9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Pricing of Multi-Sided Payment Card Networks:
Determination of Merchant and Interchange Fees
Markus Langlet
European Business School
International University Schloß Reichartshausen
Kastanienallee 34/1, 71638 Ludwigsburg, Germany
markus@langlets.eu
October 16-17, 2009
Cambridge University, UK
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9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
ABSTRACT
Payment card networks, and in particular for multi-sided systems the role of the interchange fee has been a
means of diverse discussions especially over the past decade. Even though there remains a gap in a deeper
understanding of the determination of the interchange fee as well as payment card network fees in a broader
sense. This paper is an observation focusing of three such determinants: price elasticity of demand, relative
frequency of card usage and the competitive condition of the downstream market and their impact in case of a
multi-sided payment card system. Two cases will be developed showing how the organizational structure of the
network and in particular the level of homgeneity of market shares of member banks with regard to issuing of
cards and acquiring of merchant might provoke single or as worst double marginalization with regard to card
payment fees.
Key words: two-sided markets, payment card networks, payment systems, credit card associations, platform
pricing, no surcharge rule
Methodological area: industrial organization
INTRODUCTION
Over the past decades the interchange fee of multi-party payment networks has been a means of discussion
among banks, policy makers, and economists (Armstrong, 2006, Evans, & Schmalensee, 2005, Gans, & King,
2003, Rochet, & Tirole, 2003, Rysman, 2007, Wright, 2004). One common conclusion of many of these
discussions is the fact that still there remains a significant gap in understanding, thus determining efficient
payment card pricing. Langlet (2009) introduced a model of unitary payment card network(s). This model
explores the considerations of unitary schemes in the process of determining privately optimized payment card
prices. In this paper, this model shall be adapted for the situation of multi-party payment card systems. In doing
so, two extreme cases will be discussed with the truth being somewhere in between these two extreme cases.
The first case will consider two banks with identical shares of issuing and acquisition business. Interestingly, this
second case involves only single marginalization, since for each of these banks the total of paid and received
interchange fees will be zero. Consequently, such banks’ will tend to agree on a high interchange fee in order to
transfer profits further away from merchants. The second case assumes perfect distinction between issuing and
acquiring banks. In this case there is no bank issuing payment cards and also acquiring merchants to join the
network. The result will be double-marginalization. The issuing banks being in the upstream position, therefore
expects to achieve higher profits than the acquiring banks.
Some of the results of this paper will be similar as for the model of unitary networks. Even so, it becomes
evident that generally speaking, multi-party systems involve more complexity than unitary systems. As in
Langlet (2009) during the analysis the focus is on three determinants of the merchant fee; the consumer price
elasticity of the good exchanged on the downstream market, the relative frequency of card usage, and the
competitive condition of merchants, that is the downstream market. All together, the above cited complexity
makes it hard for policy makers to find a way toward efficient payment card regulation. Anyways, the results
below promise to reduce such complexity by untangling the determination of payment card pricing.
MODEL
Langlet (2009) explored payment card pricing and in particular the determination of the merchant fee for unitary
networks (e.g. American Express). In this paper the model of Langlet will be adapted to the case of a multi-sided
payment card network, such as Visa or Mastercard are. Thus, Langlet’s assumptions, as well as Lemmas and the
second Proposition, which all concern the behaviour of merchants, will be borrowed in the below model and
therefore are summarized in the following.
Consider a homogenous product being exchanged between N identical merchants, who are indexed by n, and
consumers for a standard price p (perfect price transparency). Let Q be the total sales volume of all merchants.
Furthermore, assume constant consumer price elasticity ε, such that from the perspective of merchant n the
inverse demand curve is characterized by:
p  Q  with 0    1 .
(1)
Merchants may offer two payment methods to their customers: cash or card. Suppose γ being the portion of
consumers exclusively choosing to pay by card, i.e. there is a fixed relative frequency of card usage. Consider an
effective no-surcharge rule. Thus, merchants cannot surcharge customers who pay by card. In order to motivate
customers to use payments cards, usually are no transaction fees imposed on customers, sometimes even bonuses
are given for card usage. On the other hand, it is common practice to charge merchants a usage fee. Thus, for any
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card transaction there shall be a standard usage fee imposed on merchants. To be more precise, consider this
merchant fee to be a standard rate a in form of a percentage of the transactions volume (see figure 1).
Furthermore, let there be perfect blending with regard to this merchant fee. Furthermore let merchants incur no
further costs in regard to card transactions or any other fees than a (such as e.g. membership fees).
By assuming merchants behave according to the Cournot-Nash equilibrium, Langlet in his first lemma derived
the equilibrium sales quantity of all merchants, such that:
1


