Optimal Allocation of Social Cost for Electronic Payment System: A Ramsey Approach
 Author: Huang Pidong, Kim Young Sik, Lee Manjong
 Publish: Seoul Journal of Economics Volume 28, Issue1, p31~52, Feb 2015

ABSTRACT
Using a standard Ramsey approach, we examine the optimal allocation of social cost for an electronic payment system in the context of a dynamic general equilibrium model. The benevolent government provides electronic payment services and allocates relevant social cost through taxation on the beneficiaries’ labor and consumption. A higher tax rate on labor yields the following desirable allocations. First, it implies a lower welfare loss because of the distortionary consumption taxation. It also enhances the economy of scale in the use of electronic payment technology, reducing per transaction cost of electronic payment. Finally, it saves the cost of withdrawing and carrying around cash by reducing the frequency of cash trades. All these channels together imply optimality of the unity tax rate on labor.

KEYWORD
Cash , Electronic payment cost , Ramsey problem

I. Introduction
An electronic payment system requires recordkeeping and informationprocessing technologies for transaction clearing and verification purposes. However, from the society’s point of view, the installment and operation of such technologies incur some resource cost. Hayashi and Keeton (2012) refer to this resource cost as the “social cost” of using electronic payment methods and distinguish it from “private cost” that includes the fees imposed by one party to another. This social cost has the distinct feature of being composed largely of fixed costs that do not rely on the frequency or value of the transactions. Once the electronic recordkeeping and informationprocessing devices are installed at a substantial cost, electronic payment services could be provided with trivial marginal cost.
Considering the high social cost incurred in the installment of electronic payment technology, it is quite natural to ask how the social cost should be allocated. However, to the best of our knowledge, no study has thoroughly addressed this question. Most recent studies on electronic payment systems have focused on issues related to private cost (
e.g. , Prageret al . 2009; Rochet and Wright 2010; Verdier 2011; Lee 2014) or on issues related to the competition among payment instruments (e.g. , Li 2011; Kim and Lee 2012; Lotz and Vasseli 2013; Telyukova 2013).This paper examines an optimal allocation of the social cost for an electronic payment system by incorporating a standard Ramsey approach into a dynamic general equilibrium model where money is essential. Specifically, we adopt a standard searchtheoretic monetary model augmented with distribution of wealth. We consider wealth heterogeneity across agents to capture economy of scale in the use of an electronic payment system. At the beginning of a period, each agent becomes a buyer, a seller, or a nontrader with some probability. Each buyer is then relocated randomly to match with a seller. After the realization of such relocation shock, a buyer determines whether to carry cash or to open a checking account for subsequent pairwise trade. Carrying cash incurs disutility cost because of the inconvenience and the risk of loss. A buyer can also pay in cash by withdrawing the amount from her checking account where, for simplification, cashwithdrawing cost is assumed to be the same as its carrying cost. A buyer opening a checking account can also use an electronic payment system by shouldering its fixed installment cost.
As agents cannot commit to their future actions and trading histories are private, all trades in a pairwise meeting are
quid pro quo . Hence, either cash or checkingaccount deposit should be transferred in exchange for goods produced where the terms of trade are determined by a buyer’s takeitorleaveit offer. Electronic transactions based on checking accounts require the installment of an electronic payment system, which incurs a fixed cost to its service provider (hereinafter referred to as “the government”). The government collects the fixed cost from people who transfer and receive money via an electronic payment system. Notably, the per transaction cost of electronic payment declines as more transactions are made via the system. This captures the economy of scale in the use of electronic payment technology. As a key source of the efficiency of electronic payment, this economy of scale is determined by the choice of means of payment of heterogeneous agents with different monetary wealth.We use the model to examine an optimal allocation of the social cost for an electronic payment system. Our Ramsey problem is a standard one; to maximize social welfare, the government chooses a policy on the allocation of the social cost for an electronic payment system across buyers and sellers who use the system in pairwise trades. Under the assumed buyertakeall trading protocol, this problem is transformed into that of choosing a tax scheme on the beneficiaries (
i.e. , buyers) of electronic transactions in the form of taxation on labor or consumption. As the endogeneity of a nondegenerate wealth distribution rules out closedform solutions, the Ramsey problem is solved numerically. The main results are as follows.The choice of taxation on a buyer’s labor or consumption not only has an intensivemargin effect on the terms of pairwise trade for a given cost per electronic transaction but also an extensivemargin effect on the choice of means of payment. The welfare implications of these two effects depend crucially on a nondegenerate distribution of wealth and economy of scale in the use of an electronic payment system.
