IMES DISCUSSION PAPER SERIES

Digital Money as a Medium of Exchange andMonetary Policy in Open Economies

Daisuke Ikeda
Discussion Paper No. 2022-E-10

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IMES Discussion Paper Series 2022-E-10
July 2022

Digital Money as a Medium of Exchange andMonetary Policy in Open Economies

Daisuke Ikeda*
Abstract
The rise of digital money may bring about privately issued money that circulates
across borders and coexists with public money. This paper uses an open-economy
search model with multiple currencies to study the impact of such global money onmonetary policy autonomy -- the capacity of central banks to set a policy instrument.
I show that the circulation of global money can entail a loss of monetary policy
autonomy, but it can be preserved if government policy that limits the amount or useof global money for transactions is introduced or if the global currency is subject tocounterfeiting. The result suggests that global digital money and monetary policy
autonomy can be compatible.
Keywords: Cryptocurrency; Monetary policy autonomy; Currency counterfeiting;Government transaction policy
JEL classification: D82, E4, E5, F31*Director and Senior Economist, Institute for Monetary and Economic Studies (currently,
Financial System and Bank Examination Department), Bank of Japan (E-mail:daisuke.ikeda@boj.or.jp)
Shinnosuke Katsuki provided excellent research assistance. The author is grateful for comments
and discussions from Massimo Ferrari, Linda Goldberg, Masashige Hamano, Dinah Heybourn,
Kohei Iwasaki, Tomoo Kikuchi, Arnaud Mehl, Harald Uhlig, and seminar and conference
participants at the Bank of Japan, the European Central Bank, Waseda University, and the 2022
BOJ-IMES Conference. The views expressed in this paper are those of the author and do notnecessarily reflect the official views of the Bank of Japan.

1Introduction

Money and payment systems have evolved over time. In the ancient world, commodities
such as grains were the primary form of money and payment. These were superceded by
the use of precious metals such as gold, which in turn gave way to fiat money – currency
and coin issued by the government but with no intrinsic value – along with deposits issued
by private banks. Now a substantial change is underway. Although fiat money and bankdeposits remain dominant in many countries, technological innovations have led to a rise indigital money all over the world. Global digital money, which can be used for transactions
across borders, has yet to emerge, but it is technologically feasible and within the reach of
the future economy.
The potential adoption of global digital money would bring substantial benefits, notleast, inclusion and efficiency, but at the same time, it could pose challenges to the conductof monetary policy. Specifically, should a global digital currency substantially advance andreplace a national currency, the effectiveness of monetary policy in that economy would
be weakened (G7 Working Group on Stablecoins, 2019; IMF Staff Report, 2020). Ikeda(2020) formalizes this concern by focusing on a unit of account as a role of money in aframework of new open economy macroeconomics (NOEM). Benigno et al. (2022) reinforce
the concern by focusing on a medium of exchange as a role of money in a general setup thatcan be applied to many monetary models. They show that even in a benign scenario where
the national currency and the global currency co-exist but do not necessarily dominate
one another, monetary policy autonomy – the capacity of central banks to set a monetary
policy instrument as they see fit – could be lost. This implication could pose a serious
challenge for both policymakers and private digital money issuers.
In this paper, I revisit the issue of monetary policy autonomy in a possible future
economy in which both the national currency and the global digital currency are valued andused for transaction. Specifically, I examine the following questions. Is a loss of monetary
policy autonomy an inevitable outcome in such a future economy? If not, what mechanisms
could overturn the result of Benigno et al. (2022) and restore monetary policy autonomy?
What government policy, if any, could prevent a loss of monetary policy autonomy?
I tackle these questions by building a two-country open economy new monetarist model,
modified to incorporate a global currency. The model has search and information frictions
that make money essential for transactions. In the model, a global currency can be usedin both home and foreign countries, but each national currency can be used only in the2country of its origin. The interest of this paper is a global-money economy where both thenational money and the global money are used for transaction in each country.
I use the model and study monetary policy autonomy in such a global-money economy.
I show that if there is no additional friction, the economy features a loss of monetary policy
autonomy, which confirms the result of Benigno et al. (2022). But, if additional frictions are
introduced, it is not the only outcome. Such frictions include the combination of currency
counterfeiting and the imperfect recognizability of currencies, and government policy that
limits the type of currencies used for some transactions. Indeed, it only needs one of thesefrictions to exist for monetary policy autonomy to be restored: the central bank can set
its monetary policy instrument within a vast area where both the national money and theglobal money are valued and used for transactions. Finally, in addition to government
policy on transaction currencies, government policy that sets an upper limit on the amountof the global money used for a transaction can prevent a loss of monetary policy autonomy.
The result that monetary policy autonomy is lost in the global-money economy is anapplication of the nominal-exchange-rate indeterminacy result of Kareken and Wallace(1981). In the model with no additional friction (referred as the baseline model hereafter),an arbitrage-free condition between holding the national money and the global moneyis established in both the home and foreign countries. Thus, through holding the global
money, the opportunity cost of holding the home money is tied to that of holding the foreign
money. Hence, be it the risk-free interest rate or the money growth rate, the monetary
policy instrument, which determines the opportunity cost of holding the national money,
is tied between the two countries, resulting in monetary policy synchronization where amonetary instrument in one country is tied with that in another country. This situation isa loss of monetary policy autonomy. At the same time, the arbitrage-free condition implies
the indeterminacy of the portfolio of real money balances and thus the indeterminacy ofthe nominal exchange rate, leading to the result of Kareken and Wallace (1981).
This observation suggests that restoring monetary policy autonomy comes down toresolving the indeterminacy result of Kareken and Wallace (1981). Following the literature,
I consider two candidates:• The combination of currency counterfeiting and imperfect recognizability of currencies(Rocheteau, 2011; Li and Rocheteau, 2011; Nosal and Rocheteau, 2017, Chapter 12).
• Government policy that limits the types of money used for transaction (Li and Wright,
1998; Zhang, 2014).
3

