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Federal Reserve Bank of Chicago
A Theory of Credit Cards
Sujit Chakravorti and Ted To
WP 1999-16
A Theory of Credit Cards
Sujit Chakravorti
Ted To∗
July 2003
Abstract
Recent U.S. antitrust litigation and concerns by regulatory authorities over fees in Australia and the European Union have questioned the nature of various bilateral relationships
and associated fees underlying credit card transactions. A two-period model is constructed to
study the interactions among consumers, merchants, and a card issuer. The model yields the
following results. First, if the issuer’s cost of funds is not too high and the merchant’s profit
margin is sufficiently high, a credit card equilibrium exists. Second, the issuer’s ability to charge
higher merchant discount fees depends on the number of customers gained when credit cards
are accepted. Thus, credit cards exhibit characteristics of network goods. Third, each merchant
faces a prisoner’s dilemma where each independently chooses to accept credit cards, however
all merchants’ two-period profits are reduced because of intertemporal business stealing across
industries.
JEL Classifications: G2, D4, L2
Key Words: Credit Cards, Merchant Discount, Payment Systems, Network Effect
∗
Chakravorti: Federal Reserve Bank of Chicago, 230 S. LaSalle Street, Chicago, IL 60604. E-mail: Sujit.
Chakravorti@chi.frb.org. To: Bureau of Labor Statistics, Room 3105, 2 Massachusetts Ave., NE, Washington,
DC 20212. E-mail: To.Theodore@bls.gov. We benefited from comments on earlier drafts received from Guiseppe
Bertola, Gary Biglaiser, Catherine de Fontenay, Joshua Gans, Bob Hunt, Jamie McAndrews, Marius Schwartz, and
presentations at the 2000 Federal Reserve System Committee on Financial Structure and Regulation, the MidWest
Macroeconomics conference, the Western Economic Association International Meetings, Helsinki Conference on Antitrust Issues in Network Industries, Innovation in Financial Services and Payments Conference held at the Federal
Reserve Bank of Philadelphia, the Economics of Payment Networks held at University of Toulouse, the Chinese University of Hong Kong, European University Institute, Melbourne Business School, Reserve Bank of Australia, and
the University of Michigan. The views expressed are those of the authors and should not be attributed to the Bureau
of Labor Statistics, the Federal Reserve Bank of Chicago or the Federal Reserve System.
Today, credit cards serve as an indispensable credit and payment instrument in the United States.
In 1999, there were 14.2 billion credit card transactions accounting for $1.096 trillion (Credit
Card News, 2000). The popularity of credit cards continues to grow as evidenced by the greater
proportion of merchants that accept them and of consumers that carry them. Using a dynamic
model, we explore the costs and benefits of credit cards to consumers, merchants and the card
issuer.
Consumers find credit cards convenient for making purchases by accessing lines of credit that
they may choose to pay off at the end of the billing cycle or pay over a longer period of time. Around
thirty to forty percent of consumers pay off their balances in full every month, such consumers are
known as convenience users. In the United States, issuers seldom impose per-transaction fees and
often waive annual membership fees.1 Furthermore, issuers may provide incentives such as frequentuse awards, dispute resolution services, extended warranties and low-price guarantees to promote
usage. While revolvers usually receive the same benefits as convenience users, they are usually
charged for these card enhancements as part of finance charges on their borrowings.
Merchants also benefit from accepting credit cards. Ernst and Young (1996) found that 83
percent of merchants thought that their sales would increase and 56 percent thought that their
profits would increase by accepting credit cards. Merchants benefit from sales to illiquid consumers
who would otherwise not be able to make purchases. By participating in a credit card network,
merchants generally receive funds within 48 hours. Credit cards provide relatively secure transactions for non-face-to-face transactions as evidenced by the overwhelming use of credit cards for
online transactions. Furthermore, in today’s competitive marketplace, merchants not accepting
credit cards may lose business to other merchants that do.
However, credit cards are the most expensive payment instrument to accept. According to the
Food Marketing Institute (2000), credit cards on average cost supermarkets 72/c per transaction
compared to 34/c for online debit cards and 36/c for checks.2 A significant portion of the cost is
1
According to a recent Federal Reserve Survey, 63 percent of issuers did not charge an annual fee (Board of
Governors of the Federal Reserve System, 2000). Issuers are more likely to impose annual fees if their cards are
loaded with additional enhancements.
2
Online debit cards use automated teller networks to process transactions and are PIN-based as opposed to offline
debit cards that use credit card networks and are signature-based.
due to the merchant discount, the level of discount that each merchant receives when converting
its credit card receipts into good funds. In the United States, merchant discounts generally range
from 1.25 percent to 3 percent and are bilaterally negotiated between merchants and their financial
institutions.
