Network effects have been defined as the circumstances in which the net value of an action (e.g. consuming goods, subscribing to telephone services) is affected by the number of agents (i.e. participants in the network) taking equivalent actions [17]. Network effects appeared as a concept in early of 20^th century by Theodore Vail, the President of Bell Telephone and were popularized in early 1980s by Robert Metcalfe in the homonymous law [18]. According to Metcalfe’s law, the value of a network is proportional to the square of the number of nodes, i.e. the number of network users [10]. Moreover, Liebowitz and Margolis [9] coined the term of “synchronization value”, which is the additional value derived from the fact that the users of the network are able to interact with other users and this value is the essence of network effects. For example, Bitcoin’s value is limited if it is composed of only two users, while Bitcoin’s value raises for each user as more and more users join the system and interact with each other. Network effects are believed to be endemic in the digital economy. Digital economy experiences problems that are different in character from the problems that have been solved by markets for more ordinary commodities in the traditional economy [9] [19] [20]. In the traditional economy, the rarer a product is, the more value it has (e.g. diamonds, oil). On the contrary, in the digital economy, a digital product in abundance has greater value: the more something is demanded and the more is expected to be demanded then the more valuable it becomes (e.g. mobile phones, social media).

This feature of digital products is known as “network externalities”, which are the economic concept of the external consequences derived by the network effects and are considered the drivers of the networked economy [21]. The term was coined by Jeff Rohlfs [22]. A network externality is the effect of a transaction between two parties on a third party which is not involved in the execution of that transaction [23] and depends upon the number of other users who are in the same network [19]. Network externalities are the side effects or consequences on the user of a good/service that the existence of other users utilizing the same or similar goods/services has. The enchantment of network externalities is that they often come out as surprise and as a byproduct that was not calculated or foreseen in any way [23]. Network externalities are exhibited wherever a user enjoys benefits or suffers costs from changes in the size of an associated network [17] ; thus, they can be positive and negative. Positive network externalities are the main reason for building networks. However, the same phenomenon can be both positive and negative, depending on the role of the observer [23]. For example, in Bitcoin, as more users join the network, positive network externalities arise for exchange services since the increase in transactions will grow their profits, but these increased transactions cause congestion in the network, resulting in the augmentation of transaction fees, i.e. a negative network externality for the users.

Network effects and network externalities have been used interchangeably in the literature [16] ; however, network effects are not always network externalities [17]. The difference between them lies in whether the impact of an additional user on other users is somehow internalized [9]. Internalizing an effect means that it is no more directed towards a third party. Any network effect is an externality only if not internalized [9] [16]. When network effects are internalized, they are no longer externalities [9]. The usual ways to achieve internalization is via fines & taxes or subsidies. For example, network externalities that derive by joining a network can be internalized by subsidizing early adopters, who would not otherwise join this network if they had to face the full marginal cost of their participation in the network.

Network effects [7] [8] and externalities [24] [25] appear in the case of Bitcoin as more participants join the network. Bitcoin’s network has multiple participants, which can be grouped in two basic categories: enablers (miners, exchanges, and developers) and users (merchants, consumers, investors, and speculators) [26]. In the case of the Bitcoin’s adoption, the network effects mean the more merchants accept bitcoins, the more consumers are likely to pay with bitcoins and vice versa. However, Bitcoin faces a network effects problem, since the benefit in using bitcoins is positively correlated with the number of users: if few merchants accept bitcoins, the benefits for consumers to use products paid with bitcoins are low; if few consumers use bitcoins, merchants have little incentive to accept bitcoins [11] [12]. This benefit in using cryptocurrencies is based on their value, which is linked to the usage to buy goods and services paid with them. Because of this value, most owners of cryptocurrencies want to hold them instead using them to consume; therefore, cryptocurrencies are not widely used in commerce [14]. The utility of a cryptocurrency is based on the mutual concepts of “acceptance” and “usage” between merchants and consumers and the biggest challenge is convincing both parts to use it in exchange for goods and services [12]. In order for Bitcoin to achieve widespread adoption, people have to use it, but the problem is that people who own bitcoins prefer to hold them as an investment instead of using them for consumption because they expect a higher value [13].

