Endogenous Credit-Card Acceptance in a Model of Precautionary Demand for Money

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1 Endogenous Credit-Card Acceptance in a Model of Precautionary Deand for Money Adrian Masters University of Essex and SUNY Albany Luis Raúl Rodríguez-Reyes University of Essex March 24

2 Abstract A credit-card acceptance decision by retailers is ebedded into a siple odel of precautionary deand for oney. The odel gives a new explanation for how the use of credit-cards can differ so widely across countries. Retailers propensity to accept cards reduces the need for buyers to hold cash as the chance of a stock-out (of cash) is reduced. When retailers ake their decision with respect to credit-card acceptance they do not take into account the effect that decision has on other sellers. This externality generates ultiple equilibria over soe portions of the paraeter space.

3 1 Introduction Therearelargedifferences in the propensity to use credit-cards across countries (see Table 1). Most notably, the Japanese carry as any cards as the Aericans do but they use the far less frequently. The third colun also shows that the hardware needed to use credit-cards, Electronic Funds Transfer at Point of Sale (EFTPOS), are uch rarer in Japan than in other developed econoies. This paper considers these facts in an equilibriu odel of precautionary deand for oney and endogenous credit-card acceptance by retailers. Nuber of Nuber of Nuber of EFTPOS credit-cards Transactions per 1 per capita per capita Inhabitants Canada Japan United Kingdo United States Table 1 Country Indicators on credit-card Transactions for Casual observation ight suggest that the difference between the Japanese and Aerican outcoes could be a consequence of the low rates of personal crie and near zero noinal rates of interest in Japan. Indeed, the paper s results are consistent with this belief. However, the odel provides an additional explanation. In their choice of whether or not to accept credit-cards, individual retailers do not take account of the ipact of their decision on 1 Data are fro Statistics on payent systes in the Group of Ten countries, Bank for International Settleents, March 21. 1

4 other retailers. We show that this externality can generate ultiple equilibria. We do not incorporate any technological (network) basis for this externality. Instead, it operates entirely through the arket. 2 The basic idea is that when there is an opportunity cost of holding cash and purchasing opportunities are stochastic, a buyer ay have insufficient funds at her disposal to buy what she wants. Anticipating this possibility leads to a precautionary deand for oney. To the extent that retailers accept credit-cards, the chance of such a stock-out of funds is diinished. The ore retailers accept cards, the less buyers are inclined to carry cash which in turn increases the incentives for other retailers to accept cards. Although their results are soewhat obfuscated by the proliferation of ethods of payent, ore rigorous statistical analysis of card use across countries by Huphreys et al [1996] indicates that country specific duies and past usage of particular eans of payent are very iportant in explaining current usage. These results are at least consistent with the possibility of ultiple equilibria. Still, the ain point of this paper is to show that such ultiplicity can eerge through spillovers that operate in the arket rather then being technological. While soe other theoretical work has been done on card versus cash use (see Rochet and Tirole [22], Chakravorti and To [1999] and Markose and Loke [23]) none of the explore the source of ultiplicity of equilibriu ephasized in this paper. One feature of real payents systes that confuses the applicability of our odel is the issue as to where the odel stands with respect to debitcards. Are they cash or cards? As the facility to accept credit-cards is essentially the sae as that required for debit-cards we view the both as cards. As we have a static odel, how the buyer eets a payent ade on the card is oot. What atters is that a retailer s propensity to accept 2 See McAndrews and Rob [1996] for a discussion of network externalities in the context of electronic transaction systes. 2

5 any card depends on how uch cash shoppers carry which in turn depends on the other retailers propensity to accept cards. We use the ter creditcard because they have historically been ore iportant than debit-cards for larger purchases (for which precautionary oney holdings see relevant). In the interest of expositional clarity, Section 2 describes a preliinary versionoftheodelinwhichbuyersandsellersareex ante hoogeneous. This is the siplest environent that is able to deonstrate the precautionary deand for oney and the ultiplicity of equilibriu described above. These equilibria, however, involve either exclusively cash transactions or exclusively card transactions. As such this first odel is too stylized to provide a eaningful interpretation of Table 1. The odel of Section 3 addresses this issue by allowing for different types of retailer distinguished by the average expense of the goods in which they trade. It is shown that the results of section 2 are robust this extension and that the use of both cash and cards in realized transactions is possible in equilibriu. A further iplication of this analysis is that retailers of ore expensive ites are ore likely to accept cards. 2 The Model with Hoogeneous Buyers and Sellers The econoy coprises of a continuu of individuals divided into buyers and sellers. The ass of buyers is noralized to 1 and the ass of sellers is N. The econoy lasts for one period. For the purpose of exchange, buyers and sellers are randoly assigned to each other so that the expected nuber of buyers who enter a particular seller s establishent is η =1/N. 3 3 There is a large literature which uses the search and atching fraework to address onetary issues (e.g. Kiyotaki and Wright [1991]). A dynaical version of our baseline odel is relatively straightforward to forulate. For the purpose of the current paper, 3

