The No Surcharge Rule and Card User Rebates: Vertical Control by a Payment Network

Abstract

The No Surcharge Rule (NSR) prevents merchants from charging more to consumers who pay by card versus other means ("cash"). We consider a payment network facing local monopolist merchants that serve two consumer groups, card users and cash users. Unlike in prior work, transaction quantities are variable. The NSR raises network profit and harms cash users and merchants; overall welfare rises if and only if the ratio of cash to card users is sufficiently large. With the NSR, the network will grant rebates to card users whenever feasible. If rebates are not feasible, the NSR can harm even card users.

Full text

*Schwartz: Department of Economics, Georgetown University, Washington DC 20057-1036,
<schwarm2@georgetown.edu> Vincent: Department of Economics, University of Maryland, College Park, MD
20742 <dvincent@wam.umd.edu>. For helpful comments and suggestions, we would like to thank various seminar
participants, Andrew Dick, Patrick Greenlee, Bob Hunt, David Malueg, Alex Raskovich, George Rozanski and an
anonymous referee. We alone are responsible for the views expressed in this paper.
The No Surcharge Rule and Card User Rebates:
Vertical Control by a Payment Network
by
Marius Schwartz
Daniel R. Vincent*
January, 2006
Abstract: The No Surcharge Rule (NSR) precludes merchants from charging higher prices to
consumers who pay by card instead of other means (‘cash’). We analyze the NSR’s effects when
a payment network faces local monopolist merchants. Importantly, end-users’ demand for the
merchant’s product is elastic. The NSR raises network profit and harms cash users and
merchants, while overall welfare decreases if and only if the ratio of cash to card users is
sufficiently small. If rebates to card users are not feasible, the NSR reduces total consumer
surplus (of cash plus card users) and, if the cash market is sufficiently small, even card users
lose. When rebates are feasible, the network will grant them and raise its merchant fee,
increasing total consumer surplus and overall welfare compared to no rebates but harming cash
users. An increase in the merchant’s gross benefit from card rather than cash sales worsens the
NSR’s effects on overall welfare and total consumer surplus if rebates are not feasible, but the
reverse holds if rebates are feasible.
JEL Classification: D42, G28, L42, L8.
Keywords: No Surcharge Rule, Payment Networks, Card User Rebates, Vertical Control

1Credit cards and offline debit cards accounted for $1.5 trillion ($755 billion for Visa and $444 billion for
Mastercard—both known as bank card associations—and the rest through proprietary networks such as American
Express and Discover) and $203 billion was via online debit cards. Nilson Report, March & April 2003, issues 784
& 785; ATM Debit News EFT Data Book 2003. For a lucid description of the card industry see Hunt (2003).
2For example, there is debate over whether the joint setting of certain network fees by EPN-member banks
(as in bank card associations and regional networks) is anti-competitive (Salop 1990; Carleton and Frankel 1995,
1995a; Evans and Schmalensee 1995, 1999). Also, a common EPN requirement is that merchants must accept all of
an EPN’s cards (e.g. debit and credit cards) if they wish to accept any. This requirement was the target of a major
lawsuit by Walmart and other retailers against Mastercard and Visa, alleging anti-competitive tying. Visa and
Mastercard recently settled this suit, agreeing to relax the requirement and pay plaintiffs over $3 billion.
3Surcharges on credit card transactions were prohibited by federal statutes from 1968 to 1985 and remain
prohibited by some states (e.g., Florida). For a detailed history of the U.S. legislative and regulatory treatment of
surcharges, see Chakravorti and Shah (2001). In the U.S., Visa long had its own no-surcharge rule which it relaxed
recently. Mastercard currently prohibits its merchants from “surcharging” customers for credit purchases, though it
allows cash “discounts” (www.mastercard.com/consumer/cust_serv.html). In some European countries, card
associations prohibit both discounts and surcharges. (Rochet and Tirole 2002.)
4According to one retailer survey, fewer than 1% of merchants offer cash discounts. Chain Store Age,
Fourth Annual Survey of Retail Credit Trends, January 1994, section 2.
-1-
1. Introduction
Transactions through electronic payment networks (EPNs) in the U.S. exceeded $1.7
trillion in 2002 and are growing rapidly.1 Several practices in this important industry have
attracted controversy and antitrust scrutiny.2 One such practice involves constraints on the
ability of merchants to set different prices depending on the means of payment employed, such
as credit cards, debit cards, cash, or checks. We examine these constraints as instruments of
vertical control, assess their welfare effects, and show that their presence may explain the
phenomenon of rebates and reward programs in payment markets.
Uniform price constraints were at various times imposed by law or by EPN rules that
prohibited merchants from imposing surcharges (or adverse non-price terms) for card payments,
even though merchants may face higher costs for card transactions due to EPN fees.3 Even in the
absence of formal prohibitions, merchants are often reluctant to set different retail prices for
different means of payment.4 We refer to all these limits as the No-Surcharge ‘Rule’ (NSR). Our
analysis is relevant to several policy questions. First, it helps assess the desirability of laws or
private regulations governing surcharging; for example, prohibitions on surcharges are banned in
the United Kingdom and in Australia (Reserve Bank of Australia 2002). Second, when repeal of
the NSR is not an option — because merchants’ reluctance to surcharge derives from other

5At least in the U.S., competition is viewed as stronger on the acquiring side. Rochet and Tirole (2002, p.
552) state that acquiring is “widely viewed as highly competitive,” citing Evans and Schmalensee (1999), while
issuing “is generally regarded as exhibiting market power.”
-2-
characteristics of the trading environment — our analysis helps to explain the role of pricing
practices such as interchange fees and rebates to card users. Finally, the analysis is a necessary
step towards evaluating card tying policies (see fn. 2 above), since such tying would have no
force if merchant surcharges were unrestricted.
Any payment network intermediates between consumers and merchants. In a proprietary
(or ‘closed’) network such as American Express, the same entity deals with, and sets the fees to,
merchants and card users. In an association such as Visa (or ‘open network’ since membership is
open to multiple banks), a typical transaction involves two different banks: the cardholder’s
(‘issuer’) and the merchant’s (‘acquirer’). The issuer sets the fees to its cardholders (e.g., a per
transaction fee or, more often, a rebate) and the acquirer sets the fee to its merchants (the
‘merchant discount’— the transaction amount minus what the merchant receives from the
acquirer). The association sets an interchange fee paid by the acquirer to the issuer, which of
course affects their respective charges to merchants and cardholders. Formal economic analysis
of the interchange fee was pioneered by Baxter (1983), who showed that the socially optimal fee
must reflect the net benefits from card use on both sides of the transaction. However, in Baxter’s
analysis the association is indifferent between any levels of the interchange fee, because issuers
and acquirers are assumed perfectly competitive.
Analyses of the interchange fee under imperfect competition among association members
came considerably later. Schmalensee (2002) studies how the fee affects marketing efforts (and
pricing) by issuers and acquirers. Closer to our focus are Rochet and Tirole (2002), who also
provide a review of recent literature. In their model, consumers have unit demands for
transactions but heterogeneous private values of paying with cards versus the outside instrument,
‘cash’. Duopolist merchants are spatially differentiated and, given the interchange fee, choose
simultaneously whether to accept cards and, then, set prices. Acquiring banks are assumed
perfectly competitive, while issuers are imperfectly competitive.5 Issuers’ fee to cardholders is
represented by a reduced form, decreasing function of the interchange fee, since a higher
interchange fee lowers issuers’ net marginal cost of issuing cards. If merchants can freely
surcharge for card transactions, the level of the interchange fee is irrelevant (“neutral”) and card
diffusion among consumers is socially too low, because of the imperfect competition among

6In Rochet and Tirole, this follows because of the assumptions of unit demands and that the market is always
covered in the Hotelling competition between merchants.
7We use ‘card’ to denote any electronic payment instrument, and ‘cash’ to denote the alternative means of
payment. Also, we sometimes will refer to the EPN as the card company. We abstract from the credit role of some
electronic payments instruments and focus solely on its payment function. Chakravorti and Emmons (2001) present a
model where some consumers use cards for both functions while others use them only as a payment instrument, and
investigate the presence of cross-subsidies under an NSR from the former to the latter.
8About 24% of U.S. families do not hold cards of any kind (Federal Reserve Board, 2001, p. 25).
Presumably a large fraction of these families cannot get cards. Evans and Schmalensee (1999) characterize non-card
holders as being “on the economic fringes of society" (p. 87), with a median household income 50% below the
overall average, and more than 40% of them with incomes below the government's estimated poverty line.
-3-
issuers. With a no surcharge rule, there exists an equilibrium in which both merchants accept
cards, the net price falls to card and rises to cash users, and card diffusion either remains too low
(if so, the NSR raises overall welfare) or becomes excessive (if so, welfare may rise or fall).
Rochet and Tirole — and all other literature to our knowledge — focus on consumers’
choice between cards and cash but hold the total quantity of transactions fixed.6 Though a
sensible first approximation, this is obviously an abstraction. Our analysis is complementary, by
considering elastic demand for transactions. To focus on this dimension, we make two
simplifications.
First, we treat payment mode as exogenous: one group of consumers (of mass one) use
only cards, while others (of mass " ) use only cash.7 Transactions demand per capita is the same
for members of each group and is elastic. Thus, while the number of card users is exogenous in
our model, the per capita quantities of card and cash transactions are determined endogenously.
The assumption that some consumers must use cash can be justified partly on grounds that a
sizeable portion of the population are ineligible for cards.8 Moreover, as shown by Rochet and
Tirole (2002, Proposition 6), the key property that the NSR allows an EPN to tax cash users
continues to hold when there is imperfect substitution between cash and cards. Our assumption
that cardholders cannot use cash is strong. However, it introduces consumer benefits from cards
in a simple (albeit extreme) way and, more importantly, it captures the opposite polar case to the
one assumed in existing literature. There, a merchant who refuses cards only loses transactions to
another merchant who accepts cards (Rochet and Tirole, 2002) or converts all customers to using
cash (Wright, 2003). In practice, a merchant refusing cards can lose some transactions entirely,
most obviously because some cardholders are liquidity constrained or perhaps would only buy

