Who Pays for Card Payments? A General Model on the Role of Interchange Fees

Carlotta Mariotto; Marianne Verdier

doi:10.1515/rne-2018-0003

Article

Who Pays for Card Payments? A General Model on the Role of Interchange Fees

Carlotta Mariotto and Marianne Verdier

Published/Copyright: September 6, 2018

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From the journal Review of Network Economics Volume 16 Issue 3

Abstract

When a consumer pays by card, the merchant’s bank pays to the consumer’s bank an interchange fee. In this article, we construct a general model of a card platform that unifies the literature on interchange fees. We enrich the existing frameworks by analyzing the choice of the interchange fee when consumer demand is elastic to retail prices. We show that the difference between the privately set structure of payment card fees and the socially optimal one depends both on banks’ and merchants’ pass-through of their costs to consumers. We argue that the maturity of the payment card market impacts the redistributive effects of interchange fees (i.e. between consumers and merchants, card and cash users) and therefore, their optimal regulation.

Keywords: interchange fees; payment systems; two-sided markets

JEL Classification: E42; L1; O33

Acknowledgments

An earlier version of this article circulated under the title “Double Internalization and Interchange Fees.” We thank Emilio Calvano, Bruno Jullien, Myongjin Kim, Victor Hiller, participants to the workshop on the economics of platforms at the National University of Singapore (2015), the workshop “Industrial Organization in the Digital Economy” at the University of Liège, seminar participants at the Bank of Finland (May 2015), Norges Bank (June 2016), participants to the IOC conference (2015), ESNIE (2015), EARIE (2015) and ICT (2015) for helpful comments and discussions. We also thank two anonymous referees and the editor for their valuable comments. This paper was awarded the accessit price for the best paper at ESNIE conference in 2015.

Appendix

A Appendix

Appendix A-1: Pass-through Rates under the No Surcharge Rule

We start by studying how the retail price pGpc varies with the acquirer fee. Since from the second-order condition of merchant-profit maximization we have ∂2ΠSpc/∂2ppc|pGpc≤0, from the implicit function theorem, ∂pGpc/∂pS has the sign of ∂2ΠSpc/∂ppc∂pS|pGpc. From (2), we have

∂2ΠSpc∂ppc∂pS|pGpc=∂DBpc∂ppc|pGpc∂mS∂pS|pGpc+∂2mS∂ppc∂pS|pGpcDBpc.

Since ∂DBpc/∂ppc|pGpc≤0 and ∂mS/∂pS|pGpc≤0, the first term of the equation above is positive. If the acquirer fee is a fixed per transaction fee, since mS=ppc−pS+bS, we have ∂2mS/∂ppc∂pS|pGpc=0. This implies that

∂2ΠSpc∂ppc∂pS|pGpc≥0.

Hence, we have that ∂pGpc/∂pS≤0. This implies that the retail price pGpc increases with the acquirer fee.

If the acquirer fee is a proportional per transaction fee, since mS=ppc(1−pS)+bS, we have ∂2mS/∂ppc∂pS|pGpc=−1≤0. Therefore, if εpc(pGpc)≤1, we have

∂2ΠSpc∂ppc∂pS|pGpc=DBpc(εpc(pGpc)−1)≤0.

Therefore, if the acquirer charges a proportional per transaction fee, if εpc(pGpc)≥1, the retail price increases with the acquirer fee. Remark that a monopolistic merchant chooses a retail price such that εpc(pGpc)≥1.

We now study how the retail price pGpc varies with the issuer fee. Since ∂2ΠSpc/∂2ppc|≤pGpc0 from the second-order condition of merchant-profit maximization, from the implicit function theorem, ∂pGpc/∂pB has the sign of ∂2ΠSpc/∂ppc∂pB|pGpc. From (2), we have

Since ∂mS(bS,pS,ppc)/∂ppc≥0, ∂mS(0,0,ppc)/∂ppc≥0, ∂DBpc/∂pB≤0, ∂DBc/∂pB≥0, the sign of ∂2ΠSpc/∂ppc∂pB|pGpc is ambiguous. Therefore, the retail price may either increase or decrease with the issuer fee.

