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On the Value of Virtual Currencies

This paper develops an economic framework to analyze the exchange rate of virtual currency. Three components are important. First, the current use of virtual currency to make payments. Second, the decision of forward-looking investors to buy virtual currency (thereby effectively regulating its supply). Third, the elements that jointly drive future consumer adoption and merchant acceptance of virtual currency. The model predicts that, as virtual currency becomes more established, the exchange rate will become less sensitive to the impact of shocks to speculators' beliefs. This undermines the notion that excessive exchange rate volatility will prohibit widespread use of virtual currency.


Non-Technical Summary
This paper develops a theoretical framework to analyze the economic factors affecting the exchange rate for virtual currency. Virtual currencies, such as Bitcoin, represent both the emergence of a new form of currency and a new payment technology to purchase goods and services. These currencies may move outside the scope of current financial institutions, as they allow distant payments to be made directly between consumers and merchants without the use of any financial intermediaries. Moreover, their supply is not necessarily controlled by central banks, and speculative motives are widely believed to be an important factor for the value of virtual currencies.
Speculating on the value of currencies is not new. In the early 1900s, Fisher (1911) argued that, in certain situations, speculators may effectively regulate the money supply by withdrawing money from circulation by betting on its future value. We apply this "old" notion in a formal way by using Fisher's quantity relation and showing how the exchange rate of a virtual currency responds to changes in the speculative position of investors.
Our theoretical framework shows that three components are important for the exchange rate. First, the current use of virtual currency to make actual payments. Second, the decision of forward-looking investors to buy virtual currency (thereby effectively regulating its supply). Third, the elements that jointly drive future consumer adoption and merchant acceptance of virtual currency. On the consumer side of the market, private benefits may be large for those who frequently execute cross-border payments, such as remittances. Additionally, consumers who value privacy and anonymity more, and those who are technologically more adept are likely to gain from using virtual currencies. On the other side of the market, large merchants may experience considerable private benefits from avoiding the high fees charged by traditional payment providers. Internet stores may gain as well, since they face relatively low implementation costs when accepting virtual currencies.
The model predicts that, as virtual currency becomes more established, the exchange rate will become less sensitive to the impact of shocks to speculators' beliefs and their inflow into and outflow from the virtual currency market. This prediction undermines the notion that the current high volatility of the exchange rates of virtual currencies will prohibit their widespread usage in the long run.

"You have to really stretch your imagination to infer what the intrinsic value of Bitcoin is. I haven't been able to do it.
Maybe somebody else can." -Alan Greenspan, Bloomberg Interview, 4 December 2013

