On the Essentiality of E-money∗

Jonathan ChiuVictoria University of Wellington/ Bank of Canada

Tsz-Nga WongFederal Reserve Bank of Richmond

March 2017

Abstract

Recent years have witnessed the advances of e-money systems like Bitcion, Paypal and M-Pesa. Thispaper adopts a mechanism design approach to identify the essential features of different payment tech-nologies that implement the constrained optimal resource allocation. E-money technologies with exclusiveparticipation or discretionary transferability of balances mitigate fundamental frictions and enhance ef-ficieny, if the conditions in terms of parameters like inflation, trade frequency and bargaining powers aresatisfied. Otherwise, e-money technologies are not essential, compared to cash. An optimally designede-money system exhibits realistic arrangements including non-linear pricing, cross-subsidization, depositand positive interchange fees beyond the costs. Regulations such as a cap on interchange fees (à la theDodd-Frank Act) can distort the optimal mechanism and reduce the welfare improvement.

Keywords: money, electronic money, mechanism design, search and matching, effi ciency.

JEL Codes: E, E4, E42, E5, E58, L5, L51

∗We are grateful to Charlie Kahn, Will Roberds, Steve Williamson and Shengxing Zhang as the discussants for this paper.We have benefited from the comments and suggestions of members of the Bank of Canada E-money Working Group, Ben Fung,Scott Hendry, Tai-Wei Hu, Miguel Molico, Pedro Gomis-Porqueras, Guillaume Rocheteau, Marc Rysman, Daniel Sanches, OzShy, Robert Townsend, Neil Wallace, Zhu Wang, Warren Weber, Randy Wright and seminar participants at the Universityof Wisconsin at Madison, FRB of Philadelphia, FRB of Chicago, AMES 2014, CEA-CMSG 2014, Singapore ManagementUniversity, Chinese University of Hong Kong, Bank of Canada and Economics of Payment VII Conference at Boston Fed. Theviews expressed here do not necessarily reflect the position of the Federal Research System or the Bank of Canada.

1 Introduction

Recent years have witnessed a number of retail payment innovations known as electronic money (or e-

money).1 Well-known examples include Bitcoin, PayPal, M-Pesa and various forms of stored-value cards.2

The functioning and pricing structure of these e-money products are often quite different from those of

conventional cash. Whether these e-money technologies can improve effi ciency relative to cash? If so, how

should a desirable e-money payment system look like? If not, why? Under what condition the policy makers

should regulate e-money payment system?

To the best of our knowledge, this is the first paper that develops a micro-founded general equilibrium

model of e-money to study the above economic questions. To see the effi ciency role of e-money it is necessary

to know why money is needed in the first place. Thanks to recent developments in monetary theory, it is

now widely recognized that in the presence of such deep frictions as the lack of commitment and lack of

record-keeping, the use of money as a payment instrument improves the effi ciency of resource allocations

(Kocherlakota, 1998). In this sense, money, as a medium of exchange, is essential because it improves

effi ciency relative to an economy without money. However, modern monetary theory also teaches us that, in

a world subject to frictions that render money essential, the equilibrium allocation is typically suboptimal.

It is because the use of money requires pre-investment, giving rise to a cash-in-advance constraint. Impatient

agents acquire too little money, and hence being liquidity constrained in trading.

It is important to notice that the aforementioned frictions rendering money essential also shape the basic

functioning of monetary payment systems. Owing to its full anonymity and decentralization of trades, mone-

tary payment systems work with the following features. First, it permits non-exclusive participation: anyone

is free to hold cash without other prerequisites. Second, it allows unrestricted transferability : any amount of

cash is free to change hands at any time anywhere. In other words, non-exclusive participation means that

there is no limitation on who can use money (the extensive margin), and unrestricted transferability means

there is no limitation on how money is used (the intensive margin).

We argue that an e-money-based payment system is fundamentally different from the money-based system

because it can be free of the above-noted features via different technological breakthrough. First, e-money

issuers can exclude certain agents from participation. For example, in card-based e-money systems such

as the Octopus card in Hong Kong, only buyers who have already acquired stored-value smart cards and

1The Survey of Electronic Money Developments by the Committee on Payment and Settlement Systems (CPSS) noted that“in a sizeable number of the countries surveyed, card-based e-money schemes have been launched and are operating relativelysuccessfully: Austria, Belgium, Brazil, Denmark, Finland, Germany, Hong Kong, India, Italy, Lithuania, the Netherlands,Nigeria, Portugal, Singapore, Spain, Sweden and Switzerland. Network-based schemes are operational or are under trial in afew countries (Australia, Austria, Colombia, Italy, the United Kingdom and the United States), but remain limited in theirusage, scope and application.” (CPSS, 2001)

2There is no universal definition for e-money that can fit precisely all exisiting variants of e-money products. One definitionof e-money proposed by CPSS is the following: it is the “monetary value represented by a claim on the issuers which is storedon an electronic device such as a chip card or a hard drive in personal computers or servers or other devices such as mobilephones and issued upon receipt of funds in an amount not less in value than the monetary value received and accepted as ameans of payment by undertakings other than the issuer.” This defintion is quite broad (e.g. including debit cards), and atthe same time quite narrow (e.g. excluding Bitcoin). Similarly, the European Central Bank defines e-money as “an electronicstore of monetary value on a technical device that may be widely used for making payments to entities other than the e-moneyissuer.”For the purpose of this paper, we don’t need to stick with one specific definition of e-money. Instead, we will examineseveral features that are commonly found in e-money products.

1

Bitcoin M-Pesa Octopus card Paypal CashExclusive participation X X XDiscretionary transferability X X X

Table 1: Examples of recent e-money products.

merchants who have obtained card readers/writers from the card operator can participate in the system to

conduct payments.3 Similarly, in server-based e-money systems such as PayPal, only individual and business

users who have already signed up for an account can hold, send and receive e-money balances. In this type

of e-money scheme, non-compliance leads to exclusion from the system. Second, e-money systems can

reject balance transfers. For example, in centralized e-money schemes such as PayPal, the system operator

maintains user accounts and performs payment processing, and thus has the ability to block or restrict the

size or direction of balance transfers.4 In some decentralized systems such as Bitcoin, bilateral transactions

can be completed only after they are verified and written into a general ledger by other users (e.g. the

proof of work by Bitcoin miners). In addition, according to CPSS (2001), it is quite common globally that

the transferability of e-money balances among end-users is restricted. Specifically, 77% of e-money systems

included in that survey prohibit transferability among end-users. These two kinds of technology do not imply

each other, for example Bitcoin system cannot block any IP address from joining its peer-to-peer (P2P)

network; essentially any offl ine token-based system cannot stop any transfer of balances. Furthermore, since

balances are transferred through electronic devices, it is technically feasible to have non-zero-sum transfers:

the amount of balances transferred by the payer differs from the amount received by the payee. For example,

the payee receives only $97.1 for every $100 sent to the payee through the PayPal system. Bitcoin also has

a built-in feature that allows the individual making a transaction to include a transaction fee paid to the

Bitcoin miner. This feature of e-money can allow for charging merchants fees or other transaction fees, which

are often observed in e-money payment systems.

Of course, the fact that e-money is fundamentally different from money does not necessarily mean that it

is more essential. Our notion of essentiality borrows from Hahn (1973): e-money is more essential than money

if the use of e-money allows some implementation of socially desirable allocations that are not implementable

with the use of money only. Here, the key is the design of implementation - without a system fully exploits its

power a good technology can be useless. We use a mechanism design approach to analyze whether any of e-

money’s distinctive features are essential for the implementation. The starting point is a basic environment in

which traditional cash is used as payment. We then gradually attach to it additional features, including some

distinctive technologies of e-money. For each layer of e-money technology we formulate the set of resource

allocations that can be implemented by some payment mechanisms making using of that technology. In

3Probably the most successful adoption of e-money in the developed countries, by the end of December 2015 the totaloutstanding balances of the Octopus card (float plus deposit) amounted to 30% of the total note and coin in circulation,according to Hong Kong Monetary Authority.

4A decentralized e-money scheme is one in which the payment network is not provided or managed by a single networkprovider or operator (for example, a system in which e-money is stored and flows through a peer-to-peer computer network thatdirectly links users). In contrast, in a centralized network, there is a trusted third party that manages the payment network.One example is server-based e-money schemes such as PayPal.

2

doing so we can identify which features of e-money under what conditions of the economy can achieve better

resource allocation upon a cash economy. Under these conditions, the e-money technology is essential; we

can further compare which e-money technology more essential than other. Otherwise, cash can do the job

well and no need to introduce the e-money.

We show that only under some conditions will the introduction of e-money relax certain binding con-

straints faced by the money issuer and allow more flexible and effi cient intervention. As a benchmark, the

first main finding of our paper is that an ineffi cient allocation can arise even in an optimally designed mon-

etary system (subject to non-exclusive participation, unrestricted transferability and zero-sum transfers).

The second main finding of our paper is that certain new features of e-money are essential because they help

achieve cross-subsidization between buyers and sellers and improve effi ciency in resource allocation relative to

a payment system without these features. A mild inflation can strictly improve effi ciency under an e-money

system. Interestingly, we show that exclusive participation can be more or less essential than discretionary

transferability, depending on primitives such as bargaining powers and the frequency of trade. Finally, we

characterize some key properties of an optimally designed e-money mechanism, and provide examples of

simple direct and indirect mechanisms with membership fee, reward, deposit and interchange fee.

As discussed above, this paper identifies some welfare-enhancing features of e-money, which can be

found in certain existing schemes. We intentionally focus on examining specific technologies (e.g., limiting

participation and transferability) rather than specific e-money products (like Octopus card or Bitcoin),

because an e-money product may introduce a bundle of new features but most likely only some of them are

welfare enhancing. Even if one’s ultimate goal is a better design of e-money products, it is useful to identify

role of each essential technological element.

Finally, our paper is highly relevant to recent policy discussion. Developments in payment technologies

raise new challenges for policy-makers. The Federal Reserve System, for example, has been soliciting public

inputs on strategies and tactics for reforming the U.S. payment system.5 More specifically, in the “Survey

of Electronic Money Developments,” the Bank for International Settlements highlighted that “Electronic

money projected to take over from physical cash for most if not all small-value payments continues to

evoke considerable interest both among the public and the various authorities concerned, including central

banks.”(CPSS, 2001) Against this context, the Bank of Canada has developed an active research agenda to

understand and monitor e-money products.6 While policy-makers are definitely concerned about these new

developments, so far there has been limited guidance provided by economic theory regarding the welfare

implications of e-money adoption. To the best of our knowledge, no existing research on e-money performs

welfare analysis giving serious consideration to fundamental frictions in payment systems. While modern

monetary theory focuses on understanding the fundamental roles of conventional money and credit, the role

of e-money has not yet been explored. Our paper is also the first to apply the mechanism design approach

to study the design of retail payment systems. By uncovering essential features of a payment instrument5See the Payment System Improvement - Public Consultation Paper (https://fedpaymentsimprovement.org/wp-

content/uploads/2013/09/Payment_System_Improvement-Public_Consultation_Paper.pdf).6See, for example, the webpage http://www.bankofcanada.ca/research/e-money.

3

such as e-money, our results can provide guidance to policy-makers on how to design the future payment

systems, as well as whether and how an e-money system should be regulated. For example, our model can

be used to evaluate the effect of imposing a cap on interchange fees in an e-money system (similar to that

introduced by the Durbin Amendment to the Dodd-Frank Act7).

Literatures

So far there are not many papers about the effi ciency under an e-money system. From a different

perspective of effi ciency, Agarwal and Kimball (2015) advocate the use of e-money because it can implement

a negative interest rate and expand monetary policies beyond the zero lower bound (hence welfare improving).