Currency and its counterfeiting are two sides of the same coin. Significant and continuous effort over the ages has been made to make fiat money and bank deposits less prone
to counterfeiting and fraud. Although a global currency and its vulnerabilities have yet to
be seen, its digital feature and wide-spread use in the world would almost certainly invite
countless attacks of counterfeiting and fraud. For example, cryptocurrencies can be copied
easily (i.e., it is subject to a double-spending problem), and it would be difficult for anindividual to recognize ‘true’ cryptocurrencies.
Therefore, I introduce the counterfeiting of global digital currencies to the baselinemodel. Counterfeiting is a threat for sellers because of imperfect recognizability: theycannot distinguish between true digital currencies and counterfeits. Receiving counterfeits
in exchange for goods is equivalent to giving goods for free. To prevent such an outcome,sellers restrict the amount of digital currencies for transaction, giving rise to an endogenous
liquidity constraint. This is because while buyers can counterfeit digital currencies, doing
so is costly. And when the amount of digital currencies that can be accepted for transaction
is limited by the liquidity constraint, the cost of counterfeiting outweighs the benefits ofproducing and using counterfeits for buyers. Hence, in such a situation, buyers would not
engage in counterfeiting. When the limit of the use of digital currencies is not so high, theliquidity constraint becomes binding and the portfolio of national money and digital money
is pinned down uniquely, giving rise to determinate demand for the two kinds of money,
and thus restoring monetary policy autonomy in the global-money economy. While theseresults are obtained by adding a simple model of counterfeiting (Nosal and Rocheteau, 2017,
Chapter 12) to the baseline model, I show that the same results also hold by introducing
a model of cryptocurrency with a proof-of-work protocol, such as Bitcoin, studied by Chiu
and Koeppl (2019). The corollary of the results of counterfeiting is that government policythat restricts the amount of digital money used for a transaction can also preserve monetary
policy autonomy.
Government engages in spending and collecting taxes. The currency with which it canmake and receive payments is a choice of the government. In practice, in transactions
with the government, the permitted currency is restricted to the domestic currency in mostcountries. I formalize this idea by introducing government agents who can accept only
the domestic currency for transactions in the baseline model. These sellers can also beinterpreted as behavioral sellers who do not want to use the global money for behavioral
reasons. The presence of these sellers makes the national money and the global digitalmoney imperfect substitutes and increases the liquidity services obtained by holding the4national money. The amount of the national money demanded by private agents is adjusted
to equate the liquidity service to the cost of holding that national money, which in turn isdetermined by monetary policy. In this way, the portfolio of real balances and money demand are uniquely determined, resolving the indeterminacy result of Kareken and Wallace(1981) and restoring monetary policy autonomy.
Adding frictions, including government policy, to break an indeterminacy or irrelevanceresult has tradition in macroeconomics and finance. Modigliani and Miller (1958) clarify
conditions under which a firm’s capital structure becomes irrelevant for its value, and thevast literature that followed has explored what types of frictions would overturn the irrelevance result. In the context of monetary policy, Wallace (1981) builds on the Ricardianequivalence result emphasized by Barro (1974) and shows the irrelevance of open market
operations, and Eggertsson and Woodford (2003) the irrelevance of quantitative easing. To
resolve irrelevance results, the literature has added frictions such as imperfect asset substitution (Andrés et al., 2004) and the preferred-habitat theory of term structure (Vayanos andVila, 2021).1 As discussed above, the lineage of this paper starts from the irrelevance resultof Kareken and Wallace (1981) and aims at resolving the loss of monetary policy autonomy
shown by Benigno et al. (2022). By adding frictions to money holdings and restoring monetary policy autonomy, this paper re-affirms the trilemma of the global-money economy –the impossibility of achieving monetary policy autonomy (independence), free capital flow,
and free exchange rates at the same time, argued by Benigno et al. (2022).
Related literature This paper contributes to the emerging literature on cryptocurrency
and its implications for monetary policy. From the perspective of open economies, inaddition to Benigno et al. (2022) and Ikeda (2020), this paper is also related to Baughman
and Flemming (2020) who study global demand for a basket-based stablecoin in a twocountry open economy search model. Uhlig and Xie (2020) build a small-open economy
model with global digital currencies in the NOEM framework and study the role of shocks
that arise from the indeterminacy of exchange rates between any pair of global currencies.
Ferrari et al. (2022) examine the issuance of a central bank digital currency (CBDC) in
the NOEM framework and argue that it amplifies the international spillovers of shocks. In
the context of a closed economy, papers that study currency competition with or among
cryptocurrencies and its implications for monetary policy include Garratt and Wallace1

For the discussion of the (ir)relevance of unconventional monetary policy, see e.g. Gertler and Karadi(2013) and Christiano and Ikeda (2013).

5

(2018), Fernández-Villaverde and Sanches (2019), Schilling and Uhlig (2019), Zhu andHendry (2019), and Saito (2020). For the microeconomics of cryptocurrencies, see therecent survey by Halaburda et al. (2021).
Broadly, this paper builds upon the search-theoretic literature on dual currencies. The
earliest works on dual currencies include Kiyotaki and Wright (1989) for a closed economy
and Matsuyama et al. (1993) for an open economy. Craig and Waller (2000) surveys
the literature in the 1990s. Lagos et al. (2017) summarizes the development of a searchtheoretic model. The basic framework of the models used in this paper is the two-country
open economy model of Zhang (2014), which itself extends the closed-economy model ofLagos and Wright (2005). This paper introduces a global currency, which can be used inboth countries, and derives implications for monetary policy autonomy.
Resting on the literature on nominal exchange rate indeterminacy that starts from
Kareken and Wallace (1981), to overturn the result of a loss of monetary policy autonomy,in this paper I add currency counterfeiting and the imperfect recognizability of currencies
in a similar spirit to Rocheteau (2011), Li and Rocheteau (2011), Nosal and Rocheteau(2017, Chapter 12), and Gomis-Porqueras et al. (2017). This leads to an endogenous
liquidity constraint, which has the same form as a no-double-spending constraint, derived
by Chiu and Koeppl (2019), for a cryptocurrency with a proof-of-work protocol. As anotherversion of the model, I introduce government policy that limits the type of money used fortransaction as studied by Li and Wright (1998) and Zhang (2014).
The rest of the paper is organized as follows. Section 2 presents the main idea about aglobal currency and monetary policy autonomy in a simple framework. Section 3 articulates
the ideas by building two-country open economy search models with a global currency.
Section 4 concludes with some policy implications for monetary policy autonomy.

2

Main Ideas in a Simple Framework

This section uses a simple framework to lay out the main idea about the global-money
economy and the consequences for monetary policy. Section 2.1 describes the common
environment used in the paper. Section 2.2 shows a loss of monetary policy autonomy in
the global-money economy and clarifies the key assumptions that lead to such a result.
Section 2.3 introduces different assumptions and shows that monetary policy autonomy ispreserved in the global-money economy.
62.1

Environment

Time is discrete and continues forever. The economy consists of two countries, home andforeign, denoted as h and f , respectively. There are three currencies: a home currency,
a foreign currency, and a global digital currency, which are denoted as currency h, f, g,
respectively. In each country i ∈ {h, f }, the national money Mi denominated in currency
i is supplied by the central bank of country i. The supply of the global money Mg isexogenously given. For simplicity, I assume that there is no interest payment on any
currency. There is no aggregate uncertainty and thus the economy is deterministic.
There is a numeraire good and the price of currency l ∈ {h, f, g} in units of the numeraire good is denoted as φl ≡ 1/Pl , where Pl is the price of the numeraire good in unitsof currency l. Both countries face the same prices, so that the law of one price holds for thenumeraire good. The nominal exchange rate of currency f in units of currency h is given
by
E=

φf
Ph
=
φh
Pf

(1)

For example, an increase in E means depreciation of currency h.
Since there is no aggregate uncertainty, in this paper I focus on a stationary equilibrium
where aggregate real balances are constant in each country. In such an economy, the money
growth rate and the nominal interest rate on risk-free bonds are equivalent as a monetary
policy instrument. Without loss of generality, the central bank sets the money growth rate,
γi = Mi0 /Mi , in each country, where a variable with superscript 0 indicates the variable in
the next period. The growth rate of the global money is denoted as γg = Mg0 /Mg .
The situation of interest is such that both domestic and global currencies are valued
and held in each country: φl > 0 for all l ∈ {h, f, g}. In this paper I exclusively focus
on such a global-money economy. In addition, to shed light on the role of global money, I
follow Benigno et al. (2022) and consider a situation in which the national money can beused for transaction only in the country of its origin, while the global money can be usedin both countries.2
In this environment, the question is whether the global-money economy restricts the
conduct of monetary policy. Specifically, I am interested in whether a central bank continues
to have autonomy in setting a monetary policy instrument in the global-money economy.
2

If a national currency can be used in both countries, such a currency is regarded as an international
currency. See Zhang (2014) and Gomis-Porqueras et al. (2017) for the analysis of an international currency
in open economy new monetarist models.

