Dawen Meng1 ； Lei Sun2
1 ,2 Shanghai Institute of Foreign Trade;
2 Shanghai University of Finance and Economics
This paper has explored the design of optimal nonlinear pricing contract in the presence of network externality, in contrast with the “fulfilled expectation contract” put forth by
Sundararajan(2004),in the “optimal ex-ante contract” the allocation of both types will be distorted, and more interestingly, the consumption of different types will be distorted in different directions. The countervailing incentive problem aroused from the potential entry is also analyzed. As the competitiveness of the entrant increases, the incumbent firm will adjust its nonlinear pricing scheme accordingly.
Key words: Nonlinear Pricing; Countervailing Incentives; Type-dependent Reservation Utility
When the preferences of potential consumers are unknown to the firm, a nonlinear pricing scheme should be designed to screen their “type”. In a seminal paper, Maskin and Riley(1984) has offered a framework for the analysis of this problem. In accordance with the basic insight of contract design, their conclusion is: Under asymmetric information, the allocation of all the agents except the one at the top will be distorted downward. Sundararajan(2004) has analyzed optimal monopoly pricing under incomplete information for a good that displays positive network effects. In contrast with common goods, the utility of the consumer depends on the total quantity consumed across customers as well as his individual consumption. Our paper investigates the same problem, but different conclusions have been drawn because different solution concept has been adopted. According to the conclusion of Sundararajan(2004), if the network effect is homogenous, that means the network effect is type-independent, the presence of network externality will not distort the allocation. The solution he has presented is
“fulfilled-expectation contract”. That means the monopolist should design the optimal pricing scheme
 q
 
, Q
 
, Q
  contingent on consumer’s type
 and expected network
1 Dawen Meng 1978
—）：
Shanghai Institute of Foreign Trade; Shanghai University of Finance and Economics;
Email :devinmeng@126.com
2 Lei Sun(1977---) Shanghai University of Finance and Economics.
1

magnitude Q under the IC and IR constraints of the consumers. If the expected network magnitude equals to the fulfilled one, the solution is the “optimal fulfilled-expectation contract”, so the optimal Q can be got through solving Q
 q
 
,
  

 the optimal fulfilled-expectations contracts is.
q
*
   
, Q
   
, Q

.In contrast,in this paper we derive the “ex-ante optimal contract”, in which the allocation of both types will be distorted. That means even in the presence of homogenous network externalities the “no distortion at the top” statement should be modified. More interestingly, the consumption of different type will be distorted in different directions. Furthermore, we analyzed the optimal pricing strategy of incumbent firm when facing the potential entry of competitors. In that setting the consumer can bypass the network of incumbent firm and switch to buying from the entrant, it will result in the type-dependent reservation utility of the consumers. So when the difference of outside utility across different state is large enough, the countervailing incentive problem will arise. Laffont and Tirole(1990) has discuss the problem of optimal bypassing, in this paper we explored the more complicated problem in which network externality and countervailing incentive interwind together.
The rest of this paper is organized as follows. Section 2 specifies the benchmark
Maskin-Riley model, Section 3 presents the model with the network externalities, and section
4 discuss the countervailing incentive problem arise from threat of entry.
Let
 denote the preference parameter of consumers, which is unobservable to the monopolist. To simplify the analysis, we assume

is discrete with :
 
 
, pr
  v , pr

 

1 v .
       represents the utility of

-type consumer derived from consuming q units of goods with network magnitude Q .Where
and
   are all concave functions with
V
' 
0, V
'' 
0,

'' 
0 2 .Let t denote the price charged by the monopolist. The object of monopolist is to maximize his own revenue under the IC and IR constraints consumers.
In the case free of network externality, the problem of the monopolist is:
2 Note that we have not specify the sign of

'
, which means our conclusion is robust to both positive and negative network effect. When the network is congested, we have

' 
0
.
2

 max
 
;

,



 cq
 
1 v
 t
 cq
 s.t


IR




IR
IC
IC
   
t

0
   
t

0
   
t
   
t
   
t
 
 
t



 it is easy to determine that the binding constraints are IC and IR
 
.Through solving the above problem we can get the MR(Maskin-Riley)contract:




 
1
'
V q
 v
MR v
 
 
 c
'
V q q
MR  q
*  q
*  q
MR
 c where q
* and
* q denote the complete information solution with


'
 
 
'
 
 c

It is clear that under asymmetric information the consumption of the low-type consumer will be distorted downward, while that of the high-type consumer remain the first-best level.
In the presence of network externality the aggregate consumption among all the individuals, namely, the network magnitude. Q
 vq

1

will also affect the utility of consumers. Under complete information the monopolist’s problem is : max
  
;

,



 cq
 
1 v
 t
 cq
 s.t





 
 
 
 
 
 
-
t t


0
0

 



'
 
 
 
'
'

 vq vq
FB
FB
with q
FB  q
FB
 
 
FB
FB



 c c
Under asymmetric information the IC and IR constraints of the consumers should be satisfied,
Substitute t by

