Monetary Policy in a World of
Cryptocurrencies
Pierpaolo Benigno
LUISS and EIEF
April 14, 2019

Abstract
Can currency competition a¤ect central banks’control of interest rates andprices? Yes, it can. In a two-currency world, the growth rate of the cryptocurrency sets an upper bound on the nominal interest rate and the attainable in‡ation rate, if the government currency is to retain its role as medium of exchange.
In a world of multiple competing currencies issued by pro…t-maximizing agents,
the nominal interest rate and in‡ation are both determined by structural factors, and thus not subject to manipulation, a result hailed by the proponentsof currency competition. The article also proposes some …xes for the classicalproblem of indeterminacy of exchange rates.

I am grateful to Giorgio Primiceri for useful comments, Marco Bassetto for insightful discussion
at the NBER Monetary Economics Meeting and Roger Meservey for professional editing. Financial
support from ERC Consolidator Grant No. 614879 (MONPMOD) is gratefully acknowledged.

In recent years cryptocurrencies have attracted the attention of consumers, media
and policymakers.1 Cryptocurrencies are digital currencies, not physically minted.
Monetary history o¤ers other examples of uncoined money. For centuries, sinceCharlemagne, an “imaginary” money existed but served only as unit of accountand never as, unlike today’s cryptocurrencies, medium of exchange.2 Nor is thecoexistence of multiple currencies within the borders of the same nation a recent phenomenon. Medieval Europe was characterized by the presence of multiple media of
exchange of di¤erent metallic content.3 More recently, some nations contended withdollarization or eurization.4
However, the landscape in which digital currencies are now emerging is quite
peculiar: they have appeared within nations dominated by a single …at currency justas central banks have succeeded in controlling the value of their currencies and tamingin‡ation.
In this perspective, this article asks whether the presence of multiple currencies canjeopardize the primary function of central banking –controlling prices and in‡ation–or eventually limit their operational tools –e.g. the interest rate. The short answer
is: yes it can.
The analysis posits a simple endowment perfect-foresight monetary economy along
the lines of Lucas and Stokey (1987), in which currency provides liquidity services. For
the benchmark single-currency model, the results are established: the central bankcan control the rate of in‡ation by setting the nominal interest rate; the (initial) pricelevel instead is determined by an appropriate real tax policy.5 The combination ofthese two policies (interest-rate targeting and …scal policy) determines the path ofthe price level in all periods, and the central bank can achieve any desired in‡ation
rate by setting the right nominal interest rate.
First I add to this benchmark a privately issued currency that is perfectly substitutable for the government’s currency in providing liquidity services. The private
currency is “minted”each period according to a constant growth rate .
The …rst important result is that private currency can be worthless if it is believed
to be, while government currency always has a positive value. This result follows fromthe connection between the policy followed by the government, the levying of taxes in
the government currency and the existence of a market of interest-bearing securities
in it.
Despite these privileges, government money can lose the property of medium of
1

See BIS (2018).
See Einaudi (1936) for an analysis of the “imaginary” money from the time of Charlemagne to
the French Revolution. Loyo (2002) studies optimal choice of unit account in a context of multiple
units.
3
Cipolla (1956,1982,1990) describes several cases in the monetary history of coexistence of multiple currencies.
4
See Calvo and Vegh (1997) for an analysis of dollarization.
5
Canzoneri, Cumby and Diba (2001), Niepelt (2004), Sims (1994, 2000, 2013) and Woodford(1995, 2001, 2003), among others, present results of the benchmark single-currency case.
2

1exchange if its in‡ation is higher than that of private money. The “better money”–
with higher return and lower in‡ation –will prevail in transaction services, con…rming
the view of Hayek (1974). The loss of the property of medium of exchange can berisky for the government and threaten the very survival of the currency, unless legal
restrictions are imposed.
What happens if the government wants to crowd out private currency to avoid
getting into such dangerous territories? It has to keep in‡ation ( ) low and boundedby the growth rate of private money, i.e.
1+ : Currency competition could thus
become a useful way of keeping in‡ation low and reducing liquidity premia. But, thein‡ation bound imposed on the government can be too tight when other objectives,
such as economic stabilization, require a higher in‡ation target.
A pervasive result in this environment, along the lines of Kareken and Wallace(1981), is that the exchange rate between the two currencies is indeterminate. However, if …scal policy is set in the same way as in the benchmark single-currency
framework, this indeterminacy does not bring instability to the price level (in unitsof government currency).6
Next I extend the framework to allow for multiple currencies in a market of pro…tmaximizing issuers where entry is endogenous and subject to a …xed cost. The results
become even more striking: the nominal interest rate becomes entirely determined bystructural factors (the intertemporal discount factor, the exit rate and the …xed costof entry). As a consequence, the in‡ation rate too becomes a function of the samestructural factors. However, the (initial) price level in units of government currency is
still determinate if …scal policy is set appropriately as in the benchmark case. Multiple
currency competition, unlike the two-currency variety, can also preclude all possible
instability of the path of real money balances in each currency. The government canstill crowd out private currency by setting relatively low interest and in‡ation rates,
but these might turn out to be too low if the barriers to entry are themselves enoughstrict.
I then turn back to analyze exchange-rate determination in the two-currency economy. This problem can be solved if the private currency is issued in a centralized
system and the issuer has some taxation power. This approach has novel featureswith respect to the straightforward application of the …scal theory of the price level toa two-currency economy: only one of the two monetary authorities can issue securities
that have a solely pecuniary return.7 In general, currencies that are not the liabilities
of some agent, like cryptocurrencies, can hardly have a determined exchange rate.
The latter is therefore subject to non-fundamental sunspot shocks.
This paper is related to the recent literature prompted by the increasing number
6

It creates indeterminacy of the equilibrium path of government money supply with no consequence for the equilibrium path of consumption except but at time zero.
7
The …scal theory of the price level, instead, requires that the government issue at least somesecurities with only pecuniary value.

2

of cryptocurrencies, which has revived interest in multiple-currency monetary models. Fernandez-Villaverde and Sanches (2018) evaluate the role of competing private
currencies whose supply is determined by pro…t maximization.8 Their results di¤er
substantially from mine. They …nd that an appropriately de…ned price stability equilibrium can arise under certain restrictions on the cost function for private money
production. But, when the marginal cost goes to zero, price stability cannot be an
equilibrium.
Motivated by these results, they argue in favour of Milton Friedman’s view thata purely private system of …duciary currencies would inevitably lead to price instability (Friedman, 1960). Further they …nd that competition does not achieve thee¢ cient allocation and is socially wasteful. If anything, my currency competition
model is reminiscent of Hayek’s view that unfettered competition in the currency
market is bene…cial for society (Hayek, 1976) insofar as I …nd that the e¢ cient allocation can be reached when the …xed cost of entry goes to zero.9 Further, unlike
Fernandez-Villaverde and Sanches (2018), I …nd no equilibrium multiplicity or any
hyperin‡ationary equilibria.
The di¤erence might depend on their assumption of a …xed number of issuers
without no new entry whereas I allow for entry at a …xed cost. We share the resultthat a competing currency can restrict the set of possible equilibria. But even in thiscase the results di¤er considerably. In their model, there cannot be an equilibrium inwhich the real interest rate equals the rate of time preference unless private money
is driven out of the market, whereas in mine the real interest rate is always tied, inequilibrium, to the rate of time preference. The restrictions that I …nd are on nominal
variables –interest rate and prices.
Schilling and Uhlig (2018) also analyze coexistence and competition between traditional …at money and cryptocurrencies. But, they are more concerned with thedetermination of the price of cryptocurrencies, deriving interesting bounds and assetprice relationships. On policy, they assume that the government has always fullcontrol of the in‡ation rate. With respect to monetary policy, they emphasize theconnection between the indeterminacy of the cryptocurrency, prices and government
money supply. This result emerges also in my two-currency model but not in themultiple-currency pro…t-maximizing framework. Garratt and Wallace (2017) too areinterested in the determinacy of the exchange rate between currencies and revisit theindeterminacy result of Kareken and Wallace (1981). In my analysis, exchange-rateindeterminacy, if present, is not particularly relevant to the way in which competing
8