6 Sellers have the ability to produce any size x of a non-storable and indivisible coodity at zero cost. They cannot consue their own output, but derive utility q, if they acquire q dollars fro a buyer. At the start of the period they have to decide whether or not they will accept credit-cards. To be able to accept cards they have to incur the one-tie installation cost, Θ. Buyers have atch specific preferences. In particular, consuption of a good of size x gives utility, ( u (s) for x s U (x; s) = for x<s, where s F (s). The value of s is drawn after the buyer eets a seller. The distribution function F (.) is continuous with support [, s] and density f(.). Thefunction u(.) is increasing, concave and u() =. We will refer to the realization of s as the buyer s preferred size of purchase. Buyers are also endowed with a credit-card which is costless to carry so they keep it with the at all ties. Credit-cards have no spending liit. This fraework is eant to capture the notion that when people go shopping they do not know for sure what they are going to buy and that goods ay not be divisible. We want to odel a precautionary deand for oney so we have to allow for the possibility of buyers having insufficient cash to buy their preferred good. The easiest way to do this is to assue that retailers choose whether or not to accept credit-cards and the price at which coodities trade is exogenous to the environent. 4 Specifically, the price is noralized to 1 so that an ite of size x costs x dollars. however, all we need fro the search environent is anonyity - a one period rando assignent odel is sufficient. 4 Pricing in this environent could be the outcoe of a price-posting gae played by sellers (as in Green and Zhou [1998] or Jafarey and Masters [23]). The result of this gae would be a schedule that gives the price as a function of x. Analysis of such a odel adds unnecessary coplications and we would still have to assue that sellers had soe way of coitting to their pricing functions. 4

7 At the beginning of the period, the buyer receives a noinal incoe Y where Y s, andshedecidestheaount,, she wants to keep in cash. The reainder is allocated into a non-liquid investent that yields r utils for every dollar invested. More generally, we think of r as capturing the opportunity cost of carrying cash. This can include the possibility of theft, loss or daage. Under our assuptions, at any buyer-seller eeting a transaction will take place if either the buyer holds at least s dollars or the seller accepts creditcards. Leftover cash balances have no value but we assue that buyers spend the least aount of cash possible in order to purchase their preferred size of good. This assuption could be justified by allowing for an infinitesial value to holding unspent oney The Buyer s Proble In their decision over, buyers take into account the probability that the seller they eet accepts credit-cards. Because atching is rando, this probability is equal to the average propensity with which sellers accept creditcards, Φ [, 1]. The expected utility of being a buyer with cash holdings, given Φ, is V b (, Φ) =r (Y )+(1 Φ) Z and, the buyer s proble is to solve for u (s) df (s)+φ Z s u (s) df (s), (1) (Φ) arg axv b (, Φ). (2) The necessary condition for achieving an interior solution is r +(1 Φ) u () f () =, 5 Of course in a dynaical version of this odel, the value to holding oney at the end of the period would siply reflect its future purchasing power. 5

8 the sufficient condition for achieving an interior solution is given by u () f ()+u () f () <. To siplify the analysis we will assue differentiability of f(.) and that u () f () r > u ( s) f ( s) r < (3) f () < u () for all f () u () The first two conditions siply provide upper and lower bounds on the range of possible values of r. The firstipliesthatforφ close enough to there is an interior solution. The second condition iplies that the precautionary otive for cash holding will be active even when Φ =(i.e. () < s). The last restriction is equivalent to iposing concavity on the axiand in equation (1). It ensures that the range of perissible values of r is non-epty and that the solution is unique. Essentially, we require that high values of s are sufficiently iprobable that at soe point, even if no sellers accept cards, carrying ore cash is not worth its opportunity cost in savings. Under these assuptions (Φ) is strictly positive whenever (1 Φ) u () f () > r; otherwise (Φ) =. Straightforward analysis iplies that at an interior solution, d dφ = u () f () (1 Φ)[u () f ()+u () f ()] <. That is, buyers deand for oney reacts negatively to changes in the probability of credit-card acceptance. As the nuber of sellers accepting cards increases, the probability that the buyer will not be able to acquire his preferred good falls. This lowers the return to holding cash. Notice that when credit-cards are always accepted by sellers (Φ =1)buyers have no reason to hold oney, (1) =. 6

