Is a Secondary Currency Essential? – On the Welfare Effects of a New Currency

1 Introduction

The process of digitalization accelerates the emergence of new currencies such as cryptocurrencies, corporate currencies and central bank digital currencies. These currencies may serve as an additional medium of exchange and are new competitors on the markets for liquidity services. Kiyotaki and Wright (1993) have shown that fiat money is essential, i.e. compared to a barter economy, fiat money allows for a better resource allocation. In this paper, we put forward a similar question: Is the secondary currency essential too? Does the introduction of a new currency allow for a welfare improvement even if a fully accepted currency, i.e. cash, is in circulation and remains in circulation? To tackle this question, we use the dual currency search framework of Kiyotaki and Wright (1993). The answer we find is a conditional “yes”. Not surprisingly, the scope for a welfare improvement depends on differences in returns and costs. But in addition, the sign of the welfare effect very much depends on the proportion of cash traders who will be replaced by digital money traders, or, equivalently, the degree of substitution between the traditional and the new currency.

The focus of our model is an advanced economy with a well-functioning payment system. We have in mind the Eurozone and/or the United States, where cash is an established medium of exchange and where now a cryptocurrency such as Bitcoin is emerging. Another example is Switzerland, where the euro is accepted in most parts of the country despite the universal acceptance of the Swiss franc. We do not believe in a cashless society; our framework therefore assumes that cash as traditional currency remains in circulation even if the new currency is fully accepted. We do not model the process of currency substitution with the use of the new currency instead of cash. Such a full crowding out of government fiat money is more relevant for high-inflation countries and countries with eroding economic and political institutions; see, for instance, Jácome (2004), Noko (1993), and Rivera-Solis (2012) for Ecuador, Zimbabwe and El Salvador, respectively. The use of multiple currencies during turbulent times, studied and surveyed in, e.g. Giovannini and Turtelboom (1994) and Airudo (2014), is no equilibrium phenomenon, so that the Kiyotaki–Wright framework is not appropriate. Note, however, the different view of Colacelli and Blackburn (2009), who employ the dual currency approach to investigate the multiple currency usage during the Great Depression in the United States and the 2002 recession in Argentina. The coexistence of both the traditional and the new currency has to be an equilibrium outcome. The Kiyotaki–Wright framework shows this desirable feature. Moreover, this framework allows a distinction to be made between partial and full acceptance of the secondary currency. For different modelling approaches, we refer to the overlapping generation model of Lippi (2021), the currency competition model of Schilling and Uhlig (2019) and the New Keynesian framework of Uhlig and Xie (2020).

The economics of dual currency regimes is the topic of a wide body of theoretical and empirical literature. An excellent overview of the search-theoretic foundations of the use of multiple currencies is presented by Craig and Waller (2000). Aiyagari et al. (1996) study the coexistence of money and interest-bearing securities, Camera et al. (2004) distinguish between safe and risky fiat monies, Curtis and Waller (2000) focus on the simultaneous use of legal and illegal currencies, while Lotz (2004) addresses the question how to regulate a new currency. Ding and Puzello (2020) use laboratory experiments to explore how governmental interventions such as legal restrictions on the use of a foreign currency or a change in using costs affect the circulation of the domestic currency. Also using a laboratory experimental design, Rietz (2019) analyzes the determinants of the acceptance of a secondary currency. Surprisingly, all these studies say very little about the scale of a welfare improvement of a secondary currency. This paper aims to fill this gap. The remainder of the paper is organized as follows. Section 2 describes the model setup of our analysis. Section 3 presents the single currency regime as benchmark economy. Section 4 discusses two switching scenarios in which we distinguish between partial and full acceptance of the new currency. Section 5 concludes.

2 Framework

Our setup borrows heavily from the dual currency framework of Kiyotaki and Wright (1993). Referring to yield differences and differences in the liquidity value, Kiyotaki and Wright (1993) show that equilibria exist with both currencies in circulation. However, they do not discuss the transition from a single currency to a dual currency regime. Yet neglecting the impact of a new currency on the supply of the traditional currency turns out to be decisive for the welfare effect of a new currency. We thus modify the Kiyotaki–Wright framework in two ways: First, we take into account the interaction between the traditional currency and the new one, and second, we use an economy with a fully accepted traditional currency as initial equilibrium (benchmark).