7

To clarify the status of monetary policy, let me introduce the following two definitions.
Definition 1 : Monetary policy synchronization. Monetary policy synchronization isa situation in which the setting of a monetary policy instrument in each country is tied tothe setting of a monetary policy instrument in the other country.
Definition 2 : Monetary policy autonomy. Monetary policy autonomy is a situationin which the setting of a monetary policy instrument in each country is independent of thesetting of a monetary policy instrument in the other country.
Under monetary policy synchronization, the setting of a monetary policy instrument
in one country is restricted to a certain value that depends on the setting of a monetary
policy instrument in the other country. A special case is the equalization of the setting of amonetary policy instrument between the two countries, as studied by Benigno et al. (2022).
Under monetary policy autonomy, the central bank can set a monetary policy instrument
as it sees fit in each country. In the economy without the global currency, monetary policy
autonomy holds as in the standard two-country monetary model. The primary focus of
this paper is whether monetary policy synchronization is an inevitable outcome in the
global-money economy, or if monetary policy autonomy can be preserved.

2.2

Monetary Policy Synchronization

In this subsection, I argue that monetary policy synchronization arises in the global-money
economy with no restriction on the portfolio choice of money. In country h, a representative
individual holds both currencies h and g and has real balances of φh mhh + φg mgh , where
mjh denotes money denominated in currency j (money j for short hereafter) held by theindividual in country h. Since both currencies are valued and held, and since there is
no restriction on holding both kinds of money, the individual is indifferent about holding
money h or g. Although the liquidity service provided by money is an endogenous object,
for the sake of exposition in this subsection only, let me assume that the liquidity service
is identical between money h and g.3 Then, the return of holding money h, φ0h /φh , has tobe equated to that of holding money g, φ0g /φg . Since this arbitrage prevails in country f aswell, the returns of holding money h and f are equalized through the holding of the global
3

In Section 3.2, I will show that an identical liquidity service can be obtained as an equilibrium outcome.

8money in both countries:

φ0f
φ0g
φ0h
=
=
φh
φg
φf

(2)

In a stationary equilibrium, aggregate real balances are constant so that the the returnof holding money l ∈ {h, f, g} is equal to the inverse of the money growth rate γl . Hence,equation (2) implies that the money growth rates are equalized between the two countries:
γh = γg = γf(3)

Since equation (2) constitutes an equilibrium condition, in any stationary equilibrium inwhich the real balances of each currency is constant and positive over time, equation (2)has to hold and so does equation (3). Equation (3) shows a specific form of monetary policy
synchronization, where the setting of the monetary policy instrument is equalized between
the two countries.
Equation (2) is an application of Kareken and Wallace (1981) to the open economy withthe global currency. Actually, equations (1) and (2) imply a constant nominal exchange
rate:
E=

φ0f
φf
= 0 = E0
φh
φh

(4)

Hence, in this economy, monetary policy synchronization arises in the form of equation (3)and at the same time the level of the nominal exchange rate is indeterminate as in Kareken
and Wallace (1981).4

2.3

Monetary Policy Autonomy

Monetary policy synchronization holds in the economy considered in Section 2.2, but theeconomy imposes specific assumptions on the environment as described in Section 2.1. Suchassumptions comprise no restriction on holding money and an identical liquidity service
between the national money and the global money.
In this subsection, I suggest that monetary policy autonomy can be restored in the sameenvironment with different assumptions. The core equation that gives rise to monetary
4

Monetary policy synchronization in terms of nominal interest rates can be shown as in Benigno et al.
(2022) as follows. In the global capital market where risk-free nominal bonds in units of the national
currency i ∈ {h, f } are traded, the uncovered interest rate parity (UIP) holds: Rh = Rf E 0 /E, where Ri is
the gross nominal interest rate on the nominal bonds in units of the national currency i. Since equation(4) shows the constant nominal exchange rate, the UIP condition implies the equalization of the nominalinterest rates: Rh = Rf .

9policy synchronization is equation (2), which implies that the returns of holding different
kinds of money are identical and thereby that the portfolio of real money balances isindeterminate. If equation (2) fails, monetary policy synchronization will break down andmonetary policy autonomy will prevail.
Let zil ≡ φl mli denote the real balances of money l ∈ {h, f, g} held by a representative
individual in country i ∈ {h, f }. Then, under the presumption that the individual incountry i holds money i and g only, the total real balances held by that individual is given
by zi = zii + zig . Let me now drop the assumption of no restriction on holding money,
and assume a specific form of friction that gives rise to the following constraint on the realbalances of the global money: for i ∈ {h, f }zig ≤ z̄ig ,(5)where z̄ig is an upper bound of the amount of the global money that can be used fortransaction by an individual in country i. Here, the upper bound is exogenously given. It
can be regarded as exogenous government policy that puts the upper limit on the use of the
global money for a transaction. In Section 3.3, I will show how it can also be interpreted as
an incentive constraint, a constraint that ensures no hidden action by buyers who pay using
the global money. The intuition is that if a large volume of the global money is acceptedby a seller, the buyer has an incentive to take a hidden action, such as counterfeiting, thatwill damage the seller. Constraint (5) prevents buyers from engaging in such an action.
Suppose that constraint (5) is binding. Then, the real balances of money g are given by
zig = z̄ig for i ∈ {h, f }. Then, the global money market clearing condition, φg Mg = zhg + zfg ,determines the price of currency g as
φg =z̄hg + z̄fg
.
Mg

(6)

In addition, since the total real balances zi are determined in each country in a typicalmonetary model as will be shown in Section 3, the real balances of the national moneyi ∈ {h, f } are given by zii = zi − z̄ig . Then, the price of the national currency is determinedby the national money market clearing condition as: for i ∈ {h, f }φi =zi − z̄ig
Mi

10

(7)

Combined with equation (1), equation (7) leads to the nominal exchange rate, given byg

Mh zf − z̄fE=
Mf zh − z̄hg

(8)When the two countries are symmetric, equation (8) is reduced to E = Mh /Mf , so thatthe nominal exchange rate is simply given by the ratio of monetary aggregates between thetwo countries.
In a stationary equilibrium, the portfolios of real balances, the nominal exchange rate,
and the prices of currencies are all determinate. And nothing forces the money growth
rates to be equated between the two countries. From equations (6) and (7), the return ofholding money l ∈ {h, f, g} is given by φ0l /φl = Ml /Ml0 = γl−1 . Since the binding constraint(5) prevents these returns from being equated, the central bank in each country can set themoney growth rate without restrictions.

3

The Model

In this section, I formalize the main idea put forward in Section 2 using an open-economy
search model with multiple currencies. Section 3.1 describes the common setup of themodel and extend the two-country search model studied by Zhang (2014) to incorporate
a global currency. Section 3.2 shows that monetary policy synchronization holds in abaseline model where there is no restriction on holding money. Each of the remaining
two subsections shows that monetary policy synchronization is overturned and monetary
policy autonomy restored if a specific restriction in holding or accepting money is added.
Section 3.3 introduces the action of counterfeiting or double spending a specific currency
as in Nosal and Rocheteau (2017, Chapter 12) and Chiu and Koeppl (2019). Section 3.4
considers government transaction policy as in Li and Wright (1998) and Zhang (2014).

3.1

Common Setup

Environment In addition to the model environment of Section 2.1, two countries, hand f , are populated with a continuum of 2n and 2 agents, respectively, where n ∈ (0, 1]
denotes a relative population size of country h. Agents in each country are evenly divided
between buyers and sellers, so the size of sellers or buyers is n and 1 in country h andf , respectively. Each period consists of two sub-periods as in Lagos and Wright (2005).