With regard to the intensivemargin effect, labor taxation turns out to have a lumpsum feature in the sense that it does not affect quantity consumed in exchange for money transferred electronically, whereas consumption taxation turns out to be distortionary in the sense that it affects quantity consumed in electronic transactions. Hence, labor taxation favors relatively poor agents whose consumption is sufficiently small, whereas consumption taxation favors relatively rich agents whose consumption is sufficiently large. As the tax rate on labor increases (or the tax rate on consumption decreases), the difference between the average quantity of good consumed by buyers and that produced by sellers decreases.
Labor taxation also has an extensivemargin effect on the choice between electronic and cash payment, which then affects the cost per electronic transaction via an economyofscale channel. In particular, the frequency of electronic transactions is shown to increase with the tax rate on labor. When the tax rate on labor is zero for a given cost per electronic transaction, relatively poor buyers with sufficiently high marginal utility of consumption are willing to pay in cash rather than use electronic payment even if the cashwithdrawing cost exceeds the tax burden associated with the use of electronic payment. However, as the government increases the tax rate on labor, a greater number of poor buyers prefer electronic payment to cash payment, yielding economy of scale in the use of an electronic payment system. This economy of scale decreases per transaction cost of electronic payment, consequently enhancing both intensivemargin and extensivemargin effects of labor taxation.
These results imply that overall transaction cost, including welfare loss because of resource cost for an electronic payment system (which is raised in the form of labor taxation and distortionary consumption taxation) and cashwithdrawing cost, decreases with the tax rate on labor. As a result, the unity tax rate on labor (or zero tax rate on consumption) yields an optimal allocation of the social cost for an electronic payment system.
The paper is organized as follows. Section II describes the model economy. Section III defines a symmetric stationary equilibrium. Section IV formulates the Ramsey problem for the benevolent government and examines an optimal allocation of the social cost for an electronic payment system. Section V summarizes the paper and provides several concluding remarks.
II. Model
The background environment is in line with a standard random matching model of money (
e.g. , Shi 1995; Trejos and Wright 1995) augmented with distribution of money holdings such as Camera and Corbae (1999), Zhu (2003), Berentsenet al . (2005), and Molico (2006). A nondegenerate distribution of monetary wealth is considered to capture economy of scale in the use of an electronic payment system.Time is discrete. There are
N ≥3 number of “islands” withN types of divisible and perishable goods. In each island, there is a [0, 1] continuum of infinitely lived agents. We refer to an agent whose homeisland isn ∈ {1, 2, ...,N } as a typen agent. A typen agent potentially produces only goodn , which incurs a disutility cost ofc (q )＝q for producing q units of good, whereas a typen agent potentially consumes only good (n ＋1) (moduloN ), which gives a utility ofu (q ) for consumingq units of a good whereu” <0<u’ ,u (0)＝0,u’ (0)＝∞, andu’ (∞)＝0. Each agent maximizes expected discounted utility with a discount factorβ ∈(0, 1).Three exogenous nominal quantities describe the stock of money: upper bound on individual money holdings, size of the smallest unit of money, and average money holdings per type of agent. We normalize the smallest unit to be unity so that the set of possible individual money holdings consists of integer numbers, namely, M＝{0, 1, 2, ...,