Since the number of Bitcoin consumers is small, it would be expected that few merchants will accept bitcoins as a means of transaction. However, as it can be seen in Figure 1, which depicts the volatility of the Bitcoin’s closing price in USD [27] [28] and the number of venues that accept bitcoins [15], during a five-year time period, there is a steady increase of the venues, despite the lack of interest for consumption, which is due to investment and speculative trends that are triggered by the Bitcoin’s price volatility, as aforementioned. More specifically, there are two opposite facts, which constitute a paradox in the Bitcoin network effect: the small consumer base and the increasing number of merchants.

The essence of the paradox lies in the management of expectations, which is a feature of the networked economy. In traditional markets, equilibrium is explained by the balance between costs and demand, between marginal costs and marginal utility. In networked markets, there is also equilibrium to be achieved between actual demand and expectations of total demand [21]. Any investor or

speculator who expects Bitcoin returns is a potential consumer, while merchants expect to increase their sales turnover. When the equilibrium between those expectations will be achieved, the size of the network will be optimal for both parts, meaning that the Bitcoin’s widespread adoption as a means of transaction is achieved. At this point, it is worth clarifying the existence of the Bitcoin’s network effects paradox in the short and long term. Bitcoin price is highly volatile; however, as it is shown in Figure 1, the sudden increase or decrease (shock) in the Bitcoin price in the short term is accompanied by a long-term trend of increase or decrease in Bitcoin price. If the long-term trend of the Bitcoin price is increasing or it is expected to increase, despite the short-term shocks during this trend, the demand for bitcoins is increased, thus the number of the Bitcoin users (i.e. potential consumers) rises; therefore, an increasing number of venues are expected and the paradox is eliminated. On the other hand, if the long-term trend of the Bitcoin price is decreasing or it is expected to decrease, the demand for bitcoins is also decreased, thus the number of the Bitcoin users is reduced; therefore, an increase in the number of venues is contradictory and this constitutes a paradox.

The existence of this paradox can be explained possibly by the fact that the negative network effect of Bitcoin’s price volatility is not yet internalized. Consequently, negative network externalities exist (e.g. speculative and investment trends) and the equilibrium network size is smaller than optimal. This is the reason for which Bitcoin’s price volatility remains one of its main barriers to mass adoption [3] [4] [5] [6]. Nevertheless, Bitcoin has shown its ability to internalize network effects so far. For example, until today internalization in Bitcoin has been applied in two cases: 1) the case of early miners, who were subsidized (rewarded) to join Bitcoin by receiving greater reward for each mined bitcoin according to the Bitcoin protocol. Since the proof-of-work mining system of Bitcoin requires a great amount of computational power, miners should be financially sustainable in order to face expenses for hardware investments and electricity power, as they are the basic enablers of the Bitcoin network [29] and are dependent on the user participants, and 2) the case of early users of Bitcoin, who were also motivated to join the network, since the transaction fees were minimal compared to the transaction fees of other payment networks (e.g. credit cards, PayPal). However, as more and more users were joining the network and transactions increasing, negative network externalities derived from the fact that the network was overloaded. The transactions’ verification was delayed for a long period of time, because the size of the blocks was not enough to support the increased amount of transactions. This fact conduced to a significant increase of the transaction fees, in order miners to be motivated to accelerate the verification of the transactions, by using more computational power. These negative network externalities for users were internalized by the Bitcoin forks, which were all attempts to increase the transaction capacity of the network, thus resulting to incentivize more users to adopt Bitcoin.

The above examples show that the Bitcoin’s ability to internalize the network effects eliminates the derived network externalities and conduces to an increase of the users. When more people use Bitcoin in exchange for goods and services, the speculative trends will decrease and the volatile cryptocurrency should start to stabilize [30].

The aforementioned paradox is related with the concepts of price volatility, users’ (merchants and consumers) adoption, and network effects. For the investigation of the paradox, the following three variables were taken into consideration.

