Interchange Fee Regulation in the Credit Card Market

Transcription

1 Interchange Fee Regulation in the Credit Card Market José Ignacio Heresi August 2016 Abstract The relationship between interchange fees and the credit market has not been studied in detail in the literature, in spite of several antitrust and regulatory interventions in the debit and credit card markets. In this paper, this relationship is investigated under different market structures, and a new explanation for credit card rewards and for credit card interest rates stickiness is provided. It s also shown that the socially optimal interchange fee that considers the benefits and costs of the related credit market makes interchange fees caps implemented by antitrust authorities around the world potentially detrimental to social welfare. Keywords: Interchange fees, credit cards, interest rates. JEL Classifications: L11, E42. 1 Introduction All around the world, antitrust authorities and regulators have capped interchange fees 1 for debit and credit cards set by four party systems. For example, after a long antitrust case in Europe against Mastercard, both Visa and Mastercard have agreed to cap their interchange fees to around 0.3% for credit cards, while the Reserve Bank of Australia cut to half the interchange fees in that country in 2003 (to around 0.5%). In the United States, the Durbin Amendment of the Dodd-Frank Act in 2010 capped the debit card interchange fees 2. However, the relationship between interchange fees and per transaction fees with the credit market associated to credit cards has not been studied in detail. This article investigates this relationship by building a model where issuers offer credit cards and set both per transaction fees and interest rates. Consumers face outside options Phd Student, Toulouse School of Economics: jheresig@gmail.com. I thank Wilfried Sand-Zantman and Yassine Lefouili for their comments and suggestions on the present article. 1 The interchange fees are usually paid from the acquiring bank to the issuing bank for each transaction made through the payment system. 2 Other examples are Canada, Israel, Mexico, Spain and others. For more details see Bradford and Hayashi (2008). 1

2 for both the payment device and the credit characteristic of credit cards. They are divided in fully rational sophisticated consumers and myopic consumers who don t take interest rates into account when choosing credit cards 3. In this setting, additional factors that must be considered when regulating interchange fees arise, along with new explanations to credit card rewards and to credit card interest rate stickiness. The first insight of this article is that, even under different market structures, issuers have incentives to attract more demand to the revolving credit product associated to their credit cards by lowering per transaction fees, even to negative levels. This provides a new explanation for the offer of credit card rewards widely observed in the credit card industry. The second result is that, as long as there exists a minimal fraction of myopic consumers that don t take interest rates into account in their decisions, interest rates exhibit some stickiness: they remain unchanged in a wide range of parameters even if marginal costs, interchange fees or other market parameters vary. This happens because the existence of myopic consumers make the demand for credit less elastic than the demand for the payment device. Therefore, issuers have incentives to use per transaction fees in order to attract consumers to demand their cards and use interest rates as an exploitative device. In this model, price stickiness is a result of the multi product feature present on this market, along with the existence of myopic consumers and outside options in the credit market. Moreover, this result means that interest rates are independent from variations on the interchange fees, result that contributes to the discussion of the regulation implemented in different countries 4. Finally, the optimal per transaction fee that a social planner interested in total welfare would set is derived, along with the corresponding interchange fee that implements that fee in the market. This exercise shows new factors that should be taken into account when regulating interchange fees, namely the costs an benefits of the associated credit market affected by the aforementioned regulation. To understand why the credit feature of credit cards should be of special interest, consider that U.S consumers in 2012 made 23.7 billion of credit card payments for an amount of USD 2.2 trillion, while they possessed more than 300 millions of credit card accounts associated with more than 330 millions of credit cards in force during that year. By the end of 2015, the total revolving credit outstanding in the US was 936 billion dollars 5. On the side of issuing banks, 70% of a typical issuer income in the United States comes from interest rates revenue from credit cards 6. This fact suggests that firms are strongly influenced by incentives coming from the credit market. To analyze these issues, a model representative of a four party system is built. Consumers choose in a first stage whether to have and which credit card to use, and in a second stage 3 See Gabaix Laibson (2006) for further explanation on this behavioral assumption. 4 See The Economic Impact of Interchange Fee Regulation in the UK (2013) available at for-distribution.pdf, and The Effects of the Mandatory Decrease of Interchange Fees in Spain (2012), The Economic Impact of Interchange Fee Regulation in the UK (2013) available at final-report for-distribution.pdf, and The Effects of the Mandatory Decrease of Interchange Fees in Spain (2012), Reserve Bank of Australia Annual Report on Payment Systems available at for different arguments on this issuer 5 Almost the entire figure comes from outstanding credit from credit cards. See for more details 6 See Semeraro (2012). 2