9Wright also considers homogeneous Bertrand competition among merchants. As expected, the NSR cannot
harm cash users then either, this time because merchants will specialize in accepting cash or cards.
10 In Wright’s model, consumers vary in their benefits from cards relative to cash but have identical maximal
value v for the good if using cash. Moreover, there is a mass of consumers who will always use cash (as in our
model) because their benefits from cards are less than the marginal cost of card services. A local monopolist
merchant able to surcharge for cards will set a price v to cash users and a higher price to card users. Under no
surcharging, and if v is sufficiently large, the merchant sets a uniform price of v because any higher price entails
losing all cash transactions. The difference in our model is that cash users’ demand for transactions is smooth, so
under the NSR some price increase to cash users will be profitable. A smooth demand by cash users also arises in
Rochet and Tirole, despite individuals’ unit demands for transactions, because each merchant faces consumers that
are spatially differentiated as well as smoothly distributed in their valuations for cards relative to cash.
-4-
certain items with cards due to other bundled properties of the card such as the superior dispute-
resolution protection. Reality is likely to lie between the two polar cases.
Our second simplifying assumption is that merchants are local monopolists, so we do not
explore an important aspect studied by Rochet and Tirole (2002) and Hayashi (2005), the
interdependence between merchants’ decisions to accept or reject cards. In Rochet and Tirole,
imperfect merchant competition is the sole reason why under the NSR card use relative to cash
use can be excessive (a merchant’s gain from accepting cards comes partly by diverting sales
from the other merchant). In our model, the NSR has a different welfare-reducing aspect: it
lowers per capita transactions by cash users due to elastic demands. Our environment leads to
some interesting differences, for example, if rebates to card users are not feasible, then in our
model the NSR can harm even card users.
Our results contrast sharply with those of Wright (2003), who finds that when merchants
are local monopolists (as in our model), the NSR lowers price to card users and leaves price to
cash users unchanged.9 The absence of harm to cash users, and the strong prediction that the
NSR unambiguously increases welfare, both hinge on his assumption of unit demand by cash
users. More typically, when demand facing a merchant varies continuously as in our model and
Rochet and Tirole’s, the NSR induces the merchant (in part) to raise its price to cash users.10
Holding other fees constant, this effect induces a cross-subsidy from cash to cards. We illustrate
that this cross-subsidization plays an important role in explaining the prevalence of commonly
observed negative charges (that is, rebates) to card users.
Regarding the association members, we follow Rochet and Tirole and others in assuming
that acquirers are perfectly competitive, and we consider two polar cases regarding issuers.
Section 6 analyzes almost perfect competition — homogeneous Bertrand issuers with prices set

11 Indeed, because the NSR distorts the merchant’s behavior in the other market, it cannot be used to
completely eliminate the merchant’s card margin (as the merchant would then drop cards), so the EPN would
seemingly prefer RPM to the NSR. However, implementing RPM would be daunting for the EPN, as it would
require dictating an enormous number of retail prices.
12 In contrast, standard comparisons of uniform pricing vs. third-degree price discrimination by a monopolist
(our merchant) find, under regularity conditions that are met here, that requiring a uniform price causes at least some
price(s) to fall (Nahata et al., 1990; Malueg, 1992).
-5-
in discrete units. Our main model, however, assumes perfectly collusive issuers; coupled with
perfectly competitive acquirers, the association will behave like a monopolist proprietary EPN
that sets fees to the merchant and to cardholders. Since the merchant also is a monopolist, this
environment yields double marginalization if the merchant can surcharge card transactions.
Given double marginalization, one might expect the NSR to increase consumer surplus
and overall welfare. Like maximum resale price maintenance (RPM), that is known to be
beneficial in such situations (Tirole, 1988), the NSR curbs the merchant’s margin on card sales.
The analogy, however, is imperfect because RPM only reduces the price of the targeted product,
while the NSR squeezes the merchant’s margin indirectly, by requiring that the same price be
charged for the other product (here, cash transactions), thereby causing that price to rise.11
Despite the rise in the cash price, one might still expect the NSR to increase welfare as in
optimal taxation (or Ramsey pricing) where inefficiency is reduced by using a broader tax base
to lower the tax rate. The (imperfect) analogy is that, holding constant the EPN’s fees, an NSR
leads the merchant to set an intermediate uniform price for all transactions instead of a higher
card price and lower cash price, thus reducing misallocation in the mix of transactions. Again,
however, the analogy is flawed. Since the EPN is unregulated, it will alter its fees when allowed
to tax also non-card sales via the NSR. If EPN rebates to card users are not feasible, the EPN
raises its fee to the merchant so much that the price paid by card users also can rise.12 If card
user rebates are feasible, the NSR can induce excessive use of cards relative to cash — reversing
the no NSR bias. We show how the welfare tradeoffs depend on two parameters: the size of the
cash users’ group relative to card users, our parameter ", and the merchant’s additional benefit
from handling a card rather than a cash transaction, captured by a parameter b.
Finally, the analysis brings out sharply how the NSR, by constraining merchant pricing,
breaks the EPN’s indifference between charging to merchants or to card users. As several
authors have noted (and we show as well), if merchants can surcharge card transactions, the

-6-
division of the EPN’s charges is neutral — only the total charge matters. With the NSR,
however, the EPN prefers to load its charges on merchants, in the case of a card association by
charging a high interchange fee. Indeed, the EPN then prefers negative fees to card users —
rebates. Such rebates are often viewed as reflecting the inability of an association to prevent its
issuing banks from competing for card users and dissipating rents generated by high fees to
merchants. Our analysis reveals a different possibility: rebates allow an EPN to better exploit a
no-surcharging constraint on merchant pricing. In our model, the EPN grants rebates to card
users so as to boost their demand and raises its fee to merchants knowing that they will absorb
part of the increase, because under the NSR any price increase must apply equally to cash users.
The paper is organized as follows. Section 2 presents the model and shows that when the
merchant can set different cash and card prices, the division of the EPN’s fees between card
users and merchants (hence also the interchange fee) is neutral. Section 3 shows that with the
NSR, the EPN prefers to shift its charge away from card users. Section 4 addresses the case
where rebates to card users are not feasible. If the cash group is relatively small, the NSR harms
even card users. With a larger cash group, card users gain but aggregate consumer surplus is still
lower than under no NSR. Total surplus, however, is higher if and only if the cash group is
sufficiently large.
Section 5 allows for rebates. With the NSR, the EPN always grants rebates. Relative to
no NSR, card users gain but cash users lose; aggregate consumer surplus (of cash plus card
users) rises only if the cash group is sufficiently small whereas total surplus rises only if the cash
market is sufficiently large. A larger merchant benefit from using cards instead of cash improves
the effects of the NSR on both total surplus and overall consumer surplus when card user rebates
are feasible — but the reverse occurs when rebates are not feasible.
Section 6 considers the case of Bertrand rather than collusive issuers. An NSR again will
induce rebates, benefiting card users but harming cash users. Overall consumer surplus rises
regardless of the relative sizes of the two groups. However, provided the merchant’s benefit from
card use is not too large, overall welfare falls with the NSR. Section 7 concludes.

13 Positive b could be reinterpreted as an upward shift in card users’ inverse demand relative to cash users. To
see this, rewrite the profit function as (V
N
(x)+b-(i+t))x.
-7-
2. The Model and Pricing With Surcharging
Consumers: We consider two types of consumers. Type e consumers (‘card users’) hold
cards from the EPN. They buy units of a good only by using cards; their mass is 1. Type c
consumers buy units of a good using only an outside means of payment, call it cash. We assume
that they do not have cards; their mass is
"
. Consumers are otherwise identical and have
quasilinear preferences from purchases of goods given by
U(pc, qc)=V(qc)-pc qc,
U(pe, qe)=V(qe)-pe qe, V
N
(•)>0, V
NN
(•)<0.
Throughout, qj is the per capita number of transactions of a consumer of type j = c, e, and pj is
the net price per unit of transaction paid by such a consumer. The net price paid by cash users
equals the price charged by the merchant but the two prices may differ for card users: pe = peM+t,
where peM denotes the price charged by the merchant to a card-using consumer and t is the per
unit charge (or rebate if t < 0) imposed by the EPN on card users. For each type of consumer, the
(downward sloping) inverse demand function is given by V
N
(qj) = pj.
Merchants: Merchants are local monopolists who treat the above inverse demand curve
as the demand for their product from each type of consumer. The marginal cost of providing a
good to a cash consumer is assumed constant and is normalized to zero. The merchant may also
gain a benefit, b
$
0, from being paid by card instead of cash, reflecting potential savings on cash-
handling costs. The merchant is charged a per-unit fee i by the EPN.
The merchant’s profit is
"
pc qc from cash users and peM qe - (i-b)qe from card users, where
quantities are given by pc = V
N
(qc) and peM = V
N
(qe ) - t. For given values of i and t, the
merchant’s problem can therefore be formulated as choosing a level of x to solve
max x (V
N
(x)-(i+t-b))x.
Observe that i=t=b=0 yields the merchant’s problem vis-a-vis the cash market. Written this way,
the term i+t can be interpreted as the total tax imposed on the card market by the EPN and the
term i+t-b as the merchant’s net marginal cost in the card market.13 Thus, the merchant’s optimal
quantity in the card market is only a function of the total i+t-b and not of the composition of the
charges. For given (i,t), we denote the value of the merchant’s optimization problem by

14 However, in Section 6 we analyze the other polar case of competition among issuer banks — card user fees
are then set in a Bertrand fashion.
15 There are a variety of reasons why fully efficient two-part tariffs (or other nonlinear pricing) may not be
achievable for the EPN to eliminate such double marginalization. A typical EPN has relationships with a vast
number of merchants, and contracting costs could make merchant-specific, two-part tariffs prohibitively expensive.
Furthermore, merchants aggregated together in a single market place, such as a mall, may be able to avoid most of
the impact of a fixed fee by channeling all card purchases to a single merchant. Additionally, in the context of
asymmetric information, for example with heterogeneous merchants, the optimal two-part tariff generally yields
some surplus to the high demand merchant and pricing at levels above marginal cost.
-8-
A
M(i,t;b). The merchant’s alternative to accepting cards is to serve the cash market alone,
yielding a profit-maximizing per-capita level of transactions x0 and total profit
"
x0 V
N
(x0). The
merchant must be assured at least this amount in any equilibrium, that is, for any (i,t),
A
M(i,t;b)
$
"
x0 V
N
(x0). (IR)
We refer to this as the ‘individual rationality’ or IR constraint.
Electronic Payment Network: Our model considers a profit-maximizing agent, the EPN,
setting the charge to a merchant and, for most of the paper, also to card users. This model is most
obviously interpreted as one of a proprietary card network. It also characterizes the behavior of a
card association under two conditions: (a) acquiring banks are identical and competitive; and (b)
issuing banks are identical and collude in pricing to card users. Condition (a) implies that
variations in the interchange fee are fully passed through to the merchant discount, and the
merchant discount is effectively set by the EPN’s issuing banks through their choice of
interchange fee. Condition (b) implies that card user charges are chosen to maximize overall
profits of issuing banks.14 We, therefore, suppress the distinction between the interchange fee
and merchant discount, and simply view the EPN as setting the charge to merchants, i,
monopolistically. The timing of price setting is in a Stackelberg manner: that is, the EPN sets t
and i and commits to this profile of prices and, given t and i, the merchant sets her monopoly
price. The EPN’s marginal cost of servicing a card transaction is assumed to be zero.
We assume that two-part tariffs are not available either to the EPN or to the merchant.
What is important for our analysis is that the sequential monopoly environment between the card
company and the merchant lead to some inefficient pricing at both the merchant and EPN
levels.15 For simplicity, we assume that only linear pricing is feasible for each agent.
The first-order conditions from the merchant’s problem yields a derived inverse demand
curve for card transactions defined, implicitly, by