Introduction
This quote from the former Federal Reserve Chairman could hardly be more accurate in describing the aim of the present study, in which we attempt to answer the broad question of what drives the value of virtual currencies. Virtual currencies, such as Bitcoin, represent both the emergence of a new form of currency and a new payment technology to purchase goods and services. These currencies may move outside the scope of current financial institutions. That is, their supply is not necessarily controlled by central banks, and they allow distant payments to be made directly between consumers and merchants without the use of any financial intermediaries. The key innovation is the implementation of cryptographic identification techniques into a "distributed ledger," i.e., a digital record that allows the tracking and validation of all payments made. This allows virtual currencies to be used in a decentralized payment system while avoiding the possibility of "double spending." Bitcoin is the most well-known virtual currency. 1 For primers on the economics behind Bitcoin, see, e.g., Dwyer (2015) and Böhme et al. (2015). Bitcoin was launched in 2009 and attracted attention from the financial press, economists, central banks and governments. This paper develops an economic framework that analyzes the exchange rate of virtual currency at its early-adoption stage and its main drivers. A unique combination of the properties of virtual currencies -at least in this early stage -plays an important role in our model. First, virtual currency prices of products and services are perfectly flexible with respect to changes in the exchange rate, since merchants tend to instantly adjust price quotes in virtual currency to the latest available exchange rate. In the model, this property is key in providing a direct link between the exchange rate and the demand for virtual currency. Second, the choice for making payments with virtual currency is also a choice for an alternative transaction technology, since these payments are settled and processed through a peer-to-peer payment network associated with that virtual currency. Network economies affecting payment choice play an important role in determining the ultimate demand for virtual currency. Third, the growth of the supply of virtual currency is, to a large extent, predetermined. In line with this latter characteristic, future demand for virtual currency to execute payments is one of the main sources of uncertainty in our model.
Perfect flexibility of prices with respect to the exchange rate is not a unique feature of virtual currency in the early-adoption stage. It may also characterize a traditional currency facing extremely high and volatile inflation, when price setting occurs in practice in terms of a stable (foreign) currency (such as the US dollar); see, e.g., Dornbusch et al. (1990).
However, an important difference is that investors have no incentives to hold a highly inflationary currency as a store of value, while speculative motives are widely believed to be one of the reasons for holding virtual currency. Hence, from a monetary perspective, virtual currency in the early-adoption stage provides an interesting case study for simultaneously 2 observing speculative demand for a currency and perfect flexibility of prices with respect to the exchange rate.
Our framework combines an investor's portfolio model with a payment network model, while adding a flavour of monetary economics. In our framework, three components are important for the exchange rate: (i), the actual use of virtual currency to execute real payments; (ii), the decision of forward-looking investors to buy virtual currency (thereby effectively regulating its supply); and (iii), the elements that jointly drive future consumer adoption and merchant acceptance of virtual currency. These latter elements determine the expected long-term growth in virtual currency usage. We show that the equilibrium exchange rate is determined by both a "purely speculative" component that depends on the hypothetical price speculators would offer if not a single real transaction is settled using virtual currency, and a transaction component that depends on the actual amount necessary to facilitate real payments.
Speculating on the value of currencies is not new. In the early 1900s, Fisher (1911) argued that, in certain situations, speculators may effectively regulate the money supply by withdrawing money from circulation by betting on its future value. We apply this "old" notion in a formal way by showing that the exchange rate of a virtual currency immediately responds to changes in the speculative position of investors. Our model predicts that, as a virtual currency becomes more established, the exchange rate will become less sensitive to the impact of shocks to speculators' beliefs and their inflow into and outflow from the virtual currency market. This prediction undermines the notion that the current high volatility of the exchange rates of virtual currencies will prohibit their widespread usage in the long run.
Additionally, we borrow from the so-called "two-sided markets" literature to explain the main factors that drive future consumer adoption and merchant acceptance of virtual currency as a payment instrument; see, e.g., Armstrong (2006) and Rochet and Tirole (2006). It is shown that private benefits and cross-group externalities among merchants and consumers affect the joint demand for virtual currency to make payments for real goods and services. On The figure presents the value of a unit of the virtual currency Bitcoin. The solid black line shows the exchange rate. The grey line shows a rough estimate of the hypothetical exchange rate if no units of the virtual currency were held for speculation. The dashed line provides an impression of the potential exchange rate in the absence of speculation and is a strongly smoothed average of the grey line. For technical details, see Appendix A. Source: www.blockchain.info and authors' calculations.
one side of the market, private benefits may be large for consumers who frequently execute cross-border payments, such as remittances, that usually carry high fees. Face-to-face payments may also become easier and cheaper via application-based "wallets" on a smartphone.
Consumers who value privacy and anonymity more, and those who are technologically more adept are likely to gain from using virtual currencies. On the other side of the market, large merchants may experience considerable private benefits from avoiding the high fees charged by traditional payment providers. Internet stores may gain as well, since they face relatively low implementation costs when accepting virtual currencies. Our model tries to explain how the resulting joint demand for virtual currency affects its exchange rate.
A first impression of what the exchange rate of Bitcoin might have looked like in the absence of speculation is presented by the dashed line in Figure 1. The solid black line shows daily data on the actual exchange rate. Based on our framework, the grey line shows a downward correction of the actual exchange rate to account for the virtual currency positions 4 held by speculators. Here, the size of the speculative position is roughly proxied by the number of bitcoins that will remain in "dormant" accounts for an extended period. Economic theory, such as buffer stock models, suggest that consumers and merchants do not instantly adjust their positions to their actual liquidity needs in response to fluctuations; see, e.g., Laidler (1984) and Mizen (1997). This suggests that only a strongly smoothed version of the grey line, such as the dashed line, may provide some impression of the exchange rate in the absence of speculation.
The paper is set up as follows. In Section 2, we briefly review the literature. As a preliminary, Section 3 analyzes the exchange rate of virtual currency using Fisher's quantity equation. Section 4 describes the model, building on a two-sided payment network model and an investor's portfolio model. In Section 5, the rational expectation equilibrium results are presented and explained. Various extensions are discussed in Section 6. Finally, Section 7 provides some discussion and concluding remarks.

