Digital vs Physical Currency:
A Difference That Makes a Difference∗
Nicola Amendola†

Luis Araujo‡

Leo Ferraris§

June 2024

Abstract
This paper compares digital and physical currency, focusing on a single intrinsic difference: digital, unlike physical, currency allows to trace the monetaryflows in and out of the accounts. We show that this technological advance inrecord-keeping can be used by the monetary authority to reward active balances relative to idle balances, helping achieve efficiency in a wide range ofcircumstances.
Keywords: Cash, Digital Currency, Optimal Monetary Policy
JEL codes: E40

1

Introduction

In academic and policy circles, there is an active debate about the implications of
introducing sovereign digital currency, also known as central bank digital currency(CBDC), as payment instrument alongside or instead of physical currency. The literature has identified a number of benefits and costs of digital currency.1 Benefits∗

We would like to thank Jefferson Bertolai, Mariana Rojas Breu, Nicolas Jacquet, Ricardo Lagos,
Sebastien Lotz, Daniela Puzzello, Alberto Trejos, and Randall Wright for helpful comments. We
also thank participants at the 2023 Conference on CBDC at the Sveriges Bank, Sweden; and the2024 Search and Matching Conference at INCAE Business School, Costa Rica. Usual disclaimers
apply. Leo Ferraris thanks the Italian Ministry of Research for the research grant 2017K8ANN4.
†
Department of Economics and Finance, University of Rome, Tor Vergata‡
Department of Economics - MSU, and Sao Paulo School of Economics-FGV§
Department of Economics, Management and Statistics, University of Milan-Bicocca1
See Chapman et al. (2023) for a literature review.

1include lower transaction costs, increased competition in the banking sector, lower
tax evasion. Costs include the breach of privacy, infringement of personal freedom,risk of disintermediation. Neither the benefits nor the costs are strictly connectedwith the monetary nature of the instrument, that is often conceived simply as anintangible version of cash.
In this paper, we identify a single intrinsic difference between digital and physicalcurrency and show how it matters for optimal monetary policy. To make it transparent that this is the only driving force, we contrast two pure currency economies
that are identical except for the type of currency that is used, either physical or digital, comparing their functioning and identifying the optimal monetary policy in eachcase. The only difference between the two instruments is that, with digital currency
the monetary authority can keep track of flows of money balances in and out of theaccounts, while with physical currency this is not possible.2 The extra informationcollected through the digital technology can always be ignored, hence, any allocation
attainable with physical currency can be reached with digital currency, but there arerobust circumstances in which digital currency allows to improve strictly upon cash.
We consider a monetary search economy à la Lagos and Wright (2005), in which
the velocity of circulation of currency is endogenous since the traders choose theirsearch intensity, as in Rocheteau and Wright (2005). We characterize the equilibrium
and identify the optimal intervention by the monetary authority with physical anddigital currency, respectively. With physical currency, the optimal intervention is theFriedman rule, that, however, does not always achieve the efficient search intensity.
Moreover, any deviation from the Friedman rule always reduces search intensity andthe velocity of money.3 With digital currency, instead, the efficient search intensity
can always be achieved and in some cases the optimal policy is not the Friedman rule.
The monetary authority achieves efficiency paying a positive nominal interest rate inaccounts with a higher velocity of money and no interest in accounts with a lower
velocity of money.
We show that the result holds in a robust set of circumstances, beyond the specific trading arrangement adopted for simplicity, i.e. the Kalai bargaining protocol.
Following Gu and Wright (2016), we generalize the trading protocol to a mechanism
2This difference in the record-keeping possibilities with different types of payment instruments isin the spirit of Kocherlakota (1998).
3This occurs also in Liu, Wang and Wright (2011).

2

that subsumes among other schemes, both the Kalai and Nash protocol. Interestingly, we show that, under the Nash bargaining protocol, digital currency allows toimprove the allocation relative to physical currency, even if the search intensity is
exogenous. Although limited to the Nash protocol, this result shows that the key
for the improvement is the presence of distortions, arising from extensive or intensive
margins, that the Friedman rule alone cannot correct. We also show that the result
survives when the taxation of balances is not feasible, as in Hu, Kennan and Wallace(2009). Finally, the result continues to hold even if the agents can try to manipulate
the system at a cost, opening ”shadow accounts” to obtain interest on balances thatare not truly active.
A growing literature examines the impact of digital currencies, particularly CBDC.
For example, recent papers that adopt the Lagos and Wright (2005) framework include Williamson (2022), Chiu, Davoodalhosseini, Jiang and Zhu (2023), and Keisterand Sanches (2023). In these papers, interest can be paid with digital unlike physicalcurrency, but uniformly on all balances. We show that, if the monetary authority can
run the Friedman rule, restricting the scheme to the payment of interest uniformlyon all balances, both active and idle, digital currency cannot enlarge the set of implementable allocations relative to physical currency, as the optimal allocation coincides
with the one achieved under the Friedman rule in the corresponding physical currency
economy. However, paying interest uniformly is sub-optimal, as it does not take into
account the ability of the monetary authority to keep track of the monetary flows inand out of the accounts that is available with digital but not physical currency. This
superior ability helps stimulate the traders’ participation, through the payment of interest on active balances, allowing to achieve full efficiency on all margins, extensive
and intensive. With endogenous participation, the ability to reward active balances
is an improvement also relative to credit schemes that reward only idle balances, as inBerentsen, Camera and Waller (2007) or Ferraris and Watanabe (2008). Intuitively,
this obtains because it is preferable to pay interest in accounts with higher rather
than lower velocity to stimulate the traders’ participation.
The paper proceeds as follows. Section 2 introduces the model. Section 3 presents
the results and Section 4 compares them with the literature. Section 5 presents two
extensions and Section 6 concludes. The Appendix contains the proofs that are notin the text and the technical details of two of the extensions.

3

2

Model

The model is a version of Lagos and Wright (2005) with an endogenous participation
decision. In this section, we describe the fundamentals, the trading instruments andderive the efficient benchmark.

2.1

Fundamentals

Time is discrete and each period is divided into two sub-periods, called day andnight. There is a continuum of infinitely lived agents who discount the future at rateβ ∈ (0, 1). During the day, the agents meet in a decentralized market to trade a commodity, whose non-negative quantity is denoted with q. Half of the agents are sellersof this commodity, that, to be produced, requires a cost represented by a twice differentiable strictly increasing and convex function, c(q). The other half of the agents arepotential consumers of this commodity. Consumption of the commodity gives utility
represented by a twice differentiable strictly increasing and strictly concave function,u(q). We assume u(0) = c(0) = u′ (∞) = c′ (0) = 0, and u′ (0) = ∞. The potential
consumers choose search intensity α ∈ [0, 1] that represents the probability of meeting sellers in the day market, incurring a cost represented by a twice differentiable,
strictly increasing and strictly convex function, k(α). We assume k(0) = k ′ (0) = 0.
The buyers and sellers meet bilaterally, and the terms of trade within meetings are
determined with the Kalai protocol, where θ ∈ (0, 1) denotes the bargaining powerof the buyer.4 The traders cannot commit to future actions and are anonymous during the day, their trades being not observable by outsiders. At night, all the agents
meet in a centralized market where another commodity is traded that serves as thenumeraire and can be produced and consumed by all agents, with linear payoffs.

2.2

Currency

Lack of monitoring and commitment prevents trade during the day in the absenceof a trading instrument. We consider two alternative economies, one in which thetrading instrument is physical currency, and another with digital currency. These
economies are identical except for the use of the trading instrument. In particular,
4

See Kalai (1977). Lagos and Wright (2005) use the Nash protocol. As discussed below, a similar
logic would be at work with Nash bargaining but the algebra would be messier. We also show that
our results hold in a larger class of trading protocols.

