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Bitcoin and its mining on the equilibrium path

bitcoin Information about the efficiency of specific chips and miners, especially their introduction date and actual use, is not easily available as some mining chips are being kept secret (at least for a period of time) before being made available for public and bulk purchases. However, power consumption with respect to performed hashes is a crucial piece of the puzzle in calculating the marginal mining costs of cryptocurrencies. We use the data available at https://en.bitcoin.it/wiki/List_of_Bitcoin_mining_ASICs. These have been also checked with respect to the available information at the respective manufacturers. Mining chips for the analyzed period are listed in Tab. 1. The mining efficiency evolution is then illustrated in Fig. 1 where, in addition to the specific miners’ efficiency, we also add information about the best possible miner at the time as well as an informative hyperbolic fit to the efficiency development. The best available mining chip for a given month is then used in the marginal mining costs calculation.

Table 1. Efficiency of mining chips. Efficiency is quantified in joules per gigahash, i.e. the lower the value the more efficient the mining chip is considered. Mining chips are ordered chronologically with respect to the introduction date.

Chip Manufacturer Introduced J/Gh
BE200 ASICMiner Mar-14 0.66
BF864C55 BitFury Group Mar-14 0.5
FB1600 DigBig Mar-14 0.95
Hammer Spondoolies-Tech LTD Mar-14 0.538
A3233 Avalon Project Apr-14 0.89
BM1382 Bitmain Technologies Ltd. Apr-14 0.528
Minion Black Arrow Ltd. Apr-14 0.5
Neptune KnCMiner Jun-14 0.57
RockerBox Spondoolies-Tech LTD Jul-14 0.316
Monarch Butterfly Labs, Inc. Aug-14 0.27
BE300 ASICMiner Sep-14 0.187
A3222 Avalon Project Sep-14 0.4
BM1384 Bitmain Technologies Ltd. Sep-14 0.249
SF3301 SFARDS Mar-15 0.305
BM1385 Bitmain Technologies Ltd. Aug-15 0.181
PickAxe Spondoolies-Tech LTD Sep-15 0.14
BitShare 21 Inc. Nov-16 0.16
DW1227 Ebang Dec-16 0.14
BM137 Bitmain Technologies Ltd. May-17 0.1
DW1228 Ebang Dec-17 0.09
Fig. 1

Fig. 1. Energy consumption is measured in joules per gigahash here. The higher the energy consumption, the lower the efficiency. The chart is based on the miners data in Table 1. The circles represent specific miners, the black curve represents the most efficient miner available at the time and the red dashed curve shows the power-law fit to the miners available at the time. The estimated power-law exponent of − 0.69 suggests that the increase in mining efficiency is slower than linear in time. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3. Cointegration and vector error-correction model

We investigate the relationship between two variables – Bitcoin price and costs of mining of a single bitcoin. As these series are very likely non-stationary, the cointegration analysis is the most intuitive starting point. As it turns out, the series are cointegrated. In this section, we provide some basic notions of the cointegration analysis needed for the detailed dynamics examination.

Cointegration relationship (Engle and Granger, 1987) in its bivariate form states a relationship between two series that share a common stochastic trend and they tend to a common equilibrium. From the statistical perspective, two series are cointegrated if they are integrated of the same order d and there exists their linear combination that is integrated of order d − b,b > 0. Specifically, we have series x and y such that(1)yt=β0+β1xt+εtwith time index tt = 1,…,T where x≡{xt}t=1T and y≡{yt}t=1T are integrated of order I(d) and ε≡{εt}t=1T is an error term integrated of order I(d − b). In the very simplest form, we have d = b = 1 but the statistical properties translate into the more general case as well. If the series are cointegrated, the parameters β0 and β1 can be consistently estimated even for non-stationary series x and y. Parameter β1 then quantifies the long-term relationship between the two series. As Eq. (1) represents an equilibrium relationship, the error term ε is a deviation from this equilibrium. In the traditional setting, which is also the case for the dataset we analyze, d − b < 0.5, i.e. it is stationary and mean-reverting so that the overall relationship never systematically deviates from the long-term equilibrium.