To implement a negative interest rate, certain fraction of e-money balances will be automatically deduced

and there will be a fee to convert e-money to cash. While Agarwal and Kimball (2015) think of e-money as

reserve accounts in a central bank, in general the scheme requires centralization and a lot of administration

power to implement. Furthermore, anticipating the negative rate, people may not use the e-money in the first

place. This voluntary participation constraint is respected in our mechanism design approach. Rogoff (2016)

suggests subsidizing the use of e-money because cash is too bad - feeding tax evasion and criminal activities.

On the other hand, Fernández-Villaverde and Sanches (2016) argue against a decentralized, competitive

private issuance of fiat money, a likely scenario after the dissemination of e-money technology. Contrast to

what predicted by Hayek, they found that having private money does not improve effi ciency because the

private issuers take the price as given and the pricing scheme is linear. Here the mechanism design approach

does not impose any restriction on the pricing scheme - in doing so it takes into account all the incentive

constraints, participation constraints and resource constraints. In general we find in some situations having

a well-designed e-money system improves effi ciency, sometimes it cannot even with any non-linear scheme.

Our paper is directly related to the literature of monetary theory. In general, this literature focuses on an

economic environment in which contracts involving inter-temporal obligations are infeasible, due to frictions

such as the lack of commitment and lack of record-keeping, and in which money is the only durable object

that can serve as a means of payment. Lagos and Wright (2005) develop a tractable framework with the

presence of these frictions for studying the roles of money and monetary policy. Recent models of payment

systems building on the Lagos-Wright framework include Kahn and Roberds (2009), Li (2011) and Monnet

and Roberds (2008).8 As mentioned above, in a monetary economy, the socially optimal allocation (the

first best) typically cannot be implemented without an appropriately designed mechanism. Moreover, the

implementation of the constrained optimal allocation (the second best) is usually not unique.

There are two strands of research, both taking the payment system as given, but which focus on different

implementation mechanisms. The first strand takes an ineffi cient trading protocol as a primitive, and studies

7See, for example, Wang (2012) for details and discussion of this interchange fee regulation.8Another line of research is the two-sided-market literature in the field of industrial organization. See the surveys by Rochet

and Tirole (2003), Kahn and Roberds (2009) and Rysman (2009). This literature typically studies a partial equilibrium settingand assumes a particular form of fee structures. See Shy and Wang (2011) for a recent study on interchange fees, which usesthe two-sided-market approach to analyze a payment environment related to ours.

4

the design of monetary policy to mitigate this ineffi ciency.9 For example, Lagos and Wright (2005) and Lagos

(2010) find that the Friedman rule is optimal in these environments, but it involves taxing agents eventually.

The equilibrium allocation is not the first best because of the bargaining. With the use of a fixed fee

and linear transfers, Andolfatto (2010) illustrates how the first best can be implemented with voluntary

participation in a competitive environment.

The second strand of research, including Hu, Kennan, and Wallace (2009), Rocheteau (2012) and Wong

(2016), takes the suboptimal monetary policy as given, and designs the trading protocol to mitigate the

resulting ineffi ciency. These studies endogenize the trading protocol using a mechanism design approach, as

advocated by Wallace (2010).10 This literature finds that, under certain conditions, the first best can still

be implemented by adopting an optimal trading protocol in pairwise trades. Specifically, deviation from

Friedman’s rule can still be optimal, and the welfare cost of inflation can be zero.

Our paper is related to both strands of research. Unlike the first strand such as Lagos and Wright

(2005) and Andolfatto (2010), we do not restrict ourselves to any particular type of intervention, and use

the mechanism design approach to endogenize the payment instruments and payment system. Another key

difference from Andolfatto (2010) is that we model decentralized trade and ineffi cient bargaining, rather

than centralized trading, which tends to understate the distortions and hence overstate the power of policy,

as argued in Hu, Kennan, and Wallace (2009). Unlike the second strand such as Hu, Kennan, and Wallace

(2009), Rocheteau (2012) and Wong (2016), we take ineffi cient trading protocols as one of the primitive

inputs to the mechanism design of the payment system. Our perspective is particularly relevant for policy-

makers, such as central banks and payment system regulators, who arguably have limited influence over the

determination of terms-of-trade in a decentralized and anonymous situation.

The rest of the paper is organized as follows. We begin with a brief review of the welfare properties in

a standard monetary model in Section 2. While it takes little time, it shares some fundamental frictions in

the e-money environment, and it becomes clear to see what make e-money different and, more importantly,

essential. In Section 3 we introduce a mechanism design approach to the monetary model, which is easier

before having the e-money technologies. We also highlight the importance of a non-linear scheme and its

limitation. Section 4 designs the optimal e-money mechanism with a technology to restrict participation,

highlighting the importance of cross-subsidization as well as its limitation. Section 5 designs the optimal

e-money mechanism with a technology to restrict the transferability of e-money, illustrating the interchange

fees as an essential feature. We compare the essentiality between different e-money technologies. Section 6

extends the analysis to competitive pricing. Section 7 concludes.

9This literature is growing. See Williamson (2012) on the optimal monetary policy when public debt and private equity canbe media of exchange; Araujo and Hu (2014) on the optimal quantitative easing in an economy with money and credit; seeLagos, Rocheteau and Wright (2015) for a recent survey .10Kocherlakota (1998) is the first to use implementation theory to show the essentiality of money in a random matching

environment. Araujo, Camargo, Minetti and Puzzello (2012) show the essentiality of money in the Lagos-Wright environmentthat is used in this paper. See Kocherlakota and Wallace (1998) on money and credit; Kocherlakota (2003) on illiquid bonds;Hu and Rocheteau (2013) on money and high-yield assets; Hu and Zhang (2014) on money and capital with endogenous searchintensity.

5

2 Baseline Model

Overview. The environment follows Lagos and Wright (2005) with heterogenous agents similar to Lagos

and Rocheteau (2005). The economy is populated with unit measures of buyers and of sellers. Time is

discrete and infinite, indexed by t = 0, 1... Lagos and Wright (2005) devise a tractable model of frictional

markets mixing with frictionless markets such that the former captures the key frictions that explain the

social role of money emphasized in the literature, and the latter allows agents to rebalance after bilateral

trades. In particular, alternating in each period are subperiods of day and night: during the day a frictional

decentralized market (DM) convenes where buyers and sellers match randomly and trade bilaterally; during

the night a frictionless centralized market (CM) convenes where agents trade with each other at Walrasian

prices. In the DM, agents can only observe the actions and outcomes of their trades, and are anonymous.

There is no technology for monitoring actions, enforcing contract or coordinating global punishment. As a

result, credit is infeasible and a medium of exchange - money in this section or other payment instruments

in latter sections - is essential for trades in the DM. These essential frictions are taken into account in the

design of payment mechanisms latter.

Technology and preference. In the DM, buyers consume the DM goods q produced by sellers in the

bilateral meeting. In the CM, all agents can consume/produce the numeraire l in the centralized market

(with l > 0 denoting consumption and l < 0 production).11 The preferences of buyers and of sellers are

represented by

Vb ≡ E∞∑t=0

βt {U (qt) + lt} ,

Vs ≡ E∞∑t=0

βt {−C (qt) + lt} ,

where β ≡ 1/ (1 + r) is the discount factor with the rate r > 0, U(q) is the buyer’s utility function and C(q)

is the seller’s cost function in the DM. We assume that U ′ > 0, U ′′ < 0, U(0) = 0 and limq→0 U′(q) = ∞;

C(0) = 0, C ′ (q) ≥ 0, C ′′ (q) ≥ 0 and C ′ (0) = 0. Due to quasi-linearity of preferences, the welfare of this

economy is simply U (q)− C (q). The first best is the allocation q = q∗, where q∗ = arg max [U (q)− C (q)].

While we will introduce various payment technologies latter, in the baseline model money is the only

medium of exchange. The stock of money is denoted by Mt, which has an exogenous growth factor µ so that

Mt+1 = µMt. In this section new money is injected to agents by lump-sum transfers in the CM. Let φt be

the price of money in terms of the numeraire (the CM goods), so the real value of money m, also known as

real balances, is φm. As seen later, the real balances are the relevant state variable, which are denoted as z

for the value at the beginning of the DM and z for the value at the beginning of the CM.

In the DM, a buyer trades with a seller with probability α ∈ (0, 1], which can also be interpreted as

the frequency of transaction or of the preference shocks. Suppose the terms of trade are given by paying

d = d (zb, zs) units of real balances (money in term of goods) for q = q (zb, zs) units of goods. While details

11Our results do not depend on the linear preferences on l, which simplifies some of the analysis and notation. See Wong(2016) for a discussion on the general Lagos-Wright environment. Similarly our results are not specific to the bargaining model.

6

will be forthcoming, in general the term of trades, q (zb, zs) and d (zb, zs), are endogenous and depend on

the real balances of the buyer and of the seller, zb and zs. The DM value functions of buyer and of seller,

denoted by Vb and Vs, satisfy the following Bellman’s equation:

Vb (z) = α

∫[U [q (z, z′)] +Wb [z − d (z, z′)]] dFs (z′) + (1− α)Wb (z) , (1)

Vs (z) = α

∫[−C [q (z′, z)] +Ws [z + d (z′, z)]] dFb (z′) + (1− α)Ws (z) , (2)

where Fb and Fs are respectively the distributions of the buyers’balances and sellers’balances, and Wj is

the CM value function, j = b, s, which satisfies the following Bellman’s equation:

Wj (z) = maxz≥0,l

{l + βVj (z)} , s.t. l +φ

φ+1

z = z + T, (3)

where T is the real monetary transfers from the money issuer, and z is the real balances taken into the next

DM.

Decentralized markets and bargaining. The trading protocol follows Arouba, Rocheteau and Waller

(2007), which incorporate the proportional bargaining model of Kalai (1979) in the Lagos-Wright environ-

ment.12 Define the trade surpluses of buyer and of seller as

Sb(q, d; zb, zs) ≡ U (q) +Wb (zb − d)−Wb (zb) ,

Ss(q, d; zb, zs) ≡ −C (q) +Ws (zs + d)−Ws (zs) .

The terms of trade, q = q (zb, zs) and d = d (zb, zs), solve the following proportional bargaining problem:

maxq,d{Sb(q, d; zb, zs) + Ss(q, d; zb, zs)} , s.t. (4)

Sb(q, d; zb, zs) = θ[Sb(q, d; zb, zs) + Ss(q, d; zb, zs)], (5)

d ∈ [−zs, zb] , (6)

where θ ∈ (0, 1] is the buyer’s bargaining power that dictates the buyer a fraction θ of the total trade surplus,

Sb + Ss. Constraint (6) is the cash-in-advance constraint where the money changed hand cannot be greater

than zb or zs (in the latter case the seller pays the buyer).

Equilibrium. Define a stationary degenerate monetary equilibrium as follows:

Definition 1 A stationary degenerate monetary equilibrium consists of the price system {φt}∞t=0, the allo-

cation {q, d, zb, zs} and the policy {Mt, µ, T} , such that

a. (agents’optimization) given {φt}∞t=0, zb and zs solves (3);

b. (money markets clear) φtMt = zb + zs;

c. (bargaining) {q (zb, zs) , d (zb, zs)} solves (4) ;

d. (issuer’s budget constraint) given φt, {Mt, µ, T} satisfies 2T = (µ− 1)φtMt;

e. (monetary, stationary) φt > 0, φt/φt+1 = µ.

12We use proportional bargaining simply because it yields cleaner results than the Nash bargaining under the cash-in-advancedconstraint. As a robustness check we also consider the terms of trades determined by competitive pricing in the latter section.