 
 1    1    a  
N
N
 ,
Q*   qn   


c
n1




(2)
and the equilibrium Cournot price of the respective product:
p* 
c
 

1    1    a 
N

.
(3)
In his second lemma Langlet furthermore derived that merchants will certainly not participate in the payment
card network in any case. Much rather, there will be one maximum merchant discount rate for which merchants
make zero profit from card-paid sales, which is the case for a merchant discount fee rate according to:
a
ε
.
N  N γ  εγ
(4)
Mmerchants will participating in the network when a  a , while merchants will refrain from participating when
a  a . Finally, just as Langlet assumed, let there be only fixed costs of network operations and no variable
costs. Alike for unitary payment networks, we can expect one equilibrium N* such that for any N  N*,
merchants will make zero profit from sales paid by card. And for all N<N*, merchants will achieve profits even
from sales paid by card. Also in case of a multi-party payment card system, for all N  N* acquiring banks will
set the equilibrium merchant fee according to a*  a , since otherwise merchants would not participate in the
network.
In contrast to Langlet’s model of unitary payment card networks, consider one multi-party payment card system,
consisting of a network operator and member banks, which firstly issue payment cards and secondly acquire
merchants to join the network and accept the cards. In the below model two characteristic scenarios will be
explored, with reality being a mix of these two extreme cases. For the first scenario consider two identical banks
that to the same extent firstly issue payment cards and secondly acquire merchants to accept the cards. For the
second scenario, let the banks specialize, such that banks exclusively either issue payment cards or only acquire
merchants for the network. In order to shift profits from the acquiring side to the issuing side of the network, the
acquiring bank usually pays an interchange fee to the issuing bank. Therefore, in the blow model, similarly as for
the merchant fee, for any transaction consider the respected issuing bank to charge the acquiring bank a standard
usage interchange fee rate f in form of a percentage of the transaction volume.
Just as it is the case for e.g. Visa, consider the network operator to be a not-for-profit organization. Just as in
Langlet (2009), let there be only fixed costs for establishing and running the payment card network and
furthermore no variable costs with regard to card transactions. Therefore let the network operator only impose a
standard (fixed) membership fee on the member banks in order to cover the (fixed) costs of network operation.
Consider no other costs for either issuing or acquiring bank(s), as well as for card holders (such as e.g. annual
card holder membership fees etc.). Consider the network operator also not to influence the level of the network’s
fees other than requesting the discussed fixed membership fee. Furthermore, consider a two phase game of
negotiations of the interchange fee rate f and the merchant discount rate a: Firstly, let the issuing bank and the
acquiring bank negotiate the interchange fee f in such manner, that the issuer makes a take-it or leave it offer to
the acquirer. Secondly, consider the acquiring bank to make a take-it or leave-it offer to merchants, thus
negotiating the merchant fee rate a.
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Homogenous
product
sold for price p
Merchant n
(out of a population N)
Customer
(choosing card payment)
Pays price p
minus merchant discount
1  a p
Pays
price p
Acquiring bank
membership
fee (fixed)
Issuing bank
Pays price p
minus interchange fee
1  f  p
membership
fee (fixed)
Network operator
Payment card network
a - Merchant usage fee
f - Interchange fee
Figure 1: Payments in a multi-party payment card network (Cp. Gans, King, 2003, p. 4.)
The next two sections contain an analysis of the considerations of oligopoly member banks of the payment card
network for the two scenarios introduced above; firstly the scenario of identical banks which are executing the
issuing as well as the acquisition function of the network to an equal extent and secondly the scenario of banks
exclusively either issuing payment cards or acquiring merchants to join the network.
The Scenario of Banks Issuing and Acquiring According to Identical Proportions
Consider the member banks carry out the issuing as well as the acquiring function according to the same
proportion, i.e. each member banks market share s of issuing cards and acquiring merchants is identical. As
figure 2 illustrates, with the assumption of identical market share in issuing and acquiring, the probability pAB of
one particular bank A being the issuer and another bank B being the acquirer is identical to the probability pBA of
bank B being the issuer and bank A being the acquirer with regard to a certain transaction. Consequently, bank A
will receive the same total amount of interchange fee from bank B as bank B will have to pay to bank A. For
each of the banks the total of received and paid interchange fees will equal zero.
Issuing bank i
Acquiring bank j
s(A)
s(A)
pAA=s2(A)
s(x) – Market share of bank x∈{A,B}
s(B)
s(B)
pAB=s (A)·s(B)
s(A)
pBA=s (B)·s(A)
s(B)
pBB=s2(B)
pij – Probability of issuer i∈{A,B} and acquirer j∈{A,B}
Figure 2: Probabilities of Partner Bank Combinations for a Random Transaction
Consequently, the level of the interchange fee will be irrelevant for the profit each of the banks will gather from
card transactions. Only the level of the merchant fee matters. In contrast to the second scenario, as will be shown
below, this first scenario is characterized by only single marginalization over acquiring of merchants and the
issuing of payment cards. It is obvious to see, that with member banks maximizing profits over the merchant fee,
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the equilibria of the above scenario will follow the same logic as for unitary payment card systems. Thus, in the
rest of this section the findings of Langlet (2009) concerning unitary payment card networks will be summarized
and reinterpreted for the above scenario of a multi-party payment card network.

