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A NEW MONETARIST MODEL OF FIAT AND E-MONEY.

Despite many usurpers, cash is still king. The Economist, October 3, 2015.

Cash is one of mankind's greatest inventions; a vast improvement, one would imagine, on carting around sheep or hales of hay. Despite the proliferation of other forms of payment, cash retains qualities that alternative methods cannot match, including anonymity, instant clearing, universal acceptance and a relatively tech-free mechanism. It can be used even if the power grid goes down or the banks are all hacked. Yet a growing number of economists are now calling for cash to be phased out.

Why some economists want to get rid of cash, The Economist, August 17, 2016.

I. INTRODUCTION

Cash has always circulated in parallel with other payment instruments (e.g., checks, debit cards, credit cards); however, these forms of payment have not been perfect substitutes for cash. Some of the differences are that settlement with noncash payment instruments is delayed, acceptance can be limited, a minimum amount may be required for transactions, fees for the service are charged, transactions are traced, and usually they are used in large transactions.

Recent developments in technology have permitted alternatives to the use of cash in small transactions. The first example, launched in the 1990s in Europe, and in Asia, are electronic purses (e-purses), (1) in the form of cash cards allowing storage and transfer of electronic money (e-money) (2), followed by the reloadable prepaid cards in the United States in the 2000s. The second example is mobile payment applications such as ApplePay, GooglePay, SamsungPay, Swish in Sweden, or AliPay in China which allow multiple transactions. (3)

The overall success of e-money is mixed. Currently, in Europe, e-money does not constitute a credible substitute for cash. All the European e-purses introduced in the 1990s have disappeared. Ironically, some economists (Dowd 1998; Friedman 1999; King 1999) had predicted the imminent demise of cash used for transactions after the launch of the first European e-purses. On the contrary, e-money is successful in major Asian big cities, as well as in the United States, with the increasing trend of e-purse adoption by the under-banked. (4)

In this paper we study the competition between cash (fiat money) and e-money, in the form of prepaid cards, or e-purses, and explore why cash remains the most widely used means of payment for everyday transactions. While taking into account the lessons of the past, our intention is to better understand why it may be difficult to replace cash with e-money, or have them coexist, when agents are free to choose between two payment instruments. (5)

To study the adoption of a new payment instrument, e-money, we formalize the three most important issues developers of e-money have to tackle when they introduce e-money: security, merchants' net gain resulting from the investment, and popularity. We seek to answer the following questions. Will buyers use the new payment instrument, and merchants accept it? Is security an important element? Will a cashless society emerge in the near future, or is coexistence the norm? What are the reasons that explain the failure or the success of e-money projects? Will monetary policy, or inflation, modify the adoption process of e-money? In an attempt to answer these questions, we study the conditions under which e-money can replace or coexist with cash for payments when money is socially useful, that is, in the Lagos and Wright (2005) and Rocheteau and Wright (2005) search (or "New Monetarist") model. (6)

In our model, the new payment instrument involves a cost that merchants must incur to accept e-money (and use a dedicated terminal), while it is more secure than cash for both buyers and merchants. (7) Cash is universally accepted by sellers, but it is subject to the risk of theft. Conversely, e-money is immune from theft, yet may not be accepted as merchants have to invest in a costly payment terminal to trade with it. Therefore, both monies face different frictions making them imperfect substitutes. We illustrate that the cost of investment is important to understand merchants' e-money adoption; nevertheless, it is not the only factor. Merchants' expectations that buyers will use e-money are crucial to give them incentives to invest. The buyers' and sellers' mutual decisions create strategic complementarities in the choice of the payment instrument. Consequently, depending on the risk of theft of cash, the cost of investment, and other parameters, three payment patterns may emerge for the same fundamentals, where fiat money coexists with e-money, or where only one type of money circulates. Additionally, when only fiat money circulates, welfare may be higher or lower than when e-money replaces fiat money, or circulates in parallel. Finally, we endogenize the risk of theft of cash, which affects the cost of holding fiat money. We show that it modifies the adoption level of e-money, and the multiplicity of equilibria. Therefore, the entire replacement of cash with e-money is unfeasible. Monetary policy may also play an important role in the economy since low inflation can facilitate the adoption of e-money in parallel with fiat money.

A. Related Literature

The question of competition between payment instruments is not a recent issue. It has been studied in models based on transaction costs (e.g., Folkertsma and Hebbink 1998; Shy and Tarkka 2002; Whitesell 1992), and in search-based environments which make the choice of holding money endogenous. Specifically, the question of the establishment of one or two fiat currencies has been analyzed in the first (Kiyotaki and Wright 1991) and second generation (Trejos and Wright 1995) of dual-currency search models. These models study the problem of the acceptability of fiat currencies when there is either government intervention (Li and Wright 1998), legal tender laws, or restrictions (Lotz and Rocheteau 2002; Soller-Curtis and Waller 2000), price control, or conversion policies (Lotz 2004). Contrary to these models, there is no government intervention, policy, or law to facilitate the use of e-money.

In this paper, the focus is on the adoption of a new means of payment that is launched to eventually replace fiat money in some specific transactions (i.e., for low value purchases). Essential to the coexistence of cash and e-money in our model is that each currency has a disadvantage relative to the other as a payment instrument--in terms of security, investment cost, and acceptability. In the search literature, Kim and Lee (2010) establish the coexistence of cash and debit cards by assuming cash incurs disutility costs, while debit cards incur record keeping costs; He, Huang, and Wright (2008) show the concurrent circulation of cash and bank liabilities as media of exchange based on the assumption that cash can be stolen; Sanches and Williamson (2010) establish the coexistence of money and credit by assuming that money is subject to risk of theft, and credit is subject to limited commitment; Li (2011) demonstrates the coexistence of cash and checking deposits by considering a deposit interest rate; Lester, Postelwaite, and Wright (2012) prove the coexistence of money and other assets by assuming the assets are imperfectly recognizable; and Lotz and Zhang (2016) demonstrate the coexistence of money and credit by assuming imperfect record-keeping, and limited commitment. In an international context, Kannan (2009) proves the coexistence of international currencies by assuming agents make strategic decisions based upon invoicing of their trade in international transactions, and Zhang (2014) by assuming imperfect recognizability of assets and information costs.

This paper is based on Nosal and Rocheteau (2011, ch. 8 and 10), and closely related to He, Huang, and Wright (2008), Kim and Lee (2010), Sanches and Williamson (2010), and Li (2011). Contrary to those papers where sellers accept different types of currencies, in our paper sellers accepting e-money must first invest in technology (like in Kim 1996; Nosal and Rocheteau 2011; Lotz and Zhang 2016). (8) As a result, they do not obtain the multiplicity of equilibria that emerges in our model from the strategic complementarities between buyers and sellers. In Kim and Lee (2010) and Li (2011), in equilibrium, each currency is used for different transactions, cash for small ones and debit cards for larger transactions. A key difference here is that in equilibrium, buyers use fiat and/or e-money for the same type of transaction. Additionally, Kim and Lee (2010) assume money is indivisible, whereas in our model money is divisible, which allows us to study the effect of monetary policy on coexistence. In He, Huang, and Wright (2008) and Sanches and Williamson (2010), buyers run the risk of theft when they carry fiat money. In our model, fiat money is also subject to theft, but we assume that both buyers and sellers who hold onto cash after trade might be stolen, which affects the bargaining process, and the terms of trade as the price of goods incorporates a risk premium. Similarly to He, Huang, and Wright (2008), we endogenize the risk of theft to understand how it affects the adoption process of emoney, and the multiplicity of equilibria obtained with exogenous theft. We show that the network externalities, absent in their model, give rise to a new channel for monetary policy, which modifies monetary equilibria and the adoption level of e-money.