   
    
U
 
, where U
  denote the information rent gained by the

-type consumer. Then the optimal ex-ante contract can be got through
3

solving the following problem:



 max
   
; ,
 



 s.t






U
U
U
U
 v

 

 


U
U
 
 
  
  
 
V q


0
0
 
 cq
  
 vU

1
 
  v


    cq











Again, by applying the standard approach for solving the adverse selection problem, the IC constraint of the high-type and the IR constraint of the low-type consumer are binding. So the solution is:







1
'
V q
 v v
 
 
 

'
'
V q
 vq
SB
 
'
 vq
SB
 
SB
 
 c
SB  c compare with the case under asymmetric information we can get the following proposition.
Proposition 1: In the presence of network externalities (regardless of its sign),
Asymmetric information will distort the consumption level of different types in the opposite directions: the consumption of the low type consumers will be distorted downward, while that of the high-type consumers will be distorted upward, and the network magnitude will be downsized.
Proof : let


'
 
 
'
   
'
'

 vq vq







 c c  when FB
1
 v when SB v through differentiating the above equations with respect to parameter

we can get:
4















V
''
 
 v

''
  
1
 v
   v

''
  
V
''
   v
  


 d d q


 
 d q



 
 '
0
  d d q
 
 '
V q
''

''
V q V q

V
'' q v
 d
 
 
 
  
  

''
Q v
  
 
V
''
  
''
 



0 d q d
 
V
''
 

V
'' q vV q
  
'
  v


''
''
 
Q
 
V
''
  
''
  d Q d

 v d q d

 
1
 v


0 d q d
 
V
''
 

V
'' q
 
' '' v V q V q
   v
   
    
V
''
  
''
 

0
 q
SB  FB q , q
SB  q
FB
, Q
SB 
Q
FB
Q.E.D
Using the approach of Sundararajan(2004), the above program (3) should be solved taken the network size Q as constant firstly, then the optimal fulfilled-expectation contract:
 
 
 
1
 v v
 
'
 
 c ;

'
   c


 can be got, and Q
 vq

1

. It is as same as the solution in Maskin-Riley case. Through comparison we can find it is quite different to our “ex-ante optimal contract”.
The economic intuition behind the above solution is: In order to extract the information rent of the high type agent, the allocation of low type agent should be reduced. This is the basic trade off between allocation efficiency and rent extraction arises in almost all the adverse selection problems. While in the presence of network externality, in order to keep a fixed size of network, the high type’s consumption has to be increased to countervail the decrease of that of the low type .
In this section we assumed that the consumers can bypass the network offered by the incumbent firm and enter a competitive market made up of many homogenous firms. All of these firms is the potential entrant of the incumbent market. Let
 denote the marginal production cost of the entrant. Assuming the entrant’s network magnitude is 0,then the consumer’s utility derived from consumption of the entrant’s goods is:
G
 max q
     q
 the first order condition to the above program is :
   so the maximized utility derived network bypassing is:
G
*
   

*

 '

*
   q
*
  let
5

         then G
*
  
 q
*

 
G
*
 

G
*
  
    and d

G d

' d q
* d


' d q
* d
 note that
 '
    qV
' '
 
, d q d
 
V
1
' '
  so d

G d
 q q 0
From the above derivation we can see that when the marginal cost decrease, the difference of consumers’ utility will increase.Put it differently, the reservation utility of agents is type dependent, and the difference of reservation utility across different states dependent on the competitiveness of the potential entrant.
Considering the type-dependent participation constraints, the problem of the incumbent network supplier is :

















 max
  
; ,

IC
IC
IR
IR
 
 
 
 
:

:
:
:
 v


U


V q
U
U

G
U

G
 
  
U U



 cq vU
 
 

  v


    cq










 through solving the above program the following proposition can be got:
utility between different type of consumers, G .
(I) When

,the second contract is still available.

, q
 q SB , q
 q SB
,the gap between different types (II)When
  will be widen.
(III)When
  
 
,the first best allocation q
 q
SB
, q
 q
SB
can be
6

achieved.
(IV)When
  
, q
 q
FB
; q
 q
FB
(V)When,

, q
 q
CI
, q
 q
CI with q
CI  q
FB
, q
CI  q
FB
,where
 q
CI
, q
CI
 is determined by:





 
  v
1

  v
'
 
   c
'
   
'
   c



 when
   
G (case IV and V) the countervailing incentives problem will arises.
(I) when

, the IR
 
and IC are binding. Through solving the program (4), the second best contract can be got. It is easy to check that the neglected
IR
  and IC
  can be satisfied.
(II) When
  
, the IC
 
, IR
 
, IR
 
must be binding, so from
 

'
   