Literature
While most papers focus on the technical and computational aspects of creating virtual currencies, the literature on the economics of virtual currencies is still developing.
In a recent paper, Dwyer (2015) provides an overview of private virtual currencies with a special focus on Bitcoin (being the most prominent example). The behaviour of Bitcoin's price since it began trading on electronic exchanges is analyzed, and the author provides a comparison of the volatility of Bitcoin's price on those exchanges compared with gold and foreign exchange. He argues that a theoretical model pinning down a dynamic reputational equilibrium for the central mechanism of Bitcoin's functioning is needed for a deeper fundamental understanding of Bitcoin and similar currencies.
In another study, Böhme et al. (2015) present Bitcoin's design principles and properties, and review its past, present, and future uses. The authors point out risks and regulatory 5 issues as Bitcoin interacts with the conventional financial system and the real economy.
The authors acknowledge the deflationary risk of the fixed money-growth rule inherent in Bitcoin's design, and point to the difficulty as to whether decentralized virtual currencies can be designed with monetary policies that include feedback or discretion.
The rapid appreciation of the exchange rate and its high volatility have been posed as major concerns for the viability of Bitcoin's use as a currency. Yermack (2015) examines its historical trading behaviour to analyze whether it behaves like a traditional existing currency. Bitcoin suffers from much higher exchange rate volatility than the exchange rates of traditional currencies. Moreover, based on a purely statistical analysis of the Bitcoin exchange rate, Cheung et al. (2015) and Cheah and Fry (2015) express their concern regarding its "bubble-like" behaviour; see also Weber (2016). Moreover, Yermack (2015) documents Bitcoin's exchange rate to show virtually zero correlation with other existing currencies, reducing Bitcoin's use for risk-management purposes and making it exceedingly difficult for its owners to hedge. Bitcoin also lacks access to a banking system with deposit insurance, and it is not used to denominate consumer credit or loan contracts. In this context, Bitcoin appears to behave more like a highly speculative investment than a currency. Schuh and Shy (2016) report that, according to the 2014-2015 Survey of Consumer Payment Choice (SCPC), about half of US consumers are aware of Bitcoin. About one percent or less of US consumers have ever owned (adopted) virtual currency, and most of them used it recently to pay a person (most common) or a merchant. Expectations of future price seems to be the biggest driver of adoption, and those consumers who are especially "interested in new technologies" are more likely to adopt. See Badev and Chen (2014) for more on the technical background and empirical regularities of Bitcoin. Ali et al. (2014) examine the economic incentives of adopting virtual currencies and assess potential risks to monetary and financial stability. A key attraction of virtual currencies at present is their low transaction fees. But these fees may increase as usage grows and may eventually be higher than those charged by incumbent payment systems. They also discuss 6 various factors that influence the value of virtual currencies, such as risk-return trade-offs, transaction costs and relative benefits, and habit formation. 2 In an excellent primer on the mechanics of Bitcoin, Velde (2013) regards Bitcoin as an elegant implementation of virtual currency -controlling its creation and avoiding its duplication simultaneously -but questions whether it can truly rival or replace existing currencies. The paper argues that, should Bitcoin become widely accepted, it is unlikely that it will remain free of intervention by public authorities, if only because the governance of the Bitcoin computational code and "mining" protocol is opaque and vulnerable. 3 Network effects may play an important role in the adoption of a virtual currency. (2015) describe the emergence of virtual currency in the historical context of an ever-present competition among different forms of money. In the context of competition among virtual currencies, Gandal and Ha laburda (2014) empirically investigate the presence of a "winner-takes-all" effect for the most popular virtual currency. Their data suggest that this effect is only dominant early in the market, but this trend is reversed in a later period, which may be consistent with the use of virtual currencies as financial assets.

Ha laburda and Sarvary
Moreover, Sauer (2015) also discusses the demand for virtual currency in the context of (one-sided) network effects.
Most papers lack a formal treatment of the economics behind the exchange rate of virtual currencies, unveiling the links between speculative behaviour, currency creation, network effects, and real growth in transactions for goods and services. This paper tries to bridge that gap.