4

in both economies, currency is introduced through accounts open by the monetary
authority for each agent at the beginning of the first period. We distinguish the trading instruments by the record-keeping technology embedded in the accounts. In thephysical currency economy the monetary authority only keeps track of balance holdings in the accounts, while in the digital currency economy the monetary authority
keeps track of balance holdings and balance flows in and out of the accounts.5

2.3

Efficiency

Due to the linearity of the payoffs, utility is transferable at night and has no impact onwelfare. The efficient allocation consists of a participation rate α and a day quantity
q that maximize α[u(q) − c(q)] − k(α). The complementary slackness condition forthe optimal participation is(1 − α)[u(q) − c(q) − k ′ (α)] = 0,(1)

whereby the participation is full if the gains from trade are sufficiently large relative
to the marginal cost of participating and partial otherwise. The optimum condition
for the intensive margin isu′ (q) = c′ (q).
(2)
Let the efficient allocation that satisfies (1) and (2) be denoted by (α∗ , q ∗ ) and define′ (1)θ∗ ≡ u(q∗k)−c(q∗ ) . The next Proposition characterizes the efficient allocation.
Proposition 1 The efficient day quantity, q ∗ , equates marginal utility and cost; theefficient participation is full, α∗ = 1, if θ∗ ≤ 1; and partial, α∗ = k ′−1 (u(q ∗ ) − c(q ∗ )) ∈(0, 1), if θ∗ > 1.

3

Physical vs Digital Currency

In what follows, we first consider physical currency, and then digital currency. In both
economies, we restrict attention to stationary symmetric equilibria with constant realbalances. Throughout, we denote by ϕ the price of currency in the night market.
5

We follow Andolfatto (2010) and Wallace (2014) in assuming that the monetary authority can
implement policies that condition on balances in the physical currency economy. The implementationof these policies naturally requires the observation of balances. We do not allow the authority tocondition on the history of balances, which would be equivalent to conditioning on the flows.

5

3.1

Physical Currency

Consider a meeting between a buyer and a seller in the day market. Under Kalai
bargaining, a buyer holding an amount m of currency chooses a payment d ≤ m inexchange for a quantity q to maximize u(q) − ϕd, subject to the constraint(1 − θ)[u(q) − ϕd] = θ[ϕd − c(q)].

(3)

Note that the real balances of the seller do not affect the terms of trade. Moreover,since the seller has no use for real balances in the day market, in equilibrium it isnever the case that the seller strictly prefers to bring a positive quantity of currency.
We assume that the buyers spend all their balances during the day. Below, we willshow that this is, in fact, the optimal choice of the agents. This implies d = m, andwe can rewrite (3) as
ϕm = (1 − θ)u(q) + θc(q) ≡ g(q),(4)
which determines the quantity produced by the seller as a function of the buyer’s realbalances.
Consider now the participation decision of a buyer with ϕm real balances at the
beginning of the day. Choosing α, he incurs the cost k(α) and meets a seller withprobability α, in which case he spends all his balances. If he does not meet a seller,
he keeps his balances. Formally, the buyer chooses α ≤ 1 to maximize αu(q) + (1 −α)ϕm − k(α). Using (4), we obtain the following complementary slackness condition
for the optimum,(1 − α) {θ [u(q) − c(q)] − k ′ (α)} = 0,(5)where both terms in parentheses are non-negative. If the surplus appropriated by
the buyer is sufficiently large, he chooses full participation; otherwise, the solution isinterior and there is partial participation.
We now move to the night market, and consider the buyer’s choice of the amountof balances to bring into the day market. He chooses m to maximize −ϕm+β[−k(α)+αu(q) + (1 − α)ϕ+1 m]. Using equation (4), we obtain the inter-temporal condition forthe optimum,

 ′
u (q)
+1−α .
(6)
ϕ = βϕ+1 α ′g (q)In words, an extra unit of currency acquired presently can be spent next period on

6day consumption if the agent turns out to be a buyer or kept idle until the nightmarket otherwise.
In the economy with physical currency, the monetary authority can inject or tax
currency in a lump-sum fashion. If we let M denote the quantity of currency and τ
denote the growth rate of currency change, we have M+1 = (1 + τ )M . Stationarityof real balances implies ϕϕ+1 = MM+1 = 1 + τ , and we can rewrite (6) asu′ (q)
1 − β + τ + αβ
=
.
′
g (q)
αβ

(7)

The existence of a monetary equilibrium requires τ ≥ β − 1 to prevent an infinite
demand for money. This condition also ensures that it is never strictly optimalfor the buyers to bring balances they do not plan to use, hence vindicating our
initial assumption that the buyers spend all their balances. A stationary symmetricequilibrium is a pair (α, q) that satisfies (5) and (7). The next Proposition, whose
proof appears in the Appendix, shows the existence of a unique equilibrium.
Proposition 2 The equilibrium with physical currency exists and is unique.
We now determine the optimal policy τ . First, observe that τ does not directly
enter the participation condition (5). Also observe that, for any given α, it is optimal
to set q = q ∗ to maximize the surplus in a trade meeting. This can be achieved by
implementing the Friedman rule, i.e., setting τ = β − 1. At the Friedman rule, ifθ ≥ θ∗ , buyers choose to fully participate in trade, and the first-best is achieved. If,
instead, θ < θ∗ , buyers choose αc < α∗ , where k ′ (αc ) = θ [u(q ∗ ) − c(q ∗ )]. In this case,
the Friedman rule is optimal but it does not achieve the first-best. The following
Proposition summarizes this result.
Proposition 3 The Friedman rule is optimal. It implements the first-best if and only
if θ∗ ≤ θ < 1.
The Friedman rule drives the intensive margin to efficiency rewarding balances at
the rate of time preference to compensate for the elapse of time, but is unable, in
general, to drive the participation decision towards full efficiency, as it lacks the tools
to give the buyers the right incentive to participate.

7

3.2

Digital Currency

With digital currency, the record-keeping technology allows the monetary authority
to track the flows of balances in and out of the accounts. In what follows, we use
this information to separate balances at the end of the day market into two groups,
labeled idle and active. A balance in an account at the end of the day market isidle if it was already in the account at the beginning of the day market; while abalance in an account at the end of the day market is active if it was transferred into
the account during the day market. We consider interventions where the monetary
authority treats these balances differently, offering two distinct nominal interest rates,
ip for the idle balances and ia for the active balances. We proceed by first determining
the equilibrium and then characterizing the optimal policy.
3.2.1

Equilibrium

Consider a meeting between a seller and a buyer with m units of digital currency inthe day market. If the buyer transfers d balances to the seller and receives q unitsof goods, the surplus of the buyer is Sb = u(q) − ϕ(1 + ip )d, while the surplus of theseller is Ss = −c(q) + ϕ(1 + ia )d. Observe that the surplus of the buyer includes theinterest lost in the currency that was transferred to the seller and, correspondingly,
the surplus of the seller includes the interest gained in the process. The total surplus
is S = u(q) − c(q) + ϕ(ia − ip )d, namely the gain from trade and the net interest
payment to the transferred balances.
With Kalai bargaining, a buyer holding an amount m of currency chooses d ≤ min exchange for a quantity q to maximize u(q) − ϕ(1 + ip )d subject to the constraint(1 − θ)Sb = θSs . As with physical currency, the real balances of the seller do notaffect the terms of trade. However, the seller may want to bring real balances to the
day market, depending on the nominal interest rate paid on idle balances. We will
examine this incentive below, showing that it is never part of the optimal policy togive the sellers the incentive to bring balances. We will also show that, as in the caseof physical currency, buyers spend all their balances during the day. This allows torewrite the constraint (1 − θ)Sb = θSs as
ϕm ≡

g(q)
.
1 + θia + (1 − θ)ip

8(8)Consider now the participation decision of a buyer with balances m at the beginning of the day market. Choosing α, he incurs a cost k(α), meets a seller withprobability α, and spends all his balances. If he does not meet a seller, he keeps hisbalances until the night market, and receives an interest rate payment ip . Formally,
the buyer chooses α to maximize αu(q) + (1 − α)(1 + ip )ϕm − k(α). Using (8), we
obtain the complementary slackness condition for the optimum,(1 + ia )u(q) − (1 + ip )c(q)′− k (α) = 0,
(1 − α) θ
1 + θia + (1 − θ)ip(9)where both terms in parenthesis are non-negative. If the surplus appropriated by
the buyer is sufficiently large, he chooses full participation; otherwise, the solution
is interior and participation is less than full. Observe that, for any given q, theincentive of the buyer to participate in trade increases with the interest rate paid on
active balances and decreases with the interest paid on idle balances.
We now move to the night market. The buyer chooses m to maximize −ϕm +β[αu(q) + (1 − α)(1 + ip )ϕ+1 m − k(α)]. Using (8), we obtain the inter-temporalcondition for the optimum, ′u (q) [1 + θia + (1 − θ)ip ]+ (1 − α)(1 + ip ) .
ϕ = βϕ+1 αg ′ (q)(10)