16 This result was noted in Carleton and Frankel (1995). A generalization of the result and an explanation of
the intuition underlying it can be found in Gans and King (2003).
-9-
i+t=V
N
(x)+xV
NN
(x)+b (1)
Therefore, the EPN maximizes (i+t) x or
A
e(b)= maxx (V
N
(x)+xV
NN
(x)+b) x (2)
Since x is a function of i+t but not i or t separately, the card company varies x by varying the
sum of charges, i+t. This leads immediately to the following well-known neutrality result.16
Proposition 1: Suppose merchant surcharging for card transactions is allowed. Then
equilibrium card transactions, merchant profit and EPN profit all depend only on the EPN’s
total fee, i+t, and not on i and t individually. That is, if (i,t) maximizes the profits of the EPN,
then so too does any pair (i
N
,t
N
) where t
N
+i
N
=t+i.
Since the sum, i+t, can be viewed as a transactions tax, Proposition 1 echoes the familiar result
that the effects of a tax are invariant to whether the obligation to pay the tax is placed on buyers
or on sellers. However, the next section shows that, in the presence of an NSR, EPN profits will
vary for a given i+t depending on the relative values of i and t.
The proofs and intuition for our results are most concisely conveyed for the case of
linear demand. Thus, the remainder of the paper restricts attention to this case:
A1) Consumer per capita inverse demand is V
N
(x)=1-x and the merchant’s benefit from card
use satisfies b<1.
The assumption of linear demand enables closed form solutions for most of the relevant
variables in the analysis, and ensures that propertie P1)-P3) hold — many of our qualitative
results hold for any demand curves that satisfy these properties.:
P1) The merchant’s revenue is strictly concave in quantity and price and any increase in the
merchant’s marginal cost in serving cards is not fully passed through to consumers when
surcharging is possible.
P2) The EPN’s revenue function, x(x V
NN
(x)+V
N
(x)+b) is strictly concave in quantity.

17 Of course, a merchant may refuse and forgo card transactions. To understand the direction of EPN
incentives under the NSR, in this section we examine the structure of EPN pricing assuming the merchant’s IR
constraint does not bind. In later sections we address this constraint explicitly.
18 An implication of this observation is that, with a low card user fee (which Proposition 2 shows is desired by
the EPN), we can formulate the no-surcharge rule mathematically as the inequality constraint, peM
#
pc even if,
formally, the constraint is a ‘No Discrimination Rule’, that is, a uniform pricing rule rather than a no-surcharge on
card use rule (which would be better captured by the constraint, peM = pc). Although credit card companies, for
example, have historically imposed such rules on their merchant clients, an inequality constraint may obscure other
reasons for merchant pricing constraints. Some merchants argue that even without a formal no-surcharge rule, social
conventions make it very difficult for them to charge different prices for users of different means of payments.
Proposition 2i) shows when the effects of the two constraints are the same.
-10-
P3) With merchant surcharging of card transactions, the EPN sets a total fee, i+t>b.
P3) is a mild restriction. With linear demand, the EPN’s optimal total charge is
i+t=(1+b)/2. For this to exceed b requires b<1 — the merchant’s added benefit from card use
(e.g. saving on cash handling cost) is less than the consumers’ maximal willingess to pay for the
good itself. Under P3), the merchant’s net marginal cost is positive for serving card users, but
zero for cash users. Standard revealed preference arguments (e.g. Tirole, 1988, pp. 66-67) imply
that as marginal cost rises, the merchant’s monopoly quantity falls, hence per capita transactions
are lower for card users than for cash users.
3. Under a No Surcharge Rule the EPN Prefers Lower Card User Charges
Suppose the EPN requires any merchant that accepts its card to charge no more to card
users than to cash users, peM
#
pc.17 By P3), the EPN’s optimal aggregate fee with surcharging
allowed causes the merchant to face a higher net marginal cost of serving card users than cash
users; thus, when the two demand curves are equal — as occurs when t = 0 — the merchant
prefers a higher price to card users (peM > pc). The NSR would then bind on the merchant.
Proposition 2i) below shows that the NSR will also bind if, instead, the EPN’s fee to card users
is positive but below some threshold.18 Moreover, it is costless for the EPN to adopt such a
profile of charges pre-NSR since, by Proposition 1, only the sum of charges then matters.
Proposition 2ii) shows that if the binding NSR is accepted, the EPN would benefit. Proposition
2iii) shows that under an NSR the EPN has the incentive to continue raising the charge to the
merchant and lowering it to card users.

19 This argument also implies that, under the NSR, the total price to card users falls by more with a unit
reduction in t than in i. Let p denote the merchant’s price to card users, hence their total price is p+t. Under
surcharging, Proposition 1 implies that a unit reduction in t or in i yields the same change in p+t:
M
(p+t)/
M
t=
M
(p+t)/
M
i=
M
p/
M
i, so 1+
M
p/
M
t=
M
p/
M
i. The NSR, however, dampens the merchant’s price response to a
change in t or in i (|
M
p/
M
i| and |
M
p/
M
t| fall), because the same price change must be made also in the cash market.
Thus, cutting i yields a smaller reduction in the price to card users under the NSR than under surcharging (
M
p/
M
i is
smaller), while cutting t yields a larger reduction under the NSR (
M
p/
M
t is negative but smaller in absolute value, so
1+
M
p/
M
t is positive and larger): card users receive this cut directly, and (by P1)) the merchant responds by increasing
p by less than with no NSR.
-11-
Proposition 2: Fix i+t at the level k>b and define t*(k)
/
V
N
(x(k-b))-V
N
(x0 ) >0. For any
(i,t),with i+t =k, t < t*(k):
i) When merchant surcharges are allowed, peM > pc implying that with this profile of fees the
imposition of an NSR constrains merchant pricing;
ii) If an NSR is accepted, holding (i,t) fixed, then cash purchases fall but card purchases, and
thus EPN profits, rise;
iii) Provided the merchant continues to accept the NSR, a cut in the card user fee t and an equal
rise in merchant fee i increases per capita card transactions and EPN profits.
The intuition for Proposition 2ii) is straightforward. With a binding NSR, the merchant
will choose a uniform price between its pre-NSR card and cash prices: starting from a uniform
price equal to peM, a small move towards pc imposes a zero first-order loss in the card market
while moving closer to the optimal cash price, and similarly starting from pcM and moving
towards peM. Proposition 2ii) establishes that the EPN gains if a binding NSR is accepted: EPN
profits rise at the pre-NSR charges (i,t) and any departure from these price post-NSR, by
revealed preference, would further benefit the EPN.
The intuition behind Proposition 2iii) is as follows. A cut in t and an offsetting increase
in i would leave the EPN’s margin unchanged, and hence profit unchanged, only if card
transactions remained unchanged. This in turn would only happen if the merchant raised her
price to card users by the full increase in i, since card users’ inverse demand shifts up by an
amount equal to the fall in t (equivalently, to the increase in i). With surcharging, this indeed
would be the outcome, hence the familiar neutrality result. But since the NSR forces the
merchant to charge the same price to cash users as to card users, the merchant prefers to raise her
uniform price by less than the full increase in i and accept a lower margin on card sales.19
Proposition 2 shows that with the NSR, it becomes relevant how i+t is distributed: the
EPN prefers a lower card user charge (provided the merchant still accepts). Rebates to card

20 Gerstner and Hess (1991) obtain a similar effect in a somewhat different context. They consider a
monopolist manufacturer selling to a monopolist retailer that faces two customer groups, low demanders and high
demanders, where high demanders incur a higher transaction cost of using a rebate/coupon. In our model, the NSR
plays roughly the same role as their differential transaction costs in motivating rebates.
21 The phenomenon of card user rebates is relatively recent. While credit cards date to the late 1960s/early
1970s, money-back rebates were first offered, by Discover, in 1986. Rebate cards only became common, however, in
the early 1990s with the introduction of the GM Mastercard and other cards that offer reward points associated with
co-branding partner companies (such as frequent-flier miles). See generally, Evans and Schmalensee (1999). By the
late 1990s, roughly half of all credit volume were associated with rebates of various sorts. Faulkner and Gray (2000).
22 In (i,t) space, under the NSR the merchant’s level sets have slope strictly less than -1. Therefore, for any
given k, the line t=k-i eventually crosses the line given by
A
NSR(i,t;b)=
"
x0 V
N
( x0). Thus, if the EPN holds i+t fixed
and lowers t, it eventually runs against the merchant IR constraint.
-12-
users — negative t — are often taken as evidence of the inability of a bank card association to
control competition for card users by its member banks. Proposition 2iii) offers an alternative
interpretation: rebates can be a pricing tactic designed to better exploit the power of the NSR.20
Given the incentives for an EPN to raise i and reduce t, what determines the floor on t?
One limit may be institutional. For historical, practical or regulatory reasons, rebates to card
users may not be feasible.21 Section 4 investigates this case. A priori, the binding constraint may
be the non-negativity of t, the merchant’s IR constraint, or both.22 Proposition 3 shows, however,
that the non-negativity constraint always binds. Section 5 allows rebates, showing that the EPN
may be constrained either by the merchant’s IR constraint or by the need to ensure that the
merchant will not price so high that cash users are driven out (a type of incentive compatibility
constraint).
4. Equilibrium Under No Rebates
Let i0 be the EPN’s optimal charge given t = 0 and (for the moment) ignoring the
merchant’s IR constraint. Whether or not the IR binds depends on the relative size of the cash
market,
"
. If, at (0,i0), the IR does not bind, then, by Proposition 2, these prices are optimal for
the EPN. If the IR is violated at these prices, in Proposition 3 we provide sufficient conditions
under which setting t = 0 is still optimal for the EPN.
With an NSR, cash users and card users pay the same merchant price, p. Linear demand
then implies that per capita cash consumption is qc=1-p and per capita card consumption is
qc=1-p-t. For any given (i,t), the merchant then selects price to solve
A
NSR (i t;b)
/
maxx
"
(p(1-p)+(p-i+b)(1-p-t),

