in a World of ∗

Pierpaolo Benigno University of Bern March 17, 2021

Abstract Can competition affect central banks’control of interest rates and prices? Yes, it can. In a two-currency world with competing cash (material or digital), the growth rate of the sets an upper bound on the nominal interest rate and the attainable inflation rate, if the government cur- rency is to retain its role as . In any case, the government has full control of the inflation rate. With an interest-bearing , equilibria in which government currency loses medium-of-exchange property are ruled out. This benefit comes at the cost of relinquishing control over the inflation rate.

∗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. In recent years cryptocurrencies have attracted the attention of consumers, media and policymakers.1 Cryptocurrencies are digital , not physically minted. Monetary history offers other examples of uncoined . For centuries, since Charlemagne, an “imaginary” money existed but served only as and never as, unlike today’s cryptocurrencies, medium of exchange.2 Nor is the coexistence of multiple currencies within the borders of the same nation a recent phe- nomenon. Medieval Europe was characterized by the presence of multiple media of exchange of different metallic content.3 More recently, some nations contended with dollarization 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 fiat currency just as central banks have succeeded in controlling the value of their currencies and taming inflation. In this perspective, this article asks whether the presence of multiple currencies can jeopardize the primary function of central banking —controlling prices and inflation —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 through cash, either physical or digital. For the benchmark single-currency model, the results are established: the can control the rate of inflation by setting the nominal interest rate; the (initial) price level instead is determined by an appropriate real tax policy.5 The combination of these two policies (interest-rate targeting and fiscal policy) determines the path of the price level in all periods, and the central bank can achieve any desired inflation rate by setting the right nominal interest rate. I add to this benchmark a privately issued currency that is perfectly substitutable for the government’scurrency in providing liquidity services. The private currency is “minted”each period according to a constant growth rate µ. The first 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 from the 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

1 See BIS (2018). 2 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 mul- tiple currencies. 4 See Calvo and Vegh (1997) for an analysis of dollarization. 5 Canzoneri, Cumby and Diba (2001), Niepelt (2004), Sims (1994, 2000, 2013), Woodford (1995, 2001, 2003) and Bassetto and Sargent (2020) among others, present results of the benchmark single- currency case.

1 in it. The second result is that the perfect control of prices that the government has in the single-currency framework extends unchanged to a multiple-currency model. When liquidity services are provided by cash, currency competition does not bring any trouble to the government in controlling inflation and prices. The final result highlights a failure of the price mechanism in not conveying the “correct” information on which is the best money. This failure leads to multiple equilibria. To understand this result, consider the private issuer setting a constant growth rate µ, and the government a gross inflation rate Π through an interest rate policy. If 1 + µ < Π, there is an equilibria in which only private money is used as a medium of exchange with a lower inflation rate than government currency, Π∗ < Π. However, there is also another equilibrium, with lower welfare, in which both currencies provide liquidity services and have the same inflation rate, Π = Π∗. In this equilibrium, private money becomes less and less important over time in purchasing consumption goods. Gresham’slaw works in a way that the “bad”money crowds out the “good”money. Although the latter equilibrium is dominated in welfare, the price mechanism does not allow agents to rule out the inferior equilibrium and choose the “best”money. Their choice is only governed by the relative return between the two moneys, which is related to the inflation rates. If households expect the same return, i.e. Π = Π∗, they will hold both moneys irrespective of whether one is “good” or “bad”. The results described above can be helpful to shed light on an important debate between Friedrich von Hayek and on the desirability of currency competition. Milton Friedman’s view is that a purely private system of fiduciary currencies would inevitably lead to price instability (Friedman, 1960). In my model, currency competition does not create any trouble at all for the government to control the price level. Friedrich von Hayek’sview is instead that unfettered competition in the currency market is beneficial for society (Hayek, 1976). My model contrasts also this view showing that the relative-price mechanism that disciplines agents’choice on which currency to hold does not convey the “correct”information on which is the best money. Therefore, currency competition does not necessarily improve welfare, although it will never worsen it. On the other side, a government that risks to lose the property of medium of exchange for its currency could decide to crowd out private currency to avoid getting into such dangerous territories. In this case, it has to keep inflation, Π, low and bounded by the growth rate of private money, i.e. Π 1+µ. In this way, currency competition could thus become a useful way of keeping≤ inflation low and reducing liquidity premia. However, the inflation bound imposed on the government can be too tight when other objectives, such as economic stabilization, require a higher inflation target. Next I discuss how results change in an environment in which households can hold deposits at the central bank, which together with other default-free government

2 securities provide liquidity services in competition with an unbacked digital private money. This framework is in line with recent proposals that advocate for the creation of a central-bank digital currency by allowing the public to hold accounts at the central bank. Results change in a considerable way. First, there is never an equilibrium in which only the private money is used as a medium of exchange rate. With a central bank digital currency, the government can rule out equilibria in which it loses medium-of-exchange properties for its currency. The second result is that this benefit comes at the cost of reliquinshing the control over the inflation rate. Unless the government is able to back the supply of interest- bearing liabilities with enough taxes, multiple equilibria still arise depending on the undetermined “realization”of the exchange rate between the two currencies, and the policy followed by the private issuer. The path of the inflation rate is now depending on the “realization” of the exchange rate, in contrast with the full control that the government had in the case in which cash was the only medium of exchange. A final result is that, even in this framework, currency competition never worsens welfare. This paper is related to the seminal work of Kareken and Wallace (1981) with two important differences. In their work money is only a , while here it is also a medium of exchange to buy specific goods. Moreover, government sets policy in terms of interest rate and tax, whereas in their model monetary policies are specified in terms of money growth. These differences explain why the multiplicity of equilibria found in my work is not present in their analysis. First, note that both frameworks share a sort of Gresham’s law for which the “good” money, with lower growth rate, is crowded out by the “bad”money, in the case both moneys coexist as medium of exchange. However, in their model, the exchange rate can be any constant value, or zero, or infinite. In my model, instead, the exchange rate can also appreciate over time to support an equilibrium in which the private “good”money is the only used as a medium of exchange while being at the same time store of value together with the government currency. Therefore, there is an equilibrium in which the “good” and “bad” money are both used as medium of exchange and store of value, with a constant exchange rate between them and with the “good”money vanishing in real terms. There is also an equilibrium in which the “good”money is the only one used as a medium of exchange, with the “bad”money remaining a store of value. In the latter equilibrium, the “good”money appreciates over time. A recent literature prompted by the increasing number of cryptocurrencies has re- vived interest in multiple-currency monetary models, which share with my framework a role for money both as store of value and medium of exchange. Fernandez-Villaverde and Sanches (2019) evaluate the role of competing private currencies whose supply is determined by profit maximization.6 Their results differ substantially from mine.