This idea is integrated into the (vector) error-correction model – (V)ECM – which uses the ideal cointegration setting when d = 1 so that x and y are unit roots and their differences are by definition I(0), i.e. weakly stationary. The error-correction model is then written as(2)Δyt=ω0+ω1Δxt+η(yt−1−yt−1^)+ϵtwhere yt−1^ is a lagged fitted value from Eq. (1) and the whole term yt−1−yt−1^ shows the lagged deviation from the long-term equilibrium. Parameter η represents how fast and if at all the system tends towards the equilibrium. When η < 0, the equilibrium deviation is corrected for (hence the error-correction model label) and the system is stable. When η > 0, the system diverges.

It is more practical to rewrite the error-correction models into the vector autoregression (VAR) representation, which covers both short-term (VAR) and long-term (cointegration) dynamics of the whole system. Eq. (2) can be rewritten to(3)Δxt=ω10+ω11Δxt−1+ω12Δyt−1+η1(yt−1−yt−1^)+ϵ1t,(4)Δyt=ω20+ω21Δxt−1+ω22Δyt−1+η2(yt−1−yt−1^)+ϵ2t.

With a system of multiple endogenous variables and possibly more used lags, the representation can get naturally more complex. As the bivariate version suffices for our analysis here, we redirect an interested reader to Banerjee and Hendry, 1992Ericsson et al., 1998Hendry and Juselius, 2001Juselius, 2006, and Hoover et al. (2008) for more details on multivariate cases where different types and specifications of the vector error-correction models (un/restricted constant/trend specifications) are discussed in detail.

As the procedure of selecting a proper model specification does not need to be completely straightforward, we provide the following step-by-step procedure for transparency and possible replications:

1.

Test for the unit roots in the examined series using the augmented Dickey and Fuller (1979) test. Appropriate number of lags is selected based on the Bayesian information criterion (BIC) (Schwarz, 1978) with the maximum lag of 12 as we have monthly series. If the unit roots are not rejected, we proceed.

2.

Using the VAR representation of the VECM model, identify the optimal number of lags in the model using the BIC with the maximum lag of 12 again. Check whether the time trend is significant in this model specification. If it is, it points to either the restricted or the unrestricted version of VECM.

3.

Using the Johansen tests (both trace and Lmax) (Johansen, 1991Johansen, 1995), verify whether the series are cointegrated. Use specifications hinted during the previous steps.

4.

Estimate the final VECM specification.

5.

Test whether the error-correction term from the final model contains a unit root. If the unit root hypothesis is rejected, we have arrived at the final model for further interpretation.

4. Results

4.1. Mining costs and profitability

We examine the relationship between the mining costs and price of Bitcoin (specifically their logarithms, and logarithmic differences when appropriate) between January 20145 and August 2018. As the latter is rather trivial to obtain, we focus in more detail on the former one. Mining costs of Bitcoin are driven by several factors while the core ones are three – electricity price, power/computational demands of the mining network, and efficiency of chips/miners. Mining efficiency is represented by power demand efficiency of mining chips which is summarized in Fig. 1. In the examined period, the efficiency improved from approximately 0.65 joules per a gigahash (J/GH) at the beginning of 2014 to 0.1 J/GH in the half of 2018. Even though this is a six times improvement, the efficiency has not improved much since the second half of 2014 when it already reached the levels of 0.2 J/GH. As illustrated in Fig. 1, a simple hyperbolic fit suggests a decay of -0.7, i.e. slower than a linear increase in efficiency. The other two main factors – electricity price and network hashrate – are illustrated in Fig. 2. Electricity price remains relatively stable and even though there is an evident decreasing trend between 2014 and 2016 followed by a milder increasing trend till the end of the analyzed sample, the price ranges between $0.04 and $0.07 per kWh for the whole period. Network computational demands have experienced much more rapid development – starting at around 20 PH/s and reaching around 80 EH/s, i.e. increasing around 4000 times over the examined less than four years.