N  N    
Merchant fee a
a 
â



a*  min  ;


N

N








1 
N* 
1
1


Number of merchants N

Figure 3: Merchant Fees for the First Scenario of Identical Proportions of Issuing and Acquring (Langlet
2009, p. 10)
Assuming identical profit-maximizing member banks (i.e. identical costs and market share), the profit of one
member bank i will be
  

 1    1    a  
N

i   


c




1 


 a
I
C .
(5)
As we learned, the member banks will maximize profit Пi over the merchant discount, thus
Max  i ,
(6)
a
Assuming profit maximizing banks, that is identical merchant fee revenue of all banks… Intuition of the proof.
Solving this term leads to the merchant striving to achieve a merchant fee according to
Lemma 1. As long as merchants accept such merchant fee, that is when a  a , the oligopoly bank (in
case of the first scenario of identical banks issuing and acquiring to an equal extent), will maximize
profits by defining the merchant fee according to
â


(7)
This leads to the first roposition:
Proposition 1. As long as merchants accept such merchant fee, that is when a  a , the oligopoly bank (in
case of the first scenario of identical banks issuing and acquiring to an equal extent), will maximize
profits by defining the merchant fee according to
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9th Global Conference on Business & Economics
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


a*  min  ;
.
 N  N       
(8)
As shown in Langlet (2009), N=N* is the competitive condition of merchants, where firstly merchants make
zero profit from sales paid by card, that is, a  a (see Eq. 4) and secondly the network is able to achieve the
maximum profit by still acquiring the merchant fee â according to Eq. (7). By substitution of the two conditions,
we learn that
N* 
1 
.
1
1
(9)

In the next section the scenario of member banks exclusively either issuing payment cards or acquiring
merchants will be explored.
The Scenario of Exclusive Distinction of Issuing and Aquiring
Consider the case of a clear distinction of issuing of payment cards and the acquisition of merchants to
participate in the payment card network. Furthermore suppose there is a total of J oligopoly acquiring banks
indexed by j. Let γ·qj denote the total number of payment transactions respectively bank j processes. Thus, with
no other variable costs but the interchange fee one acquirer’s operational profit Пj will be the merchant usage fee
revenue minus the total interchange fee he will have to pay to issuing banks and fixed costs Cj, thus
 j  p  qj   a  p  qj   f  Cj .
(10)
Suppose an oligopoly acquirer maximizing profit over the usage fee:
Max  j ,
(11)
a
with the constraint of a positive profit.
a 

N  N    
Fees a, f
â  2    



fˆ 

N* 
Number of merchants N
1   2
1 
  1  2   





f *  min  ;

 N  N       




a*  min 2     ;

