()]
)]
Hhh hl
Hlh ll
hl
puw uw
puw uw



 
[(1)][()(
HH lh
HL
pp uwuw
c
pp



 

(15)
Subtracting the left-hand side of (11) from the
left-hand side of (15),
() ()()[() ()]
(1)[ ()()]0
lh hllh hl
lh hl
uw uwuw uw
uw uw


 

 
  (16)
where



u and because and . The right-hand side of (11) is the
same as the right-hand side of (15). Then, (14) holds.
Therefore, contract W satisfies the incentive compati-
bility constraint also and hereby is feasible.
0)()(   hllhwuwu
 lhhl ww 0'
Q.E.D.
By the above lemma, the optimal contract satisfies
. Then, [P1] can be simplified as
lhhl ww 
[ P2]
llHlhhlHHhhH
wwww wpwwppwp
lllhhlhh
22
,,, )1())(1(min 
In order to solve [P2], denote , which
satisfies , and because ,
and
)()( 1  uf0)
0'f0'' f0
0(f0'u
0'' u
)
0
(
u. Then, let , [P2] is
simplified as abab uw )(u
[P3]
2
min()(1)[ ()()]pfupp fufu 
,,,
2
(1 )( )
hh hllh llHhh HHhllh
uuuu
Hll
pf u
2

2
()(1 )()
(1 )(1 )()()
.()(1 )(1 )
[(1 )]()
HhhHH hllh
Hll HHhllh
HhhH hlHlhH ll
HHhllh
HL
PC p uppuu
puppuu c
st ICpupu pupu
c
ppuu
pp


 





 

 



which is a standard problem of convex programming. It
is easy to achieve the solution
*
*
'( )
'()( ,)
H
hh
H
H
hl
p
fu p
p
fu y
p


*
*
(1 )
'()( ,)
1
(1 )
'( )1
H
H
lh
H
H
ll
H
p
fu y
p
p
fu p



















(17)
where

is the Lagrangian multiplier of PC,

is that
of IC, and
[(1) ] ()(1)
(,) (1 )
HHH
HH
ppp p
ypp
 
H
  (18)
Furthermore, denote )(

k as the inverse function of
)('

f. Here, because . Then, the above
(17) equals 0'k0'' f
*
*
[]
[(,)]
,)]
H
hh
H
H
hl
p
ukp
p
uk y
p


*
*
(1 )
[(
1
(1 )
[]
1
H
H
lh
H
H
ll
H
p
uk y
p
p
uk p



















(19)
2
2
() ()
(1)[ ()()]
(1) ()
.(1)()[()()]
()[() ()]
(1)[ ()()]
[(1)][ ()()]
Hhh
HHhl lh
Hll
HHhllh
Hhh lh
Hhl ll
HHhllh
H
L
PCpu w
pp uwuw
puw
stppuw uwc
ICpuwu w
puwuw
c
ppuwuw
p
p














 


Consequently, the solution to [P1], or the optimal con-
tract for independent agents A and B with inequity aver-
sion, ))(),(),(),(( ***** l
l
lhh
l
hh ufufufufW , is defined by
(18) and (19).
5. The Joint Contracts
In case of 0




, by (18), . Furthermore,
from (19), and , by which the opti-
mal contract for multiple independent pure self-interest
agents is the independent contract and the payment to
each agent depends on only his own output. This is the
sufficient statistics result revealed by [33] and [34] in the
standard contract theory.
0)0,0( y*
ll
u
** hlhh uu *
lh
u

GUANGXING WEI, YANHONG QIN 154
In case of 0


, by (17), (18) and ,
0'f
*
(1 )'( )0
1Hll
p yfu
p


 


*
'( ) 0
H
Hhh
H
p yfu
p









(20)
From (20) and 0


, 0),( 


y. Then, by
0),( 


y, 0'
k
and (19), **h
l
hh uu  and ** l
l
lh uu .
Moreover, definitely holds by the above
lemma. Thus, if
*
lh 0
*
hl uu