3 they decide between using the revolving credit associated to the credit card (if they have one) or an alternative consumption loan with an exogenous interest rate. The model is solved under different market structures, namely a monopoly issuer, duopoly competition under single-homing and duopoly competition under multi-homing from consumers. A fraction of consumers are assumed to be myopic and don t take interest rates into account when they choose their credit cards, and the rest of consumers are assumed to be sophisticated, who fully consider both interest rates and per transaction fees when making their choices. In this model, every consumer is eligible for a credit card and will have enough credit limit if needed, while always repaying their debts. The model takes explicit account of outside options both in the payment method market as in the credit market. This article is organized as follows: in section 2, the model is explained and solved with a monopoly issuer. In section 3, a variant of the Hotelling model is defined to model competition between two firms while imposing single-homing from the consumers perspective. In section 4 this assumption is relaxed and consumers are able to hold both cards. In section 5 the privately optimal interchange fee is compared with the socially optimal one, which varies depending on the market structure. Finally in section 6 some discussion on the results is provided. Related Literature In relation to interchange fees, several papers have analyzed interchange fees in absence of credit functionality. This paper borrows closely from Rochet and Tirole (2011) the way of modeling cardholder s and retailer s benefits of using cards over cash while adding the credit feature to the model. In their paper, the authors explain why cards are often understood as must take by merchants. They also provide a benchmark for the regulation of the interchange fee, the tourist test. They show that retailers may accept cards even when these cards raise their operational costs, due to partial internalization of buyer surplus. Additionally, they show that, in many cases, the privately optimal interchange fee will be too high, producing excessive of cards as a payment device. Wright (2012) goes further in the last argument and show conditions under which the privately set interchange fee will be unambiguously biased against retailers. Beldre and Calvano (2013), in a model of usage and participation benefits heterogeneity on both consumers and merchants, also show that a network aiming to maximize profits over subsidize the usage of cards by cardholders while overcharging merchants, mainly due to the fact that consumers choose both membership and usage while retailers only choose membership. The closest papers in the literature are Chakravorti and To (2007) and Rochet and Wright (2010), who also take into account credit functionality of cards. In the first paper the authors show that merchants accept credit cards because they increase sales today in contrast with tomorrow, and thus merchants are willing to accept higher merchants discounts. They also show that in equilibrium, all merchants accept cards, but by doing so they are worse off, as in a prisoner s dilemma situation. They focus more on retailers than on issuers incentives and have no interest rate in their model. In the latter paper, the authors take explicit account of the credit functionality of credit cards and show that a monopoly network sets an interchange fee that is higher to the one maximizing consumer surplus. They also focus on the retailer side of the market by 3

4 looking for equilibria where credit cards are accepted even when stores can provide credit for themselves. They analyze the impact of interchange fees on retail prices while simplifying the issuing side of the market. Therefore, there is not explicit interest rate either in their model. In contrast with those papers, this work focus on the issuing bank and on understanding the relationship between the per transaction fees and the interest rates from banks perspectives, therefore providing a tool to explain credit card rewards and interest rate stickiness in the market, while also analyzing interchange fees in this context. Both works are thus complementary to this article. Regarding to credit card rewards, previous results in the literature show that rewards appear if the convenience benefit for merchants exceeded the costs of the transactions (Rochet and Tirole (2011), Beldre and Calvano (2013)) or due to oligopolistic competition by retailers, non surcharging by merchants and network competition (Hayashi (2008)), among others. This work provides an alternative explanation to this widely observed phenomenon through an alternative approach. 2 The Monopoly Case Consider a continuum of mass 1 of consumers demanding two products: a payment device and some form of credit for an amount of money D, to use them to purchase a basket of goods. Issuing banks offer credit cards, charging per transaction fees f for each payment made with the card and an interest rate r per unit of money lent. Consumers use the credit to purchase a basket of goods providing utility v, and they experience a convenience benefit of b b from the payment feature of credit cards, which is assumed to be heterogeneous and distributed over a closed interval [b b, b b ] following a cumulative distribution function H(.) with increasing hazard rate. Consumers also have the possibility of paying using cash, option which cost is initially normalized to 0. On the credit market, consumers choose whether to use the revolving credit associated to their credit card or whether to go to an exogenous credit market and getting the loan at rate r 0. Consumers incur in a fixed transaction cost of h b if they use the alternative form of credit (representing, for instance, within period impatience or direct transaction costs). It s assumed that every consumer will use either form of credit while always repaying their debts (no default allowed). There are two types of consumers that exhibit different behavior. A fraction (1 γ) of sophisticated consumers choose their payment device and credit form taking into account both per transaction fees and having rational expectations over future interest rates. The remainder fraction γ of myopic consumers take decisions based only on per transaction fees (while still using some form of loan and paying interest rates). It s assumed that 0 < γ < 1 for simplicity in the exposition 7. This is a behavioral assumption similar to the one used in Gabaix Laibson (2006). It s assumed that there exists an exogenous maximum interest rate r that can be charged by banks due to regulation. Given the description above, consumers have three options: 1) to pay with the card and to use the associated revolving credit; 2) to pay with the card and to use the external loan; 7 When γ = 1 the models is solved differently and the results are pretty straight forward. 4