 N  N     

Figure 4: Fees for the Second Scenario of Banks Eeither Issuing Or Acquring
The first-order condition of Eq. (10) can be solved under the assumption of profit-maximizing oligopoly
acquiring banks, thus with identical market share. Therefore each of these acquirers will process an equal
number of payment transactions, such that
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qj 
ISBN : 978-0-9742114-2-7
q j
J 1

Q
j
(12)
with γ·q-j denoting the total number of payment transactions all the other acquiring banks but bank j process. The
transformation of yield in the second-order condition of Eq. (2) proves that the below solution (Lemma X)
maximizes the profit of acquirer j.
Lemma 3. As long as merchants accept such merchant fee, that is when a  a , and with banks exclusively
either issuing or acquiring, the oligopoly acquirer will maximize profits by defining the merchant fee
according to
aˆ 

 1     f .

(13)
Furthermore, consider a total of K oligopoly issuing banks indexed by k, each of which only incurring fixed costs
Ck and no variable costs with respect to card transactions. Let γ·qk denote the total number of payment
transactions issuing bank k processes. Consequently the issuing bank’s profit Пk will be the total interchange fee
revenue minus costs of issuer k
 k  p  qk    f  Ck .
(14)
The oligopoly issuer will maximize profit over the interchange fee
Max  k .
(15)
a
The intuition of the proof is to solve the first-order condition of Eq. (14). Again assume profit-maximizing
oligopoly issuers with identical market shares, therefore processing an equal number of payment transactions.
Consequently
qk 
q k
Q

K 1 K
(16)
again with γ·q-k denoting the total number of payment transactions all the other acquiring banks but bank j
process. Similar as before, the transformation of yield in the second-order condition of Eq. (14) proves that the
following solution maximizes the profit of acquirer j:

fˆ  .
(17)

Consider the above assumption of issuing banks negotiating the interchange fee with acquirers in form of a takeit or leave-it offer. Any such offer will be accepted by acquiring banks as long as (see Eq. 4), since in that case
acquirers will be able to make profit by requesting an even higher merchant discount fee rate a from merchants,
who themselves will not accept any merchant discount rate a  a . This leads to the second proposition:
Proposition 2. In the assumed case of distinction of issuing and acquiring, as well as with the constraint
of not defining a interchange fee rate f higher than the maximum merchant discount rate, that is
f  a , oligopoly issuing banks will strive after maximizing profit according to Eq. (17), thus the
equilibrium interchange fee will be determined according to



f *  min  ;


N

N







.
(18)
This leads to the second proposition, while substituting Eq. (17) in Eq. (13):
Proposition 3. With banks exclusively either issuing or acquiring and with the constraint that merchants
will not accept a merchant fee rate a  a , the acquiring banks will strive after maximizing profits by
requesting an merchant discount according to Eq. (13), thus the equilibrium merchant discount fee a* will
be determined by
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9th Global Conference on Business & Economics
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



a*  min 2     ;

 N  N     

.
(19)
From this N*, the competitive condition of merchants where: (i) merchants make zero profit (only normal profit)
from sales paid by card , that is, a  a and (ii) the issuing as well as the acquiring bank are able to achieve the
maximum profit by acquiring the merchant fee â according to Eq. (13) and with an interchange fee fˆ according
to Eq. (17). The solution of these two conditions, that is, substituting â from term (13) in Eq. (4), brings about
the fourth proposition. (See figure 3 and details of the proof in the appendix.)
Proposition 4. With a merchant population of N=N*, the maximum merchant fee a from Eq. (13) equals
â from Eq. (21) with issuers and acquirers making maximum profits; thus,
N* 
1   2
1