23 With the exception of Propositions 3iv), 3v) and 4ii)c), Propositions 3 and 4 can be shown to hold for more
general demand functions than linear, that is, those which satisfy P1)-P3.
-13-
(3)
yielding
The EPN’s profit maximization problem when an NSR is imposed can now be expressed as
P EPN :max
i,t (i + t) qe (i,t;b)
s.t.
A
NSR (i,t;b)
$
"
x0 V
N
(x0 )(IR)
t
$
0 (No Rebates).
The objective function of the EPN is concave in (i,t) while the IR constraint is linear in (i,t).
Thus the Kuhn-Tucker first order conditions are sufficient.23
Proposition 3 (Prices): Suppose card rebates are not feasible (t
$
0). Under the NSR:
i) For any relative size of the cash market,
"
, the EPN’s optimal fee implies t = 0 (no card fees),
hence per capita card and cash transactions are equal.
ii) There exists
"
* such that the EPN choice of i is determined by the merchant’s IR constraint if
and only if
"
>
"
*.
iii) If
"
>
"
*, then i falls as
"
rises and i+t-b (=i-b) is independent of the merchant benefit b.
iv) If
"
<
"
*, i increases in
"
.
v) For all
"
, i is higher than the total charge under surcharging
Proposition 3i) shows that the EPN’s desire for lower card user fees and higher merchant
fees illustrated in Proposition 2 drives user fees to zero even if the merchant’s IR constraint
binds before the EPN achieves its optimal fee pair. An implication is that if the non-negativity
constraint is relaxed (rebates are allowed, as in Section 5) then the EPN under the NSR will set t
negative.
Proposition 3ii) shows that the merchant IR constraint binds if and only if the cash
market is not too small. (With b=0, the IR binds if
"
>
"
* = 1/3.) Since, the merchant’s profit
from serving only cash customers is increasing in the size of the cash market, as the latter
increases in the range where the merchant’s IR binds, the EPN must reduce i to maintain
merchant participation. By contrast, as
"
increases from 0 to
"
*, the EPN responds by raising i,
because in this range the IR is not binding and a larger cash market reduces the merchant’s pass-

24 Under cost and demand conditions satisfied here, prohibiting third-degree price discrimination leads a
monopolist to charge an intermediate uniform price. Sufficient conditions are that marginal cost be non-decreasing
and that demands in the various markets be independent, each yielding a quasi-concave profit function (Nahata et al.
1990, Malueg 1992).
25 The quantity effects and the role of b can be understood as follows. Under surcharging, the derived
inverse demand function facing the EPN is i(q) = b + (V’’q + V’), where V’’q + V’ is the merchant’s decreasing
marginal revenue function. With the NSR, the EPN faces iN(q) = b +(1+
α
)(V’’q + V’). Recall that x0 is the
merchant’s monopoly output for zero marginal cost (the cash-market output with surcharging), hence: V’’(x0)x0
+ V’(x0) = 0. Thus, iN(q) cuts i(q) from above at q=x0, i=b: iN(q) = i(q) = b at q = x0, while iN(q) > i(q) at q < x0
and iN(q) < i(q) at q > x0. (Intuitively, i=b makes the merchant’s net marginal cost of card transactions zero—as
-14-
through from i to the uniform retail price. Finally, observe that the NSR affects not only the
EPN’s fee structure but also the level: the EPN’s total fee is higher under the NSR for all
"
(Proposition 3iv)).
For Proposition 4ii)c), define the change in total surplus when rebates are not feasible
(
)
TSNR) to be total surplus under the NSR without rebates minus total surplus under no NSR.
Proposition 4 (Quantities and Welfare): Suppose an NSR is imposed but card user rebates are
not feasible (t
$
0). Compared to the equilibrium with no NSR,
i) If the cash market is small enough that the merchant IR does not bind (
"
#
"
*), then:
a) Cash users’ transactions and consumer surplus are lower;
b) Card users’ transactions and consumer surplus are unchanged if b = 0 and lower if
b > 0;
ii) If the merchant IR binds (
"
>
"
*), then:
a) Cash users’ transactions and consumer surplus are lower;
b) Card users’ transactions and consumer surplus are higher if
"
is sufficiently larger
than
"
*;
c) Aggregate quantity (qe+
"
qc) and aggregate consumer surplus are lower for all
"
and
all b.
d)
)
TSNR rises in
"
and falls in b. For b=0,
)
TSNR=0 at a value of
"
above
"
*.
We first explain the intuition underlying Proposition 4i) and then 4ii).
IR Not Binding (
"
<
"
*). One might have expected the NSR to raise card transactions by
inducing the merchant to choose a uniform price that lies between its card and cash prices under
surcharging. This indeed would occur if the EPN’s fees remain fixed.24 Instead, the NSR leads
the EPN to adopt a merchant fee i0 so much higher than its total fee i+t under no NSR that card
transactions remain unchanged with the NSR if b=0 and fall if b>0 (Proposition 4i)).25

for cash—so the merchant would choose equal card and cash quantities x0 under surcharging hence the NSR
would have no effect. Card quantities q < x0 correspond to the merchant facing higher marginal cost for card
than for cash sales, hence the EPN can attain such quantities at a higher i under the NSR because the merchant’s
uniform price is then pulled down by the lower marginal cost on cash sales; conversely, q > x0 requires cutting i
below b by more under the NSR than under surcharging.) The equilibrium card quantity is where the EPN’s
marginal revenue obtained from the relevant inverse demand function, iN(q) or i(q), equals its marginal cost of
zero: MR = i(q) + i’(q)q = 0 under surcharging, and MRN = iN(q) + iN ’(q)q = 0 under the NSR. Observe that
MRN =(1+
α
)MR –
α
b. Thus, for b = 0, the card quantity is the same with surcharging or the NSR. For b > 0,
MR = 0 implies MRN < 0, so the EPN’s optimal card quantity is lower under the NSR. (Note that this will be
true also if the EPN’s marginal cost were positive but not too large.)
-15-
Therefore, when the merchant’s IR is not binding, the welfare consequences of the NSR
are stark. The NSR then reduces even card transactions (leaving them unchanged only if b=0),
thus harming card users. Since (per capita) cash transactions exceed card transactions under
surcharging but are equal to them with the NSR, the NSR also reduces cash transactions. With all
quantities falling, total surplus must fall. The merchant’s profit also falls since the NSR both
leads to a higher total EPN charge and constrains the merchant’s pricing to consumers. The NSR
in this case therefore benefits only the EPN at the expense of all other parties.
IR Binding (
"
>
"
*). In this case, the NSR still reduces the merchant’s profit — since the
merchant now loses all its surplus from dealing with the EPN — and cash transactions. However,
if the cash market is sufficiently large, card transactions are higher with the NSR (Proposition
4ii)b)). To see this, observe that with the NSR and no rebates (t=0) the merchant’s profit can be
expressed as - (1+
"
)Q2V
O
(Q), where Q is the equal per-capital level of cash and card transactions.
The IR constraint is therefore -(1+
"
)Q2V
O
(Q) =
"
x0V’(x0) or
- Q2V
O
(Q) = (
"
/(1+
"
))x0V
N
(x0).(4)
As the size of the cash market,
"
, increases, to satisfy (4) the EPN must induce an increase in Q
(since concavity of the merchant’s revenue function in quantity implies merchant profit is
increasing in Q,), which requires cutting the merchant fee i. As
"
→
4
, (
"
/(1+
"
))x0V’(x0) →
x0V’(x0), so Q must approach x0, the merchant’s cash market quantity under surcharging. The NSR
therefore lowers cash transactions (since Q < x0 except in the limit), but for sufficiently high
α
it
raises card transactions (since these are less than x0 under surcharging, by P3)).
Total quantity is lower under the NSR (Proposition 4ii)c) ). Given equal per-capita linear
demands by card and cash users, imposing the NSR would leave total quantity unchanged only if
the EPN’s total charge remained constant, but in fact the EPN raises its total charge (Proposition
3iv) to exploit the decreased elasticity of demand it faces from the merchant, so total quantity
falls. Overall consumer surplus, therefore, also must fall because of the following property:

-16-
Lemma 1: Consider any pair of prices (pc, pe) to cash users and card users (where pe includes any
EPN charge t) that yield a fixed total quantity of transactions,
α
qc + qe = k. Then overall
consumer surplus of cash and card users increases with the dispersion in per capita quantities
|qc - qe|.
Lemma 1 can be understood as follows. Identical linear demands for cash and card users imply
that (a) total quantity is constant only if the weighted average price
α
pc + pe is constant, and
(b) dispersion in per capita quantities |qc - qe| is linear in |pc - pe|. Overall consumer surplus,
α
S(pc) + S(pe) =(1+
α)(
S(pc)
α
/(1+
α
) + S(pe)/(1+
α
)), is proportional to the consumer surplus of
an individual who faces pc and pe with probabilities
α
/(1+
α
) and 1/(1+
α
). An individual’s
consumer surplus S(p) is convex in price, so any mean-preserving spread of the prices will
increase the expected surplus. Since the NSR with no rebates reduces the spread in per capita
quantities (to zero) as well as total quantity, overall consumer surplus must fall (Proposition
4ii)c)).
Total surplus, however, can be higher with the NSR if the cash market is large enough.
The efficiency gain comes because the lower total quantity of transactions is allocated more
efficiently between cash and card users. To see this, consider b = 0, in which case the welfare
maximizing allocation requires equal per capita card and cash quantities. The NSR achieves this,
while surcharging does not. If the cash market is sufficiently larger than the level where the
merchant’s IR binds on the EPN (for b = 0, if
"
> 1.53 >
"
* = 1/3), then the gain from improved
allocation outweighs the loss from the fall in total quantity so the NSR raises total surplus.
There are two reasons why
)
TSNR increases in
"
for
"
>
"
* (Proposition 4ii)d). First, the
merchant’s preferred price is lower for cash than for card users. Thus, as the cash market becomes
relatively more important (as
"
increases), for given EPN fees the merchant’s optimal uniform
price falls. Second, to respect the merchant’s IR, the EPN must cut its total fee as the cash market
grows. For both reasons, a large cash market allows the NSR to curb the double marginalization
that curtails card transactions under surcharging while introducing only a small distortion in the
per capita quantity of cash transactions. (The reason why
)
TSNR falls in b, as stated in Proposition
4ii)d), is discussed in Section 5.)
5. Equilibrium When Rebates Are Feasible
Proposition 3i) shows that when the card user fee must be non-negative, the EPN cuts