M } whereM ＞0 denotes the upper bound required for compactness. We denote the average money holdings per type of agents by ＞0.At the beginning of each period, each island is randomly connected to another island, which can be interpreted as a preference shock in the sense that an agent becomes a buyer or a seller with an equal probability 1/
N or that an agent is neither a buyer nor a seller with the remaining probability 1－(2/N ). Now, a seller merely stays on his or her home island, whereas a typen buyer faces relocation shock so that she should move to a “village” ofj ∈{0, 1, 2, ...,M } of (n ＋1) island, where the villagej consists of agents holdingj amount of money. For a subsequent pairwise trade in a relocated island, each buyer determines how much to purchase and whether to carry it in cash or deposit it into a checking account. Idle money that is not necessary for immediate subsequent trade can be kept at home without any cost. The government has an intratemporal recordkeeping technology on checking accounts but not on agents’ trading histories. Other than the accountrelated tasks, the government does not engage in any other economic activities, including consumption or production of any goods.In a newly migrated island, each buyer is matched with a seller for a pairwise trade. Trading histories are private and agents cannot commit to future actions; thus, any possibility of credit trades is ruled out and a medium of exchange is essential (see, for instance, Kocherlakota 1998; Wallace 2001; Corbae
et al . 2003; Aliprantiset al . 2007). That is, either cash or checkingaccount deposit should be transferred in exchange for goods produced.In a pairwise trade, a buyer makes a takeitorleaveit offer (
q ,p ) to a seller, whereq denotes quantity of goods produced by the seller for the buyer andp denotes the amount of money transferred by the buyer to the seller.1 A buyer can payp in cash by carrying it to a bilateral meeting at a disutility cost ofηp or withdrawing it from the account at the same disutility cost.2 Then, even a buyer who is willing to purchase good in cash will deposit the relevant amount of money into a checking account and withdraw it on the spot for a pairwise trade.Transactions via an electronic payment method require the installment and operation of an electronic payment system, for which the government (
i.e. , service provider) incurs a fixed cost of Ω. In practice, setting up an electronic payment system entails substantial fixed cost, but trivial marginal cost. In order to focus on the former, the latter is assumed to be zero. From now on, a debit card will be regarded as the representative method of electronic payment because it is one of the primary means of electronic payment for instore purchases and typically lacks credit function. If a buyer uses a debit card to transferp amount of money to a seller, the government withdraws it from the buyers’s account and transfers it to the seller in cash.In order to provide debitcard service to the society, the government should raise resources Ω to pay for the fixed cost incurred in installing the debitcard payment system. First, unlike a representative agent model such as Lagos and Wright (2005), our model allows for wealth heterogeneity across agents, including those with no money. Hence, allocating the social cost for the debitcard payment system in a lumpsum fashion is not feasible. Under the assumption that cashwithdrawing cost from a checking account is the same as its carrying cost, taxation on cash traders is not feasible because they would then evade taxation by carrying around cash, which is not in the government information system. However, the government is notified of debitcard transactions because they are cleared through the debitcard payment system operated by the government. Therefore, the government can effectively collect the required resources from debitcard traders.