We now show that in the special case in which consumer demand is linear, and cash user demand does not depend on the card fee, the retail price decreases with the issuer fee. The cross-derivatives ∂2DBpc/∂ppc∂pB|pGpc and ∂2DBc/∂ppc∂pB|pGpc are equal to zero if consumer demand is linear and if the issuer charges a fixed per-transaction fee. The magnitude of ∂DBc/∂pB≥0 represents the degree of substitutability between cash and cards. In the special case in which cash user demand does not depend on the card fee (i.e. ∂DBc/∂pB=0), we have that ∂2ΠSpc/∂ppc∂pB|pGpc≤0. Therefore, the retail price decreases with the issuer fee. In other cases, the retail price may either increase or decrease with the issuer fee.

Appendix A-2: The Impact of Banks’ Prices on the Marginal Merchant

Since the marginal merchant is implicitly defined by ΠSpc(bS^,pS,pB,pGpc,pGpc)=ΠSh(pGh), from the implicit function theorem, we have

dbS^dpS=−∂ΠSpc/∂pS∂ΠSpc/∂bS|pGpc,

and

dbS^dpB=−∂ΠSpc/∂pB∂ΠSpc/∂bS|pGpc.

From (2), taking the derivative of ΠSpc with respect to p_S and b_S, respectively, we have

∂ΠSpc/∂pS|pGpc=DBpc(pB,pGpc,pGc)∂mS(bS,pS,ppc)/∂pS|pGpc,

and

∂ΠSpc/∂bS|pGpc=DBpc(pB,pGpc,pGc)∂mS(bS,pS,ppc)/∂bS|pGpc.

From the envelope theorem, we can ignore the impact of p_S and b_S on the profit-maximizing retail prices pGpc and pGc.

Since ∂mS(bS,pS,pGpc)/∂pS|pGpc≤0 and ∂mS(bS,pS,ppc)/∂bS|pGpc=1≥0, we have dbS^/dpS≥0. Therefore, the marginal merchant increases with the acquirer fee. We also have

∂ΠSpc/∂pB|pGpc=∂DBpc(pB,pGpc,pGc)/∂pB|pGpcmS(bS,pS,ppc)+∂DBc(pB,pGpc,pGc)/∂pB|pGpcmS(0,0,ppc).

Since ∂DBc(pB,pGpc,pGc)/∂pB|pGpc≥0 and ∂DBpc(pB,pGpc,pGc)/∂pB|pGpc≤0, the impact of the issuer fee on the marginal merchant is ambiguous and depends on the substitution between cash and card payments. For example, in the special case in which all consumers pay by card at a merchant who accept cards when they hold one, since ∂DBc(pB,pGpc,pGc)/∂pB|pGpc=0 and ∂DBpc(pB,pGpc,pGc)/∂pB|pGpc≤0, the marginal merchant increases with the issuer fee.

B Appendix: Proof of Proposition 1

Taking the derivative of the platform’s profit with respect to the interchange fee, we find that

dΠPFda=ρI((∂V∂pB)(pT∗−c)+V(pB,pS))+ρA((∂V∂pS)(pT∗−c)+V(pB,pS)).

This implies that at the profit-maximizing interchange fee a^π, we have

ρI((∂V∂pB)(pT∗−c)+V(pB,pS))+ρA((∂V∂pS)(pT∗−c)+V(pB,pS))=0.

Assume that the pass-through rates are symmetric, such that ρI(aπ)=−ρA(aπ). In this case, since pT∗−c>0, the first-order condition of profit-maximization can be rewritten as

ρI((∂V∂pB)−(∂V∂pS))=0.

Hence, at the profit-maximizing interchange fee, we have μB(aπ)=μS(aπ).

Assume that the pass-through rates are asymmetric. Rearranging the first-order condition we find that

(pT∗−c)(−ρI(aπ)∂V∂pB−ρA(aπ)∂V∂pS)=ρT(aπ)V,

Substituting for mT=pT∗−c, μB=−V/(dV/dpB) and μS=−V/(dV/dpS) into the equation above, if (ρI/μB)(aπ)+(ρA/μS)(aπ)≠0, we obtain that at the profit-maximizing interchange fee, the mark-up equals

mT=ρT(aπ)(ρIμB)(aπ)+(ρAμS)(aπ).

C Appendix: Relationship with the Literature

In this Appendix, we express the volume of transactions in several models of the literature according to our general model.