Merchants seldom charge more to credit card customers. Differentiated prices at the point
of sale has a long legislative and legal history (see Barron et al., 1992; Board of Governors of
the Federal Reserve System, 1983; Chakravorti and Shah, 2003; Kitch, 1990; Lobell and Gelb,
1981).3 However, studies in jurisdictions where merchants are allowed to impose surcharges find
that merchants do not usually impose them (Vis and Toth, 2000; IMA Market Development AB,
2000) confirming Frankel (1998), who suggested that merchants generally adhere to price cohesion.
While credit card surcharges are relatively rare, some merchants in California do impose surcharges
on purchases made with online debit cards suggesting that merchants may face barriers to imposing
credit card surcharges at the point of sale.
Some credit card network practices have been recently challenged in the U.S. courts. One
antitrust case questioned various credit card association rules such as governance duality, permitting
a member to serve on the Board of Directors of one association while being a member of key
committees of the other, and exclusivity, the inability of members of the associations to issue
credit and charge cards from competitor networks. The other case questioned honor-all-cards
rules requiring merchants to accept all branded payment products offered by the card association’s
members carrying its logo. In this article, we explore some of these issues in the context of the
model we develop.
Our paper provides answers to the following questions. Why do merchants accept credit cards
even though credit cards are the most expensive payment instrument to process? What conditions
are necessary for a credit card equilibrium to exist? Does the market for credit cards exhibit network
effects? Does the decision of a merchant to accept credit cards affect profits of other merchants?
We construct a two-period, three-agent model to investigate these questions. Our model yields
the following results. First, if merchants earn a sufficiently high profit margin and the cost of funds
3
The Netherlands, Sweden, and the United Kingdom prohibit credit card associations from imposing no-surcharge
rules. The Australian authorities have imposed a similar ban.
2
is sufficiently low, a credit card equilibrium exists. In other words, the issuer finds it profitable
to provide credit card services, merchants accept credit cards, and consumers use them. Second,
the discount fee that merchants are willing to pay their bank increases as the number of illiquid
credit card consumers increases. This result indicates the presence of network effects in the credit
card market. Third, a prisoner’s dilemma situation arises, where each merchant chooses to accept
credit cards but by doing so each merchant’s two period profit is lower. In other words, there exists
intertemporal business stealing among merchants across different industries. The remainder of the
article is organized as follows. In the next section, we present our model. We solve for the credit
card equilibrium in Section 2, discuss policy implications in Section 3, and conclude in Section 4.
1
The Model
In our model, the five main credit card participants—the consumer, the consumer’s financial institution or the issuer, the merchant, the merchant’s financial institution or the acquirer, and the
credit card network—have been condensed to three participants. The issuer, the acquirer, and the
network operator are assumed to be a single agent and referred to as the issuer. This may not be
unrealistic given the history of the credit card industry—when Bank of America started operating
a credit card system in 1958 it served as issuer, acquirer, and network operator.4
Assume that there is a continuum, say [0, 1], of indivisible goods exogenously priced at p.
Each good is sold by a monopolistic merchant for whom unit cost is c < p.5 Monopoly rents are
maintained because each good within this continuum is distinct from one another (e.g., car repairs,
new refrigerator, etc). By having a continuum of merchants, we focus our attention on a world
where merchants are small and have no bargaining power to set the merchant discount rate.6 Since
4
American Express and Discover operate in this manner. Although American Express is primarily known for its
charge cards, it also issues credit cards such as Optima and the Blue card. In the 1960s, Bank of America licensed
its BankAmericard product, known today as the Visa card, to other banks in large part because interstate branching
restrictions at the time hindered expansion. For more details, see Chakravorti (2000), Evans and Schmalensee (1999),
Mandell (1990) and Nocera (1994).
5
Rochet and Tirole (2002), Schwartz and Vincent (2002) and Wright (2000) also assume noncompetitive goods
markets in their credit card models. Chakravorti and Emmons (2003), Gans and King (2001), and Wright (2000)
consider competitive goods markets.
6
We do not allow merchants to issue their own credit cards. Some merchants do issue their own credit cards but
the market share of such cards is small compared to the share of third-party general-purpose credit cards. Nilson
Report (1999) reported U.S. general-purpose credit and charge card transactions in 1998 at 10.04 billion valued at
3
merchants earn rents, credit cards can be of value if they increase sales.
Each consumer’s demand for these goods is randomly determined in order to capture the notion
that consumer spending may be stochastic and that some expenditures may be unanticipated. In
particular, we assume that with probability γ a consumer does not need to consume one of these
goods and with probability 1 − γ she does. If she needs to consume then she is randomly and
uniformly matched to one of these goods and must consume or else face a utility loss of u. In either
case, consumers start at a base utility level which, for notational convenience, we normalize to 0.