Model Preliminaries: Quantity Equation
In the model in Section 4, we consider two currencies: an established currency, say e , and a virtual currency, say B. But before we proceed, we first turn to the well-known transactions version of the quantity equation popularized by Fisher (1911), i.e., where V B t denotes the velocity of virtual currency B, defined as the average number of times each unit of virtual currency is used to purchase real goods and services within period t. T B t is the quantity of real goods and services purchased with virtual currency B during period t, and P B t is the weighted average price. M B t denotes the (nominal) quantity of money defined as the number of units of virtual currency B. Eq. (3.1) holds by definition within any given period t; see, e.g., Friedman (1970). Essentially, the expression follows directly from defining the concept of the velocity of virtual currency B. It does not depend on any behavioural assumptions.
Some manipulation of the left-hand side of Eq. (3.1) gives Although (3.2) is, for simplicity, derived for a single established currency, it can easily be extended to cover multiple established currencies instead.
In the model, we essentially assume the established currency to be the unit of account. 5 Let S e /B t denote the exchange rate, i.e., the number of units of the established currency one pays to obtain a single unit of virtual currency in period t. We assume that the prices of goods and services expressed in virtual currency are completely determined by the exchange rate and their price level in the traditional currency as . This interpretation connects well with the practice of many online stores to adjust prices quoted in virtual currencies instantly to the latest available exchange rate. The prices in these stores expressed in virtual currency are perfectly flexible with respect to changes in the exchange rate. Using this assumption in Eq. (3.2) yields the exchange rate for any t as Moreover, without loss of generality, the average velocity, the V B t , can be rewritten as a weighted average of the velocity of the units of virtual currency used to settle payments for goods and services, V B * t , and the velocity of those not used to settle the payments for goods and services. Note that the latter equals zero, since velocity was defined as the average number of times each unit of virtual currency is used to purchase real goods and services. Formally, where Z B t ≥ 0 is the number of units of virtual currency not used to settle the payments for goods and services during period t. Essentially, Z B t units are "stored value" and we suggestively refer to Z B t as the "speculative position" in virtual currency. After all, speculators buy units of virtual currency in the hope of making a profit by selling them in the future. Such a strategy involves only the exchange of the established currency against the virtual currency and does not involve the use of virtual coins for the payment of real goods or services. These speculators may include both "pure" speculators, who do not use virtual currency for any real transaction, as well as merchants and consumers who hold larger positions in virtual currency than is demanded to execute payment transactions for goods and services.
Combining Eqs. (3.3) and (3.4) gives the level of the exchange rate as ( 3.5) This equation describes the effect of three factors affecting the exchange rate of virtual currency that are common in the context of the value of money: the exchange rate increases in the volume of the payments for goods and services with virtual currency, T B * t ; and it decreases in the velocity of virtual currency, V B * t , and in the total quantity of virtual currency, The expression in (3.5) also includes a fourth factor affecting the exchange rate of virtual currency, which is the quantity of virtual currency held in the speculative position, Z B t . Eq. (3.5) shows that the speculative position effectively reduces the quantity of virtual currency available to facilitate real payments and, therefore, increases the value of virtual currency. The impact of speculation on the exchange rate is shown in Figure 2. In the absence of speculation, the exchange rate for virtual currency would be determined only by the intersection between the curve and the vertical axis in Figure 2. This point on the curve corresponds to the case in which all units of virtual currency are used to facilitate payments for real goods and services. In the presence of speculation, the exchange rate is higher because fewer units of virtual currency are available to facilitate real payments.
Of course, assigning a role to speculation in determining the value of money is an old notion. For example, Fisher (1911)   where the latter is measured by a rough proxy, i.e., the units of virtual currency that will remain in dormant accounts for an extended period. For each point in time, it is subsequently possible to determine a hypothetical exchange rate in the absence of speculation by the intersection of the curve and the vertical axis in Figure 2. The intersection provides a rough estimate of the exchange rate if all units of virtual currency were in use to facilitate payments for goods and services, and is presented for each point in time by the solid grey line.

Model
The model contains two building blocks. The first building block is based on two-sided market theory that tries to identify which factors drive the uptake of virtual currency as a payment instrument. 6 The more consumers are willing to transact using virtual currency, the more merchants may wish to accept this type of currency for payment. And, vice versa, the more merchants are willing to accept virtual currency, the more likely that consumers will start using it. These two-sided cross-group externalities play an important role for optimal pricing and therefore the use of virtual currency to make payments.
The second building block concerns the behaviour of speculators. The investment decision of a speculator is modelled as a trade-off between investing in a risk-free bond denominated in the established currency or speculation on the future value of virtual currency.

Model set-up
Essentially, the model is a one-period model. Period t refers to the initial state; period in which case, the units of virtual currency will be worthless. In the case of success with probability q, the stationary equilibrium is such that the frequencies of virtual currency users (i.e., consumers and merchants) are equal to their equilibrium values. 7 Lastly, similar to the Bitcoin supply process in practice, we assume that the number of virtual currency units at t + 1 follows a predetermined growth rule, such that

Two-sided virtual currency payment network
Organized as an electronic payment network, the underlying idea is that virtual currency creates economic value by enabling transactions between merchants and consumers. Assume that consumers and merchants derive net utility from using a monopolistic virtual currency network where N j,t is the number of users from the other side who use the virtual currency network in period t, α i > 0 is the benefit that user i enjoys from transacting with each user on the other side, β i is the fixed benefit the user obtains from connecting to the network, and p i,t is the fixed "membership" fee that is levied by the network as a lump-sum charge. 8 The users of virtual currency are assumed to be heterogeneous in their fixed benefit for network services. This heterogeneity is described by a cumulative density function F i (·) with probability density function f i (·), i = m, c.
On the consumer side, the benefits from joining a virtual currency network may be larger for more technologically adaptive consumers. Such consumers may incur a lower cost when 7 In their seminal paper, Kareken and Wallace (1981) show that the nominal exchange rate between two perfectly substitutable "fiat" currencies is indeterminate. This indeterminacy result would not hold in our case since the two currencies are different in terms of their risk profiles as well as in terms of the liquidity services provided by the virtual currency payment network.
8 Although, for simplicity, it is assumed here that the utility U i is linear in the number of users N j , specifying a non-linear relation would not qualitatively change the results (Armstrong, 2006;Anderson, 2012). However, despite the linear specification, the number of consumers N c generally increases with the number of merchants N m in a non-linear way, and vice versa.