5 and 3) to use cash and to use the external loan. The utility derived from each choice is respectively given by: U 1 = v + b b f (r D) U 2 = v + b b f (r 0 D) h b U 3 = v (r 0 D) h b Where it s assumed that there is no discount factor (namely that it s equal to 1 between periods). Note that both myopic and sophisticated consumers are assumed to have the same distribution of convenience benefits b b. In contrast with Rochet and Wright (2010), retailer competition is omitted and it s only assumed that shops receive an homogeneous convenience benefit of b r from payments with cards and an homogeneous convenience benefit of h r from the credit characteristic of cards. Retailers are not allowed to surcharge payments with cards 8. The acquiring market is assumed to be competitive implying that the merchant discount rate is given by m = c a + a, where c a is the per transaction cost of the acquirer 9 and a is the interchange fee. Assume for now that there is a single issuer of credit cards and that the interchange fee a is exogenously given. The timing of the game is the following: t=1: The issuer set the per transaction fee. Then, consumers choose whether to own a card or not. t=2: The issuer set the interest rate for the revolving credit. Then, consumers choose which form of credit to use. Notice that consumers choosing to own a credit card in the first period will always use it to pay when they purchase, while in the second period they only choose which form of credit to use. The issuer can t commit to a level of the interest rate in t = 1. Define: r r 0 + h b (1) D as the interest rate that makes the sophisticated consumers indifferent between the revolving credit associated with their card and the exogenous loan choice 10. In this setting, it may only be optimal for the monopolist in t = 2 to charge either r M = r or r M = r, because given the card decision of consumers, in the second period the monopolist chooses between serving every consumer in the credit market at a low interest rate or serving only myopic consumers at a high interest rate. Note that any rate strictly greater than r will lead to only myopic consumers using the revolving credit, therefore it is in the monopolist interest to set the highest price possible and that any rate smaller or equal than r will make every consumer use the revolving credit, so it is more profitable to set r. 8 Usually there is a non-surcharge rule by networks like Visa and Mastercard. In countries where this rule has been eliminated, there is little evidence of surcharging by merchants. 9 The analysis below could be extended to acquirers having a positive margin. The important assumption is that an increase of the interchange fee will increase the merchant discount rate charged by them. 10 If they use the outside option they pay D r 0 + h b which is equal to D r. 5

6 First, assume that the monopolist charges r in t = 2 and sophisticated consumers correctly anticipate this price. Then, during the first period the demand for card payments is derived from the condition b b f M 11, for both sophisticated and myopic consumers. This implies a demand function equal to 1 H(f M ). Notice that every consumer having a card will use the revolving credit at this interest rate. The profit function in t = 1 is given by: Π(f M ; r) = (f M + a c i )(1 H(f M )) + D(r c c )(1 H(f M )) (2) Where a is the interchange fee, c i is the constant marginal cost of a payment transaction and c c is the cost per unit of debt. Taking first order conditions with respect to f M and solving for the optimal per transaction fee yields 12 : f M = c i a + 1 H(f M ) h(f M ) D(r c c ) (3) Now, assume that the monopolist charges r in t = 2 and sophisticated consumers anticipate this price. In this case, sophisticated consumers expect to use the exogenous loan. The demands for cards, for both myopic and sophisticated consumers, are again given by the condition b b f M, but now only myopic consumers will use the revolving credit. The profit function in this case is given by: Π(f M ; r) = (f M + a c i )(1 H(f M )) + γd(r c c )(1 H(f M )) (4) Solving for the optimal fees yields: f M = c i a + 1 H(f M ) h(f M ) γd(r c c ) (5) These results, an example of per transaction fees with a uniform [0, 1] distribution for b b and the condition under each equilibrium will hold are given in the following proposition: Proposition 1. Assume that b b satisfies the increasing hazard rate condition and for a given level of the interchange fee a, there are two possible equilibriums: A low interest rate equilibrium with r M = r and f M characterized by expression (3). A high interest rate equilibrium with r M = r and f M The low interest equilibrium will hold if and only if: characterized by expression (5). γ (r c c) (r c c ) Else, the high interest rate equilibrium will hold. Finally, the expression for per transaction fees assuming that b b is distributed uniformly in [0, 1] are given by: f M = c i a Dγ(r c c ) 2 11 Details on the derivation of demand functions in appendix The increasing hazard rate assumption ensures that the first order condition will provides a solution to the monopolist problem. 6 (6)

