"Rip-off" ATM surcharges.

We develop a spatial model in which we endogenize both the pricing of ATM services by banks and the choice of home bank and ATM use by consumers. The equilibrium delivers the empirical regularities: Banks set high bank account fees for their own customers but do not charge them for ATM usage;

in contrast, banks charge high ATM fees for nonmember users, fees that exceed those levels that would maximize ATM revenues from nonmembers; and larger banks set higher account fees and demand higher surcharges for ATM use than smaller banks. Paradoxically, (i) a bank's ATM revenues may fall short of its costs of ATM provision, and (ii) prohibiting banks from surcharging nonmembers, by forcing banks to charge members and nonmembers the same ATM price, leads to higher ATM prices, greater bank profits, and possibly reduced consumer welfare.

1. Introduction

* These days most people have had the experience of realizing that they are short of cash when shopping. They hunt down the nearest ATM (automated teller machine), slip their card into the slot, and seek to withdraw money from a bank account. The ATM screen pops up with the ATM surcharge that must be paid to complete the transaction; they then mutter "rip-off" to themselves, grit their teeth, and pay to get their money.

The average ATM surcharge that consumers face for using a nonmember bank's ATM rose in 1999 to $1.38 from $1.23 in 1998. In contrast, in 1998, less than 1% of banks imposed "on-us" ATM fees on member-bank customers (Federal Reserve Board, 1999). PIRG estimates that in 1998, U.S. banks reaped more than $2.1 billion in annual revenue from ATM surcharges. (1)

Large banks (2) are particularly likely to surcharge, and those that surcharge set especially high prices. In 1999, 95% of big banks set ATM surcharges, and those that surcharged extracted an average surcharge of $1.42. Surcharging rates for the 91% of small banks that surcharged were $1.30. In sharp contrast, only 42% of credit unions set ATM surcharges, and those who surcharged set average surcharges of $.98. Similarly, the Federal Reserve documented that multi-state banks charge significantly higher fees than locally owned banks.

Although bank fees have attracted less notoriety, banks, especially large banks, also set very high bank fees on bank accounts. These fees exceed $200 on average for regular checking accounts (PIRG, 1999), and they do not include the (generally greater) value of the interest rate differential between the interest rates that banks offer customers on their accounts and the risk-free market interest rate.

The goal of this article is to provide an economic grounding for this pricing phenomenon: Why do banks, especially large banks, set such high prices for ATM use by nonmembers? Why don't banks charge for ATM use by bank members? Why are bank account fees so high?

We show that these pricing outcomes are the equilibrium consequence of competition among banks. We develop a spatial model in which we endogenize pricing of ATM services by banks, as well as bank account choice and ATM use by consumers. The features of the equilibrium match the empirical regularities. Banks set high account fees for their own customers but do not charge them for ATM usage; in contrast, banks charge high ATM fees for nonmember users; and larger banks set higher account fees and demand higher surcharges for ATM use.

We first develop a spatial model in which two banks have one branch on an island. Consumers may require one of two types of service: (i) a service that any branch or ATM can provide, such as cash withdrawal; or (ii) a service such as a record of checks written in the past two months that can only be provided by the bank at which they have their account.

We show that in equilibrium, banks price discriminate across members and nonmembers in their pricing of banking services. Banks charge members the marginal cost of the ATM service, and extract all profits from members in the form of high nondiscretionary charges (e.g., account fees, bank services that only the customer's bank can provide). The economic rationale is the following: Customers internalize expected ATM use/charges and transportation costs when choosing where to establish bank accounts, so banks maximize profits from their members by encouraging efficient ATM use--marginal-cost pricing ATM use by bank members. When the ATM service is priced at its marginal cost, a member uses her bank's ATM if and only if the associated customer surplus plus bank profit exceeds customer surplus obtained from patronizing the foreign bank's ATM. Were the bank to raise its ATM price for members above marginal cost, in order to maintain its bank account base, the bank would have to reduce bank account fees by an amount that reflects not only the greater ATM payments, but also the distortion in ATM use by its members, reducing its profit from bank members.

In sharp contrast, banks can earn profits from nonmembers only by setting high ATM prices. This distorts nonmember ATM choice, causing nonmembers to sometimes patronize their home bank even though they are closer to the foreign bank. There is also a key indirect strategic effect of high ATM surcharges on nonmembers: a customer is more likely to open an account with a bank if the institution charges nonmembers higher ATM fees. Because banks inefficiently extract surplus from nonmembers but efficiently extract surplus from bank members, banks have an incentive to set higher ATM fees to increase their bank account base. Hence, equilibrium surcharges exceed the prices that would maximize expected profits from ATM use by nonmembers. We document this strategic pricing effect by contrasting the equilibrium pricing with the lower pricing of ATM services that would obtain were bank affiliation exogenously set to be equal to their equilibrium levels, so that pricing of ATM services does not alter the bank account affiliation decisions.

Paradoxically, our model is also consistent with the finding that "banks tend to lose money in their ATM operations." (3) This reflects both that

(i) ATM machines have substantial operating costs in addition to the cost of the machine itself, (4) and

(ii) banks receive no ATM revenues from their own members to cover their fixed and operating costs, and ATM surcharges are so high that nonmembers are overly reluctant from a profit-maximizing perspective (that ignores consequences for bank account choice) to use a foreign bank's ATM service, so total ATM surcharge revenues fall short of the ATM costs.

We then consider the consequences of requiring banks to set the same ATM fee for bank members and nonmembers. Recently, many U.S. communities have proposed rules to prohibit banks from imposing ATM surcharges on nonmembers. (5) Paradoxically, we show that laws requiting banks to charge each customer the same price for the ATM service appear to backfire. We show that such laws would raise ATM service prices: equilibrium prices for ATM use exceed surcharging levels! While surcharge prohibitions lead to reduced bank account fees, bank profits are sharply increased.

However, further examination reveals that surcharge prohibitions can sometimes be welfare enhancing, even for consumers. Total consumer plus producer surplus is increased because higher ATM prices just transfer surplus dollar by dollar from consumers to producers, but uniform pricing of ATM services eliminates the inefficient distortion in the choice of ATM provider. Indeed, bans on surcharging may even raise consumer surplus if consumers are sufficiently likely to be travelling when they receive bank service shocks. Although consumers are hurt by the higher ATM prices, they gain from the reduction in bank account fees and efficient use of ATMs.

The economic reasoning underlying these uniform pricing results reflects subtle tradeoffs in economic incentives. When banks can surcharge,

(i) banks have a strategic incentive to set high ATM surcharges to increase bank membership.

(ii) But there is a greater offsetting incentive to reduce ATM surcharges to increase ATM service revenues. When banks can discriminate between members and nonmembers, the marginal foreign ATM customer is closer to the foreign bank's ATM than the marginal ATM customer when banks cannot price discriminate. Consequently, with surcharging, nonmember ATM demand is more price elastic. First, if travel costs are convex, a given surcharge reduction causes a greater shift in the marginal customer location than that caused by an equal reduction in the uniform price. Second, if travel costs are linear, the measure of customer locations that switch ATM choices due to a given reduction in ATM fees does not depend on whether surcharging is possible. However, a surcharging bank's ATM serves fewer foreign customer locations, so the revenue reduction due to a price reduction is proportionately less.