26 A monopolist that faces two markets but is prohibited from 3rd-degree price discrimination will drop the low
market if the dispersion in the demands is sufficiently large (Tirole 1988, p.139).
-17-
this fee to 0. Thus, this is no longer the equilibrium when rebates are feasible (t < 0).
One obvious constraint on the EPN’s equilibrium charges remains the merchant’s IR, its
option to reject the EPN and forgo card users as discussed earlier. In addition, a less evident
constraint emerges when rebates are feasible: the merchant’s willingness to continue serving cash
customers. With large enough card user rebates and a sufficiently small cash market, the
monopoly price appropriate for card users alone will exceed the choke price of cash users, and the
merchant under the NSR will choose this price instead of cutting price enough to serve also cash
users.26 Such an outcome, however, clearly is not optimal for the EPN: since the merchant’s price
to card users is then unaffected by cash users, the NSR loses its value. This issue does not arise
with t
$
0 — (per capita) inverse demand of card users is then lower than that of cash users, so any
price that yields cash sales also yields card sales — but must be tackled under rebates.
Propositions 5 and 6 compare the equilibrium under the NSR with rebates to that under
no NSR. Proposition 7 summarizes the incremental effect of rebates by comparing the equilibria
under the NSR with and without rebates.
Proposition 5 (Prices): Under an NSR with rebates feasible:
i) For all
"
, the EPN’s optimal choice involves granting rebates (t<0);
ii) For low
"
(<.22 if b=0), the requirement that (i,t) induce the merchant to continue to sell to
cash customers is a binding constraint on the EPN; for high enough
"
(above approximately.18 if
b=0) the IR constraint binds.
iii) When the IR binds, the sum of card user and merchant charges, i+t, is the same as the EPN’s
optimal choice under surcharging ((1+b)/2). As
"
increases, i falls and t rises.
Proposition 5i) is illustrated in. Figure 1 for the case where the merchant’s IR constraint
does not bind at i0, the EPN’s optimal merchant fee conditional on t=0. Recall from Proposition 2
that, for fixed i+t, the EPN wishes to lower i in the absence of other constraints. Thus, a
movement down and to the right along the line i+t = i0 (i.e., a cut in t and an equal increase in i)
raises EPN profit. Expression (4) yields the IR constraint. Given linear demand, this constraint is
linear with slope steeper than -1. Point B in Figure 1 represents the intersection of the line i+t
with this manifold. The EPN’s solution is, then, to move down and to the right from (i0,0) to B
along the line i+t = i0, then down the IR line until it reaches an EPN indifference curve that is

-18-
tangent to the IR (point C in Figure 1). This point represents a lower total EPN charge, i+t, and a
lower t compared to (i0, 0). If, instead, the IR constraint binds at t=0 (IR cuts the horizontal axis
at i < i0), then with rebates the EPN immediately moves down and to the right along the IR
manifold to a point of tangency. In both cases, therefore, i+t is lower with rebates than in the
NSR equilibrium under no rebates.
Proposition 5ii) shows that even with rebates feasible, the EPN cannot always fully
extract the merchant’s surplus. If the cash market is sufficiently small, the floor on t is not the
merchant’s IR constraint but the need to induce the merchant to continue serving cash users.
When the cash market is large enough that EPN charges are determined by the IR
constraint, Proposition 5iii) shows that the total charge i+t under the NSR is the same as under
surcharging and is independent of the size of the cash market,
α
,; however, the spread between i
and t (which is irrelevant under surcharging) shrinks as
α
increases. (Proposition 5iii.) The EPN’s
equilibrium fees when the IR binds are (see Appendix),
The total charge, t*+i*, is therefore (1+b)/2, the same as under surcharging, but lower than under
the NSR with no rebates. The EPN prefers to grant rebates and accept a reduction in i+t, as
needed to satisfy the merchant’s IR, because rebates are a more effective way to boost card
transactions. (Recall from Proposition 2, under the NSR a reduction in t and equal increase in i
would increase card transactions. See also fn. 19) As the cash market grows, the EPN meets the
IR with the same total charge (instead of cutting it as under no rebates) by reducing the spread
between i and t: t* rises with
α
(smaller rebates) while i* falls. The merchant benefits from this
reduced spread because it gains the option of maintaining the same margin p-i on cards but at a
price p closer to the cash market optimum.
The next Proposition describes the effects of the NSR with rebates on quantities and
welfare, when the cash market is large enough that EPN charges are determined by the merchant
IR constraint (
α
> 0.22 if b=0). Define the change in total surplus when rebates are allowed
(
)
TSR) to be total surplus under the NSR with rebates minus total surplus under no NSR. The
change in aggregate consumer surplus (
)
CSR) is defined analogously.

27 The card quantity rises because the merchant’s preferred price is lower to cash than to card users, and an
increase
"
magnifies the relative importance of cash users, leading the merchant under the NSR cut its uniform price.
This effect dominates an opposing effect: that as
"
increases, the EPN reduces the spread (i-t) in its fees
(Proposition 5iv)), which by itself would reduce card transactions (by the reverse argument of Proposition 2iii)).
-19-
Proposition 6 (Quantities and Welfare): Suppose an NSR is imposed and rebates are feasible.
For
"
large enough that the merchant IR binds, compared to the equilibrium with no NSR:
i) Cash users’ transactions and consumer surplus are lower;
ii) Card users’ transactions and consumer surplus are higher;
iii) Aggregate transactions (qe+
"
qc) are unchanged;
iv)
)
TSR rises in
"
. For b=0, it is positive if and only if
"
>1/3;
v)
)
CSR falls in
"
. For b=0, it is negative if and only if
"
>1/3;
vi)
)
TSR and
)
CSR rise in b.
Parts i)-iii) of Proposition 6 follow because the EPN’s total charge i+t is equal under the
two regimes. Under surcharging, equilibrium quantities are invariant to how i+t is divided between
i and t, in particular, the same quantities would arise if one set (i*, t*) — the values that are
optimal under the NSR. Imposing the NSR while charging (i*, t*), however, constrains the
merchant’s retail pricing, causing cash transactions to fall and card transactions to rise; total
transactions remain the same because of the linearity of demand.
Now consider why
)
TSR increases with the size of the cash market (Proposition 6iv). As
"
increases, the total quantity of transactions rises under both regimes but remains equal. Thus, the
behavior of
)
TSR hinges on the change in per capita transactions of cash versus card users. The
efficient per capita levels are 1 for cash and 1+b for cards. With surcharging, the cash quantity is
1/2 and the card quantity is (1+b)/4, both independent of
"
. Under the NSR, as
"
increases both
quantities rise towards 1/2.27 As
"
increases, therefore, the allocation improves only under the
NSR, so
)
TSR rises in
"
. With zero merchant benefit from card use, b=0,
)
TSR is positive if and
only if
"
>1/3. The case of b>0 is discussed shortly.
By contrast, the change in aggregate consumer surplus when moving to the NSR with
rebates declines in
"
(Proposition 6v). The narrowing of the gap between per-capita cash and card
quantities as
"
increases under the NSR — but not under surcharging — is harmful to overall
consumer surplus, by Lemma 1. For b=0, the NSR with rebates reduces overall consumer surplus
if and only if
"
>1/3.

-20-
Positive Merchant Benefit from Cards. Moving from surcharging to the NSR with
rebates, therefore, causes opposite changes in total surplus and overall consumer surplus if b=0:
when
)
TSR>0 (
"
>1/3),
)
CSR<0. Thus, the increase in EPN profit always comes at least partly at
the expense of consumers and the merchant (recall that the merchant loses for all
"
). The case
b=0, however, presents an overly negative picture of the NSR. When there are gross merchant
benefits from card use, the NSR with rebates can increase both total surplus and overall consumer
surplus. Since both
)
TSR and
)
CSR are increasing in b (Proposition 6vi) and both equal 0 at
"
=1/3, for b>0 there will be an interval around
"
=1/3 in which
)
TSR and
)
CSR >0. These results
are illustrated in Figure 2.
The intuition for why
)
TSR and
)
CSR are increasing in b is as follows, starting with total
surplus. Since total transactions are equal under surcharging and under the NSR with rebates, the
differential effect of b under these regimes works via its effect on the mix of transactions. The
efficient per capita card and cash quantities are qe** = 1+b, qc** = 1, hence qe** - qc* = b. With
surcharging, quantities are qe = (1+b)/4, qc = 1/2, hence qe - qc = (b-1)/4. Recalling that b<1, under
surcharging qe < qc, and the gap closes at the rate
)
b/4. With the NSR and rebates, per capita
transactions are higher for card than for cash users. Moreover, the difference between the card and
cash quantities increases faster with b under the NSR than under surcharging and never exceeds
the efficient gap, b. Thus, an increase in b magnifies the allocation advantage of the NSR. Finally,
)
CSR increases in b. The total quantity of transactions is the same under surcharging and the NSR
with rebates. But the spread in per capita quantities, | qe - qc |, rises with b under the NSR with
rebates but falls under surcharging, and Lemma 1 shows that overall consumer surplus increases
with the spread.
Interestingly, the NSR’s effect on total surplus is less favorable the higher is b in the case
when rebates are not feasible and
"
in the range where the merchant’s IR binds (Proposition 4ii)d,
)
TSNR falls in b). Since the distortion from double marginalization on card transactions increases
with b under surcharging (the gap between the efficient and actual card quantities is 3(1+b)/4), and
since curbing double marginalization is what motivates the NSR in our model, one might expect
the NSR to be better for total surplus the larger is b. With no rebates, however, the EPN under the
NSR responds to an increase in b by raising its charge to the merchant enough that the net
marginal cost of card transactions, i+t-b, stays constant (Proposition 3iii), and therefore quantities