Klein (1974) is an early example of a model of currency competition with pro…t-maximizing
suppliers.
9Marimon et al. (2012) also …nd that currency competition can achieve e¢ ciency. However, they
do not model entry, so in their framework each pro…t-maximizing issuer would set its price level toin…nite to wipe out its existing stock of liabilities. Kovbasyuk (2018), instead, …nds that private
currency (tokens) can tend to in‡ate prices and harm agents who hold …at currency.

3

currencies a¤ect the …ndings of traditional monetary policy analysis. I also suggest
ways to overcome the indeterminacy problem in a centralized system of private currency creation.10
Woodford (1995) is the benchmark reference for the analysis of price determination
through interest-rate targeting and …scal rules in a single-currency economy. He alsostudies a “free banking” regime in which deposits, in units of the single currency,
compete with government money in providing liquidity services. In this case, he also…nds that the nominal interest rate and the in‡ation rate are determined by structural
factors. However, he does not analyze multiple-currency models.
The paper is structured as follows. Section 1 presents the two-currency model andsolves for the equilibrium, and Section 2 analyzes this equilibrium. Section 3 discusses
what a competing currency implies for standard monetary policy analysis. Section 4extends the model to multiple currencies with competition among pro…t-maximizing
issuers and assesses the implications. Section 5, going back to the two-currency model,shows how it is possible to solve the problem of indeterminacy of the private-currency
exchange rate. Section 6 concludes.

1

The model

We consider a two-currency economy, one issued by the government and one privately.
The model follows the lines of Lucas and Stokey (1987) and Benigno and Robatto(2018) in which there are two goods: a “cash”good and a “credit”good. The “cash”
good can only be purchased by money.
Consumers have preferences of the following form:
1
X

t t0

fln Ct + Xt g ;(1)

where is the intertemporal discount factor with 0 <the “credit”good. C can be purchased for money< 1; C is the “cash”and X

t=t0

Ct

Mt 1 Mt 1
+
Pt
Pt

(2)

in which Mt is the government-issued and Mt the privately issued money. Money canbe material (coins or banknotes) or digital, and in either form it carries no interest
payment; Pt is the price of the consumption good in terms of the government currency,
Pt is the price in the private currency.
10
In one-currency models, Sargent and Wallace (1982) and Smith (1988) have studied the coexistence, and competition, between government money and private assets and the need to separate
the credit and money markets to avoid ‡uctuations in the price level unrelated to fundamentals.

4

The cash constraint (2) assumes that the two types of cash are perfect substitutes
for the purchase of the consumption good C: This assumption simpli…es the analysis
at little cost in generality. At least, it enables the model to challenge the results thatderive from the single-currency framework to a greater extent.11
In the credit market the consumption good Xt is subject to the budget constraint
Bt
Mt Mt
+
+
+ Xt
Pt (1 + it )
Pt
Pt

Bt 1
+
Pt

Mt 1 Mt 1+
Pt
Pt

Ct + Y +Tt
T+ t ; (3)Pt P tin which B is a risk-free interest-bearing security in units of the government currency;
Y is the constant endowment of the two consumption goods; Tt are governmenttransfers in units of government currency and Tt are the private issuer’s transfers inunits of its currency.
In writing the budget constraint (3), we make two important assumptions: …rst,
interest-bearing securities provide no liquidity services; second, they are only denominated in the government currency. The signi…cance of the latter assumption is
discussed later. Bt can be positive, in which case it is an asset for the household, or
negative, in which case it is debt. I allow the private sector to borrow by issuing debt
denominated in government currency but not in the privately-issued currency. And
this debt, when issued, is paid back with certainty, being subject to an appropriate
borrowing limit.
Before addressing this limit, note that the impossibility of arbitrage has two implications for an equilibrium in which both currencies compete as a medium of exchange.
First, it requires that
P
Pt+1
= t+1
(4)
Pt
Pt
for each t t0 and therefore that the exchange rate St –the price of the privatelyissued currency in government currency (i.e. St Pt =Pt ) –is constant over time ata value S; which has to be determined. The equivalence of money returns shown in(4) follows from the assumption of perfect substitutability between the two moneys
in the cash constraint.12
The second implication of the absence of arbitrage is that the nominal interest
rate in government currency is non-negative, it 0. If it were negative, households
could make in…nite pro…ts by borrowing at negative interest rates and investing theproceed in cash.
Given these two results, the ‡ow budget constraint (3) can be written as
it Mt
it SMtWt+ C t + Xt ++(1 + it )Pt
1 + i t Pt
1 + it Pt
11

Wt 1
Tt STt
+Y +
+
;
Pt
Pt
Pt

Schilling and Uhlig (2018) also assume perfect substitutability between the two moneys.
Condition (4) and its implications hold when both currencies are used as a medium of exchange.
Later, I will discuss equilibria arising when (4) does not hold.
12

5

in which nominal wealth is de…ned in units of government currency, as
Bt + Mt + SMt :Wt

The natural borrowing limit is then written as
1
X

Wt 1
Pt
for each t

Qt;T

Y +

T =t

TT
STT
+
PT
PT

>

1

t0 given an appropriate discount factor Qt;T with Qt;t

(5)
1 and

PT Y
1
Qt;T =;
Pt j=1 1 + it+j
T

for T > t:
The natural borrowing limit (5) is the maximum amount of net debt that theconsumer can carry in a certain period of time and repay with certainty, i.e. withcurrent and future net income and assuming that future consumption and asset holdings are going to be equal to zero. The …nite borrowing limit is a requirement for
consumption to be bounded in the optimization problem. This assumption can beequivalently written as
1
X
t=t0

Qt0 ;t

Tt
Pt

1
X

<1

Qt0 ;t

t=t0

STt
Pt

< 1:

For a bounded consumption plan to exist, the following two in…nite sums musthave a …nite value
1
X
t=t0

Qt0 ;tit Mtg
1 + i t Pt

<1

1
X

Qt0 ;tt=t0

it SMt p
1 + i t Pt

< 1:

We now turn to the optimality conditions. The …rst-order condition with respect
to the consumption of the cash good is
Ct =

1
1+

(6)
t

for each t t0 in which t
0 is the Lagrange multiplier associated with the cash
constraint (2). The optimality condition with respect to holdings of the interestbearing security Bt is
1 Pt+1
1 + it =(7)
Pt
6

at each time t

t0 while those with respect to Mt and Mt are:
1+

t+1 =

1 Pt+1Pt

at each time t t0 : Combining (7) and (8), it follows that t+1 = it for each t
Finally, the intertemporal budget constraint holds with equality,1
Wt0 1 X
+
Pt0
t=t

1
X

t0 +1.

it Mt
it SMt+:1 + it Pt
1 + it Pt
t=t
0
0
(9)
Let us now specify the budget constraint of the two currency issuers. The government’s monetary authority is subject to the following budget constraint
t t0

Tt STt
Y +
+
Pt
Pt

(8)

Mtg +

=

t t0

C t + Xt +Btg
= Tt + Mtg 1 + Btg 1
1 + itwhere Mtg is the supply of cash, and Btg is the debt issued by the government, if
positive, or the assets held, if negative.
The private issuer is instead subject to the following constraint
Mt p = Tt + Mt p1 ;which can be interpreted either as the budget constraint of an agent issuing money ina centralized system or simply as an identity regulating how private money is createdin a decentralized system. The latter interpretation is akin to the way in which somecryptocurrencies, such as Bitcoins, are created nowadays.