Fig. 2

Fig. 2. Bitcoin network daily hashrate averaged over the given month in petahashes per second (left) and average electricity price in USD cents per kilowatt-hour (right). Hashrate is given in the semi-log scale, electricity price is shown non-transformed. The electricity price had started the examined period slightly below 7 c/kWh and decreased down to touch 4 c/kWh in the first quarter of 2016. Since then, the price is in a slight increasing trend but mostly varies between 4c and 6c per kWh. The network hashrate during the examined period can be split into three phases. The first one, ending around the middle of 2014, signifies the end of the ASIC miners booming introduction, followed by the second one signifying a temporary satiation of the mining network. And the third – increasing – phase is connected to the inversion of the bear market into the bull run that culminated at the end of 2017.

We thus see three intertwined forces forming the final marginal costs of mining a new bitcoin – mining efficiency, which has been improving only slightly over the analyzed period6, electricity costs, which have remained rather stable, and power demands of the mining network, which have been exploding over the studied four years7. Putting these dynamics together, we arrive at the estimated marginal costs of mining a single bitcoin shown in Fig. 3. These costs started at around $3 at the beginning of 2014 shooting up to around $3000. In the similar manner, the Bitcoin price has increased from around $1000 at the beginning of 2014 going through a bear market in 2014 and 2015 to reach the all-time highs of around $20,000 by the end of 2017 and correcting to the levels of around $6000 in the middle and second half of 2018. To better illustrate the mining profitability with respect to expenses associated with electricity consumption, we also present the mining margin simply put as a return on investment. The margin starts at rather absurd levels of around 10000% at the beginning of 2014, then going down to touch quite reasonable 200% in the second half of 2016 and the first half of 2017. This level was distorted by the explosive bull run of the second half of 2017 when the mining profitability peaked at around 2000% at the break of 2017 and 2018. Since the beginning of 2018, the mining profitability has been on a stable downward trend, currently bottoming a touch above 100%, i.e. with Bitcoin price at $6000 and mining costs between $2000 and $3000.

Fig. 3

Fig. 3. Bitcoin price in USD and marginal costs of mining a single bitcoin in USD (both left) and their margin in percent (right) defined as margin=pricecost−1. All three time series are represented in the semi-log scale. Bitcoin starts the examined period at around $1000 and stays in a bear market for around two years before starting a rally towards its all-time highs around $20000 by the end of 2017. The year of 2018 sees a correction towards levels around $6000. Marginal mining costs have been catching up the price surge the whole examined period, narrowing the gap with a rather constant rate with the exception of the second half of 2017 as represented by the margin series.

4.2. Equilibrium relationship

The mining profitability is thus in a global decreasing trend which was disturbed only by the bull run of the late 2017. This suggests that the mining industry, even though quite slowly, behaves like a standard business where the long-term profitability is slowly drained out by an influx of miners and new competition. It also seems quite clear that the increasing price motivates more miners to participate in the business which leads to higher competition and higher network hashrate which in turn increases the mining costs. To see whether this is in fact true in the statistical sense, we look at the relationship between Bitcoin price and its mining costs through the lenses of cointegration analysis which allows to study both the long-term relationship between the series but also the short-term dynamics utilizing the vector error-correction model (VECM).

The potential cointegration relationship here is the relationship between mining costs and price of Bitcoin8. Based on the dynamics observed in Fig. 3, we construct the model so that Bitcoin price is the impulse variable and mining costs represent the response variable9. Both variables are of the same level of integration as they both contain a unit root (p = 0.98 and p = 0.96 in the Augmented Dickey-Fuller test with the null hypothesis of a unit root, respectively for Bitcoin price and mining costs). Following by estimation of the long-term connection between the series, we find a highly significant (p ≪ 0.01) elasticity of 1.30 based on the Stock and Watson (1993) dynamic least squares estimator. This shows that in the long run, mining costs strongly react to the increasing price of Bitcoin and eventually catch up, the costs even overreact to the increasing prices as the estimated elasticity is above unity. Part of this effect might be due to the very cheap mining costs compared to prices at the beginning of the examined period. Further examination is needed to uncover the speed of this adjustment.