, the optimal contract surely
satisfies l
****
l
lhh
l
hh , by which the optimal
contract for multiple independent agents with inequity
aversion is the joint contract, and the payment to each
agent depends on, and further increases with, both his
own output and others’ output. Therefore, the sufficient
statistics result of the standard contract theory does not
work in the optimal contract for agents with inequity
aversion. The result that the payment to an independent
agent should depend on both his own output and others’
is not novel. It appears in the rank order tournament [35]
and the rank-order contract [36]. But what this paper
exploring is the optimal contract for multiple risk averse
agents with inequity aversion, while as pointed out by
[33], the rank order tournament is not the optimal con-
tract even for multiple risk averse agents with pure
self-interest.
uu uu
From (18), (19) and (20 ), it is easy to find that if ineq-
uity aversion is strong enough to satisfy
(1 )
1
(,) []
21
HH
HH
ypp
pp

 


 (21)
there must be . And by the above lemma,
always holds. So, for the inequity aversion
stronger than that defined by (21), . The joint
contract satisfying is defined as
egalitarian joint contract , by which the payment to
each independent agent is always equal, no matter what
output each agent achieves. On the other hand, when
inequity aversion is weaker than that defined by (21), we
can obtain from (18), (19) and (20). The joint
contract that satisfies is defined as
relative joint contract , by which the payment to
each independent agent should rest with, and further in-
crease with, both his own output and others’ output. In-
tegrating the above two aspects, the following proposi-
tion can be obtained.
** lhhluu 
*
hh
u
*
lh
u
*
hh
u
W
** lhhluu
** lhhluu 
*
ll
u
** lllh u
** lhhluu 
E
W
*
hl uu
R
*
hl
u
Proposition 1: The optimal contract under relatively
weak inequity aversion is the relative joint contract,
while that under strong, even very strong inequity aver-
sion is the egalitarian joint contract.
The relative joint contract is denoted as,
which is defined by (18) and
(19). And the egalitarian joint contract can be repre-
sented by, which is the
solution to the following [P4]. When the intensity of in-
equity aversion equals, or bigger than that defined by
(21), from (18), (19) and (20),
*(( R
Wf
m
w
****
), (), (), ())
hh hllh ll
ufufufu
),(( **** hh
EufW ))(),( **** llmufuf
hl
wlh w

 and
mlhhl uuu


. Thus, [P3] is simplified as
[P4]
2
,,
2
min( )
2(1 )()(1 )(
hh m llHhh
uuu
)
H
HmHl
pfu
ppfu pful
2
2
()2(1 )
(1 )
.() (12)
(1 )
H
hhHHm
Hll
HhhH m
Hll
H
L
PC p uppu
puc
st IC p upu
c
pu pp









The solution c an be rep rese nt e d by
**
**
**
''
()
2 '(1)'(12)
[]
2(1 )
'(1)'
[]
1
H
hh
H
HH H
m
HH
H
ll
H
p
uk p
ppp
uk pp
p
uk p
















(22)
where '

is the Lagrangian multiplier of PC while'

is
that of IC.
Obviously, different from the relative joint contract,
the egalitarian joint contract is not correlative to the in-
tensity of inequity aversion, although it is the optimal
contract under inequity aversion stronger than that of
(21).
6. Incentive Efficiency Losses
In order to analyze the incentive efficiency losses result-
ing from inequity aversion clearly, the agency cost of
joint contract, which is the optimal contract under ineq-
uity aversion is compared with that of independent con-
tract, which is the optimal contract under pure self-inter-
est. If the agency cost of joint contract is higher, the dif-
ference is the incentive efficiency losses resulting from
inequity aversion. On the contrary, if the agency cost of
joint contract is lower, the difference is the incentive
efficiency gains stemmed from inequity aversion. This is
an easy way to measure the influence of inequity aver-

GUANGXING WEI, YANHONG QIN 155
sion on incentive efficiency.
6.1 Case of Relative Joint Contract
From (19), the payment to each agent by the relative
joint contract is
*2* *
*2
()(1 )()
(1 )()(1)()
rHhhHHhl
*
H
HlhH ll
EWpf uppf u
ppfu pfu

  (23)
And from (3) and (19), the utility of each agent, ex-
cluding cost of effort, in every case, is
**
** **
*
** **
**
(
()
hh hh
hlhlhllh
ab
lhlhhl lh
ll ll
Uu
Uu uu
UUu uu
Uu



 

 



)
*
*
n
*
(24)
where when each agent achieves the same output, each
agent only obtains the utility derived from the payment,
while when the output of each agent is unequal, that is,
one agent such as A achieves high output while B
achieves low output, A suffers additional compassion
disutility and B suffers additional jealous disutility, ex-
cepting the utility d erived from the payment.
On one hand, in order to endow the pure self-interest
agent with the same utility as of (24) in every case,
the expected payment that the principa l has to make is
*
ab
U
*2** **
*** 2*
()(1)[ (())
(())](1)()
nHhhHHhlhllh
lhhl lhHll
EWpfuppf uuu
fuuupfu