In equilibrium, even if travel costs are linear, the strategic impact from competition for ATM revenues dominates that associated with competition for bank account base, so that the equilibrium ATM surcharge is less than the ATM service price charged when surcharging is banned.

Finally, we explore how bank size affects pricing. We consider two banks, one of which has more branches than the other. We show that the strategic pricing impacts are greater for the larger bank. The larger bank charges more for ATM use by nonmembers in order to increase its bank account base, and it charges substantially higher bank account fees. (6) The intuition is that because consumers are more likely to be near a large bank's ATM, an increase in the large bank's ATM surcharge leads to a greater increase in bank account base at the large bank than does a similar increase in the small bank's surcharge. This finding is consistent with PIRG's concerns that "a surcharging is part of the big bank's anti-competitive strategy to squeeze out smaller banks and credit unions by encouraging their customers to switch their accounts to banks with larger ATM networks" (PIRG, p. 9).

The article is organized as follows. Section 2 reviews related research. Section 3 develops the basic model and characterizes the equilibrium properties. Section 4 derives the consequences of banning ATM surcharges. Section 5 explores how bank size affects pricing. Section 6 summarizes our findings and details the empirical support that Hannan et al. (2000) find for our model. All proofs are in the Appendix.

2. Related theoretical research

* Our model is related to one developed by Matutes and Padilla (1994). They study the incentive of banks to share their ATMs when competing for deposits. They assume banks provide both in-branch and ATM services, and consumers receive location and service demand shocks. Their analysis focuses on banks' choices of whether to employ compatible ATM technologies; they assume that banks set uniform prices and do not discriminate across consumers in the pricing of ATM services. In contrast, we assume that ATM technologies are compatible, as is now the industry standard, and focus on price discrimination according to service and membership.

Chiappori, Perez-Castrillo, and Verdier (1995) develop a spatial competition model of the banking sector in which banks compete on providing two types of services: deposits and loans. Again, in their model, banks do not discriminate across customer types, and customers patronize but one bank.

Rochet and Tirole (2000) analyze the cooperative determination of an interchange fee by member banks in a payment card association; the interchange fee is the "access charge" paid by the merchants' banks to the cardholders' banks.

Laffont, Rey, and Tirole (1997, 1998a, 1998b) and Economides, Lopomo, and Woroch (1996) develop related spatial models of competition between two interconnected telecommunication networks. In Laffont, Rey, and Tirole (1998a, 1998b), foreign networks charge access fees to the other network for a call between networks. Access fees raise the perceived marginal cost, and networks set price equal to the average (within and across networks) perceived marginal cost if they cannot price discriminate, and equal to the perceived marginal cost if they can. If networks cannot price discriminate, subscription fees and access fees are perfect strategic substitutes, so that profits do not vary with the access fee. An analogous finding in our article is that prices of in-branch services and account fees are perfect strategic substitutes. However, we find that bank profits do depend on the surcharge, whether or not banks price discriminate. Laffont, Rey, and Tirole (1998b) show that results are subtly different if networks price discriminate and the (now exogenously set) access fee is positive so that the price for cross-network calls is higher. Utility is a strictly concave function of total calls, so that calls are reduced by variability in prices, reducing network profits.

Although our model and those of Laffont, Rey, and Tirole share common features, the basic results are very different. In Laffont, Rey, and Tirole, the networks (perceived) marginal-cost price all services, in order to avoid distorting consumer choice. In contrast, in our economy, banks set surcharges that greatly exceed marginal costs in order to distort customer choice and induce consumers to establish accounts with them. While both models predict that uniform pricing raises industry profits, the underlying economic reasoning differs. In Laffont, Rey, and Tirole, price discrimination reduces consumer demand, whereas we have no such reduction. In contrast, in our model, uniform pricing changes the mix of consumers who use their own bank's ATM. With uniform pricing, demand is less price elastic, causing banks to price ATM services less aggressively.

There is also a significant spatial literature dating back to Hoover (1948) documenting that limiting price discrimination between oligopolists may be profitable because it reduces the fronts on which they compete. As Anderson, de Palma, and Thisse (1992, p. 328) observe, "delivered pricing makes each consumer a competitive battle ground," (with the delivered price equal to the serving cost of the second lowest-cost firm): firms do better when they must offer the same price to all customers. Thisse and Vives (1988) show that if firms cannot commit to pricing strategies, they have a dominant strategy to perfectly price discriminate. In these contexts, the uniform price is below the highest delivered price and total consumer plus producer surplus is reduced. (7) Anderson and Leruth (1993) offer similar results in a multiproduct context, where duopolists can offer separate prices for each product as well as bundled prices, showing that firms have dominant strategies to both offer each component individually and offer a bundle.

The economics underlying our result that surcharging bans raise bank profits is very different, as are the ancillary predictions. In particular in our model, (i) total consumer plus producer surplus is always raised rather than lowered, and consumer surplus may even rise, and (ii) the uniform price exceeds the highest discriminatory price (here the surcharge). In our model, whether or not surcharging is prohibited, customers internalize expected ATM prices and travel costs at the ex ante stage where they are determining where to establish bank accounts, and there is only one marginal customer type, so the number of "fronts" on which banks compete is essentially unchanged. Instead, our results hinge on the implications of uniform pricing for the size of the ATM market.

3. The economy

* We consider a spatial economy in which a measure one of bank customers are uniformly distributed on the perimeter of an island with circumference one. There are two banks, E and G. Bank E is located at 0, and bank G is located on the opposite side of the island at location 1/2. The banks provide two services to consumers:

Service s = a is a service that a bank can provide to any consumer. We interpret service a as an ATM service such as a cash withdrawal.

Service s = b is an in-branch service that a consumer can obtain only from the bank at which she has an account. Service b could include updating bank books, deposits, verifying check account balances, bill payments, account transfers, etc.

Each consumer is distinguished by her initial spatial location, x. A consumer must maintain a bank account at one bank or the other. The consumer is free to choose the bank at which she establishes her account. Consumers then receive bank service demand shocks. With probability [mu] a consumer requires service a that can be provided by any bank's ATM, and with residual probability (1 - [mu]), a consumer will require service b that can only be provided by the bank at which she maintains her account. (8) With probability [phi], the consumer will be at her initial spatial location when she requires the bank service. With residual probability 1 - [phi], the consumer will receive the bank service shock when she is out shopping, in which case she is equally likely to be at any point on the island. A consumer's service demand shock is distributed independently from her location shock, and there is no aggregate uncertainty in the distribution of shocks in the economy.

Each customer receives incremental utility M from consuming the bank service that she values. M is assumed to be sufficiently large that in equilibrium, all consumers will pay for the bank service. To receive the service, a consumer must travel to a bank--any bank if she requires service a, and the bank at which she maintains her account if she requires service b. A consumer located at x who goes to a bank located at d incurs transportation costs T[(x - d).sup.2], where T > 0.