13
adopting new payment technologies. Benefits may also be larger for consumers who face higher costs from traditional payment networks for a variety of reasons. Costly cross-border payments, such as sending remittances, may be one such reason. Privacy and anonymity may also increase the private benefits from using virtual currencies. On the merchant side, heterogeneity in the benefits from joining a virtual currency network may depend on various factors, such as merchant size and distribution channels. It is well documented that large retailers pay high merchant service fees for accepting certain credit and debit cards. Regarding distribution and business models, online stores may face lower implementation costs from accepting virtual currency than traditional stores.
Based on the utility-maximizing behaviour of potential users, the number of users i that connect to the network is given by where N i is the maximum number of potential users of type i, and where β i,t = p i,t − α i N j,t .
Given costs C i that are incurred by the network when users join the virtual currency network, (per-period) total network profits are given by Substituting p i,t = α i N j,t + β i,t in Eq. (4.3), for network profits, we write: Thus, network profits can be expressed as a function of only (β c,t , β m,t ).
The interior solution (β * c , β * m ) that solves the first-order conditions in turn determines profit-maximizing numbers (N * c , N * m ) of virtual currency users and profit-maximizing fees (p * c , p * m ). Profit-maximizing network fees satisfy . (4.5) This pricing rule shows that profit-maximizing fees for each side of the market are equal to the cost of providing the service (C i ), adjusted downward by the external benefit to the other side (α j N * j ), and adjusted upward by a factor ((1 − F i )/f i ) that is related to the elasticity of participation.
To pin down the total number of transactions, the payment literature often assumes a multiplicative relation, N c,t · N m,t , relying on an "independence" assumption. For the key implications of our model, the choice of a specific functional form that describes the number of transactions is not critical. What is essential, however, is the relatively mild assumption that the value of virtual currency units necessary to make real transactions, i.e., T B * t /V B * t , increases with the adoption of virtual currency by consumers and merchants. 9 One specific form to implement this is the following. By relying on the widely used (4.7)

Speculative behaviour
We presume that the central bank is successful in the sense that changes in the price level P e in terms of the traditional currency are anticipated with certainty. Speculators are therefore assumed to choose their holdings of units of virtual currency while maximizing a second-order approximation of their utility with respect to their next-period wealth, W t+1 , where wealth is expressed in the established currency as where E(W t+1 ) and σ 2 (W t+1 ) denote period's t expectation and variance of wealth at time t + 1, respectively. For simplicity, we assume the same positive coefficient of risk aversion γ across all speculators. 10 The investment decision of the speculator is modelled as a trade-off between investing in a risk-free bond denominated in the established currency and speculation on virtual currency. 11 Under the assumption of absence of lending and borrowing in 10 This assumption can easily be relaxed by replacing the expression γ/N s,t in the solution with the expression i 1/γ i −1 > 0, where γ i = 0 denotes the coefficient of risk aversion of speculator i; see, e.g., Viaene and De Vries (1992). This modification does not change the essence of the model much. Both expressions measure, more or less, the "risk aversion aggregated across speculators." We opt for the simplest expression. 11 We refer to Hirshleifer (1988) for a more sophisticated model of speculation with multiple investment opportunities in a mean-variance framework. The results of Hirshleifer (1988) show that, given some fixed costs for entering into a speculative position, the size of the speculative position depends on systematic risk and residual risk. In our study, the size of the speculative position will depend on the total risk. virtual currency, the return on a position in virtual currency in terms of the established currency is determined only by the change in the exchange rate. Hence, next period, the wealth of a speculator investing in z B t units of virtual currency is where W t is wealth at time t, R denotes the gross return on bonds denominated in the traditional currency and whereS e /B t+1 denotes the uncertainty about the future exchange rate S e /B t+1 . We assume that individual speculators take the current exchange rate S t as given. In summary, the investor's optimization problem is the maximization of (4.8) with respect to z B t . Solving the speculator's optimization problem gives the optimal number of units of virtual currency as . (4.10) The expression in Eq.
In other words, in the aggregate, speculators choose to take a positive position in virtual currency only if the discounted expected value exceeds the hypothetical value of the current exchange rate in the absence of speculation. Otherwise, they prefer to short sell the currency, which they cannot do collectively, and their optimal aggregate position equals zero.
In summary, rewriting Eq. (4.11) gives the exchange rate at which speculators absorb The price that speculators are willing to pay for virtual currency equals the (discounted) expected future exchange rate minus a risk premium for the uncertainty in the future value of the speculative position in virtual currency.