 
(25)
From (23), (25) and , it is clear that the prin-
cipal has to make higher expected payment to the agent
with inequity aversion than that to the pure self-interest
agent in order to endow them with the same utility in
every case. The extra payment, defined as inequity rent,
is caused by inequity aversion and is the compensation
for the inequity disutility res ulting from unfair allocation.
From (23) and (25), the inequity rent can be represented
by
*
hl lh
uu
**FR r
WEWEW (26)
On the other hand, in order to endow the pure
self-interest agent with the same utility as of (24)
in every case by the independent contract, the optimal
contract for pure self-interest agent offered by the stan-
dard contract theory, the expected payment the principal
has to make is
*
ab
U
** ***
***
[(1)(())]
(1) [(())(1)]
sHHhhHhlhllh
H
Hlhhl lhHll
EWpfp upuuu
pfpuu upu



(27)
From (25), (27) and Jensen Inequality, **
s
n
EW EW
because ,
'0f'' 0f

and , by which the
relative joint contract, the optimal contract under ineq-
uity aversion, requires the principal to make more pay-
ment than that the independent contract, the optimal con-
tract under pure self-interest, requires, because the pay-
ment to each agent depends not only on his own output
but also on others’ by the relative joint contract and
hereby, each agent is confronted with higher risk. The
extra payment required by the relative joint contract, i.e.
more than that the independent contract requires, is the
compensation for the additional higher risk, which is
defined as the risk compensation for inequity aversion.
From (25) and (27), it can be denoted as
*
hl lh
uu
*
r
EW
0
*
*
s
*
s
W
**
n
WE
**FR

*
lh
uf
FL
WE
FC
WW
*
hl
u
W (28)
Both the inequity rent and the risk compensation for
inequity aversion are incentive efficiency losses resulting
from inequity aversion. The sum, defined as inequity
aversion losses, are additional compensations to agents
with inequity aversion more than those to pure
self-interest agents, by (26) and (28), can be denoted as
*FL
W E  (29)
From and , '
*0 FC
W



 and *0
*
*
)[
)]
hl
lh
u
u
FC
W
 
2*
u

*(1
()
h H
p



,
by which inequity aversion losses increase with the in-
tensity of inequity aversion. Therefo re, the more inequity
averse the agents are, the more addition al compensations,
including the inequity rent and the risk compensation for
inequity aversion, the principal has to pay. Formally, the
following conclusion can be drawn.
Proposition 2: By the relative joint contract, the prin-
cipal has to pay the inequity rent and risk compensation
for inequity aversion, which both are the incentive effi-
ciency losses resulting from inequity aversion, and in-
crease with the intensity of inequity aversion.
Inequity rent is the compensation for inequity dis-
utility resulting fro m unfair allocation of material payoff,
and the risk compensation for inequity aversion is the
compensation for the extra risk, i.e. risk increased by the
relative joint contract.
From (24), the expected utility of each agent, exclud-
ing cost of effort, is illustrated as
*2
Hh
lh
p u
**
( (1)
H
hlHll
EU p
uu p

 (30)
If the principal knows the efforts of the agen ts, he only
has to pay a certain amount, as given by

GUANGXING WEI, YANHONG QIN 156
**
2** *
** 2*
()
((1)((
())(1))
CE
HhhHH hllh
hl lhH ll
WfEU
fpupp uu
uu pu
)





(31)
Then, from (27) and (31), the risk compensation for
asymmetric information is denoted as
***
R
C
s
WEWW
CE
*
(32)
which is the so-called risk compensation discussed in
the standard contract theory.
The agency cost by the relative joint contract is the
sum of the above inequity rent, risk compensation for
inequity aversion and risk compensation for asymmetric
information. So,
***
F
RFL
AC WWW RC (33)
Comparatively, the agency cost for pure self-interest
agents includes only the risk compensation for asymmet-
ric information. Both the inequity rent and the risk com-
pensation for inequity aversion are incentive efficiency
losses arising from inequity aversion. According to
Proposition 2, the more intense inequity aversion, the
higher would be the incentive efficiency losses. But, the
incentive efficiency losses don’t increase infinitely with
the inequity aversion because when inequity aversion is
strong enough, by Proposition 1, the optimal contract is
not yet the relative joint contract, bu t the egalitarian joint
contract, by which the agency cost does not increase with
the inequity aversion and hence it is fixed and limited,
which is proved in the follo wing subsection.
6.2 Case of Egalitarian Joint Contract
From (22), payment to each agent by the egalitarian joint
contract is
** 2 **
**2 **
()
2(1 )()(1 )()
rHhh
H
HmH ll
EWpf u
ppfu pfu