7

The following lemma characterizes the bargaining solution in the equilibrium.

Lemma 1 Define D (q) ≡ (1− θ)U (q)+θC (q). In the equilibrium the bargaining solution {q = q (zb, zs) , d = d (zb, zs)}

is given by

d = min {zb, D (q∗)} ,

q = D−1 (d) .

Proof. Since Wb and Ws are linear, the bargaining problem (4) becomes

maxd≤zb,q

U (q)− C (q) , s.t. d = (1− θ)U (q) + θC (q) .

Then the solution is given by Lemma 1.

Intuitively, when the buyer brings enough money balances, i.e. z ≥ D (q∗), then the bargaining results in

first-best consumption with the terms of trade q∗ and d∗ ≡ D (q∗). In this case the buyer only spend some

of her money holding. However, when the buyer is constrained, i.e. z < d∗, she spends as much as possible

to buy q = D−1 (z) < q∗ units of goods.

A standard result in the literature, the following proposition characterizes the ineffi ciency in the equilib-

rium.

Proposition 1 Define µ ≡ β(

1 + αθ1−θ

). A monetary equilibrium exists iff µ ∈ [β, µ). If µ > β, then

q < q∗; if µ→ β, then q = q∗.

The equilibrium does not feature the first-best allocation q = q∗ when µ > β. The idea is that, to consume

q∗ in the next DM, a buyer needs to acquire the money balances zb = µd∗. So the marginal utility gain

with respect to q is βαU ′(q∗), while the marginal cost of acquiring the balance is [µ− β (1− α)]D′ (q∗) =

[µ− β (1− α)]U ′(q∗). As a result, a buyer has an incentive to marginally reduce q below q∗ when

(β − µ)U ′(q∗) < 0, (7)

which is true whenever β < µ.

To implement the first-best allocation in the baseline economy, it is necessary and suffi cient to deflate the

economy at the discount rate, i.e., µ = β < 1. Furthermore, the money issuer’s budget constraint implies

that the deflation is implemented by a positive lump-sum tax (a negative transfer T = (β − 1)φtMt/2 < 0).

If the money issuer has no taxation power, then this simple lump-sum transfer scheme cannot implement the

first best. It turns out that laissez faire, µ = 1, is the constrained optimal policy with lump-sum transfers,

which achieves the level of trade, q0, given by

C ′ (q0) =

[1− r

θ (r + α)

]U ′ (q0) ,

where we have q0 = 0 whenever the right side is negative (when the buyer bargaining power, θ, is suffi ciently

low, no trade can be sustained). A natural question is whether allocation can be improved with some

sophisticated non-lump-sum system, which is analyzed in the next section.

8

3 Optimal Money Mechanism

In this section, we adopt the mechanism design approach to study the implementation of the optimal allo-

cation. Even though there are infinitely many ways of setting up the payment system with money in this

economy, we show that one only needs to focus on a “threshold”mechanism. We then derive conditions

under which the first best can be implemented. We also characterize the constrained optimal allocations and

provide simple examples of indirect mechanism implementation.

Consider that now the money issuer can make transfers at night after the CM is closed. The money

issuer can distinguish between buyers and sellers, but cannot observe an agent’s past actions nor the money

balances - the lack of record-keeping that still renders money essential for the DM trades. The issuer cannot

produce, consume or confiscate any goods from agents. We formulate the optimal money mechanism and

set it as the (most conservative) benchmark to compare the essentiality of e-money.

A mechanism design approach. Thanks to the revelation principle, any equilibrium allocation of a

Bayesian game under a mechanism can be implemented by some direct mechanisms, where agents report

their private information to the mechanism designer (here reporting the money balances to the money issuer).

Then the mechanism designer makes the monetary transfers based on the report. For notational convenience,

we will assume that the report is about the post-transfer balance, z, and the pre-transfer balance can be

inferred straightforwardly. Notice that while it is feasible for an agent to under-report any amount z ≤ z

(i.e. hiding money), we assume that it is infeasible to over-report any amount z > z, since any over-reporting

can be easily verified (also known as the show-me-the-money constraint). The money issuer transfers the

agent Tj (z), which is based on the agent’s type j = s, b and of the report z. Formally, a money mechanism

consists of non-decreasing transfer functions for buyers, Tb (z) , and for sellers, Ts (z), and the money growth

factor µ, which is denoted asM≡ {Tb (z) , Ts (z) , µ}.

Under a money mechanismM, the Bellman equations to the DM value functions, Vj , remain the same.

The CM value functions solve the following Bellman equations, j = b, s

Wj (z) = z + maxz,z,ej∈{0,1}

{−µz + ejTj (z) + βVj (z)} , s.t. z ≤ z, (8)

where ej = 1 (ej = 0) denotes the decision (not) to participate the mechanism, z is the genuine holding of

real balances, z is the reported holding, upon which the money issuer makes the transfer Tj (z) if the agent

participates the mechanism. Let ej (z), zj (z) and zj (z) denote the solutions to (8) given his individual state,

z. Given the agent’s decision, we can characterize the set of allocation which can be implemented by some

mechanisms in the monetary equilibrium.

Definition 2 An allocation (q, d, zb, zs) is implementable under a money mechanismM if

a. ej (z) = 1 and zj (z) = zj (z) = zj solve (8);

b. q = q (zb, zs) and d = d (zb, zs) solve (4);

c. Tb (zb) + Ts (zs) = (µ− 1) (zb + zs) .

9

Notice that, as mentioned in the beginning, zb and zs are the real balances after the transfers. An

allocation (q, d, zb, zs) implementable under a money mechanism M is also an equilibrium allocation. The

constrained optimal allocation is the implementable allocation that maximizes the utilitarian welfare W ≡

U (q)− C (q).

Implementation. Consider the following threshold mechanism:

Tb (z) =

{T pb , if z ≥ z

pb ,

0, otherwise,(9)

Ts (z) =

{T ps , if z ≥ zps ,0, otherwise,

(10)

where T pb , Tps , z

pb and z

ps are some constants.

13 Under this mechanism, an agent j = b, s will receive a

fixed transfer T ρj from the mechanism as long as he carries and reports a suffi ciently high money balances.

The following proposition states that any allocation implemented by some money mechanism can also be

implemented by the threshold mechanism given by (9) - (10). The proof is provided in the Appendix.

Proposition 2 An allocation is implementable under some money mechanisms if and only if it is imple-

mentable under some threshold mechanisms given by (9) - (10).

As a result of Proposition 2, it is without loss of generality to focus on the threshold mechanism. It

greatly reduces the set of candidate mechanisms. In particular, an allocation (q, d, zb, zs) is implementable

if and only if there exists a threshold mechanism such that

D (q) = d = min (zb, d∗) , (11)

(β − µ) zb + T pb + βαθ [U (q)− C (q)] ≥ maxq′{(β − µ)D (q′) + βαθ [U (q′)− C (q′)]} , (12)

(β − µ) zs + T ps ≥ 0, (13)

T pb + T ps = (µ− 1) (zb + zs) . (14)

Equation (11) dictates the buyer’s holding of real balances required to trade q units of goods; equation (12)

is the buyer’s incentive constraint that she will join the mechanism and bring the targeted real balances,

zb, to trade; equation (13) is the seller’s incentive constraint that he will join the mechanism and bring the

targeted real balances, zs; equation (14) is the budget constraint of the money issuer. Thus, the optimal

money mechanism is the threshold mechanism that solves

maxq,d,zb≥0,zs≥0,Tpb ,T

ps ,µ≥β

{U (q)− C (q)} s.t. (11) - (14). (15)

An optimal money mechanismM implements the first best if there exist zb and zs such that (q∗, d∗, zb, zs) is

implementable under some threshold mechanisms. Otherwise some of the constraints (11) - (14) are binding

and the second best is implemented. The following proposition characterizes the implementability of the

first-best allocation under an optimally designed money mechanism.13Obviously negative values of T pb and of T ps are not incentive compatible: agents can simply skip the mechanism to avoid

paying taxes.

10

Proposition 3 Define θ ≡ rr+α

[1− C(q∗)

U(q∗)

]−1

.

a. There exists a money mechanismM that implements the first best if and only if θ ≥ θ. The first best

is implementable only if µ > 1.

b. If θ < θ, the second best allocation q = q1 is given by any µ ≥ µ and q1 solves

C (q1) =

[1− r

θ (r + α)

]U (q1) ,

where q1 ≥ q0, the inequality is strict when θ > r/ (r + α).

Opposite to the lump-sum money system, where laissez faire would be optimal, inflating the money supply

over time actually improve welfare under the optimal money mechanism. Inflation is social beneficial not

because of the redistribution role as in Chiu and Molico (2010) - distribution is degenerate in Lagos-Wright

economy. Instead, inflation is useful to finance the transfers to encourage money holding. In the previous

section, Proposition 1 shows that, without any authority to enforce taxation the first-best allocation cannot

be achieved by simple lump-sum transfers. However, a well-designed money mechanism can implement the

first best without taxation, as it must feature money creation, µ > 1, according to Proposition 3. To do

that, the transfer scheme Tb (z) has to induce buyers to carry the right amount of money balances to finance

the first-best trade. For example, a money mechanism can make a big transfer only to buyers who bring and

report a suffi ciently high money balances. In equilibrium, these big transfers are financed by printing new

money, and it generates inflation. Hence, a deviator who brings too little money and receives no transfers

will be punished by a loss in purchasing power - the first best can be implemented. However, the power of

this scheme is limited by the size of a buyer’s DM trade surplus, which is increasing in θ. That explains why

the first best can no longer be supported when θ is too low.

Notice that θ is decreasing in β and α. There is no effi ciency loss when agents tends to be extremely

patient (r = 0). In addition, θ < 1 iffα

α+ rU (q∗) > C (q∗) . (16)

We are going to assume that this condition holds throughout the paper. Intuitively, when a buyer brings

cash from the CM to finance the first-best trade q∗ with the first seller matched in the subsequent DMs, the

maximum (discounted) utility gain from this future DM trade is βαU (q∗) [1 + β(1−α) + β2(1−α)2 + ...] =

αU (q∗) / (α+ r) . The minimum price the buyer needs to pay to induce the seller to trade is C (q∗). When

the above condition is violated, the maximum gain is lower than the minimum price, and hence there is no

hope for first-best trade in a monetary economy in which agents need to bring cash to trade.

3.1 Implementation

There are many ways to carry out the optimal money mechanisms. Consider an indirect mechanism with

the following features: the money issuer imposes a fixed fee B = (1− β) d∗ on buyers, who can then collect

interest on their first K = (1 + r) d∗ units of balances at the nominal rate i ≡ µ/β − 1 at the end of the

CM. In the equilibrium the buyers will hold exactly (1 + r) d∗ units of real money balances, after paying the

11

fixed fee B and before collecting the interest. After the money interest and the inflation, the real balances

becomes d∗ in the next DM. Although money pays interest, the buyers do not hold more than that because

it pays at the nominal rate. Because interest is only paid up to the cap, K, there will not be side trading to

avoid the fixed fee.

For suffi ciently high µ, the first-best allocation can be supported if θ ≥ θ, otherwise the second-best

allocation can still be implemented. The basic idea is that the interest payment offsets the buyers’opportunity

cost of carrying money balances to the DM. This interest payment is financed by the fixed fee paid by the

buyers. In order to induce them to pay this fee, the monetary growth has to be suffi ciently high so that

non-participants’trade surplus in the DM is suffi ciently low. Notice that this scheme is piecewise linear: a

fixed fee plus a linear transfer with respect to the buyer’s money holding z. Of course we are not the first

to discover that paying interest on money by printing even more money can improve welfare; see Andolfatto

(2010) for the case under competitive pricing. But we also formally show that when implementing the first

best is impossible even allowing for any scheme.