These goods can be thought of as a critical part or service required for an unanticipated breakdown
of an appliance or a car, where the consumer has to purchase the part or service from a specific
merchant. Other formulations where consumers gain utility from consumption are possible and
yield identical results.
As is typically true in actual practice, we assume that merchants do not charge different prices
to their credit card and cash purchasers.7 Why merchants do not differentiate between credit and
cash purchases is a difficult question to answer. It may simply be the case that faced with a higher
price for credit purchases, consumers may choose to purchase elsewhere rather than pay a higher
price. For example, in the eighties, some gasoline retailers gave discounts for cash payments but
eventually returned to a policy of uniform pricing (Barron et al., 1992). In addition, national
and state laws have also restricted pricing policies in the past and some states continue to do so.
However, Federal Reserve Regulation Z allows all merchants to offer discounts for non-credit card
purchases. Given that there is no economic difference between imposing surcharges for credit card
purchases or offering discounts for non-credit card purchases, it seems puzzling that both types of
statutes coexist.
The assumption that prices are exogenously set is not entirely unrealistic as we can take the
magnitude of the markup to be dependent on the structure of the product market (i.e., the elasticity of demand, the availability of close substitutes, etc). Furthermore, franchises often make
“suggestions” to franchisees regarding many of their operating conditions, such as price. Moreover,
$986 billion whereas proprietary credit card transactions at 3.38 billion valued at $145 billion.
7
The alternative instrument that we consider need not be cash but could be an electronic substitute that are less
expensive to process than credit cards such as online or PIN-based debit cards.
4
the fact that merchants cannot increase prices to recover the additional costs associated with credit
card transactions actually strengthens any result in which they are willing to accept credit cards.
That is, because exogenously fixing prices removes a degree of freedom from the merchants, if credit
card equilibria exist under fixed prices, they would also exist if merchants were free to adjust prices.
Assume there is a continuum of consumers. Each consumer has income ωt in periods t = 1, 2.
For each consumer, ωt is independently distributed via continuous cumulative distribution function
F and associated probability density function f and has support Ω = [ω, ω̄]. Consumers have
discount factor β. Consumers may choose from two payment instruments; they can pay with cash
or with a credit card if they have sufficient credit available. Any money not spent in the first
period earns return R > 1. R will also be the issuer’s cost of funds and the interest rate earned on
merchants’ first period profits.
Assume that a monopolistic issuer offers a credit card to all consumers with credit limit L(ω).
That is, if a consumer has a first period income of ω1 , the amount of credit issued to her by the
financial institution is L(ω1 ). Since the economy only lasts for two periods, credit is only offered
in the first period. The issuer then collects debts owed (or however much is collectable) at the
beginning of the second period.8 It is important to note consumers pay no interest on credit card
purchases. This assumption is based on the observation that credit card purchases have a “grace
period” during which no interest accumulates. There would be interest due only if part of the
consumer’s debt were carried over into a third period. The agreements between the issuer and
consumers and between the issuer and the merchants is assumed to be costlessly enforceable.
For reasons of tractability, revolving credit is excluded and the economy lasts for only 2 periods.9
However, such an abstraction may not be unrealistic given the use of charge cards and industry
estimates that as much as forty percent of credit card consumers pay off their balances in full every
month. Some issuers have specifically targeted pricing policies and incentives towards this group
of consumers (see Chakravorti and Shah, 2003). Furthermore, the qualitative results of the model
would not change if an additional period of credit were added, however a monopolist issuer should
8
While there is uncertainty at the consumer level, there is no aggregate uncertainty given the large number of
consumers. Hence, the issuer’s credit risk is zero. The results would not qualitatively change if aggregate uncertainty
were introduced.
9
Chakravorti and Emmons (2003) are the only ones to consider the impact of revolvers on the credit card network.
5
Date
t=0
t=1
Agent
Issuer
Merchants
Nature
Consumers
t=2
Nature
Issuer
Consumers
Actions
Choose ω̂ and ρ
Accept/Not-Accept Credit Cards
Determine ω1 and demand
for each consumer
Buy/Not-Buy
If Buy, Use/Not-Use Credit
Determine ω2 and demand
for each consumer
Collect Debts
Make Cash Purchases
Figure 1: Sequence of actions
be able to earn rents from consumers in addition to rents that it earns from merchants.
Since the price of each good is identical, without loss of generality, the issuer can choose credit
limits among functions of the following form: L(ω) = 0 if ω1 ∈
/ Ω̂ and L(ω) = p if ω1 ∈ Ω̂ where
Ω̂ ⊂ Ω. That is, we can limit L to take only the values of 0 and p since a credit card is useful
only if it allows one to consume the good—credit limits below p or beyond p are of no use. Since
lower-income consumers will have a higher risk of default, it must be the case that the optimal L(ω)
will have Ω̂ = [ω̂, ω̄] for some ω̂ ∈ Ω. Thus, the issuer can simply choose ω̂ to maximize revenues
while minimizing defaults. Given that we examine credit limit functions which take values of 0 on
[ω, ω̂) and p on [ω̂, ω̄], we will call ω̂ the income requirement, below which consumers are not offered
credit cards.