Partial equilibrium
The level of the exchange rate is pinned down by Eqs. (3.5) and (4.13). These two equations act, respectively, as supply and demand schedules for units of virtual currency.  In contrast, if the condition in (4.12) holds because speculators are optimistic, or because virtual currency usage is still low, then the analytical solution of the equilibrium exchange rate is The equilibrium exchange rate in Eq. (4.14) is determined by two main components. The first component, S t|T B * t =0 , represents the "purely speculative" exchange rate. It is the exchange rate obtained from Eq. The second component is the current value of virtual currency that is necessary to facilitate payments, i.e., T B * t /V B * t . The larger the amount necessary for payments using virtual currency, the higher the exchange rate. The more the value of virtual currency is absorbed by payments, the lower the exchange rate risk that has to be absorbed by speculators, and thus the higher the level of the exchange rate of virtual currency. Figure 3, panel (b) reports the upward pressure on the current exchange rate from a greater number of payments using virtual currency by showing the new equilibrium after an increase in T B * t /V B * t , causing a shift in the upward-sloping curve.

Impact of speculative environment
Virtual currency has suffered from highly volatile exchange rates compared with the exchange rates of established currencies; see, e.g., Yermack (2015). The high level of volatility is often attributed to the behaviour of speculators. Speculators herding towards a new opportunity and erratic changes in their beliefs may cause large swings in the exchange rate.
Of course, the mere presence of speculators cannot explain the higher level of volatility, since speculators may invest in any currency. Nevertheless, it is possible that the exchange rate is especially sensitive to changes in speculators' beliefs in the early-adoption phase, when few real payments are settled in virtual currency.
The impact of speculators' beliefs regarding the exchange rate can be assessed by taking the derivative of Eq. (4.14) with respect to the speculators' expectations of the future exchange rate as This equation shows which determinants affect the change in the exchange rate following a shock to speculators' beliefs about the future value of virtual currency.
Shocks in the expectations of speculators as to the future value have a larger impact on the exchange rate in the early phase of a virtual currency. From Eq. (4.16), it is not difficult to derive that ∂S changes in speculators' beliefs regarding the future value of virtual currency is expected to become smaller once a virtual currency is more widely used. This can also be observed from In a similar way, it can be shown that the entry of speculators drives up the current exchange rate. In the model, this is equivalent to the entry of new speculators or having current speculators take on more risk: both correspond to a decrease in risk aversion aggregated across speculators, i.e., γ/N s,t . Given the expectations, the entry of new speculators turns the downward-sloping line in Figure 3 counter-clockwise at the intersection with the y-axis. Taking the derivative of Eq. (4.14) with respect to the speculators' aggregated risk aversion yields This expression is negative for any γ/N s,t : The first term in parentheses is positive, because ) from Eq. (3.5). In other words, an inflow of more speculators or an increased risk appetite are both expected to increase the exchange rate of virtual currency.
The inflow and outflow of speculators in the virtual currency market have a larger impact on the exchange rate in the early-adoption phase of virtual currency. Although the derivative in (4.17) remains negative, it gets closer to zero as the transaction volume T B * t /V B * t increases.
In other words, the model suggests that the exchange rate becomes less sensitive to the inflow and outflow of speculators as the use of virtual currency by merchants and consumers increases in intensity. To summarize, the impact of speculative behaviour on the exchange rate of virtual currency becomes smaller as virtual currency is used more frequently to make real payments.