To summarize this section, we learned that the first best cannot be achieved by any money mechanism

when the buyer’s bargaining power is too low, the buyer is too impatient or trades are too infrequent. A non-

linear transfer scheme and monetary expansion are essential features of a money mechanism to implement

the first best, which can be implemented by a simple indirect mechanism with interest on buyers’balances,

financed by fixed fees and monetary expansion.

4 Electronic Money with Exclusive Participation

In this section we introduce e-money to the monetary economy considered in the last section. Different

from the public recording of agents’ identity and actions in Kocherlakota (1996), we think of e-money as

balances of anonymous accounts in the issuer’s ledger.14 The only power of the account holder is to debit his

own accounts by crediting other accounts, i.e., sending the e-money. Agents could open multiple anonymous

accounts. In this environment e-money shares all the attributes of money: divisible (balance of any amount),

durable (balances can be transferred but not destroyed), portable (no physical cost of reading or writing

the ledger) and cannot be counterfeited (cannot credit his own accounts without debiting others). Different

e-money technologies allows the e-money issuer various management of the ledger. In this section we first

consider that the e-money issuer can destroy accounts. Effectively, it can prohibit agents from holding

e-money balances. We will call this the participation technology.

In this economy, money still exists and is injected by lump sum transfers at the exogenous growth factor

µ, or equivalently under an exogenous nominal rate i = µ/β − 1. We assume that the supply of e-money

balances, Nt, is fixed at the same growth rate of money, i.e., Nt+1 = µNt. Nevertheless, the e-money issuer

14While some e-money systems allow the issuer to track the identity, it can be diffi cult to implement in most of the anonymoussystems (e.g., in Bitcoin and prepaid card). In reality anonymity can be a desirable feature; there is a demand for privacy whichis absent in our model. One future extension of this paper is to explore the welfare implication of introducung an expandedrecord-keeping technology into this environment. Nevertheless, one would expect that giving an additional technology to thee-money mechanism designer should only make it easier to achieve desirable allocations.

12

can choose the initial supply of e-money, or equivalently relative supply of e-money (to the total supply),

denoted as λ ≡ N0/ (N0 +M0). This constraint is to capture the fact that in reality e-money products often

involve this feature a priori or by law (e.g. denominated in cash or required to be convertible at par), and

the fact that the e-money issuer, private or public, often takes inflation as given.

Decentralized markets and bargaining. Decentralized markets with e-money are similar to the one

with money, except that now agent can pay in money and e-money, denoted as the vector d = (dz,dn).

Denote n ≡ φN the real balances of e-money, and the vector a as the portfolio held at the beginning of the

DM, and a as the one at the beginning of the CM. The DM value functions solve a similar Bellman equation:

Vb (a) = α

∫[U [q (a,a′)] +Wb [a− d (a,a′)]] dFs (a′) + (1− α)Wb (a) , (17)

Vs (a) = α

∫[−C [q (a′,a)] +Ws [a + d (a′,a)]] dFb (a′) + (1− α)Ws (a) , (18)

where Fj is the joint distribution of the type-j agents’portfolio, and q (ab,as) and d (ab,as) are the terms

of trade in the bilateral meeting between a buyer holding ab and a seller holding as. Like in the previous

section, the trade surpluses are given by Sb(q,d; ab,as) ≡ U (q)+Wb (ab − d)−Wb (ab) and Ss(q,d; ab,as) ≡

−C (q) + Ws (as + d) − Ws (as). The terms of trade, q = q (ab,as) and d = d (ab,as), solve the same

proportional bargaining problem:

maxq,d=(dz,dn)

{Sb(q,d; ab,as) + Ss(q,d; ab,as)} , s.t. (19)

(1− θ)Sb(q,d; ab,as) = θSs(q,d; ab,as)], (20)

−as ≤ d ≤ ab, (21)

where x ≤ y means that x1 ≤ y1 and x2 ≤ y2.

Mechanism design with participation technology. Thanks to the revelation principle again, in

the CM agents report their portfolio of the real balances of money and e-money, denoted as the vector a.

While it does not matter for our results, we maintain the "show me the e-money" constraint: the agents

can hide (probably in other anonymous or unregistered accounts, or by intertemporal side trading) any

e-money balances reported to the e-money issuer, but they cannot over-report. All the results remain the

same if the e-money issuer can observe agent’s e-money balances in the end of CM. An e-money mechanism

MP consists of two e-money transfer functions based on the portfolio reported, and the relative supply of

e-money, denoted asMP ≡ {Tb (a) , Ts (a) , λ}.

With the participation technology of e-money, the CM value functions solve the following Bellman equa-

tion:

Wj (a) = z + n+ T + maxa=(z,en),a=(z,n),e∈{0,1}

{−µ (z + en) + eTj (a) + βVj (a)} , s.t. a ≤ a. (22)

The e-money issuer can prohibit agents from holding e-money in the next period if they do not participate

the mechanism in the CM (e = 0). In particular, if e = 0, then DM e-money holdings become en = 0. Only

13

participating agents (e = 1) can hold e-money in the following period. Let ej (a), aj (a) and aj (a) denote

the solution to (22). Again, the constraint a ≤ a captures that agents cannot over-report their portfolios.

Since the CM value functions given by (22) are still linear, by extending Lemma 1, the bargaining solution

{q (ab,as) ,d (ab,as)} satisfies

D (q) = dz + dn = min {ab,z + ab,n, d∗} ,

dz ≤ ab,z,

dn ≤ ab,n.

Notice that the payments are given by dz = zb and dn = nb if zb + nb ≤ D (q∗); otherwise the total payment

is given by dz+dn = D (q∗) but its composition is indeterminate: any dz ≤ zb and dn = D (q∗)−dz ≤ nb can

be the bargaining solution. Finally, the definition of implementability is modified to the portfolio of money

and e-money, as follows.

Definition 3 An allocation (q,d,ab,as) is implementable under an e-money mechanismMP with the par-

ticaption technology if

a. ej (a) = 1, aj (a) = aj (a) = aj solve (22);

b. q = q (ab,as) and d = d (ab,as) solve (19);

c. Tb (ab) + Ts (as) = (µ− 1) (nb + ns);

d. φM = zb + zs.

4.1 Essentiality

The following proposition characterize the optimal allocation that can be implemented by an optimal e-money

mechanism with the participation technology.

Proposition 4 Define

Θ ≡ β + maxq′

βαU (q′)− ψD (q′)

, (23)

ψ ≡ [βα− (1− β) (1− θ)]U (q∗)− [βα+ (1− β) θ]C (q∗) . (24)

a. An e-money mechanismMP implements the first best with the participation technology if and only if

µ ≥ Θ.

b. If µ < Θ, then the second best allocation, q = qP , is given by[1− r

α(1− θ)

]U (qP )−

(1 +

rθ

α

)C (qP ) = max

q′

{U (q′)−

(1 +

i

α

)D (q′)

}. (25)

c. Θ ∈ (1, µ) and Θ is decreasing in α. qP and welfare are increasing in µ.

This proposition shows that, the first best can be implemented using this e-money when inflation is

suffi cient high, µ ≥ Θ. An increase in µ facilitates the e-money to implement the first best because it

reduces the outside option of non-participants who use money only as their means of payment and do not

receive any transfer in e-money.15 A higher µ could brings higher revenue by creating more e-money balances.15 Interestingly, this is consistent with a popular view that inflation induces agents to adopt some e-money products. For

example, Bitcoin is considered by some to be a safe haven from inflation.

14

When the first best cannot be implemented, an increase in inflation also raise the second best allocation.

On the other hand, given the inflation rate, the first best is more likely implemented by the participation

technology under when agents trade and use e-money more often (higher α). It is similar to the economy of

scale effect, although neither the matching or the production technology exhibits increasing return to scale.

We now show that e-money cannot be less essential than money.

Proposition 5 a. If there exists a money mechanism M that implements the first best with µ, then there

also exists an e-money mechanismMP that implements the first best with the participation technology under

the same µ;

b. If there exists a money mechanism M that implements the second best q = q1 with µ, then there also

exists an e-money mechanismMP that implements the second best q = qP with the participation technology

under the same µ such that qP > q1.

This proposition implies that, fixing the inflation rate, an optimal e-money mechanism with the partic-

ipation technology is at least as good as an optimal money mechanism in implementing the first best, or

implementing a better second best. But if e-money is equally essential as money, there is still no need to

introduce e-money on top of money. Consider that the first best cannot be implemented by the e-money,

then it is neither by money too so both implement the second best. In this case e-money is strictly more

essential than money, following Proposition 5b. In the case of the first best, the following corollary gives

conditions under which e-money is strictly more essential than money.

Corollary 1 Define θ ≡(1 + α

r

)θ − α

r . If θ ∈[θ, θ), then the first-best allocation

a. cannot be implemented by any money mechanism;

b. can be implemented by some e-money mechanisms with the participation technology under some µ.

This corollary makes a case for e-money that restrict participation. Part a, implied by Proposition 3,

states that no money mechanism can implement the first best when θ < θ. At the risk of being repetitive,

we reproduce here the intuition: a money mechanism uses a non-linear transfer scheme to induce buyers

to “cooperate”and to carry suffi cient money balances. This scheme relies on the “punishment”of eroding

deviating buyers’DM trade surplus by not giving them a transfer. However, the power of this scheme is

limited by the size of the buyers’trade surplus, which depends on θ. When θ < θ, the buyers’trade surplus

is insuffi cient for inducing them to carry the right money balances. In the extreme case of θ → 0, buyers

have no surplus to be extracted.

The ability of the e-money issuer to limit participation provides an additional tool. By threatening to

exclude agents from participating in the e-money system, the issuer can extract extra resources (especially

from sellers), and use those extra resources to induce buyers to bring the right balances. How much resources

can be extracted from buyers and sellers? The answer is equal to the difference between the trade surplus

of an e-money user and that of a money user. The power of this scheme is maximized when money users’

trade surplus is zero, and this will happen when µ ≥ µ (from Proposition 1). In this case, the threat to

15

exclude deviators allows the issuer to extract the whole of the (discounted) trade surplus, which equals to

βα[U(q∗)− C(q∗)]. This explains why e-money with the participation technology is essential.

The power of this scheme is still insuffi cient to achieve first best when θ is too low (i.e. θ ≤ θ). To

illustrate by an extreme example, suppose that µ = 1 and θ → 0. In this case, the buyer has no bargaining

power and thus the price for q∗ is D (q∗) = U(q∗). So to induce buyers to bring D (q∗) in the previous CM,

the transfer Tb has to be suffi ciently negative that they still have a positive payoff:

Tb − [µ− β (1− α)]D (q∗) + βαU(q∗)−maxq{− [µ− β (1− α)]D (q) + βαU (q)} ≥ 0.

However, the above discussion implies that the maximum transfer the e-money issuer can make is Tb =

βα[U(q∗)− C(q∗)], which is the whole of the (discounted) trade surplus. Plugging µ = 1, D (q) = U(q) and

Tb = βα[U(q∗)− C(q∗)] into the LHS, the buyers’payoff becomes

− [1− β (1− α)]U(q∗) + βα[U(q∗)− C(q∗)] + βαU(q∗)

= −(1− β)[U(q∗)− C(q∗)] + {βαU(q∗)− [1− β (1− α)]C(q∗)} ≥ 0.