Merchants must decide whether or not to accept credit cards as payment for first period purchases. The issuer imposes a per-sale transaction fee, ρ ≥ 0, for each credit card purchase. The
issuer pays the merchant the difference between the sales receipts and the amount corresponding
to the merchant discount. Since the mix of customers matched to each merchant is the same, each
merchant faces an identical profit maximization problem. As will later be verified, in equilibrium
all merchants will either accept credit cards or none will accept them.
The structure of the model is illustrated in Figure 1. At time 0, the issuer chooses the transaction
fee, ρ, and the income requirement, ω̂, for consumers to qualify for a credit card. The merchants
then decide whether or not to accept credit. At the beginning of periods 1 and 2, consumer incomes
6
and desired consumptions are randomly determined. In period 1, each consumer decides whether
to purchase her desired consumption good and then if she has access to credit, how she should pay
for it. Any outstanding debts are collected after the realization of second period income. If she has
sufficient funds in period 2, she may choose to consume but can only pay with cash, because the
issuer extends no credit in period 2.
2
Equilibrium
2.1
Consumers
Starting with the second period, a consumer will always purchase the good she desires if she can
afford it. If the consumer had consumed with credit in the first period then she can afford to
consume in the second period if Rω1 + ω2 ≥ 2p. That is, she earns a return R on her first period
endowment, ω1 . Her total money balances in the second period must be used to pay off her debt
from the first period, namely p. If Rω1 +ω2 < p then she defaults and the sum Rω1 +ω2 is seized by
the issuer. Before the realization of her second period income, the probability that she can afford
to consume is Pr(ω2 ≥ 2p − Rω1 ). Similarly, given that she consumed with cash in the first period,
the probability that she can afford to consume in the second period is Pr(ω2 ≥ p − R(ω1 − p)).
Finally, if she did not buy at all in the first period, the probability that she can afford to consume
in the second period is Pr(ω2 ≥ p − Rω1 ). Given some target second period wealth level, x, the
probability that ω2 is at least x can be written in terms of the cumulative distribution function as
Pr(ω2 ≥ x) = 1 − F (x).
We can now calculate a consumer’s first period discounted expected utility from purchasing
with her credit card and from purchasing with money. With probability 1 − γ, a consumer needs
to consume. If she has access to and buys on credit, she receives discounted expected utility of:
U c (ω1 ) = −β(1 − γ) Pr[ω2 < 2p − Rω1 ]u.
(1)
By purchasing, she prevents a utility loss of u in the first period and, if necessary (with probability
1 − γ), if she can afford to, will consume in the second period. Second period consumption is
7
discounted by β. If a consumer consumes with cash, she receives discounted expected utility of:
U m (ω1 ) = −β(1 − γ) Pr[ω2 < p − R(ω1 − p)]u.
(2)
If a consumer does not consume in the first period, she gets utility of:
U 0 (ω1 ) = −u − β(1 − γ) Pr[ω2 < p − Rω1 ]u.
(3)
Finally, with probability γ, the consumer simply does not need to consume at all and receives utility
of:
U ∅ (ω1 ) = −β(1 − γ) Pr[ω2 < p − Rω1 ]u.
(4)
It follows that all consumers, given the opportunity, will consume and that if they have credit
available will prefer to purchase on credit rather than pay cash. Because consumers are not explicitly
charged for using their credit cards and earn interest on their funds for one period, credit card
payments dominate cash payments.
If merchants were to set different prices based on the payment instrument used, liquid consumers
would choose to use cash over credit cards given the appropriate price difference. However, there are
different views in the academic literature about the welfare effects of setting different prices based
on the payment instrument used (see Chakravorti, 2003). Chakravorti and Emmons (2003) suggest
such a pricing strategy improves welfare given competitive markets when issuers offer incentives
to convenience users who do not share in the cost of providing payment services. Schwartz and
Vincent (2002) suggest that consumer surplus may be lower if merchants charge the same price for
credit card and cash purchases. Rochet and Tirole (2002) and Wright (2000) suggest that allowing
merchants to set different prices may not be welfare enhancing. We will discuss the effect of such
a pricing strategy on other participants in the context of our model below.
8
2.2
Merchants
We are primarily interested in and will derive conditions for the existence of equilibria in which
merchants accept credit cards. Since prices and costs are exogenously specified, and accepting
credit cards is costly, merchants will be willing to accept credit cards only if they increase sales
volume. In a credit card equilibrium, it must be that ω̂ < p since otherwise merchants who accepted
credit cards would not increase their sales and would also be required to pay fee ρ on all credit card
sales.