9 Also, d dr = 1 (1 Φ)[u () f ()+u () f ()] <. As should be expected, the deand for oney varies in the opposite way to the opportunity cost of holding cash. 2.2 The Seller s Proble Let φ denote the propensity with which an individual seller decides to have the card-reading equipent installed. In general, sellers each choose a value of φ [, 1] taking the distribution of other sellers choices as given. 6 The value of φ represents a randoization the outcoe of which is realized prior to atching with buyers. That is, buyers will only eet sellers that either accept cards or not. After the realization of the randoization and the installation (or not) of the equipent, a seller s original choice of φ is irrelevant. Card acceptance cannot be ade contingent upon the oney holdings of any buyers that show up. This sees reasonable, buyers with insufficient cash are unlikely to wait around while a seller gets hooked up to a credit-card network. In their choice of φ, sellers take as given,, theaountofoneybeing carried by the buyers. In principle, Φ, the propensity with which other buyers accept credit-cards could directly influence an individual seller s choice of φ through a network externality (e.g. Θ could depend on Φ). However, here Φ, will only affect the choice of φ indirectly through its ipact on. Let V s (φ, ) represent the value to being a seller who decides with probability φ to install the card reading equipent given all buyers carry units of oney. Then, Z Z s V s (φ, ) =η (1 φ) sdf (s)+φ η sdf (s) Θ, 6 As we seek a syetric equilibriu, we allow Φ to suarize the (degenerate) distribution of other sellers values of φ. 7

10 Sellers solve for Ṽ s () ax V s (φ, ). φ [,1] This partial analysis of the odel fro the sellers perspective reveals that φ is weekly decreasing in. That is, if we define c such that then 2.3 Equilibriu Z s sdf (s) Z c sdf (s) = Θ η > c φ = = c φ [, 1] < c φ =1 Definition 1 A Market Equilibriu is a pair (,φ ) that solves 1. =argaxv b (, φ ), and, 2. φ =argaxv s (φ, ). φ Three types of equilibriu are possible: pure onetary, pure credit and ixed. In a pure onetary equilibriu, sellers do not accept credit-cards, φ = and = (). It exists if accepting oney (weakly) doinates credit-card acceptance when no other sellers accept cards. That is whenever Θ exceeds the critical value Θ such that Θ η Z s sdf (s) η Z () sdf (s) (4) A pure credit equilibriu has the for φ =1, =and exists whenever accepting cards (weakly) doinates not accepting the when all other sellers are accepting cards. That is, Θ is less than Θ c where Θ c η Z s 8 sdf (s) (5)

11 In any ixed strategy equilibriu sellers are ex ante indifferent between accepting cards and not accepting the. Each seller is randoly chosen, with probability φ (, 1), to accept cards and = (φ ) such that η Z s sdf (s) η Z (φ ) sdf (s) =Θ where (.) is definedinequation(2). As Θ < Θ c the regions of existence of the pure credit and the pure onetary equilibria overlap. Moreover, as (φ) (), ixed strategy equilibria only exist for values of Θ between Θ and Θ c. This deonstrates, in the context of this highly stylized environent, the principal result of the paper. That despite buyers all being endowed with credit-cards, the propensity of their use can vary widely across siilar econoies. Multiplicity of equilibriu occurs because of the inability of sellers to coordinate on a particular adoption strategy. If no seller adopts the credit-card syste, buyers carry a lot of cash and it is in no single seller s interest to deviate toward credit-card acceptance. On the other hand if all other sellers accept cards, buyers carry no cash and credit-card adoption is a doinant strategy. While the ixed strategy equilibriu is ore consistent with real econoies in that both cash and cards are used in realized transactions, it is unstable under heuristic dynaics 7 and has pathological coparative statics (increases in Θ lead to ore credit-card acceptance). The next section therefore provides an adaptation of this basic odel in which the use of both cards and cash is observed in a pure strategy equilibriu. 7 This is a static odel, the heuristic dynaics referred to are as follows. Suppose for soe reason soe infinitesially sall but strictly positive subset of buyers is expected to bring too uch oney shopping with the. Then, sellers would no longer be indifferent between credit-card acceptance and rejection and the equilibriu would breakdown. The other two equilibria are robust to such considerations. 9