[] Service costs and prices. Bank j charges customers a fee [F.sup.j] to maintain an account at the bank, j = E, G. This fee [F.sup.j] is meant to capture such nondiscretionary charges as the standard account fees, fees for checks, the value of the difference between the bank and market interest rates offered on accounts, etc. Let [c.sub.a] be the cost to a bank of providing the ATM service to a customer. Let [c.sub.b] be the cost of providing the in-branch service b.

Let [[delta].sub.j] [member of] {0, 1} capture the customer's bank account status: [[delta].sub.j] = 1 if the consumer establishes an account with bank j, and [[delta].sub.j] = 0 if the consumer does not. Bank j charges a customer with affiliation status [[delta].sub.j] price [p.sup.j.sub.a] ([[delta].sub.j]) for the ATM service, and it sets price [p.sup.j.sub.b] for the in-branch service. Let [P.sup.j] = [[p.sup.j.sub.b], [p.sup.j.sub.a](1), [p.sup.j.sub.a](0), [F.sup.j]] be the vector of prices set by bank j.

[] Timing of the game.

Stage 1. The banks set service prices.

Stage 2. Given their location and prices set by each bank, each consumer chooses a bank at which to establish an account.

Stage 3. Each consumer receives a location and bank service demand shock and chooses a bank at which to obtain the service.

[] Equilibrium. We solve for equilibrium outcomes recursively, beginning with the choice of bank by a consumer with a given affiliation who requires a bank service.

Stage 3. Let [y.sub.a](1) represent the location of the customer affiliated with bank E who is indifferent between getting the ATM service a from bank E and G:

(1) M - [[p.sup.E.sub.a](1) + T[y.sub.a][(1).sup.2]] = M - [[p.sup.G.sub.a](0) + T [(1/2 - [y.sub.a(1)).sup.2]].

Solving,

(2) [y.sub.a](1) = [[p.sup.G.sub.a](0) - [p.sup.E.sub.a](1) + T/4]/T.

Customers with accounts at bank E who are located closer to bank E than [y.sub.a](1) obtain ATM service a from bank E, while the others obtain theirs from bank G.

Analogously, the consumer affiliated with bank G who is indifferent between getting service a from bank E and G is located at

(3) [y.sub.a](0) = [[p.sup.G.sub.a](1) - [p.sup.E.sub.a](0) + T/4]/T.

Stage 2. At this stage, consumers choose where to establish bank accounts. A consumer located at x who establishes an account at bank E expects at the next stage to receive utility

(4) E[[u.sup.E](y, s, [P.sup.E], [P.sup.G]) | x] = [PHI] [[mu](M - [p.sup.E.sub.a](1) - T[chi square]) + (1 - [mu])(M - [p.sup.E.sub.b] - T[chi square]) + (1 - [PHI]) [K.sub.E],

where

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is the expected utility of a consumer who requires bank services when not at her initial spatial location. The first term corresponds to those times when she travels to obtain in-branch service b; the second term corresponds to when she obtains the ATM service a from her own bank; and the last term captures those occasions when she uses foreign bank G's ATM.

If she instead sets up a bank account at bank G, she expects to receive utility

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Given account fees [F.sup.E] and [F.sup.G], the consumer who is indifferent between establishing an account at banks E and G is located at

(8) [x.sup.E] = 1/T [[[F.sup.G] - [F.sup.E]]/[PHI] + (1 - [mu])([p.sup.G.sub.b] - [p.sup.E.sub.b]) + [mu] ([p.sup.G.sub.a](0) - [p.sup.E.sub.a](1)) + T/4 + [(1 - [PHI])([K.sub.G] - [K.sub.E])]/[PHI]].

Consumers located closer to bank E than [x.sup.E] optimally establish bank accounts at E, while those located closer to G prefer to establish bank accounts at G. Thus, [N.sup.E] = 2[x.sub.E] consumers establish accounts at bank E, and [N.sup.G] = 2[(1/2) - [x.sup.E]] establish accounts at bank G.

Stage 1. Bank E's expected profit is

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first line is the bank's expected profit from affiliated customers (total bank account fees plus in-branch service profits plus the profits from affiliated customers who acquire the ATM service); the second line corresponds to the profits from nonaffiliated customers who acquire the ATM service from the bank. This formulation of expected profits reflects an equilibrium result that a customer who is at her initial location and requires ATM services always chooses, along the equilibrium path, to go to the bank at which she established an account. The customer's own bank charges a lower price for ATM use than the foreign bank.

[] Equilibrium solution. To solve for equilibrium outcomes we derive the first-order conditions for profit maximization and then solve for the symmetric pricing outcomes. The analysis was done using MAPLE (a mathematical software program built by the University of Waterloo, Ontario, Canada). Here, we detail two key observations that offer intuition for our equilibrium characterization, and allow a simplified presentation.

Observation 1. Consumers expect to spend F +(1 - [mu])[p.sub.b] in total on bank account fees and charges for the in-branch service, b, which they must acquire at their bank. Consequently, consumers are indifferent among linear combinations of account fee and in-branch service charges that yield the same expected expenditure. As a result, there are multiple payoff-equivalent equilibria in which bank account fees and the price of the in-branch service b are linearly related on a range such that [p.sub.b], is not so high that some consumers prefer to forgo the service. (9)

This nonuniqueness is due solely to the fact that the demand for in-branch services is completely inelastic, so that expenditures on in-branch services are essentially nondiscretionary. If one introduces as an equilibrium refinement the possibility that demand for in-branch service was not perfectly inelastic, then in equilibrium, banks charge members the marginal cost of the in-branch service, setting [p.sub.b] = [c.sub.b]. If banks charge the marginal cost of the in-branch service, then customers use the service if and only if doing so raises total (consumer plus producer) surplus. We describe the equilibrium in which [p.sub.b] = [c.sub.b], in what follows.

Observation 2. A similar argument underlies the equilibrium result that banks charge member-customers the marginal cost of the ATM service, [p.sub.a](1) = [c.sub.a]. (10) As highlighted in the Introduction, a bank wants members to use its ATM service if and only if the profit it would receive plus the associated consumer surplus would exceed the customer surplus obtained from patronizing the foreign bank's ATM. Customers internalize expected ATM use/charges and travel costs when choosing where to establish bank accounts, so that raising ATM charges for bank members above marginal cost causes them to inefficiently reduce use of the bank's ATM. To keep bank membership unchanged, the bank must reduce bank account fees by an amount that reflects not only the greater ATM payments, but also the distortion (from the bank's perspective) in ATM use by its members, reducing total bank profit.

Exploiting these observations, bank E's profits become

(10) [[PI].sup.E] = [F.sup.E]2[x.sup.E] + [mu](1 - [PHI])(1 - 2[x.sup.E])[[p.sup.E.sub.a](0) - [c.sub.a]] 2[y.sub.a](0),

where

[x.sup.E] = 1/T ([[[F.sup.G] - [F.sup.E]]/[PHI]] + T/4 + [[(1 - [PHI])([K.sub.G] - [K.sub.E])]/[PHI]])

and

[y.sub.a](0) = [[p.sup.G.sub.a](1) - [p.sup.E.sub.a](0)]+ [T/4]/T.