2

Equilibrium

Equilibrium in the goods market implies that the sum of the consumption of the twogoods is equal to the constant endowment
Ct + Xt = Y:
Equilibrium in the market for the interest-bearing security in government currency
requires that
Bt = Btg ;while equilibrium in the cash market for the two currencies implies that supply anddemand equalize for each currency
Mt = Mtg ;7

Mt = Mt p :We can now summarize the set of equilibrium conditions concisely.
The …rst equation to consider is the Euler equation (7) derived from household
optimality condition with respect to the interest-bearing security B :1 + it =1 Pt+1Pt(10)which holds for each t t0 : The Euler equation links the nominal interest rate to thereal rate and the future gross in‡ation rate.
The second equilibrium condition is the consumer’s intertemporal budget constraint (9) which, using equilibrium in the good and asset markets, can be written
as1
Mtg0 1 + Btg0 1 + SMt0 1 X
+
Pt0
t=t

1
X

Mtg SMt p
:
+
P
P
t
t
t=t0
0
(11)
The mirror image of this constraint is an aggregate intertemporal budget constraint
consolidating both currency issuers. In (11), the initial value of the real liabilities ofboth agents plus the present discounted value of real transfers should be equal to theseigniorage revenues accruing to both suppliers from issuance of non-interest bearing
liabilities that have non-pecuniary bene…ts for the consumer.
In equilibrium, for a bounded consumption path to exist, it should be the casethat all in…nite sums have a …nite value, i.e.
1
X
t=t0

1
X
t=t0

t t0

t t0 Tt

Pt

t t0

Tt STt
+
Pt
Pt

1
X

< 1

t=t0
1
X

it Mtg< 1
1 + it Pt=

t t0

t t0 STt

< 1;

(12)

it SMt p< 1:
1 + i t Pt

(13)

Pt
t t0

t=t0

it
1 + itAlthough there is an aggregate intertemporal budget constraint that pools thetwo issuers together, each has a di¤erent ‡ow budget constraint on policy decisions.
The following constraint applies to the government
Mtg +

Btg
= Tt + Mtg 1 + Btg 1
1 + it(14)

given initial conditions Mtg0 1 and Btg0 1 :Private issuance of currency instead follows the law of motion
Mt p = Tt + Mt p1 ;8

(15)

given an initial condition Mt0p 1 .
From the optimality condition of the household, consumption of good C is inversely related to the nominal interest rate
1
Ct =
;
(16)
1 + it 1
for each t

t0 + 1, and1
1 + t0at time t0 . The cash-in-advance constraint implies that
Ct0 =

Mtg 1
Mt p1
+S
Ct
(17)
Pt
Pt
for each t t0 with equality whenever it 1 > 0, but for each t > t0 .
To close the set of equilibrium conditions, I must specify the monetary policy
regime.13 There are three degrees of freedom according to which one can set monetary
policy for the two issuers. I assume that the government authority eventually sets thepath fit ; Tt g1
t=t0 as a function of other variables: In the ‡ow budget constraint (14),this means that given initial conditions Mtg0 1 and Btg0 1 , the government is setting
the size of the balance sheet Mtg +Btg =(1+it ) and leaving the private sector to allocate
its wealth optimally between the two government securities. I assume that the privatepissuer sets the transfer fTt g1
t=t0 which, given the initial condition on Mt0 1 ; directly
implies the path of its money supply according to (15): This speci…cation is perfectlyin line with how such cryptocurrencies, as Bitcoins, are now being issued, namely by
a proof-of-work system that determines the new units to be emitted at each point intime.
De…nition 1 An equilibrium with two competing currencies is a set of sequences
fCt ; Pt ; it ; Mtg ; Mt p ; Btg , Tt , Tt g1
0 and Mtg ; Mt p > 0 at each time
t=t0 with Ct ; Pt ; itt t0 and a positive S and non-negative t0 that are consistent with the monetarypolicy regime and that satisfy (10), (14); (15); (17) (with equality whenever it 1 > 0)for each t
t0 and (16) for each t
t0 + 1 together with the intertemporal budget
constraint (11), the bounds (12), (13) and the constraint Ct0 = (1 + t0 ) 1 , given
initial conditions Mtg0 1 ; Btg0 1 ; Mt0p 1 :Analysis will be restricted to a certain class of policy rules.
De…nition 2 Assume the following speci…cation of the monetary policy regime. The
government sets a constant interest rate policy it = i at each t t0 and the following
transfer policy
i Mtg
Tt
=
(1
)
(18)
Pt
1 + i Pt
13

Analysis of all possible regimes is beyond the scope of this work.

9

for each t t0 and for some di¤erent from zero: The private currency issuer sets
Tt = Mt p1 at each t t0 with > 1, to achieve a constant growth rate; 1 + , ofits money supply.
Let us focus on the implications of such a monetary policy regime. First, by the
Fisher equation (10), the constant-interest-rate policy implies that in‡ation too isconstant
Pt+1= (1 + i):(19)Pt
The in‡ation rate is positive whenever (1 + i) > 1 , negative when (1 + i) < 1 .
A price stability policy is in place when the nominal interest rate is set equal to theinverse of the consumer’s rate of time preference, i.e. (1 + i) = 1 .
The government also sets a transfer policy in which it rebates its entire seigniorage
revenue to the consumers –the …rst element on the right-hand side of (18) –minus aconstant proportional to a non-zero value . In the terminology of the literature on
price-level determination, the transfer rule (18) is an “active”transfer policy.
Finally, it is assumed that the private issuer sets a constant growth rate of supply;
growth can also be negative, where the issuer is able to destroy money by someelectronic algorithm.
The assumptions on the monetary policy regime carry important implications forthe conditions of possible equilibrium. Assuming i > 0, the cash constraint (17) holdswith equality. After plugging in the private issuer’s supply rule and Ct = 1=(1 + i);
we obtain(1 + )t+1 t0 Mt0p 1
1
Mtg
+S
=
Pt+1
Pt+1
1+i
into which one can substitute (19) to obtain
Mtg
+S
Pt+1

1+
(1 + i)t+1 t0

Mt0p 1
1
:
=
Pt0
1+i

From this it follows that for an equilibrium with two currencies as medium of exchange
to exist it must be the case that 1 +(1 + i) : otherwise the government money
supply will be negative within some …nite period of time, which is not feasible.
Note, however, that when i = 0, then the cash constraint (17) holds with inequality
so it is possible for real money balances to grow without bound.
Moreover, under the monetary policy regime de…ned above, we explicitly write
the following summations
1
X
t=t0

t t0

STt
Pt

=

1
X
t=t0

t t0

S Mt p1
Pt

101
SMt0p 1 X=
Pt0 t=t
0

1+
1+i

t t0

(20)

1
X
t=t0

t t0

it
1 + itSMtPt

1pi(1 + ) SMt0 1 X=
1+i
Pt0 t=t
0

1+
1+i

t t0

(21)

where in (20) the transfer rule of the private issuer is substituted into the …rst equality
and the equilibrium condition (19) into the second equality. Similarly in (21). The…rst sum is …nite whenever < i for any i 0, while (21) requires that < i only
when i > 0.
The results of the above discussion are grouped in the following Lemma.
Lemma 3 Given the monetary policy regime speci…ed in De…nition 2, a necessary
condition for an equilibrium with competing currencies to exist is: i) 1 +(1 + i)when i > 0 and ii) < 0 when i = 0.
We now characterize the properties of the equilibrium and discuss whether given
the above speci…cation of the policy regime the price level and the exchange rate canbe determined.