Specifically, according to the widespread agreement on fixed cost per debitcard transaction (see, for instance, Wang 2010), the government is assumed to raise resource cost Ω by levying
ω per debitcard trade whereω satisfies the following government’s budget constraint:Here, S denotes the frequency of debitcard transactions;
τ ∈[0, 1] is the share of the cost allocated to a buyer using a debit card; andτ^{c} ＝(1－τ )∈[0, 1] is the share of the cost allocated to a seller accepting a debit card. As discussed in the Introduction, the installment cost in the real world is a onetime cost. In our model, however, the government raises Ω in each period. Our interpretation of this is as follows: the government first finances total installment cost Φ and then raises its annuity value Ω＝(1－β )Φ in each period to smooth out the taxes over time.For a given (Ω,
τ ), S captures the economy of scale in the use of the debitcard payment technology in the sense that a higher S implies a lower cost per debitcard transaction (ω ). We assume that debitcard traders pay the cost to the government by producing their specialized types of goods at the point of a bilateral trade before returning to their home islands. We will show in Section IV (Optimal Cost Allocation) that, with the bargaining rule of a buyer’s takeitorleaveit offer, the cost shareτ ∈[0, 1] allocated to a buyer can be interpreted as taxation on the buyer’s labor required to produce output for tax payment. On the other hand, the cost shareτ^{c} ∈[0, 1] allocated to a seller can be interpreted as taxation on the buyer’s consumption.For a type
n agent, perperiod utility is given bywhere
q _{n＋1} is the consumption of goodn ＋1, is the production of goodn (including the share of costτ^{c} ω borne as a seller by accepting debit cards); I is an indicator function that is equal to 1 if a typen agent as a buyer meets a type(n ＋1) agent and trades via a debit card; andc is the amount of cash withdrawn by a typen agent for a pairwise trade.After a pairwise trade, a buyer returns to her home island and the information on each checking account is completely wiped out.3 An agent goes on to the next period with endofperiod money holdings.
1As shown by Head and Kumar (2005) and Rocheteau and Wright (2005), for instance, this trading protocol can mimic an allocation in a competitive market (or a market with competitive price posting). Suppose that typen buyers in village j are matched collectively with type(n＋1) sellers in village k and then they trade in a competitive market (or in a market with competitive price posting). An equilibrium allocation would be reminiscent of an allocation with a buyertakeall bargaining, where a buyer has j amount of money and a seller has k amount of money. 2In Section IV, we consider the case in which the proportional cost η is replaced with a fixed cost. We also consider the case in which cashcarrying cost is borne by a seller. 3Note that unsecured credit transactions are available and money is inessential if the government is assumed to have an intertemporal recordkeeping technology rather than an intratemporal one.
III. Equilibrium
We study a symmetric (across “islands” or specialization types) and stationary equilibrium. For a given (Ω,
η ) and government policy onτ , a symmetric steady state consists of functions (v ,π ) andω that satisfy the conditions described in the following. The functionsv : M→R andπ : M→[0, 1] pertain to the beginning of a period and prior to the realization of preference shock such thatv (m ) is the expected discounted value of having monetary wealthm andπ (m ) is the fraction of each island with monetary wealthm .Consider a generic pairwise meeting between a buyer with
b ∈M and a seller withs ∈M. We letThat is, Γ(
b ,s ) is the set of feasible wealth transfers from the buyer to the seller. After the realization of relocation shock, the buyer determines how much to offer (p ) to the seller and how to pay it, cash (I＝0) or debit card (I＝1). Noting that the seller accepts all offers that leave her no worse off (tiebreaking rule), the buyer’s problem can be expressed aswhere
Let
p_{d} (b ,s ,v ) andp_{c} (b ,s ,v ) be the solutions to the first bracket and second bracket of the righthand side in (4), respectively; that is,Now, let
p ^{*}(·) be the associatedp_{i} (·) fori ∈{c ,d } such thatp ^{*}(b ,s ,v )＝p_{d} (b ,s ,v ) if the buyer chooses to pay using a debit card (I＝1) andp ^{*}(b ,s ,v )＝p_{c} (b ,s ,v ) if the buyer chooses to pay in cash (I＝0).4p (b ,s ,v ) is discrete and thus,p ^{*}(b ,s ,v ) can be multivalued, in which case we allow for all possible randomizations over them. Let Δ(b ,s ,v ) be the set of measures that represents those randomizations. Then Δ(b ,s ,v ) can be described aswhere
δ (m ;b ,s ,v ) is the probability that in a pairwise trade, the buyer withb ∈M offersb －m to the seller withs ∈M, ending up withm .Now we can describe the evolution of wealth distribution induced by pairwise trades as follows:
The first probability measure on the righthand side of (10) corresponds to pairwise trades, whereas the second corresponds to all other cases.