The first-order conditions for profit maximization are

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[[[delta][x.sup.E]]/[[delta][p.sup.E.sub.a](0)]] = [[2[micro](1 - [PHI]]/[T.sup.2][PHI]] [[[c.sub.a] - [p.sup.E.sub.a](0)]+[T/4].

Recognizing that the equilibrium features symmetric pricing strategies, [F.sup.E] = [F.sup.G] = F, [p.sup.E.sub.a](0) = [p.sup.G.sub.a](0) = [p.sub.a](0), we solve equations (11) and (12) for the equilibrium values of F and [p.sub.a] (0).

Proposition 1 summarizes equilibrium outcomes.

Proposition 1. A pure-strategy equilibrium exists if and only if [PHI] [greater than or equal to] [mu]/(18 + [mu]). In equilibrium,

(i) Banks charge affiliated ATM users the marginal cost of the service: [p.sub.a](1) = [c.sub.a].

(ii) Banks set high bank account fees: F = [T[mu](1 - [PHI])/36] + T[PHI]/4.

(iii) Banks impose high surcharges on nonaffiliated ATM users: [p.sub.a](0) = [c.sub.a] + T/6.

(iv) Bank profits are [PI] = [T[mu](1 - [PHI])]/36 + T[PHI]/8.

(v) Consumer surplus is

CS = (1 - 1/2[PHI][mu]) M - (1 - [mu])[c.sub.b] - [mu] (1 - 1/2[PHI]) [c.sub.a] - [T/48] [4 + 12[PHI] + [mu](1 - 21[PHI]/8)].

Note that if [PHI] is very small and [mu] is very large, then customers are very likely to receive ATM service shocks when travelling, so that there is very little difference between banks ex ante. Given the posited symmetric equilibrium strategy by one bank, the other bank could profitably deviate by ceasing to compete for ATM revenues by imposing a prohibitively high surcharge, inducing most consumers to establish a bank account with it, and then efficiently extracting surplus from them. Consequently, a pure-strategy equilibrium does not exist. At [PHI] = [mu]/(18 + [mu]) a bank is indifferent between the posited symmetric strategy and the account-poaching strategy. Also, note that since the fixed fee strictly exceeds the expected ATM fee payment, F > [mu][p.sub.a](1) - [mu](1 - [PHI])[p.sub.a](0), it is never optimal for a consumer to establish accounts at both banks (provided that the in-branch service b--obtaining a record of checks written, etc.--can only be acquired at one bank).

The qualitative predictions of our model match the empirical regularities. Banks impose high bank account fees and high surcharges on nonmember ATM use, (11) but minimal ATM charges for bank members. Banks set high bank account fees and marginal-cost price ATM use by members for the same reason that a monopolist imposes a positive fixed fee and a per-unit price equal to marginal cost. (12) This causes the accountholder to use the ATM service if and only if there is surplus to be claimed by the bank, and the bank extracts this entire surplus through the fixed bank account fee.

In sharp contrast to the marginal-cost pricing of ATM services for bank members, banks impose especially high ATM surcharges on nonmembers because each bank wants to encourage more customers to establish accounts with it (the first term of the bank's first-order condition, equation (12)). These surcharges raise the costs of nonaffiliation. Banks want larger account bases because they more efficiently extract rents from these customers with fixed fees than with high charges on ATM services, where they face price-elastic demand.

To illustrate how the competition for greater account bases induces banks to raise their surcharges for ATM use by nonaffiliated customers, we contrast the equilibrium ATM surcharge with the surcharge set when bank account selection and all prices save those for ATM use by nonaffiliated customers are set exogenously to be equal to their equilibrium levels above. Then the banks set ATM surcharges that maximize profits from nonmembers.

Proposition 2. If affiliation choices are exogenously fixed to correspond to those choices when affiliation is endogenous, which is denoted by (* | ex), then ATM surcharges on nonmember customers, [p.sub.a](0 | ex), are reduced:

(13) [p.sub.a](0 | ex) + T/8 + [c.sub.a] < [p.sub.a](0) = T/6 + [c.sub.a].

When affiliation is endogenous, banks set high ATM surcharges for two reasons: To encourage affiliation and to earn profits from the provision of ATM services. This causes banks to set ATM surcharges for nonmembers above the level that maximizes profits from nonmembers. When affiliation is exogenous, banks price solely to maximize profits from provision of ATM services.

The comparative static properties of the equilibrium outcomes are straightforward. As T rises, it becomes more costly to travel, so the demand for ATM service becomes less price elastic. This allows banks to exploit their greater market power by charging higher nondiscretionary fees and ATM surcharges. In turn, this raises the proportion of consumers who obtain ATM services from the bank at which they establish accounts.

More important, if consumers are more likely to be at their home locations, banks can better exploit the ex ante heterogeneity in location (which is zero in expectation conditional on moving). Hence, as [PHI] rises, banks impose higher fixed account fees. So, too, as the probability of requiring the in-branch service rises, affiliation elasticities fall, inducing banks to set higher account fees.

4. Uniform pricing of ATM services

* This section is motivated by the recent attempts of some communities to ban banks from imposing ATM surcharges on nonmembers. Our goal is to determine how such a government intervention would affect both bank profitability, consumer welfare, and total welfare.

Let [p.sup.u.sub.a] be the common equilibrium price of the ATM service when banks cannot price discriminate between members and nonmembers, and index the other equilibrium variables by u.

If banks charge bank members and nonmembers the same price for ATM use, then only nondiscretionary bank charges and the possibility of requiring in-branch service affect affiliation decisions. Without loss of generality, we consider the equilibrium in which banks set the price of the nondiscretionary in-branch service equal to its marginal cost, [p.sup.u.sub.b] = [c.sub.b]. Bank E's expected profit simplifies to

(14) [[PI].sup.Eu] = [F.sup.Eu]2[x.sup.Eu] + 2[micro](1 - [PHI])[[p.sup.Eu.sub.a] - [c.sub.a]([x.sup.Eu] + [y.sup.u.sub.a]),

where [y.sup.u.sub.a] = [[p.sup.Eu.sub.a] - [p.sup.Eu.sub.a] + (T/4)]/T is the location of the consumer who is indifferent between getting the ATM service a from bank E or G, and [x.sup.Eu] = [([F.sup.Gu] - [F.sup.Eu]) + [PHI][mu]([p.sup.Gu.sub.a] - [p.sup.Eu.sub.a])]/T[PHI] + 1/4 is the location of the consumer who is indifferent between establishing a bank account at banks E and G.

The first-order conditions for profit maximization are given by

(15) 2[x.sup.Eu] + [[[delta][x.sup.Eu]]/[[delta][F.sup.Eu]]](2[F.sup.Eu] + 2[micro](1 - [PHI])[[p.sup.Eu.sub.a] - [c.sub.a]]) = 0

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[delta][x.sup.Eu]/[delta][F.sup.Eu] = -1/T[PHI], [delta][y.sup.u.sub.a]/[delta][p.sup.Eu.sub.a] = -1/T and [delta][x.sup.Eu]/[delta][p.sup.Eu.sub.a] = -[mu]/T.

Substituting for the symmetric equilibrium pricing and solving yields the following proposition.

