Payment System Self-Regulation through Fee Caps

Fabian Griem

doi:10.1515/rne-2020-0015

Article

Payment System Self-Regulation through Fee Caps

Fabian Griem

Published/Copyright: August 12, 2020

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From the journal Review of Network Economics Volume 18 Issue 3

Abstract

This paper considers the organization of a single (domestic) payment system. When card issuers that are members of a payment system set their fees individually, this gives rise to a free-riding problem, as in providing access to different customers, card issuers are complements from the perspective of each merchant. When payment systems can threaten to exclude, in particular, card issuers with a smaller customer base that do not adhere to a common cap on fees, this allows to restore the full internalization outcome, leading to lower fees but higher profits and higher welfare. When payment systems cannot threaten to exclude card issuers, the full internalization outcome arises only when card issuers are sufficiently symmetric.

Keywords: payment systems; fee caps; antitrust

JEL Classification: L11; L14; L40

Corresponding author: Fabian Griem, Goethe University Frankfurt, Chair of Finance and Economics, Theodor-W.-Adorno-Platz 3, 60323Frankfurt am Main, Germany, E-mail: f.griem@em.uni-frankfurt.de

Appendix A Nash Bargaining over Individually Set Fees

I now consider an axiomatic Nash bargaining game with i∈ℐ={1, 2}. The two issuers negotiate within a single payment scheme over their fees (p₁, p₂). In what follows, I show that both agree under the threat of “mutual exclusion”. Recall that

π1=v1(p1−c)H(vs−v1p1−v2p2),

while π2 is defined analogously. These define the mapping (p1,p2)→(π1,π2), where I can restrict consideration to the Pareto non-dominated outcome. The resulting bargaining frontier can now be constructed as follows. Suppose that i = 2 must leave i = 1 with a certain value of π1≥π˜1. The “contract” on the frontier then realizes

π2=arg maxp1,p2v2(p2−c2)H(vs−v1p1−v2p2) s.t. π1=v1(p1−c1)H(vs−v1p1−v2p2)≥π˜1.

It is obvious that the constraint always binds, otherwise p₁ could be decreased which would increase π2 due to a higher acceptance probability. Then, I can write the bargaining frontier as π2=π2*(π1).

The standard procedure to derive the Nash bargaining solution is now as follows. To obtain a unique outcome, it is sufficient that the bargaining frontier is concave. The Nash bargaining solution (π1, π2) maximizes the Nash product (π1−π1O)(π2−π2O), where the factors are the differences with respect to the bargaining frontiers and the outside options.

I now apply immediately the uniform distribution to H(k) and assume that c_i = c. From the binding constraint, I obtain

π2*(π1)=π1v2v1arg maxp1,p2p2−cp1−c.

This suggests that on the bargaining frontier (p₂ − c)/(p₁ − c) must be highest, however, note that I clearly cannot freely choose both p_i. Turning again to the uniform distribution so that

π1=v1(p1−c)1K(sv−v1p1−v2p2),

which I can rewrite as

p2=1v2(sv−v1p1−Kπ1v1(p1−c)).

Now, simplifying by setting c = 0, I thus want to maximize

p2p1=svv1p1−(v1p1)2−Kπ1v1v2p12.

The sign of the derivative with respect to p₁ is then given by

v1v2p12(svv1−2v12p1)−2v1v2p1(svv1p1−v12p12−Kπ1)=2Kπ1−svv1p1,

from which I obtain

p1=2Kπ1sv1(v1+v2).

Inserting yields

p2p1=svv1p1−(v1p1)2−Kπ1v1v2p12=s(v1+v2)v12Kπ1sv1(v1+v2)−(v12Kπ1sv1(v1+v2))2−Kπ1v1v2(2Kπ1sv1(v1+v2))2=14Kπ1v2(s2v13+2s2v12v2+s2v1v22−4Kπ1v1),

so that

π2*(π1)=π1v2v1arg maxp1, p2p2p1=π1v2v114Kπ1v2(s2v13+2s2v12v2+s2v1v22−4Kπ1v1)=14Kv1(s2v13+2s2v12v2+s2v1v22)−π1.

I have thus shown that at the optimally chosen fees (p₁, p₂) the bargaining frontier is in fact linear (i.e., π1+π2*(π1) is the same and thus independent of how profits are distributed). I can thus indeed apply the Nash bargaining solution, which now simplifies to the requirement that

(π1−π1O)=(π2−π2O).

It remains to specify the outside options, where each issuer operates a separate payment system. Hence, with π1=v1(p1−c)1/K(sv1−v1p1) and now c_i = 0, I obtain p₁ = s/2 and thus π1O=1/(4K)s2v12. This holds analogously for i = 2. Now, I substitute for the outside options and for π2=π2*(π1) to obtain

(π1−π1O)=(π2−π2O)

π1−14Ks2v12=14Kv1(s2v13+2s2v12v2+s2v1v22)−π1−14Ks2v22.

This yields the equilibrium profits

π1=14Ks2v1(v1+v2)

and

π2=14Kv1(s2v13+2s2v12v2+s2v1v22)−π1=14Kv1(s2v13+2s2v12v2+s2v1v22)−14Ks2v1(v1+v2),

which simplifies to

π2=14Ks2v2(v1+v2).

The corresponding values of p₁ and p₂ on which they thus agree must jointly solve

v1p11K[s(v1+v2)−v1p1−v2p2]=14Ks2v1(v1+v2),

v2p21K[s(v1+v2)−v1p1−v2p2]=14Ks2v2(v1+v2),

leading to the symmetric full internalization outcome because p₁ = p₂ = s/2, independent of any asymmetries in size.

I thus have shown that with two parties Nash bargaining over the fees (p₁, p₂) and the threat of exclusion the full internalization outcome is obtained.

Acknowledgments

I thank my supervisor Roman Inderst for valuable support and continuous guidance.

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Received: 2020-02-26

Accepted: 2020-04-27

Published Online: 2020-08-12

Published in Print: 2020-09-25

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Articles in the same Issue

https://doi.org/10.1515/rne-2020-0015

Keywords for this article

payment systems; fee caps; antitrust