Noting that in (9),
δ is defined over the posttrade money holdings of the buyer, the buyer’s posttrade money holdings (b －m ＋s ) corresponds to the seller’s posttrade wealthm (＝b －(b －m ＋s )＋s ). Note also that the dependence of Π onv comes from the dependence ofδ onv in (9).Finally, let
g (b ,s ,v ) be the maximized value of (4). Note that the payoff of a seller withm ∈M is simplyβv (m ), and thus, the expected value of holdingm at the beginning of a period,v (m ), can be written asDefinition 1:
For given (Ω,η τ ),a symmetric stationary equilibrium is a set of functions (v ,π )and ω such that (i) the value function v satisfies (11); (ii) the probability measure π of wealth distribution satisfies π ∈Π(v )where Π (v )is given by (10); and (iii) w satisfies the government’s budget constraint, .The existence of a symmetric stationary equilibrium for some parameters is a straightforward extension of the existence results in Zhu (2003) and Lee
et al . (2005). That is, if (,M /) are sufficiently large respectively and (Ω,η ) are not too large respectively, then a monetary symmetric stationary equilibrium (v ,π ,ω ) exists withv strictly increasing and strictly concave.4Without the loss of generality, we can disregard the case in which a fraction of p is paid in cash and the remainder using a debit card because once a debit card is used, cost per debittransaction (ω) is imposed regardless of the amount of debitcard transaction.
IV. Optimal Cost Allocation
We now examine an optimal allocation of the social cost for an electronic payment system using a standard Ramsey taxation approach.
> A. The Ramsey Problem
In order to formulate a Ramsey problem for the benevolent government as a provider of debitcard payment services, we first define welfare as the lifetime expected discounted utility of a representative agent before the assignment of wealth according to a stationary distribution. Let W
_{τ} denote the welfare of a stationary equilibrium (v_{τ} ,π_{τ} ,ω_{τ} ) for a given policyτ . Then W_{τ} can be expressed as follows:Here, the element in row
b ∈M and columns ∈M of the matrix U iswith
c (·) denoting the amount of cash withdrawn for a pairwise trade (·)＝q (·)＋τ^{c}ω I(·) and where the second term,τ^{c}ω I(·), captures the disutility from the production of output for taxation borne as a seller by accepting debit cards.Under the buyertakeall trading protocol, the benefit principle is implementable in the sense that the social cost for the debitcard payment system is borne by its beneficiaries (
i.e. , buyers) regardless ofτ . That is, from (4), in a pairwise trade between a buyer withb ∈M and a seller withs ∈M, the buyer’s net payoff for I=1 iswhere
q (b ,s ,v_{τ} )＝βv_{τ} (s ＋p_{d} )－βv_{τ} (s ) andτ ＋τ^{c} ＝1. Therefore, the choice ofτ can be interpreted as allocating the social cost to the beneficiaries of debitcard transactions in the form of labor taxation (τ ＝1,τ^{c} ＝0) or consumption taxation (τ ＝0,τ^{c} ＝1) or a certain combination of the two taxations [τ ,τ^{c} ∈(0, 1)]. In the above equation, labor taxation (τ ＝1,τ^{c} ＝0) has a lumpsum feature in the sense that it does not affect quantity consumed in exchange for money transferred via a debit card. On the other hand, the consumption taxation (τ ＝0,τ^{c} ＝1) is distortionary in the sense that it affects quantity consumed with the debitcard transactions.Now the Ramsey problem for the government is to choose the tax rate on labor (
τ ) and the implied tax rate on consumption (τ^{c} ＝1－τ ) to maximize welfare, taking into account its effect on the equilibrium reactions of buyers and sellers in pairwise trades.Definition 2:
The Ramsey problem for the benevolent government is to choose a symmetric stationary equilibrium (v_{τ} ,π_{τ} ,ω_{τ} )in Definition 1 that maximizes (12) or equivalently choose τ ^{*}＝arg max_{τ∈[0, 1]}W_{τ} .In order to compare the magnitude of welfare loss across different policies, we calculate the welfare cost of
τ policy relative to that of an infeasible lumpsum taxation.5 More specifically, we first find Δ that solveswhere is the welfare with lumpsum taxation. The element in row
b ∈M and columns ∈M of U_{Δ} isThat is, Δ is an additive consumption compensation that makes the welfare with
τ policy equal to that with lumpsum taxation ().6 The welfare cost ofτ policy is then calculated as the ratio ofΔ to the average consumption in the stationary equilibrium withτ policy.Although labor taxation (