In Rochet and Tirole (2003) or Wright (2004), banks charge fixed per transaction fees and consumers’ utility is linear. All assumptions (A1)–(A7) hold. There is no modelling of the product market [(A4) is irrelevant]. The marginal merchant is defined by mS(bS^,pS,p)=mS(0,0,p). Therefore, the marginal consumer and the marginal merchant are given respectively by bB^=pB∗ and bS^=pS∗. From (A3), the transaction volume is the product of consumers’ and merchants’ quasi-demands, that is, the probability that each user wants to use/accept the card, that is, we have V(pB∗,pS∗)=DB(pB∗)DS(pS∗).
Wright (2012) relaxes (A3) by assuming that merchants internalize consumers’ benefit of making a transaction on the platform. The marginal merchant equals the merchant fee minus the expected surplus that a consumer obtains from card usage, that is, we have bS^=pS∗−vB(pB∗), where vB(pB∗)=E(bB−pB∗∣bB≥pB∗). Therefore, the transaction volume is given by
V(pB∗,pS∗)=DB(pB∗)DS(pS∗−vB(pB∗)).
Creti and Verdier (2014) relax (A3) by assuming that consumers bear a share L of fraud losses that depends on merchants’ investment in fraud prevention. The volume of transactions is given by
V(pB∗,pS∗)=∫bS^(pS∗)bS¯DB(bB^(pB∗,pS∗))hS(bS)dbS,
where the marginal consumer bB^ is such that bB^(pB∗,pS∗)≡pB∗+L(pS∗,pB∗,bS). The losses in case of fraud are related to the cardholder fee and the merchant fee, because a merchant chooses how much to invest in fraud prevention according to the volume of card transactions and its transaction costs.
Bourguignon et al. (2014) relax (A6) and drop (A2). There is only one monopolistic merchant that is able to steer consumers to pay by card by using surcharges or cash discounts. The marginal consumer yB^(pB∗,pS∗) is defined as the consumer who obtains the same utility ex ante whether he chooses to visit or not the merchant’s store. The merchant’s optimal surcharge is denoted by τ∗(pB∗,pS∗) and the merchant’s optimal cash discount is denoted by σ∗(pB∗,pS∗). The volume of transactions is then given by
V(pB∗,pS∗)=DB(pB∗+τ∗(pB∗,pS∗)+σ∗(pB∗,pS∗))DG(yB^(pB∗,pS∗)).
In Section 5 of their online Appendix, Bedre-Defolie and Calvano 2013 relax (A7) by assuming that banks charge both membership and usage fees. The volume of transactions is given by
V(pB∗;pS∗)=∫bS^bS¯T(pB∗;p∗(pB∗;pS∗−bS))hS(bS)dbS,
where
T(pB∗;p∗)=(1−FG(p∗))DB(pB∗)+∫y¯p∗DB(pB∗+p∗−y)fG(y)dy
is the total volume of card transactions of a store accepting cards and p∗≡p∗(pB∗,pS∗−bS) denotes the price chosen by a profit-maximizing monopolistic merchant. The result of our Proposition 1 does not hold (see our extension section).
Wang (2010) relaxes (A7) by assuming that payment card fees paid by end-users are proportional to the value of the transaction. He also assumes that merchants are homogenous. A consumer pays by card if and only if bB≥ppc(1+pB∗)−pc. In his framework, the transaction volume is expressed as
V(pB∗,pS∗)=DB(ppc(1+pB∗)−pc),
where depending on the modelling of competition between merchants, p^pc and p^c may depend on pB∗ and pS∗.

D Appendix: Consumer and Merchant Surplus

Appendix D-1: Merchant Surplus

A merchant who accepts only cash makes profit ΠSh(pGh), whereas a merchant who accepts cards makes profit ΠSpc(bS,pS∗,pB∗,pGpc). Therefore, merchants’ surplus equals

(11)SS=∫0bS^ΠSh(pGh)hS(bS)dbS+∫bS^bS¯ΠSpc(bS,pS∗,pB∗,pGpc)hS(bS)dbS.

Since the marginal merchant makes the same profit if he accepts cash or if he accepts cards, we have that ΠSh(pGh)=ΠSpc(bS^,pS∗,pB∗,pGpc,pGpc). Therefore, the impact of the interchange fee on merchant surplus is given by

dSSda=∫bS^bs¯dΠSpcdahS(bS)dbS.