  1  2   



.
(20)
DISCUSSION AND CONCLUSION
Langlet (2009) analyzed the determination of unitary payment card networks with the focus on three
determinants of the merchant fee: consumer price elasticity, the relative frequency of card usage and the
competitive condition of the downstream market, indicated by the number of oligopoly merchants.
With the same focus, in this paper the model of Langlet was transferred to the case of a multi-party payment card
network. In doing so, two key cases were analyzed: firstly a network with identical banks that issue payment
cards to the same extent as they acquire merchants to join the network and secondly the case of specialized
member banks that either issue payment cards or acquire merchants for network participation. In this study a
two-phase game was considered. In the first phase issuers and acquirers negotiate the level of the interchange
fee, while in the second phase acquirers and merchants negotiate the level of the merchant usage fee. For
parsimony such negotiations were assumed to be take-it or leave-it offers of the respected upstream partners, i.e.
issuers in phase 1 and acquirers in phase 2.
Generally speaking the above three determinants of unitary payment card system pricing were found to similarly
influence the fee equilibria of multi-sided payment card systems. For the first case of identical member banks,
the outcome was even practically the same as for the above cited study of unitary networks. For these situations
the fee equilibria are characterized by single marginalization. Interestingly, for this case the level of the
interchange fee was even found to be irrelevant for the merchant usage fee level or the bank’s profit which has to
do with the assumed negociation set-up and will be discussed lateron as an extention of the foregoing model.
In case of banks either issuing payment cards or acquiring merchants to join the network, there will be double
marginalization in the network with a good chance of leading to inefficient price equilibrium. This supports for
example policymakers approach to regulate the interchange fee, with a high potential of abuse of the upstream
player’s position.
In the following I discuss three possible extensions of the above study. First, alike Langlet (2009) for unitary
payment card networks, we could extend the above model by giving some bargaining power to the downstream
party. Consider the first case of identical member banks and allow Nash-bargaining over the merchant discount
rate with αi denoting the bargaining power of the member banks. The bargaining will happen over the profit
margin of the acquiring bank. In contrast to the case of a unitary network, for the multi-sided payment system
this profit margin of the acquirer is influenced by the prior negotiated interchange fee and the market share of the
bank denoted by si, thus
a N   i  a * 1  si   f  .
(21)
Still the total of paid and received interchange fee is zero (see discussion above), thus still the level interchange
fee remains irrelevant in regard to the bank’s profit as long as the merchant usage fee is kept constant. Only that
with the bargaining situation the merchant fee depends on the interchange fee. Thus, by agreement of a high
interchange fee, the banks can in advance negotiate away the profits from the acquiring to the issuing side of the
network. In fact the banks would define the interchange fee so high that the profit margin of acquiring will be
zero. In this situation even though merchants may try to negotiate with the banks over the merchant discount
rate, they will achieve no different outcome than in case of the take-it or leave-it offer. Only by changing the
game, that is with merchants that would also influence the level of the interchange fee, a bargaining between
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merchants and banks would influence the outcome, i.e. the equilibrium merchant usage fee. In this context
interchange fee regulation appears particular efficient, as it limits the banks’ ability to negotiate away profits
before entering into negotiation with merchants over the merchant usage fee.
In contrast to the first case, in case of banks either issuing cards or acquiring merchants Nash bargaining would
simply impact the level of the negotiated fees, such that the parties would agree accordingly, thus
a N   i  a * respectively a   i  f .
N
(22)
Apparently there would be no back-loop-effect influencing the prior negotiations, since the different levels of
negotiation partners are clearly separated organization, that is merchants, issuing banks and acquiring banks.
Second, in the foregoing model two natural cases were observed with reality somewhere in between these two:
identical banks and banks that either issue cards or acquire merchants for the network. A deeper analysis is
needed to disentangle the reality in between of these two key cases. In particular how the equilibrium fees
develop on the interval between these two cases and the implication on fee regulation might be.
A third natural extension would consider other genders of two-sided markets, such as for example video
consoles, shopping malls or supplier platforms (as they exist in the automotive industry). Transfers are possible
for example in regard to the findings concerning organizational levels of platforms, which apparently can be
organized as natural one-level organization (i.e. unitary network), as two-level systems (multi-party network
with members either operating as issuers or acquirers) or an organizational mix somewhere on the scale of the
foregoing extreme cases.
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References
Armstrong, M. (2006). Competition in two-sided markets. The Rand Journal of Economics, 37 (3), 668-691.
Evans, D., & Schmalensee, R. (2005). The economics of interchange fees and their regulation: An Overview.
MIT Sloan Working Paper 4548-05 (July 8, 2005), accessed April 14, 2008, available at http://dspace.mit.edu/
Gans, J., & King, S. (2003). The neutrality of interchange fees in payment systems. Topics in Economic Analysis
& Policy, 3 (1). 1-16.
Langlet, M. (2009). Pricing of Unitary Payment Card Networks: The Relationship between Consumer Price
Elasticity and Merchant Fees. Forthcoming in: Oxford Journal.
Rochet, J., & Tirole, J. (2003). An economic analysis of the determination of interchange fee in payment card
systems. Review of Network Economics, 2 (2), 69-79.
Rysman, M. (2007). An empirical analysis of payment card usage. Journal of Industrial Economics, 55 (1), 1-36.
Wright, J. (2004). The determinants of optimal interchange fees in payment systems. Journal of Industrial
Economics, 52 (1), 1-26.
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APPENDIX