12 3 Two Types of Seller Here we relax the assuption of ex ante hoogeneity aong sellers. Specifically, they now differ according to the expected expense of the ites buyers want fro the. Buyers who are assigned to a sall seller draw s fro F s, buyers who are assigned to a large seller draw s fro F l. We assue F l first-order stochastically doinates F s. We also assue that restrictions (3) apply for each distribution. A proportion λ of sellers are sall. 3.1 The Buyer s Proble Buyers have to decide how uch of Y they want to keep in cash bearing in ind the propensity, Φ i, for a type i seller to accept credit-cards, i = s, l. Let ˆV b (, Φ s, Φ l ) represent the expected utility fro holding units of oney given Φ s, and Φ l,then Z Z ss ˆV b (, Φ s, Φ l ) = r (Y )+λ u (s) df s (s)+φ s u (s) df s (s) Z Z sl +(1 λ) u (s) df l (s)+φ l u (s) df l (s). (6) where s i is the supreu of the support of F i,i= s, l. The buyer s proble is to solve for (Φ s, Φ l ) arg ax ˆV b (, Φ s, Φ l ). (7) The necessary condition for achieving an interior solution, < s l, is λ (1 Φ s ) u () f s ()+(1 λ)(1 Φ l ) u () f l () =r. It should be clear that conditions (3) iply existence and uniqueness of an interior solution. 8 8 It is possible that can exceed s s. In this range, we take f s (s) =. 1

13 3.2 The Seller s Proble A aintained assuption is that Sellers utility is linear in the size of object they sell. We use φ i,i= s, l to represent the probability with which an individual seller has card reading equipent installed. As in the previous odel, sellers individually choose φ i considering the cost of accepting credit-cards, Θ, and the buyer s oney holdings. Again, because there is no iposed network externality, other sellers propensity to accept creditcards will only enter the private seller s decision proble through the oney holding of buyers. Atypei seller s expected utility is V si (φ i,) where, V si (φ i,) = η (1 φ i ) +φ i η Z Z si sdf s (s) sdf i (s) Θ. Again, for any given value of Θ, this iplies the existence of ci where η Z s ci sdf i (s) =Θ so that > ci φ i = = ci φ i [, 1] < ci φ i =1 Stochastic doinance of F l over F s eans that cs cl. 3.3 Equilibriu Definition 2 A Market Equilibriu, with two types of fir is a triple, (,φ s,φ l ), such that 1. =argaxˆv b (, φ s,φ l ), 11

14 2. φ i =argaxv si (φ i, ) for i = s, l φ i [,1] We will focus the discussion on pure strategy equilibria of which there are, potentially, four - each seller type can either accept or reject credit-cards. That is, pure strategy equilibria can be suarized by (φ s,φ l ) {, 1} {, 1}. We will posit an equilibriu and look for the range of acceptance of each equilibriu in Θ space given the other paraeters and the functional fors of u, and F i,i= s, l. First, notice that cs cl eans an equilibriu of type (1,), in which sall sellers accept cards and large ones do not, cannot exist. This leaves three possibilities for pure strategy equilibria: (1) Pure credit equilibriu, type (1,1) Fro (7) (1, 1) =, so (φ s,φ l )=(1, 1) is an equilibriu whenever it is individually rational accept cards given no one is carrying any oney. As F l stochastically doinates F s, this is true for all Θ Θ 1 where Θ 1 η Z ss sdf s (s). (2) Pure onetary equilibriu, type (,) Fro (7) (, ) ˆ where λu (ˆ) f s (ˆ)+(1 λ) u (ˆ) f l (ˆ) =r. As type l sellers have the ost to gain fro credit-card adoption their decision to switch out of accepting oney is critical to the existence of this equilibriu. In particular, for a type l seller to confor to this equilibriu we need Θ Θ 2 where Z sl Θ 2 η sdf l (s) ˆ (3) Hybrid Equilibriu, type (,1) Fro (7) (, 1) 1 where λu ( 1 ) f s ( 1 )=r. 12