6 Klein (1974) is an early example of a model of currency competition with profit-maximizing

3 They find that an appropriately defined price stability equilibrium can arise only un- der certain restrictions on the cost function for private money production. But, when the marginal cost goes to zero, price stability cannot be an equilibrium. Further they find that competition does not achieve the effi cient allocation and is socially wasteful. Schilling and Uhlig (2019) also analyze coexistence and competition between tra- ditional fiat money and cryptocurrencies. But, they are more concerned with the determination of the price of cryptocurrencies, deriving interesting bounds and asset- price relationships. On policy, they assume that the government has always full control of the inflation rate. With respect to monetary policy, they emphasize the connection between the indeterminacy of the cryptocurrency, prices and government . In my analysis, exchange-rate indeterminacy, if present, is not par- ticularly relevant to the way in which competing currencies affect the findings of traditional monetary policy analysis.7 Marimon et al. (2012) find that currency competition can achieve effi ciency relying on a seigniorage function which is maximized at a deflation rate equal to the rate of time preferences. Kovbasyuk (2018), instead, finds that private currency can tend to inflate prices and harm agents who hold fiat currency. Whereas all the above mentioned works share some similarities in the way of modelling currency competition when cash is the only medium of exchange, none has considered an environment in which government’sinterest-bearing liabilities can compete with an unbacked private money, to provide liquidity services. Woodford (1995) is the benchmark reference for the analysis of price determina- tion through interest-rate targeting and fiscal rules in a single-currency economy. He also studies a “”regime in which deposits, in units of the single currency, compete with government money in providing liquidity services. In this case, he finds that the nominal interest rate and the inflation rate are determined by structural fac- tors. However, he does not analyze multiple-currency models. Similarly, Marimon et al. (2003) find that also inside money puts downward pressure on inflation. However, the reason for limitations to policy, in this literature, originates from the way private securities are backed rather than from direct competition for medium-of-exchange properties coming from unbacked private currency, as in my framework. The paper is structured as follows. Section 1 presents the two-currency model. Section 2 discusses the equilibrium conditions and Section 3 characterizes it. Section 4 discusses the extension of the model by allowing government securities to provide liquidity services. Section 5 concludes. suppliers. 7 In one-currency models, Sargent and Wallace (1982) and Smith (1988) have studied the coex- istence, and competition, between government money and private assets and the need to separate the credit and money markets to avoid fluctuations in the price level unrelated to fundamentals.

4 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 (2019) 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: ∞ t t0 β − U(Ct) + Zt , (1) t=t { } X0 where β is the intertemporal discount factor with 0 < β < 1; the function U( ) is an · increasing and concave function of its argument, moreover Uc(0) where Uc( ) is the derivative of the function U( ) with respect to its argument; C→is ∞ the “cash”and· Z the “credit”good. C can be purchased· for money

Mt 1 Mt∗ 1 Ct − + − (2) ≤ Pt Pt∗ in which Mt is the government-issued and Mt∗ the privately issued money. In this model, I label money what has the property of being medium of exchange. It can be 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 cur- rency, Pt∗ is the price in the private currency. At time t, as standard in models of this type, the household splits into a shopper and a seller. The shopper brings the money acquired in period t 1 to purchase goods while the seller brings the constant endowment, Y . − The cash constraint (2) assumes that the two types of cash are perfect substitutes for the purchase of the consumption good C when they are both used. This assumption simplifies the analysis at little cost in generality. At least, it enables the model to challenge the results that derive from the single-currency framework to a greater extent. The reader should note that the degree of substitution between the two moneys is in this model endogenous depending on the relative price between the two currencies. Indeed, I will discuss equilibria in which one of the two currencies is not used as medium of exchange. After the goods market, seller and shopper meet in the household bringing, on the one side, the revenues from selling the goods and, on the other side, the remaining moneys. The consumption good Zt is subject to the budget constraint

Bt Mt Mt∗ Bt 1 Mt 1 Mt∗ 1 Tt Tt∗ + + + Zt − + − + − Ct + Y + + , (3) P (1 + i ) P P ≤ P P P − P P t t t t∗ t  t t∗  t t∗ in which B are default-free interest-bearing securities in units of the government currency; Tt are government transfers in units of government currency and Tt∗ are the private issuer’stransfers in units of its currency.

5 In writing the budget constraint (3), we make two important assumptions: first, interest-bearing securities provide no liquidity services; second, they are only de- nominated in the government currency. The significance of the latter assumption is discussed later. At this stage, I am writing the budget constraint (3) in a com- pact way with Bt that denotes the overall net asset position of households with re- spect to default-free interest-bearing securities. These securities can be issued by the households itself and/or by the government, treasury and/or central bank. However, absence of arbitrage opportunities implies that all non-defaulted interest-bearing se- curities carry the same interest rate since they all deliver the same payoff. Bt can be positive, in which case it indicates a net asset position for the household, or negative, in which case it is net debt. Debt, when issued, is paid back with certainty, being subject to an appropriate borrowing limit

∞ TT TT∗ t Qt,T Y + + > (4) W ≥ − PT P −∞ T =t T∗ X   for each t t0 given the discount factor Qt,T with Qt,t 1 and ≥ ≡ P T 1 Q = T , t,T P 1 + i t j=1 t+j Y for T > t. In (4), the real wealth at time t is defined as

Bt 1 Mt 1 Mt∗ 1 t = − + − + − . W Pt Pt Pt∗ The natural borrowing limit (4) is the maximum amount of net debt that the consumer can carry in a certain period of time and repay with certainty, i.e. with current and future net income and assuming that future consumption and asset hold- ings are going to be equal to zero. The finite borrowing limit is a requirement for consumption to be bounded in the optimization problem. The consumer chooses sequences Ct,Zt,Bt,Mt,M ∗ ∞ with Ct,Zt,Mt,M ∗ 0 { t }t=t0 t ≥ to maximize (1) under the constraints (2), (3) and (4) for each t t0, given initial ≥ conditions Bt0 1,Mt0 1, Mt∗0 1. We now turn− to− the optimality− conditions. Given the linearity of utility with respect to the credit good, the Lagrange multiplier associated with the constraint (3) has a unitary value. The first-order condition with respect to the consumption of the cash good is therefore Uc(Ct) = 1 + λt (5) for each t t0 in which λt 0 is the Lagrange multipliers attached to the con- straint (2).≥ The optimality condition≥ with respect to holdings of the interest-bearing

6 securities Bt is 1 1 = β(1 + it) (6) Pt Pt+1 at each time t t0 while those with respect to Mt and M ∗ are: ≥ t 1 1 + λt+1 = β + ξt (7) Pt Pt+1 and 1 1 + λt+1 = β + ξt∗, (8) Pt∗ Pt∗+1 respectively, at each time t t0, in which ξ and ξ∗ are the non-negative Lagrange ≥ t t multipliers associated with the constraints Mt 0 and M ∗ 0, respectively. The ≥ t ≥ Kuhn-Tucker conditions are ξtMt = 0 and ξt∗Mt∗ = 0. We can derive already some implications by inspecting the above first-order conditions. First at least one of the Lagrange multipliers ξt and ξt∗ should be zero at each point in time, since the marginal utility of consumption good C is assumed to be infinite at zero consumption. By inspecting (7) and (8), both currencies are positively demanded at a generic time t if ξt = ξt∗ = 0 which implies that P P t+1 = t∗+1 . (9) Pt Pt∗

Therefore the exchange rate St —the price of the privately-issued currency in govern- ment currency (i.e. St Pt/Pt∗) —is constant between period t and t + 1 to a value that has to be determined.≡ The equivalence of money returns shown in (9) follows from the assumption of perfect substitutability between the two moneys in the cash constraint, when they are both used.

Government money is the only demanded if and only if ξt = 0 and ξt∗ > 0, in which case the inflation in private currency is higher than that on government currency P P t∗+1 > t+1 . (10) Pt∗ Pt Viceversa when only private money is used P P t∗+1 < t+1 . (11) Pt∗ Pt The reader should note at this point that what matters for the household in choosing one money rather than the other is only their exchange-rate variation across time. This observation is going to be critical to explain some results.

