J. Service Science & Management, 2009, 2: 329-333
doi:10.4236/jssm.2009.24039 Published Online December 2009 (www.SciRP.org/journal/jssm)
329
The Dynamic Multi-Task Supply Chain
Principal-Agent Analysis
Shanliang LI1,2, Chunhua WANG3, Daoli ZHU1
1Management School, Fudan University, Shanghai, China; 2Management School, Soochow University, Suzhou, China; 3Information
School, Shanghai Ocean University, Shanghai, China.
Received August 18, 2009; revised September 25, 2009; accepted November 5, 2009.
ABSTRACT
In the supply chain by the composition of the supplier and the retailer, the supplier offers products to the retailer for
sales while the retailer affects the sales outcome by his effort which is divided into two dimensions. One is for the
short-term sales task and th e other is for the long-term sales task. Fo r the long-term development of the enterprise, the
supplier wants to inspire th e retailer to make more effort for the long-term task. However, due to the asymmetric infor-
mation, the supplier can’t observe the retailer’s action and the moral hazard will come into being. To deal with this
problem, we construct the dynamic multi-task supply chain principal-agent model, by which we analyze the impact of
the information asymmetry to the supply chain contract. Furthermore, by comparing the contracts between the sin-
gle-term multi-task and two-term multi-task, we have analyzed their different effect on the commission rate.
Keywords: Supply Chain Management, Multi-task Principal-agent, Dynamic Incentive, Moral Hazard
1. Introduction
In the supply chain system, there exists the conflict be-
tween the local interests of the supply chain members
and the overall performance of the supply chain, which
leads to the system inefficiency. At present, one of the
most important research areas of supply chain is to de-
sign the suitable coordination mechanism in order to ob-
tain the global optimization of the supply chain per-
formance. In this process, the information plays a very
important position. As the supply chain members tend to
hide their private information to maintain information
superiority, this will lead to “Adverse Select” and “Moral
Hazard” in various fields [1].
In the recent decade, scholars have studied on the issue
of the supply chain coordination from various angles.
These studies can be roughly divided into two categories.
One is to resolve the “double marginalization” problem
by contract design in the situation of the full information
[2–4]. Such contracts do not involve the information in-
centive. The other is to study the supply chain incentive
problem in the situation of the asymmetric information.
Corbett etc. studied that the optimal quantity discounts
incentive contract between the supplier and the retailer
[5]. Basu etc. studied the incentive issues of the sales
force under asymmetric information based on agency
theory [6]. Lal etc. [7,8] and Chen [9] extended the above
studies. Many Chinese academics are also carried out
research on this issue [10–14]. For the supply chain co-
ordination, the author’s research team had a systemic
research on the issue earlier. Some relevant research re-
sults have been published [15–22]. This paper is the im-
portant one of the systemic study. In the simple princi-
pal-agent model, the agent is engaged in one job and the
agent’s effort is one-dimensional. However in many
cases of the real life, agents are engaged in the job of
more than one. Or, even if there is only one job, it in-
volves more than one dimension. Furthermore, it exits
conflict in the distribution of the same agent’s energy
between the different jobs. To deal with this problem, we
construct the dynamic multi-task supply chain princi-
pal-agent model, by which we analyze the impact of the
information asymmetry to the supply chain contract.
Furthermore, by comparing the contracts between the
single-term multi-task and two-term multi-task, we ana-
lyzed their different effect on the commission rate.
2. Assumptions and Parameters Set
We make the following assumptions for the tractable
analysis. Considering a Stackelberg model between a su-
pplier S who is the principal and a retailer R who is the
agent, the supplier offers the retailer products to sale and
pays the retailer according to sales outcome which is

The Dynamic Multi-Task Supply Chain Principal-Agent Analysis 330
affected by the retailer’s effort and the random factors.
Set is the retailer’s expected profit whose own-
ership belongs to the supplier. denotes the retailer’s
effort for the short sales goal. denotes the retailer’s
effort for the long sales goal. denotes the cost
of the retailer’s effort, satisfying
112
(,Baa)
1
a
2
a
(,Ca
12
)a
1,2
0
C
a

，
2
2
1,2
0
C
a

，
i.e. the cost of the effort increases and the marginal cost
increases. For the simplicity, Assume 22
12
12
(,Ca
2
a
)22
aa
a.
The supplier can’t observe and , but can observe
and verify the sales outcome x, which is affected by the
retailer’s effort together with the random variables, de-
noted by
1
a
12
(, )xaa

 , where 1 (,aa
2
)

is the out-
put function of the effort, satisfying
1,
a2
0


, which
means the marginal sales outcome of the effort is positive.
i.e. more efforts mean more sales; 2
2
1,2
0
a
 
, which mea-
ns marginal sales outcome decrease (When the equal sign
is set up, marginal sales unchanged). Set 12
,()
T


 
12
,()
,
which is the random variable of Normal distribution and
satisfy ; Set
22
12
, ;0, ;)Nr  (0T x
x
x, For the
sake of simplifying the calculating, assume that
11
()xa
111
a