As βαU(q∗)− [1−β (1− α)]C(q∗)→ 0, the LHS becomes negative, which is a contradiction. Therefore, this

example shows that, when θ is small and βαU(q∗)− [1− β (1− α)]C(q∗) is small (but remains positive, as

assumed), the first-best allocation is not implementable by any e-money mechanism with the participation

technology. This explains part b of the above corollary.

Overall, Proposition 3 and Corollary 1 characterize the implementability using the money mechanism

and e-money mechanism. When buyers’bargaining power is high (θ ≥ θ), an optimal money mechanism

can implement the first best. Hence, e-money is not needed in this region. When buyers’bargaining power

is moderate (θ > θ ≥ θ), only e-money with the participation technology can implement the first best, given

suffi ciently high money growth µ. Hence, e-money is essential relative to money in this region. Finally, when

buyers’bargaining power is too low (θ > θ), even an e-money with the participation technology may not

implement the first best, but implement a better second best than money.

Essential features. What does the e-money system look like when it is essential?

Proposition 6 If there does not exist any money mechanism M = {Tb (z) , Ts (z) , µ} but some e-money

mechanisms MP = {Tb (a) , Ts (a) , λ} that implement the first best with the participation technology under

some µ ≥ 1, then it is necessary that Ts (as) < 0 and Tb (ab) > 0.

As mentioned above, when θ is too low, extracting trade surplus from buyers alone cannot raise enough

resources to support the first best. The power of the participation technology helps implement the first best

by extracting surplus from sellers (i.e. Ts (z, n) < 0) to cross-subsidize buyers’holding of e-money balances

(i.e. Tb (z, n) > 0). The key benefit of limiting participation is allowing cross-subsidization from sellers to

buyers, which is infeasible under a money mechanism.

16

4.2 Implementation with Deposit

Suppose that µ ≥ Θ and θ ∈ [θ, θ). According to Corollary 1, e-money with the participation technology

is essential; according to Proposition 8 it is essential to have cross-subsidization from sellers to buyers with

the help of inflation. We will illustrate how to implement the first-best allocation with a simple indirect

mechanism. The e-money issuer requires at the end of the period minimal e-money balances L in each buyer

account, which will pay interest after K balances at the nominal rate i:

K = (1− βα) d∗ + βαC(q∗),

L = K + βi−1 [(i− r + α) d∗ − αC(q∗)]] .

The account with be terminated without maintaining at least L balances, of which K balances will be the

deposit. Similarly, the sellers have to pay a maintenance fee B to continue their accounts:

B = βα[d∗ − C(q∗)].

The idea is that K and L implement enough balances to trade at the first best such that the issuer’s budget

is balanced after collection the seller’s maintenance fee, B. In this example, B is set to maximal such that

sellers get zero ex-ante trade surplus, but more general cases can be similarly constructed. At the nominal

rate, the buyers will not hold more than L despite of the fact that e-money pays interest. And there will be

no incentive to side trading in order to avoid deposit becuase any buyers’account with balances less than L

will be terminated. The first best is implementable if the inflation is high enough that µ ≥ Θ.

To summarize this section, we learned that when buyers have moderate bargaining power and inflation is

high, an e-money mechanism with the participation technology is essential to implement the first best. Given

the inflation rate, an optimal e-money mechanism with the participation technology is never less essential

than any money mechanism. In this case, cross-subsidization from sellers to buyers is an essential feature

to implement the first best. The first best can be implemented by a simple indirect mechanism with fixed

maintenance fees on sellers’account, and requiring deposit on buyers’account.

5 Electronic Money with Discretionary Transferability

In this section we consider an e-money technology that can restrict transferring of e-money balances in the

DM. We will call it the transferability technology. Suppose that the e-money issuer can no longer destroy

anonymous accounts in the public ledger, instead it can reject any credit-debit in the DM.16 Notice that

transferability technology is different from participation technology: in a mechanism with the transferability

technology, agents first match in the DM and then decide whether to use e-money as the means of payment,

may be subject to some charges or subsidy in e-money on the spot.17 To use e-money, both the buyer

16Think of the scenario where it is up to the Bitcoin miners to put transactions in their mining block for approval. Of course,no one can eliminate IP address from sending and recieving Bitcoin.17 In the Bitcoin scenario the payer specifies the transaction fee sent to the miner who puts the transaction in his mining

block, if it is successfully mined.

17

and the seller have to obtain the issuer’s promise not to eliminate the transferring record involving their

anonymous accounts (the issuer can always eliminate all the transfers without permission). It is without loss

of generality to consider that the permission can be obtained if the buyer and seller simultaneously transfer

∆b and ∆s units of e-money to the issuer (says, to its public account; receiving from the public account if

∆ negative). In the CM, e-money is assumed to be freely transferable among agents.18

Decentralized markets and bargaining. Due to the potential transfer in the DM, the e-money

payment made by the buyer can be different from the e-money received by the seller in the DM, denoted as

the vectors db = (dz, db,n) and ds = (dz, ds,n) respectively. Notice that the e-money issuer, however, cannot

charge fees on agents’money holdings. The DM value functions solve the following Bellman equation

Vb (a) = α

∫[U [q (a,a′)] +Wb [a− db (a,a′)]] dFs (a′) + (1− α)Wb (a) , (26)

Vs (a) = α

∫[−C [q (a′,a)] +Ws [a + ds (a′,a)]] dFb (a′) + (1− α)Ws (a) , (27)

where [q (ab,as) ,db (ab,as) ,ds (ab,as)] is the bargaining solution in the bilateral meeting. The bargaining

solution is the same as before if e-money is not used. Define ∆ ≡ ∆b + ∆s. Due to the linearity of CM value

function, it is straightforward to show the following lemma, which characterizes the bargaining solution.

Lemma 2 The bargaining solution [q (ab,as) ,db (ab,as) ,ds (ab,as)] satisfies:

D (q) = min {ab,z + In (ab,n − θ∆) , d∗} , (28)

In =

1if ab,n + as,n ≥ ∆ and

U (q)− C (q)−∆ ≥ U [qz (ab,z)]− C [qz (ab,z)]0 otherwise

(29)

db,z = ds,z = min (ab,z, d∗) , (30)

db,n = In (D (q)− db,z + θ∆) , (31)

ds,n = In (db,n −∆) . (32)

Unlike the previous e-money technology, the buyer and seller now coordinate a pecking order of payment

in the presence of ∆: using money before e-money to avoid paying ∆. To induce the use of e-money, ∆ has

be to be low enough that ∆ ≤ U (q)−C (q)−U [qz (zb)] +C [qz (zb)]. The key difference from the bargaining

solution in the previous section is that the buyer and the seller now have to share ∆ according to their

bargaining power. A higher buyer’s bargaining power θ will result in a higher pass-through of ∆ on the total

payment made by the buyer.

Mechanism design with transferability technology. An e-money mechanism MT consists of two

e-money transfer functions based on the portfolio reported and ∆, denoted asMT ≡ {Tb (a) , Ts (a) ,∆, λ}.

With the transferability technology of e-money, the CM value functions solve the following Bellman equation:

Wj (a) = z + n+ T + maxa=(z,n),a=(z,n),e∈{0,1}

{−µ (z + n) + eTj (a) + βVj (a)} , s.t. a ≤ a. (33)

18Allowing the issuer to have the additional power to restrict transferability in the CM can only strengthen our findings.

18

Unlike in the case of the participation technology, the e-money issuer can no longer prohibit non-participating

agents from holding e-money. The definition of implementability is modified as follows.

Definition 4 An allocation (q,db,ds,ab,as) is implementable under an e-money mechanism MT with the

transferability technology if

a. ej (a) = 1, aj (a) = aj (a) = aj solve (33);

b. q = q (ab,as), db= d (ab,as) and ds= d (ab,as) solve (28) - (32);

c. Tb (ab) + Ts (as) = (µ− 1) (nb + ns) + α∆;

d. φM = zb + zs.

To summarize, here the novelty is that the transferability technology will change the terms of trade

[q,db,ds] with the presence of ∆. It also brings an additional revenue α∆ to the issuer’s budget constraint.

5.1 Essentiality

The following proposition characterizes the constrained optimal allocation under the e-money mechanism

with the transferability technology.

Proposition 7 Define

Φ = β + maxq′

βαU (q′)− ζD (q′)

,

ζ ≡{

[βαθ + (β − 1) (1− θ)]U (q∗)− (βα− β + 1) θC (q∗) , if θ ≥ α(1+r)r+α

(β + α− 1)U (q∗)− αC (q∗) , otherwise

}.

a. An e-money mechanism MT that implements the first best with the transferability technology if and

only if µ ≥ Φ.

b. If µ < Φ, the second-best allocation, q = qT , is given by

[max [α (1 + r)− (r + α) θ, 0] + αθ − r (1− θ)] [U (qT )− C (qT )]− rC (qT ) = maxq{αU (q)− (i+ α)D (q)} ,

c. Φ ∈ (1, µ) and Φ is increasing in θ but decreasing in α. qT and welfare are increasing in µ.

This proposition shows that, to implement the first best using this e-money mechanism, inflation need to

be high enough that µ ≥ Φ. The idea is that an increase in µ reduces the value of the buyers’outside option

of non-participation. A higher µ could brings higher revenue by creating more e-money balances. This result

is parallel to Proposition 4, with different inflation thresholds (Θ v.s. Φ) and different levels of the second

best (qP v.s. qT ).

We first derive conditions under which an optimal mechanism with the transferability technology is at

least as good as one with the participation technology, and vice versa.

Proposition 8 Φ ≤ Θ if and only if θ ≤ α. For the second best allocation, qT ≥ qP if and only if θ ≤ α.

19

Proposition 8 gives a simple condition θ ≤ α that is suffi cient and necessary for the transferability

technology to be at least as essential as the participation technology. Suppose that θ ≤ α. If there exists an

e-money mechanism MP that implements the first best with the participation technology, then there also

exists an e-money mechanismMT that implements the first best with the transferability technology under

the same µ. IfMT cannot implement the first best, then qT ≥ qP for the second best. Suppose that θ > α.

If there exists an e-money mechanismMT that implements the first best with the transferability technology,

then there also exists an e-money mechanism MP that implements the first best with the participation

technology under the same µ. IfMP cannot implement the first best, then qP > qT for the second best.

The economy does not need both e-money technologies when they are equally essential. When the

transferability technology is strictly more essential than the participation technology, and vice versa?

Corollary 2 a. If α > θ and θ ≤ min(α(1+r)r+α , θ

)then the first-best allocation can be implemented by

an e-money mechanism with the transferability technology for some µ. The first-best allocation cannot be

implemented by any money mechanism.

b. If α > θ and θ < θ, then the first-best allocation can be implemented by an e-money mechanism with

the transferability technology for some µ. The first-best allocation cannot be implemented by any e-money

mechanism with the participation technology.

c. If θ ∈(α, θ

)and α < θ, then the first-best allocation can be implemented by an e-money mechanism

with the participation technology for some µ. The first-best allocation cannot be implemented by any e-money

mechanism with the transferability technology.