In order to make their decision, merchants forecast the current and future demand for their
product. First period demand is based on the distribution of first period income, F (ω1 ), and the
credit limit offered by the issuer, L(ω1 ). Second period demand depends on the distribution of total
wealth, net of cash purchases or credit repayments, at the beginning of the second period. This in
turn depends on the equilibrium and as a result, the credit limit function L. Let the distribution
of second period net total income be given by the cumulative distribution function, H(x; ω̂).
Provided that p ≥ ω̂, each merchant’s discounted expected profits from accepting credit cards
will be proportional to:
π c = [1 − F (ω̂)](p − c − ρ) +
1
[1 − H(p; ω̂)](p − c).
R
(5)
The same merchant’s discounted expected profits from not accepting credit cards will be proportional to:
π nc = [1 − F (p)](p − c) +
1
[1 − H(p; ω̂)](p − c).
R
(6)
Notice that since individual merchants are massless, a single merchant’s decision of whether or not
to accept credit has no effect on second period sales and as a result, a merchant will accept credit
cards when:
[1 − F (ω̂)](p − c − ρ) ≥ [1 − F (p)](p − c).
and will not when the opposite is true. As long as ρ is sufficiently small, if ω̂ < p, the merchant
will choose to accept credit cards. By accepting credit, a merchant sells an additional F (ω̂) − F (p)
9
units—all credit sales come at the additional unit cost of ρ. Finally, as we will see, the fact that
the merchant’s problem does not depend on H(p; ω̂) (and thus its credit acceptance decision) will
have important implications for merchant welfare.
2.3
The Issuer
The issuer maximizes profits through choice of ρ and ω̂. The question then is, under what conditions
will the issuer choose ρ and ω̂ such that merchants are willing to accept credit as a form of payment.
To solve the issuer’s problem, it needs to be able to forecast the gross income, x = Rω1 + ω2 , of
consumers to whom they extend credit, ω1 ≥ ω̂. The distribution of x conditional on the realization
of ω1 is G(x | ω1 ) = Pr[Rω1 + ω2 ≤ x] = F (x − Rω1 ). Conditional on ω1 ≥ ω̃ for some ω̃ < ω̄, the
distribution of x is:
Zω̄
G(x | ω1 ≥ ω̃) =
F (x − Rω1 )
f (ω1 )
dω1 .
1 − F (ω̃)
(7)
ω̃
When all merchants accept credit, the issuer’s profits can be written as:
(
Π = (1 − γ)[1 − F (ω̂)]
− (p − ρ)+
"
1
p(1 − G(p | ω1 ≥ ω̂)) +
R
Zp
#)
xg(x | ω1 ≥ ω̂)dx
, (8)
min{Rω̂+ω,p}
where G( · | ω1 ≥ ω̂) is given by (7) and g( · | ω1 ≥ ω̂) is the associated conditional probability
density function. The first term is the amount lent to consumers for first period credit purchases,
less the sales fee charged to merchants. The terms within the brackets are repayments from the
consumers who do not default and those from consumers who do. Notice that as long as F is
continuous, Π is continuous. Since (ρ, ω̂) must belong to the compact set [0, p − c] × [ω, ω̄], Π has
a global maximum so that there exists at least one equilibrium. We now investigate some of the
properties of these equilibria.
Notice that ρ’s purpose is to extract rents from the merchants and has no effect on the gross
rents available. It is therefore straightforward to derive the optimal ρ by setting π c = π nc and
solving. When ω̂ > p, no additional sales are generated by the acceptance of credit cards and
10
ρ = 0. When ω̂ ≤ p, this is given by:
ρ(ω̂) =
F (p) − F (ω̂)
(p − c).
1 − F (ω̂)
(9)
In other words, ρ is a fraction of the additional first period revenues generated by the acceptance
of credit cards. Differentiating ρ(ω̂) with respect to ω̂ yields:
f (ω̂)(1 − F (p))
∂ρ
=−
(p − c).
∂ ω̂
(1 − F (ω̂))
(10)
and the following proposition:
Proposition 1 The fee that the issuer can charge merchants falls as credit becomes more restrictive.
In a broader sense, the ability to increase merchant fees is directly related to the number of consumers that have access to credit cards. Credit cards in our model display characteristics of a
network good because as the number of illiquid cardholders increases the value of accepting them
also increases (e.g., Economides, 1996; Katz and Shapiro, 1985).10
The variable ω̂ determines the magnitude of available rents. Note first that if (1 + R)ω ≥ p then
even the poorest consumers can afford to repay p and there would be no defaults. As a result, if the
bank issues any credit, it issues it to everyone (i.e., ω̂ = ω). On the other hand, if (1 + R)ω < p and
ω̂ < p then depending on ω̂, some consumers may default. We investigate this more interesting,
case. In particular, if ω̂ is such that Rω̂ + ω < p, a positive measure of consumers default with
certainty.