We analyze the impact of the interchange fee on the profit of a merchant who accepts cards. From the envelope theorem, we can ignore the effect of the interchange fee that impacts the profit-maximizing price chosen by the merchant pGpc at stage 3. Therefore, the total derivative of the profit of a merchant that accepts cards with respect to a is given by

(12)dΠSpcda=∂ΠSpc∂pB|(pB∗,pS∗)ρI+∂ΠSpc∂pS|(pB∗,pS∗)ρA,

where from (2)

(13)∂ΠSpc/∂pB|(pB∗,pS∗)=mS(bS,pS∗,pGpc)∂DBpc/∂pB|(pB∗,pS∗)+mS(0,0,pGpc)∂DBc/∂pB|(pB∗,pS∗),

and from (2)

(14)∂ΠSpc/∂pS|(pB∗,pS∗)=DBpc∂mS/∂pS|(pB∗,pS∗)≤0.

Appendix D-2: Consumer Surplus

We denote the average surplus of card users and cash users who buy from a merchant who sets a retail price p by ϕBpc(p,pB∗) and ϕBc(p), respectively. The surplus of cash users who buy from a merchant who refuses cards is ϕBh(p). We denote by pGpc~ the retail price evaluated at the marginal merchant.

The surplus of cash users The surplus of cash users is given by

SBcash=∫0bS^ϕBh(pGh)hS(bS)dbS+∫bS^bs¯ϕBc(pGpc)hS(bS)dbS.

Since the price of cash transactions chosen by a merchant who accepts only cash pGh is independent of b_S, we have

(15)SBcash=ϕBh(pGh)HS(bS^)+∫bS^bs¯ϕBc(pGpc)hS(bS)dbS.

The impact of the interchange fee on cash users surplus is expressed as follows

dSBcashda=(bS^)′(a)hS(bS^)(ϕBh(pGh)−ϕBc(pGpc~))−∫bS^bs¯∂ϕBc∂ppc|pGpc∂pGpc∂ahS(bS)dbS,

where ∂pGpc/∂a=ρBρI+ρSρA.

The surplus of card users The surplus of card users is given by

(16)SBcard=∫bS^bs¯ϕBpc(pGpc,pB∗)hS(bS)dbS.

From (16), the impact of the interchange fee on card users surplus is given by

dSBcardda=−(bS^)′(a)hS(bS^)ϕBpc(pGpc~,pB∗)−∫bS^bs¯(∂ϕBpc∂ppc|pGpc∂pGpc∂a+ρI∂ϕBpc∂pB|pGpc)hS(bS)dbS.

E Appendix: Our Illustration

Appendix E-1: The Merchant’s Prices and the Marginal Card User

The first-order condition From (4), the first-order condition of the merchant’s profit-maximization is given by

(17)β(DG(yB^)−fG(yB^)(bS−pS+p−d))+(1−β)(DG(p)−fG(p)(p−d))=0.

The retail price pGpc is implicitly defined by (17).

Mature market If β = 1, from (17), since bS−pS+p−d=yB^−(pT+d−bS−bB), the marginal card user yB^ is implicitly defined as a function of pT+d−(bB+bS). Let

(18)yB^≡δ(pT+d−(bB+bS)).

The fact that δ is increasing follows from the second-order condition of profit-maximization and the implicit function theorem. Replacing for

κ(bS,pB,pS)≡ε(yB^)(yB^+bS+bB−pT−d)/yB^

into (17), where ε=fG(p)p/DG(p), we find that if β = 1, we have κ(bS,pB,pS)=1.

Appendix E-2: The Pass-through Rates of the Merchant’s Price

The pass-through of banks’ fees to consumers From Appendix A-1 and (4), since ∂2ΠSpc/∂p∂pS|pGpc=−∂2ΠSpc/∂p∂pB|pGpc=βfG(pGpc+pB−bB), we have ∂pGpc/∂pS=−∂pGpc/∂pB. Therefore, we have ρS=−ρB.

Mature market If β = 1, since yB^=pGpc+pB−bB and yB^=δ(pT+d−(bB+bS)) from (18), we have ∂yB^/∂pB=∂yB^/∂pS, which implies that ρB+1=ρS. Since ρS=−ρB, we have ρS=−ρB=1/2.

The marginal merchant in a mature market If β = 1, from Appendix E-1, at the equilibrium of stage 3, the profit of a merchant who accepts cards only depends on the total net cost of selling the good and we have ΠSpc(pGpc)≡ΠSpc¯(d+pT−bB−bS). The profit of a merchant who accepts only cash only depends on the marginal cost of selling the good and it is given by ΠSh(pGh)=ΠSh¯(d). Note that ΠSpc¯ and ΠSh¯ are necessarily the same functions in equilibrium, because they represent the profit of a monopolist that faces the same demand from consumers, but with different marginal costs. We denote this function by ΠS¯. The marginal merchant of type bS^ makes the same profit if he accepts only cash or both payments instruments. Therefore, we have ΠS¯(d+pT−bB−bS^)=ΠS¯(d). This implies that bS^=pT−bB.