One variable is the closing prices of Bitcoin, whose variances indicate the Bitcoin’s price volatility and represents the negative network effect that affects the observed paradox. Price volatility is a negative network effect, whereby too many investors and speculators can cause the network to be useless for payments, thus reducing the benefit in using bitcoins in exchange for goods and services for the other users of the network. Bitcoin’s price volatility is internally driven, meaning that the Bitcoin’s market returns are driven by its market participants and this feature indicates that Bitcoin’s is still in an early life-cycle stage [31]. An internal and structural reason underlying Bitcoin’s price volatility is its perfectly inelastic supply (due to its fixed supply, as it is defined by the system’s protocol), which results in small changes in demand to cause large price movements [32]. These large price movements in turn conduce to expectation for returns. Therefore, speculative demand is considered to be the primary driver of Bitcoin’s price volatility [31] [32], which hinders consumption and should also be expected to hinder the interest for merchants to accept bitcoins, while an increase in the number of venues is observed, as aforementioned.

Another variable is the number of venues that accept bitcoins, whose upward trend indicates the merchants’ adoption and represents a positive same-side network effect [33]. This network effect in conjunction with consumers’ adoption forms a positive cross-side network effect. However, this network effect can be actually evaluated only by the side of merchants, since there are recorded data only about the venues that accept bitcoins, while the number of Bitcoin consumers is not officially recorded. According to Suomi [23], measuring networks is difficult and overall measures that might allow for comparing different types of networks are rare, if any. This study is akin to other empirical studies [34] but it was considered not to use, as variables, the number of unique transactions or addresses [7] [34] nor the number of digital wallets and nodes. Because, since one user can have more than one digital wallet, which in turn may consist of many unique addresses, and conduct many transactions of low value, but the user is still only one with a given total amount of bitcoins, these variables are perhaps a way to measure the size of the network in the absence of other recorded data but may lead to misconceptions about the actual size of the network. Also, the number of unique transactions, unique addresses, and the number of digital wallets refer to all types of “users” (merchants, consumers, investors, and speculators) without distinction. Therefore, these variables are not suitable to evaluate the observed paradox of this study. Finally, in the literature the number of network nodes is considered the only actual comparable figure [23]. However, in the case of Bitcoin, since the number of nodes only refers to mining bitcoins, transactions’ verification and even transactions between them (e.g. for remittances or just for sending bitcoins from one wallet to another of the same node), it does not reflect the interaction between merchants and consumers.

The last variable is market capitalization, which is related to the first aforementioned variable as it is calculated by multiplying the closing price by the circulating supply of bitcoins [27]. Therefore, Bitcoin’s market capitalization follows its price volatility and it is expected to affect the observed paradox similarly. Market capitalization is the economic result that derives from the trading interaction of Bitcoin’s users. Thus, market capitalization represents the additional value of interaction among the users of the Bitcoin network, i.e. the “synchronization value”, which derives by the existence of network effects, according to Liebowitz and Margolis [9], as mentioned above. Market capitalization determines the total value of the Bitcoin network [35] and is used to rank its relative size compared to other cryptocurrencies or traditional forms of payments (e.g. credit cards, fiat currencies); thus, network externalities derive, e.g. more digital wallets, exchange services appear for a cryptocurrency with bigger market capitalization than others. The examination of the relations among the above three variables will help to elucidate the observed network effects’ paradox of this study.

For the integration test of the series of the examined model, we use the Dickey-Fuller [38] [39] tests, as well as Phillips-Perron [40] test. Dickey-Fuller [38], through Monte-Carlo simulation, found a suitable asymmetric distribution that used in unit root test. This distribution can be used to separate an AR (1) model from an integrated series. In the case that a time series follows an autoregressive model of higher order, AR (p) then we use the Augmented Dickey-Fuller (ADF) [39] test, which comprises the lags of dependent variable that corrects the autocorrelation of the residuals. Phillips [41] and Phillips-Perron [40] suggest a non-parametric test for the estimation of coefficients’ model making some changes to t-statistic. Phillips-Perron test (PP) differs from the ADF test mainly on the examination of autocorrelation and heteroscedasticity of errors. In other words, Phillips-Perron methodology deals with a non-randomness of the residuals, modifying t distribution using non parametric methods [42].

Johansen [43] and later on, Johansen and Juselius [44] [45] proposed a technique using VAR models (Vector Autoregression), where the maximum number of cointegrating vectors among a group of variables can be defined. This technique is known as Johansen maximum likelihood cointegration test. Maximizing likelihood function with the constraint that there are r cointegrating relationships among variables, we can proceed to cointegration tests applying the criterion of maximum likelihood with the following assumptions:

H₀: r cointegrating vectors

H₁: r + 1 cointegrating vectors

Johansen and Juselius [44] for the above test created two statistical criteria, namely trace statistic and maximum eigenvalue. The asymptotic critical values for these tests can be found in MacKinnon, Haug, and Michelis [46] tables.