3

Characterization of equilibria

Before analyzing the consequences of a multiple-currency environment, let me …rst
outline the results of the benchmark one-currency model. Assume that there is only
the government currency and that the government sets the same policies as in De…nition 2: constant nominal interest rate and appropriate real transfers. First, by
setting the nominal interest rate at a target i it can also set the gross in‡ation rate,; to any desired constant number given by = (1 + i); which is a function of theinterest rate chosen. Second, the initial price level and the entire price path are fully
determined by the …scal policy rule (18). The central bank thus has full control of theprice level and its rise. Note that in order to determine the price level it is critical forthe government to issue some liabilities with only pecuniary return, such as security
B.14 Moreover, given interest-rate targeting, the money supply path is endogenously
determined by the cash-in-advance constraint (2).
The presence of a competing currency alters these conclusions substantially. This
Section is divided into three parts. The …rst characterizes the equilibrium outlined in
the previous Section in which the two currencies compete as medium of exchange; thesecond examines equilibria in which only one of the two currencies serves as medium of
exchange; and the third summarizes the results in order to draw policy implications.
14

Benigno (2017) discusses price determination when the net asset position of the government is
positive.

11

3.1

Equilibrium with competing currencies

In a world of competing currencies, Lemma 3 states the …rst result. If the central
bank sets a positive nominal interest rate, then, in order for the two currencies tocoexist, the rate of growth in private money should be bounded by 1 +(1 + i).
An alternative intuition for this upper bound uses the Fisher equation (10) to obtain
1+: Thus, given the in‡ation rate set by the government, the growth rateof private currency should be capped by that number. Note that in equilibrium thein‡ation rates in the two currencies are the same, = , because of (4), so there is
not necessarily any linkage between the growth and in‡ation rates of private money.
If, instead, the government has a zero interest rate policy, i = 0; with a de‡ation
rate of = , then for an equilibrium with competing currencies to exist the gross
growth rate of private currency can also be higher than but must in any case belower than 1, i.e.
< 0: In this equilibrium=
always,being the private
currency in‡ation rate.
We now investigate whether a competing currency can create problems for thedetermination of the price level. The following Proposition summarizes these results.
Proposition 4 Given the monetary policy regime of De…nition 2, an equilibriumwith competing currencies exists if and only if: (i) the government sets a positive
interest rate and (1 + )(1 + i); or (ii) the government sets i = 0 and < 0: In
either case, the path of the price level in government currency is determined, but theexchange rate S and consumption at time t0 are not:
Proof. See Appendix A.1.
Lemma 3 and Proposition 4 carry two interesting implications. First, the second
currency causes no problem for the determinacy of the price level. The potential
problem would be the absence of a single intertemporal budget constraint for eachcurrency supplier. But given that i > in any equilibrium, an intertemporal budget
constraint does not hold for the supplier(s) of private currency. To see this, note thatgiven i > it follows that
lim

T !1

SMt0p 1
=
PT +1
Pt0

T t0 SMT

p

1+
1+i

T +1 t0=0which, together with the ‡ow budget constraint (15), implies an intertemporal budget constraint for the private currency issuers (see the Appendix for details). This,
together with the aggregate intertemporal budget constraint (11), implies that theconstraint holds for the government in exactly the same form as in the benchmark
case of a single currency. Applying the real transfer policy (18), we get price determination.
The second result, where multiple equilibria arise, is the indeterminacy of theexchange rate, as in Kareken and Wallace (1981) and recently restated by Garratt
12and Wallace (2017) and Schilling and Uhlig (2018). In my model, this results in theindeterminacy of consumption at time t0 and of the equilibrium path of government
money supply starting in period t0 + 1:15
To see the …rst result, consider the cash constraint at time t0
Mt0 1 SMt0 1
+
Pt0
Pt0

Ct0

and note that since Pt0 is determined and Mt0 1 , Mt0 1 are given, variations in S
translate into variations in Ct0 unless consumption is at the e¢ cient level. To see thesecond result, consider the cash constraint (17) when i > 0 and use (16) and (19) toobtainMtg
M p+S t =Pt
Pt
for each t
t0 . Since Pt is determined and Mt P is also given by the process ofprivate money creation, the indeterminacy of the exchange rate S translates directly
into indeterminacy of the path of government money supply. By contrast, prices ingovernment currency are completely insulated from the indeterminacy of the exchange
rate.

3.2

Equilibria with a single currency as medium of exchange

So far we have characterized equilibria in which multiple currencies coexist. For thisto happen, the return on the two moneys should be the same, which is condition (4).
But when (4) is violated in either direction, there can be equilibria in which agents
hold only one of the two currencies.
In what follows, we maintain the monetary policy regime of De…nition 2. The …rst
equilibrium discussed here is one in which government money is the only medium of
exchange. This outcome does not depend on the value of . To intuit this, assume
that consumers believe the return on private money is lower than that on government
money, i.e. Pt+1 =Pt > Pt+1 =Pt . In this case no one will rationally hold any privatemoney. The value of private money in terms of goods is going to be zero –its pricein…nite–validating the consumer’s initial belief.
The reverse result –that there is always an equilibrium without government money–does not hold. There are two reasons for this. First, is the trade in interest-bearing
securities issued in government currency. Second, is the government policy of interestrate peg and real transfer. Setting the nominal interest rate …xes the in‡ation rate
as well, while the real transfer policy pins down the price level. In this way, the valueof government money is never zero.
15

The path of consumption from period t0 + 1 on, instead, is fully determined by the path of
nominal interest rates set by the government.

13Nonetheless, there can be equilibria in which only private money is used as medium
of exchange. For this to be so the return on private money must be higher than thaton government currency, i.e. Pt+1 =Pt < Pt+1 =Pt : In this equilibrium, the “cash”
constraint simpli…es to:
Mt p1
1
Ct =
;
Pt
1+ t
for each t t0 given that only private money is used to purchase consumption good
C, where the equality follows from (6) and t is the Lagrange multiplier attached to
the “cash”constraint, with t 0. The optimality condition with respect to privatemoney holdings is1
1= (1 + t+1 )Pt
Pt+1for each t
t0 : Combining the two foregoing conditions to eliminate the Lagrange
multiplier, we get
Mt p1
Pt
1+
for each t
t0 : The above holds with equality whenever t > 0, in which case it
determines the price level in units of private money and its growth rate at Pt+1 =Pt == 1 + . Since >and = (1 + i), for this equilibrium to exist it shouldbe that (1 + i) > 1 + :The two moneys coexist and prices are determined in both, but the private currency appreciates over time at the rate = .16 For this reason, only private money
is used as a medium of exchange. Government money still maintains a role in theeconomy for two reasons: it is the unit of account for taxes and the denominationof government debt, which private agents are willing to hold. This equilibrium isfragile. Government money is losing value with respect to private money, and agents
could decide to replace it with the latter either in credit markets or in tax payments
unless some form of legal constriction is in place. The economy could become entirely dominated by the cryptocurrency, and the government currency could become
worthless.