In words, an extra unit of currency acquired presently can be spent next period on
day consumption if the agent turns out to be a buyer or kept idle until the nightmarket otherwise. In either case, the agent receives an interest payment for using
actively the balances or keeping them idle.
As in the economy with physical currency, the monetary authority can inject ortax currency in a lump-sum fashion. It can also inject or tax currency by using thenominal interest rates ip and ia . If we let τ denote the rate at which money is injected
or taxed in the economy in a lump-sum way, we have M+1 = [1+τ +(1−α)ip +αia ]M .
Stationarity of real balances implies ϕϕ+1 = 1 + τ + (1 − α)ip + αia , and we can rewrite(10) asu′ (q)
τ + 1 − β + αβ + αia + (1 − α) (1 − β) ip
=
.
(11)
′
g (q)
αβ [1 + θia + (1 − θ)ip ]The existence of a monetary equilibrium requires ϕ ≥ βϕ+1 (1 + ip ), otherwise anagent would have the incentive to demand an infinite amount of currency. We can9rewrite this condition asτ + α (ia − ip )≥ β − 1.
1 + ip(12)

A stationary symmetric equilibrium is a pair (α, q) that satisfies (9) and (11). The
next Proposition, whose proof appears in the Appendix, shows the existence of aunique equilibrium.
Proposition 4 The equilibrium with digital currency exists and is unique.
As in the physical currency economy, an increase in τ has no direct impact on
the participation of buyers; but it reduces the surplus, with a negative impact on
the quantity produced in trade meetings. In contrast, changes in the nominal interest
rates on active and idle balances impact both the extensive and the intensive margins.
We start with the intensive margin. To provide intuition, fix the extensive margin.
An increase in interest rates leads to money creation, which negatively impacts thereal rate of return on balances. This reduces the demand for real balances, and thequantity in trade meetings, as captured in the numerator on the right-hand side of(11). However, an increase in interest rates increases the quantity in trade meetings,
for any given real balances ϕm. This is so because an increase in interest rates
increases the relative surplus of the seller (an increase in ia increases the surplus of
the seller, while an increase in ip reduces the surplus of the buyer) which must bebalanced by an increase in the quantity produced in the meeting. This positive effectis captured in the denominator on the right-hand side of (11). The overall effect isambiguous.
Consider now the extensive margin. To provide intuition, fix the intensive margin.
An increase in the net interest payment ia − ip has a positive impact in the participation of buyers. It does so either by increasing the region of parameters where fullparticipation is optimal or by increasing the participation of buyers in the region ofparameters with partial participation. This can be seen by observing that the termin brackets in (9) is strictly increasing in ia and strictly decreasing in ip . Intuitively,
an increase in ia − ip encourages participation due to its direct positive effect on thesurplus in a trade meeting.

10

3.2.2

Optimal Policy

In the digital currency economy, in addition to impacting the overall return on currency through changes in τ , the monetary authority can use ia and ip to target how
the return on currency will be distributed between agents holding idle balances andagents holding active balances. In what follows, we will show that the availability ofthese additional instruments allows the monetary authority to always implement thefirst-best.
We proceed using θ∗ and θ to divide the parameter space into three regions.
First, if θ∗ ≤ θ < 1, Proposition 2 shows that the first-best can be achieved in thephysical currency economy. Since any policy in the physical currency economy canbe replicated in the digital currency economy if ia = ip = 0, the first-best can also be
achieved if the authority implements the Friedman rule. The next Lemma summarizes
the result.
Lemma 1 Let θ∗ ≤ θ < 1. The first-best involves full participation of the buyers andis implemented by the Friedman rule, i.e., τ = β − 1 and ia = ip = 0.
Next, consider θ < θ∗ ≤ 1. In this region, if an intervention implements the firstbest, the efficient quantity is produced in trade meetings, and there is full participationof buyers. Thus, the right-hand-side of (11) evaluated at α = 1 must be equal to one,which requiresτ = (β − 1) (1 + ip ) − (1 − βθ) (ia − ip ) .
(13)
Using (9), we also needk ′ (1) ≤ θ(1 + ia )u(q ∗ ) − (1 + ip )c(q ∗ ),1 + θia + (1 − θ)ip

(14)

to ensure that buyers want to fully participate. Finally, under full participation, theexistence of the monetary equilibrium requires τ + ia − βip ≥ β − 1, which, using (13),can be rewritten as ia ≥ ip .
A natural policy candidate for the implementation of the first-best involves setting
ip = 0. This is consistent with a zero lower bound on interest rates and it provides thebuyers with the incentive to participate in trade while vindicating our initial claimthat agents have no incentive to bring balances into the day market that they do notplan to use. Since θ∗ ≤ 1 implies k ′ (1) ≤ u(q ∗ ) − c(q ∗ ), a sufficient condition for the11buyers to fully participate is that u(q ∗ ) − c(q ∗ ) equals the right hand side of (14), i.e.,u(q ∗ ) − c(q ∗ ) = θ(1 + ia )u(q ∗ ) − (1 + ip )c(q ∗ ),1 + θia + (1 − θ)ip

which can be rewritten asi∗a =1 − θ u(q ∗ ) − c(q ∗ ).
θ
c(q ∗ )The positive nominal interest rate on active balances compensates for the insufficientbargaining power of the buyer. In fact, note that i∗a is strictly decreasing in θ, and it
goes to zero when θ goes to one. Using (13), we also obtainτ ∗ = − [1 − β + (1 − βθ) i∗a ] .
Note that i∗a > ip = 0 implies ϕ > βϕ+1 and the real rate of return on currency islower than the discount rate. In this sense, the optimal intervention is away from theFriedman rule. In fact, there is money creation under the optimal policy if τ ∗ +i∗a > 0,
which can be rewritten asβ>c(q ∗ ).
(1 − θ) u(q ∗ ) + θc(q ∗ )The next Lemma summarizes our result.
Lemma 2 Let θ < θ∗ ≤ 1. The first-best involves full participation of the buyers andit can be implemented by the policy τ = τ ∗ , ia = i∗a , and ip = 0, which deviates fromthe Friedman rule.
We now turn to the region where θ∗ > 1. In this case, the first-best involves
efficient production of q ∗ in trade meetings but partial participation of buyers, i.e.,
the efficient search intensity is given by α∗ < 1, wherek ′ (α∗ ) = u(q ∗ ) − c(q ∗ ).