11

In the first sub-period, agents meet bilaterally and at random in decentralized markets(DM) and trade the DM good. In the second sub-period, all agents meet in a friction-lesscentralized market (CM), trade the CM good, and exchange money. The CM good is taken
as a numeraire. Both the DM and CM goods are perishable. In the DM, agents lack
commitment and individual histories are private information, rendering unsecured credit
unusable and making money essential as a medium of exchange.
Preferences and technologies In the DM, a seller in each country can produce theDM good, q, with the cost c(q), but does not have utility over consumption. A buyer, on
the other hand, has the utility, u(q), by consuming q but cannot produce. In the CM, allagents get utility U (y) by consuming the CM good, y, which is produced according to alinear one-to-one technology in hours worked x. Periodic utilities for a buyer and a seller,
U B and U S , are assumed to be additively separable between the DM and CM goods andquasi-linear in hours worked:
U B = u(q) + U (y) − x(9)U S = −c(q) + U (y) − x(10)

I assume that u(·) and U (·) are strictly increasing and strictly concave, and satisfy the
Inada conditions, c(·) is weakly convex, u(0) = c(0) = c0 (0) = U (0) = 0, and c(q̄) = u(q̄)
for some q̄ > 0. Agents discount the future between adjacent periods with a discount factorβ ∈ (0, 1).
Currencies As described in Section 2.1, three currencies are available: a home currency,
a foreign currency, and a global currency. The corresponding monetary aggregate is denotedas Ml and the associated growth rate is denoted as γl = Ml0 /Ml for l ∈ {h, f, g}. I assume
that holding money is costly: γl > β for all l. Changes in the money supply are implemented
through lump-sum monetary transfers in the CM to buyers. The way how lump-sum
monetary transfers are distributed to buyers does not affect the results of this paper, butPfor simplicity, it is assumed that each buyer receives τ ≡ l∈{h,f,g} (γl − 1)φl Ml /(1 + n) in
both countries.
Matching In the DM, agents in both countries meet bilaterally and at random. Buyers
are mobile while sellers are immobile. With probability α ∈ [0.5, 1], a buyer from one

12country stays in the country, and with probability 1 − α, the buyer visits the other country.
As in Zhang (2014), the number of matches in the DM of country i ∈ {h, f } is given by
the matching function
Mi ≡ M (Bi , Si ) =B i SiB i + Si

where Bi and Si are the measures of buyers and sellers, respectively, in the DM of country
i. Then, the probability that a buyer from country i meets a seller in country j, denotedby µij , is derived as5
1−α
1 + α + (1 − α)nαµf f =
1 + α + (1 − α)nα,
1 + α + (1 − α)/n1−αµf h =,
1 + α + (1 − α)/n
µhh =

µhf =

If country h is smaller than country f and some buyers move to the different country, i.e.,
if n < 1 and α < 1, the probability of matching becomes asymmetric in the two countries.
In the special case of α = 1 in which all buyers stay in their country, the probabilityof matching with a seller in the same country is reduced to µhh = µf f = 0.5, and theprobability of matching with a seller in the different country is zero: µhf = µf h = 0.
Consumption and labor decisions in the CM Let WiB (zi ) and ViB (zi ) denote a valuefunction of a buyer from country i ∈ {h, f } with a portfolio of real balances zi ≡ (zih , zif , zig )in the CM and the DM, respectively. In the CM, the problem of a typical buyer from country
i can be written as
WiB (zi ) =
B 0
max0 U (y) − x + βVi (zi ){y,x,zi }(11)subject to the flow budget constraint
f0
g0
f
g
h
y + φh mh0
i + φf mi + φg mi = x + z i + z i + z i + τ(12)The first-order condition with respect to y yields
y = y ∗ ≡ U 0−1 (1)(13)

Since the real balances of money l purchased today, φl ml0i , in the budget constraint (12)will be valued as zil0 = φ0l mil0 tomorrow, today’s real balances can be written as φl ml0i =5

See Appendix A.1 for the derivation of these probabilities.

13

(φl /φ0l )zil0 = γl zil0 . By using this relationship and equation (13), substituting the budget
constraint (12) into the problem (11) yields
WiB (zi ) = U (y ∗ ) − y ∗ + zih + zif + zig + τ + max
z0in−γh zih0 − γf zif 0 − γg zig0 + βViB (z0i )

o

(14)

Equation (14) implies that the CM value function is linear in real balances, i.e., WiB (zi ) =WiB (0) + zih + zif + zig , and that the portfolio choice in the CM is independent of the current
holdings of real balances.
Since holding money is costly, γl > β for all l ∈ {h, f, g}, sellers have no incentive tocarry real balances either in the DM or the CM. Hence, the CM value function of a typicalseller in country j is given by
WjS (zj ) = U (y ∗ ) − y ∗ + zjh + zjf + zjg + βVjS (0)(15)

Terms of trade in the DM Upon matching with a seller in country j, a buyer fromcountry i makes a take-it-or-leave-it offer to the seller.6 The offer consists of a pair, (dij , qij ),where dij ≡ (djij , dgij ) is a portfolio of real balances of monies j and g that the buyer pays tothe seller, and qij is the quantity of the DM good that the seller produces in exchange forreceiving the payment. The total payment is dij ≡ djij + dgij . This offer takes into accountthe assumption that a seller in country j does not receive the currency of the other country.
Consider a buyer from country i who enters the DM with a portfolio of real balances,zi ≡ (zih , zif , zig ). Since money is the only means of payment, the offer has to satisfy thepayment constraint
dlij ≤ zil

for l ∈ {j, g}(16)

Since holding money is costly, the constraint (16) holds with equality in equilibrium. Also,
the seller’s value function (15) in the CM implies that the value function in the DM will
also become linear in real balances, so that the seller accepts the offer if and only if thepayment is no less than the cost of producing the DM good:djij + dgij − c(qij ) ≥ 0
6(17)Here, a take-it-or-leave-it offer is assumed for simple exposition. The main results of this paper will holdfor a general terms-of-trade mechanism, studied by Gu and Wright (2016). The general mechanism includes
e.g. Kalai’s proportional bargaining and generalized Nash bargaining in addition to a take-it-or-leave-it
offer.

14

Constraint (17) holds with equality in equilibrium, otherwise, the buyer would be strictly
better off by decreasing djij or dgij , or increasing qij .

3.2

The Baseline Model

Now I consider a case with no additional frictions, which is referred as the baseline model.
In this model, only requirements for trade in the DM are constraints (16) and (17) that holdwith equality. I show that monetary policy synchronization ensues in this global-money
economy, articulating the idea presented in Section 2.2.
Value function in the DM Consider a buyer from country i who enters the DM witha portfolio of real balances zi ≡ (zih , zif , zig ). The buyer is matched with a seller in country
j with probability µij . Upon the match, the buyer makes the take-it-or-leave-it offer of(dij , qij ) that satisfies constraints (16) and (17) with equality, and receives qij in exchange
for payment dij = djij + dgij = zij + zig . The buyer leaves the DM with utility u(qij ) and thereal balances of ziĵ unused for ĵ 6= j. Since the value function of a buyer in the CM is linear
in real balances as implied by equation (14), given the match, the buyer’s life-time utility
in the DM can be written as u(qij ) + WiB (0) + ziĵ . With probability 1 − µih − µif , the buyer
is not matched with a seller. In this case, the buyer leaves the DM with periodic utilityu(0) = 0 and all real balances zi unused. Then, the buyer’s life-time utility in the DM canbe written as WiB (0) + zih + zif + zig . To summarize, the value function of the buyer in the
DM is given by
ViB (zi ) = µih [u(qih ) + zif ] + µif [u(qif ) + zih ] + (1 − µih − µif )(zih + zif + zig ) + WiB (0) (18)Since constraints (16) and (17) hold with equality, the quantity of the DM good in equation(18) is given by
qij = c−1 (zij + zig )(19)