Finally, like us, Arifovic, Duffy, and Jiang (2017) study the competition between cash and e-money by conducting a laboratory experiment, based on a Kiyotaki and Wright (1991) model where money and goods are indivisible objects. They do not establish the coexistence of cash and e-money that we have in the Rocheteau and Wright (2005) framework, and they cannot study the effects of monetary policy on the choice between payment instruments.

The remainder of our paper is outlined as follows. In Section II, we present the environment. In Section III, we endogenously determine both the buyers portfolio (cash vs. e-money) and the sellers investment decision in e-money technology. In Section IV, we illustrate the existence of different types of monetary equilibria, and study welfare. In Section V, we endogenize the risk of theft, and study its effect on the multiplicity of equilibria. We also explain how monetary policy influences the adoption of e-money. Section VI concludes.

II. ENVIRONMENT

Our model is based on the monetary search model proposed by Lagos and Wright (2005), and developed by Nosal and Rocheteau (2011). There is a [0, 2] continuum of infinitely lived active agents in the economy, evenly divided between buyers (consumers) and sellers (producers), and time is discrete. Active agents visit two markets during each period, which correspond to two subperiods. In the first subperiod, these agents enter a decentralized market, called DM. Anonymous buyers and sellers, specialized in consumption or production, are matched bilaterally and randomly. Sellers can produce an output, in quantity q [member of] [R.sup.+], but do not consume, while buyers want to consume but cannot produce. In this model, barter is impossible, credit is ruled out (agents are anonymous), and goods are perishable. The fact that sellers do not want to consume and that buyers cannot produce generates a problem of double coincidence of wants that justifies the use of a means of payment. As a result, money is essential to trade.

At the beginning of the DM, a buyer meets a seller, and vice versa, with probability [sigma]. After a meeting, if an agreement is reached, active agents exchange an amount of a specific DM good, called search good, for money. Here, agents can either hold fiat money, or a new payment instrument, e-money. Thus, in the DM, q units of goods are traded either against fiat money (f), e-money (e), or both depending on the composition of the buyer's portfolio, that is, q = q(f,e).

At the end of the DM, we assume that active agents (buyers and sellers) may encounter some inactive people whose sole purpose and activity is to steal cash in the DM (we call them thieves) to purchase goods in the following subperiod. Contrary to cash, e-money is assumed to be fully secured, and therefore immune from theft. We note [alpha] [member of] (0,1) the probability for an active person to encounter a thief at the end of the DM, which is exogenous for now. Consequently, active agents' units of fiat money are exposed to risk of theft with probability [alpha] [member of] (0,1), contrary to agents' units of e-money which are fully secured. The money stolen corresponds to a transfer of resources from active to inactive agents, meaning that the total amount of fiat money brought in, and spent in the following market remains constant.

At the end of the first subperiod, agents enter a frictionless centralized market that corresponds to the second subperiod, called CM since it is a competitive market. There, all individuals can consume, but only active agents can produce a general good x [micro] [R.sup.+] by supplying labor (h). Buyers produce the general good in order to increase their monetary balances spent in the previous market, and sellers consume the general good to reduce their money holdings. Additionally, buyers choose the composition of their portfolio for the following period, which may contain one or both types of money.

The utility functions of buyers ([U.sup.h]) and sellers ([U.sup.s]) for the entire period, assumed separable between the two subperiods, are described as follows:

[U.sup.b] (q,x,h) = u (q) - h

[U.sup.s] (q,x, h) =-c (q) + x.

In the CM, buyers' and sellers' utility functions are linear in labor and production, respectively, with no loss in generality. In the DM, the utility and cost functions are assumed to respect the following properties: u (q) > 0, u (q) < 0, c'(q)> 0, c'(q)> 0, u(0) = c(0) = c'(0) = 0, and ;/(0) = +[infinity]. The optimal quantity produced and exchanged corresponds to the level of production and consumption that maximizes active agents trade surplus u(q) - c(q). This quantity q solves u(q) = c'(q*). Finally, all agents discount the future between each period, but not between subperiods, at the discount factor [beta] = 1/[1+r] [member of] (0,1).

As mentioned above, there are two means of payment in the economy, fiat money (cash) and e-money, that are perfectly divisible and storable. E-money is a new payment instrument that is a potential substitute for cash in the DM. In the CM, buyers choose which type of currency to utilize in the following DM. E-money is 100% backed by fiat money in the CM, meaning that one unit of cash can be exchanged for one unit of e-money, and reversely. This service is provided by e-money issuing institutions. Therefore, to obtain e-money, buyers exchange (one for one) some or all of their units of fiat money with e-money, transferred and stored into an e-purse (or prepaid card) obtained for free. Then, in the DM, the sellers must have a payment terminal to accept e-money, which is costly for them.

The utility cost to invest in the payment terminal at the beginning of the DM, before agents are matched, is I > 0. A fraction [LAMBDA] [member of] [0,1] of sellers may decide to invest in this technology which will allow them to accept e-money in addition to fiat money, whereas the remaining sellers will only be able to accept cash in exchange for goods. The fraction A, which is an endogenous variable, will be determined at the equilibrium by the buyers' portfolio choice, and the sellers' investment decision. After trade, at the beginning of the CM, all the units of e-money held by buyers or obtained by sellers in the previous DM can be redeemed for fiat money (for free) from the e-money-issuing institutions.

At the beginning of the period, each buyer enters the DM with a portfolio of money balances (m = [m.sub.f] + [m.sub.e]) consisting of [m.sub.f] nominal units of fiat money and/or me nominal units of e-money, with [m.sub.f] [greater than or equal to] 0, and [m.sub.e] [greater than or equal to] 0. Monetary units are valued at [[empty set].sub.t] = 1/[p.sub.t], [[empty set].sub.t] being the price of money in terms of the general good, at time t. Consequently, the real value of the nominal portfolio m is: [empty set]([m.sub.f] + [m.sub.e]) = f + e, where f = [empty set][m.sub.f], and e = [empty set][m.sub.e]. The aggregate stock of money in the economy is M, and the monetary authority can increase or decrease M at a constant gross rate [gamma] = [M.sub.t+1]/[M.sub.f]. Fiat money is injected or withdrawn from the economy through lump-sum transfers T to the buyers in the CM. E-money-issuing institutions cannot create money; their role is to transform fiat money into e-money, and reversely (i.e., they have the responsibility to safeguard that instrument). Finally, we study stationary equilibria such that the real value of the money supply is constant over time: [[empty set].sub.t][M.sub.t] = [[empty set].sub.t+1][M.sub.t+1]. The timing of events is described in Figure 1.

III. EQUILIBRIUM

A. The Centralized Market

We begin with the centralized market (CM). All active agents can produce, consume, and readjust their money balances, that is, choose the type, and quantity of money to hold before entering the following DM.

First, we consider the buyer's maximization problem. The value function of a buyer [W.sup.b.sub.t] ([omega]) who enters the CM with wealth to is:

[mathematical expression not reproducible]

where [V.sup.b.sub.t+1] is the buyer value function in the following DM. The second equation is the buyer budget constraint. He finances his next period money holdings ([m'.sub.f]+[m'.sub.e]) with labor (h), his wealth ([omega]), and lump sum transfers from the government (T). From the previous equations, the buyer value function at the beginning of the CM can be rewritten:

(1) [mathematical expression not reproducible]

According to this equation, the buyer lifetime expected utility when he enters the CM is the sum of his wealth, the lump-sum transfers from the government, and his value function at the beginning of the following DM minus the cost ([gamma]) of holding real balances (f' + e;). As in Lagos and Wright (2005), the choice of the buyer portfolio (f',e') is determined independently of his current wealth.