Proposition 3. Prohibiting banks from charging members and nonmembers different ATM prices

(i) raises the price for ATM use above the ATM surcharge:

[p.sup.u.sub.a] = T/4 + [c.sub.a] > T/6 + [c.sub.a] = [p.sub.a](0).

(ii) Reduces the equilibrium bank account fees:

[F.sup.u] = T[PHI]/4 - [mu][PHI]T/4 < [PHI]T/4 + T[mu](1 - [PHI])/36 = F.

(iii) Raises bank profits:

[[PI].sup.u] = T[PHI]/8 + T[mu](1 - [PHI])/8 > T[PHI]/8 + T[mu](1 - [PHI])/36 = [PI].

(iv) Reduces consumer surplus if and only if [PHI] < 4/7, i.e., if and only if consumers are unlikely to be travelling when requiring ATM services:

C[S.sup.u] - CS = T[mu]/6 (1 - 7[PHI]/4).

(v) Raises total consumer plus producer surplus:

[surplus.sup.u] - surplus = T[mu]/36 (1 + 7[PHI]/2).

At first blush, this result is very surprising. The uniform ATM price does not lie between the ATM prices for members and nonmembers, but rather exceeds the ATM surcharge on nonmembers. The economic reasoning underlying this result is complex, because banks face offsetting economic incentives when pricing ATM services. When banks can discriminate, they have a strategic incentive to set high ATM surcharges for nonmembers and marginal-cost members for ATM use in order to raise the cost of not establishing bank accounts with them, and thereby increase their base of bank account customers (the first term of the first-order condition, equation (12)). This incentive exists because banks extract surplus efficiently from members without distorting their ATM choice by setting high fixed bank account charges and by marginal-cost pricing ATM use; but to profit from nonmembers, they must distort ATM choice by setting high ATM charges. When banks must set uniform ATM service prices, ATM prices cease to affect bank account decisions, so this incentive to set high ATM prices for nonmembers no longer exists. Now bank competition can be decomposed into competition over ATM services (via [p.sup.u.sub.a]) and competition over nondiscretionary bank services (via the bank fee, [F.sup.u]).

Offsetting this incentive to set high ATM surcharges in order to increase the bank account base is the impact on competition for ATM profits. When banks surcharge, the marginal ATM customer is closer to the surcharging bank, and the surcharging bank's share of nonmember ATM customers is smaller than when banks do not surcharge. Hence, the surcharging bank's ATM demand from nonmember customers is more price elastic: (i) when a bank has a smaller ATM customer base, it loses fewer revenues from lowering price to attract customers; and (ii) convex travel costs imply that a price cut gains more customers relative to when they are further away. Thus, if banks can surcharge, a slight reduction in the surcharge for nonmember ATM use loses revenues on fewer existing ATM customers and wins more customers. The more elastic demand associated with surcharging raises the incentives to cut ATM prices to win a greater ATM market share.

As Proposition 3 details, the impact of a more elastic demand on ATM pricing (leading to a lower ATM surcharge) exceeds the impact of the incentive to increase customer base (leading to a higher ATM surcharge). Further, when surcharging is banned, the set of customers from which the bank earns profit from ATM use is increased dramatically because consumers now patronize the closest ATM rather than favor their own banks' ATMs. As a result, when ATM surcharging is banned, bank profits rise sharply, even though bank account fees decline.

The fact that banks earn greater profits need not imply that consumer welfare falls, as total consumer travel costs are reduced when banks price uniformly--the marginal consumer does not travel as far to obtain ATM service from her own bank.

We find that only when consumers are likely to receive service shocks at their home location, i.e., [PHI] < 4/7, are ATM expenditures so much higher (and are now also incurred when going to their own banks' ATMs), that consumer surplus is reduced by a prohibition against ATM surcharging.

It follows immediately, however, that uniform pricing raises total surplus. Firm profit rises by more than consumer utility falls. Fixing consumer actions, higher prices just transfer surplus dollar by dollar from consumers to producers, but uniform pricing eliminates the inefficient distortion in choice by consumers of ATM provider.

Proposition A1 in the Appendix shows that the finding that the uniform ATM price exceeds the surcharge for nonmembers holds even if travel costs are linear rather than quadratic, so that a consumer located at x who goes to a bank located at d incurs transportation costs T(x - d). With linear transportation costs, a pure-strategy equilibrium exists if [PHI] [greater than or equal to] [mu]/(18 + [mu]). Banning surcharging again leads to (i) higher ATM prices, (ii) higher bank profits, (iii) higher total surplus, and (iv) higher consumer surplus if and only if consumers are sufficiently likely to be travelling when requiring ATM services.

The result that the uniform ATM price exceeds the surcharge for nonmembers is more surprising when travel costs are linear. It is only the fact that a surcharging bank has a smaller base of (nonmember) ATM customer locations that gives rise to the more competitive ATM service pricing. That is, if travel costs are linear, the measure of customer locations that switch ATM choices due to a given reduction in ATM fees does not vary with whether surcharging is possible. However, a surcharging bank serves fewer foreign ATM customer locations than when banks cannot price discriminate according to bank membership, so that the revenue reduction due to the ATM fee reduction is proportionately less.

5. Heterogeneous bank sizes

* This section explores how pricing differs across banks of different sizes. We modify our model in the simplest way. The large bank B has two branches, located at 0 and 2/3, and the small bank S has one branch located at 1/3. The other features of the model remain the same.

We interpret bank size as corresponding to the (local) number or the concentration of ATM branches. Empirically, Hannan et al. (2000) find that surcharging rises with both institution size and share of the market ATMs. While we take the location of bank branches as exogenous, the qualitative predictions do not depend on their particular location. Massoud and Bernhardt (2000) endogenize the choice of ATM location and service pricing in a model featuring a novel spatial structure in which a consumer has an idiosyncratic bank-specific location, so that her spatial location relative to one bank's ATMs is independent of her spatial location relative to the other bank's ATMs. This renders tractable the otherwise impossible problem of solving jointly for a firm's choices of both product locations and pricing.

The setup of the economy mirrors that of the homogeneous bank framework and is relegated to the Appendix. The different bank sizes lead to asymmetries in bank strategies, so the analysis is somewhat more complicated. An equilibrium exists as long as transportation costs, T, and the probability of requiring bank services when at her initial location, [PHI], are sufficiently high ("sufficiently" depends on the economy's parameters--if [mu] is smaller, then T and [PHI] can be smaller). Equilibrium outcomes are given by the solution to a sixth-order polynomial, which we solve numerically. Figures 1-4 graph equilibrium outcomes, illustrating how varying travel costs and the probability of requiring bank services when travelling affect outcomes. The other parameters are set so that the marginal cost of ATM services is [c.sub.a] = $.01; the marginal cost of the in-branch service is [c.sub.b] = $.10; and the probability of requiring the ATM service is [mu] = .8.

[FIGURE 1-4 OMITTED]

[] Qualitative features of the equilibrium.

Feature 1. Both banks charge bank members the marginal cost of the ATM service.