Appendix E-3: The Impact of the Interchange Fee on Merchant Acceptance

Replacing for DBpc=βDG(p+pB−bB) and DBc=(1−β)DG(p) in the formula given in Appendix A-2 gives

(19)dbS^dpS=1,

and

(20)dbS^dpB=κ(bS^,pB,pS).

The total derivative of the marginal merchant with respect to the interchange fee is given by

(bS^)′(a)=ρA(∂bS^/∂pS)+ρI(∂bS^/∂pB).

Replacing for ∂bS^/∂pS and ∂bS^/∂pB given by (20) and (19), we have that

(bS^)′(a)=ρA+κ(bS^,pB∗,pS∗)ρI.

If β = 1, since κ(bS^,pB∗,pS∗)=1, we have (bS^)′(a)=ρT.

Appendix E-4: The Impact of the Interchange Fee on the Transaction Volume

Taking the derivative of (10) with respect to p_B and p_S, since ∂yB^/∂pB=ρB+1 and ∂yB^/∂pS=ρS, from (20), we find that

∂V(pB∗,pS∗)∂pB=−βκ(bS^,pB,pS)hS(bS^)DG(yB~)+β∫bS^bS¯(ρB+1)fG(yB^)hS(bS)dbS,

and

∂V(pB∗,pS∗)∂pS=−βhS(bS^)DG(yB~)+β∫bS^bS¯ρSfG(yB^)hS(bS)dbS,

where yB~ denotes the indifferent card user evaluated at the marginal merchant.

If β = 1, since yB^=δ(d+pT−bB−bS) and κ(bS^,pB,pS)=1 from Appendix D-1, we have ∂V/∂pB=∂V/∂pS.

Appendix E-5: The Impact of the Interchange Fee on Social Welfare

The impact of the interchange fee on merchant surplus The impact of the interchange fee on merchant surplus is given by (12) and (11) as in our general setting. From (2), since DBpc=βDG(yB^), DBc=(1−β)DG(p), and yB^=p+pB−bB, we have ∂ΠSpc/∂pB|yB^=−β(yB^+bS+bB−pS−pB−d)fG(yB^) and ∂ΠSpc/∂pS|yB^=−βDG(yB^). Therefore, the impact of the interchange fee on the profit of a merchant who accepts cards is given by

dΠSpcda=−β(ρI(yB^+bS+bB−pS−pB−d)fG(yB^)+ρADG(yB^)).

Replacing for κ(bS,pB,pS) given by (9), we find that

(21)dΠSpcda=−βD(yB^)(ρIκ(bS)+ρA).

From (21), the impact of the interchange fee on merchant surplus depends on the sign of ρIκ(bS)+ρA . If ρIκ(bS)+ρA>0, merchant surplus decreases with the interchange fee, whereas the reverse is true if ρIκ(bS)+ρA<0.

If all consumers hold a card (β = 1) and since κ(bS^)=1 from Appendix E-1, we have that dΠSpc/da=−D(yB^)ρT. Therefore, we have that

(22)dSSda=−βρT∫bS^bS¯D(yB^)hS(bS)dbS.

The impact of the interchange fee on consumer surplus In our illustration, the average card user surplus is

ϕBpc(pGpc,pB∗)≡ϕB(yB^)=E(y−yB^|y≥yB^),

where yB^=pGpc+pB∗−bB represents the marginal card user. This implies that ϕB′(x)=−DG(x). If all consumers hold a card, since (yB^)′(a)=ρSρT and pGh=yB~, we have

(23)dSBda=−ρT∫bS^bS¯ρSDG(yB^)hS(bS)dbS.

Proof of Proposition 3 From (23) and (22), we have

dSBda|aπ+dSSda|aπ=−ρT(aπ)(vpc(aπ)+∫bS^bS¯ρSDG(yB^)hS(bS)dbS).

Since ρ_S ≥ 0, since vpc(aπ)≥0, the derivative of social welfare evaluated at a=aπ has the opposite sign of ρT(aπ). Since W is concave in a by assumption, if ρT(aπ)≥0, we have aπ≥aW. If ρT(aπ)≤0, we have aπ≤aW. If ρT(aπ)=0, we have aπ=aW.