The economy consists of a continuum of infinitely lived agents with population size normalized to unity. We follow Matsuyama et al. (1993) and assume that agents of type i ∈ {1, …, I} with I ≥ 3, consume only goods of type i, but produce goods of type i + 1 (modulo I). As a consequence, there is no double coincidence of wants and no pure barter in the economy. Money is necessary for trading.^[1]

In accordance with Kiyotaki and Wright (1993) and, again, Matsuyama et al. (1993), we assume that goods production requires a consumption good as input and that agents cannot produce until they have consumed. An agent produces one unit of output according to a Poisson process with constant arrival rate, α, where α measures output per unit of time. We will focus on the limiting case, α → ∞, so that production is instantaneous. The share of the population who are producers degenerates to zero, all agents are traders (see Appendix A for the dynamic structure of the model).

In addition to the commodities, the economy is endowed with two types of money, a traditional and a new secondary currency. The traditional currency is fiat money (cash) issued by the government. Since we primarily have a cryptocurrency or a central bank digital currency in mind, we call the new currency digital money. However, the secondary currency may also of course be fiat money issued by a foreign government. We distinguish between three trading states: Agents are cash traders C, digital money traders D or commodity traders (sellers S). We use the following notation: In a dual currency regime, let μ _C and μ _D be the share of agents endowed with one unit of cash and digital money, respectively. The share of commodity traders, μ _S, then is μ _S = 1 − μ _C − μ _D. In a single currency regime, there is no digital money, μ _D = 0, the share of cash traders is μ C s , the share of commodity traders is μ S s = 1 − μ C s . The superscript s stands for single currency regime.

Meetings are pairwise and occur according to a Poisson process with constant arrival rate, β, with β I = 1 .^[2] This assumption ensures that a money trader meets exactly one seller within one period who is able to produce the preferred good of the money trader. The probability of a seller meeting a cash (digital money) trader is μ _C (μ _D). Given such a meeting, the seller decides whether to accept cash (digital money) and to switch the status from a commodity trader to a cash (digital money) trader. The decision is captured by the probabilities π _C and π _D. The return of switching the state is given by V _C − V _S respective V _D − V _S, where V _j , j = S, C, D, are the value functions for a trader of type j. If r > 0 denotes the rate of time preference, the sellers’ expected return to search is given by the Bellman equation

If the return of switching the state is positive (negative), a seller always accepts (rejects) the currencies and sets the optimal response, π _C respective π _D, to unity (zero). If sellers are indifferent between states, they flip a coin with 0 < π _C, π _D < 1, a currency is partially accepted.^{[3] , [4]} Since, by assumption, there is no pure barter and no consumption of the own production, a positive flow return to a seller requires a switch of status from a commodity to a money trader.

The flow return to a cash trader is given by the probability of meeting a seller, μ _S, times the probability that a random commodity trader accepts cash (overall acceptance rate), Π_C, times the utility from consumption, U, minus a transaction fee, η _C, minus the loss of switching from C to S, (V _S − V _C). Apart from that, we assume that cash has some nonuse value (monetary benefit), γ _C: Cash has some pleasing aesthetics, holding financial assets in the form of cash has the advantage of anonymity, cash may serve as safe haven. If there are some storage and/or transportation costs, we have γ _C < 0. For a digital money trader, the line of argument is very much the same. Then the Bellman equations are

Note that although we call the secondary currency digital money, we do not model any specific feature of digital currencies. Transaction fees, a high rate of return or degree of volatility, a more speedy settlement of payments etc. are subsumed under γ _D and η _D. Moreover, a trade between cash and digital money trader does not make both agents better off. In case of such a meeting, both agents continue with their own money. To put it another way, we rule out side payments, see also Aiyagari et al. (1996).