Identically, the value function of a seller with wealth [omega] at the beginning of the centralized market is given by [W.sup.s.sub.t]([omega]) = [omega] + [beta][V.sup.s.sub.t+1], where [V.sup.s.sub.t+1] is the value function of a seller without money at the beginning of the subsequent DM. Indeed, sellers are specialized in production, and hence do not need money since they do not consume in the DM.

B. The Decentralized Market

We now describe the determination of the terms of trade in a bilateral match between a buyer holding the portfolio f + e in the DM, and a seller without money. Agents bargain over (q, d) where the buyer receives q [greater than or equal to] 0 units of the search good produced by the seller in exchange for d [member of] [0,f + e] units of money. The buyer may transfer to the seller [d.sub.f] units of fiat money and/or [d.sub.e] units of e-money. In order to give sellers an incentive to invest in the e-money technology, we adopt the proportional bargaining solution so they can extract a positive fraction of the total surplus. According to Kalai's bargaining solution (see Aruoba, Rocheteau, and Waller 2007), the buyer receives the fraction [theta] of the total surplus and the seller receives the remaining fraction (1 - [theta]), where [theta] corresponds to the buyer's bargaining power. (9)

The quantity of goods exchanged depends on the total amount of monetary units in the buyer's portfolio, that may be composed of fiat, electronic, or both monies. By assumption, after trade takes place, and before entering the CM, fiat money may be stolen whereas e-money is secured by technology, and therefore immune from theft. (10) 11 Indeed, if electronic purses are lost or stolen after a trade, buyers immediately report it to the issuer. In this case, the buyer is redeemed for the total amount stolen as the funds recorded on the electronic device will no longer be activated. (11)

The determination of the terms of trade, when buyers can enter the market with fiat and emoney, results from the following maximization program:

(2) [mathematical expression not reproducible]

According to this program, the buyer maximizes his expected trade surplus (i.e., his utility of consuming the DM goods, net of the expected monetary units he transfers to the seller), and subject to the constraint that the buyer's trading surplus is equal to [theta]/(1 - [theta]) times the seller's trading surplus. Additionally, the buyer cannot transfer to the seller more fiat or e-money than he has. From this program, we obtain the nonlinear pricing rule:

(3) (1-[alpha])[d.sub.f] + [d.sub.e] = z[q(f,e)]

where z[q(f,e)] = (1 - [theta])u[q(f, e)] + [theta]c[q(f,e)].

Equation (3) indicates that when the risk of cash being stolen (a) increases, the buyer must give a higher amount of fiat money ([d.sub.f]) for the same quantity of goods than without the risk of theft. The additional amount of fiat money that the buyer gives to the seller corresponds to a risk premium that compensates the seller for the transfer of risk before entering the CM. Notice that this risk premium is only associated with the transfer of fiat money, but not with the transfer of e-money as it is secured. Consequently, in an unsafe economic environment, buyers obtain more goods in exchange for e-money than in exchange for fiat money. Substituting the terms of trade (Equation (3)) into the buyer's objective function (Equation (2)), the buyer's problem becomes:

(4) [mathematical expression not reproducible]

The buyer's quantities consumed depend on his portfolio composition, and are determined so that he obtains the share 0 of the total surplus. If the buyer carries enough monetary balances, Equation (4) is not binding, and he consumes the optimal quantity q*. However, if

Equation (4) is binding, he purchases the quantity q(f, e) < q*. Thus, the terms of trade satisfy z[q(f,e)] = mm[z(q*),(l - [alpha])f + e].

In the following, we study buyers' portfolio choices, and sellers' decisions to invest or not in the electronic payment terminal.

C. The Buyer's Choice of Money

Before entering the DM, according to Equation (1), the buyer chooses the amount, and the type of money he wants to hold in his portfolio, from his DM value function that satisfies (see Appendix A):

(5) [V.sup.b.sub.t](f,e) = [sigma] (1 - [LAMBDA])[theta] [u([q.sub.f]) - c ([q.sub.f])]

+[sigma][LAMBDA][theta] [u ([q.sub.f,e])-c([q.sub.f,e])]

+(1 - [alpha])f + e + [W.sup.b.sub.t](0,0)

where [q.sub.f] = q(f) is the quantity of search goods exchanged for fiat money, and [q.sub.f,e] = q(f,e) is the quantity exchanged for both cash and e-money. Indeed, a buyer may be matched with a seller who does not accept e-money (with probability 1 - [LAMBDA]), or a seller who accepts the two monies (with probability [LAMBDA]). The last two terms result from the linearity of the value function, and correspond to the continuation value in the CM when no transaction occurs. The buyer does not expect to enter the CM with his initial portfolio, but only with the expected units of fiat money that will not be stolen, yet he will bring all of his units of e-money in the CM.

Substituting Equation (5) into Equation (1), the buyer's real balance choice is given by:

(6) [mathematical expression not reproducible]

where i = [gamma]-[beta]/[beta], and [i.sub.[alpha]] - [delta]-(l-[alpha])[beta]/[beta] = i + [alpha] represent the holding cost of electronic, and fiat money, respectively. We note [S.sub.f] = u([q.sub.f]) - c([q.sub.f]) the total surplus obtained with fiat money, and [S.sub.f,e] = u([q.sub.f,e]) - c([q.sub.f,e]) the total surplus obtained with both fiat and e-money.

The first two terms of Equation (6) correspond to the costs incurred by the buyer when he carries f real units of fiat money, and e real units of e-money in the subsequent DM. The costs of holding fiat or e-money are different. The third term represents the buyers' share of the expected trade surplus if he enters the market with fiat money only, or if he enters with both monies.

Both surpluses depend on the fraction A of sellers who invested in the e-payment technology.

When the risk of theft is strictly positive ([alpha] > 0), the holding cost of fiat money ([i.sub.[alpha]]) is higher than that of e-money as it is the sum of two costs: the opportunity cost (i) of both fiat and e-money, plus the insecurity cost of trading with cash ([alpha]).

When i > 0, both monies are costly to hold, so buyers will never accumulate more real balances than they need in the DM (z[q(f,e)] [less than or equal to] z(q*)). According to Equation (6), the buyer chooses his money balances (f [greater than or equal to] 0,e [greater than or equal to] 0) to maximize his share of the expected trade surplus in the DM, net of the holding costs of each type of money. The buyers' objective function is strictly concave, so a solution exists. The first-order (necessary and sufficient) conditions associated with Equation

(6) are:

(7)-[i.sub.[alpha]]/[sigma][theta] + [L.sub.f]([q.sub.f]) + [L.sub.f]([q.sub.f,e]) [less than or equal to] 0

(8)-i/[sigma][theta] + [L.sub.e]([q.sub.f,e]) [less than or equal to] 0

where [L.sub.f]([q.sub.f]) [equivalent to] (1 - [LAMBDA]) (1-[alpha])/(1-[theta])u'([q.sub.f])+[theta]c'([q.sub.f]) and [L.sub.f]([q.sub.f,e]) [equivalent to] = [LAMBDA](1-[alpha])[S'.sub.f,e]/ [(1-[theta])u'([q.sub.f,e])+[theta]c'([q.sub.f,e])] correspond to the marginal return from bringing one additional unit of cash when holding cash only ([L.sub.f]([q.sub.f])), or when holding both cash and e-money ([L.sub.f]([q.sub.f,e])); [L.sub.e]([q.sub.f,e]) [equivalent to] [S'.sub.f,e]/(1-[theta])u'([q.sub.f,e])+[theta]c'([q.sub.f,e]) COrresponds to the marginal return from bringing one additional unit of e-money when holding both cash and e-money.

According to Equations (7) and (8), a buyer brings positive real balances into the DM up to the point where he equalizes the holding cost of a marginal unit of each type of money with the expected increase in the trade surplus. The marginal return of each currency depends both on the risk of theft of fiat money ([alpha]), and the degree of acceptability of e-money by sellers (A).