(15)

If a policy implements the efficient quantity, the right hand side of (11) evaluated atα∗ must be equal to one, i.e.,τ = (β − 1) (1 + ip ) − (1 − βθ) α∗ (ia − ip ) .
12

(16)

In turn, in order to give buyers the incentive to search efficiently, the term in brackets
in (9) must be satisfied at q ∗ and α∗ with equality, i.e.,k ′ (α∗ ) =θ[(1 + ia )u(q ∗ ) − (1 + ip )c(q ∗ )].
1 + θia + (1 − θ)ip

(17)

Finally, the existence of the monetary equilibrium requires (12) which, using (16),can be rewritten as ia ≥ ip .
As in the region where θ∗ ≤ 1, a natural policy candidate for the implementationof the first-best has ip = 0. Using ia to equate the right-hand side of (15) with theright-hand side of (17), we obtain ia = i∗a . In turn, using (16), we obtainτ ∗∗ = − [1 − β + α∗ (1 − βθ) i∗a ] .
As in the case where θ < θ∗ ≤ 1, i∗a > ip = 0 implies ϕ > βϕ+1 , and the real rateof return on balances is lower than the discount rate, so the economy is away fromthe Friedman rule. The next Lemma summarizes our results.
Lemma 3 Let θ∗ > 1. The first-best involves partial participation of the buyers andit can be implemented by the policy τ = τ ∗∗ , ia = i∗a , and ip = 0, which deviates fromthe Friedman rule.
The previous three Lemmas together imply the following Proposition.
Proposition 5 If θ < θ∗ , there exist policy schemes that deviate from the Friedman
rule and implement the first-best rewarding active balances with positive interest andidle balances with zero interest.
This is the main result of the paper. When there are distortions that would imply
inefficiencies in both intensive and extensive margins, the optimal monetary policy inthe physical currency economy, which is the Friedman rule, achieves efficiency along
the intensive but not the extensive margin. In the corresponding digital currency
economy, the optimal policy, which is not necessarily the Friedman rule, achieves efficiency also along the extensive margin stimulating the participation of the buyers in
the process of trade through the payment of interest on active but not idle balances.
Unlike in the physical currency economy, this is feasible in the digital currency economy, since the technology allows to observe the flows of balances in and out of theaccounts, which allows to distinguish active and idle balances.
13

3.3

Velocity of circulation

It is well known from the work of Levine (1991), Kehoe, Levine and Woodford (1992)
and Wallace (2014) that extensive margins play a key role in monetary economies.
In our framework, the extensive margin is captured by the endogenous participation
decision of the buyers. The ability to observe the flows of money balances allows to
implement intervention schemes that improve welfare by giving extra incentives tothe buyers to participate in trade, thus, increasing the velocity of money. To see this,compare the velocity of circulation of currency across economies under the optimal
intervention. In both economies, there is a proportion α of traders who end up inmeetings in which currency changes hands for sure, with unit velocity of circulation,
and a fraction 1 − α of traders who keep their currency in the account, in which casethe velocity is nil. Therefore, the velocity of circulation of either physical or digitalcurrency is given by v = α × 1 + (1 − α) × 0 = α. If θ∗ ≤ θ < 1, both economies
implement the first-best and velocity is the same. If, instead, θ < θ∗ , the velocity ishigher in the digital currency economy because participation is inefficiently low in thephysical currency economy. Thus, the key message of our paper can be interpreted
as follows. The observation of balance holdings, which is feasible in the physicalcurrency economy, impacts the intensive margin, as it allows to set a positive real
rate of return on currency, thus reducing the cost of carrying balances across periods.
In turn, the observation of balance flows in and out of the accounts, which is feasible
in the digital currency economy, allows to set a spread between interest rates on active
and idle balances that impacts the extensive margin, giving the buyers the incentive
to participate in trade and, as a consequence, increasing the velocity of money.

4

Constrained Intervention

In this section, we compare our results with the literature by exploring constrainedintervention schemes. First, we examine interventions that pay the same interest rate
on idle and active balances. Second, we consider interventions that pay interest only
on idle balances. Finally, we examine interventions that do not involve the taxationof balances.

14

4.1

Uniform Interest on Balances

The potential impact of the introduction of CBDC on financial intermediation has
spurred a large literature. The idea is that CBDC may compete with banks on theliability side, thus affecting their ability to fund investments, as, for instance, in
Andolfatto (2021), Williamson (2022), Keister and Sanches (2023), and Chiu et al.
(2023). For the most part, this literature distinguishes CBDC and cash assumingthat they are imperfect substitutes as means of payment. In particular, CBDC is a
closer substitute to bank issued debt than cash. Apart from this difference, CBDC
is treated as a digital form of cash, and the only notable difference between them isthe fact that interest can be paid on CBDC.
In our model, treating digital currency as physical currency in digital form means
assuming that the former embeds the same record-keeping technology as the latter,
which implies that the monetary authority can no longer distinguish between idle andactive balances, and must set ia = ip . Letting ia = ip = i in (11), we obtainτ

1 − β + 1+i + αβu′ (q)=.
g ′ (q)αβ(18)Note that the only difference between (18) and (7) is in the first-term in the numeratorof these equations. In fact, while in the physical currency economy, the rate τ atwhich money is injected or taxed in the economy in a lump-sum way is the only
policy instrument; in the digital currency economy with uniform interest the policy
instrument is given by τ divided by the gross nominal interest rate.
Consider now the extensive margin of the digital currency economy. If ia = ip = iin (9), we obtain(1 − α) {θ [u(q) − c(q)] − k ′ (α)} = 0,
which coincides with (5), the extensive margin in the physical currency economy. We
have immediately the following proposition.
Proposition 6 Let τc (resp. τd ) be the rate at which money is injected or taxed ina lump sum way in the physical currency (resp. digital currency) economy, and let
i be the uniform interest payment on balances in the digital currency economy. An
outcome (α, q) is an equilibrium in the physical currency economy if and only if it isalso an equilibrium in the digital currency economy.

15

This result highlights that, if one restricts attention to intrinsic properties, a key
fundamental difference between physical and digital currency is the fact that thelatter embeds a technology which allows to keep track of flows and holdings acrossaccounts. In turn, this allows the monetary authority to implement interventions thatcondition on this information. In other words, if one assumes that digital currency
is simply a digital form of physical currency, differences between these instruments
must be related to extrinsic features of the economy, i.e., frictions that make physicaland digital currencies imperfect substitutes as means of payments.

4.2

Rewarding Idle Balances

The literature has examined the role of banks and liquidity markets to reallocate idlebalances in a model based on Lagos and Wright (2005), as in Berentsen, Camera andWaller (2007), Ferraris and Watanabe (2008) and Geromichalos and Herrenbrueck(2017). These models usually feature an exogenous intensive margin and competitivemarkets, so the Friedman rule implements the first-best. Away from the Friedman
rule, it is shown that banks and liquidity markets can improve the allocation bytransferring balances from the agents without to those with a consumption opportunity. This transfer is beneficial as it allows the payment of positive interest to
agents without a consumption opportunity, who would otherwise keep their balances
idle. Intuitively, paying interest on idle balances is beneficial because it insures agentsagainst the risk of not participating in trade.
To replicate the role of banks and liquidity markets in reallocating balances withdigital currency, consider an intervention by the monetary authority that pays positive
interest on idle balances but no interest on active balances. To make the insurance
effect of paying interest on idle balances relevant, we restrict attention to the regionof parameters where θ∗ > 1 and full participation is not efficient.
Let ia = 0. We can rewrite (11) as
1 − β + αβ + τ + (1 − α) (1 − β) ip
u′ (q)=,′g (q)αβ [1 + (1 − θ)ip ](19)

and we can rewrite (9) ask ′ (α) = θu(q) − (1 + ip )c(q).
1 + (1 − θ)ip
16

(20)