Our focus will be on symmetric equilibria with π _C = Π_C and π _D = Π_D. In accordance with Kiyotaki and Wright (1993), welfare is defined by the expected utility of all agents before the initial endowment of money and commodities is randomly distributed among them. In terms of expected flow returns, the welfare criterion can be expressed as (see Appendix A):

3 Single Currency Regime

Despite the truism that the welfare effect of a new currency depends to a large extent on the starting point (or initial equilibrium), the literature has neglected this issue. Since we are primarily interested in developed economies with a well-functioning payment system, our starting point will be a single currency regime in which only cash is in circulation, and cash is fully accepted. In the initial equilibrium, there is no digital money, μ _D = 0. Full acceptance of cash, π _C = Π_C = 1, requires that the gain of accepting cash and switching the state from S to C must be positive, V C s − V S s > 0 . For the single currency regime the Bellman equations simplify to

By combining these equations it is easy to show that the condition V C s − V S s > 0 is equivalent to

Here, ρ C s is the expected per period return of cash. If the sum of the expected net utility from buying and consuming a good minus the storage costs (or plus the monetary benefit) is positive, cash will be universally accepted. Inserting (5) and (6) into (4), and observing μ _D = 0, delivers the level of welfare in the single currency regime:

Note that the welfare effects of switching the status (from sellers to cash traders and from cash traders to sellers) add up to zero. By switching from S to C, the group of sellers improve their welfare by μ _S μ _C(V _C − V _S). By switching from C to S, the group of cash traders face a loss of μ _C μ _S(V _S − V _C). These effects add up to zero. For the cash traders, the loss of switching is overcompensated by the increase in welfare due to consumption. On aggregate, welfare is thus positive, see (8).

An increase in the money supply, in our model captured by an increase in the share of cash traders, has two (well-known) effects on welfare. A higher μ C s facilitates trade and sellers find a trading partner more easily (liquidity effect). But a higher μ C s means a lower μ S s , the number of commodities (sellers) declines. The welfare-maximizing share of cash traders, μ C s * , balances these effects. Observing (7) as well as μ S s = 1 − μ C s , the derivation of (8) with respect to μ C s yields

Equation (9) extends Kiyotaki and Wright (1993), who focus on the special case γ _C = 0 with μ C s * = 1 / 2 . Depending on the sign of γ _C (monetary benefit versus storage costs), μ C s * exceeds or falls short of 1/2.

4 Two Switching Scenarios

Besides the initial equilibrium, the welfare effect also depends on the acceptance of the new currency. We distinguish between two scenarios. First, cash is fully accepted and digital money is partially accepted (Section 4.1), and second, both currencies are fully accepted (Section 4.2).^[5]

4.2 Both Currencies Fully Accepted

Our second switching scenario assumes Π_C = Π_D = 1. Full acceptance of cash requires V _C > V _S, full acceptance of the digital money requires V _D > V _S. Rearranging the Bellman Eqs. (1)–(3) shows that these constraints are fulfilled if and only if

(19) V C f > V S f ⇒ ρ C f > μ D ν D f ρ D f

(20) V D f > V S f ⇒ ρ D f > μ C f ν C f ρ C f

hold. Here, ρ D f ≡ γ D + μ S f ( U − η D ) is the expected per period return of digital money, and ν D f ≡ 1 / r + μ S f + μ D . The superscript f denotes the dual currency regime with full acceptance of the new currency. If the expected per period return of cash does not exceed threshold (19), cash will no longer be fully accepted. Similarly, if the expected per period return of the digital money does not exceed threshold (20), the digital money will not be fully accepted. In other words, the existence of an equilibrium requires that

(21) μ C f ν C f − 1 < ρ D f − ρ C f ρ C f < 1 μ D ν D f − 1

holds. The relative spread between ρ D f and ρ C f must not be too big, otherwise either digital money or cash is no longer fully accepted. Kiyotaki and Wright (1993: 75) report a similar result, but they do not specify the interval.

Welfare in the regime of two fully accepted currencies can be computed as

(22) r W f = μ C f ρ C f + μ D ρ D f .

To sign the net welfare effect of the introduction of a universally accepted new currency, we have to compare (22) with (8). Again, we need a hypothesis on the replacement of sellers and cash traders by the digital money traders. We adapt Eqs. (13) and (14) by assuming μ S f = μ S s − λ μ D and μ C f = μ C s − ( 1 − λ ) μ D . The condition for a positive net welfare effect is

(23) r W f − r W s > 0 ⇒ − Γ 1 λ 2 + Γ 2 λ + Γ 3 > 0

with Γ₁ ≡ μ _D(U − η _C), Γ 2 ≡ γ C + μ D + μ S s − μ C s ( U − η C ) − μ D ( U − η D ) and Γ 3 ≡ γ D + μ S s ( U − η D ) − γ C + μ S s ( U − η C ) . Suppose digital money and cash are very close substitutes, so that digital money traders replace only cash traders, while the number of sellers remains constant, λ = 0. In this case, (23) boils down to Γ₃ > 0. The cash traders who switch status from C to D switch to a currency with the same liquidity value (acceptance rate), they gain γ D + μ S s ( U − η D ) , they lose γ C + μ S s ( U − η C ) . If the former exceeds the latter, the economy yields a payoff. If digital money and cash are bad substitutes, digital money traders replace only sellers, λ = 1, and condition (23) simplifies to ρ D f > μ S s ( U − η C ) . The sellers, who switch status from S to D, gain ρ D f . But the cash traders face a loss. Since there is a smaller number of sellers, the probability of exchange and consumption declines.