This corollary provides some parameter regions where we can rank the essentiality of e-money technolo-

gies. In general, the transferability technology is more powerful than the participation technology under low

θ and high α. The opposite is true under high θ and low α. What is the intuition? On the one hand, the

amount of interchange fees passed through to the buyer is θ∆, so a low value of θ means that the buyer

bears a small interchange fee burden. On the other hand, recall that the participation technology allows

the issuer to use exclusion from period t + 1 DM as a threat to enforce fees in period t. That is why the

maximum surplus extractable from a seller, αβ[D (q∗)−C(q∗)], is discounted, since the fee is paid a period in

advance. In contrast, the ability to limit transferability allows an issuer to extract the seller’s trade surplus

in period t DM by enforcing interchange fees in the same period. The maximum surplus extractable from a

seller becomes α[D (q∗) − C(q∗)], without discounting. Therefore, the gain from postponing fee collection,

which relaxes the seller’s participation constraint, is stronger when α is high.19 In sum, for high α the buyer

can be rewarded a large sum financed by the seller’s surplus; for low θ the buyer only bears a small inter-

change pass-through. As a result, under high α and low θ the technology limiting transferability can help

induce buyers to bring suffi cient balances to support the first-best allocation, which cannot be done by the

technology limiting participation. But will postponing fee collection tighten the issuer’s budget constraint?19The idea is also related to "double marginalization" effect found by Shy and Wang (2012): imposing the interchange fee

∆ on sellers to reduce their monopoly/ bargaining power. Here the issuer can futher improve the welfare by redistributing theinterchange fee to the buyers, which overcomes the liquidity constraint.

20

No, because the issuer can always create more e-money balances when needed. From the issuer’s point of

view, collecting the fee in the CM or in the following DM does not matter, as long as the money growth

rate between two CM markets can be maintained at µ. In particular, the issuer can temporarily create extra

e-money balances in CM, and undo them later when interchange fees are collected in the following DM.

Therefore, the transferability technology allows the e-money issuer to postpone fee collection, maximizing

surplus extraction, without tightening its budget constraint.20

Essential features. After establishing the essentiality of transferability technology, we now characterize

the essential features of different e-money systems.

Proposition 9 Given µ,

a. if there exists an e-money mechanism with the transferability technologyMT = {Tb (z, n) , Ts (z, n) ,∆, λ},

but not any e-money mechanism with the participation technology MP , that implements the first best, then

it is necessary that ∆ > 0, Ts (zb, nb) ≥ 0 and Tb (zb, nb) > 0.

b. If there existsMP = {Tb (z, n) , Ts (z, n) , λ} but not anyMT that implements the first best, then it is

necessary that Ts (zs, ns) < 0 and Tb (zb, nb) > 0.

As discussed above, to implement the first best, the issuer has to extract trade surplus in the DM (∆ > 0),

which is then used to induce buyers to carry suffi cient e-money balances in the CM (Tb (zb, nb) < 0). Note

that this scheme requires the issuer to temporarily expand the e-money supply in the CM (to pay buyers

Tb (zb, nb)) and later undo it in the DM (by charging fees ∆), ensuring constant money growth across periods.

5.2 Implementation with Interchange Fee

Suppose that µ ≥ Φ and α > θ < θ. In this case the e-money with transferability technology is the most

essential. We will illustrate how to implement the first best allocation with a simple indirect mechanism

featuring interchange fee. Instead of charging maintenance fee in the case of particpation technology, here

the e-money issuer imposes an interchange fee ∆ on the payee in the DM:

∆ = U (q∗)− C(q∗).

To implement the cross-subsidization, interest is paid at the nominal rate i after the minimal holding K and

up to L:

K = (1− α)U (q∗) + αC (q∗) ,

L = K + i−1 [(µ− 1 + α)U − αC(q∗)] .

In this example, the interchange fee, ∆, is set to maximal such that sellers get zero ex-ante trade surplus,

but more general cases can be similarly constructed. To trade at the first best under the interchange fee,

20Note that ∂θ/∂β > 0, implying that the transferability technology is more essential relative to the participation technologywhen the discount factor is low. A real-world interpretation is that charging interchange fees at the time of the transaction ismore desirable relative to charging a membership fee in advance, when the frequency of membership fee payment is low (e.g.annual membership paid a year in advance).

21

∆, the buyers are induced to hold U (q∗) units of balances in the DM (after interest). At the nominal rate

agents will not hold more e-money than that, and there will not be side trading beacuse the interest is paid

up to L. The first best is implementable if the inflation is high enough that µ ≥ Φ. This scheme features

cross-subsidization from sellers to buyers, with piecewise linear pre-trade transfers to buyers, and post-trade

fees on sellers.

To summarize this section, we learned that given inflation an optimal e-money mechanism with the

transferability technology is never less essential than any money mechanism. When buyers have low bar-

gaining power and high frequency of trade, an e-money mechanism with the transferability technology is

more essential than any e-money mechanism with the participation technology (and vice versa). In this case,

cross-subsidization using interchange fees is an essential feature to implement the first best.

6 Extension on Competitive Pricing

The above analysis only considers an environment with bilateral trading under the pricing protocol of pro-

portional bargaining, which is convenient for capturing buyers’and sellers’bargaining powers and two-sided

externalities. One may wonder if our result is robust in other trading environments. In particular, if agents

conduct monetary trades in a centralized market and take competitive prices as given, they do not consider

other agents’balances and adoption decisions, and do not have any bargaining power. Even in that environ-

ment, however, the curvatures of U and C imply that buyers’and sellers’surpluses remain positive. As a

robustness check, we show in the Online Appendix that all the main results on essentiality of various money

and e-money mechanisms still hold, if we reinterpret agents’bargaining powers appropriately as the relative

trade surplus at the first best under competitive pricing, θ = [U (q∗)− C ′ (q∗) q∗] / [U (q∗)− C (q∗)].

7 Discussion and Conclusion

Using the mechanism design approach, we have identified several essential features of e-money that help

improve the effi ciency of a monetary economy. First, unlike conventional cash, e-money systems can exclude

participation. Second, unlike cash, e-money systems can restrict and block balance transfers and these trans-

fers are not necessarily zero-sum bilaterally. Our model then predicts that an optimally designed e-money

system with the above technologies can exhibit several features, including non-linear pricing, membership

fees, interchange fees and rewards to buyers. This prediction does have some empirical support, since several

successful real-world e-money systems also possess these features. For example, M-Pesa sets a non-linear fee

structure. The Octopus card imposes fixed and variable fees on merchants, and offering rewards and dis-

counts to consumers.21 PayPal charges merchants a fee on accepting payments while Bitcoin miners charge

a transaction fee for appending transactions to the Blockchain.22 According to our model, these pricing

21Merchants are subject to a fee structure involving a fixed deposit, a fixed monthly fee and a variable fee proportional totransaction value. Individual buyers need to pay a fixed deposit to obtain an Octopus card. Rewards are offered to cardholders,such as Octopus reward (at least 0.5% of spending) and discounts on selected products (e.g. transportation).22Under the basic arrangement, PayPal charges a 2.9% merchant fee plus $0.30 per transaction, with volume discounts

applied. For Bitcoin, the average transaction fee is 0.0088129% in 2015. It is expected that the fee has to go up substantially

22

features are important components for incentivizing participants and cross-subsidizing across different types

in order to support effi cient economic outcomes.

Notice that while other payment systems (such as credit cards and large-value settlement systems) may

also exhibit the above-mentioned features, the operation of an e-money system typically requires less in-

formation and enforcement. For example, a credit card system usually requires monitoring performed by

banks and credit card issuers who possess a richer information set and/or a stronger enforcement technology

than a typical e-money issuer. It is not clear (i) whether these arrangements (e.g. credit) are feasible in

the current environment, and (ii) whether money and e-money remain co-essential with these arrangements.

In this regard, the model constructed in this paper may not be good enough for analyzing these systems.

In addition, the functioning of many traditional payment systems relies on accounts with identified owners,

making it inaccessible for some populations or undesirable in some situations. In contrast, e-money such

as prepaid cards and Bitcoins can function even in a setting with anonymous users —a setting that renders

cash essential.

Our model implications provide useful lessons for policy-makers. First, e-money is fundamentally different

from conventional money. Improved information and technologies as a result of the introduction of e-money

allow more general fee structures and can increase effi ciency. Second, pricing arrangements such as merchant

and interchange fees can be essential components of an optimal payment system. Hence, fee regulation may

distort the optimal mechanism and reduce welfare. For example, the Durbin Amendment to the Dodd-

Frank Act limits the maximum permissible interchange fees for a debit card transaction based on issuers’

costs associated with processing, clearance and settlement. Our theory suggests that imposing this type of

regulation on e-money can be welfare reducing because the optimal fee is positive even in an environment in

which the physical cost of payments is zero. Third, our theory suggests that different payment instruments

emerge to mitigate different economic frictions. For example, there is a fundamental difference between

money (including e-money) and credit because consumers need to acquire balances in advance in the former

but not the latter case. The optimal design of a money-based payment system is different from that of a

credit-based system, since they are subject to different incentive and feasibility constraints. For instance,

limiting interchange fees can be optimal for some specific payment instruments but not all.

While our paper has provided novel economic and policy insights, we have abstracted from some in-

teresting aspects, leaving those extensions for future research. Below we briefly discuss a few potential

extensions.

First, we apply the mechanism design approach to examine some commonly observed features of e-

money. Naturally, one can use our framework to evaluate the essentiality of other e-money features. For

example, many cryptocurrencies base on the blockchain technology to keep track of the whole history of

past transactions. In contrast, cash, as a record keeping technology, captures only the current distribution

of balances. It is interesting to investigate whether using a transaction-based ledger can help improve the

effi ciency of a monetary system. Similarly, one can easily introduce additional features of e-money such as

as the coinbase rewards converge to zero in the long run .

23

convenience, online payments, and transactions speed into our framework. While these features can enhance

effi ciency, they may not be essential for mitigating fundamental frictions in a monetary economy.23

Second, the paper focuses on the technological advantages of e-money, but one can easily incorporate

some disadvantages of e-money into the model. For example, we have assumed that there is no cost of

operating an e-money system to highlight the result that the optimal fee on sellers can be positive even in

this extreme setting. In a more general environment where a positive operating cost is incurred, we expect

that a similar pricing arrangement would remain optimal because it could help raise resources effi ciently

to finance the operation of the system. Similarly, one can study a setting in which the usage of e-money

involves a higher cost of record keeping (e.g. mining cost in Bitcoin) or the digital nature of e-money induces

a double-spending problem. This will naturally lead to a trade-off between using cash and e-money.

Third, our model environment can be generalized. Our model focuses on a simple pricing protocol —

proportional bargaining, because it can easily capture the split of trade surplus between the buyer and

the seller. As shown, our finding can be generalized to other cases, including competitive pricing, where the

parameter θ can be mapped to the share of trade surplus allocated to the buyer under the first-best allocation.

Moreover, we have assumed that trading status (i.e. buyers and sellers) is permanent because this is more

realistic given the frequency of trade captured by the model. However, our findings will remain unchanged

when types are random (especially when agents know their types before portfolio choice is made). Overall,

our model builds on a very standard environment used in the money search literature. Many alternative

model variations (such as endogenous entry and endogenous matching) can be explored, but we leave those

interesting analyses for future work. Finally, we have not studied the equilibrium outcome when e-money is

issued or operated by private profit-maximizing agents. In a companion paper, Chiu and Wong (2014), we

investigate the ineffi ciency of privately issued e-money.

References

[1] Agarwal, R. and Kimball, M., 2015. "Breaking through the Zero Lower Bound." IMF Working Paper

WP/15/224. Washington: International Monetary Fund, October.

[2] Andolfatto, D. 2010. “Essential interest-bearing money.” Journal of Economic Theory, 145, 4, 1495—

1507.

[3] Araujo, L., Camargo, B., Minetti, R., and Puzzello, D. 2012. “The essentiality of money in environments

with centralized trade.”Journal of Monetary Economics, 59(7), 612-621.

[4] Araujo, L. and T. Hu. 2014. “Optimal monetary interventions in credit markets.”Manuscript.

[5] Aruoba, S.B., Rocheteau, G. and Waller, C., 2007. "Bargaining and the Value of Money." Journal of

Monetary Economics, 54(8), pp.2636-2655.