To determine whether or not banks are willing to extend credit to consumers, take the first
order condition for the issuer’s profit maximization problem with respect to ω̂. Since optimal ρ is
given by (9) for any ω̂, we can first substitute (9) into (8) before taking the first order condition.
10
For a discussion of network effects in credit card markets see Katz (2001).
11
The first derivative is given by:
1
∂Π
= (1 − γ)f (ω̂) c + p(F (p − Rω̂) − 1) −
∂ ω̂
R
Zp
min{Rω̂+ω,p}
xf (x − Rω̂)dx ,
(11)
where ω̂ must lie in [ω, ω̄].
Proposition 2 If R and c2 are not too large then in every equilibrium, the issuer extends credit
(i.e., the ω̂ ∈ [ω, p2 )), almost all merchants accept credit and some consumers default.
Proof: To evaluate the first order condition we need to evaluate it for ω̂ where 1) Rω̂ + ω ≥ p and
2) Rω̂ + ω < p.
Consider ω̂ such that Rω̂ + ω ≥ p. In this case there will be no defaults—the minimum income
required to qualify for a credit card, including the return it earns, and the minimum income in the
second period is sufficient to fully repay p. In this case,
Zp
F (p − Rω̂) =
xf (x − Rω̂)dx = 0,
min{Rω̂+ω,p}
so that (11) simplifies to:
n
∂Π
po
= (1 − γ)f (ω̂) c −
.
∂ ω̂
R
(12)
If R = 1 or c = 0, this is strictly negative so that as long as R and c are not too large, an equilibrium
must have ω̂ < p so that there is credit.
Now, notice that at Rω̂ + ω = p − ε, for ε sufficiently small this derivative is still negative. Since
(11) is strictly negative for ω̂ ∈ [(p − ω − ε)/R, ω̄], any maximum must satisfy Rω̂ + ω < p. Thus, in
every equilibrium, a positive measure of the consumers who purchased on credit, F (p)−F (Rω̂ +ω),
default.
Finally, the analysis so far has assumed that all merchants accept credit cards. Suppose that
this were not the case and that in equilibrium some proportion ζ accept credit cards while some
proportion 1 − ζ do not accept credit cards. In this case, the issuer’s profits would be given by
ζΠ where Π is as given in (8) and optimal ω̂ is still characterized by (11). However, if the issuer
12
lowered ρ by ε, then all merchants would strictly prefer to accept credit. Since we can take ε to
be small, the issuer can discontinuously increase her profits by lowering ρ and in equilibrium the
set of merchants who do not accept credit is of measure zero. As a result, in equilibrium almost all
merchants accept credit.
That is, as long as there are sufficient rents available and the cost of funds is not too large, in
every equilibrium of this model the issuer offers credit, merchants accept credit cards, and consumers
use credit cards to make purchases when possible. The issuer chooses the income requirement, ω̂,
above which consumers can purchase on credit, such that a non-zero mass of consumers will be
unable to pay off their first period debt and default. If the objective function is concave, then the
equilibrium is unique.
Combined with the fact that merchants are massless, the set of any individual merchant’s
repeat purchasers will be of measure zero. This implies that merchants will not consider the effect
of current decisions on future revenues. Since merchants’ second period revenues are not directly
impacted by their decision over whether or not to accept credit and since all of the additional first
period rents are extracted, they must be worse off because the aggregate second period distribution
of wealth and therefore sales are lower.
This effect comes about because merchants face an externality much like that in the Prisoner’s Dilemma. As a group, merchants realize group acceptance of credit cards reduces second
period profits and that first period rents generated by the acceptance of credit cards will be fully
extracted—they therefore recognize that, as a group, they would be better off not accepting credit.11
Individually, however, a merchant’s decision of whether or not to accept credit cards has no effect
on net total consumer incomes and the issuer can choose ρ such that all merchants find it in their
best interest to accept credit cards. Thus, merchants accept credit despite the fact that they are
made worse off.
One can also think of this externality as an intertemporal business stealing effect. Since merchants are unlikely to face the same customer in the future, the acceptance of credit cards allows
individual merchants to capture sales which might otherwise be made by another merchant in the
11
The effect of one merchant’s decision on other merchants is discussed by Katz (2001).
13
second period. Note that this externality is not a feature of our assumed finite horizon. In any
given period, the merchant faces a decision over whether to accept credit. With almost no repeat
purchasers, their current decision has no impact on future sales so that their decision is based only
on the additional revenues generated today, leading to the externality described. Note that business
stealing in our model occurs across industries and across time unlike any other model of payment
card networks.