τ ) has a lumpsum feature as in (13), whether a higherτ improves or deteriorates welfare is not obvious at all. The marginal gain from a higherτ is (∂u /∂τ )＝u’ (q )ω , whereas the marginal labor cost of increasingτ is justω . Hence, for a givenω a higherτ is beneficial to relatively poor buyers whose consumption is sufficiently small so thatu’ (q )＞1, whereas it is detrimental to relatively rich buyers whose consumption is sufficiently large so thatu’ (q )＜1. This implies that relatively poor buyers prefer labor taxation, whereas relatively rich buyers prefer consumption taxation. Notice that, for a givenω , a higher t increases the intensive margin (output per unit of money), which then increases the fraction of buyers withu’ (q )＜1.Labor taxation (
τ ) also has an extensivemargin effect on the choice between electronic and cash payment. For a simple illustration of this point, consider a pairwise trade between a buyer and a seller who have money holdings ofb ands , respectively. Suppose money is divisible and the value functionv is differentiable. Now, for given (η ,ω ,τ ), let (b ,s ) denote the amount of money transfer at which a buyer is indifferent between cash and debitcard payment; that is,7It can be shown that pe105(
b ,s ) decreases withτ and debitcard payment is preferred forp (b ,s )＞(b ,s ). This is particularly true of relatively poor buyers withu’ (q )＞1. Hence, asτ increases, a greater number of poor buyers are willing to use debit cards, and the fraction of debitcard traders withu’ (q )＞1 would consequently increase.This extensivemargin effect will lower per transaction cost (
ω ) of the debitcard payment system via an economyofscale channel, which in turn will enhance both intensive and extensive margin effects of a higherτ . For instance, a lowerω not only increases output produced per unit of money (i.e. , intensive margin) but also decreases in (15), implying that a larger fraction of buyers use debitcard payment (i.e. , extensive margin).The above discussion suggests that the effect of
τ on welfare depends crucially on a nondegenerate distribution of wealth across agents and economy of scale in the use of debitcard payment system. The endogeneity of a nondegenerate wealth distribution rules out closedform solutions and hence, we tackle our question based on numerical solutions in the following subsections.> B. Parameterization
We parameterize the basic environment to solve the model numerically as follows. We first assume
N ＝3, the smallest number of types of agents consistent with no doublecoincidence meeting, andβ ＝0.99. We set (,M )＝(40, 3) to ensure that the indivisibility of money and the upper bound on money holdings are not too severe.8We let
u (q )＝θ ln(1＋q ) whereθ is chosen, together with (Ω,η ), to make the model fit the U.S. data. Based on the 2011 Survey of Consumer Payment Choice, Shy (2012) reports that S＝0.511 where S is the fraction of debitcard transactions out of cash and debitcard transactions. In addition, based on the 2010 Diary of Consumer Payment, Stavins (2012) reports that the average value of debitcard transactions relative to that of cash, denoted by D, equals to 1.65. In our model, (θ , Ω,η )＝ (2.1, 2.0×10^{－3}, 8.42×10^{－4}) withτ ＝0 generates S＝0.512 and D＝1.50. Here, we considerτ ＝0 because in the U.S. almost all costs of debitcard transactions are imposed on the sellers on the surface. Note that Ω＝2.0×10^{－3} corresponds to 0.18% ofq ^{*}＝argmax [u (q )－q ]＝1.1, which is close to the estimate of Aiyagariet al . (1998).> C. Labor Taxation versus Consumption Taxation
Figure 1 shows welfare level and welfare cost as a function of the tax rate on labor (
τ ). The welfare increases withτ and the solution to the Ramsey problem turns out to beτ ^{*}＝1. That is, the unity tax rate on labor (or zero tax rate on consumption) attains the highest welfare. The welfare cost with zero tax rate on consumption (τ ＝1) is 0.03%, but increases up to 0.12% with zero tax rate on labor (τ ＝0).The underlying mechanism that renders the unity tax rate on labor (or zero tax rate on consumption) optimal can be explained in terms of (i) its intensivemargin effect on the quantity consumed in exchange for money transferred via a debit card for a given cost per debit transaction and (ii) its extensivemargin effect on the choice of means of payment between debit card and cash, which affects the cost per debit transaction. The welfare implications of these two effects hinge on a nondegenerate distribution and economy of scale in the use of debitcard payment system.