28 Under the NSR and no rebates, when the merchant’s IR binds, per capita transactions Q are determined by
(3), whose right hand side is independent of b — since the merchant’s outside option of serving only cash customers
is independent of b, so too is the profit, and thus quantity, Q, that the EPN must leave to the merchant.
-21-
are unchanged.28 Under surcharging, the EPN responds to an increase in b by reducing i+t-b, and
therefore card transactions rise. Thus, an increase in b raises card transactions under surcharging
but not under the NSR with no rebates. Under the NSR with rebates, however, the EPN allows
card transactions to rise faster as b increases than under no NSR, so
)
TSR rises in b.
Proposition 7 draws on previous results to compare the outcomes under the NSR if rebates
are or are not feasible. For simplicity, we focus on the case when the cash market is large enough
that the merchant’s IR determines EPN pricing with or without rebates.
Proposition 7 (NSR, Rebates vs. No Rebates): Suppose the NSR is imposed, the merchant’s IR
binds, and rebates are feasible. Compared to the outcome under the NSR with no rebates:
i) Card users’ consumer surplus is higher, cash users’ consumer surplus is lower, and aggregate
consumer surplus is higher with rebates.
ii) For relative sizes of the cash market
"
that make the merchant’s IR constraint bind in both
cases, total transactions and total surplus are higher with rebates.
The superiority of rebates for total output and overall welfare (Proposition 7ii) reflects the
ability of rebates under the NSR (and only then) to more effectively reduce double marginalization
than by only cutting the fee to the merchant. Recall that with the NSR an increase in i by
)
and an
equal cut in t would lower the net price to card users, because the merchant would raise its uniform
price by less than
)
. (In (3),
)
p =(
)
i-
)
t)/2(1+
"
) = 1/((1+
"
).) Moreover, when cutting t below 0,
the EPN raises i by less than the rebate amount to respect the merchant’s IR constraint. Since the
aggregate EPN charge i+t is lower with rebates than without, total transactions are higher. Overall
consumer surplus therefore also is higher with rebates: total quantity is higher and so is the spread
in per capita quantities (under the NSR, the spread is positive with rebates but zero without),
which benefits consumers by Lemma 1. Finally, total surplus is also higher with rebates: overall
consumer surplus is higher, the EPN’s profit is higher (by revealed preference), and the merchant’s
profit is the same (for values of
"
that make the IR bind under rebates or no rebates).
Cash users, however, lose from the NSR even with no rebates to card users (Proposition
4), and lose further if rebates are feasible. Granting rebates increases the inverse demand of card

29 Gerstner and Hess (1991) cite empirical evidence that retailers indeed raise their prices in response to
manufacturers’ granting of rebates to consumers.
-22-
users, prompting the merchant to raise its retail price.29 In addition, when the EPN grants rebates, it
also raises its fee to the merchant somewhat, putting further upward pressure on the merchant’s
price (see (3) where p increases in i and decreases in t, while qc does the reverse).
6. Competitive Card Issuers
To this point, our analysis applies most directly to the case of proprietary networks where
the EPN is a single card issuer. Alternatively, it describes outcomes when despite multiple card
issuers, the issuing industry behaves as if were maximizing issuing banks’ joint profits. How do
the results change if the EPN is an association of competitive issuing banks? In this scenario,
member banks issue the cards, and they, rather than the network, set most of the terms to
cardholders, including prices (annual fee, interest rate, rebates). This section explores the effects of
an NSR when the EPN is unable to control t.
A sequential/simultaneous game emerges. First, through their partnership with the EPN,
banks set the merchant discount fee i and commit to it. Merchants continue to set prices taking i as
given but recognizing that t is determined through competition for card users by issuing banks. If
bank member W of the EPN is one of m banks charging the lowest card user fee, it obtains sales of
qW = x/m, where x is derived from equation (1) and is given by x=1/2-(i+t-b)/2. If the fee of bank
W is not among the lowest, qW is zero. That is, taking i as given, banks compete as Bertrand price
setters to card users and each of the banks that charge the lowest fee t obtains 1/m of total
transactions, where the latter quantity x is determined by the equality of the merchant’s marginal
revenue function from card transactions with its net marginal cost. Suppose that banks can only set
fees in discrete units,
0.
0. By the standard Bertrand logic, the equilibrium tW satisfies
tW = -i +
0
. Card issuers compete away (virtually) all their rents by offering rebates that are close
to the interchange fee.
As before, the constraints on (i,t) are to ensure the merchant continues serving cash users,
and continues participation with the EPN (IR). In both cases, equilibrium quantities are obtained
by substituting tW
.
-i into expressions (3) that show the merchant’s quantities as functions of i
and t. Since the quantities under no NSR are qc = 1/2, qe = (1+b)/2, the changes are

30 The Bertrand assumption implies t
.
-i. Linear demand implies that, under surcharging, the merchant’s card
price is (1-b-2t)/2. This exceeds the cash price (1/2): that is, the NSR binds only if t<-b/2.
31 The theorem is shown for b = 0, however, given the continuity of the environment, quantity and welfare
results will continue to hold for b small and positive. They may not hold for b large since, even with competitive
issuers, there is then a significant bias away from cards under no NSR. (The efficient quantities are 1 for cash and
1+b for cards while the no NSR levels are 1/2 and (1+b)/2, so only the card underprovision rises with b.)
-23-
)
qc=(2t+b)/(2(1+
"
))<0,
)
qe=-
"
(2t+b)/(2(1+
"
))>0.
The inequalities follow since the NSR binds on the merchant only if t < - b/230. The changes in
equilibrium quantities imply that with competitive issuers total transactions under the NSR with
rebates is the same as under no NSR. For b=0, the per-capita card quantity exceeds the cash
quantity under no NSR (because, with competitive issuers, the markup is only at the merchant
level), but exceeds it under the NSR with rebates.
As long as the EPN’s issuing banks enjoy some profits from transactions (
,
> 0), the EPN
will wish to generate the largest possible quantity of such transactions. Since card transactions are
decreasing in t, the EPN will fix a high i, inducing its competing issuers to offer large negative
values of t (large rebates). Proposition 8 summarizes the effects of this incentive on equilibrium
quantities under the NSR and competitive issuers. 31
Proposition 8: Assume b=0. With perfectly competitive issuers, in the equilibrium under the NSR:
i) If
"
<1, the EPN sets i until merchants are just indifferent between selling to cash customers or
not; if
"
>1, the merchant’s IR constraint binds;
ii) Cash transactions are lower than with no NSR, card transactions are higher, but total
transactions are the same;
iii) For all values of
"
, overall consumer surplus is higher than with no NSR but merchant profit
and total surplus are lower;
iv) In the limit as the mass of cash users becomes large, the per-capita cash quantity approaches
the single monopoly level and the per capita card quantity approaches the competitive level.
Proposition 8i) illustrates that, with competitive issuers, the constraint that the EPN
ensures that the merchant continues to serve the cash market binds for a larger size of the cash
market (
"#
1 rather than
"#
.22). This is because the stronger tendency to offer rebates under
competition among card issuers makes the option of pricing cash users entirely out of the market

-24-
relatively more attractive to merchants. Total quantity remains the same as under no NSR
(Proposition 8ii)) because demand is linear and the total EPN fee remains the same (
,
). Card and
cash quantities therefore move in opposite directions because only card users get rebates.
Given the same total quantity and b=0, total surplus must fall under the NSR with rebates,
since per capita quantities of cash and card users are then different, while efficiency calls for equal
levels as occurs with competitive issuers and no NSR. This divergence of quantities only with the
NSR implies, however, that overall consumer surplus rises (Lemma 1).
Result 8iv) shows that as the cash market becomes large relative to cards, the NSR in
conjunction with competitive rebates by card issuers succeed in eliminating the distortion in the
pricing of card transactions due to the monopolist merchant. The merchant charges a uniformly
high (monopoly) price to both card and cash users, but card users receive a rebate and therefore
obtain a net price close to the competitive price. However, the net price to cash users is the
(uniform) price charged by the merchant. When the cash market is large, the merchant’s price is
driven by the cash market and thus will approach the simple monopoly level.
7. Conclusion
The complex cycle that makes up a typical payment network offers a rich field for
economic analysis, with prices playing important roles at every link of the cycle. Our principal
model analyzed the No Surcharge Rule as an imperfect instrument of vertical control by a card
payment network (EPN) facing a merchant in an environment of double marginalization, where the
merchant also serves outside consumers—‘cash’ users. By requiring the merchant’s card price to
equal its cash price, the NSR leads the EPN to prefer a higher fee to the merchant and a lower fee
to card users (whereas the EPN is indifferent to how it allocates its total fee when the merchant can
set the card price independent of the cash price). Throughout, the NSR benefits the EPN but harms
the merchant and cash users. Other welfare effects depend on the ratio of cash to card users, the
merchant’s benefit from card versus cash transactions, and whether rebates (negative fees) to card
users are feasible.
If rebates are not feasible, the EPN charges card users zero but raises its merchant fee
above its no-NSR total fee. This increase in total fee reduces total transactions and aggregate
consumer surplus; with a sufficiently small cash market, even card users pay more under the NSR.
Despite the fall in total quantity, overall welfare increases if (and only if) the cash market is large
enough, because the rise in per capita card quantity combats the pre-NSR bias from double

-25-
marginalization, at the cost of a relatively small distortion in per capita cash quantity.
If rebates are feasible the EPN grants them, benefiting itself and card users while harming
cash users and (weakly) the merchant. However, the EPN’s total fee is lower with rebates (as
needed to maintain merchant participation), so total quantity is higher than under the NSR with no
rebates, as are aggregate consumer surplus and overall welfare. Relative to no NSR, the NSR with
rebates leaves total quantity unchanged but reverses the gap between the card and cash quantities
from negative to positive. Overall welfare rises if and only if the cash market is sufficiently large,
because the per capita cash distortion then is small (though a larger cash market makes the NSR
less favorable to total consumer surplus). A larger merchant benefit from card compared to cash
transactions increases the pre-NSR distortion from double marginalization in card pricing and
improves the effects of the NSR with rebates on overall welfare and total consumer surplus;
interestingly, the reverse occurs if rebates are not feasible.
Our emphasis has been on the impact of limits on merchant pricing flexibility when there
exists some power over price. To highlight this effect, we assumed monopoly pricing at both the
merchant and EPN levels, but we conjecture that similar effects will arise whenever there remains
a significant margin between price and marginal cost at both levels. We also analyzed a case
where the EPN margin is almost zero, because the EPN’s card issuing banks behave as Bertrand
competitors. In that case, pre-NSR there is no significant bias against cards (if merchant benefit
from cards is low), so by encouraging card transactions at the expense of cash the NSR with
rebates reduces welfare, though it increases overall consumer surplus. Our analysis abstracted
away from consumers’ choice of the means of payment in order to focus on the impact on the level
of transactions per consumer. Extensions of this research would include analyzing the effects of
the NSR under a broader class of merchant market structures and with endogenous consumer
choice of the means of payment (e.g., Rochet, 2003). Another direction will be to examine how the
NSR influences the competition among rival payment networks both in pricing and in other
practices such as the tying of multiple cards (e.g., Rochet and Tirole, 2003).