'
 vq
 we can get: q
 q
SB
, and q
 q
SB
because
   c dq dq
  v

''
 vq

V
''
   v
 
 
''
 vq

  
.As a special case, when

, q
 q
SB
, q
 q
SB
.
(III)when
  
As

G increases furtherly, it’s suboptimal for the monopolist to increase the consumption level of the low-type consumers ensuring the high type agent getting enough rent. In that case, the main task for the firm toward the high type consumers is to prevent them from quitting the incumbent market instead of preventing them from misreport. So the binding constraints are IR , IR .
From program (4) we can get the first best contract:


 
'
V q
'
 
 
 
 
'
'

 vq vq
FB
FB
 
 
FB
FB



 c c
(III) when
 
V q

V q , the high difference of benefit will induce the low type consumers pretend to be the high type one, from which the countervailing
7

incentive arises. The IC
So we can get
  

 
'
 
  
 
'
 
  
 c it is easy to check that q
 q FB ; q
 q FB
, IR
  in program (4) is binding.
(IV) When

the rising of

G distorted the consumption level furtherly, so the participation constraint of the low-type has to be slackened. That means
IC , IR
  is binding, the low-type consumer will get information rent which is equal to
   
. From program (4), we can get: let






'
V q

1 v

 v

'
 
 
'
 c
    '
   c








'
V q
'
 
 
 
 
'
'
 
 

 c c

 note that :


CI q , q
CI

FB ; differentiate these two equations with respect to
 we can get : v
1
 v
CI



V
''
 
 v



 

 d d d d q
 q





 v

''
 
V
V
''
''
''
 
 
 

 
 
''


 
1
1

 v v

  
V
''
   v
  
  
V
''
  
'
 




 d d d q




 
 d q

 v

 '

''  
''
V q V q v Q
''
     
  
V
''
   v



V
V
''
''
 

 
 



''
''
'
0
 
 
 




0
0
 q
F B  CI q , q
FB  q
CI remark: CI denotes “countervailing Incentive” Q.E.D
From the above analysis the binding and slack constraints can be summarized in the following table:
8

constraints Case I Case II Case III
Binding
Case IV
Binding
Case V
IR
 
IR
 
IC
 
IC
 
Binding
Slack
Slack
Binding
Binding
Binding
Slack
Binding
Binding
Slack
Slack
Binding
Binding
Slack
Slack
Binding
Binding
Slack
In this paper we have explored the same problem as Sundararajan(2004), but because we have adopted the optimal ex-ante contract, the result is different. In the presence of network externality, even if it affects the utility of all the consumers in the same way, the allocation will be distorted, and more interestingly, the distortion of different types are in different directions. Furthermore we explore the countervailing issue originated from outside competition and consumer’s type-dependent reservation utilities. As the marginal cost of competitors decrease, that means their competitiveness has been increased, the reservation utility difference between different type will increase, then the optimal nonlinear pricing scheme should be altered accordingly. When the reservation utility is large enough the countervailing issue will arise. That means the principal should prevent the high type consumers from quitting the market and the low type consumers from misreporting their type, which is dramatically contrary to the traditional insight in adverse selection settings.
Maskin, E., Riley, J, 1984. Monopoly with incomplete information. Rand Journal of Economics 15,
171–196.
Jullien, B. Participation Constraints in Adverse Selection Problems. Journal of Economic Theory Vol.
93 (2000), pp. 1-47.
Rochet, J., and Stole, L. Nonlinear Pricing with Random Participation. forthcoming,
Review of Economic Studies (2001).
Lewis, Tracy R. & Sappington, David E. M., 1989. Countervailing incentives in agency problems,
Journal of Economic Theory, Elsevier, vol. 49(2), pages 294-313, December.
Sundararajan.A, Nonlinear pricing and type-dependent network effects Economics Letters 83 (2004)
107–113
JJ Laffont, and J Tirole Optimal Bypass and Cream Skimming. The American Economic Review,
Vol. 5, 1042-1061. Dec., 1990
JJ Laffont, and David Martimort The Theory Of Incentives
——
The Principal-Agent Model 2002
9

Princeton University Press
t

G

B

A q
SB q
FB q
FB q
SB
Figure 1(Case 1:

) q
10

t
B

G
A q
SB q q
FB q
FB q q
SB


Figure 2(Case 2:
  
) q t

G
B
A q
SB q
FB q
FB
Figure 3(Case 3:
  q
SB



 
) q
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t

G

B
A q
SB q q
FB q
FB q q
SB

Figure 4(Case 4:
   t

G
B
A q
SB q
IC q
FB q
FB q
IC q
SB
Figure 5( Case5 :

 
)


) q q
12

q q SB
CI q
FB q q
FB q
CI q
SB
 
V q
 
V q
 
 
 
 

G
Figure 6
作者：
通讯作者：孟大文（ 1978 ——）上海对外贸易学院教师，研究方向为博弈论、合约理论等。
邮件： devinmeng@126.com
电话： 13816629495 021-67703846
通讯地址：上海市松江区文翔路 1900 号上海对外贸易学院国际经贸学院
邮编： 201600
第二作者：孙蕾（ 1977 ——）上海财经大学国际工商管理学院博士研究生，研究方向为博弈
论、国际经济学等。
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