23For example, in a companion paper, Chiu and Wong (2014), we study the introduction of e-money when heterogenousbuyers and merchants face differential adoption incentives, capturing some e-money features unmodelled in the current paper.

24

[6] Chiu, J., and M. Molico. 2010. "Liquidity, redistribution, and the welfare cost of inflation." Journal of

Monetary Economics, 57(4), pp.428-438.

[7] Chiu, J., and T. Wong. 2014. “E-Money: Effi ciency, Stability and Optimal Policy.”Bank of Canada

Working Paper No. 2014-16.

[8] Committee on Payment and Settlement Systems (CPSS). 2001. “Survey of Electronic Money Develop-

ments.”Bank for International Settlements.

[9] Fernández-Villaverde, J. and D. Sanches, 2016. "Can currency competition work?" National Bureau of

Economic Research No. w22157.

[10] Hahn, F. 1973. "On Transaction Costs, Inessential Sequence Economies and Money." The Review of

Economic Studies 40, 449-462

[11] Hu, T., Kennan, J. and N. Wallace. 2009. “Coalition-proof trade and the Friedman rule in the Lagos-

Wright model.”Journal of Political Economy, 117, 116-137.

[12] Hu, T. and G. Rocheteau. 2013. “On the coexistence of money and higher-return assets and its social

role.”Journal of Economic Theory, 148(6), 2520-2560.

[13] Hu, T. and C. Zhang. 2014. “Responding to the inflation tax.”Manuscript.

[14] Kahn, C. M., and Roberds, W. 2009. “Why pay? An introduction to payments economics.”Journal of

Financial Intermediation, 18(1), 1-23.

[15] Kocherlakota, N. 1998. “Money is Memory.”Journal of Economic Theory 81, 232—251.

[16] Kocherlakota, N. 2003. “Societal benefits of illiquid bonds.”Journal of Economic Theory, 108, 179-193.

[17] Kocherlakota, N., and N. Wallace. 1998. “Incomplete record-keeping and optimal payment arrange-

ments.”Journal of Economic Theory, 81(2), 272-289.

[18] Lagos, R. 2010. “Some results on the optimality and implementation of the Friedman rule in the Search

Theory of Money.”Journal of Economic Theory, 145(4), 1508-1524.

[19] Lagos, R., Rocheteau, G. and Wright, R., 2015. "Liquidity: A new monetarist perspective." Journal of

Economic Literature, forthcoming, pp.1-108.

[20] Lagos, R. and R. Wright. 2005. “A Unified Framework for Monetary Theory and Policy Analysis.”

Journal of Political Economy 113(3), 463-484.

[21] Li, Y. 2011. “Currency and checking deposits as means of payment.”Review of Economic Dynamics,

14(2), 403-417.

25

[22] Monnet, C. and W. Roberds. W. 2008. “Optimal pricing of payment services.” Journal of Monetary

Economics, 55(8), 1428-1440.

[23] Rochet, J. C., and Tirole, J. 2003. “Platform competition in two-sided markets.”Journal of the European

Economic Association, 1(4), 990-1029.

[24] Rocheteau, G., 2012. “The Cost of Inflation: A mechanism design approach.” Journal of Economic

Theory, 147 1261-1279.

[25] Rogoff, K.S., 2016. The Curse of Cash. Princeton University Press.

[26] Rysman, M. 2009. “The economics of two-sided markets.”The Journal of Economic Perspectives, 125-

143.

[27] Shy, O., and Z. Wang. 2011. “Why do payment card networks charge proportional fees?”The American

Economic Review, 101(4), 1575-1590.

[28] Wallace, N. 2010. “The mechanism-design approach to monetary theory.”Handbook of Monetary Eco-

nomics (Forthcoming).

[29] Wang, Z. 2012 “Debit card interchange fee regulation: some assessments and considerations.” FRB

Economic Quarterly, 98(3), 159-183.

[30] Williamson, S.D., 2012. "Liquidity, monetary policy, and the financial crisis: A new monetarist ap-

proach." The American Economic Review, 102(6), pp.2570-2605.

[31] Wong, T.N., 2016. "A tractable monetary model under general preferences." The Review of Economic

Studies, 83(1), pp.402-420.

26

Appendix

A Proof of Proposition 2

The "if" is straightforward since any threshold mechanism is also a money mechanism. To prove the

"only if"part, consider an allocation (q, d, z′b, z′s) is implementable under a money mechanism with trans-

fers {T ′b (z) , T ′s (z) , µ}. The value functions W ′b and W ′s are given by

W ′j (z) = z +{−µz′j + ejT

′j

(z′j)

+ β[αSj (z′b, z

′s) + z′j +W ′j (0)

]}. (A.1)

Consider the threshold mechanism {Tb (z) , Ts (z) , µ} given by

Tb (z) =

{T ′b (zb) , if z ≥ z′b,

0, otherwise,(A.2)

Ts (z) =

{T ′s (zs) , if z ≥ z′s,

0, otherwise.(A.3)

The values functions Wb and Ws under {Tb (z) , Ts (z) , µ} are

Wb (z) =

[z + max

z,z,e∈{0,1}{−µz + eTb (z) + β [αSb (z, zs) + z +Wb (0)]}

], s.t. z ≤ z, (A.4)

Ws (z) =

[z + max

z,z,e∈{0,1}{−µz + eTs (z) + β [αSs (zb, zs) + z +Wb (0)]}

], s.t. z ≤ z. (A.5)

Since the allocation z = z = z′j and e = 1 are always feasible, it is necessary that Wj (z) ≥ W ′j (z). Let

zj (z) , zj (z) and ej (z) denote the solutions to (A.4). What we need to prove are that zj (z) = zj (z) = z′j

and ej (z) = 1. First, notice that from the definition of Tb (z), for any eb, zb and zb such that zb ≤ zb < z′b

or zb < z′b ≤ zb, we have

−µzb + ebTb (zb) + β [αSb (zb, z′s) + zb] = −µzb + β [αSb (zb, z

′s) + zb] ,

≤ −µz′b + T ′b (z′b) + β [αSb (z′b, zs) + z′b] ,

= −µz′b + Tb (z′b) + β [αSb (z′b, zs) + z′b] .

Second, since T ′b (z) and T ′s (z) are non-decreasing, for any eb, zb and zb such that z′b ≤ zb ≤ zb we have

−µzb + ebTb (zb) + β [αSb (zb, zs) + zb] = −µzb + ebT′b (z′b) + β [αSb (zb, zs) + zb] ,

≤ −µz′b + ebT′b (z′b) + βα [Sb (z′b, zs) + z′b] ,

= −µz′b + Tb (z′b) + β [αSb (z′b, zs) + z′b] .

Concluding these cases, we must have zb (z) = zb (z) = z′b and eb (z) = 1. Applying the same proof to the

sellers, we establish Proposition 2.

B Proof of Proposition 3

First, notice that, after substituting T ps of (14) into (13), it is always optimal to have

zs = T ps = 0,

27

T pb = (µ− 1) zb.

Then the problem of the money issuer becomes

maxq,µ{U (q)− C (q)} s.t.

(β − 1)D (q) + βαθ [U (q)− C (q)] ≥ maxq′{(β − µ)D (q′) + βαθ [U (q′)− C (q′)]} , (B.1)

where the money issuer can implement zb = D (q). Notice that µ only matters the right side of (B.1), which

is decreasing in µ and equal to zero for any µ ≥ µ. Thus it is always optimal to set µ ≥ µ. Then the first

best can be implemented if and only if (B.1) is not binding for any µ ≥ µ

(β − 1)D (q∗) + βαθ [U (q∗)− C (q∗)] ≥ 0,

which is equivalent to θ ≥ θ. On the other hand, since q′ 6= q∗ for any µ ≥ 1, (B.1) holds with q = q∗ only if

µ > 1. Thus, we have established Proposition 3a.

Suppose (B.1) is binding for any µ ≥ µ, then the solution to the above money issuer’s problem is simply

the value of q = q1 solving (β − 1)D (q1) + βαθ [U (q1)− C (q1)] = 0, which can be rearranged to

C (q1) =

[1− r

θ (r + α)

]U (q1) .

Thus we have established Proposition 3b.

C Proof of Proposition 4

Using the similar argument in the proof of proposition 2, we have that any allocation (q, d,ab,as) implemented

by some e-money mechanism with the participation technology can also be implemented by a threshold

mechanism, given by

Tb (a) =

{T pb , if az ≥ ab,z, and an ≥ ab,n0, otherwise.

(C.1)

Ts (a) =

{T ps , if az ≥ as,z, and an ≥ as,n0, otherwise.

(C.2)

Then, following the similar argument to proposition 3, an allocation (q,d,ab,as) is implementable under

some threshold mechanism if and only if

D (q) = d = min (ab,z + ab,n, d∗) , (C.3)

T pb ≥ 0, (C.4)

(β − µ) (as,z + as,n) + T ps + βα (1− θ) [U (q)− C (q)] ≥ βα (1− θ) [U [q (ab,z)]− C [q (ab,z)]] , (C.5)

T pb + T ps = (µ− 1) (ab,n + as,n) , (C.6)

ab,n + as,n = λ (ab,z + as,z + ab,n + as,n) . (C.7)

28

The e-money issuer’s problem is given by

maxq,d,ab≥0,as≥0,Tpb ,T

ps ,λ∈[0,1]

{U (q)− C (q)} s.t. (C.3) - (C.7). (C.8)

After substituting T ps of (C.6) into (C.5), it is always optimal to have as,z = as,n = 0 and T ps = −βα (1− θ) [U (q)− C (q)].

Then the problem of the e-money issuer becomes

maxq,λ∈[0,1]

{U (q)− C (q)} s.t.(β − µλ

+ µ− 1

)ab,n + βα [U (q)− C (q)] ≥ max

q′{(β − µ)D (q′) + βαθ [U (q′)− C (q′)]} . (C.9)

Since µ ≥ 1 > β, it is optimal to set λ = 1. Thus the first best is implementable if and only if (C.9) is not

binding, i.e.,

ψ ≥ maxq′{(β − µ)D (q′) + βαθ [U (q′)− C (q′)]} , (C.10)

where ψ ≡ (β − 1)D (q∗) + βα [U (q∗)− C (q∗)]. Thus the above inequality holds if and only if for all q′ we

have

ψ ≥ (β − µ)D (q′) + βαθ [U (q′)− C (q′)] ,

⇔ µ ≥ β +βαθ [U (q′)− C (q′)]− ψ

D (q′),

⇔ µ ≥ Θ ≡ β + maxq′

{βαθ [U (q′)− C (q′)]− ψ

D (q′)

}.

If (C.9) is binding then the second best is given by q = qP solving

(β − 1)D (qP ) + βα [U (qP )− C (qP )] = maxq′{(β − µ)D (q′) + βαθ [U (q′)− C (q′)]} . (C.11)

Notice that r = β−1 − 1 and i = µ/β − 1. The left side of (C.11) is decreasing in qP if βα ≤ (1− β) (1− θ),

and otherwise inverted U-shaped in qP . In the first case qP is decreasing in µ. In the second case, since

qP < q∗ is the social welfare is increasing in qP , so it is optimal to implement the greater qP on the inverted

U-shaped portion, which is also decreasing in µ.

Finally to show part c, notice that

Θ = β + maxq′

{βαθ [U (q′)− C (q′)]− ψ

D (q′)

}> β +

βαθ [U (q∗)− C (q∗)]− ψD (q∗)

,

= 1 + βα (1− θ) U (q∗)− C (q∗)

D (q∗),

≥ 1.