-26-
Figure 1

-27-
Figure 2

-28-
References
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perspectives.” Journal of Law and Economics 26 (October): 541-588.
Carleton, Dennis and Alan Frankel (1995a). “The antitrust economics of credit card networks.”
Antitrust Law Journal 63 (2): 643-668.
Carleton, Dennis and Alan Frankel (1995). “The antitrust economics of credit card networks:
Reply to Evans and Schmalensee Comment.” Antitrust Law Journal 63 (3): 903-915.
Chain Store Age, Fourth Annual Survey of Retail Credit Trends, January 1994, section 2.
Chakravorti, Sujit (2003). “Theory of credit card networks: A survey of the literature.” Review of
Network Economics , Vol. 2, Issue 2 (June): 50-68.
Chakravorti, Sujit and Williams Emmons (2001). “Who pays for credit cards”. Federal Reserve
Bank of Chicago. EPS-2001-1.
Chakravorti, Sujit and Alpah Shah (2001). “A study of the interrelated bilateral transactions in
credit card networks.” Federal Reserve Bank of Chicago. EPS-2001-1.
Evans, David and Richard Schmalensee (1995). “Economic aspects of payment card systems and
antitrust policy toward joint ventures.” Antitrust Law Journal 63 (3): 861-901.
Evans, David and Richard Schmalensee (1999). Paying with Plastic: The Digital Revolution in
Buying and Borrowing. MIT Press.
Faulkner and Gray (2000), Card Industry Directory, 2000 edition, Chicago.
Federal Reserve Board (2001). Recent Changes in Family Finances: Evidence from the 1998 and
2001 Survey of Consumer Finances.
http://www.federalreserve.gov/pubs/oss/oss2/2001/bull0103.pdf
Gans, J. and King, S. (2003). “The Neutrality of Interchange Fees in Payment Systems.” Topics in
Economic Analysis & Policy: Volume 3, Issue 1. Article 1.
Gerstner, Eitan, and James D. Hess (1991). “A theory of channel price promotions.” American
Economic Review 81 (September): 872- 886.
Hayashi, Fumiko (2005). “A puzzle of card payment pricing: why are merchants still accepting
card payments?” Federal Reserve Bank of Kansas City. Payments System Research
WP04-02, December 2004, revised September 2005.

-29-
Katz, Michael L. Reserve Bank of Australia. Reform of Credit Card Schemes in Australia II:
Commissioned Report, www.rba.gov.au , August 2001.
Malueg, David (1992). “Direction of price changes in third-degree price discrimination:
Comment.” Freeman School of Business, Tulane University, Working Paper 92-ECAN
95.
Nahata, Babu, Krzysztof Ostaszewsi, and P.K. Sahoo (1990). “Direction of price changes in third-
degree price discrimination.” American Economic Review 80 (December): 1254- 1258.
The Nilson Report, May, 2003.
Reserve Bank of Australia (2002). Reform of Credit Card Schemes in Australia IV: Final Reforms
and Regulation Impact Statement. 27 August.
Rochet, J.C. (2003). “The theory of interchange fees A synthesis of recent contributions.” Review
of Network Economics , Vol. 2, Issue 2 (June): 97-124.
Rochet, Jean-Charles and Jean Tirole (2002). “Cooperation among competitors: Some economics
of payment card associations,” Rand Journal of Economics, Vol. 33, 549-570.
Rochet, Jean-Charles and Jean Tirole (2003). “Platform competition in two-sided markets.”
Journal of the European Economic Association, forthcoming.
Salop, Steven (1990), “Deregulating self-regulated shared ATM networks,” Economics of
Innovation and New Technology volume 1, numbers 1-2 (December): 85-96.
Schmalensee, Richard (2002). “ Payment systems and interchange fees.” Journal of Industrial
Economics, Vol. 50, 103-122.
Tirole, Jean (1988). The Theory of Industrial Organization. MIT Press.
Wright, Julian (2003). “Optimal card payment systems.” European Economic Review 47: 587-612.

-30-
Appendix
Proof of Proposition 2: (i) Since merchant sales are decreasing in marginal cost, x(k-b) < x0 and
therefore, V
N
(x(k-b))>V
N
(x0). Since t*
/
V
N
(x(k-b))-V
N
(x0), for all t,i such that i+t=k, x(k-b) remains
constant and t<t* implies V
N
(x(k-b))-t=peM>V
N
(x0)=pcM.
(ii) Consider a choice of (qe, qc) that solves the merchant’s profit maximization problem with an
NSR. Suppose that qe < x(k-b). The pair (x(k-b),qc) is also feasible for the merchant since V
N
(qc)+t
$
V
N
(qe) implies V
N
(qc)+t
$
V
N
(x(k-b)) by the concavity of V(
C
). But the choice of (qe, qc) over (x(k-
b),qc) then implies that
qe (V
N
(qe)-k+b)
$
x(k-b)(V
N
(x(k-b))-k+b)
which violates the definition of x(k-b). A similar proof shows qc
#
x0 . Now suppose qe=x(k-b). The
merchant’s first order condition with respect to qe under the NSR constraint is qeV
O
(qe)+V
N
(qe)-i-
t+b-
8
V
O
(qe) where
8
> 0 is the multiplier on the constraint imposed by the NSR. Evaluating this
expression at x(k-b) yields -
8
V
O
(x(k-b))>0 since the first terms are the merchant’s first order
condition with no NSR and equal zero at x(k-b). Therefore, merchant profits are strictly increasing
in qe at qe=x(k-b).
(iii) Let f(p) be the demand curve of cash users with pf(p) concave. The demand curve of card
users is f(p+t). Let i+t = k and let p denote the optimal (uniform) price charged by the merchant
under an NSR when the card user fee is t (so i = k-t). Similarly, let p
N
denote the optimal uniform
price charged by the merchant when the card user fee is t
N
< t. Finally, for convenience, set t-t
N
/
)
> 0. By definition of p, charging a price p under the fee profile, (k-t,t) yields higher merchant
profits than charging a price p
N
-
)
. Note that this second price implies a net price to card users of
p
N
+t
N
. Thus,
"
pf(p)+(p+t-(k-b))f(p+t)
$
"
(p
N
-
)
)f(p
N
-
)
)+(p
N
+t
N
-(k-b))f(p
N
+t
N
).
Similarly, under the fee profile, (k-t
N
,t
N
), p
N
raises more profits than charging a price p+
)
.
"
p
N
f(p
N
)+(p
N
+t
N
-(k-b))f(p
N
+t
N
)
$
"
(p+
)
)f(p+
)
)+(p+t-(k-b))f(p+t).
Adding the two inequalities and eliminating the common terms which denote revenues in the card
market and dividing by
"
, yields
p f(p)-(p+
)
) f(p+
)
)
$
(p
N
-
)
)f(p
N
-
)
)-p
N
f(p
N
).
Recall that p and p
N
are higher than the price which maximizes p f(p). Suppose that p
N
+t
N
>p+t. This
implies p
N
-
)
>p. But this violates the assumption of concavity of p f(p) since the slope of the
revenue function must become steeper as we move further to the right of the maximum point. 2

-31-
Proof of Proposition 3: (i) When the IR constraint does not bind, the result follows from
Proposition 2iii). Now suppose the IR binds and consider (t,i) space. At t = 0, and i such that the
merchant IR curve binds, we show that the slope of the EPN level set is a lower negative number
than the slope of the merchant IR curve which has slope with absolute value less than one. This
implies that this point is a constrained maximum.
Under an NSR, the Lagrangian representing the merchant’s profit maximization problem is
L(qc qe ,
8
;i,t)=
"
qc V
N
(qc )+ qe (V
N
(qe )-i-t+b)+
8
(V
N
(qc )+ t - V
N
(qe )),
where
8
>0 is the lagrangian on the No Surcharge constraint. The EPN’s profit function is given by
A
e=(i+t)qe(t,i;b).
Now consider the level sets of the merchant and the EPN in (t,i) space. The slopes at t = 0 are given
by
Note that the denominator in the first expression is non-negative since the EPN’s profit function is
quasi-concave along the line t=0 and the fact that the IR constraint binds implies it is constrained to
select an i less than its unconstrained optimum. Equation (3) which provides the merchant’s optimal
choice of qe then implies
The first order conditions for the merchant’s optimal choice of quantity imply that i = b +
8
(1+
"
)/
"
. This yields
and

-32-
The first inequality comes because the IR is binding on the EPN and the second inequality follows
because qe is decreasing in i. Combining these results yields
Now consider the level sets of the EPN and the merchant in (t,i) space. Subtracting the second from
the first yields, after substituting the inequality from above,
Recalling that Q -
8
/
"
is positive, 2i) follows.
ii) Note that at t = 0, the first order conditions for the merchant’s problem imply that
(1+
"
)(Q2 V
O
+QV
N
) = (i-b) Q
(Note that this implies that the merchant’s choice of Q is a strictly decreasing function of (i-b).)
Substituting in for (i-b) Q the IR constraint is equivalent to
(1+
"
) QV
N
(Q) - (i-b)Q = -(1+
"
) Q2 V
O
(Q)
$
"
x0 V
N
(x0)
or (Equation (4) in the text)
- Q2 V
O
(Q)
$
"
x0 V
N
(x0)/(1+
"
) . (1A)
The right side is increasing in
"
. Concavity of the merchant revenue function in quantity implies the
left side is increasing in Q. For low
"
, the constraint does not bind when the EPN selects its
globally optimal Q at (i0 ,0). As
"
rises, the constraint binds and the EPN must offer a successively
higher Q (lower i) in order to induce the merchant to participate.
iii) The merchant choice of Q is strictly decreasing in i-b, so, holding b fixed, 3ii) implies i
decreasing in
"
. When the IR binds, Q is determined by (1A) which, in turn, determines i-b.
iv) If the IR does not bind, then the EPN optimal choice of i is (1+
"
+b)/2 which is increasing in
"
.
If the IR binds, then the optimal choice of i is determined solely by the merchant’s IR constraint (at
t=0) and is given by