The existence of the monetary equilibrium requires ϕ ≥ βϕ+1 (1 + ip ), which canbe rewritten as τ ≥ (β − 1) (1 + ip ) + αip ≡ τ (ip ). It follows immediately that the
optimal policy if ia = 0 is the Friedman rule, given by τ = β − 1 and ip = 0. This
policy achieves the efficient quantity and it encourages participation by offering zerointerest on idle balances. As shown in the previous section, this policy is dominatedby an intervention that sets positive interest on active balances and zero interest on
idle balances.
Now, let ia = 0 and consider the scenario where ϕ > βϕ+1 (1+ip ) and the economy
is away from the Friedman rule. In particular, fix τ = τ (ip ) + δ. We can rewrite (19)as
δ
1 + ip + αβu′ (q)=.
g ′ (q)
1 + (1 − θ)ip
and an increase in ip leads to an increase in the quantity produced by the seller ifαβθ, and the economy is sufficiently away from the Friedman rule.
and only if δ > 1−θIntuitively, an increase in ip insures buyers against the risk of not meeting a seller.
However, unlike banks, the insurance provided by the monetary authority requires
the injection of currency into the economy, which reduces its real rate of return.
The overall effect on the intensive margin is thus ambiguous. If the economy is
not too far away from the Friedman rule, the insurance benefit is less relevant and ahigher interest rate on idle balances reduces the quantity in trade meetings. Since theincrease in ip also reduces participation in trade, welfare decreases. However, if theeconomy is farther away from the Friedman rule, the insurance effect dominates andan increase in ip increases the quantity in trade meetings. However, since participation
goes down, the overall welfare effect is ambiguous.
Summarizing, digital currency may replicate the insurance role of banks in improving the intensive margin by paying interest on idle balances, but it does so at
the cost of creating currency to fund the payment of the interest. Allowing for thepayment of interest on active balances, instead, digital currency can improve boththe intensive and the extensive margins of trade.

4.3

No taxation of Balances

The taxation of balances may not be feasible in pure currency economies, as pointed
out, among others, by Hu, Kennan and Wallace (2009), Andolfatto (2010), Wallace(2014), and Bajaj et al. (2017). The idea is that the same frictions on commitment
17and monitoring that render money essential prevent also the working of lump-sum
taxation schemes. In our case, this translates into a restriction to policies with τ ≥ 0.
In Appendix B, we show that, even if balances cannot be taxed, there still exists anopen region of parameters that replicates the results obtained in the previous section,i.e., the observation of balance flows still allows digital currency to dominate physicalcurrency and implement the first-best. The intuition is the following.
In the physical currency economy, we consider a scheme where the monetary
authority makes a transfer of real balances to each account that conditions on the realbalance in the account. As in Bajaj et al. (2017), the optimal scheme compensates
for the inflation tax accounts with real balances larger than or equal to the socially
optimal amount, i.e., the amount of real balances that induces the sellers to produce
the efficient quantity. This ensures that the efficient quantity is produced. However,since this transfer scheme cannot separate between idle and active balances, it fails to
directly give the buyers the incentive to participate. In the digital currency economy,
instead, since flows in an out of accounts are also observed by the authority, theoptimal scheme compensates for the inflation tax only active balances. This ensures
not only an efficient production in trade meetings but also an efficient participationof buyers.
Finally, the only difference between these schemes and the optimal interventions
considered in the case where real balances can be taxed, is that agents must be
sufficiently patient, both in the physical currency and in the digital currency economy.
This is so because, unlike the scenario where real balances can be taxed and the
optimal intervention can condition on the agent’s discount factor, here the injectionof real balances must be sufficiently large to convert the decision of a buyer on how
much real balances to bring, into a binary choice between bringing the socially optimal
amount and bringing zero balances. Patience is required for the former option todominate.

5

Robustness

In this section, we extend the result to a larger class of bargaining procedures thanjust Kalai bargaining and show that the scheme with the payment of interest on
active balances is robust to manipulation attempts that involve the opening of shadow
accounts.
18

5.1

Bargaining

In the search and matching monetary literature, since trade is not mediated by the
Walrasian auctioneer, the pricing mechanisms are key. In our setting, trade meetings
are bilateral and we have adopted the Kalai bargaining procedure for simplicity, as
it makes computations easier. Other bargaining schemes could be adopted without
altering the main results. In Appendix C, we adapt to our setting the monetarymechanisms proposed by Gu and Wright (2016) and show that our results remain
valid. Particular cases of their mechanism include Kalai bargaining, Nash bargaining,and competitive price taking. In what follows, we consider Nash bargaining in moredetail, since it unveils an additional channel through which the observation of balance
flows allows the implementation of policies that improve welfare.
It is well known from Aruoba, Rocheteau and Waller (2007) that, even without
the endogenous participation decision, i.e. when the extensive margin plays no role,
the Friedman rule implements the efficient quantity within trade meetings under
Kalai but not Nash bargaining, unless the buyers have full bargaining power. Thisoccurs because the traders’ payoffs are non-monotonic in real balances under Nashbargaining.6 Hence, under Nash bargaining, there may be room for digital currency
to allow the implementation of the efficient allocation even without the endogenous
participation decision.
Consider the digital currency economy with an exogenous extensive margin, so
a buyer meets a seller in the day market with an exogenous probability α. It is
straightforward to show that, under Nash bargaining, if a buyer with an amount mof currency meets a seller, terms of trade are given byϕm =(1 − θ)u(q)c′ (q) + θc(q)u′ (q)≡ N (q, ia , ip ).
θ(1 + ia )u′ (q) + (1 + ip )(1 − θ)c′ (q)(21)

In turn, in the previous night market, the buyer chooses m to maximize −ϕm +β[αu(q) + (1 − α)ϕ+1 (1 + ip )m]. Using the derivative of (21) with respect to q, we
obtain the intertemporal condition for the optimum,u′ (q)
1 − β + αβ + τ + αia + (1 − α)(1 − β)ip=.
Nq (q, ia , ip )
αβ
6

(22)

See Hu and Rocheteau (2020) for a unified strategic foundation of the Kalai and Nash protocols
that shed further light on their differences.

19

We can replicate the allocation in the economy with physical currency by letting
ia = ip = 0, which delivers
1 − β + αβ + τu′ (q)=.
Nq (q, 0, 0)
αβ

(23)

It is straightforward to show that the solution qc to (23) is strictly lower than q ∗ ,even if τ = β − 1 and the economy is at the Friedman rule. Moreover, qc converges
to q ∗ as θ goes to one. These are well known results in the literature. Notice that(21) implies that an increase in the interest rates allows the buyer to consume thesame quantity with lower real balances, since an increase in ia increases the surplus
in trade meetings, while an increase in ip increases the outside option of the buyer.
This suggests that the buyer may be encouraged to bring more real balances to the
day market if he expects positive interest rates. However, positive interest rates
require currency creation, which reduces the real rate of return on currency andlowers the incentive to bring real balances. The overall effect is ambiguous. Should
the monetary authority set ia = ip = i > 0, (22) would collapse into (23), leading
to the same allocation as in the physical currency economy. Thus, under uniform
interest rates, it is still the case that the surplus of the buyer may decrease while thetotal surplus increases, which is at the root of the inefficiency under Nash bargaining.
In the following Proposition, whose proof is in the Appendix, we show that setting
different interest rates on active and passive balances allows the monetary authority
to achieve the efficient quantity.
Proposition 7 Fix α and assume the Nash bargaining protocol, with θ < 1. Settingτ = β − 1 and ip = 0, there exists a unique ia > 0 that achieves the first best, q = q ∗ .
Summarizing, the observation of balance flows offers an additional instrument to
the monetary authority, i.e., the spread between the interest rate on active and idlebalances. If the extensive margin is endogenous, this additional instrument can beused to improve welfare by increasing the participation of buyers and the velocity
of money. If the extensive margin is exogenous and terms of trade are determinedby Nash bargaining, this additional instrument can be used to improve welfare byaligning the surplus of the buyer and the total surplus in trade meetings.