7 for the low interest rate equilibrium and: f M = c i a D(r c c ) 2 for the high interest rate equilibrium. Proof. See appendix 1. The following comparative statics results will be useful for discussion. Lemma 1. In the optimal solution for the monopolist problem: The per transaction fees charged by the monopolist are strictly decreasing on the interchange fee. The interest rates charged by the monopolist are independent of the interchange fee. The probability of having a high interest rate equilibrium is increasing on the regulated exogenous interest rate r. Proof. The per transaction fees being strictly decreasing on the interchange fees follows directly from the implicit function theorem and the increasing hazard rate assumption for H( ). The probability of having a high interest equilibrium is increasing on r because, as this exogenous rate is greater, the condition for the existence of the high interest rate equilibrium is more likely to be fulfilled and the condition for the low interest rate equilibrium is less likely to hold. Some remarks on the monopolist problem: first, the per transaction fee is decreasing on the profits made on the credit market. Thus, if the credit market is profitable enough, the per transaction fee may be negative, meaning that the monopolist is offering rewards to consumers for the use of the credit card. This is a new explanation for the widely observed credit card rewards observed in the credit card market. The monopolist induces more demand to the credit market by charging lower, or even negative, per transaction fees. Second, whether the high or low interest equilibrium hold, depends on the relationship between the fraction of myopic consumers in contrast with the relative margins of setting a low and a high interest rate. Therefore, if there exist many myopic consumers or if the exogenous interest rate is too high, it s more likely that high interest rates will be observed in equilibrium. The condition leading to each equilibrium is independent of the interchange fee, meaning that interchange fee regulation doesn t affect interest rates in a monopoly market structure. The only effect of interchange fee regulation in this context is an increase in per transaction fee (or a decrease of credit card rewards). Moreover, changes in the marginal cost of credit that do not switch the equilibrium condition for high or low interest rates, won t have any effects on the equilibrium interest rates. This means that for a very wide set of values for c c, interest rates are sticky even for changes in the marginal cost of credit. Finally, downward regulation of the exogenous interest rate can directly reduce the rate payed by myopic consumers and can also increase the probability of a low interest rate equilibrium by making the high interest strategy less profitable. 7

8 3 Pure Single-Homing Competition Now, consider two firms, A and B, that compete for consumers in a Hotelling line. The total amount of consumers is normalized to M + 2K and it is assumed that the card market is covered, meaning that every consumer will use a card 13. Further, consider that there are two exogenous firms, A and B offering only the payment product at per transaction fee f These firms may represent alternative payment cards, such as debit cards, that also provide the benefit b b to consumers, but without giving the credit option. Firms A and B are located symmetrically in the middle of the line at distance M from each other, while A and B are located in the extremes of each side of the line, at a distance K from the main firms, as it is showed in the following figure: Figure 1: Hotelling Competition with Outside Options The linear transport cost is given by t and consumers still have the choice of getting a loan at rate r 0 at the exogenous credit market, incurring in a transaction cost of h b. It s imposed in this section that consumers can only choose to have only one card between the offers of firm A and B, representing a pure single-homing case. The timing of the game is the following: t=1: Issuers A and B set per transaction fees simultaneously. Then, consumers choose between the cards from firm A, firm B or one of the outside option firms. t=2: Issuers A and B set interest rate for the revolving credit simultaneously. Then, consumers choosing outside option firms use the exogenous loan, while consumers with cards from A or B choose between their firm s revolving credit and the exogenous loan. 13 This can be justified assuming that b b being high enough. 14 This is a variation of the Hotelling model, similar to that in Rochet and Tirole (2003). 8

9 Suppose that a given consumer chooses the card from firm A or B. In the second period, this consumer can only use the revolving credit associated with his card or the exogenous loan. At this stage, firms stop competing with each other and only compete with the outside option. Therefore, the only relevant options for them in t = 2 are to serve all of their consumers at interest rate r or serve only their myopic consumers in the credit market at r. Focusing on symmetric equilibria 15, suppose in a first step that both firms charge the low interest rate in t = 2. In this case, demands for cards in t = 1 for firm i is given by 16 : D i (f i ; ri e ) = M + K f i 2 t + f i + f 0 D(1 γ)(2re i r i e r) (7) 2t 2t Where i {A, B}, and i represents the competing firm. At this expected interest rate, every consumer choosing firms A or B expect to use the revolving credit associated to their card. Therefore, the profit function for each firm is: Π i (f i ; r) = D i (f i ; r e i )(f i + a c i ) + D i (f i ; r e i )D(r c c ) (8) Taking first order conditions for each firm with respect to f i and using rational expectation from consumers yields: fi = t(m + K) + f 0 2(a c i ) 2D(r c c ) (9) 3 Analogously, assuming that issuers will charge r in t = 2, sophisticated consumers expect to go to the exogenous loan. Therefore, the profit functions is in this case: Π i (f i ; r) = D i (f i ; r)(f i + a c i ) + γd i (f i ; r)d(r c c ) (10) Solving for optimal per transaction fees: fi = t(m + K) + f 0 2(a c i ) 2γD(r c c ) (11) 3 These results and the condition under which each equilibrium will hold are given in the following proposition: Proposition 2. In the described Hotelling model, when consumers single-home, and for a given level of the interchange fee a, there are two possible symmetric equilibriums: A low interest rate equilibrium with r i = r and f i given by (9), for i {A, B}. A high interest rate equilibrium with r i = r and f i The low interest equilibrium will hold if and only if: given by (11), for i {A, B}. Else, the high rate equilibrium will hold. γ (r c c) (r c c ) (12) 15 In fact, an asymmetric equilibrium where one firm charges the high interest rate and the other the low interest rate does not exist in this game. 16 See details on the derivation of demands in appendix 2. 9