Equations (18) and (19) imply that the value function of a buyer in the DM depends only
on real balances brought from the CM in the previous period.
Portfolio choice in the CM With ViB (zi ) on hand, consider the buyer’s portfolio choiceproblem in the CM in equation (14). By pushing one period back and using equations (18)15and (19), the problem can be written as
max{zih ,zif ,zig }− γh zih − γf zif − γg zig + β hiµih u c−1 (zih + zig ) + zif


h 

i
g
f
g
−1 f
h
h
+ µif u c (zi + zi ) + zi + (1 − µih − µif )(zi + zi + zi )(20)

Given that the matching probabilities, µih and µif , are strictly positive, an interior solution
has to satisfy the first-order conditions with respect to zih , zif , and zig , which are given,
respectively, as"#
u0 c−1 (zih + zig )γh −1− 1 = µih 0 −1 hβc c (zi + zig )

 

g
−1 f
0
c
(z
+
z
)
u
i
i
γf
 − 1
− 1 = µif  βc0 c−1 (z f + z g )iγg− 1 = µih
β

"

(21)

(22)

i

u0 c−1 (zih + zig ) −1
c0 c−1 (zih + zig )


#








u c
 − 1
+ µif  fg
0−1c c (zi + zi )0−1

(zif + zig )(23)

where u0 (·) and c0 (·) denote the derivatives of functions u(·) and c(·), respectively. In eachequation, the left-hand-side is the opportunity cost of holding money and the right-handside captures the liquidity service of money – the expected marginal benefit of holding realbalances, u0 (·)/c0 (·), in excess of the benchmark of 1 = u0 (q ∗ )/c0 (q ∗ ).
Monetary policy synchronization Combining equations (21)-(23) yields
γh + γf = γg + β(24)This condition requires that the sum of the growth rates of money in the two countries
is constant for given γg . The setting of the monetary policy instrument in one country
is dependent on the setting of the monetary policy instrument in the other country, andhence this global-money economy features monetary policy synchronization.
To obtain the intuition of condition (24), let il ≡ γl /β − 1 denote the opportunity costof holding money l. Then, condition (24) can be written as
ig = ih + i f

16

(25)

This equation implies that the cost of holding the global money is equated to the sum of thecosts of holding money h and money f . Since the global money can be used in both countries
while the national money j ∈ {h, f } can be used only in country j, holding the global money
brings the sum of the benefits of holding the two kinds of money, which corresponds to theright-hand-side of equation (23). The benefit of holding money l ∈ {h, f, g} is equated toits opportunity cost in equilibrium, leading to equation (25).
Special case To map this model and the main idea described in Section 2.2, consider aspecial case of α = 1 in which all buyers stay in their country: µhf = µf h = 0. In this case,buyers from country i hold money i and g only, as assumed in Section 2.2. Then, given aninterior solution, the first-order conditions of the problem (20) are reduced to 0 −1 iγiu (c (zi + zig ))− 1 = µii 0 −1 i−1βc (c (zi + zig )) 0 −1 iγgu (c (zi + zig ))− 1 = µii 0 −1 i−1βc (c (zi + zig ))for i ∈ {h, f }

(26)
(27)

These equations imply the equalization of the money growth rates, γh = γg = γf , as derived
in equation (3). In addition, the liquidity service of money, which is captured by the righthand-side of equations (26) and (27), is identical between monies h, f , and g, as assumed
in Section 2.2. Hence, this special case provides solid micro foundations for the idea ofmonetary policy synchronization presented in Section 2.2.

3.3

Currency Counterfeiting

In this subsection, I add a specific friction in the form of currency counterfeiting andimperfect recognizability to the common setup of Section 3.1 and show that, unlike theeconomy considered in Section 3.2, monetary policy autonomy is preserved in this globalmoney economy.
Recognizability problem The national currencies are perfectly recognizable by sellers
so that there is no counterfeiting of the national currencies as in the previous subsection.
But here I assume that the global currency can be counterfeited by buyers and is imperfectly recognizable for sellers.7 Specifically, sellers are unable to distinguish genuine global
7This asymmetric treatment between the national currencies and the global currency is not critical forthe result of monetary policy autonomy. Even if the national currencies are subject to counterfeiting as

17money from counterfeit global money in the DM, and buyers can produce counterfeit digital currencies of the global currency at the end of each period – between the closure of
the CM and the opening of the DM. For simplicity, following Nosal and Rocheteau (2017,
Chapter 12), I assume that producing counterfeit digital currencies entails a fixed cost of
ψ > 0 only and the variable cost is zero. Then, by spending ψ > 0, a buyer can produce any
amount of counterfeit global digital currencies. In addition, counterfeits produced in theprevious period are all detected and confiscated when agents enter the CM today. Hence,in each period a buyer will incur the fixed cost to enter the DM with counterfeits in hand.
Incentive constraint In general, a buyer chooses a portfolio of genuine global money
and counterfeit global money, but this choice can be simplified as follows. Since increasing
the holding of genuine money is costly but increasing the holding of counterfeit money is
free if the fixed cost is already paid, a buyer holds either genuine global money or counterfeit
global money, but not both. In addition, holding more counterfeits than those used in the
DM generates no benefit since all the unused counterfeits will be confiscated in the following
CM. Hence, a buyer from country i accumulates either the real balances zig of the genuine
global money or the real balances zig of the counterfeit global money. The associated costsof doing so are γg zig and ψ, respectively.8 Then, the buyer has no incentive to counterfeit
if γg zig ≤ ψ, or
zig ≤

ψ
γg

(28)

Since the payment constraint (16) holds with equality, a take-it-or-leave-it offer madeby the buyer satisfies dij ≡ (djij , dgij ) = (zij , zig ). Then, the incentive constraint (28) can be
written as
dgij ≤

ψγg(29)

Upon receiving the offer, if the offer satisfies the incentive constraint (29), the seller findsthat the global money used for the payment is genuine. If the constraint (29) does nothold, then I assume that the seller believes the global money used for the payment to be
counterfeited. In this case, the seller rejects the offer even if it satisfies the participation
constraint (17) with equality, since the constraint fails to hold ex-post once the seller has
well, applying the model of symmetric counterfeiting considered by Gomis-Porqueras et al. (2017) to thismodel would generate the same result – the preservation of monetary policy autonomy – although such amodel would be more complicated than that presented here.
8
From the portfolio choice problem in the CM value function (14), the costs of accumulating zig aregγ g zi .