3.3

Summing up

We can now summarize the results in the following Proposition and comment on thepolicy implications.
16

To see that prices are determined in government currency, note that real money balances inprivate currency are bounded, so an intertemporal budget constraint holds for the government.
Therefore the price level P is determined using the transfer rule (18).

14

Proposition 5 Given the monetary policy regime of De…nition 2: 1) there is always
an equilibrium in which private currency is worthless and government currency is themedium of exchange, in which case<; 2) if (1 + i) = 1 + ; in which case
= , then there is only one additional equilibrium with both currencies competingas medium of exchange; 3) if either i > 0 and (1 + i) > 1 + ; in which case > ;or i = 0 and< 0, in which case=, there are two additional equilibria,
one with both currencies competing and one with only private money as medium of
exchange.
Some important conclusions can be drawn from Proposition 5 and the foregoing
discussion. As in the benchmark one-currency model, the government has full controlof the interest rate and in‡ation, since it has taxation power and there exists a market
in interest-bearing securities denominated in government currency. These privileges
are not shared by the private currency, so that if consumers believe it to be worthless,it becomes such.
Despite these advantages, the presence of an alternative currency can limit theproperties of government money, in particular the function of medium of exchange.
In this respect, an important lesson of Proposition 5 is that the best money –the one
with higher return or lower in‡ation –prevails as medium of exchange, following theprinciple set out, among others by Hayek (1974).
It is worth repeating that government money can get into unknown and dangerous
territories if the return on it is less than on other currencies and can therefore lose
all medium-of-exchange function. It could eventually lose its other properties as welland become worthless, unless it is bolstered by legal restrictions.
Another way to see the results of Proposition 5 is to ask how government should
set policy so as to absolutely preclude equilibria in which private currency is used.
The government could …x the interest rate in the range 0 < i < ( =1); keeping
in‡ation in the interval < < (1+ ). Interest rates and in‡ation are then bounded
above by the growth rate of private money. To intuit these bounds, one should observethat in this model currency competition acts in the direction of facilitating certaintransactions. To compete and exclude other currencies, the government should o¤er abetter money, one with higher return and lower in‡ation. In line with Hayek (1974),
currency competition can improve the welfare of consumers if the growth rate ofprivate money is su¢ ciently low and the competing money correspondingly good.
However, government objectives for monetary policy are generally broader thanjust merely liquidity premia in money markets. Central banks also seek to moderatethe ‡uctuations of the economy. Along these lines, the literature o¤ers justi…cations
for a positive in‡ation rate, such as the need to avoid the zero lower bound on nominalinterest rates or to “grease the wheels”of the economy in the presence of downward

15nominal wage rigidity.17 Taking these considerations into account in the simple modelof this paper would imply that the bound on in‡ation can be too stringent for theattainment of other objectives, whenever the growth rate of private money is low.
The desired or optimal in‡ation target could be high enough to enter the regionof multiple equilibria where government money might lose its medium of exchange
function and be at survival risk.
The conclusion of this simple model – that a competing currency can limit theaction of monetary policy –is further reinforced in the next section, which analyzes
competition by pro…t-maximizing private issuers.

4

Multiple-currency competition

Now let us extend the above framework to a multiple-currency environment. Eachcurrency issuer operates in a centralized system and chooses its money supply tomaximize pro…t. Entry into the market is subject to a …xed cost, and the number ofissuers is endogenously determined by the zero-rent condition. Government currency
is supplied as in the benchmark model, with a monetary policy of a constant nominalinterest rate and an appropriate real transfer policy.
A concise summary of the results is that nominal interest rate and in‡ation rate are
now both determined by structural factors. The price level and real money balancesin each currency are also determined.
Assume that at time t0 1 there is only government money; therefore at t0 the
cash-in-advance constraint is:
Mt0 1
:
Ct0
Pt0
Starting from time t0 other issuers can enter and the liquidity constraint generalizes
tot 1
Mt 1 X Mt;t 1
+
Ct
Pt
Pt;t
t=t0

for each t t0 + 1; in which Mt;t 1 is the aggregate money supply chosen for periodt 1 by the issuers that entered the market in period t, with t0 t t 1; Pt;t isthe price of goods in terms of the currency of these issuers. The aggregate balances
Mt;t 1 also include
the money issued at t 1 by issuers that are still in the market,
Ri.e. Mt;t 1 = j2Nt;t 1 Mt;t 1 (j)dj where the mass of issuers remaining in the market
is given by Nt;t 1 = (1)t 1 t Nt;t in which Nt;t is the mass entering at time t:
Entering at time t, a generic issuer j pays a …xed cost > 0 and commits to thesequence fMt;t (j)g1
t=t so as to maximize its liquidity rent (speci…ed below) subject to17
On the …rst justi…cation, see Eggertsson and Woodford (2003); on the second, among others,
Benigno and Ricci (2011) and Dupraz et al. (2018).

16

an exit rate of > 0. In the case of exit at time t + 1, the money holdings Mt;t (j)chosen for period t still provide liquidity services in period t + 1 and only become
useless at the end of the period. Therefore, the …rst-order condition with respect to
Mt;t (j) is1
1= (1+ t+1 );
(22)
Pt;tPt;t+1since the currency has unitary payo¤ with probability 1
and zero otherwise andalways provides liquidity services, receiving the premium t+1 . The above condition
holds for any t t:Given that …rst-order condition (8) still applies, comparing it with (22), we getthat
Pt;t+1
Pt+1
=
(23)
Pt;t
Pt
which implies that the returns on the two moneys are equalized when the exogenous
exit rate is taken into account. Given the latter, the number of private currencies Jt
follows the law of motion Jt = Nt;t + (1)Jt 1 with initial condition Jt0 1 = 0.
Each issuer is subject to the ‡ow budget constraint
Mt;t 1 (j) + Tt;t (j) = Mt;t (j);starting from its entry at time t with Mt;t 1 (j)= 0: Real pro…ts at time t are givenby the liquidity premium t+1 in period t + 1 multiplied by the real money balances,i.e. t+1 Mt;t (j)=Pt;t+1 discounted by the rate of time preference . It is essential toobserve that the liquidity premium is a function of the money supply chosen by thesingle issuer j. In fact when t > 0
Ct =

1
1+

Mt 1 X Mt;t 1
+
:
Pt
Pt;t
t 1

=
t

(24)

t=t0

We model a market of monopolistic competition, in which each supplier exerts somein‡uence on the liquidity premium, but not much with respect to the overall market
or with respect to the total set of suppliers entering at the same time. The latterassumption implies that the issuer takes prices, and in particular Pt;t ; as given in
its optimization problem. Each supplier maximizes the discounted present value ofreal pro…ts by choosing the sequence fMt;t (j)g1
t=t ; conditional on its still being in themarket, and therefore it maximizes:
1X[(1) ]t t t+1 (Mt;t (j))

t=t

Mt;t
Pt;t+1(25)

starting from its entry period t where t+1 shows its dependence on the choice Mt;t (j)through (24).
In an equilibrium with competing currencies, then, the following results obtain.
17

Proposition 6 Given a …xed entry cost
> 0 and an exogenous exit probability
0 < < 1, in an equilibrium with currency competition and pro…t-maximizing issuers:(i) consumption is constant and equal to1