Rational Equilibrium Exchange Rate Analysis
The (partial) equilibrium for the exchange rate derived in Eq. (4.14) does not rely on an assumption of rational expectations on the part of speculators. In this section, we show how the rational equilibrium exchange rate can be obtained by combining this specification with the payment network equilibrium outcome.
When the number of transactions has reached its stationary equilibrium (and no more growth can be expected), the condition in (4.12) implies that speculators will no longer have an incentive to invest in virtual currency. Hence, all available units will be used to purchase real goods and services, i.e., Z B t+1 = 0. While assuming rational expectations among speculators, Eq. (4.7) implies that the expected future exchange rate in period t In other words, the larger the success probability q, the larger the expected exchange rate, and, conditional upon q > 1/2, the lower the volatility of the future exchange rate. Substitution of Eqs. (4.7), (5.1) and (5.2) into the partial equilibrium for the current exchange rate in Eq. (4.14) gives 3) The first term in Eq. (5.3) is the expected future exchange rate. Its value depends on the success probability q, and on the -conditional upon success -expected long-term growth in the adoption of virtual currency towards N * c .
The second term in Eq. (5.3) is the "discount factor" applied by investors. The δ t represents the hypothetical discount factor in case no real transactions are settled using virtual currency, i.e., if N c,t = 0. This hypothetical discount factor is calculated as The full equilibrium equation in (5.3) shows that the actual discount factor is an increasing function of current adoption of virtual currency N c,t . 14 Actual adoption increases the exchange rate of virtual currency via this channel. Moreover, the exchange rate of virtual currency does not suffer from "money illusion." Given all other parameters in Eqs. (5.3) 14 It is also possible to derive a, albeit somewhat less elegant, closed-form solution for the exchange rate under a (CRRA) specification assuming log-utility for speculators. Such an alternative specification leaves our results qualitatively unchanged, but requires a mild condition of having sufficient current (aggregate) wealth W t in the hands of speculators relative to the current value of virtual currency necessary to facilitate payments T B * t /V B * t .

Discussion
In this section, we discuss several extensions of the model.

Speculating consumers
Speculators and users of virtual currency making actual payments are assumed to be different agents in the model. In practice, speculators and users are sometimes the same agents; see, e.g., Johnson (1960). This does not change the essence of the model. We show this by considering the case of speculation on the consumer side. Suppose that the decision to join the network provides consumers with the opportunity to speculate on the value of virtual currency. Presuming that users face the same investment decision as speculators, then their net utility derived from using the monopolistic virtual currency payment network would be where ∆ c,t is the additional utility derived from the opportunity to speculate on the value of virtual currency, which equals for the optimal level of the speculative position z B t , which is derived in Eq. (4.10). 15 The steady-state equilibrium will not be changed by giving consumers the opportunity to speculate on the value of virtual currency. The reason is that the additional utility derived from the opportunity to speculate on the value of virtual currency ∆ * c = 0 equals zero in the steady state, since no further appreciation of virtual currency is to be expected. In the steady state, both speculators and consumers will reduce their positions held for speculation to zero, and therefore the expression in (6.2) will equal zero. If the utilities of the users are unchanged, the equilibrium number of users, N * c and N * m , will also be unchanged. Therefore, the future exchange rate will be unchanged.
Nevertheless, the opportunity for consumers to speculate on the value of virtual currency will affect the current exchange rate. This is because the aggregate speculative position of both speculators and consumers will be equal to z B t (N s,t + N c,t ). When working through the entire equilibrium solution, this basically means that the N s,t in the denominators in Eqs.
(5.3) and (5.4) will be replaced by N s,t + N c,t . 16 In other words, the additional speculative demand from the consumer side will increase the discount factor and, therefore, provide upward pressure on the level of the current exchange rate.
The opportunity of consumers to speculate may change the dynamics of the model. In terms of comparative statics for the current exchange rate, an increase in the consumer base of the virtual currency network will not only increase the value of the transactions using virtual currency, but will also increase speculative demand. In other words, both curves in the diagrams in Figure 3 will shift upward as a consequence of more consumers using virtual currency. Such an increase in consumer usage is illustrated in Figure 4. Besides the upward shift of the upward-sloping curve as a result of higher transactional demand stemming from more consumers using virtual currency, the inflow of new speculating consumers also turns the downward-sloping schedule representing speculative demand counter-clockwise. This further increases the current exchange rate. The counter-clockwise turn of the speculative demand schedule shown in Figure 4 does not presume higher expectations regarding the exchange rate as new consumers enter the virtual currency network. Such a change in Moreover, allowing speculating consumers results in a higher level of utility derived from joining the network for early adopters, since these consumers do not only benefit from using virtual currency, but also from a non-zero speculative position in virtual currency. Therefore, higher expectations regarding the future value of virtual currency will not only increase the positions of speculators, but also draw more consumers towards the network, which may result in a faster convergence to the steady-state use of virtual currency.

Intra-group benefits
Positive intra-group externalities due to, e.g., peer-to-peer payments, such as remittances, do not change the essence of the model. Although consumers are not directly affected by other consumers in the original specification, in equilibrium, consumers exert a positive (in-27 direct) externality on each other through their impact on merchant acceptance. To explicitly account for positive intra-group externalities, we may modify the net utility from joining the virtual currency network to Here, the parameter α ii > 0 denotes the benefit that user i enjoys from interacting with other (same type) users on the network. Essentially, given prices p c,t and p m,t , the extension in Eq.
(6.3) yields two demand equations, N c,t = N c (p c,t , N m,t , N c,t ) and N m,t = N m (p m,t , N c,t , N m,t ), in two unknowns (N c,t , N m,t ). Solving this system gives N * c,t (p c,t , p m,t ) and N * m,t (p c,t , p m,t ). Substituting these optimal demand functions into the network's profit function in Eq. (4.3) and maximizing then yields optimal fees p * c,t and p * m,t . Qualitatively, this extension will not alter our results.