18

found the global money to be counterfeited. Hence, the incentive constraint (29) constitutes
a necessary condition for the seller to accept the offer.
No double spending While counterfeiting with the fixed cost of ψ is a parsimonous
way to derive the liquidity constraint (29) endogenously, it can be derived as a no-doublespending constraint for a cryptocurrency that employs a consensus protocol based on proofof-work. Here, double spending by a buyer is equivalent to faking payment dgij = zig to aseller without using the actual real balances of digital money zig , while receiving the good
produced by the seller. As shown by Chiu and Koeppl (2019), for a cryptocurrency with aproof-of-work protocol, a buyer has no incentive to double spend if and only if the amountof the payment in the cryptocurrency is limited as
dgij ≤ R(N + 1)N(30)

where R is a reward for mining – updating a record of transactions (blockchain) – and N isa confirmation lag – the number of confirmations in updating blockchains, where the goodis produced and transferred to the buyer after the N th confirmation.
The no-double-spending constraint (30) is intuitive. The left-hand-side of (30) is thebenefit obtained when suceeding in double spending, and the right-hand-side of (30) captures the cost of double spending. It is increasing in the reward R, since the incentives ofminers to compete for updating blockchains increase as R increases. It is increasing exponentially in N , since winning the competition of mining for N consecutive times, which is
required for double spending, becomes much more difficult as N increases.
The reward R and the confirmation lag N can be endogenized as in Chiu and Koeppl(2019), but doing so would not affect the main results of this paper as long as the constraint(30) is binding. For example, R can be financed by seignorage by assuming γg > 1. In
this case the lump-sum monetary transfers do not include the seignorage and are givenPby τ ≡ l∈{h,f } (γl − 1)φl Ml /(1 + n). An important point here is that the parsimonious
incentive constraint (29) can be obtained by considering a cryptocurrency with a proof-ofwork protocol as a global digital currency in this model.
Portfolio choice in the CM Given that the offer is accepted by the seller, the DM
value function of the buyer is given by equation (18) as in the baseline model in Section
3.2. With the DM value function on hand, the buyer’s portfolio choice problem in the CM

19

is given by (20) – the same maximization problem as in the baseline model – but subject
to the incentive constraint (28).9 Let ξi denote a Lagrange multiplier on the constraint.
Given that the holding of real balances is positive for all kinds of money, the first-orderconditions with respect to zih , zif , and zig are given, respectively by equations (21) and (22),
and
"

0

u
γ g + ξi
− 1 = µih 0
βcc (zih + zig ) −1
c−1 (zih + zig )
−1



#




u0 c−1 (zif + zig ) − 1
+ µif  fg
0−1c c (zi + zi )(31)

The only difference between condition (31) in this economy and condition (23) in thebaseline model is the presence of the multiplier ξi in equation (31).
Monetary policy autonomy Combining equations (21), (22), and (31) yields
ξi = γ h + γ f − γ g − β(32)This condition shows that as long as ξi > 0, monetary policy autonomy holds. In otherwords, given γg > β, any pair of (γh , γf ) that satisfies ξi > 0 and γh , γf > β constitutes anequilibrium, given that the cost of counterfeiting is not too high.10 If the right-hand-side ofequation (32) is zero, condition (32) is reduced to condition (24) as in the baseline modelstudied in Section 3.2 and monetary policy synchronization arises. If the right-hand-sideof equation (32) is negative, then the global money is not held in equilibrium: zig = 0.
To illustrate, let me set β = 0.97 and γg = 1.01 and draw Figure 1, which plots
the areas of monetary policy autonomy, monetary policy synchronization, and no global
money on the (γh , γf ) plane. In this economy with the possibility of counterfeiting of the
global money, the region of monetary policy autonomy is vast, as shown in Figure 1. For
example, given the setting of the money growth rate in the foreign country, the home
central bank is free to set the money growth rate in the home country within the area ofmonetary policy autonomy. But, if the incentive constraint (28) is slack, an equilibriumwith positive holdings of all kinds of money is restricted to the line indicated as “monetary
policy synchronization” in Figure 1. This is the case when the right-hand-side of equation(32) is zero, i.e., when the opportunity cost of holding the global money is just equal to
the sum of the costs of holding the home and foreign money.
9

In the case of a cryptocurrency with a proof-of-work protocol as a global digital currency, the constraintg(28) is replaced by zij≤ R(N +1)N , where the right-hand-side is the same as that of the no-double-spending
constraint (30).
10
For the condition that ensures the positive holdings of all kinds of money, see Appendix A.2.

20

Figure 1: Regions of monetary policy

Special case As in Section 3.2, consider a special case of α = 1 in which all buyers stayin their country. Then, given that the holding of each money is positive, the first-orderconditions of the buyer’s portfolio choice problem in the DM are reduced to equation (26)and
 0 −1 iγ g + ξiu (c (zi + zig ))− 1 = µii 0 −1 i−1βc (c (zi + zig ))(33)

Combining equations (26) and (33) yields
ξi = γi − γ g(34)This equation implies that, as long as γi > γg – the return on holding the national money issmaller than that on holding the global money – and the cost of counterfeiting is relatively
small, the incentive constraint (28) is binding and monetary policy autonomy holds. In
this economy, the upper bound z̄ig in the ad-hoc constraint (5) in the simple framework
discussed in Section 2.3 corresponds to ψ/γg in the incentive constraint (28), and the total
real balances, zi = zii + zig , are determined by equation (26). Hence, the price of the global
currency, the price of each national currency, and the nominal exchange rate are determinedby equations (6), (7), and (8), respectively, as in Section 2.3.

21

3.4

Government Transaction Policy

I now consider a different friction in the DM: government transaction policy. I add such afriction to the common setup of Section 3.1 and show that monetary policy autonomy canbe preserved in the global-money economy.
Government sellers Thus far I have assumed that sellers are private sellers who behave in their own interests. In this subsection, I introduce government sellers who follow
an exogenous policy over which currency they should accept. Specifically, I assume thatgovernment sellers in country j accept the national currency of country j only. In eachcountry j ∈ {h, f }, a fraction of government sellers is assumed to be 0 ≤ µ̃j ≤ 1. Then,for a buyer from country i, the probability of matching with a private seller in country j,
who can accept both the national currency and the global currency, is µij (1 − µ̃j ), and theprobability of matching with a government seller in country j, who can accept the national
currency only, is µij µ̃j .
The terms of trade with a government seller in the DM differ from those with a private
seller since a government seller can accept the national currency only. A take-it-or-leave-it
offer made by a buyer from country i to a government seller in country j consists of a pair(d˜j , q̃ij ), where d˜j is the payment using the national currency j in country j and q̃j is theij

ij

quantity of the DM good produced by the government seller. This offer has to satisfy thepayment constraint similar to (16),d˜jij ≤ zij(35)

and the modified participation constraint,d˜jij − c(q̃ij ) ≥ 0(36)This constraint reflects the fact that the government seller can accept the national currency
j only. The terms of trade with a private seller remain the same as in the common setup
of Section 3.1.
Value function in the DM Consider a match in the DM for a buyer from country iwith a portfolio of real balances zi ≡ (zih , zif , zig ). For the purpose of exposition, I focus ona case where the payment constraint (16) is binding for all matches.11 In this case, with
11

For the case of non-binding payment constraints, see Appendix A.3.