  (34 )
From (3) and (22), the utility in every case, excluding
the cost of effort, of each agent, is
** **
**** **
** **
hh hh
abm m
ll ll
Uu
UUu
Uu






(35)
It is clear that payment to each agent is always equal
by the egalitarian joint contract and, therefore, even
agents with inequity aversion don’t suffer any inequity
disutility.
Furthermore, similar to the former subsection, it is
easy to conclude that the risk compensation for inequity
aversion by the egalitarian joint contract can be denoted
as
** ******
(( (1))
FL
WEWpfpupu 
**2 ****
2(1 )
** **
(1 ) ((1)))
rH HhhHm
HHm Hll
pf
pu pu (36)
From (22), (34) and (36), the risk compensation for
inequity aversion that each agent receives by the egali-
tarian joint contract, is fixed and doesn’t change with the
intensity of ineq uity aversion. Thus,
Proposition 3: By the egalitarian joint contract, the
principal need not pay the inequity rent, and only pay the
fixed and hereby limited risk compensation for inequity
aversion, which is the incentive efficiency loss resulting
from inequity aversion.
From (35), the expected utility of each agent, exclud-
ing cost of effort, is represented by
2* *
(1 )
H
hhHH m
Hll
pu
EUpuppu
****2 ****
()(2(1)
CE
(37)
If the principal knows the effo rts of all agents, he only
has to pay a certain amount as
2* *
(1 ) )
H
hhHH m
Hll
pu
WfEUfpuppu 
(38)
Then, from (34) and (38), the risk compensation for
asymmetric information can be derived as
****** CE
r
RC WEWW  (39)
The agency cost by the egalitarian joint contract
equals the sum of the above risk compensation for ineq-
uity aversion and the risk compensation for asymmetric
information.
**** **
F
L
AC WW RC (40)
Integrating the above analysis, it is clear that the
agency cost increases with the intensity of inequity aver-
sion within an upper limit defined by (40). By the egali-
tarian joint contract, even agents with inequity aversion
don’t suffer any inequity disutility, an d th e principal only
need pay the fixed and hereby limited risk compensation
for inequity aversion. However, by the relative jo int con-
tract, agents with inequity aversion suffer inequity dis-
utility in face of unfair allocation, and the principal has
to pay inequity rent, besides the risk compensation for
inequity aversion. Whether the egalitarian joint contract
or the relative joint contract is optimal depends on their
respective agency costs. The one with lower agency
costs is optimal. In case of relatively weak inequity aver-
sion, agency cost of the relativ e joint contract is less and,
therefore, it is the optimal, while in case of strong, even
very strong inequity aversion, agency cost of the egali-

GUANGXING WEI, YANHONG QIN 157
tarian joint contract is smaller and, hence, it is the opti-
mal.
7. A Numerical Example
In order to explain the above theoretical analysis clearly
and fix ideas easily, the following offers a numerical
example. The agent can achieve high output
H
x
with
probability of if he takes high effort H,
while with probability of
0.8
H
pe
0
L
p

if he takes low effort
L
e. Low effort
L
e costs 0, while high effort
H
e costs
, privately to him. Further, denote
10c()uw
ab
w
(,
hh
ww
with the reverse function , by which
2
wu

can be transformed as . ,,
lh
ww )
ll
w
hl ), ll
uu ,,lhhl u(hhab uu
7.1. Benchmark: Pure Self-Interest
In the case of pure self-interest, 0


. Therefore,
[P3] is simplified as
[P5]
2
,,,
22
min 0.64
0.160.16 0.04
hh hllh llhh
uuuu
hllh ll
u
uu
2
u
( )0.640.16
0.160.0410
.. ()0.8 0.2
0.8 0.2 12.5
hh hl
lh ll
hh hl
lh ll
PC uu
uu
st IC uu
uu