10 Proof. The argument follows the same steps as the ones for proposition 1 in appendix 1, using the demand functions for the competition case. And the comparative statics results on single-homing competition: Lemma 2. In the competitive solution for single-homing consumers: The per transaction fees charged by firms A and B are strictly decreasing on the interchange fee. The interest rates charged by firms A and B are independent of the interchange fee. The probability of having a high interest rate equilibrium is increasing on the regulated exogenous interest rate r. Proof. The first statement is concluded directly from expressions (9) and (11). The interest rates are independent of the interchange fee as the equilibrium condition defining whether the high or the low interest rate will hold in equilibrium is independent of the interchange fee. Finally, if the exogenous interest rate r increases, the fraction of myopic consumers needed to sustain a high interest rate equilibrium is lower, and therefore this equilibrium holds for a wider range of parameters in the model. As in the monopoly case, we have two possible equilibriums, one with a high interest rate and one with a low interest rate. The condition under which these equilibriums hold is the same than in the monopoly case, because when consumers are assumed to single-home, they are captive to their own credit card supplier in t = 2. Therefore their credit choice reduces to using their card revolving credit or the credit market outside option. At this stage, each firm faces the same trade-off as the monopolist: whether to serve every consumer at a low interest rate or only to myopic consumers at a high interest rate. This trade-off is again defined by the relative amount of myopic consumers in the population relative to the interest margins with high or low interest rates. This result shows that, even in a competitive environment, if the credit market is profitable enough, per transaction fees may be negative. Therefore, the explanation to credit card rewards extends to competition when single-homing is imposed. Moreover, as the condition leading to each equilibrium is the same, the result for interest rate stickiness also remains unchanged in this competitive environment. 4 Pure Multi-Homing Competition Consider a variation of the Hotelling model of last section, in which sophisticated consumers hold both cards after the first period, and in the second period choose which card to use observing both per transaction fees and interest rates. In the case of myopic consumers this assumption is innocuous because they decide only observing per transaction fees just as in the last section. The timing is now the following: 10

11 t=1: Issuers A and B set per transaction fees simultaneously. Both types of consumers hold cards from firms A and B. t=2: Issuers A and B set interest rate for the revolving credit simultaneously. Then, consumers observe both per transaction fees and interest rates and choose whether to pay using on of the firms cards or the outside option and which form of credit to use. The only restriction for consumers in this section is that in order to use the revolving credit associated to a card, they must use that card as a payment device. Any other combination of card and outside options is allowed. The main difference with the single-homing case is that sophisticated consumers will choose considering the best combination of fees and observed interest rates in t = 2. Therefore, different interest rates charged by issuers in t = 2 may influence sophisticated consumers choice of their payment device. Demands for card payments are the same as in the last section but considering observed interest rates instead of expected ones. The set of strategies for issuers is now broader, as now they can attract more consumers by charging an interest rate lower than r. It s still never profitable for issuers to charge anything between r and r, because sophisticated consumers will go to the outside option and myopic consumers could be furthered exploited by charging r. However, lowering the interest rate to levels below r will attract demand from sophisticated consumers, in contrast with the single-homing case. Therefore, firms choose interest rates either in the set [0, r] or equal to r. Suppose first that issuers set interest rates in the interval [0, r]. Then, the demand for cards for firm i in t = 2 is given by: D i (r i ; f i ) = M + K f i 2 t + f i + f 0 D(1 γ)(2r i r i r) (13) 2t 2t When interest rates belong to the mentioned interval, every consumer demanding the card of firm i will also use the revolving credit associated to that card. Therefore, the problem in t = 2 is to maximize the following profit function with respect to r i for any value of per transaction fees f i, with the restriction that ri [0, r]: Π(r i ; f i ) = D i (r i ; f i )(f i + a c i + D(r i c c )) (14) Omitting the restriction for the moment and solving for r i as a function of per transaction fees of both firms yields: r i = 5 (15 8γ)f i + 2γf i 15D(1 γ) with (t(m + K) + f 0 ) + 5D(1 γ)(r + 2c c ) 10(1 γ)(a c i ). Now, the maximization problem of the firms in t = 1 is to choose per transaction fees to maximize: (15) Π(f i, r i, r i) = D i (f i, r r, r i)(f i + a c i + D(r i c c )) (16) This profit function is strictly decreasing on the per transaction fees, and the second order conditions for maximization are not conducive to a maximum, so the first order condition 11