20

5.2

Manipulation

We would like to find out whether our results are robust to the possibility of manipulation, whereby agents try to obtain interest on idle balances equal to the interest
paid on active balances. For example, if an agent is allowed to open more than one account with the monetary authority, she can transfer unused balances across accounts
and be paid an interest ia on those balances. We prevent this possibility by assumingthat each agent has only one account with the monetary authority.
In what follows, we introduce the possibility of manipulation by assuming thatagents can open a shadow account that the authority cannot associate with the ownerof any existing account. However, in order to do so, they incur a cost γ > 0 during
the night market. This account can then be used in the next day market to transfer
balances in order to receive the interest ia .
Consider the incentive of a single buyer to deviate and incur the cost γ. To
make it transparent the benefits of incurring this cost, we consider the region ofparameters where θ∗ > 1 and full participation is not efficient. In this case, Lemma
3 implies that the optimal policy is given by ip = 0, ia = i∗a and τ = τ ∗∗ . If thedeviant buyer holds an amount m̃ of currency, he chooses q̃ and d˜ ≤ m̃ to maximizeu(q̃)−ϕ(1+i∗a )d˜ subject to the constraint (1−θ)S̃b = θS̃s , where S̃b = u(q̃)−ϕ(1+i∗a )d˜˜ Note that the buyer receives interest i∗ on his passive
and S̃s = −c(q̃) + ϕ(1 + i∗a )d.
ag(q̃)
balances. This allows to rewrite the constraint (1−θ)S̃b = θS̃s as ϕm̃ = 1+i∗ . Considera
now the participation decision. Optimization leads to the complementary slackness
condition for the participation rate {θ[u(q̃) − c(q̃)] − k ′ (α̃)} (1 − α̃) = 0, where bothterms in parenthesis are non-negative. Finally, moving to the night market, theintertemporal condition for the optimum can be written asu′ (q̃)1 + τ + α∗ i∗a − β(1 + i∗a ) (1 − α̃)=.
g ′ (q̃)β α̃(1 + i∗a )(24)

If the buyer chooses not to incur the cost, his participation is given by (9) and theintertemporal condition is given by (11), both evaluated at the optimal policy. Since
the optimal policy implements the first-best, the net benefit of not incurring the costis
1 + τ + α∗ i∗a − β(1 − α∗ ) ∗β [αu(q ∗ ) − k(α∗ )] −g(q ),(25)
1 + θi∗a21

while the net benefit if the buyer chooses to incur the cost is−γ + β [α̃u(q̃) − k(α̃)] −1 + τ + α∗ i∗a − β(1 − α̃)(1 + i∗a )g(q̃).
1 + i∗a(26)

Provided (25) is not smaller than (26), the buyer has no incentive to incur the cost.
The next Proposition, whose proof appears in the appendix, shows that there exists
an open region of parameters under which the buyer has no incentive to deviate andcreate a shadow account.
Proposition 8 Suppose the system can be manipulated at a cost, γ ≥ 0. For any
γ > 0, there exists an open interval of values of θ in which the optimal policy withdigital currency identified above is incentive compatible.
The optimal policy that reproduces the Friedman rule and achieves the best allocation attainable with physical currency is always available. Hence, for any positive costof manipulation, the optimal policy with digital currency can always at least replicate the outcome with physical currency and, in an open set of economies, strictly
improve upon it. Intuitively, this is achieved when the interest payments that induce
efficiency can be kept small. If the digital system can be manipulated at no cost,obviously, the digital technology cannot constitute an effective improvement over traditional currency. If it is even slightly costly to game the system, however, there arerobust circumstances in which the extra information provided by the digital technology can be exploited to achieve the fully efficient allocation, while discouraging themanipulation.
In principle, the agents could try to manipulate the system forming partnerships
with other agents in order to transfer balances between the accounts and get undue
interest payments. However, in our economy, by assumption, the agents meet randomly and bilaterally and cannot commit to future actions. These frictions, that are
key to generate an essential role for currency as a medium of exchange, prevent theformation of partnerships of this type as well as market trade and group interactions
in general. Hence, the only feasible manipulation schemes are individual.

22

6Conclusion

We have argued that the key difference between digital currency and cash consists inthe technological ability to trace monetary flows in and out of the accounts. Since
the extra information collected with this superior technology can be ignored, any
allocation achieved with cash can be attained also with digital currency. However,if the extra information is used to stimulate the velocity of circulation of money,
there are robust circumstances in which digital currency can help attain the efficient
allocation that would be unattainable with cash. The availability of such information
does indeed make the difference for the design of optimal policy.7
The payment of interest on active balances outperforms other reward schemes thathave been identified as welfare improving in physical currency economies, such as thepayment of interest on idle balances, obtained through monetary loans. This occurs
because the positive interest differential between active and idle balances stimulates
the velocity of circulation through its effect on the extensive margin, while rewarding
idle balances affects positively the intensive but not the extensive margin.
Interestingly, in some cases, notably when the trading protocol is Nash bargaining,
the interest payment on active balances in the digital economy can be used to correct
intensive margin distortions, that optimal policy would be unable to redress in thecorresponding physical currency economy. Hence, this type of intervention can play arole in addressing inefficiencies that arise from both extensive and intensive margins.
This is ultimately due to the availability of a further policy instrument that can bebrought into play thanks to the extra information obtained through the technologythat underlies the digital payment system relative to the information made available
when the payment system is based on cash.
The banking system plays no special role in our environment. Indeed, we showthat the efficient allocation can be achieved without banks. However, this does notmean that the adoption of CBDC would necessarily lead to disintermediation. On
the contrary, banks could play a key role in the actual implementation of the system,as the digital payments could continue to work exactly as they currently do with theinterest on active balances being paid through the banking system.

7

In systems theory and cybernetics à la Wiener and Shannon, information is defined as negentropy or ”the difference that makes a difference”, e.g. Bateson (1972), a definition we alluded to
playfully in the title.

23

7

Appendix

In Appendix A, we provide the proofs omitted in the main text. In Appendix B,
absent taxation of balances, we adapt the transfer scheme of Bajaj et al.(2017) to our
setting. In Appendix C, we show that our results are valid in a more general class of
trading mechanisms than the Kalai bargaining procedure adopted in the text.

7.1

Appendix A: Omitted Proofs

Proof of Proposition 1. Suppose α∗ > 0. Define Φ(q) ≡ u′ (q) − c′ (q), a continuous function with Φ(0) = ∞, Φ(∞) < 0 and Φ′ (q) < 0 by the properties of thefundamentals. Hence, a q ∗ ∈ (0, ∞) that satisfies Φ(q) = 0 exists and is unique. The
function k ′ (α) is monotonic in α, hence, invertible. By equation (1), if θ∗ ≤ 1, α∗ = 1,otherwise α∗ = k ′−1 (u(q ∗ ) − c(q ∗ )) ∈ (0, 1). In both cases, α∗ > 0. QEDProof of Proposition 2. Solve (7) for the quantity q = f −1 (α, τ ) and plug it
into (5). Define θ̂(τ ) as the unique value of θ that satisfies the first term at equality
in equation (5) with α = 1. Such a cutoff that exists and is unique by the propertiesof the fundamentals is at least as large as θ∗ . There are two possible cases. Suppose,first, θ ≥ θ̂(τ ), then, α = 1 and q = f −1 (1, τ ). The existence and uniqueness of thesolution of (7) for any τ ≥ β − 1 in this case follows immediately from the continuityof the function f (·), with f (∞) = 0, f (0) = ∞ and f ′ (q) < 0. Second, supposeθ < θ̂(τ ), then α < 1 solves ∆(α, τ ) ≡ θ[u(f −1 (α, τ )) − c(f −1 (α, τ ))] − k ′ (α) = 0 and
q = f −1 (α, τ ). The solution exists for any τ ≥ β − 1 since ∆(·, τ ) is continuous in α,with ∆(0, τ ) = 0, ∆(1, τ ) < 0 and ∆α (0, τ ) = ∞. Uniqueness can be guaranteed forτ not too large. Once α is determined, equation (7) gives uniquely q. Let α(τ ) andq(τ ) denote the equilibrium variables as function of policy, τ . Implicit differentiationshows that q ′ (τ ) < 0 and α′ (τ ) < 0 when α < 1. QEDProof of Proposition 4. Define ι ≡ (τ, ia , ip ). Solve (11) for the quantity
q = f −1 (α, ι). and plug it into (9). Define θ(ι) as the unique value of θ that satisfies
the first term at equality in equation (9) with α = 1. Such a cutoff that exists andis unique by the properties of the fundamentals is at least as large as θ∗ . There are
two possible cases. Suppose, first, θ ≥ θ(ι), then, q = f −1 (1, ι) and α = 1. The
existence and uniqueness of the solution of (11) for any τ that satisfies (12) in thiscase follows immediately from the continuity of the function f (·), with f (∞) = 0,
f (0) = ∞ and f ′ (q) < 0. Second, suppose θ < θ(i), then α < 1 solves Γ(α, ι) ≡24