D. The Seller's Investment Decision

At the beginning of the DM, sellers choose whether to invest in the technology (at utility cost I) to accept e-money in parallel with cash. (12) This choice results from the following maximization program:

(9) max {[sigma](l - [theta])[S.sub.f],-I + [sigma](l - [theta])[S.sub.f,e]}.

According to Equation (9), if a seller does not invest in the reading terminal, he can only sell his goods in exchange for fiat money. In this case, he receives the fraction (1 - [theta]) of the total surplus obtained with fiat money. On the contrary, a seller who chooses to invest in the new technology can exchange goods for fiat and e-money, and receives the fraction (1 - [theta]) of the total surplus obtained with both monies. From Equation (9), the fraction [LAMBDA] of sellers who decide to invest in the e-payment terminal satisfies:

(10) [mathematical expression not reproducible]

where G [equivalent to] [sigma](1 - [theta])[[S.sub.f,e - [S.sub.f]] is the gain (in terms of additional surplus) obtained by a seller if he accepts both fiat and e-money instead of fiat money only.

Now, we are able to study the different equilibria of the model that arise from buyers' and sellers' interactions. A steady-state equilibrium is a list (q, f, e, [LAMBDA]) that satisfies Equations (3), (7), (8), and (10).

IV. TYPES OF EQUILIBRIA

In this section, we describe the different Nash equilibria when buyers choose the composition of their portfolio based on their expectations of the fraction [LAMBDA] [member of] [0,1 ] of sellers who will accept e-money, and sellers decide to invest in the payment technology based on their expectations about the type of money buyers will use.

We consider three successive economic environments: one where all sellers accept e-money ([LAMBDA] = l), one where all sellers refuse e-money ([LAMBDA] = 0), and one where only a fraction of sellers accept e-money ([LAMBDA] [member of] (0,1)). In each of these environments, we determine the optimal portfolio choice of the buyer, and then according to this choice, we check if it is optimal for the seller to invest in the new payment technology. This second step makes A an endogenous variable of our model.

A. All Sellers Accept E-Money

First, we study the equilibrium such that all sellers have invested in the e-money technology ([LAMBDA]=l). Consequently, two means of payment can be accepted to settle transactions: fiat money and e-money. From Equations (7) and (8) with [LAMBDA]=1, the equilibrium output, if traded, is the solution to [i.sub.[alpha]]/[sigma][theta] = [L.sub.f]([q.sup.1.sub.f,e]) and 1/[sigma][theta] = [L.sub.e]([q.sup.1.sub.f,e]), where the exponent "1" refers to an equilibrium where all sellers accept e-money. These two equations establish the conditions for fiat money and e-money to be valued. (13) Buyers are indifferent to both monies when these equations are equal, that is, if and only if [alpha] = 0; otherwise e-money is preferred to cash. Indeed, when there is no risk of theft of fiat money, and when both currencies are accepted by all sellers, buyers are indifferent to holding cash or using e-money. An additional unit of one or the other type of money involves the same marginal increase in the buyers' surplus. However, when the risk of theft of cash ([alpha]) is strictly positive, it is not rational for a buyer to use fiat money given that e-money, which is safer, is always accepted. Since e-money has a higher return, and a lower holding cost than fiat money, the equilibrium is such that only e-money is used. In this equilibrium, fiat money is no longer used and valued, and the quantity exchanged with e-money is higher than the quantity that would have been exchanged with cash. However, when i > 0, the quantity traded is less than the optimal quantity (q*).

The monetary authority can drive the holding cost of e-money to zero. Indeed, when [gamma] = [beta], the opportunity cost of holding money (i) is zero, whereas the cost of insecurity of fiat money remains positive for all [alpha] > 0. Therefore, if the opportunity cost of e-money is zero (i = 0), at the Friedman rule buyers will hold enough e-money to buy the optimal quantity of output [q.sup.*].

When [LAMBDA] = 1, and [alpha] > 0, the buyers choice of money holdings, and the quantities traded, are such that:

[mathematical expression not reproducible]

We now analyze the sellers' investment decisions when buyers use e-money. According to Equation (10), when [LAMBDA] = 1, it is optimal for a seller to invest in the e-payment technology if:

I < [G.sup.1]

where [G.sup.1] [equivalent to] [sigma](l-[theta])[S.sup.1.sub.f,e] - [S.sup.1.sub.f] is the gain obtained by a seller if he accepts both fiat and emoney instead of fiat money only, in an environment where all other sellers accept both monies.

We note [q.sup.1.sub.f] the output produced by a seller who did not invest in e-money equipment while all the other merchants have invested. Sellers who invested will exchange the quantity [q.sup.1.sub.f,e] as they can accept both monies. The quantity [q.sup.1.sub.f] satisfies (1 - [alpha])[f.sup.1] = z ([q.sup.1.sub.f]) and since from Equations (7) and (8), [f.sup.1] = 0 when [LAMBDA] = 1, the quantity traded [q.sup.1.sub.f] = 0. Consequently, since only e-money is valued and used, [q.sup.1.sub.f,e] = [q.sup.1.sub.e] and [G.sup.1] = [sigma](l - [theta])[S.sup.1.sub.e] > 0. Therefore, it is optimal for a seller to invest if the cost is strictly smaller than the gain resulting from the use of e-money (see Figure 2). (14) The seller's best response function is:

[mathematical expression not reproducible]

From the buyers' and sellers' choices, there exists an equilibrium (pure e-money equilibrium) such that e-money becomes the universally accepted means of payment whereas fiat money is not used. This equilibrium corresponds to point E (for "Electronic") in Figure 3A.

PROPOSITION 1. When [alpha] > 0 and I is low enough (I < [G.sup.1]), if all sellers accept both fiat and e-money ([LAMBDA] = 1), the only monetary equilibrium is a pure e-money equilibrium such that cash does not circulate in the economy.

B. No Sellers Accept E-Money

We now study the equilibrium such that no seller has invested in the reading terminal ([LAMBDA] = 0). Consequently, sellers can only trade if buyers use fiat money. From Equations (7) and (8), with [LAMBDA] = 0, e-money has no value, and the equilibrium output exchanged for cash is the solution to [i.sub.[alpha]]/[sigma][theta] = [L.sub.f]([q.sup.0.sub.f]) where the exponent "0" refers to an equilibrium where no seller accepts e-money. Rearranging this first-order condition, the necessary condition for fiat money to be valued when no seller accepts e-money is i < [[bar.i].sub.f] [equivalent to] [sigma][theta](1-[alpha])-[alpha](1-[theta])/(1-[theta]) or, equivalently, the critical value [bar.[alpha]] above which buyers no longer use cash satisfies:

[bar.[alpha]] = [sigma][theta]-i(1-[theta])/(1-[theta])+[sigma][theta]

with [bar.[alpha]] > 0 [??] i < [sigma][theta]/(1-[theta]) so for fiat money to be valued, the risk of theft must be sufficiently small ([alpha] < [bar.[alpha]]).

FEMMA 1. When [LAMBDA] = 0, there is a critical value [bar.[alpha]] for the risk of theft such that if [alpha] [greater than or equal to] [bar.[alpha]], then fiat money is no longer valued.

Buyers' portfolio choice depends on the level of risk of using fiat money, but also on buyers expectations about the number of sellers accepting e-money. When no seller accepts it, e-money cannot be used as a medium of exchange, even if the e-purse is obtained for free.

In an environment where no seller accepts e-money, and where the risk of theft is small enough, the only equilibrium is a pure fiat money equilibrium. Indeed,

[mathematical expression not reproducible]

Furthermore, Equation (10) implies that it is optimal for sellers not to invest in the e-money technology if:

I > [G.sup.0]

where [G.sup.0] = [sigma](l - [theta])[[S.sup.0.sub.f,e] - [S.sup.0.sub.f] is the gain for a seller accepting both monies when all other sellers accept cash only.