Feature 2. Both banks impose high ATM surcharges on nonmembers (Figure 2) and set high bank account fees (Figure 1), reflecting that they again extract compensation from ATM use by bank members through the bank account fee.

Feature 3. The large bank imposes much higher ATM surcharges and account fees than the small bank (Figure 2).

Feature 4. As the probability [PHI] that a customer obtains services when at her initial location rises, both banks set sharply higher account fees (Figure 1); the large bank's ATM surcharge falls, but the small bank's surcharge rises (Figure 2).

Feature 5. Bank account fees and surcharging at both banks rise sharply as the distance between ATMs grows, as Hannan et al. (2000) document.

Feature 6. Profits of both banks rise with [PHI] and T. The ratio of large- to small-bank profits falls as the probability [PHI] that a customer requires bank services at her initial location rises (Figures 3 and 4).

As when banks were symmetrically situated, banks charge their members the marginal cost of the ATM service to avoid inefficiently distorting their selection of ATM provider because they can earn it back (efficiently) in higher bank fees. So, too, ATM surcharges on nonmembers exceed what they would be were banks to care about profits from ATM provision, but not about the effects of ATM surcharges on the choice of bank affiliation.

The large bank sets higher prices than the small bank because (i) the large bank internalizes the "competition" between its branches for customers who are located between them, very far from the small bank's branch; and (ii) the large bank provides a service more valued than that provided by the small bank--most consumers are located closer to a large bank branch both initially and after a location shock.

The large bank offers a better service to most, but not all, consumers who receive bank service shocks at their initial location--most, but not all, consumers are located closer to a large bank branch. In contrast, when consumers change location, all, in expectation, are closer to a large bank branch. As the probability [PHI] that consumers receive a bank service shock at their initial location rises, so does the ex ante expected heterogeneity in location across consumers, so that price elasticities fall. This leads to higher prices and profits for both banks, but relative bank profits rise faster for the small bank because it can better exploit its initial locational monopoly with those consumers who are initially located closer to it. That is, the small bank's profits are more sensitive to [PHI] than are the large bank's (Figures 3 and 4).

When a consumer is more likely to change location, ATM surcharges become a more important factor in the consumer's choice of affiliation, so that it becomes more important to be affiliated with the larger network. Because relocated consumers are more likely to be near a large bank branch, the large bank's ATM surcharges have a greater marginal effect on affiliation as [PHI] falls. Consequently, as [PHI] falls, the large bank raises its ATM surcharges on nonaffiliated customers to encourage more customers to establish bank accounts with it.

In contrast, the small bank's ATM surcharges fall as [PHI] falls. As its surcharge elasticity of bank account affiliation falls, it seeks instead more nonaffiliated ATM business.

In summary, large banks exploit their larger networks of ATMs by setting higher ATM surcharges for nonmember customers and higher fees to establish bank accounts. Larger banks set higher ATM surcharges to induce consumers to establish accounts with them. The comparative advantage of a large bank rises when customers are more likely to want bank services when they go out shopping. These findings are consistent with PIRG's concerns that "a surcharging is part of the big bank's anti-competitive strategy to squeeze out smaller banks and credit unions by encouraging their customers to switch their accounts to banks with larger ATM networks" (PIRG, p. 9).

6. Conclusion

* This article endogenizes both the pricing of ATM services by banks and bank account choice and ATM use by consumers. Our spatial model reconciles the observed pricing of services by banks: (i) banks set high fixed fees on customers who establish accounts but set minimal charges for their ATM use; (ii) banks impose very high surcharges on ATM use by nonmembers; and (iii) large banks set far higher account fees and ATM surcharges than do small banks.

Because banks extract surplus efficiently from members but not nonmembers, they set high ATM surcharges on nonmembers in order to increase bank account base. Finally, we show that banning ATM surcharging paradoxically leads to higher prices for ATM use, higher bank profits, and possibly reduced consumer welfare.

Hannan et al. (2000) find significant empirical support for the predictions of our model. Although they do not structurally estimate our model, they "draw upon [our] framework to guide [their] empirical specification and interpretation of the coefficient estimates" (p. 5). One issue that Hannan et al. must confront is how to interpret our model within a dynamic context in which customers regularly use ATMs but only occasionally change the bank at which they establish accounts because of the transaction costs involved. (13) They find that surcharging rises sharply with rates of local population inflow, which captures the number of customers looking to establish new accounts, concluding that "banks use the surcharge strategically to provide an incentive for newcomers to establish deposit accounts with the surcharging institution in order to avoid paying surcharges every time they use an ATM belonging to that institution" (p. 20). They also find that surcharging rises with both institution size and share of area ATMs (i.e., within a local metropolitan statistical area) and declines with the number of ATMs per square mile in the market. In short, the data confirm key predictions of our theory.

Appendix

* Proofs of Proposition 2 and Proposition A1 follow.

Proof of Proposition 2. The expected profit for bank E is

(A1) [[PI].sup.E](* | ex)= R + [mu](1 - [phi]) [p.sup.E.sub.a](0 | ex) - [c.sub.a]] 1\2 {s[y.sub.a](0 | ex)}],

where R is the profit from providing the services to members, and

(A2) [y.sub.a](0 | ex) = [p.sup.G.sub.a](1 | ex) - [p.sup.E.sub.a](0 | ex)+T/4 /T

is the location of the customer affiliated with bank G who is indifferent between getting service a from bank E or G.

The first-order condition for profit maximization is

(A3) [mu](1 - [phi])[y.sub.a](0 | ex) [mu](1 - [phi])[[p.sup.E.sub.a](0 | ex) - [c.sub.a]][differential][y.sub.a](0 | ex) / [differential][p.sup.E.sub.a](0 | ex) = 0,

where

[differential][y.sub.a](0 | ex) / [differential][p.sup.E.sub.a](0 | ex) = -1 / T.

Recognizing that the equilibrium features symmetric pricing strategies, [p.sup.E.sub.a](0 | ex) = [p.sup.G.sub.a](0 | ex) = [p.sub.a](0 | ex) and [p.sup.E.sub.a](1 | ex) = [p.sup.G.sub.a](1 | ex) = [c.sub.a], we solve (A3) for [p.sub.a](0 | ex):

(A4) [p.sub.a](0 | ex) = T / 8 + [c.sub.a].

Q.E.D.

Proposition A1. If transportation costs are linear, then, in equilibrium, prohibiting banks from charging members and nonmembers different ATM prices

(i) raises the price for ATM use above the ATM surcharge:

[p.sup.uL.sub.a] = [c.sub.a] + T / 2 > [c.sub.a] + T / 3 = [p.sup.L.sub.a](0).

(ii) Reduces bank account fees:

[F.sup.uL] = (T[phi]) / 2 - 1 / 2 [phi][mu]T < [T[phi] / 2 + T[mu](1 - [phi]) / 18 = [F.sup.L].

(iii) Raises bank profits:

[[PI].sup.uL] = ([phi]T / 4 + T[mu](1 - [phi])) / 4 > (T[phi]) / 4 + T[mu](1 - [phi] / 18 = [[PI].sup.L].