Combining (6) and (7) when government money is used, i.e. ξt = 0, then λt+1 = it for each t t0 + 1 and since λt+1 0 it also follows that the nominal interest cannot ≥ ≥ 7 be negative, i.e. it 0. When, instead, only private money is used, i.e. ξ∗ = 0, then ≥ t λt+1 0 implies in (8) that P ∗ /P ∗ β and since in this case Pt+1/Pt > P ∗ /P ∗ it ≥ t+1 t ≥ t+1 t follows that it > 0 by using (6). Finally, the intertemporal budget constraint holds with equality,

∞ ∞ t t0 Tt Tt∗ t t0 it Mt 1 St+1 t0 1+ β − Y + + = β − Ct + Zt + + Mt∗ . W − Pt Pt∗ 1 + it Pt Pt∗ − (1 + it)Pt t=t0   t=t0     X X (12) Let us now specify the budget constraint of the two currency issuers. The government is subject to the following budget constraint g g Bt g g Mt + = Tt + Mt 1 + Bt 1 (13) 1 + it − − g g where Mt is the supply of cash, and Bt is the debt position of the government with g T respect to default-free interest-bearing securities. Government debt, Bt = Bt + Xt, T includes treasury’sdebt, Bt , and central bank’sreserves, Xt. It is important to specify that the central bank can also issue interest-bearing reserves since I am going to set monetary policy for the central bank in terms of the interest rate on reserves. In the baseline case, indeed, I assume that the government sets the path of interest rates and transfers, i.e. the sequences i ,T ∞ and let the household decides how much to t t t=t0 hold of money and bonds.8 It{ is important} to note that, differently from households, there is no additional requirement to impose on the government to ensure its solvency. g Indeed, Xt and Mt , being central bank’s liabilities are default-free by definition. A claim at the central bank is always paid since the central bank can “print” its own currency. Moreover, I am assuming that this default-free property also extends to the treasury through an implicit backing by the central bank.9 The private issuer is instead subject to the following constraint p p Mt∗ = Tt∗ + Mt∗ 1, − which can be interpreted either as the budget constraint of an agent issuing money in a centralized system or simply as an identity regulating how private money is created in a decentralized system. The latter interpretation is akin to the way in which some cryptocurrencies, such as , are created nowadays. The private-currency issuer sets the transfer policy at each point in time, T ∞ . Note, again, that there is no t∗ t=t0 need to impose any solvency condition. The private{ } issuer —in principle —can print its money without any boundary. Equilibrium in the market for the interest-bearing securities denominated in gov- ernment currency requires that g Bt = Bt ,

8 g T Since Bt = Bt + Xt, the central bank also needs to specify the path of reserves. 9 See Benigno (2020a,b) and Woodford (2000).

8 while equilibrium in the cash market for the two currencies implies that supply and demand equalize for each currency

g Mt = Mt ,

p Mt∗ = Mt∗ . Equilibrium in the goods market implies that the sum of the consumption of the two goods is equal to the constant endowment

Ct + Zt = Y.

Using equilibrium in the goods and asset markets, the intertemporal budget con- straint of the household can be written as

g g p M + B M ∗ ∞ ∞ t0 1 t0 1 t0 1 t t0 Tt Tt∗ t t0 it Mt 1 St+1 − − + − + β − + = β − + Mt∗ . Pt0 Pt∗ Pt Pt∗ 1 + it Pt Pt∗ − (1 + it)Pt 0 t=t0   t=t0     X X (14) The mirror image of the intertemporal budget constraint of the consumer is an ag- gregate intertemporal budget constraint consolidating both currency issuers. In (14), the initial value of the real liabilities of both agents plus the present discounted value of real transfers should be equal to the seigniorage revenues accruing to both suppliers from issuance of non-interest bearing liabilities that have non-pecuniary benefits for the consumer. We conclude the presentation of the model by specifying policies for the two currency issuers.

Definition 1 The government sets a constant interest rate policy it = i at each t t0, with i 0, and the following transfer policy ≥ ≥ T i M g t = t (1 β)τ (15) Pt 1 + i Pt − −

p for each t t0 and for some τ positive. The private currency issuer sets Tt∗ = µMt∗ 1 ≥ − at each t t0 with µ > 1, to achieve a constant growth rate, 1 + µ, of its money supply. ≥ −

The above definition underlines some important differences in the way the two monetary policies are modelled. Key is that the government can issue cash and interest-bearing securities, which include central bank’sreserves, whereas the private issuer only supplies (digital) cash. For this reason, the government can set its policy in terms of the interest rate while the private issuer can only discipline the growth rate of its money supply. Moreover, the treasury can run a real tax policy, which is

9 going to be critical for determining the price level in government currency. The reader should note that the reason for why the treasury can be stubborn in a real tax policy is because the central bank is guaranteeing treasury’s solvency at any equilibrium prices. Before turning to the characterization of equilibria, I discuss what this model can or cannot capture in terms of currency competition and its practical relevance. First, this is a closed-economy model with two currencies: one is a government currency, which is backed by taxes, and the other is a private currency, completely unbacked. The framework is therefore different from the old literature on currency competition in an international context, see Calvo and Vegh (1996) for a comprehensive survey. That literature analyzed competition between different government currencies. The key difference between that framework and mine is in who gets transfers of central bank’s profits and in who is, eventually, taxed to back the currency. Therefore the private currency in this model cannot be interpreted as a foreign government cur- rency, let’ssay dollars, that competes with a , because profits and taxes accrue the foreign economy and not the domestic one, as in the current framework. Can it be thought as a cryptocurrency? The process of creation of cryptocurrencies is quite convoluted, but the simplification that their growth rate follows some de- termined path is not far from reality. Here, it is a constant one. There are some noteworthy settlement costs of cryptocurrencies, which we abstract from. The envi- ronment described here is then suitable to characterize a limiting case in which all these costs are negligible.

2 Equilibrium

I first write down in a synthetic way all the relevant equilibrium conditions to de- termine the endogenous variables of interest. Prices in government currency should satisfy Pt+1 = ΠPt (16) in which Π is the inflation rate in government currency determined by the government through the interest-rate targeting policy it = i in equation (6) at Π = β(1 + i). In equilibrium consumption of good C is bounded by the sum of real money balances in the two currencies g p Mt 1 Mt∗ 1 − + − Ct. (17) Pt Pt∗ ≥

The liquidity constraint holds with strict equality when i > 0 and Pt∗/Pt∗ 1 > β. Therefore the complementary slackness condition holds −

g p Mt 1 Mt∗ 1 Pt∗ − + − Ct min i, β = 0, (18) Pt Pt∗ − · Pt∗ 1 −    −  10 for each t t0 + 1. The value of government money should not be lower than its marginal benefits≥ 1 U (C ) β c t+1 (19) Pt ≥ Pt+1 with the complementary slackness condition is given by

1 U (C ) β c t+1 M g = 0. (20) P − P t  t t+1  Similarly for private currency

1 U (C ) β c t+1 (21) Pt∗ ≥ Pt∗+1

1 Uc(Ct+1) p β M ∗ = 0. (22) P − P t  t∗ t∗+1  The following intertemporal restriction is derived from (14) having used (15), it = i, p and Tt∗ = µMt∗ 1. − g g p p M + B M ∗ ∞ M ∗ ∞ t0 1 t0 1 t0 1 t t0 t 1 t t0 1 St+1 p − − + − +µ β − − = τ + β − Mt∗ . Pt0 Pt∗ Pt∗ Pt∗ − (1 + it)Pt 0 t=t0   t=t0   X X (23) Finally, due to the finiteness of the borrowing limit in (4), in equilibrium the following sum should be finite ∞ t t0 Tt Tt∗ β − Y + + < P P t=t t t∗ ∞ X0   for each t t0 requiring ≥ g p ∞ M ∗ t t0 i Mt t 1 β − + µ − < , (24) 1 + i P P t=t t t∗ ∞ X0   having substituted in the fiscal rule (15) and the private-issuer transfer policy. g p The non-negative sequences Pt,P ∗,M ,M ∗ ,Ct,St ∞ satisfy (16)—(17) and (19)— { t t t }t=t0 (22) at each t t0, equation (18) at each t t0 + 1 and equations (23) and (24), ≥ ≥ p p given the definition St = Pt/Pt∗, the private-money supply rule Mt∗ = (1 + µ)Mt∗ 1, g − the additional boundary conditions Pt∗+1/Pt∗ β, i 0 and initial conditions, Mt0 1, g p 10 ≥ ≥ − ∗ Bt0 1 and Mt0 1. − − 10 Note that there are overall thirteen conditions but six restrictions in the six endogenous variables, since there are three slackness conditions, an intertemporal condition and three boundary conditions.