We note [q.sup.0.sub.f,e] the output produced by a seller who has invested in the e-money technology when all the other merchants have not. Sellers who only accept fiat money will exchange the quantity [q.sup.0.sub.f]. From Equations (7) and (8), since [e.sup.0] = 0 when [LAMBDA] = 0, the quantity traded [q.sup.0.sub.f,e] = [q.sup.0.sub.f]. Consequently, [S.sup.0.sub.f,e] = [S.sup.0.sub.f] and [G.sup.0] = 0 (see Figure 2). Therefore, it is optimal for a seller not to invest if the cost is strictly positive. The seller's best response function is (15):

[mathematical expression not reproducible].

Following the same fundamentals in the previous example, there also exists an equilibrium in which e-money is not used, and fiat money keeps its role of universally accepted means of payment (pure fiat money equilibrium). (16) The quantities traded in a pure fiat money equilibrium are lower than in the pure e-money equilibrium given the risk of theft associated with cash. This equilibrium corresponds to point F (for "Fiat") in Figure 3A.

In summary, if no seller accepts e-money then, even if e-money is less costly to hold, buyers neither hold nor trade e-money.

PROPOSITION 2. When [alpha] [member of] (0, [bar.[alpha]]), and I > 0, there exists a pure fiat money equilibrium such that sellers do not accept e-money ([LAMBDA] = 0), and buyers do not hold it, although e-money is less risky or costly to hold than fiat money.

C. Some Sellers Accept E-Money

We have studied the existence of pure monetary equilibria such that only one type of money is used as a medium of exchange. However, is it possible to have the coexistence of fiat, and e-money in the economy, such that a fraction [LAMBDA] [member of] (0,1) of sellers accept both fiat and e-money, while a fraction 1 - [LAMBDA] of sellers only accept fiat money?

Buyers' expectations about the sellers' level of investment are crucial to encourage them to use e-money in parallel with cash. From Equation (8), the equilibrium output [q.sup.[LAMBDA]]/f,e exchanged for both cash and e-money satisfies 1/[sigma][theta] = [L.sub.e]([q.sup.[LAMBDA].sub.f,e]) where the exponent "[LAMBDA]" refers to an equilibrium where only a fraction [LAMBDA] [member of] (0,1) of sellers accept e-money. From Equation (7), the quantity traded when only cash is accepted ([q.sup.[LAMBDA].sub.f]) is the solution of [i.sub.[alpha]]/[sigma][theta] = [L.sub.f]([q.sup.[LAMBDA].sub.f,e]) + [L.sub.f]([q.sup.[LAMBDA].sub.f,e]) After rearrangements, the necessary condition for e-money to be valued when some sellers accept e-money, and fiat money is valued, is i < [LAMBDA][sigma][theta]/(1-[theta]). Therefore, buyers decide to enter the DM with e-money if and only

[LAMBDA] > [bar.[LAMBDA]] [equivalent to] i(1-[theta])/[theta]

with [bar.[LAMBDA]] < 1 [??] i < [sigma]0/(1-[theta]). Below this threshold, there would be an insufficient number of sellers accepting e-money to encourage buyers to use it. This critical value [bar.[LAMBDA]] is an increasing function of the interest rate i.

When [LAMBDA] > [bar.[LAMBDA]], Equations (7) and (8) must be satisfied simultaneously for a mixed monetary equilibrium to exist. Inserting Equation (7) into Equation (8), we show that fiat money is valued if and only if:

[LAMBDA] < [??] [equivalent to] 1-[alpha](1+i)(1-[theta])/[sigma][theta](1-[alpha])

where [??] is a decreasing function of [alpha] and i. Moreover, [bar.[LAMBDA]] < [??] if and only if:

i < [[bar.i].sub.f] [??] [alpha] < [bar.[alpha]].

LEMMA 2. Both fiat and e-money can be valued if and only if a fraction [LAMBDA] [member of] ([bar.[LAMBDA]],[??]) of sellers accept e-money.

In summary, for any [LAMBDA] [member of] ([bar.[LAMBDA]],[??]), there always exists a quantity [q.sup.[LAMBDA].sub.f,e] > 0, and [q.sup.[LAMBDA].sub.f] > 0 such that Equations (7) and (8) are both satisfied.

Consequently, when [LAMBDA] [member of] ([bar.[LAMBDA]], [??]), some buyers may enter the market with e-money, as well as cash. The buyers' portfolio composition in a mixed monetary equilibrium is defined by:

[mathematical expression not reproducible]

Additionally, according to Equation (10), a seller is indifferent about trading the quantity [q.sup.[LAMBDA].sub.f,e], or [q.sup.[LAMBDA].sub.f] if:

(11) 1 = [G.sup.[LAMBDA]]

where [G.sup.[LAMBDA]] [equivalent to] (1 - [theta])([S.sup.[LAMBDA].sub.f,e] - [S.sup.[LAMBDA].sub.f] is the gain for a seller if he accepts both monies instead of cash only, when a fraction of sellers accept emoney (see Figure 2). Equation (11) determines the unique value of A such that a mixed monetary equilibrium exists (see Rocheteau and Nosal 2017, and Lotz and Zhang 2016, for a similar approach).

Since z = ([q.sup.[LAMBDA].sub.f]) = (1-[alpha])[f.sup.0] when [LAMBDA] = 0, z([q.sup.1.sub.e])=[e.sup.1] when [LAMBDA]=l, and z([q.sup.[LAMBDA].sub.f,e]) = (1-[alpha])[f.sup.[LAMBDA]]+[e.sup.[LAMBDA]] when [LAMBDA] [member of] ([bar.[LAMBDA],[??]), we have z([q.sup.0.sub.f]) < z([q.sup.[LAMBDA].sub.f,e]) < z([q.sup.1.sub.e]) and 0 < [G.sup.[LAMBDA]] < [G.sup.1]. Moreover, [LAMBDA] < [??], so the seller's gain [G.sup.[LAMBDA]] cannot be higher than [G.sup.[??]]. Consequently, [G.sup.[LAMBDA]] [member of] (0, [G.sup.[??]]), where [G.sup.[??]] is a decreasing function of the risk of theft ([dG.sup.[??]]/d[alpha] < 0) as long as [alpha] < [bar.[alpha]]. Some comparative statics are summarized in Table 1.

For any given I [member of] (0, [G.sup.[??]]), there exists an equilibrium such that a fraction [LAMBDA] [member of] ([bar.[LAMBDA]), [??]) of sellers invest to accept e-money as conditions (7), (8), and (11) can all be satisfied. Therefore, sellers' best response function is:

[mathematical expression not reproducible]

PROPOSITIONS 3. When 0 < I < [G.sup.[??]], there exists an equilibrium such that some sellers accept only fiat money whereas others accept both fiat and e-money, and buyers use both.

The fact that a mixed monetary equilibrium exists is quite an intuitive result. Payment with e-money allows exchanging a larger amount of goods than cash. Therefore, if buyers anticipate that e-money may be accepted by some sellers, all of them will decide to possess e-money in addition to cash. This equilibrium corresponds to point M (for "Mixed") in Figure 3A.