Ct = 1(26)z 2 > 0;for each t t0 + 1 with z (1 (1) ) = (assuming z < 1); (ii) the supply ofreal money balances of each private issuer is constant and equal to
1
Mt
= z2
Pt+1

(27)

z > 0;and (iii) the number of private suppliers of currency is given by1

Jt = z 2

1

Mtg
1
Pt+1 1 z 12

(28)

:

Proof. See Appendix A.2.
Some interesting implications derive from Proposition 6: (i) the nominal interest
rate and (ii) the in‡ation rate in each currency are constant and are functions ofstructural parameters.18 To see the …rst result, combine the constant consumption
level of equation (26) with the equilibrium condition Ct = 1=(1 + it 1 ) at each timett0 + 1: It follows that the nominal interest rate is constant and determined by
the entry cost , the exogenous exit rate and the intertemporal discount factor
through the parameter z
1(29)
1 + it =1 ;1 z2
for each t t0 . One implication of currency competition is therefore that the government loses control of the nominal interest rate.19 Moreover, to see that structural
factors also determine the in‡ation rate in each currency, combine (29) with the Euler
equation (10) to obtain
Pt+1
=
Pt
1

z

1
2

Pt+1
=
Pt
1

1

z2

;

(30)

where the second equality follows from (23).
We collect these results in the following Corollary.
18
With appropriate quali…cations a similar result could arise in a model of competition among
securities all denominated in the same currency, see Woodford (1995).
19
The real interest rate, however, is always tied to the rate of time preference. This is di¤erent
from the …nding of Fernandez-Villaverde and Sanchez (2018), namely that there is no equilibrium
in which the real interest rate is equal to the rate of time preference.

18

Corollary 7 Given a …xed entry cost > 0 and an exogenous exit probability 0 << 1, in an equilibrium with currency competition and pro…t-maximizing issuers:(i) the nominal interest rate and (ii) the in‡ation rates are determined by structural
factors:
1
1 + it =
1 ;
1 z2
Pt+1
Pt+1
=
=
:
1
1
Pt
Pt
1 z2
1 z2
Note that in the special case of zero entry cost there are an in…nite number ofissuers and the competition drives liquidity premia to zero. The e¢ cient allocation
is attained.20 Therefore the nominal interest rate is at the zero lower bound, whichis equivalent to have full satiation of liquidity. Prices in government currency fallat rate and all other prices – in private currencies – at (1): These results
are perfectly in line with Hayek (1974), who advocated private money creation way
to achieve e¢ ciency in the liquidity provision. The loss of control over the nominalinterest rate and in‡ation too is consistent with his position that government should
not have monopoly power to manipulate interest rates and prices. Rather, unfetteredcompetition can work to keep in‡ation under control.
An important further result of Proposition 6 is that equilibrium real money balances for each of private currency are uniquely determined and constant.21 They are
not a¤ected by exchange rate indeterminacy, which was instead pervasive in the twocurrency model (Section 1). As a consequence, real money balances for government
currency are also determined again departing from the two-currency model.
The key remaining question is whether there is some indeterminacy of the pricelevel, in government currency, now that interest and in‡ation rates are determined byexogenous factors. The answer is no, the argument follows the foregoing discussion
with some caveats. For one thing, we need to appropriately rede…ne the monetary
regime.
De…nition 8 Assume the following speci…cation of the monetary policy regime. The
government sets the sequence of money supply fMtg g1
t=t0 and the following transfer
policyTt
it Mtg
=
(1
)
(31)
Pt
1 + i t Pt
for each t t0 and for some di¤erent from zero:
20

Fernandez-Villaverde and Sanchez (2018) instead …nd that private money creation can be sociallywasteful.
21
This is again in contrast with Fernandez-Villaverde and Sanches (2018), who …nd multiple
equilibria, including hyperin‡ationary equilibria with real money balances that converge to zero.

19With respect to the benchmark monetary policy regime (De…nition 2), here theinterest-rate policy is replaced by a money supply policy.
Proposition 9 Given the monetary policy regime of De…nition 8 and the results of
Proposition 6, in an equilibrium under currency competition the path of the price levelin government currency is determined as is consumption at time t0 :Proof. See Appendix A.3.
This result contradicts Friedman (1960), who held that purely private …duciary
currencies would inevitably produce price instability. Fernandez-Villaverde and Sanchez(2017) reach a similar conclusion: “in a monetary system with competitive issuers the
supply of each brand becomes unbounded when the marginal cost goes to zero...Private
entrepreneurs always have an incentive to mint just a little bit more of the currency.”
In my model, what eliminates this is the possibility of entry into the market, which
drives all rents to zero. Moreover, an appropriate real transfer policy fully determines
the path of prices.
To this point the equilibrium characterized is one in which multiple currencies
coexist as medium of exchange, with striking results on what government policy isleft to do. However, along the lines of the discussion in Section 3.3, in theory thegovernment could set a higher interest rate and so achieve higher in‡ation, but in thiscase government money will not serve as medium of exchange. Or else, the government
could crowd private money out by setting the interest rate lower than (29): However,this interest rate could be very low where entry barriers are negligible, i.e. close to
zero. In line with our previous conclusions, then, a market of competitive currency
suppliers can place signi…cant limits on monetary policy action.

5

Determinacy of the exchange rate

Let us return to the two-currency model to see whether it is possible to solve theproblem of indeterminacy of the exchange rate of the cryptocurrency. I have alreadyobserved that this is not the main reason for monetary policymakers to be concernedover the existence of other currencies.
We need to explore alternative monetary regimes for private issuance to address
the indeterminacy problem. A characteristic of some cryptocurrencies is that they are
not the liability of any single agent (decentralization), while government currency is
the liability of the central bank (centralization). The rule of constant money-supply
growth for private currency (De…nition 2) can describe the behavior of a decentralized
as well as of a centralized system. On the other hand, transfer policy (18) can be morenaturally instituted by an agent that has full control of its balance sheet and that canback the value of its liability by a real transfer/tax policy. To obtain determinacy,
a one possibility is to posit that the private currency is issued by an agent in a20

centralized system that at some point can switch to an “active”transfer policy of thesame kind as (18).
De…nition 10 Assume the following speci…cation of the monetary policy regime. The
government sets a constant interest rate policy it = i at each t t0 and the following
transfer policy
Tt
i Mtg
=
(1
)
(32)
Pt
1 + i Pt
for each t t0 and for some di¤erent from zero: The private currency issuer sets
Tt = Mt p1 for each t0
t < t~ with > 1, to achieve a constant growth rate;
1 + , of its money supply and switches to a real transfer policy
i Mt pTt=
Pt
1 + i Pt
for each t

t~ and some

(1

)

(33)

> 0.