11 3 Equilibrium characterization

Before analyzing the consequences of a multiple-currency environment, let me first 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: constant nominal interest rate and appropriate real transfers (15). First, by setting the nominal interest rate at a target i it can also set the gross inflation rate, Πt, to any desired constant number given by Π = β(1 + i) for each t t0, which is a function of the interest rate chosen. Second, the initial price level and≥ the entire price path are fully determined by the fiscal policy rule (15), as it can be seen from equation (23) setting p Mt∗ = 0 at all times g g Mt0 1 + Bt0 1 − − = τ, (25) Pt0 g g therefore Pt0 is determined by τ given initial conditions Mt0 1, Bt0 1. The central bank thus has full control of the price level and its rise. Moreover,− given− interest-rate targeting, the money supply path is endogenously determined by the cash-in-advance constraint (17), when i > 0, g Mt 1 = Uc− (1 + i), Pt+1 for each t t0. Note that rate of money growth and of inflation are the same. Finally ≥ consumption at time t0 is determined by g Mt0 1 Ct0 = − Pt0 g given the initial condition Mt0 1 and the value of Pt0 determined in (25). The following analysis is divided− into three parts. The first characterizes the equilibrium in which the two currencies compete as medium of exchange; the second 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.

3.1 Equilibrium with competing currencies In a world of competing currencies, the exchange rate should be constant, let’s say S, as condition (9) shows. The inflation rate in government currency is fully under control of the government by setting a constant nominal interest rate policy. Therefore Π = β(1 + i). Assume first that i > 0, then the liquidity constraint holds with equality and combining it with equation (19) with equality, and using Π = β(1 + i), we can get

g p Mt 1 SMt∗ 1 1 − + − = Uc− (1 + i). (26) Pt Pt

12 g p Since in equilibrium real money balances Mt 1/Pt should be non-negative, then Mt∗ 1 − − cannot grow at a rate higher than Pt. Therefore the following restriction should hold, 1+µ Π = β(1+i). Thus, given the inflation rate set by the government, the growth rate of≤ private currency should be capped by that number. Note that in equilibrium the inflation rates in the two currencies are the same, Π = Π∗, because of (9), so that there is not necessarily any linkage between the growth rate and the inflation rate in private currency. Most interesting is the fact that growth rates of the two moneys can be different in equilibrium. When 1+µ < Π, real money balances in private currency are shrinking over time and converge to zero in the long run, therefore real money 1 balances in government currency are rising over time to reach in the limit Uc− (1 + i). The growth rate of government money is higher than Π and, therefore, higher than that of private money. We have an example of Gresham’s law in which the “bad” money, the one with higher growth rate, crowds out the “good”money. Restriction (26) does not apply when i = 0 since the liquidity constraint can hold with inequality. In this case, condition (24) provides an additional restriction given that the present-discounted value of real money balances in private currency should be bounded. Since prices are falling at the rate β when i = 0, then µ 0 ensures that the boundary condition (24) holds. Note that in this case while inflation≤ falls at rate β, the gross growth rate of private money can also be higher than this inflation rate, and can also be disconnected from the growth rate of government money.

Proposition 2 In an equilibrium with competing currencies: (i) if the government sets a positive interest rate then the growth rate of private money should satisfy (1 + µ) β(1 + i); or (ii) if the government sets i = 0 then µ 0. In either case, the path of the≤ price level in government currency is determined, but≤ the exchange rate S and consumption at time t0 are not.

Proposition 2 carries two interesting implications. First, the second currency causes no problem for the determinacy of the price level. To see this, consider that restrictions (i) or (ii) imply that equation (23) collapses to (25) and therefore Pt0 is set at the same value as in the single-currency framework.11 The second result, where multiple equilibria arise, is the indeterminacy of the ex- change rate, as in Kareken and Wallace (1981) and recently restated by Schilling and Uhlig (2019). In my model, this indeterminacy brings indeterminacy of consumption at time t0 and of the equilibrium path of government money supply starting from pe- 12 riod t0 + 1. However, as underlined above, the indeterminacy of the exchange rate

11 The intuition is that in equilibrium under the restrictions (i) and (ii) of Proposition 2 there are no Ponzi schemes associated to the issuance of private currency, so an intertemporal budget constraint holds for the private issuer. 12 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 does not bring any trouble to the government in controlling the price level in govern- ment currency, so that it is not really a worrying element for traditional monetary policy in this analysis. To see that consumption at time t0 is not determined, consider the cash-in- advance constraint at time t0

Mt0 1 SMt∗0 1 − + − Ct0 Pt0 Pt0 ≥ and note that since Pt0 is determined and Mt0 1, Mt∗0 1 are given, variations in S − − translate into variations in Ct0 . To see the second result, consider the cash constraint (17) when i > 0 and use (19) and Π = β(1 + i) to obtain g p Mt Mt∗ 1 Π + S = ΠU − P P c β t t   P for each t t0. Since Pt is determined and Mt∗ is also given by the process of private money≥ creation, the indeterminacy of the exchange rate S translates directly into indeterminacy of the path of government money supply, but not of its growth rate.

3.2 Equilibria with a single currency as medium of exchange So far we have characterized equilibria in which multiple currencies coexist. For this to happen, the return on the two moneys should be the same, which is condition (9). But when (9) is violated in either direction, there can be equilibria in which agents hold only one of the two moneys. The first equilibrium discussed here is when government money is the only medium of exchange. This outcome does not depend on the value of µ. 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 private money. The value of private money in terms of goods is going to be zero — its price infinite—validating the consumer’sinitial belief. An unbacked currency can be worthless. 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 interest- rate peg and real transfer. Setting the nominal interest rate fixes the inflation rate as well, while the real transfer policy pins down the price level. In this way, the value of government money is never zero. An appropriately backed currency has always a positive value. Before proving the above result more formally, we study under which conditions there can be equilibria in which only private money is used as medium of exchange.

14 For this to be so the return on private money must be higher than that on government currency, i.e. Pt∗+1/Pt∗ < Pt+1/Pt. In this equilibrium, the “cash”constraint simplifies to: p Mt∗ 1 − Ct, (27) Pt∗ ≥ for each t t0 given that only private money is used to purchase consumption good ≥ C. Note that (27) holds with equality whenever Pt∗+1/Pt∗ > β. Using equation (21) p with equality, since Mt∗ > 0, we can write it as

p p Mt∗ 1 β Mt∗ − = Uc (Ct+1) Pt∗ 1 + µ Pt∗+1 and therefore using (27) as

p p p Mt∗ 1 β Mt∗ Mt∗ − Uc (28) P ≥ 1 + µ P P t∗  t∗+1  t∗+1 for each t t0. In what follows, I assume log utility, U(C) = ln C to simplify the exposition,≥in which case (28) determines uniquely the price level in units of private money and its growth rate at Pt∗+1/Pt∗ = Π∗ = 1 + µ. Indeed the left-hand side is p ∗ always equal to β/(1+µ) and given an initial condition Mt0 1 the initial price level is 13 − determined. Since Π > Π∗ and Π = β(1 + i), for this equilibrium to exist it should be that β(1 + i) = Π > 1 + µ. The growth rate of private money should be less than the inflation rate in government currency. To see that prices are determined in government currency, note that real money balances in private currency are bounded, and therefore

p p M ∗ ∞ M ∗ ∞ t0 1 t t0 t 1 t t0 1 St+1 p − + µ β − − = β − M ∗ P P P (1 + i )P t t∗0 t=t t∗ t=t t∗ − t t X0   X0   holds identically for any equilibrium path P ,P ∞ . Therefore equation (23) boils t∗ t t=t0 down to (25), and the price level in government{ currency} is determined in the same way as in the single-currency case. The fact that government currency is not used as a medium of the exchange does not create any trouble for the government to control the price level. The two currencies coexist and prices are determined in both, but the private currency appreciates over time at the rate Π∗/Π. For this reason, only private money

13 1 ρ In the more general case of isoelastic utility of the form U(C) = C − /(1 ρ), with ρ > 0, p 1/ρ − equation (28) has a stationary solution in which Mt∗ 1/Pt = (β/(1 + µ) , three different cases arise: i) when ρ < 1 solutions for real money balances− around the stationary one are diverging; ii) when ρ = 1, there is only one stationary solution, as discussed in the text; iii) when ρ > 1, solutions around the stationary one converge to it.