In summary, when the risk of theft is low enough ([alpha] < [bar.[alpha]]), different multiplicities may appear, depending on the value of the investment cost. Of course, the equilibrium where only fiat money is accepted always exists when [alpha] < [bar.[alpha]]. In Figure 2, the surplus in a mixed monetary equilibrium is lower than the surplus in the pure e-money equilibrium, which means that when the value of I satisfies the sellers' investment condition for a mixed equilibrium to exist (point M), then the sellers' investment condition for a pure e-money equilibrium to exist (point E) is also satisfied. Therefore, three monetary equilibria may coexist, where no sellers, all sellers, or a fraction of them choose to invest in the new technology, as long as the investment cost is not too high (I < [G.sup.[LAMBDA]]). There is also a region where the investment cost is within an intermediate range ([G.sup.[??]] < I < [G.sup.1]) such that a mixed monetary equilibrium does not exist, but such that a pure fiat money and a pure e-money equilibrium coexist. However, if the investment cost is too high for the pure e-money equilibrium to exist (I > [G.sup.1]), then the economy will end up in a pure fiat money equilibrium (point F). If the investment cost is zero, then e-money dominates fiat money, and the only equilibrium is with e-money.

PROPOSITION 4. When the risk of theft is high ([alpha] [greater than or equal to] [bar.[alpha]]), only a pure e-money equilibrium can exist if the investment cost is not too high (I < [G.sup.1]). When the risk of theft is not too high ([alpha] < [bar.[alpha]]), there may be a multiplicity of equilibria. If the investment cost is high (l > [G.sup.1]), there only exists a pure fiat money equilibrium. If I [member of] (0, [G.sup.[??]]), a pure fiat money equilibrium, a pure e-money equilibrium, and a mixed monetary equilibrium can coexist. If 1 [member of] [G.sup.[??]],[G.sup.1]], only a pure fiat money equilibrium ana a pure e-money equilibrium can coexist.

D. Welfare

In the presence of a multiplicity of equilibria, it is interesting to study the question of social welfare. Below, we compare the three steady-state equilibria we have previously obtained in terms of welfare when the risk of theft of cash ([alpha] < [bar.[alpha]]) varies. Social welfare is measured as the steady-state sum of the buyers' and sellers' utilities in the CM and the DM, and is given by:

W [equivalent to] [sigma][[LAMBDA][S.sub.f,e]+(1-[LAMBDA])[S.sub.f]] - [LAMBDA]I.

In our environment, social welfare across the three types of equilibria is shown in Table 2.

We study the effects of changing exogeneously the probability of cash being stolen ([alpha]) on social welfare, for a given interest rate (i), and investment cost (I), such that an equilibrium with both fiat and e-money exists. Figure 4 illustrates the change in social welfare in the three types of equilibria. When only fiat money is accepted ([LAMBDA] = 0), welfare ([W.sub.f]) is strictly decreasing with [alpha], and reaches 0 when [alpha] = [bar.[alpha]]. When all sellers have adopted e-money technology ([LAMBDA] = 1), and all buyers use e-money, welfare ([W.sub.e]) is independent of a, and constant. The [W.sub.e] line is always below [W.sub.f] at [alpha] = 0 when I > 0, and above [W.sub.f] when I = 0 and [alpha] > 0. When some sellers accept both fiat and e-money, the number of sellers accepting e-money ([LAMBDA]) adjusts so that the sellers' gain from adopting the new technology always equals the fixed investment cost (see Equation (11)). In a mixed monetary equilibrium, when [alpha] increases, [S.sup.[LAMBDA].sub.f,e] - [S.sup.[LAMBDA].sub.f] remains constant whereas [q.sup.[LAMBDA].sub.f], [S.sup.[LAMBDA].sub.f], and [LAMBDA] decrease. Consequently, when [alpha] increases, [W.sub.f,e], decreases. Furthermore, [mathematical expression not reproducible], meaning [W.sub.f,e] is below [W.sub.e], as shown in Figure 4.

We can compare welfare in a pure fiat money economy versus welfare in a pure e-money and a mixed monetary economy. (17) From Figure 4, we can see there are three different regions: one where a is low (below [??]) such that [W.sub.f] > [W.sub.e] > [W.sub.f,e], one where [alpha] is in the middle range (between [??] and [??]) such that [W.sub.e] > [W.sub.f] > [W.sub.f,e], and one where a is high (above [??]) such that [W.sub.e] > [W.sub.f,e] > [W.sub.f]. This result suggests that in economies where the risk of theft of fiat money is low, and the investment in the e-money technology is costly, the use of fiat money only improves welfare. Conversely, in economies where the risk of theft of fiat money is high, carrying and investing in e-money improves welfare. Therefore, the equilibrium involving only the new payment instrument is not always socially optimal, contrary to Arifovic, Duffy, and Jiang (2017). In the presence of a multiplicity of equilibria in a laissez faire environment, coordination defaults are possible. Agents may end up in "bad" equilibrium as welfare could be increased by switching to another equilibrium. However, there are no external forces in our model, like government intervention for example, that could facilitate the transition to a better equilibrium in terms of welfare. (18)

V. ENDOGENOUS THEFT

To endogenize the total number of thieves in the economy, we assume that among the inactive population, a measure n of individuals decide to enter into the DM as thieves, at a flow cost [kappa] > 0. Thieves, whose only activity is to steal fiat money at the end of the DM, spend all of it in the CM by purchasing the general goods, and since they belong to the inactive population, they do not produce, or pay tax, nor do they receive any transfer from the government in the CM.

The matching probability of an active person (a buyer or a seller) and a thief becomes [alpha](n), and we assume that [alpha](n) is increasing and concave, with [alpha](0) = 0, [alpha]([infinity]) = 1, [alpha]'(0) = [infinity], and [alpha]'([infinity]) = 0. Therefore, at the end of the DM, an active person may encounter a thief with probability [alpha](n), and a thief may encounter an active person with probability 2[alpha](n)/n since he may either encounter a buyer or a seller. In an encounter between a thief and an active person holding fiat money, the former steals all the units of fiat money held by the latter, whose real value (f) now depends on the total number of thieves in the DM: f=f[[alpha](n)]=f(n).

From the linearity of the value function in the CM, the payoff of a thief who enters the DM is [V.sup.T] =-[kappa]+f(n)2[alpha](n)/n + [beta]max{[V.sup.T],0}. (19) We assume free entry of thieves, meaning inactive people enter the DM as thieves until the expected profit from entering the DM as a thief is zero, or [V.sup.T] = 0. Consequently, the number of thieves among the inactive population solves the following equation:

(12) f(n) = [kappa] n/2[alpha](n)

where n/2[alpha](n) is an increasing function of n.

According to Equation (12), inactive people enter the DM as thieves until the expected benefit of stealing fiat money (f(n)2[alpha](n)/n) equals the cost of entering as a thief ([kappa]). The gain from stealing fiat money decreases with the number of thieves in the economy: f'(n) < 0. Subsequently, there is a negative relationship between n and fin), as shown in Figure 5. Conversely, the expected cost of entering the DM as a thief increases with n. The decision rule (Equation (12)) endogenizes the number of thieves in the economy as shown in Figure 5, where the two curves intersect.

A. Equilibria

With endogenous theft, a steady-state equilibrium is now a list (q, f, e, [LAMBDA], n) that satisfies Equations (3), (7), (8), (10), and (12). In the following, we verify whether the previous types of equilibria with exogenous theft still exist with the free entry of thieves.

When no seller accepts e-money, or [LAMBDA] = 0, e-money is not valued, and the value of fiat money [f.sup.0](n) is a decreasing function of n. As shown in Figure 5, for Equation (12) to be satisfied, the optimal number of thieves in the economy must adjust to [n.sup.0]. The real value of fiat money that agents hold, and thieves can steal, is therefore [f.sup.0]([n.sup.0]) (point 1 in Figure 5). Note that because buyers refuse to hold cash when the risk of theft is too high ([alpha] [greater than or equal to] [bar.[alpha]]), there is an upper boundery for the number of thieves ([bar.n]) above which buyers do not enter the DM with cash. Consequently, above [bar.n] there is no fiat money to steal in the market, and no thief enters the market.