First, with regard to the intensivemargin effect, a higher tax rate on labor implies relatively low tax rate on consumption, thereby mitigating consumption distortion. Hence, the higher the tax rate on labor (consumption), the more favorable to the relatively poor (rich) whose consumption is sufficiently small (large). Figure 2 shows that the gap between the average quantity of good consumed by buyers and that produced by sellers shrinks as the tax rate on labor increases. As a result, welfare loss due to distortionary consumption taxation decreases and eventually converges to zero as the tax rate on labor approaches 1.
Second, a higher tax rate on labor also has an extensivemargin effect on the choice of payment method. Figure 3 shows that the frequency of debitcard transactions (S) increases with
τ , which eventually lowers cost per debitcard transaction (ω ＝Ω/S). The two extreme policies,τ ＝1 andτ ＝0 for a given (ω ,η ), are useful in understanding the frequency of debitcard transactions (S) increases withτ . In the case ofτ ＝1, all transactions are completed using debit cards as long asp ＞(ω /η ). However, whenτ ＝0, transactions can be made in cash even ifp ＞(ω /η ). That is,p exceeding (ω /η ) will be paid in cash rather than using a debit card as long asu {βv [s ＋p (·)]－βv (s )}－u {βv [s ＋p (·)]－βv (s )－ω }＞ηp (·). The latter happens to a relatively poor buyer in a pairwise trade whenβv [s ＋p (·)]－βv (s )≡q_{c} is sufficiently small. A higher frequency of debitcard transactions with a higher tax rate on labor enhances economy of scale as it subsequently lowers cost per debitcard transaction, as shown in Figure 3. Also, Figure 4 shows that the extensivemargin effect reduces the fraction of buyers with [u’ (q )＜1] who prefer consumption taxation.Third, Figure 5 shows that the total cost of withdrawing cash decreases with the tax rate on labor. This is because the frequency of cash transactions decreases as
τ increases. More transactions are subject to labor taxation and hence, the cost for the debitcard payment system allocated in the form of labor tax also increases with the tax rate on labor.In sum, Figure 6 shows that overall transaction cost, including welfare loss due to social cost for the debitcard payment system (which is raised in the form of labor taxation and distortionary consumption taxation) and cashwithdrawing cost, decreases with the tax rate on labor. This immediately implies the results shown in Figure 1.