Appendix E-6: Uniform Distribution When β ≠ 1

In this Appendix, we look at the special case in which β < 1, b_S is uniformly distributed on [0,1], y is uniformly distributed on [0,y¯] and d = 0. A merchant that accepts cards maximizes its profit with respect to the price pGpc. From (3), we have

(24)pGpc=12(y¯+β(bB−bS−pB+pS)),

A merchant that accepts cash maximizes its profit with respect to the price pGh and we have

(25)pGh=y¯2.

Therefore, we have ρB=−β/2 and ρS=β/2, which implies that the maturity of the card market impacts the pass-through rates of banks’ fees to consumers by merchants. By plugging in Eq. (24) into the marginal merchant bS^, we have

bS^=1β(−2bB+2pB−y¯+β(bB−pB+pS)+X),

where X=(2bB−2pB+y¯)2−4β(bB−pB)(bB−pB+y¯). The impact of the platform fees p_B and p_S on the marginal merchant is not symmetric, and we have

dbS^dpS=1,

and

dbS^dpB=2(β−1)(2bB−2pB+y¯)+(2−β)X)X.

Therefore, the price structure impacts merchant acceptance and consumer demand. Plugging in (24) into the marginal card-user y^B=pGcard+pB−bB, we obtain that

y^B=12(y¯+β(bB−bS−pB+pS)+2(pB−bB)).

At stage 3, the volume of transactions is given by

V(pB,pS)≡∫bS^1(1−yB^y¯)dbS=4(pB−bB)(pS−1)−2(pB+pS−bB−1)y¯+(bB+pS−pB−1)2β4y¯.

To check whether the volume of card transactions is impacted by the price structure pB/pS, we look at the difference of the derivatives of the volume with respect to the issuing and acquiring fees, and we obtain

dVdpB−dVdpS=−(bB+pS−pB−1)(β−1)y¯.

Since this difference is different from zero, the transaction fees have different impact on the volume of transactions, and therefore the price structure plays a role if β ≠ 1.

We are now able to determine the profit-maximizing prices and levels of interchange fees according to the market structure on the issuing and the acquiring side.

(a) Perfectly competitive acquirers and monopolistic issuer If the acquirers are perfectly competitive and if there is a monopolistic issuer, we have pS∗=cA+a and

(26)pB∗=13β(−4(−1+a+cA)+2y¯+β(a+2(−1+bB+cA)+cI)+Z,

where

Z=4(−2(−1+a+cA)+y¯)2−2β(2(−1+a+cA)(−4(cA−1)+5a+bB−cI)+(7(cA−1)−8a−bB+cI)y¯))+(−1+2a+bB+cA−cI)2β2.

From (26), we see that the pass-through rate on the issuing side is different from −1.

At stage 1, the platform sets the profit-maximizing interchange fee a^π. We simplify our computations by taking c_I = 0 and cA=0. We are able to find the profit-maximizing interchange fee a^π and the welfare-maximizing interchange fee a^W for different maturities of the card market β, different values of the card usage benefit b_B and for different values of ȳ. We denote by a^S the interchange fee that maximizes merchant surplus, and by a^B the interchange fee that maximizes consumer surplus. Furthermore, we are able to check that the platform’s profit and social welfare are concave in a in our set of parameters. We define the total user surplus TUS=SBTOT+SSTOT.

We obtain the following results for y¯=1:

b_B = 0.1	β = 0.2	β = 0.5	β = 0.8	b_B = 0.8	β = 0.2	β = 0.5	β = 0.8
a^π	0.69	0.69	0.69	a^π	0.48	0.48	0.48
a^W	0.65	0.57	0.5	a^W	0.4	0.28	0.175
SBcash\|aπ	0.107106	0.0823713	0.0609	SBcash\|aπ	0.092156	0.0481017	0.0126196
SBcash\|aW	0.107456	0.0849939	0.0646636	SBcash\|aW	0.0932248	0.0552486	0.0229958
SScash\|aπ	0.122819	0.115608	0.106285	SScash\|aπ	0.0137139	0.0349231	0.0137139
SScash\|aW	0.115298	0.100512	0.0951409	SScash\|aW	0.00629235	0.00354033	<0
SBcard\|aπ	0.143105	0.164501	0.177597	SBcard\|aπ	0.181204	0.24933	0.293163
SBcard\|aW	0.141907	0.157823	0.169448	SBcard\|aW	0.178217	0.2328	0.271818
SScard\|aπ	0.269922	0.30159	0.335972	SScard\|aπ	0.318688	0.411351	0.522765
SScard\|aW	0.271533	0.308903	0.343633	SScard\|aW	0.318696	0.440742	0.554339
SBTOT\|aπ	0.137929	0.155471	0.170966	SBTOT\|aπ	0.168366	0.226156	0.275248
SBTOT\|aW	0.137834	0.155189	0.171055	SBTOT\|aW	0.168106	0.224699	0.274358
SSTOT\|aπ	0.269922	0.30159	0.336266	SSTOT\|aπ	0.312075	0.412063	0.523319
SSTOT\|aW	0.271533	0.308903	0.343633	SSTOT\|aW	0.318696	0.440742	0.554339
TUS\|aπ	0.407851	0.457061	0.507232	TUS\|aπ	0.480441	0.638219	0.798567
TUS\|aW	0.409367	0.464092	0.514688	TUS\|aW	0.486802	0.665441	0.828697

We obtain the following results for y¯=10:

b_B = 0.1	β = 0.2	β = 0.5	β = 0.8	b_B = 0.8	β = 0.2	β = 0.5	β = 0.8
a^π	0.721	0.721	0.721	a^π	0.54	0.54	0.54
a^W	0.67	0.6	0.53	a^W	0.46	0.34	0.23
SBcash\|aπ	0.0921904	1.10768	0.896927	SBcash\|aπ	0.689571	0.66276	0.3222345
SBcash\|aW	0.091807	1.10825	0.692512	SBcash\|aW	0.692512	0.669554	0.3303
SScash\|aπ	0.0137139	1.1255	1.11429	SScash\|aπ	0.280329	0.252102	0.222364
SScash\|aW	0.006292	1.10345	1.10068	SScash\|aW	0.243371	0.194446	0.186771
SBcard\|aπ	1.26406	1.27949	1.28792	SBcard\|aπ	1.28041	1.33087	1.35432
SBcard\|aW	1.26195	1.27144	1.2796	SBcard\|aW	1.28322	1.30959	1.33236
SScard\|aπ	2.5156	2.5392	2.56305	SScard\|aπ	2.54257	2.60732	2.6732
SScard\|aW	2.51583	2.54	2.56382	SScard\|aW	2.54351	2.61085	2.67654
SBTOT\|aπ	1.25811	1.27002	1.28178	SBTOT\|aπ	1.27264	1.30561	1.33772
SBTOT\|aW	1.25808	1.27004	1.28184	SBTOT\|aW	1.27252	1.30553	1.33796
SSTOT\|aπ	2.5156	2.5392	2.56305	SSTOT\|aπ	2.54257	2.60732	2.6732
SSTOT\|aW	2.51583	2.54	2.56382	SSTOT\|aW	2.54351	2.61085	2.67654
TUS\|aπ	3.77371	3.80922	3.84483	TUS\|aπ	3.81521	3.91293	4.01092
TUS\|aW	3.77391	3.81004	3.84566	TUS\|aW	3.81603	3.91638	4.0145

Note that a^π seems invariant with respect to β since we rounded the interchange fee to the second digit. However there are small variances, and in particular, with this set of parameters, the profit maximizing interchange fee slightly increases with the maturity of the card market β.

(b) Perfectly competitive issuers and monopolistic acquirer In this second case, we assume perfectly competitive issuers and a monopolistic acquirer, we have that pB∗=cI−a and

(27)pS∗=23(2a+2bB+y¯β+(2−2bB−a))+T,

where T=β2(4(a+bB)+y¯)2−2(10a2−y¯+2a(4y¯+9bB−1)+bB(7y¯+8bB−2))β+(2a+bB−1)2β2. From (27), we see that the pass-through rate on the acquiring side is different from −1.