12 are not useful in this case. The equilibrium per transaction fees (details of the derivation in appendix 3) are given by: fi = t(m + K) + f 0 2(a c i ) 2D(r c c ) (17) 3 Note that this per transaction fee is exactly the same than in the single homing case. This means that the low interest equilibrium remains unchanged. This result follows the following intuition: firms, given how interest rates will be affected by per transaction fees in t = 2, always have incentives to lower per transaction fees in t = 1. Increasing them would lead to lower interest rates, lower demand and lower profits. However, when they reach the per transaction fee consistent with the restriction of the maximization problem and with expression (15), firms still have incentives to lower per transaction fees, even if they can t keep increasing the interest rate. This process goes on until both firms set per transaction fees equal to expression (17) in equilibrium, which is the same expression for equilibrium per transaction fees than in the single homing case. The explanation of this result comes from the existence of an outside option in the credit market and the participation of myopic consumers in the market. Even a very low fraction of these consumers give banks incentives to always use per transaction fees in order to attract demand, while setting the highest interest rate possible. The outside option in the credit market sets a cap on this interest rate at the level r. This two factors make the multi homing case equal to the single homing case in the low interest rate equilibrium. Assume now that the firm charges a high interest rate in t = 2. This means that sophisticated consumers will use the outside option credit even when choosing the card of either firm A or B. This means that firms maximize in t = 1 the following profit function: Solving for both firms: Π(f i ; r) = D i (f i ; r)(f i + a c i ) + D i (f i ; r)γd(r c c )) (18) And r i = r. f i = t(m + K) + f 0 2(a c i ) 2Dγ(r c c ) 3 (19) Proposition 3. In the described Hotelling model, when consumers multi-home, and for a given level of the interchange fee a, there are two possible symmetric equilibriums: A low interest rate equilibrium with r i = r and f i A high interest rate equilibrium with r i = r and f i given by (17), for i {A, B}. given by (19), for i {A, B}. Then, the low interest equilibrium holds if and only if: Else, the high interest equilibrium holds. γ (r c c) (r c c ) (20) Proof. See appendix 3. 12

13 Proposition 3 extends the intuition on both credit card rewards and interest rate stickiness to the multi-homing competition case. Finally: Lemma 3. In the competitive solution for multi-homing consumers: The per transaction fees charged by firms A and B are strictly decreasing on the interchange fee. The interest rates charged by firms A and B are independent of the interchange fee. The probability of having a high interest rate equilibrium is increasing on the regulated exogenous interest rate r. Proof. Same proof as in Lemma 2. 5 Socially and Privately Optimal Interchange Fees 5.1 Privately Optimal Interchange Fees Rochet and Tirole (2011) show that under different competitive environments, retailers are willing to accept a higher merchant discount rate m than their direct convenience benefits, due what they call merchant internalization, meaning that merchants partially internalize a part of cardholders benefits of their card acceptance policy. Following their work, define v b E[b b f b b f] as the net cardholder benefit per card payment. These authors show that retailers accept cards if and only if: m b r + v b (f) Where f is the per transaction fee charged to consumers. In this model, there are some additional factors that must be considered. First, retailers also get a convenience benefit from the revolving credit characteristic, namely h r, and therefore it s logical to assume that they are willing to pay even more now as merchant discount, in addition to b r. Second, consumers incur in a transaction cost of h b when going to the exogenous loan, which can be additionally internalized from merchants in addition to v b (f). Finally, note that in the model studied above, issuers profits are always increasing on the interchange fee (as it is an exogenous source of income), just as in Rochet and Tirole (2011). Therefore, a network aiming to maximize issuers profits, will set the highest interchange fee subject to merchants accepting cards, that is: m b r + v b (f) + (h b + h r ) Assume for simplicity that we are always in a low interest rate equilibrium, meaning that credit card users always use the revolving credit from their cards. Given the competitive acquiring market assumption, this implies an interchange fee equal to: a p = (b r + h r ) + (v b (f) + h b ) c a (21) In this model, retailer s benefits are simplified by assuming an homogeneous convenience benefit h r coming from the revolving credit characteristic of cards. This can come from fear of 13