15 Existence requires that when buyers are carrying 1 units of oney, Θ is sufficiently large that sall sellers do not choose to accept cards but is also sufficiently sall that large sellers do choose to accept cards. That is, this equilibriu exists whenever Θ is in the range [Θ 3, Θ 4 ] where Z ss Θ 3 η sdf s (s), 1 Θ 4 η Z sl sdf l (s). 1 Stochastic doinance iplies 9 Z sl sdf l (s) Z ss sdf s (s) for all This eans that Θ 3 Θ 4. Each of the equilibria therefore exists for soe range of Θ. Fro (3) it follows that ˆ 1 so Θ 3 Θ 1 and Θ 4 Θ 2. This eans that: at least one equilibriu exists for each value of Θ; the pure-credit and the hybrid equilibria coexist for values of Θ between Θ 3 and in{θ 1, Θ 4 }; the pure-onetary equilibriu and the hybrid equilibriu coexist between ax{θ 2, Θ 3 } and Θ 4. If Θ 1 < Θ 2 (which happens if the seller types are sufficiently different), between Θ 1 and Θ 2 the hybrid equilibriu is the unique pure-strategy equilibriu. Otherwise the hybrid equilibriu coexists with at least one other equilibriu and all three equilibria coexist between ax{θ 2, Θ 3 } and in{θ 1, Θ 4 }. The upshot fro this section is that by allowing for ultiple types of seller, pure strategy (i.e. dynaically stable) equilibria in which buyers anticipate aking purchases with either cash or credit-cards can be supported. 9 Integration by parts yields Z si Z si sdf i (s) = s i F i () F i (s)ds The result follows as s l s s and F l () F s (). 13

16 These equilibria always exist over a positive portion of the paraeter space and involve larger sellers accepting cards while saller ones do not. Equilibria in which larger sellers only accept cash while saller ones accept cards are ruled out. 4 Concluding Rearks We have developed a odel that gives a new explanation for how the use of credit-cards can differ widely across countries. Stochastic purchasing opportunities lead to a precautionary deand for holding oney. Holding credit-cards eans that, to the extent sellers accept the, buyers can avoid stocking-out of funds to eet their purchases. Buyers propensity to hold cash is decreasing the probability that any seller accepts cards. This generates an externality between sellers because the incentive to accept cards increases as the aount of cash held by buyers falls. The siplest environent exhibits two extree pure-strategy equilibria in which either cards are never used or cash is never carried. As these outcoes are clearly counterfactual the environent was extended to include two-types of seller. In this odel, it is shown that pure-strategy equilibria exist in which all buyers hold oney even though soe transactions are carried out using cards. In such equilibria, sellers of large ites accept cards while sellers of sall ites require cash. As such, this extended environent also provides a testable epirical prediction: that, all else equal, retailers of ore expensive ites (such as furniture) are ore likely to accept credit-cards than retailers of cheaper ites (such as newspapers). In order to focus on the above results the odel was necessarily abstract. One direction to extend this analysis is the incorporation of a strategic role for the credit-card issuer. A ore coplete odel should also allow for price effects. For instance, here we assue that the seller bears all the cost of the 14

17 credit-card syste. In realty, it is likely that sellers are able to pass soe of the cost of the syste on to buyer through increased prices. This could be true even if sellers are banned fro explicitly charging different prices for goods bought with cards. Such explorations are left for future work. References Chakravorti, S. and T. To, [1999] Toward a Theory of Merchant Creditcard Acceptance, Federal Reserve Bank of Chicago, Working Papers Series, WP Green, E. and R. Zhou [1998] A rudientary rando-atching odel with divisible oney and prices Journal of Econoic Theory 81: Huphrey, D., L. Pulley, and J. Vesala, [1996], Cash, Paper and Electronic Payents: A Cross-Country Analysis, JournalofMoney,Creditand Banking, 28: Jafarey, S. and A. Masters [23] Output, Prices and the Velocity of Money in Search Equilibriu Journal of Money Credit and Banking, 35: Kiyotaki, N. and Wright, R. [1991] A Contribution to the Pure Theory of Money, Journal of Econoic Theory 53: Markose, S. and Y. Loke [23] Network effects on cash-card substitution transactions and low interest rate regies, The Econoic Journal, 113: McAndrews, J. and R. Rob [1996] Shared ownership and pricing in a network switch International Journal of Industrial Organization, 14:

18 Rochet, J.-J. and J. Tirole, [22] Cooperation aong copetitors: soe econoics of payent card associations, Rand Journal of Econoics, 33:

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