Net welfare is a quadratic function in λ. Depending on λ, the sign of the net welfare effect may change. Figure 1 illustrates this, we assume Γ₂ > 0 and Γ₃ = 0. For λ = 0, digital money is neutral with respect to welfare. As λ increases, so does the sum of cash and digital money (aggregate money supply). Therefore, an increase in λ very much resembles an increase in money supply in the Kiyotaki and Wright (1993) framework. Endowing more agents with money facilitates exchange and improves welfare; the net welfare effect becomes positive. But endowing more agents with money is equivalent to endowing fewer agents with commodities, and consumption and welfare go down. If the replacement parameter, λ, exceeds a critical value, λ _crit = Γ₂/Γ₁, the net welfare effect switches the sign and turns into negative.

Figure 1:

Solution to r W ^f  − rW ^s > 0.

Two remarks are in order: First, the higher the share of cash traders in the initial equilibrium, μ C s , the lower is the welfare-enhancing liquidity effect of a new currency, and the more important is the negative effect of the lower number of commodities, λ _crit declines, the probability of a negative net welfare goes up. Second, λ _crit may be larger than one, see the dashed line in Figure (1). In this case, we observe a net welfare gain for all λ ∈ 0,1 . The welfare effects of a relaxation of the assumption Γ₃ = 0 are straightforward, in Figure 1 the net welfare curve shifts up (Γ₃ > 0) or down (Γ₃ < 0). Since there are no novel and crucial insights, we skip the discussion.

In a world with two fully accepted currencies, the welfare-maximizing share of cash traders is given by

(24) μ C f * = 1 − μ D 2 + γ C 2 ( U − η C ) − μ D ( U − η D ) 2 ( U − η C ) = μ C p * − μ D ( U − η D ) 2 ( U − η C ) .

As shown in (24), the optimal response to the introduction of a partially accepted currency is a reduction in the supply of cash (share of cash traders) by 0.5μ _D. If instead digital money is fully accepted, its liquidity value is even higher, so that the decline in the optimal supply of cash is even stronger, μ C f * < μ C p * . We get

Proposition 2:

Suppose that both cash and the new currency (digital money) are fully accepted. (i) The existence of an equilibrium requires that the relative spread between ρ D f and ρ C f fulfills (21). (ii) If digital money and cash are very close substitutes (λ → 0), a positive spread ensures a net welfare gain. (iii) The lower the degree of substitution between digital money and cash (increasing λ), the higher the probability of a negative net welfare effect. (iv) A new fully accepted currency lowers the welfare-maximizing supply of cash more than the introduction of a partially accepted currency.

5 Conclusion

Money is essential. The use of fiat money relaxes information constraints and thus promotes trade and allows for a better resource allocation. This paper shows that, for plausible parameter constellations, a secondary currency is essential too, even if the traditional currency remains in circulation. Given the fact that digital monies are on the rise, this result is noteworthy.

How should the government and/or monetary policymakers respond to this development? Most important from our point of view: Policymakers should not obstruct digital monies by, for instance, a legal ban. Such a ban probably hinders welfare improvements. Similarly, the monetary authority should accept the emergence of private providers of liquidity. The best policy response is a decline in the supply of the traditional currency. Moreover, the government should pay more attention to regulations concerning the market for payment systems. The new competitors on this market must not be a threat to payment security. Our framework is too simple to draw more far-reaching policy conclusions, and therefore extensions are necessary. However, we are at the starting point of a fruitful discussion of the economic consequences of digital monies. Two promising lines of research are the impact on financial intermediation, for an overview see Thakor (2020), and the macroeconomic consequences of a central bank digital currency, see, e.g. Barrdear and Kumhof (2022) or Fegatelli (2022).