Finally, we check the robustness of our main results in different settings. We first consider the fixed cashcarrying cost for sellers. That is, a seller accepting cash incurs a fixed handling cost
κ .9 In computing a stationary equilibrium for this case, we setκ to make the model fit the U.S. data on the fraction of debitcard transactions (S＝0.511) and the ratio of average value of debitcard transactions to that of cash (D＝1.65). The model parameterized with (θ , Ω,η ,κ )＝(2.1, 2×10^{－3}, 8×10^{－4}, 1.15×10^{－4}) implies S＝0.512 and D＝1.50 As reported in Figure 7, this variation does not change the optimality of the unity tax rate on labor.As another robustness check, we consider the case in which the cashwithdrawing cost for a buyer is fixed, whereas the cashcarrying cost for a seller is proportional. That is, withdrawing cash incurs a fixed cost for a buyer, whereas cashhandling cost for a seller increases with the amount of cash transaction at a rate of . In computing a stationary equilibrium for this case, we again choose (, ) to make the model fit the data. The model parameterized with (
θ , Ω, , )＝(2.1, 3.5×10^{－3}, 1.01×10^{－3}, 1×10^{－3}) implies S＝0.568 and D＝1.56. Figure 8 shows that optimality of the unity tax rate on labor is also immune to this variation.5If the social cost for debitcard payment system is raised by imposing a lumpsum tax on each agent, all trades are made using debit cards, and trading behaviors are not affected. However, as discussed in Section II, lumpsum tax cannot be forced effectively in our model. 6Consumption compensation is calculated as an addition to the consumption with τpolicy rather than as a multiple of consumption with τpolicy, because consumption is zero in some pairwise meetings. 7Note that the lefthand side of (15) is decreasing in p, whereas the righthand side is increasing in p. The lefthand side is also greater than the righthand side when p is sufficiently close to 0. Hence, (b, s) in (15) is well defined. 8In this type of model, almost all monetary offers are either 0 or 1 if the indivisibility of money is too severe. However, it is not the case in our examples. In addition, M＝3 is large enough so that almost no one is at the upper bound in a stationary equilibrium and hence, the result would be hardly affected even if a larger M were assumed. 9Recent empirical studies such as GarciaSwartz et al. (2006) and Schmiedel et al. (2012) suggest that both buyers and sellers incur cashcarrying cost.
V. Concluding Remarks
In this paper, we explored an optimal allocation of the social cost for an electronic payment system by incorporating a standard Ramsey taxation approach into an offtheshelf matching model of money. Our results suggest that economy of scale in the use of an electronic payment technology is enhanced with the tax rate on labor. This then decreases not only per transaction cost of electronic payment and cashwithdrawing cost, but also welfare loss because of the distortionary consumption taxation. As a result, the unit tax rate on labor (or zero tax rate on consumption) yields an optimal allocation of the cost for an electronic payment system.
In order to capture an economyofscale channel thoroughly, we here adopt a TrejosWright model. Alternatively, we can also consider a rather tractable model such as a version of Lagos and Wright (2005). For the model, tractable distribution of wealth can be generated in many ways, allowing us to obtain a closedform solution. We believe, however, that even in this case, our main mechanism would still work and the main results would remain intact.
Finally, we focused on an allocation scheme of the social cost for an electronic payment system without considering its implications from the viewpoint of an industrial organization. For instance, we do not deal with the issue on how to impose fees on merchants and consumers by a profitmaximizing card issuer. We leave the generalization of our model to future research, in which private fees together with the resource cost can be explored systematically.

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[FIGURE 1] WELFARE LEVEL AND WELFARE COST

[FIGURE 2] WELFARE LOSS DUE TO CONSUMPTION TAXATION

[FIGURE 3] FREQUENCY OF DEBITCARD TRANSACTIONS AND COST PER TRANSACTION

[FIGURE 4] FRACTION OF BUYERS WITH u’(q)＜1

[FIGURE 5] CASHWITHDRAWING COST AND LABOR TAX PAYMENT FOR A DEBITCARD SYSTEM

[FIGURE 6] OVERALL TRANSACTION COST

[FIGURE 7] FIXED CASHCARRYING COST FOR SELLERS

[FIGURE 8] FIXED CASHWITHDRAWING COST AND PROPORTIONAL CASHCARRYING COST