If there exists a cointegration vector between the variables under consideration, there is causality among these variables at least in one direction [47]. Granger [48] proposed a time-series data based approach in order to determine causality.

A disadvantage of VAR models is that parameters’ estimators of the system cannot be explained from economic point of view. This problem can be solved transforming a VAR model in a moving average one (Vector Moving Average—VMA). So, the responses can be calculated deriving from a random shock of errors of the values in endogenous variables.

To calculate these variations, impulse response functions are used. This function determines the responses of endogenous variables in shocks coming from certain variables. Thus, the response from a sudden shock of one or more standard deviations on current and future values in endogenous variables is calculated.

The explanation of impulse response functions is done with errors’ orthogonalization where variance-covariance matrix is transformed to a lower triangular matrix. This transformation is called Cholesky decomposition. We have to point out that Cholesky decomposition depends on the sequence of the functions written on VAR model. Changing the sequence, we have different impulse response functions.

Variance decomposition indicates the proportion of the variance of disturbance terms of a series in comparison to the proportion of the disturbance terms of other series. If the disturbance term does not explain any of the forecast error variance, then this series is the exogenous variable.

In Table 2, the results from the two unit root tests used, namely ADF test and PP test, are depicted.

Given that the variables are integrated order I (1), we may proceed to determine whether there exists a long-run relationship using the Johansen cointegration test. The first step involves determining the optimal number of lags k to apply the VAR model. Two information criteria were used to determine the optimal number of lags that is the Schwarz Information Criterion (SBC) and Hannan-Quinn Criterion (HQ). The optimal length of time lags in each variable is two. Thus, a VAR second order model—VAR (2) arises.

Cointegration test, using Johansen methodology, is applied examining the long-run relationship among the variables. The Johansen technique examines the number of cointegrating vectors. In Table 3, the results of Johansen cointegration test are depicted.

In Table 3, we can see that both trace statistic and maximum eigenvalue present a cointegrating vector. Thus, it is concluded that there is a long-run relationship among the variables of the examined model. The cointegrating vector is

a. *indicates 10% level of significance. b. The numbers within parentheses followed by ADF statistics represent the lag length of the dependent variable used to obtain white noise residuals. c. The lag lengths for ADF equation were selected using Schwarz Information Criterion (SIC). d. MacKinnon [49] critical value for rejection of hypothesis of unit root applied. e. The numbers within brackets followed by PP statistics represent the bandwidth selected based on Newey-West [50] method using Bartlett Kernel. f. C = Constant, T = Trend, L, LN = Logarithms, Δ = First Differences.

a. Trace and max-eigenvalue tests indicate one cointegrating equation at 5% level. b. MacKinnon, Haug, and Michelis [46] p-values.

the following:

ln Market t = 13 . 563 ln Price t − 1 − 42 . 454 ln Venues t − 1                                         ( 8 . 374 )                                 ( 6 . 773 )

For the causal relationship between time series, the VAR Granger causality/block exogeneity Wald tests, developed by Enders [51], are applied. According to Enders, in VAR systems an endogenous variable can be used as exogenous with time lags. For this causality test, we use chi-square (Wald) statistics, which examines the joint significance of each variable with the lagged endogenous variables in every equation of VAR model [52].

In Table 4, in the first part where LMARKET is the dependent variable, we can see that the probabilities of the variables LNPRICE and LNVENUES (0.0062 and 0.0605 respectively) are less than 5% and 6% level of significance. Therefore, we can reject the null hypothesis meaning that LMARKET is endogenous and there is a causal relationship from LNPRICE and LNVENUES to LMARKET.

In the second part, where LNPRICE is the dependent variable, the probability of LMARKET (0.6083) is larger than 5% level of significance whereas the probability of LNVENUES (0.000) is less than 5% level of significance. So, there is causality of LNVENUES on LNPRICE.

Finally, in the third part where LNVENUES is the dependent variable, the probability of LMARKET (0.6930) is larger than 5% level of significance, while probability of LNPRICE (0.0033) is less than 5% level of significance, concluding causality of LNPRICE on LNVENUES.