It is easy to find that the solution is ,5.12()0,0(

ab
u,0(
ab
w , and hereby the optimal contract is
, which is surely an independ-
ent contract.
12.5,0,0)
0) (156.2 5,156.25,0,0)
On one side, the expected payment to each agent is
12502.025.1568.0)0,0( 
r
EW
On the other side, the expected utility of each agent,
excluding the cost of effort, is equivalent to
1002.05.128.0)0,0(

EU . If the principal can
observe the efforts, he only has to pay certain as
10010))0,0(()0,0( 22  EUW CE
So, the risk compensation for asymmetric information,
also the agency cost, can be denoted as
(0,0)(0,0) (0,0)
RC
AC WEW 

(0,0)125 10025
r
CE
W

7.2 Common Case: Average Inequity Aversion
The average inequity aversion is denoted as 8.0


and 3.0

[7]. Then, [P3] is simplified as
[P6]
22
min 0.640.16uu
,,,
22
0.16 0.04
hh hllh llhh hl
uuuu
lh ll
uu
( )0.640.016
hh hl
PC uu

0.3360.04 10
.. ()0.8 0.78
1.380.2 12.5
lh ll
hh hl
lh ll
uu
st IC u u
uu








The solution rounded to two decimal digits is
)07.0,21.4,50.9,65.13()3.0,8.0(

ab
u
(0.8
ab
w
0)
, and hereby the op-
timal contract with integer is
, which is surely a relative joint contract.
,0.3)(186,90,18,
On one hand, the expected payment to each agent is
(0.8,0.3)0.8 0.81860.80.2 90
0.20.818136.32
r
EW

 

On the other hand, the utility of each agent in every
case, excluding cost of effort, is denoted as
()
(0.8,0.3) ()
13.65
9.50 0.3(9.5 4.21)
hh hh
hlhlhllh
ab lhlhhl lh
ll ll
Uu
Uu uu
UUuuu
Uu




 

 





4.21 0.8 (9.54.21)
0.07
13.65
7.91
0.02
0.07












(0.8,0.3)
ab
U
2
2
(0.8,0.3)0.80.813.65
0.8 0.2 7.910.2
0.2 0.2 0.07129.26
n
EW 
 

For a pure self-interest agent, when endowed with the
same utility as in every case, the expected
payment he receives is
22
0.8 0.02
where the negative payment is mapped to a negativ e util-
ity. Therefore, the inequity rent is
(0.8,0.3)(0.8,0.3) (0.8,0.3)
136.32 129.267.06
FR rn
WEWEW

While in order to endow him with the same utility as
by the independent contract, the expected
payment the principal has to make equals
(0.8,0.3)
ab
U

GUANGXING WEI, YANHONG QIN 158
(0.8,0.3)(0.8,0.3)(0.8,0.3)
FL
WEWEW 

RC CE

RC FC
 
2
2
(0.8,0.3)0.8(0.813.650.27.91)
0.2 (0.80.020.2 0.07)125
s
EW  
 
where a negative payment is also mapped to the neg ative
utility. Then, the risk compensation for inequity av ersion
is illustrated as
129.26 1254.26
ns
 
Then, the inequity aversion losses are
(0.8,0.3)(0.8,0.3) (0.8,0.3)
7.06 4.26 11.32
FCFR FL
WWW

The expected utility of each agent, excluding effort
cost, equals
(0.8,0.3)0.80.813.65
0.8 0.2 7.91 0.8 0.2 0.02
0.2 0.20.0710
EU 


If the principal knows the effo rts of all agents, he only
has to pay a certain amount as
22
(0.8,0.3)((0.8,0.3))10100
CE
WEU.
consequently, the risk compensation for asymmetric in-
formation is
(0.8,0.3)(0.8,0.3) (0.8,0.3)
125 10025
s
WEWW

Summarily, the agency cost is
(0.8,0.3)(0.8,0.3)(0.8,0.3)
25 11.3236.32
AC WW
 
7.3. Extreme Case: Infinite Inequity Aversion
The above theoretical analysis has found that wh en ineq-
uity aversion is stron g eno ugh, th e op timal co ntract is th e
egalitarian joint contract, by which payment to each in-
dependent agent is always equal, no matter what output
each agent achieves. In order to illustrate it, the follow-
ing four extreme cases,(10,

0.4)

,( 100,


0.4)