Furthermore, the exogeneity of prices is not an issue when the issuer is a monopolist. At first
glance, one might suppose that an ability to set prices would allow the merchant to retain a share
of any additional rents. However, since the issuer can rationally anticipate any price increase, it can
still choose ρ to completely extract any additional rents. In other words, given full extraction, if the
merchant can raise its price in response to the merchant discount, it would still be unable to retain
any of the additional first period rents from illiquid consumers. Allowing merchants to set prices
based on the underlying cost of the payment instrument used would not qualitatively change our
results since the issuer would still fully extract rents from illiquid consumers. Similarly, additional
issuer revenues garnered directly from consumers in the form of fees or potential finance charges
from borrowing for longer periods, would have no bearing on the issuer’s incentive to extract rents
from the merchant.
Finally, we would like to discuss introducing revolving credit. One way to introduce revolvers
is to add a third period during which consumers receive further income but do not consume.
Consumers who are unable to pay their remaining balances in the second period must pay interest
on the remaining balances in the third period. Allowing for revolvers in this manner involves some
additional difficulties; it complicates the consumer’s problem and the rate of interest charged to
the consumer should ideally be endogenously determined by the issuer. However, as far as the
choice over the availability of credit goes, the problem remains essentially unchanged. Given some
interest rate on consumer debt, one can calculate the future distribution of income. As long as this
distribution is reasonably well behaved, maximizing issuer profits will yield qualitatively similar
results.
14
3
Policy Implications
Our model provides a benchmark for policymakers to consider when setting polices regarding credit
card networks. We consider a credit card market where consumers are given rebates in the form of
an interest-free short-term loan and merchants do not impose surcharges on credit card purchases.
We also consider a monopolistic issuer and merchants that are monopolists but no single merchant
has any bargaining power with the issuer. Under these conditions, we find that a credit card
equilibrium exists and certain participants benefit while others may not benefit. In this section, we
consider the recent policy debate that is occurring in numerous countries regarding certain common
market practices.
In October 2001, a verdict was given on an antitrust case against the two largest credit card
networks—MasterCard and Visa—that claim over 75 percent of the U.S. general-purpose credit
card market.12 The U.S. Department of Justice (DOJ) charged the two associations with colluding
resulting in adverse effects on competition and consumer welfare. The court ruled in favor of the
associations regarding dual governance, but ruled in favor of DOJ regarding exclusivity. The card
associations are appealing the decision to allow member banks the right to issue other credit and
charge card products belonging to other networks. The underlying premise of this case is that
market power at the network level may affect certain participants adversely.13
A key issue raised by the Australian authorities is the nature of the cooperative arrangement in
setting certain fees such as the interchange fee—the fee collected by the issuer from the acquirer.
These fees are set at the network level based on certain merchant characteristics and other factors.
The Reserve Bank of Australia have imposed formulas to determine interchange fees in credit card
networks and have prohibited no-surcharge rules.
Rochet and Tirole (2002) and Schmalensee (2002) examine the setting of the interchange fee.
12
In another case, over 5 million retailers accused the two credit card associations of imposing illegal tying arrangements that forced merchants accepting one of the credit card association’s branded payment products to accept all
of their branded products. This case addressed competition among payment networks and the ability of merchants
to accept certain products and decline other products from the same card association. Recently, the case was settled
immediately before the scheduled trial date. As part of the settlement, MasterCard and Visa agreed to pay the
merchants $3 billion and remove their honor-all-cards rules.
13
Hausman et al. (2001) argue that if these joint-ventures are not-for-profit, collusion among the networks yields
productive efficiency. While Visa remains a non-profit joint venture, MasterCard has recently changed its governance
structure to a private share company.
15
Rochet and Tirole conclude that there are conditions under which the equilibrium interchange fee
is second best optimal and that lifting the no-surcharge rule in these cases reduces welfare. When
the equilibrium interchange fee is sub-optimal, lifting the no-surcharge rule may either increase or
reduce welfare. Schmalensee does not model the product market and as a result cannot conduct
welfare analysis. Instead, he looks at interchange fees which maximize the weighted sum of issuer
and acquirer profits under various competitive regimes. Both Rochet and Tirole, and Schmalensee
conclude that interchange fees need not, as argued by some, be zero.14 Wright (2001) finds under
various simplifying assumptions that the optimal interchange fees are increasing in the issuers’ costs
and profitability and decreasing in the acquirers’ costs and profitability.
Our model differs in two respects from the models of interchange discussed above. First, rather
than taking a reduced form approach where the costs and benefits of credit cards are exogenously
assigned functional forms, we specify a model which endogenously yields costs and benefits to the
involved parties. Second, we use a dynamic setting in which there are intertemporal tradeoffs
for all of the parties involved. Using this approach, we identify an intertemporal externality that
merchants impose on one another because their credit acceptance decision has no (or little) impact
on future earnings.