θ[u(f −1 (α, ι)) − c(f −1 (α, ι))] − k ′ (α) = 0 and q = f −1 (α, ι). The solution exists forany τ that satisfies (12) since Γ(·, ι) is continuous in α, with Γ(0, ι) = 0, Γ(1, ι) < 0
and Γα (0, ι) = ∞. Uniqueness can be guaranteed for τ not too large. Once α isdetermined, equation (11) gives uniquely q. Let α(ι) and q(ι) denote the equilibrium
variables as function of policy. Implicit differentiation shows that qτ (ι) < 0 andατ (ι) < 0 when α < 1; qia (ι) > 0 and αia (ι) > 0; the signs of the derivatives of q(ι)
and α(ι) wrt ip are ambiguous. QEDProof of Proposition 7. Consider the equation (22) evaluated at q ∗ , τ = β − 1and ip = 0. Manipulating such a condition, we obtain a quadratic equation of theinterest on active balances, ia , Φ(ia ) = 0. Since Φ(0) < 0, Φ(∞) > 0, Φ′ (ia ) > 0 for
all ia ≥ 0, we conclude that there exists a unique ia > 0 that induces the efficient
allocation. QEDProof of Proposition 8. Consider the case θ∗ > 1. Let the authority adopt
the optimal policy that implements the first-best. Use (25) to define the function
Ψ(θ) and use (24) and (26) to define Ψ̃(θ). The function Ψ(θ) − Ψ̃(θ) is continuousin θ and Ψ(1) − Ψ̃(1) = γ. Hence, for γ > 0, by continuity there is an interval ofvalues of θ ≈ 1 so that Ψ(θ) − Ψ̃(θ) ≥ 0. Finally, to guarantee that the agents do
not create shadow accounts even if they do not want to participate in trade, it has≥ 1 + ia , that is satisfied for γ > 0 if θ ≈ 1. Consider the caseto be that (1+τ )(1+γ)β∗∗θ ≤ 1 with θ < θ . Let the authority adopt the optimal policy that implements thefirst-best with τ = β − 1, ia > 0 and ip < 0. Taking θ → θ∗ from below, an analogous
continuity argument applies for θ ≈ θ∗ . QED

7.2

Appendix B: No Taxation

Suppose taxation of balances is not feasible. For simplicity, we assume that θ < θ∗ ≤1, implying that the first-best involves full participation of buyers, and it cannot be
achieved in the physical currency economy. We begin with physical currency. The
transfer scheme we use is adapted from Bajaj et al. (2017). Assume that the monetary
authority transfers τc (ϕm) ≥ 0 real balances to each account with ϕm real balances atthe end of the day market, for all m ≥ 0.8 Thus, if the buyer gives ϕm real balances
8

This transfer scheme is feasible because balances can be observed in a physical currency economy.
In Bajaj et al. (2017) transfers are given to agents at the beginning of the day market, while we areconsidering a slightly modified version, where agents receive their transfers at the end of the day
market. As it will become clear, this change is immaterial in the physical currency economy but it
matters in the digital currency economy.

25

to a seller with zero balances during the day market, the seller anticipates that she
will receive a transfer of τc (ϕm) real balances at the end of the day market. Under
Kalai bargaining, the quantity produced in the meeting satisfies
ϕm + τc (ϕm) = (1 − θ)u (q) + θc(q) ≡ g(q).

(27)

Bajaj et al. (2017) propose a transfer scheme that compensates for the inflation tax
the buyers who bring at least the socially optimal amount of real balances, i.e., theamount of real balances that induces sellers to produce the efficient quantity. In our
setting, this translates into the following transfer scheme(τc (ϕm) =∗)0 if ϕm < g(q1+τ,∗)τ g(q ∗ ), if ϕm ≥ g(q
1+τ
1+τ

(28)

where M+1 = (1 + τ )M . Note that (27) implies that this transfer scheme gives the∗)seller the incentive to produce the efficient quantity if the buyer brings g(q
real
1+τ
 ∗ 
g(q )
g(q ∗ )
balances, since 1+τ + τc 1+τ = g(q ∗ ). Moreover, since τc (ϕm) > 0 is a step
∗

∗

)
)
function at g(q, carrying balances above g(qis costly and the seller does not receive
1+τ
1+τ
∗)
additional transfers if she receives more than g(qfrom the buyer. As a result, the1+τ
∗)real balances into the day market. In
buyer has no incentive to bring more than g(q1+τ
the day market, the buyer with ϕm real balances makes his participation decision,
solvingmax {αu(q) + (1 − α) [ϕm + τc (ϕm)] − k(α)} .
αUsing (27) and (28), the optimum satisfies
k ′ (α) = θ [u(q) − c(q)] ,(29)

so that the participation decision of the buyer only depends on the surplus in the
trade meeting. In the night market, the buyer chooses the money balances. There

26

∗

)
are two scenarios. If the buyer brings g(qreal balances, his payoff is9
1+τ

Uc+ = −g(q ∗ ) + β [αc u(q ∗ ) + (1 − αc )g(q ∗ ) − k(αc )] ,where αc solves (29) evaluated at q = q ∗ . Since the efficient quantity is produced in
trade meetings, and αc < 1 solves (5), we obtain the same allocation achieved under
the Friedman rule when the balances can be taxed. If, instead, the buyer brings∗), his payoff is
ϕm < g(q1+τU − = −ϕ−1 m + β [αu(q) + (1 − α)g(q) − k(α)] ,where α solves (29) evaluated at q. Note that, since ϕ → 0 when τ → ∞, (27) implies
that q → 0 when τ → ∞. Thus, there exists τ such that U − ≤ 0 for all τ ≥ τ . As∗)
a result, if τ ≥ τ , a sufficient condition for the buyer to bring g(qreal balances is
1+τ
∗)
g(q
that U + ≥ 0, which can be rewritten as β ≥ β (αc ) ≡ g(q∗ )+αc k′ (αc )−k(αc ) . Therefore,
if θ < θ∗ ≤ 1. If τ ≥ τ and β ≥ β (αc ), there exists a transfer scheme that replicatesthe allocation achieved under the optimal policy in the physical currency economy
with taxation of balances.
Consider now the digital currency economy. The analysis is similar to the one in
the physical currency economy, but with a modified transfer scheme,
∗





)
0 if ϕm < g(q1+τ∗).
τd (ϕm) =0, if ϕm ≥ g(qand the balance is passive
1+τ

∗)
 τ
g(q
g(q ∗ ), if ϕm ≥ 1+τ and the balance is active
1+τThis scheme only gives transfers to active balances that are at least equal to thesocially optimal amount. This is the case of the balances held by the seller that participated in a trade meeting where the buyer brought the socially optimal amount ofreal balances. As in the physical currency economy, the sellers reciprocate by producing the efficient quantity in exchange for these balances. Moreover, since τd (ϕm) > 0∗)g(q ∗ )
is a step function at g(q
,
the
buyer
has
no
incentive
to
bring
more
than
real
1+τ
1+τ
balances into the day market. Consider now the participation decision of the buyer. If
9

∗

∗

)
g(q )
In more detail, if ϕm = g(q1+τ , then m = ϕ(1+τ ) and the cost of acquiring m units of currency∗

ϕ−1M−1 g(q )∗in the previous night market is ϕ−1 m = ϕϕ(1+τ) = g(q ), since ϕ = M−1 = 1 + τ . In turn, since
∗

∗

g(q )
)
the buyer receives a transfer τ 1+τ
if he holds g(q1+τ real balances at the end of the day market, his
total balances if he does not participate in a trade meeting is given by g(q ∗ ).