  2 ; 2222 a()ax

 
1
. i.e. dif-
ferent efforts result in different information (However,
different information may be relevant if

and 2

are
relevant. ): 1
x
reflects ，
1
a2
x
reflects . The owner-
ship of the sales profits belongs to the supplier, and the
supplier offers the linear salary to pay the retailer.
2
a
1122
() T
x
xsx x



 (1)
where ()
s
x is the wage paid to the retailer if the total
sales outcome is
x
.

denotes the one-off wealth
transfer which doesn’t affect the incentive intension
（Called Salary）; 12 (,)
T

 which denotes the
incentive intension（Called Commission Rate）, that
means if
x
increase by one unit , the wage of the re-
tailer increased by

unit.
3. The Single-Stage Multi-Task Model
In the single-stage model, the supplier offers a one-time
wage schedule, ()
s
x, according to which the retailer is
rewarded. Assume the supplier is risk-neural, the ex-
pected utility function is as follows:
12 12
1211 22
,) (,) (
(,)
T
SaEa
Ba
EB
a
a
aa
U 
 



Assume the retailer is risk-averse, and the utility is
that ()
x
Vx e


 , where

is the risk aversion coef-
ficient. When 0


, the retailer is risk-neural. When
0

, the retailer is risk-averse. When 0


, the
retailer is risk preference. The retailer’s expected util-
ity is as follows:
12
(()( ,))
R
EUEVsC aax

 (3)
To make the analysis simple, we use the certainty
equivalent (CE) instead of the expected utility of the
retailer [18].
12 12
1
(, )(, )
2
T
RT
aaC aCaE    (4)
where is the expected wage,
12
(, )
Taa
 

is
risk aversion coefficient, T



is the income variance,
is the risk cost. is the covariance matrix
of
/2 
T
 
1

and 2

, denoted by 2
112
2
12 2
,
,
r
r








 .
The supplier is the leader in the Stackelberg model,
who has first-step advantage in the game. However,
when he/she pursuits the profit maximization, he/she
must consider the incentive compatibility constraint and
participation constraint. Thus, the principal-agent model
between the supplier and the retailer can be rewritten as
the following optimization p rogramming.
1211 22
,,
() (,)
aS
PMax aaBaEaU




  (5)
112 2
222
112 112
12 22
12 2
.. (IR)
,,
1()
22
,
R
saa
ra
E
r
C
a
t
0
2


 
 
 











 (6)
12
,
(IC) argmax
R
aaCE (7)
where (6) is participation constraint (IR), and (7) is in-
centive compatibility constraint (IC).
3.1 The Full Information Benchmark
In this section, let’s begin with the full information case
where the retailer’s effort is observable and verifiable.
Then the supplier can assign an effort level to the retailer
by designing a forcing contact. Under this condition, the
incentive compatibility (7) is invalid and we only con-
sider the participation constraint (6), which is binding.
Namely, can be rewritten as follows:
()P
1211 22
,,
() Max(
..
,
0
)
aS
R
aPUBa
sC
a
tE

a

 


 (8)
a
(2)

The Dynamic Multi-Task Supply Chain Principal-Agent Analysis 331
3.2 The Asymmetric Information Case
Solving , we can obtain that: ()P
Generally, the supplier can’t observe the retailer’s action
, and only can observe outcome a
x
. In this case, the
incentive compatibility constraint (7) is valid. Substitut-
ing (7) by the first-order condition, we can obtain the
equivalent programming. i.e. (7) is equal to that
1212 1
11
122 2
2
1
2
,) ,)
,)
(
(,
(
()
BaCa a
aa
BaCa a aa
a
a
a
a





(9)
1
22
a
a
1




(10)
The Equation (9) is the class condition of the Pareto
optimality: the expected marginal profit of the effort is
equal to the expected marginal cost. That is similar to the
single-task case. We have the following conclusion. Solving the model
()P
Proposition 1：Under the condition of full information,
by designing the linear incentive contract, the game be-
tween the supplier and the retailer can achieve the Pareto
optimality when t he retailer has m ulti -dimensional effort.
222
112 112
121 22
,2
12 2
,
(,
1
,) (,)
22
r
Max Baaa ar


 
 
2











 



Substituting by (10), we get
22
2222 12
1 211121222
,
1()
2
(, )(222 )Max Br



    (11)
Solve the first order derivative of (13), (14) about 1

,
obtain
2
11212 1
1
1)(2
2
(2
Br
 


)0
 (12)
Solving the above equation, we obtain that
1212
12
1
1
Br







 (13)
Similarly, we get
112
22
2
2 (A)
1
r
B





 (14)
By (13), (14), we get the following conclusion.
Proposition 2： When0

, the retailer is risk-neural,
then ii （ ），whic h m eans the gam e
can get the Pareto optimization just as the full information
case. When
ii
BC
a
a