Concluding Remarks
This paper proposes an economic framework for analyzing the exchange rate of a virtual currency and its key drivers. The paper documents three main determinants. First, the current use of virtual currency to make real payments. Second, the decision of forward-looking investors to buy virtual currency (thereby effectively reducing its supply) given expected growth in virtual currency transactions. Third, the elements that jointly drive consumer adoption and merchant acceptance of virtual currency, which determine expected long-term growth in usage. In particular, we show that the equilibrium exchange rate depends on both a purely speculative component pinning down a "floor" under the exchange rate and a transaction component that affects the exchange rate risk absorbed by speculators. More widespread use of virtual currencies by merchants and consumers lowers the impact of speculative behaviour and therefore stabilizes the exchange rate.
Our analysis illustrates that a steep increase in the exchange rate due to speculative 28 motives is exactly what one can expect at the introduction of a potentially successful virtual currency. Moreover, the current high levels of volatility seem to be a symptom of early development: theoretically, volatility is expected to drop if the adoption by consumers and merchants increases. The future will show how much volatility will drop. The fixed supply of virtual currency may suggest its volatility will reflect that of other commodities (see, e.g., Selgin, 2014) rather than that of traditional currencies whose quantities are managed by central banks. Moreover, the model also shows that, conditional upon survival, deflationary virtual currency prices may be expected during the early-adoption stage.
Our study is only a first step in trying to understand the underlying economics of virtual currencies. Empirically, little is known about the actual number of payments in virtual currency for goods and services. Reliable time series for user and acceptance statistics are still lacking and need to be developed, as these will be -without a doubt -an important input for studying the behavioural aspects of using virtual currencies. Competition among virtual currencies raises a further issue of whether only a few virtual currencies will ultimately dominate a global market where network effects play an important role. Our framework suggests that the exchange rates of various virtual currencies may diverge widely, depending on adoption, transactional demand, the quantity and growth rate of monetary units, speculative demand and network stability (survival probabilities). Finally, analyzing the empirical determinants of switching behaviour from traditional means of payment to virtual currencies may lead to interesting cross-fertilizations between the payment choice literature and the literature on currency substitution. Much more research remains to be done, and we are planning to go down that route.

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Appendix A: Estimate of the Speculative Position The hypothetical exchange rate in the absence of speculation in Figure 1 is based on a very rough estimate of the magnitude of the speculative position. Theoretically, the magnitude of speculative position is defined as the number of units of virtual currency not used to settle payments for goods and services. A rough estimate of this amount is obtained as follows. At each day t, we obtain from the public ledger the number of bitcoins associated with addresses that are not involved in any transactions over the next three months. This is the empirical proxy for Z t in Eq. (3.5). Given this estimate of the speculative position on day t, the total number of bitcoins in circulation on day t, and the actual exchange rate on day t, it is not difficult to calculate from Eq. (3.5) the hypothetical exchange rate for a speculative position with a zero balance, while assuming a fixed ratio T B * t /V B * t . Figure 1 presents this level as the grey line. The dashed line is based on an exponentially weighted average with a daily smoothing parameter of 0.997 (this corresponds to an observation one quarter ago receiving a 75 percent weight of the weight of the most recent observation).
We described our estimate of the size of the speculative position as a rough estimate. The reason is that there are several issues with this estimate. Our approach relies on subtracting the bitcoins not involved in transactions for real goods and services from the total amount.
Even though all transactions are recorded in the public ledger, this does not directly reveal which units of virtual currency have not been used to settle payments for real goods and services. All recorded transactions represent a larger set than the transactions for processing payments for real goods and services, since they may include, among others, transactions with exchanges, transactions with other speculators, and transactions between addresses without a change in ownership (i.e., someone moving bitcoins from the left to the right pocket). In this respect, our approach is likely to result in an overestimation of the number of bitcoins involved in processing payments for real goods and services, and, therefore, in an underestimation of the size of the speculative position. In this context, the level of the grey line in Figure 1 is probably too high.

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Moreover, the raw data on the total number of bitcoins in circulation is not corrected downward for units that are effectively permanently withdrawn from circulation. This refers to units that are permanently lost because users lost access to the cryptographical keys that are necessary to transfer bitcoins from one address to another. Of course, the number of Bitcoin units that are truly permanently lost is difficult to measure. However, if such data were available, following Eq. (3.5), a correction based on such data would result in an upward adjustment of the exchange rate in the absence of speculation.