,(300,0.4)


and (500,0.4)

 are
examined. The results in every case are given in Table 1.
From Table 1, the following conclusions can be drawn.
(1) The optimal contract under weak inequity aversion is
the relative joint contract, and trends towards the egali-
tarian joint contract as the intensity of inequity aversion
increases. (2) Pure self-interest only requires risk com-
pensation for asymmetric information, while inequity
aversion results in inequity aversion losses, which in-
crease with the intensity of inequity aversion within an
upper limit. (3) Risk compensation for asymmetric in-
formation under pure self-interest is the same as that
under inequity aversion because it arises from asymmet-
Figure 1. Relationships of inequity aversion, contract types
and agency cost
ric information and is irrelative with the intensity ineq-
uity aversion. It is consistent with the former theoretical
findings. A simple illustration of intuitive explanation is
shown in Figure 1.
Under pure self-interest represented by point O, the
optimal contract is the independent contract, and the
agency cost includes only the risk compensation for
asymmetric information. Under inequity aversion weaker
than that of point S, the optimal contract is the relative
joint contract, and the agency cost includes both the risk
Table 1. Computational results of optimal contract under different scenarios
),(


ab
u ab w Expected
payment Inequity aversion losses Risk compensation for asymmetric information Agency costs
(0,0) (12.5,12.5,0,0) (156,156,0,0) 125 0 25 25
(0.8.0.3) (13.86,9.50,4.21,0) (186,90,18,0) 136.32 11.32 25 36.32
(10,0.4) (13.65,7.32,6.65,0) (192,54,44,0) 138.56 13.56 25 38.56
(100,0.4) (13.65,6.98,6.92,0) (192,49,48,0) 138.86 13.86 25 38.86
(300,0.4) (13.65,6.96,6.93,0) (192,48,48,0) 138.88 13.88 25 38.88
(500,0.4) (13.65,6.95,6.94,0) (192,48,48,0) 138.88 13.88 25 38.88

GUANGXING WEI, YANHONG QIN 159
compensation for asymmetric information and inequity
aversion losses, which increase with intensity of inequ ity
aversion. Under inequity aversion stronger than that of
point S, the optimal contract is the egalitarian joint con-
tract and the agency cost includes both risk compensa-
tion for asymmetric information and inequity aversion
compensation, which doesn’t increase with the intensity
of inequity aversion any more.
8. Concluding Remarks
The above designs the optimal contract for multiple
agents with inequity aversion and then analyzes the in-
centive efficiency losses resulting from inequity aversion .
Incorporating inequity aversion into optimal contract
design can improve our understanding of the real world
incentives. If agents exhibit an aversion towards unfair
allocations, the optimal contract is the joint contract by
which payment to each independent agent depends on
both his own output and others’. The optimal contract
under relatively weak inequity aversion is the relative
joint contract by which payment to each independent
agent increases with both his own output and others’,
while that under strong, even very strong, inequity aver-
sion is the egalitarian joint con tract by which payment to
each independent agent is always equal, no matter how
many outputs an agent achieves independently. While
the optimal contract designed in the standard contract
theory balances insurance and incentives, the joint con-
tract incorporating inequity aversion balances insurance,
incentives and fairness. Therefore, the inequity aversion
adds an additional incentive constraint because the prin-
cipal has to pay inequity rent and risk compensation for
inequity aversion by the joint contract besides risk com-
pensation for asymmetric information by the independent
contract investigated in the standard contract theory.
So, the inequity aversion results in incentive efficiency
losses, which include inequity rent and risk compensa-
tion for inequity aversion, increase with the intensity of
inequity aversion by the relative joint contract, while
only include risk compensation for inequity aversion, is
fixed and hereby is limited by the egalitarian joint con-
tract. Therefore, in order to design the optimal contract
for agents with inequity aversion, whether the relative
joint contract or the egalitarian joint contract, the princi-
pal must screen and evaluate the intensity of inequity
aversion. In this way, some new theoretical insights are
obtained by incorporating inequity aversion into the
standard frame of optimal contract design and hence real
economic behaviors can be explained more properly.
However, there remain many open questions to be
answered. Firstly, how to measure the intensity of ineq-
uity aversion? There are few economic literatures that
discuss evaluation and screening of inequity aversion.
Secondly, how to change the preferences of the agents on
behalf of the principal? To be more frank, how can an
employer change the intensity of inequity aversion of his
employees? Although most activities of human resource
management are targeted at shaping preferences of em-
ployees, there are few concrete feasible solutions avail-
able. Finally, what is the scope of the reference group?
What is the time horizon? The right framing of social
comparison is surely another important task. All these
questions are worth exploring further.
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