22

probability µij (1 − µ̃j ), the buyer is matched with a private seller in country j and makes
an offer (dij , qij ) that satisfies conditions (16) and (17). The resulting life-time utility ofthe buyer is u(qij ) + WiB (0) + ziĵ for ĵ 6= j as in the baseline model in Section 3.2. With
probability µij µ̃j , the buyer is matched with a government seller in country j and makes anoffer (d˜j , q̃ij ) that satisfies conditions (35) and (36). The buyer leaves the DM with monies
ijĵ and g unused, so that the buyer’s life-time utility is given by u(q̃ij ) + WiB (0) + ziĵ + zig .
With probability 1 − µih − µif , the buyer is unmatched with a seller. Then, the DM valuefunction of the buyer can be written as
ViB (zi ) = µih (1 − µ̃h )[u(qih ) + zif ] + µih µ̃h [u(q̃ih ) + zif + zig ]+ µif (1 − µ̃f )[u(qif ) + zih ] + µif µ̃f [u(q̃if ) + zih + zig ]+ (1 − µih − µif )(zih + zif + zig ) + WiB (0)Since the constraints (16), (17), and (36) hold with equality, the quantity of the DM
good produced by a private seller and a government seller is given by equation (19) andq̃ij = c−1 (zij ), respectively.
Portfolio choice in the CM Similar to the formulation in Section 3.2, the buyer’s
portfolio choice problem in the CM can be written as
max{zih ,zif ,zig }− γh zih − γf zif − γg zig + β
h


h
i

µih (1 − µ̃h ) u c−1 (zih + zig ) + zif

h 

i
g
−1 f
h
+ µif (1 − µ̃f ) u c (zi + zi ) + zi
+ µih µ̃h u c

h 

i
g
f
g
−1 f
h
h
+ µif µ̃f u c (zi ) + zi + zi + (1 − µih − µif )(zi + zi + zi )−1(zih )

+ zif + zigi

The first-order conditions with respect to zih , zif , and zig for their interior solution are given,
respectively, as
#
"
#
"


u0 c−1 (zih )u0 c−1 (zih + zig )γh − 1 + µih µ̃h −1− 1 = µih (1 − µ̃h ) 0 −1 hβc c (zi + zig )c0 c−1 (zih )


 

 

g
0
−1 f
0
−1 f
u
c
(z
+
z
)
u
c
(z
)
i
i
i
γf
 − 1 + µif µ̃f   − 1
− 1 = µif (1 − µ̃f )  
f
g
f
β
0
−1
0
−1
c c (z + z )c c (z )
i

i

i

23

(37)

(38)


 

"
#

g
−1 f
0
g
0
−1 h
c
(z
+
z
)
u
i
i
u
c
(z
+
z
)
γg
i
 − 1 + µif (1 − µ̃f )   − 1 (39)
− 1 = µih (1 − µ̃h ) 0 −1 higfgβc c (zi + zi )0−1c c (z + z )iiCompared to equations (21)-(23) in the baseline model, holding the national money has
additional benefits as appeared in the second term in the right-hand-side of equations (37)and (38) due to the presence of government sellers who accept the national money only.
If there are no government sellers, i.e., µ̃h = µ̃f = 0, equations (37)-(39) are reduced toequations (21)-(23).
Monetary policy autonomy For given γh , γf , and γg , the three equations (37)-(39)provide a solution for zih , zif , and zig . For given γg , a pair of (γh , γf ) that ensures a solution
constitutes an area of monetary policy autonomy. For simplicity, assume that a fractionof government sellers is identical between the two countries: µ̃ = µ̃h = µ̃f . By combining
equations (37)-(39), a necessary condition for the existence of an equilibrium with positive
holdings of the national money and the global money is derived as12
γh + γf − γg − β >µ̃(γg − β)
1 − µ̃

(40)

If there are no government sellers, µ̃ = 0, the left-hand-side of this inequality should bepositive, but it does not hold since conditions (37)-(39) are reduced to condition (24) ifµ̃ = 0. This observation highlights the role of government sellers for the existence ofequilibrium with monetary policy autonomy. Condition (40) suggests that given γf and
γg , the money growth rate of the home country within the range of γh > γ h can support a
global money equilibrium in which all the currences are valued and circulated, where thelower bound is implied by condition (40) as γ h ≡ γg − γf + β − µ̃(γg − β)/(1 − µ̃).
Special case Consider a special case of α = 1 in which buyers are matched with sellers
in the same country only. In this case, equations (37)-(39) are reduced to 0 −1 i 0 −1 iγiu (c (zi + zig ))u (c (zi ))− 1 = µii (1 − µ̃i ) 0 −1 i
− 1 + µii µ̃i 0 −1 i − 1βc (c (zi + zig ))c (c (zi )) 0 −1 igγgu (c (z + zi ))− 1 = µii (1 − µ̃i ) 0 −1 ii−1βc (c (zi + zig ))12

for i ∈ {h, f } (41)(42)

The proof of this claim can be found in Appendix A.4. The appendix also derives necessary andsufficient conditions for the existence and shows that condition (40) is a part of such conditions.

24

Combining equations (41) and (42) yields
 0 −1 iγi γgu (c (zi ))−= µii µ̃i 0 −1 i − 1ββc (c (zi ))(43)

If there is no government seller, µ̃i = 0, this condition is reduced to the equalization of
the money growth rates as in Section 3.2. Thus, the presence of government sellers playsa critical role in preventing the loss of monetary policy autonomy in the global-money
economy.
Given that there are government sellers, µ̃i > 0, and given that the growth rate of thenational money is greater than that of the global money, γi − γg > 0, the real balancesof the national money is determined by equation (43). The total real balances of money,zi ≡ zii + zig , can be solved from equation (42). The solution constitutes an equilibrium inthe global-money economy if zig = zi − zii > 0. From equations (42) and (43), this is thecase if and only if
γi − γg >µ̃i (γg − β)
1 − µ̃i(44)

In the region that satisfies condition (44), monetary policy autonomy prevails in the global
money economy, which features the positive holdings of the national money and the global
money. Since the national money provides a greater liquidity service – the right-hand-sideof equation (41) – than the global money, in equilibrium the cost of holding the national
money – the left-hand-side of equation (41) – also has to be greater than the cost of holding
the global money. Thus, the growth rate of the national money has to be high enough tosatisfy condition (44) in the global-money economy.

4

Conclusion

This paper has considered a possible future economy in which privately issued digital money
is circulated across borders. It has explored the impact of such global money on monetary
policy autonomy using a two-country open economy search model with a global currency.
The theoretical analysis has shown that the emergence of global money will cause a lossof monetary policy autonomy if there is no limit on using global money for transaction,
as in Benigno et al. (2022); but if using global money is subject to additional frictions
such as currency counterfeiting and government transaction policy, then monetary policy
autonomy will be preserved.

25

The result suggests that a loss of monetary policy autonomy is not an inevitable outcomeof the circulation of global digital money and that monetary policy autonomy and global
digital money can be compatible. Specifically, if an incentive mechanism or government
policy limits the amount of global money used for transaction, central banks can set theirmonetary policy instrument as they see fit. This result points to the critical role of thedesign and regulation of global digital money in preserving the capacity of central banks
to conduct monetary policy.
I would like to close the paper with a few caveats. First, this paper focuses exclusively on
the impact of global digital money as a medium of exchange on monetary policy autonomy.
Thus, it is silent about the effectiveness of monetary policy. Ikeda (2020) explores theimpact of global digital money as a unit of account on the effectiveness of monetary policy.
For analytical purposes, distinguishing the different roles of money, and between autonomy
and effectiveness are useful in deriving sharp theoretical implications, but in practice theroles of money, autonomy, and effectiveness need to be considered simultaneously. Second,although this paper considers currency counterfeiting and government policy, there could beother important frictions such as imperfect financial markets (Gabaix and Maggiori, 2015)that could also help preserve monetary policy autonomy. Finally, this paper abstracts
from the detailed design of global digital money, everything other than its amount used
for transaction and its growth rate. In the case of stablecoins, for example, safe assetsthat support the value of stablecoins would need to be considered explicitly. Despite thesecaveats, I hope that this paper provides some insight into answering the questions posedby the rise in digital money.