14 missed sales by retailers. For a much more detailed discussion on this issue see Bourguignon et al. (2014). 5.2 Socially Optimal Interchange Fees To define the social welfare function, it s important to mention the sources of social benefit and cost in this model. Social benefits are given by convenience benefits b b, b r and h r and the social costs are given by c i, c c, c a and h b. Note that prices are not relevant in the total welfare function as they represent transfers from consumers to producers. Assume again that we are in a low interest rate equilibrium. Also, assume that the exogenous loan has a cost of c l per unit of credit given. Then, the social welfare function as a function of per transaction fees is given by: SW (f i ) = f i (b b + b r (c i + c a + (D(c c c l ) h b )))h(b b )db b (22) Maximizing with respect to the equilibrium per transaction fee: f W i = c i + c a + D(c c c l ) h b b r (23) This optimal per transaction fee can be implemented by the regulator by setting the appropriate interchange fee depending on whether the market structure is a monopoly, singlehoming competition or multi-homing competition. Using expressions for per transaction fees as a function of interchange fees for each market structure gives as a result the socially optimal interchange fee. For example, assuming that b b is distributed uniformly in [0, 1], means that the optimal interchange fee for the monopoly case is: a W M = h b + b r c i 2c a 2D(c c c l ) (24) When comparing privately optimal and socially optimal interchange fees, the literature has argued in the past few years that the privately optimal one is too high and therefore it should be reduced. This analysis shows that, if the transaction costs related to the exogenous credit market are high, it may be the case that reductions in the interchange fee are detrimental for total welfare. This happens because issuers are inducing more demand for their revolving credit product through their per transaction fees, and therefore the extra income of interchange fees is partially used by issuers to incentivize more revolving credit use by consumers. If this form of credit is socially desirable, for example due to high transaction costs of going to the exogenous credit market, capping interchange fees may be detrimental for welfare. 6 Discussion and Final Remarks The study of the relationship between the payment market and its transaction fees and the credit market and its interest rates is important for several reasons. First, debit and credit card interchange fee regulation has been implemented due to some agreement on the fact that privately set interchange fees would be too high, generating an 14

15 over usage of payment cards and excessive costs for merchants. However, this regulation may have significant influence on the credit market associated with payment cards, and these implications should be taken into account when evaluating this policy. Interchange fee caps reduces the incentives for issuers to induce more demand to their revolving credit associated with credit cards, and therefore it increments the number of consumers going to the exogenous loan market. If the transaction costs of going to the outside option market are too high or if the convenience benefit from retailers from the revolving credit is also high, reductions on interchange fees have the potential to reduce total welfare. Which effects dominates in practice depends on the consumers and retailers behavior of a given market, and antitrust authorities should take this additional effects into account when regulating interchange fees. Second, this work found an alternative explanation to credit card stickiness, which has been a long studied problem 17. Even if a behavioral component in the model is necessary in order to obtain this result, only a very small fraction of myopic consumers is needed to generate this result. Moreover, any alternative regulation based on the education of consumers could fail in the attempt to generate lower interest rates in the market, while according to this model, direct regulation of the maximum interest rate could have positive effects in the market 18 Finally, this analysis provides some new insights on why credit card rewards are offered and whether they are a good practice for consumers and for overall welfare. Even if some consumers are getting benefited by these rewards, issuers may be implementing them in order to exploit myopic consumers in the credit market, generating a loss for this group of consumers. Appendix 1: Monopoly Section Demand For Cards Myopic consumers do not observe or value differences on interest rates, valuing only per transaction fees. They will demand the card offered by the monopolist if and only if: While sophisticated consumers only when: v + b b f M v v + b b f M Dr v Dr 0 h if they expect an interest rate equal to r, meaning they will use the revolving credit, or: v + b b f M Dr 0 h v f 0 Dr 0 h 17 See Ausubel (1991) for a much earlier discussion of this issue. 18 The model has assumed implicitly that the margins of firms in equilibrium are always positive, that is, firms are making profits even in low interest rate equilibriums. If this wasn t the case, direct regulation could also be detrimental for the market. 15

16 if they expect an interest rate strictly higher than r, meaning they will use the outside option credit. In both cases, the condition defining the demand function is given by: Proof of Proposition 1 b b f M The expressions for both per transaction fees and interest rates are already derived above. To prove when each equilibrium holds, name fm L to the per transaction fee charged in the low interest rate and fm H to the fee charged in the high interest equilibrium. Suppose first, that the low interest equilibrium is holding. Then, we need: Π(f L M, r) Π(f L M, r) (25) Meaning that the monopolist won t deviate to charge the high interest rate in the second period. This condition is equivalent to: (f L M + a c i )(1 H(fM L )) + βd(r c c ) (fm L + a c i )(1 H(fM L )) + βγd(r c c ) (26) Which is equivalent to γ (r cc) (r c c). Finally, for the expressions assuming b b distributed uniform on the interval [0, 1], it suffices to replace h(f) by 1 and 1 H(f) by 1 f and solve for the optimal fees. Appendix 2: Single-Homing Section Demand For Cards Call x 1 the consumer indifferent between A and B, x 2 the consumer indifferent between A and A and x 3 the consumer indifferent between B and B. Assume that a consumer to the left of A will never go to B and that a consumer to the right of B will never go to A. The conditions to calculate x 1, x 2 and x 3 are, for sophisticated consumers, respectively: b b f A Dr e A x 1 t = b b f B Dr e B (M x 1 )t b b f 0 Dr 0 h x 2 t = b b f A Dr e B (K x 2 )t b b f B Dr e B x 3 t = b b f 0 Dr 0 h (K x 3 )t Assuming that both ra e and re B are weakly smaller than r, meaning that sophisticated consumers using A or B expect to use the revolving credit. If the expected rate is higher than r, consumers expect to go to the exogenous loan, meaning that these relationships change to: b b f A Dr 0 h x 1 t = b b f B Dr 0 h (M x 1 )t b b f 0 Dr 0 h x 2 t = b b f A Dr 0 h (K x 2 )t 16