Proof of (13):
Assuming supernormal profits of merchants, the oligopoly acquirer j will be able to freely maximize profit Πj over the usage fee a, while considering the interchange
fee f as costs
1
(11)
Max  j with  j  q 1
a


 
 1    1    a  

 N 

  a  f   C j , and q  q C  
 .
c
J






 
 1    1    aˆ  
N


Thus, Max 

a
c




1



J
 aˆ  f   C j .
The first order condition is
 

   1  

 N  1 
j 

c



 
 1    1    aˆ  
 N 



c




1 2



 
 1    1    aˆ  

 N 

  aˆ  f   

J
c




 
 


   1  
1    1    aˆ 
N
N
1










  aˆ  f  
 0
c

J
c
J
  
1 

 aˆ  f   1    aˆ   0
   aˆ      aˆ    f      f        aˆ  0
October 16-17, 2009
Cambridge University, UK
11
1



J
!
0
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
 aˆ  f    f 
 (13)
aˆ 

0


 1    f with 0    1 , as well as 0    1 , and 0  f  aˆ  1 .

The second order condition proves this solution to be a maximum:
2

 

    1   
 
 N   1   1  2
j 
    
c




2
 
 
   1   
   N   1   1  2
j 
    
c






 
 1    1    aˆ  
 N 



c






 
 1    1    f  
 N 



c






 
 1    1    f  
N
1






j 


 
c




1 3


1 3

 

  1  

 N  1 
  aˆ  f   2 

J
c



 
 1    1    aˆ  
 N 



c




 

  1  

     f 
 N  1 

 2

J
c

1 2



 
 1    1    f  
 N 



c






.
J
1 2


2
 
 
 
    

   1   
  1    1    1    f   
1   N   1  2
 
 N   N 
  

       f   2 


  
J
c

c
c



 



 


 
 1    1    f  
N
1






j 


 
c




October 16-17, 2009
Cambridge University, UK
1 3
1 3

2
 
 
   1   
1   N 
 
  1  2   1    f   2  1    f 
J
c




12

J
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
j
1  
 

 1  
  N

with
and as long as
as long as
1 
1  2


1

 1    f 
1 

1 2

1
 
c
as long as
1 



1 2
   1  2    1
J
 0 , since 0    1 ,
 
1

 1 , thus   with 1  
2
 N
 1 , thus  
1
1

 0 , since 0    1 and N  1 ,
1 2
1
with 1    f    0 , since 0    1 and 0  f  1 ,
3
1
1
 1 , thus   with  
2
c
1

 0 , since 0  c ,
1
 0 , since J  1 ,
J
with  2  0 , and last but not least 1  2  0 , since   0 .
1
1

Consequently,  j  0 for sure with 0    and most likely with    1 .
3
3

Proof of (17):
Assuming supernormal profits of merchants, i.e. a  aˆ , issuer k will be able to freely maximize profit Πk over interchange fee f
(15)
October 16-17, 2009
Cambridge University, UK
Max  k
f
13
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
1
with  k  q 1


 
 1    1    aˆ  


 N 

  fˆ  C k , as well as q  q C  
and aˆ   1     f .

c
K







2


 1     1    fˆ 
 N 

Thus, Max 

f
c






1



K
 fˆ  C k .
The first order condition is


2


 

 1     1    fˆ 
   1  


 N  1    N 
k 



c
 
c





2
1 2




 
 


   1  
1    1    fˆ

 N  1   ˆ  N 


 f
 0
c
 K
c
K
2
2


 1        fˆ    1    fˆ  0
   fˆ    0
 (17 )
The second order condition proves this solution to be a maximum:
October 16-17, 2009
Cambridge University, UK
14

2


 1     1    fˆ 
 N 


  fˆ  

K
c







fˆ  .