-33-
This is decreasing in " for all ".2
Proof of Proposition 4: i) Proposition 3 yields the optimal solution t = 0 which implies that per
capita cash and card purchases are the same. This gives the first order condition of the merchant, i
= b+(1+
"
)( V
N
(x)+xV
O
(x)). Define q
"
= argmaxx x(b+(1+
"
)(V
N
(x)+xV
O
(x))) to be the quantity of
card-user transactions which maximizes EPN profits with the NSR. Concavity of the EPN’s profit
function implies this is unique. Note that q0 maximizes profits with no NSR. If b=0, then the
definition indicates that q0 = argmaxx (1+
"
) x((V
N
(x)+xV
O
(x))) and so q0 also solves the EPN’s
problem with the NSR. Thus, card transactions are unchanged with the NSR (and b=0) but cash
transactions are lower (since they exceeded card transactions without the NSR). Now consider b >
0. By definition,
q0 (b+V
N
(q0)+q0 V
O
(q0)) $ q
"
(b+V
N
(q
"
)+q
"
V
O
(q
"
))
and
q0 (b+(1+
"
)( V
N
(q0)+q0 V
O
(q0))) # q
"
(b+(1+
"
)( V
N
(q
"
)+q
"
V
O
(q
"
))).
Subtract the two inequalities and divide by -
"
to get
q0(V
N
(q0)+q0 V
O
(q0)) # q
"
(V
N
(q
"
)+q
"
V
O
(q
"
)).
Suppose that q
"
> q0. Then b q0 < b q
"
. This implies
q0(b+V
N
(q0)+q0 V
O
(q0)) < q
"
(b+V
N
(q
"
)+q
"
V
O
(q
"
))
which violates the definition of q0. The EPN first order conditions with the NSR, evaluated at q0,
indicates that EPN profits are strictly declining in quantity at that point:
so q
"
<q0 given b>0. Thus, card transactions are lower with the NSR for b>0 and so, too, are cash
transactions.
ii) a) - b) The limit of the right side of (1A) as
"
becomes large is x0V
N
(x0) so Q must approach x0
Before reaching the limit, though, Q < x0 so cash users’ purchases and surplus fall with the NSR.
P3) implies that eventually cardholder purchases and surplus are higher with the NSR.
c)-d) With an NSR and linear demand, merchant profit is (1+
"
)Q2 so (4) can be written Q2=
"
/(4(1+
"
)). With no NSR, the total quantity of transactions is (1+b)/4+
"
/2. For
"
>
"
*, the IR
constraint binds and determines Q=(
"
/(4(1+
"
)).5. The NSR thus raises total quantity if and only if,

-34-
This is impossible so total quantity falls. The limit as
"
goes to infinity of the difference in total
quantity is
Applying L’Hopital’s Rule to the limit (the term in the limit is 1/2) yields a value for the difference
in total quantity as
"
goes to infinity is -b/4.
For any two pairs of per capita transactions, (qc,qe),(Qc,Qe), and defining,
)
Qc = Qc - qc,
)
Qe = Qe - qe,
)
QT =
")
Qc +
)
Qe,
the change in total surplus when moving from the first outcome to the second is
)
TS =(1-.5(Qc+qc))
)
QT +(b-((Qe-Qc)+(qe-qc))/2)
)
Qe . (2A)
Thus, let qc =1/2, qe=(1+b)/4 be the per capita transactions when surcharging is allowed and
Q=Qe=Qc the (common) per capita quantity under the NSR without rebates. The IR constraint
yields Q2=.25
"
/(1+
"
), so the limit as
"
goes to infinity of Q is ½. Thus, as
"
goes to infinity,
)
Qc
= 0,
)
Qe = (1-b)/4,
)
QT = -b/4 and we have
lim
"64
)
TSNR = (1+b)2/32 -b2/4.
So the limit is decreasing in b. Furthermore, using the fact that Q and qc are independent of b and
M
qe/
M
b=1/4, we have
M)
TSNR /
M
b=-
M
qe/
M
b+qe
M
qe/
M
b+Q-qe-b
M
qe/
M
b.
Therefore,
M
{
M)
TSNR /
M
b} /
M"
=
M
Q /
M"
>0.
Direct computation shows that
)
TSNR is increasing in
"
for b=0. Thus, it is increasing in
"
for all
b>0. Since, for
"
large enough, b
N
> b implies
)
TSNR(
"
,b
N
)<
)
TSNR(
"
,b) (the limit is decreasing in
b), and since
M)
TSNR /
M"
is increasing in b, we have
)
TSNR(
"
,b
N
)<
)
TSNR(
"
,b) for all
"
.
(Computations show that total surplus exceeds total surplus with no NSR at
"
> 1.53 for b=0.).

-35-
(4A)
(5A)
(6A)
The change in consumer surplus is
)
CS =.5(Qc+qc)
)
QT +.5((Qe-Qc)+(qe-qc))
)
Qe , (3A)
and in this case,
)
QT <0, Qe-Qc=0, and (qe-qc)
)
Qe <0. 2
Proof of Proposition 5:i) If
"
<
"
*, then the IR does not bind under no rebates and, by Proposition
2iii), the EPN increases profits by holding i fixed and lowering t. If
"$
"
*, from the Proof of
Proposition 3i) the slope of the EPN indifference curve in (i,t) space is steeper than the slope of the
merchant’s IR curve at the constrained optimal solution, t=0. Thus, again, EPN profits are strictly
higher as (i,t) is varied by lowering t below zero and raising i so as to stay on the IR curve.
ii) With t<0, the merchant’s demand curve for card transactions is strictly above the (per capita)
cash demand curve. With sufficiently high rebates, the monopoly price from serving the card
market exceeds the choke price for the cash market. In this case, the merchant’s profit function has
two local maxima. If the merchant serves only the card market, the per capita card transaction is the
same as with surcharging allowed but cash transactions are now zero. Repeating (3), if the merchant
prices to serve both markets, the solution is
The merchant chooses to serve both markets and thus selects prices and quantities as in (4A)
if and only if
Using the values for p, qc,qe from (4A) in (1A) gives the values of (i,t) for the EPN when the IR
binds. Maximizing with respect to (i,t) yields
Equation (6A) implies that (5A) is violated as
"
gets small.
iii) Follows from Equation (6A). 2

-36-
Proof of Proposition 6:i)-iii) Follows from (4A) and noting that under surcharging, per capita cash
quantity is 1/2 and card quantity is (1+b)/4.
iv)-v): Total surplus is affected by total quantity and by the differences in quantities. The NSR and
linear demand imply qcNSR=qeNSR+t. Utilizing Equation (2A) for the change in total surplus moving
from surcharging to an NSR with rebates, along with the fact that total quantity is unchanged yields
)
TSNSR =(b-(-t-(1-b)/4)/2)
)
Qe =(7b+1+4t)
)
Qe /8.
The change in total surplus is positive if and only if (7b+1+4t)>0. It is increasing in
"
if
(7b+1+4t)>0 since
)
Qe and t are increasing in
"
. Note that if b=0, then TSNSR-TSSUR =0 at
"
=1/3.
(At that point, t=-1/4 and qeNSR exceeds qcNSR by exactly the same amount that qcSUR exceeds qeSUR.)
Given constant total quantity and linear demand, aggregate consumer surplus depends on
the split between the types of consumers. Equation (3A) yields
)CSR =-(qeNSR -qeSUR)[(qcNSR- qeNSR ) + (qcSUR -qeSUR)].
Direct computation yields that )CSR is increasing in
"
for all b. Since )TSR depends on
"
as )CSR
depends on -
"
, we also have )TSR falls in
"
for
"
such that (7b+1+4t)<0.
vi) Direct computation shows that )TSR and )CSR rise in b.2
Proof of Proposition 7:i) Equation (4A) shows that qc falls as i-t rises and qe rises if i+t and t fall.
Propositions 3 and 5 reveal that compared to the NSR with no rebates, when rebates are feasible,
i+t and t are lower and i-t is higher. If the IR binds, then Propositions 4 and 6 imply total quantity is
higher under the NSR with rebates and since, with no rebates, per capita quantities are always
identical, (qe=qc) and with rebates, Qe > Qc, Equation (3A) implies total consumer surplus must
rise.
ii) Suppose that the IR binds at the optimal solution with t=0. The optimal solution with the t
$
0
relaxed is at a point downward and to the right of this point. Part i) implies total consumer surplus
rises. Revealed preference implies that EPN profits rise and, since we remain on the merchant’s IR
curve, merchant profits stay the same. Thus, total surplus rises when the t
$
0 constraint is relaxed
from a point at which the IR constraint binds.
iii) Shown by computation.

-37-
Proof of Proposition 8: i): Solving the merchant participation constraint simultaneously with the
constraint t=-i, yields
The constraint that the merchant continue to be willing to serve the cash market is t>i-(1+
"
).5/2.
Using t=-i, this yields a value
The lowest value for icomp is the binding constraint. The second one is lower than the first if and only
if
"
<1.
ii),iii): Use the quantity equations from Equation (4A) for per capita purchases and t=-i to get
"
qc=
"
(.5-i/(1+
"
)) and qe=(.5+
"
i/(1+
"
)). Summing the two yields total quantity (1+
"
)/2 which is
independent of i and is equal to the total quantity of purchases with competitive issuers and no
NSR.. Conditional on total quantity remaining constant, social surplus is maximized when the cash
and non-cash quantities are the same. Any value of t strictly less than zero along with the NSR,
violates this condition, so social surplus must fall. Consumer surplus rises because, holding total
quantity fixed, the loss to cash consumers from the higher price is more than compensated by the
gain to EPN consumers from the lower price. Using qe=(.5+
"
i/(1+
"
)) and letting
"
grow large
yields i approaches 1/2, EPN quantity approaches 1 and cash quantity approaches 1/2.2