The di¤erence with respect to the monetary policy regime set in De…nition 2 is theassumption that the private issuer switches to a real transfer policy at or after time t~;which may coincide with the initial period t0 or be postponed far into the future. Thischange to the monetary regime is su¢ cient for an equilibrium to have determinacy of
all the variables, and in particular the price level (in government currency) and thenominal exchange rate. A necessary condition, in the following Proposition, is thatthe government-set nominal interest rate be positive.
Proposition 11 Given the monetary policy regime of De…nition 10; in an equilibrium with a positive nominal interest rate it = i > 0 for each t t0 , the equilibriumis determinate.
Proof. See Appendix A.4.
It is worth highlighting some of the key elements of the proof of Proposition 11.
First is the requirement of a positive interest rate. In this case the liquidity constraint(17) holds with equality, and it follows that in equilibrium real money balances arebounded and therefore limt!1 t Mt p1 =Pt = 0. This condition is su¢ cient to separate the aggregate intertemporal budget constraint into two components, one forthe government and the other for the private issuer. It then follows that given thebacking implied by the regime of De…nition 10 through transfer rules (32) and (33),the path of prices in each currency is determined and so is the exchange rate.
The proposal of De…nition 10 might seem to be a straightforward extension of the…scal theory to a two-currency setting, but there is one key di¤erence. While the …scal
theory applied to a single currency requires that the monetary authority issue at least
some securities with only pecuniary return, like B, in the two-currency extension only
21

one currency issuer should supply such securities. Indeed, supposing that the private
issuer of currency also issues interest-bearing securities with only pecuniary return,let’s say B p , the fact that limt!1 t Mt p1 =Pt = 0 in any equilibrium with positive
interest rate no longer implies the possibility of de…ning two separate intertemporalbudget constraints. Consequently prices and the exchange rate cannot be determined
separately.
Instead, the proposal of De…nition 10 could be interpreted as a combination of the…scal theory approach for the government and the proposal of Obstfeld and Rogo¤(1983) for private money, since currency is exchanged for goods (a stream of goods)
at a certain point in time. However, in my model, the purpose of this exchange isdi¤erent. In their case, it serves to rule out multiple equilibria and in particular hyperin‡ationary solutions. In mine, explosive equilibria are ruled out and the purposeof exchanging currency for goods is to determine the constant exchange rate between
the two currencies.
Note, further, that under the monetary regime proposed the path of private money
supply is determined by the private issuer’s policy, while that of the government isendogenously determined by the cash constraint (17).
Finally, note that the monetary policy regime of De…nition 10, given a positive
nominal interest rate, does not necessarily imply the existence of an equilibrium. But
if an equilibrium does exist under that regime, it features determinate prices andexchange rate.

6

Conclusion

This paper analyzes models of coexistence between government and privately-issued
currencies. Competing currencies can limit the ability of the central bank to use
the interest rate as a policy instrument; they can also restrict the attainable equilibrium in‡ation rate. In a market with free entry for multiple currencies with pro…tmaximizing issuers, the central bank completely loses control of the interest rate andthe in‡ation rate, which both come to be determined by structural factors (rate oftime preference, entry cost, exit rate).
I have kept the analysis as simple as possible in order to focus on this important
topic, which has recently received a good deal of attention. In particular, private
and government currencies are assumed to be perfect substitutes, delivering the sameliquidity services. However, an extension to imperfect substitutability does not alter
the results signi…cantly. Most interesting would be to devise a model in which theacceptance or non-acceptance of currencies is endogenous not only for the medium of
exchange function but also for other properties of money – a task I leave to future
research. Another interesting avenue would be extension to a multi-country world, as
a way of studying competition among international reserve currencies and national
22

currencies.

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26

A

Appendix

A.1

Proof of Proposition 4.

Proof. Consider …rst the case in which i > 0: Lemma 3 implies that in an equilibrium(1 + i) > 1 (1 + ). Since 0 < < 1; then i > . When instead i = 0, Lemma 3
also implies that i > : Therefore in any equilibrium i > . Consider the ‡ow budget
constraint (15) and iterate it forward to obtain1
SMt0p 1 X
+
Pt0
t=t
0

t t0 STt

=

Pt

1
X

t t0

t=t0

it SMt p+ lim
T !1
1 + i t Pt

T t0 SMT

p

PT +1

:

(1)

Note that we can write
lim

T !1

T t0 SMT

p

PT +1

=

SMt0p 1
Pt0

1+
1+i

T +1 t0=0

using in the …rst equality PT +1 = [(1 + i) ]T +1 t0 Pt0 and MT p = (1 + )T +1 t0 Mt0p 1 .
The second equality follows since i > in any equilibrium. Moreover, under thisinequality the two sums in (1) are also …nite. We can therefore write
1
SMt0p 1 X
+
Pt0
t=t

t t0 STt

Pt

0

1
i X
=
1 + i t=t

t t0 SMt

Pt

0

p

(2)

:

Substituting (2) into (11) we obtain:1
Mtg0 1 + Btg0 1 X
+
Pt0
t=t
0

1
i X
=
Pt
1 + i t=t

t t0 Ttgt t0 Mt

0

Pt

:

(3)

Plugging in the active monetary policy (18), (3) simpli…es toMtg0 1 + Btg0 1
= ;
Pt0

(4)

which determines the price level Pt0 . Note that in the case Mtg0 1 + Btg0 1 > 0, i.e.
when government is a net debtor, then it should be that > 0. Conversely, when
Mtg0 1 + Btg0 1 < 0 it should be that < 0. Given a constant interest rate policy i, it
follows from (10) that the entire path of prices fPt g1
t=t0 is determined; further, using(16) the path of consumption starting from period t0 + 1 is also determined.
Consider …rst the case i > 0; this implies
Mtg
M p+S t =
Pt
Pt
27

for each t

t0 and in particular at time t0 + 1 in which case
Mt0p 1 (1 + )
Mtg0
+S
= :
Pt0
Pt0

(5)

Moreover, substituting (18) into the government budget constraint we get
Mtg0 1 + Btg0 1 = (1

)Pt0

Mtg0
Btg0
+
+
:
1+i 1+i(6)

Since Pt0 is determined in (4), the two restrictions (5) and (6) are insu¢ cient todetermine the three variables S, Btg0 and Mtg0 : Note also that the liquidity constraint
at time t0 cannot determine the variables of interest. Indeed,
Mtg0 1
Mt0p 1
+S
Pt0
Pt0

Ct0

cannot disentangle S and Ct0 :Consider now the case in which the government sets i = 0 provided
Equation (6) also applies when i = 0; but now (5) is replaced by

(7)

< 0:

Mt0p 1 (1 + )
Mtg0
+S
>1
Pt0
Pt0
which cannot determine S and Mtg0 . Note instead that (7) still applies, so that Ct0 too
is not determined. To see the result of indeterminacy from a di¤erent angle, suppose
instead that S is determined, so that (5) determines Mtg0 ; (6) determines Btg0 ; andso forth using the foregoing two equations in the subsequent periods. Further, given
initial conditions Mtg0 1 and Mt0p 1 ; the values of S and Pt0 determine the amount ofinitial real money balances. If the left-hand side of (7) is greater than the unitary
value, then Ct0 = 1 and t0 = 0. If it is less, then Ct0 is constrained by this value and11.
t0 = Ct0

A.2

Proof of Proposition 6

Proof. Consider the optimization problem of a generic supplier of private currency.
Each supplier internalizes that it has some power on the rents given by t . If liquidity
rents are zero, i.e. t = 0, the supply of currency will be in…nitely elastic. But if
t > 0; rents are inversely related to the overall supply of liquidity, as shown by thefollowing equation
t 1
1
Mt 1 X Mt;t 1
Ct =
=
+
:
(8)
1+ t
Pt
Pt;t
t=t0

28

Inserting the above constraint into the objective function (25) and deriving it withrespect to the sequence fMt;t (j)g1
t=t taking prices and others’money supply as given,
one gets
Mt;t (j)Ct+1 (1 Ct+1 ) =;(9)
Pt;t+1for each t t in which t is the entry time of a generic issuer j of private currency.
Therefore, the above equality holds for every t
t0 : Since the left-hand side isindependent of j, so too is the right-hand side, and the equilibrium is therefore
symmetric. De…ning Mt;t (j)=Pt;t+1 = Mt =Pt+1 for each j and t; we can write (9) as
Ct+1 =Mt
M+ Jt+1 t
Pt+1
Pt+1