1



K
!
0
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
2
2


    1    
 
 N   1   1  2
k  

 
c










2


 1     1    fˆ 
 N 



c






1 3

 

  1  
N


 1 
  fˆ  2 

K
c

2


2


 1     1    fˆ 
 N 



c






1 2



with fˆ 

2
2


    1    
 
 N   1   1  2
 k  

 
c








2


 1     1    

N
 
 k   

c






1 3

2


 1     1    
 N 



c







 

  1  
N


 1 
  2

K
c

2
2


 1     1    
 N 



c






2
2
2


 

    1    
  1  
1 
 N    1    1  2    2   N   1  
  

K 
c


c







2


 1     1    

 1    N 
 k 


 
c





 k
October 16-17, 2009
Cambridge University, UK
1 3

1  
 

 1  
  N
1 3

2

 1   
15
1 3

1
 
c
1


1 2
    1
K



K
2



 1     1    

 N 
  

 
c

 K






2


    1    

 N  1

   1  2  2  1   
c

 K




2  2
1 2

K
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
with
and as long as
2  2

as long as
1 

 0 , since 0    1 ,
2
 

 1 , i.e.   with 1  
3
 N
1  3

as long as
 1 , i.e.  
1 

2  2

 0 , since 0    1 and N  1 ,
13
1
with 1      0 , since 0    1 ,
4
1
1
 1 , i.e.   with  
2
c
1

 0 , since 0  c ,
1
 0 , since K  1 ,
K
and last but not least  2  0 .
1
1

Consequently,  k  0 for sure with 0    and most likely with    1 .
4
4
Q.e.d.
October 16-17, 2009
Cambridge University, UK
16
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
Assuming normal profits of merchants, i.e. a  a , issuer k will be able to freely maximize profit Πk over interchange fee f
(15)
Max  k
f
1
with  k  q 1


 
 1    1    a  


 N 

  fˆ  C k , as well as q  q C  
and a 
.

c
K
N  N    










  
1


 1  N   1  N    N

1 


 



Thus, Max 

f
c










1



K
 fˆ  C k .
The first order condition is
 

   1  
N


 1 
k 

c

2


2


 1     1    fˆ 
 N 



c






1 2




 
 


   1  
1    1    fˆ

 N  1   ˆ  N 


 f
 0
c
 K
c
K
2
2


 1        fˆ    1    fˆ  0
October 16-17, 2009
Cambridge University, UK
17

2


 1     1    fˆ 
 N 


  fˆ  

K
c






1



K
!
0
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
   fˆ    0

fˆ  .
 (17 )

The second order condition proves this solution to be a maximum:
2
2


    1    
 
 N   1   1  2
k  

 
c










2


 1     1    fˆ 
 N 



c






1 3

 

  1  

 N  1 
  fˆ  2 

K
c

2


2


 1     1    fˆ 
 N 



c






1 2



with fˆ 

2
2


    1    
 
 N   1   1  2
 k  

 
c








2


 1     1    

  N 
 k  

c






October 16-17, 2009
Cambridge University, UK
1 3

2


 1     1    
 N 



c






1 3

 

  1  

 N  1 
  2

K
c

2
2


 1     1    
 N 



c






2
2
2


 

    1    
  1  
1 
 N   1   1  2
 N  1 
  

  2 

 
K 
c


c






18
1 2



K
2



 1     1    

 N 
  

 
c

 K







K
9th Global Conference on Business & Economics
ISBN : 978-0-9742114-2-7
2


 1     1    

 1    N 
 k 


 
c





 k

1  
 

 1  
  N
with
and as long as
2  2

as long as
1 3

2
2


    1    

 N  1

   1  2  2  1   
c

 K




2  2
1 


 1   


as long as
 1 , i.e.  
1 



1 2
    1
K
2  2

 0 , since 0    1 and N  1 ,
13
1
with 1      0 , since 0    1 ,
4
1
1
 1 , i.e.   with  
2
c
1

 0 , since 0  c ,
1
 0 , since K  1 ,
K
and last but not least  2  0 .
1
1

Consequently,  k  0 for sure with 0    and most likely with    1 .
4
4
Q.e.d.
October 16-17, 2009
Cambridge University, UK
1
 0 , since 0    1 ,
 
2

 1 , i.e.   with 1  
N
3


1  3
1
 
c
1 3
19