15 is used as a medium of exchange. Government currency still maintains a role in the economy for two reasons: it is the unit of account for taxes and the denomination of government debt, which private agents are willing to hold.

3.3 Summing up We can now summarize the results in the following Proposition and comment on the policy implications. Proposition 3 Given the monetary policy regime of Definition 1: 1) there is always an equilibrium in which private currency is worthless and government currency is the medium of exchange, in which case Π < Π∗; 2) if i = 0 and µ 0 or if i > 0 and ≤ β(1 + i) = 1 + µ, in which cases Π = Π∗, then there is one additional equilibrium with both currencies competing as medium of exchange; 3) if i > 0 and β(1 + i) > 1 + µ, there are two additional equilibria, one with both currencies competing, in which case Π = Π∗, and one with only private money as medium of exchange, in which case Π > Π∗. In all equilibria the inflation rate in government currency is given by

Π = β(1 + i) and the price Pt0 is determined by (25). Some important conclusions can be drawn from Proposition 3 and the foregoing discussion. As in the benchmark one-currency model, the government has full control of the interest rate and inflation, since it has taxation power and there exists a market of 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. The second important result is that there are multiple equilibria. Note that this multiplicity is not related at all with the indeterminacy result of Kareken and Wallace (1981). Indeed their analysis is limited to the case in which both currencies are simultaneously used as a store of value in an OLG model, differently from here where money can be at the same time store of value and means of exchange. Indeed, the multiplicity in my framework arises along the different possible usage of the two moneys. To clarify this, set the interest-rate policy at i > 0 and the monetary policy of the private issuer to 1 + µ < β(1 + i). There are three equilibria: one in which only government money is used as a medium of exchange, one in which both coexist and another one in which only private money serves as a medium of exchange. The reader should remember that in the equilibrium in which both currencies coexist as a means of exchange the “bad”government money crowds out the private money. The three equilibria can be also ranked with the one with only private money as medium of exchange to give the highest welfare. Under log utility, consumption is equal to β/(1 + µ) with only private cash, and to 1/(1 + i) in the other two equilibria. But, why can’tthe households coordinate on the best equilibrium? The failure is in the relative price mechanism (exchange rate variations over time) which is the solo

16 incentivizer for households to use one money rather than another. This mechanism fails because it does not convey information on the growth rates of money supplies. The household takes the exchange-rate path as given. Given the policies i > 0 and 1 + µ < β(1 + i), there are three possible paths of the exchange rate consistent with the same policies: in the first the exchange rate is zero because private money is worthless, St = 0; in the second the exchange rate is constant across time at a value not determined, St = S > 0; in the third, the exchange rate appreciates over time at the rate St+1/St = (1 + µ)/β(1 + i) < 1. Therefore the price mechanism fails to direct the household to choose the best equilibrium. To do so, households should be able to understand that the lower growth rate set by the private issuer could lead to a better equilibrium, with lower inflation and higher purchasing power of money. However, this information is not in the discovery process of the price mechanism and therefore, if households expect a constant exchange rate between the two currencies they would hold both moneys irrespective of the lower growth rate of the private one. Disregarding the equilibrium in which St = 0 that depends on the possibility that an unbacked money can be worthless, the reason for the multiplicity of equilibria is in the Gresham’slaw I have discussed when both moneys coexist. The two moneys can indeed be simultaneously used as a means of exchange even if the have different growth rates. They would have exactly the same return, but the good one, with lower growth rate, is disappearing in real terms providing over time lower and lower purchasing power. Kareken and Wallace (1981) finds a similar Greshman’slaw, but they do not have the same implications for equilibrium multiplicity. In their framework, money has only the property of store of value. Here, instead, money can also be medium of exchange and, for this reason, there can be an equilibrium in which one money serves only the role of store of value while the other is also a medium of exchange. Indeed, in the equilibrium in which only private money is used as a medium of exchange, government currency is still valued because it is backed by taxes and serves only the role of store of value. In Kareken and Wallace (1981), the exchange rate can be either a positive S, in which case the two moneys coexist, S = 0 or S = in which only one money exists. There is no equilibrium in which the exchange∞ rate varies over time. I am now able to comment on Hayek (1974)’s view that forces of competition should bring the best money —the one with higher return or lower inflation. Hayek’s intuition in many of his writings leans on the discovery mechanism that the informa- tion provided by prices should bring about. On the contrary, the above analysis has shown that prices do not convey all the relevant information to households in choos- ing the best money. The correct discovery mechanism should also include knowledge about the rate of growth of private money and on how this would result into a lower inflation rate, if all agents adopt it. A coordination problem arises like in multiple- equilibria context. It should be clear, however, that adding more currencies, and then

17 more competition, does not change results and cannot eliminate the multiplicity of equilibria. Another important result is that welfare doesn’t decrease with currency compe- tition. To have a possibility to coexist, the unbacked money should be at least as good as the government money. When it is a better money, there could be also an equilibrium with higher welfare than in the single government-currency model. Looking from a welfare perspective, the government should not combat the adop- tion of new fiat currencies and not even be worried in terms of control of its own interest rate and inflation rate. In this respect, the path of prices in government cur- rency would be exactly the same independently of the existence or not of the private currency. However, there is a subtlety here connected to one important assumption we made on the existence of private and public debt in government currency and on taxes levied in that currency. I have not modelled currency competition in those markets. Since the existence of government currency depends on some contracts to be settled in government currency, risking to lose medium-of-exchange properties could open the way to lose other markets for government currency putting at risk its exis- tence. Government currency 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 well and become worthless, unless it is bolstered by legal restrictions. Another way to see the results of Proposition 3 is to ask how government should set policy so as to absolutely preclude equilibria in which private currency is used. The government could fix the interest rate in the range 0 < i < (µ/β 1), keeping inflation in the interval β < Π < (1+µ). Interest rates and inflation are then− bounded above by the growth rate of private money. To intuit these bounds, one should observe that in this model currency competition acts in the direction of facilitating certain transactions. To compete and exclude other currencies, the government should offer a better money, one with higher return and lower inflation. In this way, the framework could be in line with Hayek (1974): currency competition can improve the welfare of consumers if the growth rate of private money is suffi ciently low and the competing money correspondingly good. However, government objectives for monetary policy are generally broader than just merely liquidity premia in money markets. Central banks also seek to moderate the fluctuations of the economy. Along these lines, the literature offers justifications for a positive inflation rate, such as the need to avoid the zero lower bound on nominal interest rates or to “grease the wheels”of the economy in the presence of downward nominal wage rigidity.14 Taking these considerations into account in the simple model of this paper would imply that the bound on inflation can be too stringent for the

14 On the first justification, see Eggertsson and Woodford (2003); on the second, among others, Benigno and Ricci (2011) and Dupraz et al. (2018).

18 attainment of other objectives, whenever the growth rate of private money is low. The desired or optimal inflation target could be high enough to enter the region of multiple equilibria where government money might lose its medium of exchange function and be at survival risk.