Like Rochet and Tirole (2002) we can also model both the merchant discount and the interchange
fee by assuming that the market for acquirers is competitive.15 Given such market structures,
interchange fees serve as a lower bound for the merchant discount. Industry estimates indicate
that the market for acquirers is fairly competitive in the United States given that certain classes
of merchants are basically charged close to the interchange fee. Our results suggests that given the
market structures we consider, if there is significant market power to set these fees, both consumers
and the issuer must be better off—if it were not the case then either consumers would refuse to use
credit cards or the issuer would refuse to issue them.
However, whether merchants gain in a credit card equilibrium vis-à-vis a no credit card equilibrium depends on several factors. In our model, there is a single monopolistic issuer/network,
merchants have no bargaining power and an individual merchant’s current sales have no effect on
14
15
For a discussion of credit card interchange fees, see Evans and Schmalensee (1999) and Balto (2000).
Gans and King (2001) and Wright (2001) consider the setting of interchange fees for different market structures.
16
future sales. To see the effect of relaxing these assumptions, first consider the extreme case where
there is instead a single merchant who still has no bargaining power. This single merchant will
face all future consumers so that any change in next period’s distribution of consumer wealth will
directly impact its future sales. As a result, the merchant will fully internalize the impact of its
current credit acceptance decision. However, since the card issuer still holds all of the bargaining
power, all rents will still be extracted and therefore the merchant must be indifferent between accepting credit and not. Now suppose that the single merchant does have some bargaining power.
In this case, the merchant must garner a share of the rents and must therefore be better off. In
general, we can think of a model in which both the number of merchants and their bargaining
power can vary with the equilibria ranging from the merchants being worse off (infinite number
of merchants and zero bargaining power) to the merchants being better off (one merchant with
positive bargaining power). To summarize, the merchants’ welfare depends on:
1. The degree of concentration in the market for credit cards.
2. The amount of bargaining power held by merchants.
3. The impact of a single merchant’s decision on its volume of future sales.
Our model raises several policy questions regarding the regulation of credit card networks.
First, to what degree are the credit card networks able to collude? Second, what is the impact of
a single merchant’s decision to accept credit cards on the future distribution of consumer wealth
and therefore own future sales? Third, how much bargaining power do merchants have in the
determination of the merchant discount? It seems reasonable to believe that for most merchants,
the decision to accept credit cards or not will have little impact on future income distribution
and consequently will have little impact on future sales. Moreover, bargaining power appears to
differ between merchants ranging from almost no bargaining power to large chains with strong
bargaining power. Merchants with strong bargaining power is evidenced by merchant discounts
being extremely close to the appropriate interchange fee. Thus, under our framework, whether
or not some merchants are worse off depends on the degree to which the credit card network is
monopolistic.
17
4
Conclusion
We constructed a model along the lines of Baxter (1983) where we consider the various bilateral
relationships in a credit card network. With the exception of Chakravorti and Emmons (2003),
the literature on credit card networks ignores the cost and benefits of the extension of credit to
network participants. We explain why merchants accept credit cards using the most restrictive
possible environment—a single issuer, massless merchants, and no cost sharing by consumers either
directly in the form of fees or finance charges or indirectly in the form of higher prices. Credit
increases sales because both purchases and incomes vary over time and with credit cards ‘credit
worthy,’ liquidity-constrained consumers are able to purchase—all else equal, merchants prefer to
make a sale today rather than tomorrow. We demonstrate that a credit card equilibrium can exist
if the cost of funds is relatively low and the merchant’s profit margin is sufficiently high. We also
show that using the merchant discount, the issuer will be able to fully extract rents from merchants
resulting from sales to illiquid consumers.
Furthermore, the equilibrium interaction between the merchant discount and the accessibility
of credit has network effects. If the card issuer makes credit more widely available, the merchant
increases its sales to illiquid customers. This in turn allows the card issuer to increase the discount
the merchant is charged. In other words, merchants are willing to pay higher merchant discounts
if credit cards generate greater sales. That is, credit card services exhibit network effects.
Finally, we show that there is an externality where merchants find themselves in a prisoner’s
dilemma situation. In equilibrium, each merchant chooses to accept credit cards. However, when
all merchants accept credit cards, they are all worse off. This result is dependent on the degree
of market power held by the issuer, the amount of bargaining power held by merchants, and the
ability of merchants to internalize the effect of their current credit acceptance decision on their own
future sales. This result is unique to our model.
To summarize, we constructed a dynamic model of credit cards where the benefits to various
participants are endogenously determined. Furthermore, in addition to explaining why merchants
are willing to accept credit cards, the explicit dynamic nature of the model allows us to identify
an important, intertemporal externality which exists in the market for credit cards. The existence
18
of this externality may have important antitrust implications—what conclusion one draws depends
on the degree to which the credit card market is monopolistic.
References
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21
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