 1, 2i
0

,i

（ ）is in inverse ratio with 1, 2i

, the risk aversion coefficient will reduce the incentive
intensity i

; i

（）is in inverse ratio with
variance
1, 2i
2
i

（ ）; in inverse ratio with the Co-
variance
1, 2i
12
r

 ，i.e. the random factors also reduce the
incentive intensity of i

. 1

is also in inverse ratio
with 2

. More 2

means less 1

, and vice versa.
4. Two-Stage Multi-Task Game Model
In the two-stage multi-task model, suppose the retailer’s
effort for the long task in the first stage will affect
the profit in the second stage of the supply chain. Set
denotes the expected effort profit of the first
stage of the retailer，denotes the ex-
pected profit of the second stage. Where and
denote the effort for the short task and long task respec-
tively. Because the effort in the first stage will affect
the profit in the second stage, it will be the variable of the
output function of the second stage. The ownership of
and belongs to the supplier,
the supplier offers the wage schedule according to the
two-stage outcome. Similarly to the assumption of one-
stage, the observed outcome in the second stage is that
2
a
2
)a
11
(,Ba
221222
(,,)Ba aa
2
a
222
,)a a
12345
,,,,)
21
a22 a
112
(, )Baa 221 (,Ba
2(
T
x
xxxxx
1
(15)
where11 xa 222 xa




, , 3213 xa

,
422
xa 452 xa 5


. Assume 345
T
 (,,) 


2
)
 ,


,
which is the random variable in the second stage, Inde-
pendent with 1
(,
T

 
11
22
()
xa
, the random variable in the
first stage. The supplier offers the two-stage payoff con-
tract according the observed outcome as follows.
1122
3344
x x
xx 55
()s
s
x
x




 
222 21
()( ,,xB aaa


2 1
)Es

(16)
The supplier’s expected utility is that:
211 2
(, ()a ax)Es
S
EU B

 (17)
Now, the certainty equivalent of the retailer in the first
stage and the second stage is that
2
1
a

2
12
22
1
2
RT
CE a  
11 22
a   1a   (18)
22
2
R
CE 21 22
23221 1
1
222
Taa
   
4 5221
a a a   12 
2

(19)

The Dynamic Multi-Task Supply Chain Principal-Agent Analysis 332
where is covariance matrix
1
3

，4

and 5

. De-
noted by
2
1112213
2
1122 323
2
213 3233
1
,,
,,
,,
r
r
rr
r
r



 



 







(20)
For obtaining the retailer’s optimal effort of the second
stage, solve
21 22
,
2
argmax
R
aa CE , and get
21 3
22 4
a
a



 (21)
Considering the participation constraint and incentive
compatibility constraint, the supplier need solve the fol-
lowing program m i ng:
21 11 2221 222
1
,
12
2
() MaxE(,)(,,)
..
0
(IC) argmax
(IR)
s
s
P
a
PUBaaEBaaa
s tCECE
CE CE

 


E
(22)
Instead （IC） in (22) by the first-order condition and
substitute by (10), (21), Solve the programming P

 and
get
22
2222 12
11211121222
22
223113412235121 23
22
134124 234523
2351334523
2
1
(, )(2)
2
1
(,,) [(
2
+
+
Baa r
Ba aarr
rr
rr




 








 


2
2



22
22 34
53
)+
22

]

 
(23)
Solve 1

by derivate (23) about 1

and get
2
12
11
11 212
(1 )0
Br B

 


 
(24)
2212
1
1
1
1
1
12
1
r
BB
a







  (25)
Solve 2

by derivate (23) about 2

and get
2
12
22 112
22
(1 )0
Br B




  (26)
12 112
22
22 2
2
(B)
1
r
BB
a






  (27)
Because doesn’t involve the vari-
able ,
2221 22
(,,)Ba aa
1
a2
10

B


. The Equation (13) is the same to
the Equation (25). Comparing (14) with (27), because
2

0
B


, it is evident . Thus, we have the fol-
lowing conclusion.
B>A
Proposition 3: By designing dynamic multi-task con-
tract, the supplier can inspire the retailer to pay more
effort for the long-term goal without the premise of cha-
nging the retailer’s effort for the short goal. It shows that
the dynamic contract design is conducive to maintain the
long-term supply chain partnership.
5. Conclusions
The supply chain contract design is the important means
of the supply chain coordination. For different environ-
ment, it will greatly improve the level of supply chain
collaboration by the design of appropriate contract. In
this paper, we have studied the incentive contract be-
tween the supplier and the retailer. Because of asymmet-
rical information, the supplier can’t observe the effort
level of the retailer. Therefore, the supplier can only in-
spire the retailer’s different effort level by the incentive
mechanisms design. The major study of the paper is on
how to design the dynamic incentive contract to stimu-
late retailers to pay more efforts for the long-term under
asymmetric information and multi-task environment,
which has the guiding role for establishing the supply
chain dynamic alliance. At the same time, our study ex-
tends the existing research results of the principal-agent.
In our research work, for the sake of simplifying the
technical analysis and the calculating, we focused on the
second-term multi-task game. In the future, we will ex-
tend our research to multi-term multi-task model, which
would be challenging and meaningful.
6. Acknowledgements
The authors would like to thank the referees for their
helpful suggestions. The work is supported by China
Postdoctoral Science Foundation under Grant No.
20080430075 and Special Grade of Financial Support
from China Postdoctoral Science Foundation under
Grant No. 200902199 and National Natural Science
Foundation under Grant No. 70171010 .
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