4 Competing with a central bank digital currency

I now extend the analysis to consider a different environment in which central bank’s reserves provide also liquidity services to households. We can think at this new framework as one in which households are allowed to keep interest-bearing deposits at the central bank in line with recent proposals of central bank digital currency. The other way to introduce central bank digital currency through tokenization will not change the analysis of Section 1 once Mt is understood to capture digital rather than (only) material cash. The cash-in-advance constraint now becomes

g Mt 1 Bt 1 Mt∗ 1 − + − + − Ct, (29) Pt Pt Pt∗ ≥ which also includes interest-bearing securities issued by the government. I am as- suming that all government’sliabilities provide liquidity services and this assumption is justified by the non-default properties that central bank’sliabilities have, which I assume they are extended to treasury’ssecurities, too. The credit-market constraint is now written as g ˜g g ˜g Bt Bt + Bt Mt Mt∗ Bt 1 Mt 1 Bt 1 + Bt 1 Mt∗ 1 + g + + + Xt − + − + − − + − Ct Pt(1 + it) Pt(1 + it ) Pt Pt∗ ≤ Pt Pt Pt Pt∗ − ! T T +Y + t + t∗ . (30) Pt Pt∗

The reader should note that I have separated private non-defaulted bonds, Bt, and government bonds and among the latter ones those that the households decide to g bring to the goods market, Bt , and those that are left to be only store of value, ˜g Bt . It could be argued that it is arbitrary to exclude private non-defaulted bonds from providing liquidity services. This assumption can be justified on the ground that there is a subtle difference between private and government non-defaulted debt securities. Private bonds need to respect a solvency condition for them to be free of default, whereas government bonds are default-free by definition since the central bank is always solvent with respect to its own liabilities. Constraint (29) includes only securities that are default free without need to satisfy a solvency condition, in their respective currency. In the constraint (30), bonds are issued at discount g and the interest rate on government bonds, it , can be now different from that on

19 private bonds, it. By inspecting the above two constraints, it should be noted that g cash in government currency is now dominated by reserves unless it = 0, therefore the economy will be cashless in equilibrium, but only for what concerns government g currency. However, the possibility that cash can be used restricts it to be non- g g negative, i.e. it 0. Note moreover that it it otherwise arbitrage opportunities ≥ g ≥ arises. Therefore it it 0. Furthermore, to simplify the analysis, I will assume log utility, U(C) = ln C.≥ ≥ g ˜g The first-order conditions with respect to Ct, Bt,Bt , Bt and Mt∗ are now, respec- tively, given by 1 = 1 + λt (31) Ct 1 1 = β(1 + it) (32) Pt Pt+1

1 g 1 + λt+1 = β(1 + it ) + ξt (33) Pt Pt+1 1 g 1 ˜ = β(1 + it ) + ξt (34) Pt Pt+1

1 1 + λt+1 = β + ξt∗, (35) Pt∗ Pt∗+1 in which I have used the same notation for the Lagrange multipliers as in Section ˜ ˜g 1 and where ξt is the Lagrange multiplier associated to the constraint Bt 0. Us- ing equations (33) and (35), we can study the conditions under which one≥ currency prevails as a medium of exchange. Government and private moneys coexists whenever

St+1 g = (1 + it ). (36) St The exchange rate is no longer constant, as in the Kareken-Wallace world, because g of a positive and possibly time-varying it . When the above equality becomes an in- equality with the sign <, only government’sbonds are used as a medium of exchange, viceversa with the opposite inequality sign. I can also draw some further interesting g observations by inspecting the first-order conditions with respect to the securities Bt ˜g and Bt , the liquid and illiquid portfolio components of government bond. When gov- ernment bonds are used as a medium of exchange, i.e. ξt = 0, then households will not hold them just as a store of value, i.e. ξt∗ > 0, unless λt+1 = 0. When liquidity is not satiated, it is not optimal to invest in government securities and not use them for goods exchange. On the other side, when government money is not used as medium of exchange because dominated by the private money, i.e. ξt = 0, then it will be kept in portfolio just as a store of value, i.e. ξt∗ > 0.

20 In what follows, I do not repeat all derivations done in Section 1, but I will just remark the important changes. Starting with the intertemporal budget constraint of the household, it can be now written as g p g g B M ∗ ∞ ∞ t0 1 t0 1 t t0 Tt Tt∗ t t0 it it Bt 1 St+1 p − + − + β − + = β − − g + Mt∗ . Pt0 Pt∗ Pt Pt∗ 1 + it Pt Pt∗ − (1 + it)Pt 0 t=t0   t=t0     X X (37)

Accordingly to the new framework, I also appropriately modify the policies for the two currency issuers. The government sets now the interest-rate policy in terms g of the interest-rate on reserves it . g g Definition 4 The government sets a constant interest rate policy it = i at each g t t0, with i 0, and the following transfer policy ≥ ≥ g g Tt it i Bt = − g (1 β)τ (38) Pt 1 + i Pt − − p for each t t0 and for some τ positive. The private currency issuer sets Tt∗ = µMt∗ 1 ≥ − at each t t0 with µ > 1, to achieve a constant growth rate, 1 + µ, of its money supply. ≥ − I first characterize the equilibrium of the benchmark case in which there is no p currency competition, i.e. Mt∗ = 0 at all times. Combining the first-order conditions g g (32) and (33), I obtain that λt+1 = (it i )/(1 + i ) 0 saying that the liquidity con- g − ≥ straint is not binding when it = i . Using the first-order condition (31), consumption of good C is given by 1 + ig Ct+1 = , (39) 1 + it p for each t t0. Combining it with (29), assuming M ∗ = 0, we get ≥ t Bg 1 + ig t = (40) Pt+1 1 + it for each t t0. We can use (38) into (37) to obtain ≥ g Bt 1 − = τ (41) Pt for each t t0. The above restriction determines the price level at time t0 given the ≥ g initial condition Bt0 1. But it is also valid at all future times and can be used in (40) to obtain that 1 + i− = (1 + ig)/τ. Using the first-order condition (32), this result implies that the inflation rate is constant and given by Π = β(1 + ig)/τ. Note that when τ 1, there is full satiation of liquidity. It is now the tax policy, and not the interest-rate≥ policy like in Section 3.1, that determines the degree at which liquidity is satiated in the economy. In what follows, it is meaningful to assume that real taxes cannot be that high, and therefore τ < 1.

21 4.1 Equilibrium with one money I consider first equilibria in which only one money is used as a medium of exchange. Results are striking. As before, there is always an equilibrium in which the unbacked currency is worthless. However, there is never an equilibrium in which only the private money is used as a medium of exchange, unlike Section 3.2. Proposition 5 When interest-bearing government liabilities provide liquidity ser- vices, they will be always used as medium of exchange. To understand this results, remember that for the private money to crowd out the g g public money it should be that St+1/St > 1 + i which requires that Πt+1/(1 + i ) > Πt∗+1. Assume now that public money is not used for liquidity purposes, it will g be still used as a store of value and therefore it = i . Using (32), I obtain that g Πt+1 = Π = β(1 + i ) and therefore for this equilibrium to occur it should be that Πt∗+1 < β, which cannot happen in equilibrium otherwise it will violate equation (35). By paying an interest-rate on liquid securities, like it could be by giving the possibility to citizens to hold deposits at the central bank, the central bank can rule out to be crowded out by an unbacked private money. To intuit the result, consider that for g government money not to be held as a medium of exchange its return, (1 + i )/Πt+1, should be lower than that on private money, 1/Πt∗+1. But, when government money does not serve any liquidity role its return is just 1/β which cannot be beat by the return on any other non-interest-bearing money. As it will be clear in the next section, however, the return on the government money is lower than 1/β when both moneys coexist.