26

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Appendix
A

Proof and Derivation

A.1

Matching probabilities

As described in Section 3.1, a buyer stays in the domestic country with probability α ∈[0.5, 1], and a buyer visits the foreign country with probability 1 − α; the number of trade
matches in the DM of country i is given byMi ≡ M(Bi , Si ) =B i Si,B i + Si

where Bi and Si denote the measures of buyers and sellers in the DM of country i, respectively. In country h, the number of sellers is Sh = n, while the number of buyers is
Bh = αn + 1 − α. In country f , the number of sellers is Sf = 1, while the number ofbuyers is Bf = α + n(1 − α). Then, the number of matches in countries h and f is given,
respectively, byn(αn + 1 − α)n + αn + 1 − α
α + n(1 − α)
Mf =1 + α + n(1 − α)Mh =The number of matches in country j where the matched buyers are from country i, denotedby Mij , is given by1−α,αn + 1 − αα
Mf f = Mf.
α + n(1 − α)αn,αn + 1 − αn(1 − α)Mhf = Mf,
α + n(1 − α)Mhh = Mh

Mf h = Mh

Conditional on being in country j ∈ {h, f }, a buyer meets a seller with probability
1
Mh
=
Bh
1 + α + (1 − α)/n
Mf
1
af ≡
=
Bf
1 + α + n(1 − α)ah ≡Then, the probability that a buyer from country h meets a seller in country j ∈ {h, f },denoted by µhj , is given byµhh = αah ,µhf = (1 − α)af

30

Similarly, the probability that a buyer from country f meets a seller from country j ∈{h, f }, denoted by µf j , is given byµf h = (1 − α)ah ,A.2µf f = αaf

Condition on the cost of counterfeiting

In the model of currency counterfeiting in Section 3.3, the equilibrium conditions regarding
the portfolio choice of real balances consist of equations (21), (22), and (31), which are
reproduced here for convenience:"#u0 c−1 zih + zig
 − 1
ih = µih 0 −1 hc czi + zig u0 c−1 zif + zig − 1
if = µif  
g
f
0
−1
c c
zi + zi
ξi= i h + if − i g
β

(45)

(46)

(47)

where ih , if , ig > 0 by assumption and ξi is a Lagrange multiplier on the incentive constraint(28): zig ≤ z̄ g ≡ γψg .
Assume ih + if − ig > 0 so that ξi > 0 and the constraint (28) is binding: zig = z̄ g . Let
zihg ≡ zih + zig denote a solution for equation (45). Then, since zig = z̄ g , the real balancsof the home money is positive, zih > 0, if zihg > z̄ g . Similarly, the real balances of theforeign money is positive, zif > 0, if zif g > z̄ g , where zif g ≡ zif + zig is a solution to equation(46). Hence, both the real balances of the home money and foreign money are positive,
zih , zif > 0, if min{zihg , zif g } > z̄ig or the cost of counterfeiting is not too large, satisfying
ψ < γg × min{zihg , zif g }This condition ensures the existence of an equilibrium in the global-money economy forany pair of (γh , γf ) that satisifies ξi > 0 and γh , γf > β for given γg > β.

A.3

Non-binding payment constraints in Section 3.4

There are four types of match regarding the types of seller: a home private seller, a homegovernment seller, a foreign private seller, and a foreign government seller. Consider a buyer
from the home country and matches with these sellers. It is straightforward to show thatthe payment constraint (16) is binding for a match with a government seller. Otherwise,31

the payment constraint is non-binding for a match with a private seller. The non-binding
payment constraints for matches with both private and government sellers are inconsistent
with γh , γf > β. Hence, the payment constraint may be non-binding for a match witha private seller. Suppose that the payment constraint is non-binding for matches withprivate sellers in both countries h and f . This implies that the marginal utility of holding
the global money is zero, which is inconsistent with the positive cost of holding the global
money, γg > β. Hence, the payment constraint may be non-binding for a match with a
private seller either in the home country or in the foreign country, and not in the bothcountries.
Without loss of generality, consider a case in which the cash constraint is not binding fora match between a buyer from the home country and a private seller in the same country. In
this case, the real money balance held by the buyer is great enough to satisfy zhh + zhg > z ∗ ,where the satiation amount of the real balances, z ∗ , are given by
u0 (c−1 (z ∗ ))=1
c0 (c−1 (z ∗ ))Given zhh + zhg > z ∗ , the first-order conditions of the portfolio choice problem with respect
to zhh , zhf , and zhg are given, respectively as#"u0 c−1 zhh
ih = µhh µ̃h 0 −1 h  − 1
c c
zh


 
 


 
f
g
−1
0
−1
0
zh + zhzhf
u c
u c
 − 1 + µhf µ̃f    − 1
if = µhf (1 − µ̃f )  
f
g
0
−1
0
−1
c c
zh + zh
c c
zhf


 

u0 c−1 zhf + zhg
 − 1
ig = µhf (1 − µ̃f )  
f
g
0
−1
c c
zh + zhEquation (48) can be solved for zhh . Combining equations (49) and (50) yields
 

 
u0 c−1 zhf
  − 1
if − ig = µhf µ̃f  
c0 c−1 zhf

(48)

(49)

(50)

(51)

Given if > ig , equation (51) can be solved for zhf . With zhf on hand, equation (50) can besolved for zhg . The solution has to satisfy zhh + zhg > z ∗ and zhf + zhg < z ∗ . If these inequalities
are satisfied, the solution of zhh , zhf , and zhg constitutes an equilibrium. Since the portfolioof real balances is determined, monetary policy autonomy is preserved in this economy.
32

A.4

Proof of monetary policy autonomy in Section 3.4

I show that condition (40) is a necessary condition for the existence of a stationary equilibrium with positive holdings of currencies h, f , and g in the economy with government
sellers. I also derive necessary and sufficient conditions for the existence. Assume thata fraction of government sellers is identical between the two countries: µ̃ = µ̃h = µ̃f .
The equilibrium conditions that characterize positive holdings of the three currencies areequations (37)-(39).
From equation (37), zih can be written as a function of zig , which is denoted as zih =Zh (zig ). This function is decreasing but the degree of a decrease in less than unity, −1 <Zh0 (·) < 0. Let z̄ih and z hi be defined asz̄ih = Zh (0),

g
−1 h
0
c
(z
u
+
z̄
)
i
i
 =1
z hi = Zh (z̄ig ), where 0 −1 hc c (z i + z̄ig )Similarly, from equation (38), function zif = Zf (zig ) can be derived and it has the same
properties as zih = Zh (zig ). Define z̄ g = min{z̄hg , z̄fg }.
By substituting zih = Zh (zig ) and zif = Zf (zig ), equation (39) can be written as
 0 −1
 0 −1u (c (Zh (zig ) + zig ))γg − βu (c (Zf (zig ) + zig ))µih 0 −1
− 1 + µif 0 −1
−1 −
=0
g
g
g
g
c (c (Zh (zi ) + zi ))c (c (Zf (zi ) + zi ))β(1 − µ̃)(52)

Let Zg (zig ) denote the left-hand-side of this equation as a function of zig . Since −1 < Zj0 (·) <0 for j ∈ {h, f }, Zj (zig ) + zig is increasing in zig . Since u00 (·) < 0 and c00 (·) ≥ 0, the function
Zg (zig ) is strictly decreasing in zig . Then, equation (52) can be formulated as a fixed-point
problem for zig as
Zg (zig ) = 0(53)

Since Zg (zig ) is strictly decreasing, a unique interior solution to (53) exists if and only
if Zg (0) > 0 and Zg (z̄ g ) < 0. From equations (37)-(38) with µ̃h = µ̃f = µ̃, condition
Zg (0) > 0 can be derived as condition (40), which is necessary but not sufficient. Necessary
and sufficient conditions for the existence are condition (40) and Zg (z̄ g ) < 0.

33