In Figure 2, we can see that there is unidirectional causality from LNPRICE

on LMARKET and from LNVENUES on LNMARKET. Also, there is a bidirectional causality between LNPRICE and LNVENUES.

The absence of causality from LMARKET on LNPRICE and LNVENUES means that there is no causal relationship with direction from LMARKET to LNPRICE and LNVENUES. These findings are reasonable because 1) Bitcoin’s market capitalization does not cause its price, since the market capitalization is calculated by multiplying the price by the circulating supply of bitcoins [27], thus Bitcoin’s price causes its market capitalization, but the contrary cannot occur and 2) venues cannot cause Bitcoin’s market capitalization, since merchants are only one type of Bitcoin’s users [26], while other types of users (i.e. investors, speculators) conduct the majority of the transactions, thus forming the majority of the market capitalization [53].

Granger causality/block exogeneity Wald test shows the existence and direction of causal relationship but it doesn’t show the sign of this relationship, if it is positive or negative, as well as the time period which is required in order this interaction to occur.

The impulse response function describes the effects on endogenous variables for future periods. In other words, through this technique we examine the response of a variable to a shock which occurs in another variable. A sudden shock influences directly not only the variable itself but also the rest of the endogenous variables of the system via the dynamic structure of a VAR model. Dynamic model reflects the intertemporal evolution of a dependent variable in relation to its lagged values. Figure 3 plots the impulse responses of LMARKET, LNPRICE, and LNVENUES over a horizon of 10 weeks. Standard errors are calculated by the Monte Carlo method, with 100 repetitions (of ±2 standard deviations). The curves in the nine graphs indicate the forecasting of how each endogenous variable (LMARKET, LNPRICE, and LNVENUES in the first, second, and third row respectively) will respond to a shock which occurs in another variable. A flat curve shows that there is a stable effect on the endogenous variable, when there is a shock on the other variables. Also, a curve that is not significantly different from the X-axis shows that there is not significant effect on the endogenous variable, when there is a shock on the other variables.

Thus, in the first graph (row 1 column 1) causing a shock to endogenous variable LMARKET, LMARKET will have a slight increase in the first weeks and afterwards a slight decrease until the next 10 weeks. The curve in row 1 column

2 shows that a shock of LNPRICE has a positive effect on LNMARKET and the graph in row 1 column 3 shows that a shock to LNVENUES has a negative effect on LMARKET.

In the first graph of the second row, the flat curve indicates that a shock to LMARKET will have a stable effect on the endogenous variable LNPRICE. The graph in row 2 column 2 shows that a shock to LNPRICE has a positive permanent effect on LNPRICE, and the graph in row 2 column 3 shows that a shock to LNVENUES has a decrease on LNPRICE for two weeks and for the rest period LNPRICE remains steadily decreased.

Finally, the graph in row 3, column 1, where the curve is not significantly different from the X-axis, indicates that a shock to LMARKET has no significant variation on the endogenous variable LNVENUES. The graph in row 3 column 2 shows that a shock of LNPRICE has a positive effect in the first two weeks and then becomes stable on LNVENUES for the rest of the period. The last graph in row 3 column 3 shows that a shock of LNVENUES has a decreasing effect on LNVENUES.

In Table 5, Table 6, and Table 7, the forecast error variance decompositions between the examined variables, across the 10-week forecast period for each of the variables set, are presented. Shocks are defined with Cholesky methodology. Standard errors are generated through Monte Carlo simulations (100 repetitions).

So, in Table 5 we can see that in the fifth and tenth week period, LNPRICE accounts for 0.09% and 1.25% respectively of the variation in LMARKET and the variable LNVENUES accounts for 1.45% and 4.26% respectively of the variation in LNMARKET.

In Table 6, we notice that for the same time period, LNMARKET accounts for 8.41% and 8.16% respectively of the variation of LNPRICE, whereas LNVENUES accounts for 22.12% and 28.03% of the variation of LNPRICE.

In Table 7, we can see that LMARKET accounts for 0.23% and 0.14% in the fifth and tenth week period of the variation of LNVENUES and LNPRICE accounts for 17.77% and 22.91% for the same time period of the variation of LNVENUES.