4.2 Equilibrium with competing currencies I now discuss equilibria when both moneys are used as a medium of exchange. Con- sumption is again determined by equation (39). When both moneys coexist, the liquidity constraint can be written as g p g B M ∗ 1 + i t + t , (42) Pt+1 Pt∗+1 ≥ 1 + it g which holds with equality whenever it > i . What complicates the analysis, with respect to the simple model of Section 1, is the fact that the interest-rate policy set by the government is now no longer suffi cient to determine whether the liquidity constraint is or is not binding. I start from characterizing equilibria in which both moneys coexist and the first best with full satiation of liquidity is reached. Therefore g it = i ,Ct = 1. In equilibrium the following summation should be finite p ∞ M ∗ t t0 t 1 µ β − − < . P t=t t∗ ∞ X0 22 g First note that when it = i , the inflation rate in government currency is determined g by the interest-rate on reserves, Πt = Π = β(1 + i ). Therefore, using (36), inflation in the private currency is constant and equal to beta, Πt∗ = Π∗ = β. The above p summation can be finite only when µ 0, in which case limt Mt∗ 1/Pt = 0 implying in (37) together with the flow budget≤ constraint of the private→∞ money− supplier that (41) holds for each t t0. Therefore, I can write (42) as ≥ p M ∗ t 1 τ Pt∗+1 ≥ − for each t t0 saying that real money balances in the private currency should com- plement the≥ supply of government liquidity in order for liquidity to be completely satiated. Recall that I am assuming τ < 1. In this type of equilibrium, the above inequality should hold at all times and in particular at time t0

p S M ∗ t0 t0 β(1 τ). (43) Pt0 ≥ − The inequality (43) can be satisfied only for certain realizations of the exchange rate p ∗ above a threshold. But, the exchange rate is not determined! Note that Mt0 is given by the policy rule of the private issuer and Pt0 is determined by the fiscal policy of the p p ∗ ∗ government. Using (41) and Mt0 = (1 + µ)Mt0 1, the lower bound on the exchange − rate can be written as g 1 τ Bt0 1 S∗ β − p− . (44) ∗ ≡ τ(1 + µ) Mt0 1 − We can collect the above results in the following Proposition.

Proposition 6 When interest-bearing government liabilities provide liquidity ser- vices, given the monetary policy regime of Definition 4 and assuming τ < 1, there is an equilibrium with full satiation of 1iquidity and currency competition if µ 0 for g ≤ any i 0 provided St S∗. ≥ 0 ≥ There are important differences with respect to the case in which only cash (or digital cash) provides liquidity services, see Proposition 2, item ii. In that case, the government has full control, with its interest-rate policy, of the conditions under which the economy is satiated with liquidity. Setting the nominal interest rate to zero achieves the first best. Government cash endogenously adjusts to fulfill liquidity at any level of the exchange rate S, and the increase in cash can be absorbed at the same tax rate by reducing the supply of “illiquid” bond securities. Here, the government can still achieve the first best but not anymore with the interest-rate policy. It needs to raise enough taxes to back all its liabilities. If not, private money can now complement public money to satiate the economy. Although the requirement

23 on the growth rate of private money, µ 0, is the same comparing Propositions 2 and 6, here, it is no longer suffi cient. Whether≤ there is going to be satiation of liquidity depends on the exchange rate to be above the threshold S∗, which is something none has control on in this economy. I will now analyze equilibria in which both moneys coexist but where liquidity is not satiated, at least to start with. Use (32) and (36) to write (42) as g p B M ∗ t + (1 + ig) t = β(1 + ig). Pt Pt∗ The above equation is key to characterize the equilibrium and holds with strict equal- g ity whenever it > i . First note that since reserves are in positive supply, then p 0 M ∗ /P ∗ β at all times, which used into (37) together with the flow budget ≤ t t ≤ constraint of the private money supplier implies that (41) holds for each t t0.I g ≥ have already discussed that when it = i equation (41) holds for each t t0 provided µ 0. Therefore, µ 0 is going to be a restriction to take into account≥ for equilibria ≤ ≤ g that converge to the first-best in the case it starts above i . Considering that in any case Pt0 is determined by (41), we can write the above equation as p g Mt∗ g τΠt+1 + (1 + i ) = β(1 + i ) (45) Pt∗ which holds for each t t0. Consider it at time t0 ≥ g p ∗ 1 + i St0 Mt0 Πt +1 = β , (46) 0 τ − P  t0  p ∗ and note that Mt0 is determined by the supply of the private issuer of money while

Pt0 is determined by the fiscal policy. As in the previous case, there is nothing that determines the exchange rate at time t0 and its future path. This is again the classical result of indeterminacy of the exchange rate which now translates into indeterminacy of the inflation rate at time t0 + 1 and onward. There are bounds however on the g exchange rate to be consistent with this equilibrium since it should be that it0 i g ≥ and therefore Πt +1 = β(1+it ) β(1+i ). Using this inequality into (46) we obtain 0 0 ≥ that the exchange rate should be such that St0 (0,S∗] in which S∗ is defined by (44). In what follows it is useful to define a threshold∈ Bg β ˜ t0 1 S p− 1 . ∗ ≡ Mt0 1 τ(1 + µ) − −  

Note that 0 < S˜ < S∗ provided β < (1+µ) < β/τ and that S˜ 0 when (1+µ) β/τ. We can write (45) as ≤ ≥

g g g β(1 + i ) (1 + µ)(1 + i ) Πt+1 = (1 + µ)(1 + i ) + 1 τ − Π  t  24 which characterizes the path of the inflation rate for each t t0 + 1 given the initial ≥ l g condition Πt +1 and considering the boundary condition Πt+1 Π = β(1 + i ) for 0 ≥ each t. Whenever 0 < S˜ < S∗, there are two stationary equilibria: one in which ˜ g h g ˜ Πt = Π = (1 + µ)(1 + i ) and another in which Πt = Π = β(1 + i )/τ. If Πt0+1 > Π h ˜ l inflation is converging to Π , if Πt0+1 < Π inflation is reaching the barrier Π . Note ˜ that whether Πt0+1 is higher or lower than Π, it depends on the whether the exchange rate at time t0 is lower or higher than S.˜ Whenever S˜ 0 there is only one stationary h ˜ ≤ l equilibrium at Π and if Πt0+1 < Π inflation is reaching the barrier Π which happens for any realization of the exchange rate below S∗. Note that whenever the inflation rate reaches the first-best level for that path to be an equilibrium µ 0. ≤ Proposition 7 In the model in which interest-bearing government liabilities provide liquidity services, given the monetary policy regime of Definition 4 and assuming τ < 1, there are the following equilibria in which both moneys coexist as a medium of exchange: i) if β/τ 1 µ 0 the inflation rate in government currency reaches l − ≤ ≤ Πt = Π in a finite period of time, for any St0 (0,S∗); ii) if β 1 < µ < β/τ 1 : ˜ ∈ ˜ − − a) the inflation rate is constant at Π if and only if St0 = S; b) the inflation rate l ˜ reaches Π in a finite period of time, if and only if St0 (S,S∗), provided µ 0; c) h ∈ ≤ the inflation rate converges to Π if and only if St (0, S˜). 0 ∈ Another way to read the results of the Proposition is the following. Suppose that τ is suffi ciently high so that β τ < 1, meaning that the economy with a single currency is suffi ciently close to the≤ first best. Then equilibria with competing currency exists if and only if the growth rate of private money is non-positive, µ 0. There are multiple equilibria depending on the realization of the exchange rate at time≤ t0. However, if St (0,S∗) all these equilibria converge to the first-best allocation 0 ∈ in a finite period of time . Whenever St S∗, the first best is achieved starting 0 ≥ from period t0, as shown in Proposition 6. On the contrary, when τ is low so that 0 < τ < β, meaning that the economy with a single currency is relatively far from the first best, there are equilibria with competing currencies if and only if (1 + µ) < β/τ 1 with µ that may be now positive. However, there are multiple equilibria in this− case too, and of different characteristics. Equilibria converges to the first-best ˜ allocation in a finite period of time only if St0 (S,S∗) and µ 0, requiring again a non-positive growth rate of money. The other∈ equilibria do not≤ necessarily require a non-positive growth rate of private money and they will display a constant inflation ˜ ˜ rate Π if and only if St0 = S, or an increasing path of inflation rate converging h ˜ to Π if and only if St0 (0, S). The latter equilibrium is another example of the Gresham’s law in which∈ the “bad” money crowds out the “good” money since real money balances in private money shrink to zero. I can draw some further conclusions comparing the results of this Section with those of Section 3.3. In both cases, there are multiple equilibria. However, when

25 interest-bearing government liabilities provide liquidity services there cannot be equi- libria in which only private money is held as a medium of exchange. This result comes at some costs. In all the equilibria with competing currencies, the government does not have anymore full control of the inflation rate in its currency, unlike Section 3.3, which now depends on the policy followed by the private issuer of currency and on the random realization of the exchange rate. In both cases, however, welfare with currency competition is never below the single-currency case. To see this result in the analysis of this Section, note that consumption (39) is decreasing with the infla- tion rate in government currency through the nominal interest-rate on illiquid bonds. Since (45) shows that in any equilibrium with currency competition inflation is never below the single-currency level, the result follows. Another difference is in the bound that gross growth rate of private money should respect to hope to compete with the government money. In Section 3.3, this was given by the maximum between 0 and the inflation rate in government currency, Π = β(1 + i) which was under tight control of the government through the nominal interest rate. Here it is given by the maximum between 0 and β/τ which involves the taxation rate.