(iv) Reduces consumer surplus if and only if [phi] < 4/7, i.e., if and only if consumers are unlikely to be travelling when requiring ATM services:

C[S.sup.uL] - C[S.sup.L] = -T/3 [mu](1 - 7/4 [phi]).

(v) Raises total consumer plus producer surplus:

[surplus.sup.uL] - [surplus.sup.L] = T[mu]/18 (1 + 7/2 [phi]).

[] Equilibrium solution when transportation costs are linear. We present the analysis for the special case where individuals only use ATM services, i.e., [mu] = 1, and the marginal cost of the ATM service is [C.sub.a] = 0. Accordingly, bank profits are

(A5) [[PI].sup.E] = [F.sup.E]2[x.sup.E] + 2[x.sup.E][p.sup.E.sub.a](1) ([phi] + (1 - [phi])2[y.sub.a](1))+(1 - [phi])(1 - 2[x.sup.E])[p.sup.E.sub.a](0)2[y.sub.a](0),

where

[x.sup.E] - 1 / 4 + [F.sup.G] - [F.sup.E] / 2[phi]T + 1 - [phi] / 4[phi]T ([p.sup.E.sub.a](0) - [p.sup.G.sub.a](0))+ (1 - [phi]) / 4[phi][T.sup.2] ([p.sup.G.sub.a][(0).sup.2]] - [p.sup.E.sub.a][(0).sup.2]]) + 1 + [phi] / 4[phi]T ([p.sup.G.sub.a](1) - [p.sup.E.sub.a](1)) + (1 - [phi] / 4[phi][T.sup.2] ([p.sup.E.sub.a][(1).sup.2] - [p.sup.G.sub.a][(1).sup.2]) + 1 - [phi] / 2[phi][T.sup.2] ([p.sup.E.sub.a](0) [p.sup.G.sub.a](1) - [p.sup.E.sub.a](1) [p.sup.G.sub.a](1)),

[y[.sub.a](0) = 1 / 4 - [p.sup.E.sub.a](0) - [p.sup.G.sub.a](1) / 2T,

and

[y[.sub.a](1) = 1 / 4 - [p.sup.E.sub.a](1) - [p.sup.G.sub.a](0) / 2T,

To simplify presentation, we first assume the result that in equilibrium, banks charge member-customers the marginal cost of the ATM service, and then verify. Then profits become

(A6) [[PI].sup.E] = [F.sup.E]2[x.sup.E] + (1 - [phi]) (1 - 2[x.sup.E]) [p.sup.E.sub.a](0)2[y.sub.a](0)2[y.sub.a](0),

where

[x.sup.E] = 1 / 4 + [F.sup.G] - [F.sup.E / 2[phi]T + 1 - phi / 4[phi]T ([p.sup.E.sub.a](0) - [p.sup.G.sub.a](0)] + [1 - [phi]) / 4[phi][T.sup.2] / [p.sup.G.sub.a][(0).sup.2]] - [p.sup.E.sub.a][(0).sup.2])

and

[y.sub.a](0) = 1 / 4 - [p.sup.E.sub.a](0)] / 2T.

The first-order conditions for profit maximization are

(A7) [differential][[PI].sup.E] / [differential][F.sup.E] = 2[x.sup.E] + 2[F.sup.E][differential][x.sup.E] / [differential][F.sup.E] - 4(1 - [phi])[differential][x.sup.E] / [differential][F.sup.E] [p.sup.E.sub.a](0)[y.sub.a](0) = 0

and

(A8) [differential][[PI].sup.E] / [differential] [p.sup.E.sub.a](0) = 2 [F.sub.E][differential][x.sup.E] / [differential][p.sup.E.sub.a](0) +2(1 - [phi])(1 - 2[x.sup.E]) [y.sub.a](0)+2(1 - [phi]) (1 - 2[x.sup.E] ([p.sup.E.sub.a](0)[differential][y.sub.a](0) / [differential][p.sup.E.sub.a](0) -(4(1 - [phi]) [differential][x.sup.E] / [differential] [p.sup.E.sub.a](0) [p.sup.E.sub.a](0)[y.sub.a](0) = 0.

Substituting

[differential][x.sup.E] / [differential][F.sup.E] = -1 / 2[phi]T, [differential][x.sup.E] / [differential][p.sup.E.sub.a](0) = 1 - [phi] / 4[phi][T.sup.2] (T - 2[p.sup.E.sub.a](0)), [differential][y.sup.a](0) / [differential][[F.sup.E] = 0, and [differential][y.sub.a](0) / [differential][p.sup.E.sub.a](0) = - 1 / 2T

into the above first-order conditions, we get

(A9) 2[x.sup.E] - 2[F.sup.E] / 2[phi]T + 1 - [phi] / T[phi] [p.sup.E.sub.a](0)2[y.sub.a](0 = 0

and

(A10) 2[F.sub.E](1 - [phi] / 4[phi][T.sup.2] (T - 2[p.sup.E.sub.a](0)+2(1 - 2[x.sup.E]) ([y.sub.a](0) - [p.sup.E.sub.a](0) / 2T) - (1 - [[phi].sup.2] / [phi][T.sup.2] [y.sub.a](0)[p.sup.E.sub.a](0) (T -2[p.sup.E.sub.a](0)) = 0.

Recognizing that the equilibrium features symmetric pricing strategies, [F.sup.E] = [F.sup.G] = [F.sup.L], [p.sup.E.sub.a](0) = [p.sup.G.sub.a](0) = [p.sup.L.sub.a](o), yields

[x.sup.E] = 1/4, [y.sub.a](0) = 1/4 - [p.sup.L.sub.a](0) / 2T.

We substitute for [x.sup.E] and [y.sub.a](0) into (A9) to obtain

(A11) 0 = 2 1/4 - [F.sup.L] / [phi]T + 1 + [phi]/T[phi] (p.sup.L.sub.a](0)2(1/4 -[p.sup.L.sub.a](0) / 2T)

(A12)= 1/2 - [F.sup.L] / [phi]T + 1 - [phi] / 2[phi][T.sup.2](T[p.sup.L.sub.a](0) - 2[p.sup.L.sub.a][(0).sup.2])

Solving for [F.sup.L] yields

(A13) [F.sup.L] = [phi]T / 2 + (1 - [phi]) / 2T(T[p.sup.L.sub.a](0) - 2[p.sup.L.sub.a][(0).sup.2]).

Similarly, substituting for [x.sup.E] and [y.sub.a] (0) into (A10) yields

(A14) [F.sup.L](1 - [phi]) / 2[phi][T.sup.2] (T - 2[p.sup.L.sub.a](0)) + (1 - [phi]) / 4T (T - 4[p.sup.L.sub.a](0)) + (1 - [[phi].sup.2] / 4[phi][T.sup.3] (T - 2[p.sup.L.sub.a](0)) (T[p.sup.L.sub.a](0) - 2[p.sup.L.sub.a][(0).sup.2]) = 0.

Substituting (A13) for [F.sup.L] into (A14) and solving for [p.sup.L.sub.a](0) yields the following equilibrium outcome:

(A15) [p.sup.L.sub.a](0) = T / 3.

Substituting for [p.sup.L.sub.a](0) = T/3 into (A13) and simplifying yields

(A16) [F.sup.L] = ([phi]T / 2 + T(1 - [phi]) / 18.