We conduct the same numerical analysis as above for y¯=1:

b_B = 0.1	β = 0.2	β = 0.5	β = 0.8	b_B = 0.8	β = 0.2	β = 0.5	β = 0.8
a^π	0.2	0.2	0.2	a^π	−0.28	−0.28	−0.28
a^W	0.19	0.185	0.12	a^W	−0.3	−0.31	−0.34
SBcash\|aπ	0.107106	0.0823713	0.0609048	SBcash\|aπ	0.0921561	0.0481017	0.0126196
SBcash\|aW	0.10742	0.0831178	0.0634888	SBcash\|aW	0.0926702	0.0496122	0.015139
SScash\|aπ	0.122819	0.115608	0.106285	SScash\|aπ	0.0498075	0.0349231	0.0137139
SScash\|aW	0.121845	0.114525	0.103821	SScash\|aW	0.0483889	0.032854	0.0109227
SBcard\|aπ	0.0308232	0.0730994	0.110083	SBcard\|aπ	0.0762229	0.178099	0.262676
SBcard\|aW	0.0301496	0.0715837	0.105437	SBcard\|aW	0.0747186	0.173948	0.256381
SScard\|aπ	0.147103	0.185982	0.229687	SScard\|aπ	0.26188	0.376428	0.509051
SScard\|aW	0.148079	0.187106	0.232204	SScard\|aW	0.263319	0.378715	0.5125
SBTOT\|aπ	0.139931	0.159148	0.17406	SBTOT\|aπ	0.170283	0.231397	0.282137
SBTOT\|aW	0.140121	0.15956	0.175471	SBTOT\|aW	0.170552	0.23221	0.283767
SSTOT\|aπ	0.279987	0.31874	0.348221	SSTOT\|aπ	0.350587	0.479327	0.572942
SSTOT\|aW	0.279942	0.318641	0.347757	SSTOT\|aW	0.350485	0.47899	0.572447
TUS\|aπ	0.419918	0.477888	0.522281	TUS\|aπ	0.52087	0.710723	0.855079
TUS\|aW	0.420063	0.478201	0.523228	TUS\|aW	0.521038	0.7112	0.856215

We obtain the following results for y¯=10:

b_B = 0.1	β = 0.2	β = 0.5	β = 0.8	b_B = 0.8	β = 0.2	β = 0.5	β = 0.8
a^π	0.179	0.179	0.179	a^π	−0.34	−0.34	−0.34
a^W	0.125	0.08	0.065	a^W	−0.42	−0.49	−0.525
SBcash\|aπ	1.10768	0.896927	0.689571	SBcash\|aπ	1.01227	0.66276	0.322345
SBcash\|aW	1.10943	0.90218	0.693535	SBcash\|aW	1.01638	0.675521	0.332799
SScash\|aπ	1.13634	1.1255	1.11429	SScash\|aπ	0.280329	0.252102	0.222364
SScash\|aW	1.13119	1.11987	1.11168	SScash\|aW	0.267986	0.237886	0.215227
SBcard\|aπ	0.150425	0.373112	0.592208	SBcard\|aπ	0.260376	0.642845	1.01538
SBcard\|aW	0.148516	0.367461	0.58798	SBcard\|aW	0.255627	0.628455	1.00377
SScard\|aπ	1.37925	1.4137	1.44877	SScard\|aπ	2.26224	2.35522	2.45084
SScard\|aW	1.38439	1.41927	1.45132	SScard\|aW	2.27455	2.36922	2.45777
SBTOT\|aπ	1.25909	1.27138	1.28251	SBTOT\|aπ	1.27646	1.31096	1.34073
SBTOT\|aW	1.25934	1.27201	1.28288	SBTOT\|aW	1.27736	1.31329	1.34226
SSTOT\|aπ	2.51689	2.54125	2.56439	SSTOT\|aπ	2.54812	2.61619	2.67902
SSTOT\|aW	2.51687	2.54117	2.56433	SSTOT\|aW	2.54804	2.61587	2.67877
TUS\|aπ	3.77598	3.81263	3.84689	TUS\|aπ	3.82457	3.92715	4.01976
TUS\|aW	3.77621	3.81318	3.8472	TUS\|aW	3.8254	3.92916	4.02103

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Article note:

Carlotta Mariotto [European Commission DG COMP – Chief Economist Team, 1210 Sint-Joost-ten-Noode, Belgium]: The essential part of this research was conducted at ESCP Europe, LabEx ReFi and MINES Paris Tech, PSL Research University, CERNA-Centre for industrial economics, i3, CNRS. The subsequent employment of the author by the European Commission is unrelated to this work that is based entirely on publicly available information and not related to any activity of the Commission. The views expressed in this article are solely the author’s and do not reflect or represent the oflcial position of the Commission.

Published Online: 2018-09-06

Published in Print: 2017-09-26

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Articles in the same Issue

https://doi.org/10.1515/rne-2018-0003

Keywords for this article

interchange fees; payment systems; two-sided markets