When some sellers accept e-money, or [LAMBDA] [member of] (0,1), fiat money is always less valued than when [LAMBDA] = 0 for any given n > 0, implying [f.sup.[LAMBDA]](n)< [f.sup.0](n) with [f.sup.[LAMBDA]](n) a decreasing function of [LAMBDA]. This has two consequences. First, it can be seen in Figure 5 that when e-money is accepted by some sellers in parallel with fiat money, the number of thieves in the economy at the equilibrium ([n.sup.[LAMBDA]]) is lower than when only fiat money is accepted, and valued ([n.sup.0]). The reason is that the expected cost of entering the DM as a thief must decrease when the gain from stealing fiat money decreases, so n has to adjust downward. Secondly, we illustrate that the number of sellers ([LAMBDA]) who must accept e-money for a mixed monetary equilibrium to exist is different with exogenous and endogenous theft. With exogenous theft, at n = [n.sup.0], for example, the value of fiat money in a mixed monetary equilibrium would be determined at point 2 in Figure 5, where the fraction of sellers who accept e-money ([LAMBDA]) would satisfy Equation (10). However, with endogenous theft, for this given [LAMBDA], the number of thieves must decrease to point 3 in Figure 5 to satisfy Equation (12), but as the reduction in the number of thieves increases the real value of money, the number of sellers who accept e-money ([LAMBDA]) must increase for Equation (10) to continue to be satisfied when fiat and e-money coexist, implying a downward shift of the [f.sup.[LAMBDA]](n) curve toward its equilibrium position (point 4 in Figure 5), where the number of thieves is [n.sup.[LAMBDA]]. The intuition is that because the disadvantage of carrying fiat money decreases with a reduction in the number of thieves, the relative advantage of holding e-money must increase to maintain the coexistence of fiat and e-money in the economy, so more sellers must accept it in equilibrium ([LAMBDA] must increase). Therefore, to obtain the coexistence of fiat and e-money, the number of sellers who accept e-money in equilibrium must be higher with endogenous theft than with exogenous theft if the number of thieves in the economy is initially higher than its equilibrium value, as in our example. (20)

Finally, one characteristic of the model is that the pure e-money equilibrium ([LAMBDA]= 1) obtained with exogeneous theft disappears with endogenous theft: If buyers expect that all sellers accept e-money ([LAMBDA] = 1), their best response is to enter the DM with e-money only. Hence, thieves expect no gain from entering the DM and no thief enters; the risk of theft ([alpha]) becomes zero. However, since buyers know that the best decision of sellers when [alpha] = 0 is not to invest to receive e-money when it is costly, they do not enter the DM with emoney, and carry fiat money which in turn gives an incentive for thieves to enter. Consequently, an equilibrium with [LAMBDA]= 1, where only e-money circulates, is not feasible with the free entry of thieves, which means that it is impossible with endogenous theft to eliminate cash totally from circulation when it is accepted in trade by merchants. (21)

In summary, the adoption process of e-money by both buyers and sellers ([LAMBDA]), and the multiplicity of equilibria of our model, are affected by endogenous theft, as shown in Figure 3B, where the e-money equilibrium (E), and the equilibria with [alpha] > [bar.[alpha]], are not feasible. Only a pure fiat money equilibrium (F), and an equilibrium with both fiat and e-money (M) can coexist with the free entry of thieves.

B. Monetary Policy

The last point we study is the effect of monetary policy on the adoption process of e-money. When the nominal interest rate (money growth, or inflation rate) is modified, the opportunity cost of holding fiat or e-money is affected, which modifies agents' portfolio choices, the real value of both fiat and e-money in equilibrium, and in return the expected benefit of entering the DM as a thief. Therefore, the number of thieves in the economy will adjust to the new interest rate. This last result is similar to He, Huang, and Wright (2008), although here there are strategic complementarities between buyers and sellers, absent in their model, which gives rise to a new transmission channel of monetary policy, as explained below.

Similar to He et al., an increase in the nominal interest rate reduces the real value of fiat money, and the number of thefts in the economy, but here it also affects the number of sellers who must invest in the e-money technology ([LAMBDA]) for a mixed monetary equilibrium to exist. In particular, when the interest rate increases, the number of sellers who accept e-money must increase for Equation (10) to be satisfied, meaning it is more difficult to sustain an equilibrium where both fiat and e-money circulate in the economy when inflation increases. The intuition is the following: when the opportunity cost of holding money increases, the value of fiat money (e-money also) decreases. Therefore, it becomes more costly for inactive agents to be a thief, and fewer thieves enter the market, which increases the value of fiat money relative to e-money. Consequently, the inconvenience of holding fiat money relative to e-money is lowered as the number of thieves in the market (and the probability to be stolen) is reduced, which affects the mixed monetary equilibrium. To increase the relative advantage of holding e-money in parallel with fiat money, and therefore allow for the existence of a mixed monetary equilibrium, the number of sellers accepting e-money must increase in equilibrium to satisfy both Equations (10) and (12).

In summary, when inflation, or the nominal interest rate increases, it makes the buyers' adoption of e-money more difficult with endogenous theft than with exogenous theft as it increases the fraction of sellers ([LAMBDA]) who must accept e-money in equilibrium for fiat and e-money to coexist. Conversely, a decrease in the nominal interest rate (or inflation) improves the buyers' adoption process of e-money as it decreases the fraction of sellers (A) who must accept e-money in equilibrium for both fiat and e-money to circulate in the economy. Thus, with the endogenous risk of theft, a low interest rate policy facilitates the buyers' adoption of e-money in the mixed monetary equilibrium.

VI. CONCLUSION

In order to better understand why the launch of a new type of payment instrument may succeed or fail, it is important to analyze the crucial role of buyers' and sellers' decisions. In our model, we examine the policy of substituting fiat money with e-money by considering the direct and indirect costs and benefits of each type of money. More accurately, we determine monetary equilibria as a function of three main variables: the safety level of the monetary instrument that can be used as a medium of exchange, the cost of investment in a new e-payment terminal, and the sellers' e-money adoption rate.

Different types of equilibria can emerge. When all sellers accept the two payment instruments, the unique equilibrium is such that only e-money is used, meaning fiat money is no longer valued. We also demonstrate the existence of an equilibrium where both fiat and e-money are used as a payment instrument, although only a few sellers accept e-money. This equilibrium coexists with a pure fiat money and pure e-money equilibrium. The adoption of e-money may improve welfare compared to the exclusive use of fiat money, or reduce it, depending on the risk of theft, the investment cost, and the number of sellers who accept e-money. Additionally, we endogenize the risk of theft of cash. We study its effect on the adoption level of e-money, the multiplicity of equilibria, and show that it is easier to sustain an equilibrium where both fiat and e-money circulate when inflation decreases.

Finally, past experiences of e-money success or failure can be explained by our model. In Japan or Singapore, where e-purses are widely used, we can see the importance of prescribed uses in the adoption process of e-money. In these countries, not only the purchase of mass transit tickets must be made with e-money, but it is the payment instrument issuer that fully incurs the investment cost. In the United States, the recent success of prepaid cards use is linked to the fact that these cards are compatible with the payment terminals used by the bank industry for debit and credit card transactions. In this situation, merchants have no additional investment cost to incur. Finally, in Europe, the failure of the electronic purse is probably due to the lack of existence of a captive market, high investment costs, and no "critical mass" of users necessary for a new payment instrument to be widely adopted.

ABBREVIATIONS

CM: Centralized Market

DM: Decentralized Market

e-money: Electronic Money

e-purses: Electronic Purses

doi: 10.1111/ecin.12714

APPENDIX

The buyer's DM value function is:

[mathematical expression not reproducible].

From the linearity of the value function in the CM, it can be written:

[mathematical expression not reproducible]

and from the terms of trade (Equation (3)), we obtain Equation (5).