5 Conclusion

This paper analyzes models of coexistence between government and privately-issued currencies both in an environment in which government or private cash compete as a medium of exchange and in a context in which interest-bearing government securities compete with privately-issued (digital) cash. Depending on the environment restrictions might arise on government policies and on the possible equilibria. In any case, welfare is never decreased by currency competition. 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 same liquidity services. However, an extension to imperfect substitutability does not alter the results significantly. Most interesting would be to devise a model in which the acceptance 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 currencies. Benigno, Shilling and Uhlig (2019) investigate the consequence for the international monetary system of trading a global currency, finding restrictions on cross-country interest rates and on the exchange rate.

26 References

[1] Bassetto, Marco and Thomas Sargent. 2020. “Shotgun Wedding: Fiscal and Monetary Policy.”NBER Working Paper No. 27004.

[2] Benati, Luca. 2019. “Cagan’sParadox Revisited.”Unpublished manuscript, Uni- versity of Bern.

[3] Benigno, Pierpaolo. 2020a. “A Central-Bank Theory of Price Level Determina- tion.”American Economic Journal: Macroeconomics 12(3): 258-83.

[4] Benigno, Pierpaolo. 2020b. Lectures on Monetary Economics.

[5] Benigno, Pierpaolo and Luca Ricci. 2011. “The Inflation-Output Trade-Off with Downward Wage Rigidities.” The American Economic Review 101(4): 1436- 1466.

[6] Benigno, Pierpaolo and Roberto Robatto. 2019. “Private Money Creation, Liq- uidity Crises, and Equilibrium Liquidity.”Journal of Monetary Economics 106: 42-58.

[7] Benigno, Pierpaolo, Linda Schilling and Harald Uhlig. 2019. “Cryptocurrencies, Currency Competition, and the Impossible Trinity.”NBER Working Paper No. 26214.

[8] Bank for International Settlements. 2018. Annual Economic Report. Chapter V. Cryptocurrencies: looking beyond the hype. Basel.

[9] Canzoneri, Matthew, Robert Cumby and Behzad Diba. 2001. “Is the Price Level Determined by the Needs of Fiscal Solvency?”American Economic Review 91(5): 1221-1238.

[10] Calvo, Guillermo and Carlo Vegh. 1996. “From Currency Substitution to Dollar- ization and Beyond: Analytical and Policy Issues.”In Guillermo A. Calvo, ed., Money, Exchange Rates and Output. Cambridge, MA: MIT Press.

[11] Cipolla, Carlo Maria. 1956. Money, Prices, and Civilization in the Mediterranean World. Princeton University Press, Princeton, NJ.

[12] Cipolla, Carlo Maria. 1982. The Monetary Policy of Fourteenth-Century Flo- rence. University of California Press, Berkeley, CA.

[13] Cipolla, Carlo Maria. 1990. Il governo della moneta a Firenze e a Milano nei secoli XIV—XVI. Il Mulino, Bologna.

27 [14] Dupraz, Stéphane, Emi Nakamura and Jón Steinsson. 2018. “A Plucking Model of Business Cycles.”Unpublished manuscript, University of California at Berke- ley.

[15] Eggertsson, Gauti and Michael Woodford. 2003. “The Zero Bound on Interest Rates and Optimal Monetary Policy.” Brookings Papers on Economic Activity 2003(1), 139-211.

[16] Einaudi, Luigi. 1936. “Teoria della moneta immaginaria nel tempo da Carlo- magno alla rivoluzione francese.”Rivista di Storia Economica 1, 1—35.

[17] Fernandez-Villaverde, Jesus and Daniel Sanches. 2019. “Can Currency Compe- tition Work?”Journal of Monetary Economics 106: 1-15.

[18] Friedman, Milton. 1960. A Program for Monetary Stability. New York: Fordham University Press.

[19] Hayek, Friedrich. 1976. The Denationalization of Money. Institute of Economic Affairs, London.

[20] Kareken, John and Neil Wallace. 1981. “On the Indeterminacy of Equilibrium Exchange Rates.”The Quarterly Journal of Economics, vol. 96, issue 2, 207-222.

[21] Klein, Benjamin. 1974. “The Competitive Supply of Money.”Journal of Money, Credit and Banking, vol. 6, no. 4, pp. 423—453.

[22] Kovbasyuk, Sergey. 2018. “Initial Coin Offering and Seigniorage.”Unpublished manuscript. EIEF.

[23] Loyo, Eduardo. 2002. “Imaginary Money Against Sticky Relative Prices.”Euro- pean Economic Review, Vol. 46, Issue 6, pp. 1073-1092.

[24] Lucas, Robert and Nancy Stokey. 1987. “Money and Interest in a Cash-in- Advance Economy,”Econometrica, Vol. 55, No. 3, 491-513.

[25] Marimon, Ramon, Juan Pablo Nicolini and Pedro Teles. 2003. “Inside-Outside Money Competition.”Journal of Monetary Economics 50, 1701-1718.

[26] Marimon, Ramon, Juan Pablo Nicolini and Pedro Teles. 2012. “Money is an Experience Good: Competition and Trust in the Private Provision of Money.” Journal of Monetary Economics 59, 815-825.

[27] Niepelt, Dirk. 2004. “The Fiscal Myth of the Price Level.”The Quarterly Journal of Economics 119 (1): 277-300.

28 [28] Obstfeld, Maurice, and Kenneth Rogoff. 1983. “Speculative Hyperinflations in Maximizing Models: Can We Rule Them Out?” Journal of Political Economy 91: 675-687.

[29] Sargent, Thomas J., and Wallace, Neil. 1982. “The Real-Bills Doctrine versus the Quantity Theory: A Reconsideration.” Journal of Political Economy 90: 1212-1236.

[30] Schilling, Linda and Harald Uhlig. 2019. “Some Simple Economics.” Journal of Monetary Economics 106: 16-26.

[31] Sims, Christopher. 1994. “A Simple Model for Study of the Determination of the Price Level and the Interaction of Monetary and Fiscal Policy.” Economic Theory 4: 381-399.

[32] Sims, Christopher. 2000. “Fiscal Aspects of Central Bank Independence.” Un- published manuscript, Princeton University.

[33] Sims, Christopher. 2013. “Paper Money.” American Economic Review 103(2): 563-584.

[34] Smith, Bruce D. 1988. “Legal Restrictions, “Sunspots,” and Peel’s Bank Act: The Real Bills Doctrine versus the Quantity Theory Reconsidered.”Journal of Political Economy 96(1): 3-19.

[35] Woodford, Michael. 1995. “Price Level Determinacy without Control of a Mon- etary Aggregate.” Carnegie-Rochester Conference Series on Public Policy 43: 1-46

[36] Woodford, Michael. 2000. “Monetary Policy in a World without Money.”Inter- national Finance, 2(3): 229-260.

[37] Woodford, Michael. 2001. “Fiscal Requirements for Price Stability.”Journal of Money, Credit and Banking 33 (1): 669-728.

[38] Woodford, Michael. 2003. Interest and Prices. Princeton, NJ: Princeton Univer- sity Press.

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