17 b b f B Dr 0 h x 3 t = b b f 0 Dr 0 h (K x 3 )t Finally, demands for sophisticated consumers are given by: D s A(f A ; r e A) = x 1 + K x 2 D s B(f B ; r e A) = M x 1 + x 3 These relationships along with the definition of r yield the demands used in section 3. Demands for myopic consumers are derived analogously without taking into account interest rates: DA m (f A ) = M + K f A 2 t + f B + f 0 (27) 2t Finally, total demand for firms A and B is given by D i (f i, ri e ) γdi m (f i )+(1 γ)di s (f i ; ri e ). Appendix 3: Multi-Homing Section Demand For Cards Demands for cards are calculated just as in appendix 2 but with observed interest rates instead of expected interest rates. Therefore, the conditions to calculate x 1, x 2 and x 3 are: b b f A Dr A x 1 t = b b f B Dr B (M x 1 )t b b f 0 Dr 0 h x 2 t = b b f A Dr B (K x 2 )t b b f B Dr B x 3 t = b b f 0 Dr 0 h (K x 3 )t If observed interest rates are lower than r and: b b f A Dr 0 h x 1 t = b b f B Dr 0 h (M x 1 )t b b f 0 Dr 0 h x 2 t = b b f A Dr 0 h (K x 2 )t b b f B Dr 0 h x 3 t = b b f 0 Dr 0 h (K x 3 )t If observed interest rates are higher than r. The rest of the calculations follows the same steps as in the appendix 2. Proof of proposition 3 Solving the problem of the firms in t = 2, means maximizing: Π(r i ; f i ) = D i (f i, r i )(f i + a c i + D(r i c c )) (28) With respect to r i, which first order condition is: td i (f i, r i ) = (1 γ)(f i + a c i + D(r i c c ) (29) 17

18 For each firm. Solving this system yields the following interest rates as a function of per transaction fees: r i = 5(t(M + K) + f 0) + 5D(1 γ)(r + 2c c ) 10(1 γ)(a c i ) (15 8γ)f i + 2γf i 15D(1 γ) (30) Replacing these expressions in demand functions and solving the maximization in t = 1 of the following profit function: Π(f i, r A, r B) = D i (f i, r A, r B)(f i + a c i + D(r i c c )) (31) This profit function is strictly decreasing on per transaction fees, therefore a candidate equilibrium is the largest interest rate and the corresponding per transaction fee, that is : f i = C + D(1 γ)(r + 2c c) 2(1 γ)(a c i ) 3D(1 γ)r (3 2γ) And RA = R B = R. However, at even if firms can t keep increasing interest rates (because they would loose all sophisticated consumers), the still have incentives to lower per transaction fees. This will happen until the following equilibrium condition hold, for any positive : Which means that per transaction fees are given by: (32) Π(f i, r) Π(f i, r) (33) fi = t(m + K) + f 0 2(a c i ) 2D(r c c ) (34) 3 At this value for per transaction fees, firms haven o incentives to deviate either setting slightly higher or lower per transaction fees. Now, in t = 2 assume you are in the low interest rate equilibrium. Firms are playing best responses to per transaction fees, but they can still deviate to a high interest rate. They won t do so if and only if: γ r c c r c c (35) By the same argument as in the single-homing case. Note that for any given per transaction fees charged by firms in the first period, this condition makes setting a low interest rate a dominant strategy. Now, assume the high interest rate equilibrium holds. Firms won t deviate to r if and only if: γ r c c r c c (36) By the same reasoning. However, now firms can deviate to even lower interest rates because, in contrast with the single homing case, they can obtain more demand by lowering 18

19 interest rates. Suppose the firm deviates and less thanr in t = 2. The profits of sticking to the Nash equilibrium strategies and deviating are given by: This condition is equivalent to: (f i + a c i )D i (f i, r) + γd(r c c )D i (f i, r) (37) (f i + a c i )D i (f i, r ) + γd(r c c )D i (f i, r ) (38) 0 (3 2γ)D(r r) (39) Which can never hold, so this deviation is not profitable, if r > r. Finally, note that deviations in t = 1 and in t = 2 by some firm can t occur in any of both equilibriums because, given any per transaction fees charged in the first period, the equilibrium condition for γ defines the interest rate charged in the second period and charging this interest rate will be a dominating strategy for each firm. Therefore, firms won t have incentives to deviate in the first period either. 19

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