And since ψ ≥ 0, we have

Θ ≤ β + maxq′

{βαθ [U (q′)− C (q′)]

D (q′)

}= β + βαθmax

q′

{U (q′)− C (q′)

(1− θ) [U (q′)− C (q′)] + θC (q′)

}< β

(1 +

αθ

1− θ

)= µ.

29

Thus we have Θ ∈ (1, µ). Also, an envelope theorem implies

dΘ

dα= β

θ [U (q′)− C (q′)]− [U (q∗)− C (q∗)]

D (q′)< 0,

D Proof of Proposition 5

Part a. From (B.1) in the proof of proposition 3, if the first best is implementable with some µ under some

money mechanism then we have

(β − 1)D (q∗) + βαθ [U (q∗)− C (q∗)] ≥ maxq′{(β − µ)D (q′) + βαθ [U (q′)− C (q′)]} .

Since θ ≤ 1, the fact that the above holds implies that (C.10) is also holds. Thus we have established

proposition 5.

Part b. Since the money mechanism implements the second best q = q1, from proposition 3b we must

have µ ≥ µ. Thus, from (C.11) the second best q = qP under the e-money mechanism with the same µ solves

(β − 1)D (qP ) + βα [U (qP )− C (qP )] = 0, which can be rearranged to

C (qP ) =

[1− r

α (1− θ)1 + rθ

α

]U (qP ) .

Recall that q1 is given by

C (q1) =

[1− r

θ (r + α)

]U (q1) .

Given θ < 1 (the first best cannot be implemented under the money mechanism in this case), we have1− r

α (1−θ)1+ rθ

α

> 1− rθ(r+α) . Compare the above two formulas of q1 and of qP , we must have qP > q1.

E Proof of Proposition 6

Suppose that this is not the case, i.e., there does not exist any money mechanism but an e-money mech-

anism MP = {Tb (a) , Ts (a) , λ} (not necessarily threshold mechanism) that implements some first-best

allocation (q∗, d∗,ab,as) with some µ and Ts (as) ≥ 0. Consider the following money threshold mechanism

M = {T ′b (z) , T ′s (z) , µ}:

T ′b (z) =

{Tb (ab) +A, if z = ab,z + ab,n0, otherwise,

T ′s (z) = 0 for all z,

where

A ≡ (µ− 1) (ab,z + ab,n + as,z + as,n) + Ts (as) .

Notice that A ≥ 0 due to the premise Ts (as) ≥ 0. Then it is straightforward to verify that (13) is satisfied

under the first-best allocation with zb = ab,z + ab,n and zs = 0 withM. Also, notice that

T ′b (zb)− [µ− β (1− α)] d∗ + βαU (q∗)

= Tb (ab)− [µ− β (1− α)]D (q∗) + βαU (q∗)−A

≥ maxq′{− [µ− β (1− α)]D (q′) + βαU (q′)} −A,

30

where the last inequality comes from the fact that (q∗, d∗,ab,as) is incentive compatible to buyers under

MP . So (12) is satisfied under the first-best allocation with M. Finally, it is straightforward to verify

that (14) is satisfied under the first-best allocation (q∗, d∗, zb, zs) with M. This leads to a contradiction

to the premise as there exists a money mechanism M that implements the first best. Thus, we must have

Ts (as) < 0, and hence Tb (ab) < 0 given the e-money issuer’s budget (i.e. definition 3(c)) under µ ≥ 1.

F Proof of Proposition 7

Using Lemma 2, e-money is used only if

D (q) = d = min (ab,z + ab,n − θ∆, d∗) , (F.1)

∆ ≤ U (q)− C (q)− U [qz (ab,z)] + C [qz (ab,z)] , (F.2)

∆ ≤ ab,n + as,n, (F.3)

What are novel are the constraints (F.2) and (F.3): ∆ cannot be too high in order to induce the agents to

transfer the e-money. Consider a single deviation that agents do not join the mechanism in the current CM

(but unlike the previous technology, in the DM they can still pay ∆ to transfer the e-money), but will always

join the mechanism in all the CM after. Obviously, the deviating buyers do not hold e-money in the current

CM, since money will do all the same without paying ∆ in the DM, hence the continuation value after a

single deviation is just the continuation value of using money as in the previous section. On the other hand,

the deviating seller has to choose between holding the minimal e-money to pay ∆, a′s,n = max(∆− ab,n, 0),

such that the transfer of buyer’s e-money will go through, or not holding any to give up the e-money revenue.

Thus the participation constraints become

(β − µ) (ab,z + ab,n) + T pb + βαθ [U (q)− C (q)−∆] ≥ maxq′{(β − µ)D (q′) + βαθ [U (q′)− C (q′)]} , (F.4)

(β − µ) (as,z + as,n)+T ps +βα (1− θ) [U (q)− C (q)−∆] ≥ max

{(β − µ) a′s,n + βα (1− θ) [U (q)− C (q)−∆] ,

βα (1− θ) [U [qz (ab,z)]− C [qz (ab,z)]]

}.

(F.5)

Altogether with the issuer’s budget constraint and the market clearing conditions:

T pb + T ps = (µ− 1) (ab,n + as,n) + α∆, (F.6)

ab,n + as,n = λ (ab,z + as,z + ab,n + as,n) . (F.7)

Like the previous mechanism it is always optimal to have as,z = ab,z = 0. From (F.5), it implies optimal

to have as,n = T ps = 0. Thus, the constraint (F.2) becomes ∆ ≤ U (q) − C (q), and the constraint (F.3)

becomes ∆ ≤ D (q) + θ∆ following D (q) = ab,n+ θ∆ from (F.1), where the second inequality always implies

the first one. The problem of the e-money issuer becomes

maxq,∆{U (q)− C (q)} s.t.

31

[α (1− θ)− (1− α) (1− β) θ] ∆+(β − 1)D (q)+βαθ [U (q)− C (q)] ≥ maxq′{(β − µ)D (q′) + βαθ [U (q′)− C (q′)]} .

(F.8)

∆ ≤ U (q)− C (q) . (F.9)

If α (1− θ) − (1− α) (1− β) θ ≤ 0 then the issuer wants to set ∆ = 0; if α (1− θ) − (1− α) (1− β) θ > 0

then the issuer wants to maximize ∆ such that 0 ∆ = U (q)−C (q). Combining these two cases the issuer’s

problem becomes

maxq{U (q)− C (q)} s.t.

[max {α (1− θ)− (1− α) (1− β) θ, 0}+ βαθ] [U (q)− C (q)]+(β − 1)D (q) ≥ maxq′{(β − µ)D (q′) + βαθ [U (q′)− C (q′)]} .

(F.10)

The rest of the proof follows the argument of the proof to Proposition 4.

To compare the second best, recall that the second best allocations, q = qP and q = qT , are given by

[βα− (1− β) (1− θ)] [U (qP )− C (qP )] + (β − 1)C (qP ) = maxq′{(β − µ)D (q′) + βαθ [U (q′)− C (q′)]} ,

(F.11)[max {α (1− θ)− (1− α) (1− β) θ, 0}

+βαθ − (1− β) (1− θ)

][U (qT )− C (qT )]+(β − 1)C (qT ) = max

q′{(β − µ)D (q′) + βαθ [U (q′)− C (q′)]} .

(F.12)

Notice that the left side of (F.11) is the higher value function than the left side of (F.12) if and only if α < θ.

From the proof of Proposition 4 and of Proposition 7 both are decreasing functions. So we have qP > qT if

and only if α < θ.

G Proof of Proposition 8

To prove the "if" part of a, suppose thatMP implements the first best. Then from the proof of Proposition

4 it is necessary to have

maxq{− [µ− β (1− α)]D (q) + βαU (q)}

≤ − (1− β)D (q∗) + βα [U (q∗)− C (q∗)] ,

= − (1− β) [(1− θ)U (q∗) + θC (q∗)] + βα [U (q∗)− C (q∗)]

− (β + α− 1)U (q∗) + αC (q∗) + (β + α− 1)U (q∗)− αC (q∗) ,

= − [[1− βα− (1− β) θ] + (β + α− 1)] [U (q∗)− C (q∗)]− βC (q∗)

+ (β + α− 1)U (q∗)− αC (q∗) ,

≤ −βU (q∗) + (β + α− 1)U (q∗)− αC (q∗) ,

≤ (β + α− 1)U (q∗)− αC (q∗) ,

where the second-last inequality has used the premise that θ ≤ α. Thus the last line implies that µ ≥ Φ.

Then from the proof of Proposition 7 there exists an e-money mechanismMT that implements the first best

for the given µ.

32

To proof the "only if" part, suppose thatMT implements the first best. Then from the proof of Propo-

sition 7 we have

maxq{− [µ− β (1− α)]D (q) + βαU (q)}

≤(θ − θ

)[1− β (1− α)] [U (q∗)− C (q∗)] + [α− θ [1− β (1− α)]] ∆,

≤(θ − θ

)[1− β (1− α)] [U (q∗)− C (q∗)] + βα (1− θ) [U (q∗)− C (q∗)] ,

=[βα (1− θ) + [1− β (1− α)]

(θ − θ

)][U (q∗)− C (q∗)] ,

where the last inequality uses the fact that ∆ ∈ [0, U (q∗)− C (q∗)] and the fact that θ > α implies

βα (1− θ) > α − θ [1− β (1− α)]. Thus, we have µ > Θ. Then by Proposition 4 there exists MP im-

plementing the first best for the given µ as well.

Finally to show the "if" part of b, we replace q = q∗ in the first part with q = qP under equality, then

we reach the conclusion with replacing q = q∗ with q = qT . Similar argument for the "only if" part of b.

H Proof of Proposition 9

In the interest of brevity, we show only part a and part b follows a similar argument. Suppose that this is

not the case, i.e., there exists an e-money mechanismMT = {Tb (z, n) , Ts (z, n) ,∆b,∆s, λ} that implements

some first-best allocation (q∗, zb, zs, nb, ns) with the transferability technology under ∆ = 0 and some µ, but

there does not exist any e-money mechanism MP = {Tb (z, n) , Ts (z, n)} that implements the first-best

allocation with the participation technology under the same µ. Given ∆ = 0, consider a money mechanism

M = {T ′b (z) , T ′s (z) , λ} where

T ′s (z) =

{Ts (zs, ns) , if z = zs + ns0, otherwise

T ′b (z) = Tb (z, n) + (µ− 1) (zb + zs) .

Then it is straightforward to verify that (13) is satisfied under the first-best allocation (q∗, z′b, z′s) withM,

where z′b = zb + nb and z′s = zs + ns, since (13) and (??) are the same. Also, notice that

Tb (zb)− [µ− β (1− α)]D (q∗) + βαU (q∗)

= Tb (zb, nb)− [µ− β (1− α)]D (q∗) + βαU (q∗) +(1− µ−1

)(zb + zs)

≥ maxq{− [µ− β (1− α)]D (q) + βαU (q)} ,

where the last inequality comes from the fact that (q∗, zb, zs, nb, ns) is incentive compatible to buyers under

MT . So (12) is satisfied under the first-best allocation (q∗, z′b, z′s) withM. Finally, it is straightforward to

verify that (14) is satisfied under the first-best allocation (q∗, z′b, z′s) with M. Thus (q∗, z′b, z

′s) is incentive

compatible to buyers and sellers under M, and M is self-financed. This contradicts Proposition 5, since

there exists a money mechanism M that implements the first best but there does not exist any e-money

mechanism MP = {Tb (z, n) , Ts (z, n) , λ} that implements the first-best allocation with the participation

technology under the same µ. Therefore, we establish ∆ > 0. Finally, notice that (8) is satisfied only if

Tb (zn, nn) > 0. Thus, we prove Proposition 9.

33