27

∗

)
he brings less than g(qreal balances, his participation solves (29), as in the physical1+τ
∗). In this case, he solves
currency economy. Things are different if a buyer brings g(q1+τg(q ∗ )
max αu(q ) + (1 − α)− k(α) ,
α
1+τ


∗

and the optimum satisfiesg(q ∗ )∗′
(1 − α) u(q ) −− k (α) = 0,
1+τwhere both terms in parenthesis are non-negative. Full participation requiresk ′ (1) < u(q ∗ ) −g(q ∗ ).
1+τSince θ∗ ≤ 1 implies k ′ (1) ≤ u(q ∗ ) − c(q ∗ ), a sufficient condition for buyers to fully
participate is(1 − θ) [u (q ∗ ) − c(q ∗ )]
τ≥.
c(q ∗ )A higher rate of money growth gives the incentive for full participation by imposing
an inflation tax on idle balances, even if the balances are equal to the socially optimal
amount. In the night market, the buyer chooses his balances. There are two scenarios.
∗), his payoff is the same as in the physical currency
If the buyer brings ϕm < g(q1+τ
−economy, given by U . Thus, since ϕ → 0 when τ → ∞, (27) implies that q → 0
when τ → ∞ and there exists τ such that U − ≤ 0 for all τ ≥ τ . If, instead, the buyer∗)
real balances, his payoff isbrings g(q1+τUd+ = −g(q ∗ ) + β [u(q ∗ ) − k(1)] .
∗)real balances is that
Thus, if τ ≥ τ , a sufficient condition for the buyer to bring g(q1+τ+Ud ≥ 0, which can be rewritten asβ(1) ≥g(q ∗ ).
g(q ∗ ) + k ′ (1) − k(1)Therefore, in the case θ < θ∗ ≤ 1, if τ ≥ τ and β ≥ β (1), there exists a transfer
scheme that replicates the allocation achieved under the optimal policy in the digitalcurrency economy with the taxation of balances.
28

7.3

Appendix C: General Mechanism

Gu and Wright (2016) propose a trading mechanism that satisfies a few desirable axioms, namely feasibility, individual rationality, monotonicity and bilateral efficiency,
and subsumes, as particular cases, both Kalai and Nash bargaining, and competitive
price taking, among other solution concepts. In what follows, we first provide a general form of their mechanism that is consistent with the record keeping technology
under digital currency. We then show that our main result holds, i.e., the observation of balance flows allows to achieve the first best. Consider the following tradingmechanism Γ = (Γp (z), Γq (z)), where Γp (z) sets the real balances transferred fromthe buyer to the seller, and is given by(Γp (z) =z if z < p∗,p∗ if z ≥ p∗

where p∗ is defined as the minimum payment required for a buyer to receive q ∗ unitsof goods from the seller. In turn, Γq (z) sets the quantity produced by the seller andis given by(p∗v −1 ((1 + ia )z) if z < 1+iaΓq (z) =,p∗∗
q if z ≥ 1+ia
where v(·) is a strictly increasing, twice continuously differentiable function, withp∗
. This is the same trading mechanism as in Gu and Wrightv(0) = 0 and v(q ∗ ) = 1+ia(2016) if ia = 0. In the digital currency economy, we need to take into accountthat, if the buyer transfers z real balances to the seller in the day market, at the
beginning of the night market, the seller will have (1 + ia )z real balances. Note thatthe terms of trade do not depend on the real balances of the seller, and the efficientquantity is produced if (1 + ia )z ≥ p∗ . In turn, without loss in generality insofar as
optimal allocations are concerned, henceforth, we restrict attention to interventions
that pay zero interest on idle balances. This implies that, in equilibrium, the sellers
do not bring balances and the buyers do not bring balances they are not planning touse. We start with the participation decision of a buyer with real balances z at the
beginning of the day market. He chooses α to maximize αu(q) + (1 − α)z − k(α). The
complementary slackness condition for the optimum is



v(q)
′
(1 − α) u (q) −− k (α) = 0,
1 + ia29

where both terms in parenthesis are non-negative. Note that, given q, an increase in
ia increases the incentive of the buyer to participate in trade. Consider now the nightmarket. The buyer chooses z to maximize −ϕ−1 z/ϕ + β[αu(q) + (1 − α)z − k(α)].
The intertemporal condition for the optimum isu′ (q)
1 − β + αβ + τ + αia
=
.
′
v (q)
αβ(1 + ia )where we used M+1 = (1 + τ + αia )M and the stationarity of real balances. We now
determine the optimal intervention. First, let θ∗ > 1, so the first-best involves q = q ∗and α∗ < 1, where(30)u(q ∗ ) − c(q ∗ ) = k ′ (α∗ ).
If a policy implements the first-best, we must haveu′ (q ∗ )1 − β + α∗ β + τ + α∗ ia=,v ′ (q ∗ )α∗ β(1 + ia )andu (q ∗ ) −v(q ∗ )
= k ′ (α∗ ).
1 + ia(31)(32)

If ia = 0, efficient participation requires v(q ∗ ) = c(q ∗ ), which is not generically satisfied. For example, if v(q) is determined by Kalai bargaining or Nash bargaining, thisis only possible if θ = 1, and the buyer has full bargaining power. Thus, the first-best
cannot be implemented if the authority cannot distinguish between active and idlebalances. If, instead, ia > 0, we can equate the left hand side of (30) and (32) andachieve efficient participation by settingi∗a =v(q ∗ ) − c(q ∗ ).
c(q ∗ )

Using (31), we determine the optimal τ , which is given byτ ∗ = β − 1 + α∗ β′ ∗′ ∗u′ (q ∗ ) − v ′ (q ∗ )∗ βu (q ) − v (q ) ∗+αia .
v ′ (q ∗ )v ′ (q ∗ )30

The existence of the monetary equilibrium requires τ ∗ ≥ β − 1 − α∗ i∗a . Thus, thefirst-best can be implemented by the policy (τ ∗ , i∗a ) if and only if(1 + i∗a )u′ (q ∗ )≥ 1.
v ′ (q ∗ )(33)Consider Kalai bargaining, as in the text. In this case, we havev(q) =∗(1 + i∗a ) [(1 − θ)u(q) + θc(q)],1 + θi∗a∗u(q )−c(q )which implies i∗a = 1−θ, i.e. the optimal policy considered in the text. In thisθc(q ∗ )
case, we have proved that the first-best can be implemented. Given the regularity ofthe solution, a transversality argument shows that the same occurs for an open anddense set of mechanisms that are close to the Kalai protocol. Consider now the Nashbargaining protocol. We havev(q) =(1 + i∗a ) [(1 − θ)u(q)c′ (q) + θc(q)u′ (q)],
θ(1 + i∗a )u′ (q) + (1 − θ)c′ (q)∗∗u(q )−c(q ), as in the case of Kalai. Computing v ′ (q ∗ ) andwhich implies i∗a = 1−θθc(q ∗ )u(q ∗ )−c(q ∗ )inserting it into (33) together with i∗a = 1−θ, we obtain the condition Γ (θ) ≥θc(q ∗ )
0. Since Γ (1) > 0, by continuity there exists a non-empty interval of values of θ suchthat (33) is satisfied. Finally, under competitive pricing, we have v(q) = pq, where p
is taken as given but is equal to c′ (q) in equilibrium. In this case, (33) becomesc′ (q ∗ )q ∗≥ 1,c(q ∗ )
which is always satisfied under our assumptions on the fundamentals.

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