Mt
Pt+1

=1

M
Mt
+ Jt+1 P t
Pt+1
t+1

;

(10)

which holds for every t t0 and where Jt+1 is the overall mass of suppliers active attime t+!: Free entry proceeds until there are no rents left in the market, i.e. at eachpoint in time the discounted present value of the pro…t of each issuer entering themarket is equal to the …xed cost1X) ]t t t+1

[(1

t=t

for every t

Mt;t (j)= ;Pt;t+1(11)

t0 . We can further write (11), using (8) and (9), as1
X[(1) ]t t (1Ct+1 )2 =

:

(12)

t=t

Since (12) applies at each t

t0 ; this implies that1

Ct = 1for each t

(13)

z2;

t0 + 1. Moreover, using (9) and (13), we get
1
Mt
= z2
Pt+1

z;

for each t
t0 . Combining this with (10) and (13), the number of supplier in themarket, at each point in time, is given by1

Jt = z 2

1

Mt
1
Pt+1 1 z 12

29

:

A.3

Proof of Proposition 9

Proof. To determine the price level in government currency, consider …rst the consumer’s ‡ow budget constraint, which in this more general case can be written as
Bt
Mt
M
+
+ J t t + Xt
Pt (1 + it )
Pt
Pt

Bt 1
Mt 1
+
+ (1
Pt
Pt
Tt
T
+Y +
+ Jt t :
Pt
Pt

)Jt 1

Mt 1
Pt

Ct

+

The consumer’s intertemporal budget constraint together with goods market equilibrium implies the following aggregate resource constraint:
1
Mtg0 1 + Btg0 1 X
+
Pt0
t=t

t t0

0

Tt
T
+ Jt tPt
Pt

Using the law of motion Jt = (1

=

t t0

t=t0

t=t0

t t0 Tt

Pt

0

Mtg

=

it MtgMt+ Jt
1 + it Pt
Pt+1

t t0

t+1

:

)Jt 1 + Nt;t , this can be written as1
Mtg0 1 + Btg0 1 X
+
Pt0
t=t

1
X

1
X

it
+
1 + it Pt

1
X

+

1
X

t t0

Nt;t

t=t0

t t0

Nt;t

t=t0

1
X

1
X

) ]t t

[(1

t=t

[(1

) ]t t

t+1

t=t

Tt
Pt

Mt
:
Pt+1

(14)

Consider now the ‡ow budget constraint of a generic private currency issuer, expressedin real terms,
Mt 1 TtM
M+= t = (1+ t+1 ) t 1
Pt
Pt
Pt
Pt
in which in order to obtain the second equality we have used (22). The above budget
constraint can be reiterated forward starting from the entry period t to obtain
1
X
[(1
t=t

t t Tt

) ]

Pt

=

1
X

[(1

) ]t t

t=t

Mt
Pt+1

t+1

(15)

using the fact that Mt =Pt+1 is bounded in all periods given (27) and therefore
limt!1 (1)t t Mt =Pt+1 = 0. Note, further, that the right-hand side is also …nite
and equal to
given the zero-rent condition. Pro…t maximization under currencycompetition and free market entry therefore implies that an intertemporal budget
constraint holds for each private issuer. We can then plug (15) into (14) to obtain1
Mtg0 1 + Btg0 1 X
+
Pt0
t=t
0

t t0 Tt

Pt

=

30

1
X
t=t0

t t0

it Mtg;
1 + it Pt

which reduces toMtg0 1 + Btg0 1= ;Pt0

having used (31). Therefore the price level at time t0 is determined by the above
equation. Given the cash constraint Ct0
Mtg0 1 =Pt0 consumption at time t0 is
determined by Ct0 = min(1; Mtg0 1 =Pt0 ). Given that the interest rate is constant at(29), the path of the price level is determined by Pt+1 = (1 + i)Pt for each t t0 .
Note from Proposition 6 that since the government monetary authority sets fMtg g1
t=t0
and since fPt g1
isdetermined;itfollows
that
J
too
is
determined.
Moreover,
using
t
t=t0
the ‡ow budget constraint of the government and the real transfer policy (31), we
obtainMtg 1 + Btg 1
Mtg + Btg
= (1
) +
;
Pt
Pt+1
which determines the sequence fBtg g1
t=t0 given that all other variables are determined.

A.4

Proof of Proposition 11

Proof. When the nominal interest rate is positive, (17) holds with equality and
Mtg 1 Mt p1
1
+
=
Pt
Pt
1+i
holds in equilibrium, which implies that the real money balances are non-negative andbounded above at every point in time. Therefore in equilibrium limt!1 t Mt p1 =Pt =0; which implies limt!1 t (Mtg 1 + Btg 1 )=Pt given the transversality condition of theconsumer’s problem.22 Using limt!1 t (Mtg 1 + Btg 1 )=Pt , the ‡ow budget constraintof the government (14) and Pt+1 = Pt (1 + i); we obtain1
Mtg0 1 + Btg0 1 X
+
Pt0
t=t
0

1
i X
=
Pt
1 + i t=t

t t0 Tt0g
t t0 Mt

Pt;

in which (32) can be used to get
Mtg0 1 + Btg0 1=
Pt0
and to determine the price level Pt0 at time t0 . Using Pt+1 = Pt (1 + i) startingtfrom t = t0 ; one can determine the sequence fPt g1
Mt p1 =Pt = 0;
t=t0 . Using limt!1
22

When i = 0, it is not necessarily the case that limt!1

31t

Mt p1 =Pt = 0:

Pt+1 = Pt (1 + i) and (15) starting from period t = t~, we obtainMt~ p1
Pt~

+

1
X

i X
=
Pt
1+i ~
1

t t~ Ttt=t~t t~ Mt

t=t

p

Pt

into which (33) and Mt~ p1 = (1 + )t~ t0 Mt0p 1 can be substituted to obtain(1 + )t~ t0 Mt0p 1
=
Pt~

(16)

;

which determines Pt~ : Using (4) the entire sequence fPt g1
t=t0 can be determined, andtherefore also the nominal exchange rate S from S = Pt =Pt . Using Pt+1 = Pt (1 + i),(15) for time t t~ and (33) we obtain
Mt p1
= (1
Pt

)

+

Mt p
;
Pt+1

(17)

which can determine the path of private money supply from period t~ onward, whereas
in previous periods the path of private money issuance is determined by the transfer
rule Tt = Mt p1 for each t0
t < t~ given initial condition Mt0p 1 . Consumption isdetermined by (16) for each t t0 + 1; whereas at time t0 it is determined by
Ct0 = minMtg0 1 Mt0p 1
+
;1 :
Pt0
Pt0

The sequence of government money issuance is determined by
Mt p
1
Mtg
+
=
Pt+1 Pt+1
1+i
for each t
t0 given the path of all the other variables. Using Pt+1 = Pt (1 + i),
equation (14) for time t t0 and the policy rule (32) we obtain
Mtg 1 + Btg 1
= (1
Pt

) +

Mtg + Btg
Pt+1;

which determines the sequence {Btg g1
t=t0 given that all the other variables are alreadydetermined. Note that the existence of an equilibrium imposes restrictions on theparameter and the time t~. Indeed, it should be the case that1+
(1 + i)t~+1 t0

Mt0p 1
Pt0

given the value Pt0 determined in equilibrium.
32

11+i