To see indeed that banks in fact marginal-cost price the ATM fee for members, derive the first-order condition for [p.sup.E.sub.a](0):

(A17) [differential][[PI].sup.E] / [differential] [p.sup.E.sub.a](1) = 2[F.sup.E] [differential][x.sup.E] / [differential][p.sup.E.sub.a](1) - 4(1 - [phi]) [differential][x.sup.E] / [differential][p.sup.E.sub.a](1) [p.sub.E.sub.a](0)[y.sub.a](0) + 2([phi]+ (1 - [phi]) 2[y.sub.a](1))[x.sup.E]

+ 2[p.sup.E.sub.a](1)[phi]) + (1 - [phi])2[y.sub.a](1)) [differential][x.sup.E] / [differential][p.sup.E.sub.a](1) + 2[p.sup.E.sub.a](1)(1 - [phi])2[differential][y.sub.a](1) / [differential][p.sup.E.sub.a](1) x.sup.E

= 0.

Substituting for the equilibrium values of [p.sup.L.sub.a](0) = T/3 and [F.sup.L] = [phi]T/2 + T(1 - [phi])/18, the above first-order condition simplifies to

(A18) [differential][[PI].sup.E]/[differential][P.sup.E.sub.a](1) = 2[phi] - [[phi].sup.2] - 1 / [phi][T.sup.3] [p.sup.L.sub.a][(1).sup.3] + -[phi] - [[phi].sup.2] / [phi][T.sup.2] [p.sup.L.sub.a][(1).sup.2] + -12[phi] + 5[[phi].sup.2] - 5 / 12[phi]T [p.sup.L.sub.a](1) = 0.

Solving the above first-order condition for [p.sup.L.sub.a](1) yields

(A19) [p.sup.L.sub.a](1) [member of] { 0 T(2[phi] + 4 [square root of 19[[phi].sup.2] - 20[phi] + 1 / 6(1 - [phi], T(2[phi] + 4 + [square root of 20[phi] + 1 / 6(1 - [phi]}.

The two latter roots are real only for [phi] very small; for those small values of [phi], were we to substitute these values of [p.sup.L.sub.a](1) into the first-order conditions for [F.sup.L] and [p.sup.L.sub.a](0), real solutions would not obtain. Hence, the unique equilibrium solution is [p.sup.L.sub.a](1) = 0. Q.E.D.

[] Equilibrium solution of the heterogeneous bank model.

Stage 3. Let [p.sup.j.sub.a]([[delta].sub.j]) denote the ATM price set by the bank of size j, j = S, B. The bank B member who is indifferent between using the large bank's ATM and the small bank's is located at

(A20) [y.sub.a](1) = 3[[p.sup.S.sub.a](0) - [p.sup.B.sub.a](1) + T/3 / 2T.

The bank S member who is indifferent between using each bank's ATM is located at

(A21) [y.sub.a](0) = 3[[p.sup.S.sub.a](1) - [p.sup.B.sub.a](0) + T/3 / 2T.

Stage 2. A large bank member located at x expects at the next stage to receive utility

(A22) E[[u.sup.B] (y, s, [P.sup.B], [P.sup.S] | x] = [phi][M - [mu][p.sup.B.sub.a](1) - (1 - [mu]) [p.sup.B.sub.a] - T[x.sup.2]]+ (1 - [phi])[K.sub.B],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If she instead sets up an account at bank S, she expects to receive utility

(A23) E[[u.sup.S](y, s, [p.sup.B], [p.sup.S]) [ x] = [phi] [M - [mu][p.sup.S.sub.a](1) - (1 - [mu])[p.sup.S.sub.b] - T [(1/3 - x).sup.2] + (1 - [phi])[K.sub.s],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let 1/3 - [x.sup.B] denote the distance from the small bank of a consumer who is indifferent between establishing a bank account at S and B:

(A24) [x.sup.B] = 3/2T {[F.sup.2] - [F.sup.B] / [phi] + (1 - [mu]) [p.sup.S.sub.b] - [p.sup.B.sub.b] + [mu][[p.sup.S.sub.a](1) - [p.sup.B.sub.a](1)] + T/9 + (1 - [phi]k / [phi]},

where k -- KB -- Ks. The number of customers with bank accounts at bank j is

(A25) [N.sup.B]= 2([x.sup.B] + 1/6), [N.sup.S] = 2(1/3 - [x.sup.B]).

Stage 1. The large bank's expected profit is

(A26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Analogously, the small bank's expected profit is

(A27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If an equilibrium exists, then it is characterized by the solution to the first-order conditions for the choice variables in (A26) and (A27). Q.E.D.

(1) The fourth U.S. Public Interest Research Group (PIRG) national survey of ATM surcharging rates.

(2) A large bank is one of the 300 largest in the country.

(3) Litan (1999, p. 3), citing the latest data from the Federal Reserve Board. In 1997, institutions with under $150 million in assets lost an average $.50 per ATM transaction, while institutions with more than $150 million in assets averaged a $.20 loss per ATM transaction. These statistics appear to ignore the opportunity benefit of ATM use for banks--reduced teller demand--something that this article also does not consider.

(4) Figures from the Consumer Bankers Association indicate that new ATMs cost at least $7,000 and have monthly operating costs of $700-$1,200.

(5) In the fall of 1999, voters in both Santa Monica and San Francisco backed initiatives that would ban surcharges.

(6) PIRG figures indicate that large banks set higher account fees--large banks charge an average of $234.87 for a regular checking account, versus $202.79 for smaller banks and $111.59 for credit unions.

(7) Consumer expenditures including travel costs may be lower for all consumers when firms can price discriminate than when firms must set a uniform price.

(8) Our qualitative findings extend even if [mu] = 1, so that consumers never require in-branch services.

(9) See Laffont, Rey, and Tirole (1998a) for a related result.

(10) The interested reader can peruse the explicit derivation of equilibrium outcomes when transportation costs are linear in the Appendix.

(11) When a customer uses another bank's ATM, two fees are charged: a switch fee, which ranges from $.02 to $. 15 per transaction to cover the costs of deploying and servicing the ATM, and an interchange fee, which ranges from $.30 to $.60 per transaction, paid to the ATM owner. Both fees are paid by the cardholder bank. The marginal cost of providing the ATM service to nonmembers is essentially zero. The other costs of ATM provision are largely fixed or operating in nature (including the regular costs of servicing the machines, etc.).

(12) See Coyte and Lindsey (1988) for a related result for duopolists in a spatial context.

(13) Kiser (2000) finds that households rarely switch banks, generally switching only if they move from one market to another.

References

ANDERSON, S.P. AND LERUTH, L. "Why Firms May Prefer Not to Price Discriminate via Mixed Bundling." International Journal of Industrial Organization, Vol. 11 (1993), pp. 49-61.

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Nadia Massoud *

Dan Bernhardt **

* University of Alberta; nadia.massoud@ualberta.ca.

** University of Illinois; danber@uiuc.edu.

The suggestions by the Editor, Joseph Harrington, and two anonymous referees have greatly improved the article.