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SEBASTIEN LOTZ and FRANQOISE VASSELIN *

* We would like to thank Guillaume Rocheteau, Yiting Li, and Cathy Zhang for helpful discussions, and the three referees for their comments. We are also grateful to Jeremie Cabessa. This paper first circulated in 2013 under the title "Electronic Purse versus Fiat Money: A Harsh Competition," and has been conducted as part of the project Labex MME-DII (ANRI l-LBX-0023-01). The usual disclamer applies.

Lotz: Professor, LEMMA, University of Paris II, 75006 Paris, France. Phone +33170232004, E-mail lotz@uparis2.fr

Vasselin: Dr, LEMMA, University of Paris II, 75006 Paris, France. E-mail f.vasselin@orange.fr

(1.) Namely "Proton" in Belgium, "Gelskarte" in Germany, "Chipnick" in the Netherlands, "Mondex" in the United Kingdom, "Moneo" in France, "Octopus" in Hong Kong, "Suica" in Japan, or "EZ-link" in Singapore.

(2.) Electronic money is defined as "a stored value or prepaid product in which a record of funds or value available to the consumer for multipurpose use is stored on an electronic device in the consumers' possession" (CPSS 2004). The device acts as a prepaid bearer instrument which does not necessarily involve bank accounts in transactions. It is different from a debit card which is linked to a bank account.

(3.) The principal advantage of e-purses, or mobile payment applications, is that transfers are instantaneous and free of charge for individuals, contrary to debit, or credit cards. They are also easier, and safer than using cash. Additionally, no minimum amount is required for transactions. In a sense, they compete with cash, and debit cards.

(4.) These successful examples have occurred due to some unusual circumstances. The use of e-purses have been strongly promoted and subsidized in Hong Kong, Singapore, and Tokyo, while in the United States, e-money provides a new service to a portion of the population that have no access to payment methods other than cash.

(5.) Kenneth Rogoff argues, in his book, The Curse of Cash (Rogoff 2016), that there are good reasons to phase out cash: it would lower tax evasion and criminal activity. Other arguments suggest that cash has bad qualities as well. Notes and coins can be lost or stolen, are heavy, inconvenient, can be counterfeited, and are costly to manage.

(6.) See Williamson and Wright (2010), Rocheteau and Nosal (2017), and Lagos, Rocheteau, and Wright (2017) for the main topics and issues studied in New Monetarist models.

(7.) e-Money collected by merchants is electronically recorded in the reading terminal before being sent by communication networks to their bank account to be credited (Lacker 1996). For buyers, banks can guarantee the refund of e-money lost or stolen if e-purse holders have chosen a safe electronic means of storage which is personalized. For a discussion on the fact that cash is less secure than other payment methods, see He, Huang, and Wright (2008).

(8.) The technology differs from record keeping technology since it allows the transfer of units of e-money between buyers and sellers, but not their trading histories, or identity. Hence, our agents remain anonymous, credit is not possible, and money is essential. For additional literature on money and credit, and dual-currency search models, see Liu (2016). On the question of the essentiality of money, see Gu, Mattesini, and Wright (2016).

(9.) In Appendix 4 of our working paper version, we revisit the analysis with Nash bargaining, and illustrate that the results are similar to the ones described in the paper with the proportional solution.

(10.) In Appendix 5 of our working paper version, we show that the results would be similar if cash were stolen before trade, although it would complicate the analysis.

(11.) American Express, Mastercard, and Visa advertise their reloadable cards as "safer than cash," therefore allowing registered e-purse consumers to recover money if lost or stolen; "you can't say that about cash" (mastercard.us/prepaid-card).

(12.) We assume that the investment cost is identical for all sellers, for simplicity. Sellers could also be heterogeneous according to their investment cost, like in Lotz and Zhang (2016).

(13.) A necessary condition for money to be valued (q > 0) is i <-[sigma][theta]/1-[theta], which is assumed in the following analysis. Note that there always exists a nonmonetary equilibrium driven by beliefs when inflation is low.

(14.) The total trade surplus S(q(z)) = u(q(z)) - c(q(z)) is trictly concave if z < [theta]c(q*) + (1 - [theta])u(q*) = z(q*), which is satisfied when i > 0.

(15.) When i = 0, and 0 < [alpha] < [bar.[alpha]], e-money is not costly to hold, whereas cash is. Therefore, buyers should always hold e-money, meaning an equilibrium where [LAMBDA] = 0 is impossible as long as I is not too high. The only equilibrium in this case is a pure e-money equilibrium (see Section A).

(16.) Siu (2008) compares the success and failure of two electronic payment systems in Hong Kong. "Mondex was endowed with the full legal status of money, launched by a mammoth banking group, with Mondex cards given away for free to consumers. Yet the system went into oblivion within five years. Octopus started as a modest stored value transport ticket that required a deposit. It ended up as a city-wide multipurpose payment card used by 95% of the adult population. (...) The two systems sought to overcome user resistance in different approaches: Mondex relied on voluntary uptake, but Octopus imposed a compulsory switch upon a large base of commuters." Therefore, social consensus, network effects, and point-of-sales devices are important aspects of the success of a new payment instrument.

(17.) The analysis does not take into account some private and public costs linked to the use and management of cash in practice, which would reduce welfare in the pure fiat money economy.

(18.) The question of the transition from one equilibrium to another is not analyzed since we assume rational expectations. However, this question could be interesting in a different environment, with heterogenous beliefs and learning, like in Branch and McGough (2016), for example. Arifovic, Duffy, and Jiang (2017) study the question of the equilibrium selection by conducting a laboratory experiment. They find that a lower fixed fee for sellers favors very quick adoption of the new payment instrument, while for a sufficiently high fee, sellers gradually learn to refuse to accept it.

(19.) When considering similar reasoning with the free entry of banks, see Rocheteau, Wright, and Zhang (2018).

(20.) We would have the opposite result if the number of thieves were initially lower than its equilibrium value.

(21.) A similar result is in He, Huang, and Wright (2008). The only way to obtain here a pure e-money equilibrium with free entry of thieves would be to assume that sellers' investment is free, or that once sellers accept e-money, they refuse cash transactions (which is currently what Swedish merchants do, although cash is legal tender).

Caption: FIGURE I Timing

Caption: FIGURE 2 Real Balances and Total Surplus

Caption: FIGURE 3 Monetary Equilibria. (A) Exogenous Theft and (B) Endogenous Theft

Caption: FIGURE 4 Welfare Comparison

Caption: FIGURE 5 Endogenous Theft
TABLE 1

Comparative Statics

[partial               [partial                   [partial derivative]
derivative][??]/       derivative][G.sup.[??]]/   [LAMBDA]/[partial
[partial               [partial                   derivative]I
derivative][alpha]     derivative][alpha]

 -                                 -                    +

[partial               [partial
derivative][bar.       derivative][??]/
[LAMBDA]]/[partial     [partial
derivative]i           derivative]i

+                                 -

TABLE 2
Social Welfare Across the Three Types of
Equilibria

Equilibrium        Steady-State Welfare

Fiat money         [W.sub.f] [equivalent to] [sigma][S.sup.0.sub.f] =
                   [sigma]S([q.sup.0.sub.f])

e-money            [W.sub.e] [equivalent to] [sigma][S.sup.1.sub.e] =
                   [sigma]S([q.sup.1.sub.e])-I

Fiat and e-money   [W.sub.f,e] [equivalent to]
                   [[LAMBDA][S.sup.[LAMBDA].sub.f,e] + (1-
                   [LAMBDA])[S.sup.[LAMBDA].sub.f]]-[LAMBDA]I =
                   [sigma][LAMBDA][theta][[S.sup.[LAMBDA].sub.f,e]-
                   [S.sup.[LAMBDA].sub.f] +
                   [sigma][S.sup.[LAMBDA].sub.f]
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Author:Vasselin, Sebastien Lotz Francoise
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