Following the procedure described in the previous sections, we estimate the vector autoregression representation of the possible cointegration relationship and identify the optimal number of lags based on the Bayesian information criterion. For both types of models (with and without a time trend), the optimal number of lags is a single one. Detailed inspection into the model uncovers that the time trend is statistically significant for the mining costs equation (t = 2.7481 and p = 0.0084). We then proceed by testing for cointegration relationship using the set of Johansen tests with this model specification. Out of two possibilities – cointegration with a restricted trend, and an unrestricted trend – the testing procedure selects the specification with the restricted trend, which is further confirmed by the Engle and Granger (1987) testing procedure (p = 0.0934). We can thus proceed to the final VECM estimation.

The resulting vector error-correction model is summarized in Tab. 2. The estimated effects suggest that the short-term dynamics is not dominant in the relationship. Apart from a relatively weak auto-correlation element in the Bitcoin price dynamics, there are no significant effects. Importantly, the negative and statistically significant effect of the error-correction term in the marginal costs equation and the insignificant error-correction term in the price equation validate the equilibrium relationship between Bitcoin price and its mining costs. Even though the significance is quite lacking in this VECM system, we see that the coefficient of determination for the marginal costs equation is decent for practically financial data and price evidently plays an important role in setting marginal costs in future. In the next section, we further discuss the interactions and causality between the variables and show that the model still delivers reasonable implications.

Table 2. Vector error-correction model. Results are shown for separate marginal costs and price equations. Estimates, standard errors, t-statistics, and p-values are reported and accompanied by (adjusted) coefficients of determination and serial correlation statistics (ρ^ and Durbin-Watson statistics).

Estimate SE t-stat p-value
Marginal costs equation
Constant −0.1468 0.0893 −1.6451 0.1064
ΔCostst−1 −0.0627 0.1196 −0.5239 0.6027
ΔPricet−1 0.0606 0.1538 0.3943 0.6951
ECt−1 −0.2477 0.0857 −2.8898 0.0057
R2 0.2568 R¯2 0.1962
ρ^ 0.0456 D-W stat 1.8725
Bitcoin price equation
Constant 0.0038 0.1009 0.0373 0.9704
Δ Costst−1 0.0644 0.1353 0.4759 0.6363
ΔPricet−1 0.3154 0.1739 1.8141 0.0758
ECt−1 −0.0186 0.0969 −0.1919 0.8486
R2 0.1081 R¯2 0.0353
ρ^ 0.0049 D-W stat 1.9799

As we work with only 56 observations, it is crucial to check the essential assumptions of such time series analysis, specifically remaining serial correlation, heteroskedasticity, and normality of error terms (residuals) of the final model. There is no remaining serial correlation (according to the Rao-type serial correlation test (Lutkepohl, 2005) with F(48,52) = 1.428 and p = 0.1044 for up to 12 lags), no heteroskedasticity (according to the Lagrange Multiplier test (Engle, 1982) with χ2(108) = 117.86, p = 0.2430 up to 12 lags), and normality is not rejected either (according to the Doornik and Hansen (2008) test with χ2(4) = 6.9673 and p = 0.1376). As the model passes all these tests, the results can be considered valid and reliable.

4.3. Interactions and causality

To further examine the interaction between Bitcoin price and its mining costs, we test for Granger causality between the series in two dimensions – short-term and long-term. The short-term Granger causality shows no causality from either side (t = 0.3943 and p = 0.6951 for “price not causing mining costs” null hypothesis, and t = 0.4759 and p = 0.6363 for “mining costs not causing price” null hypothesis). These testing statistics are the t-stats in Tab. 2. In the long-term, the Toda and Yamamoto (1995) approach uncovers strong causality from price towards mining costs (F(2,49) = 4.6883 and p = 0.0137) and mild causality from the other side (F(2,49) = 2.9354 and p = 0.0625). These results are further supported and illustrated by the impulse-response functions and variance decompositions in Fig. 4.

Fig. 4

Fig. 4. On the top, the generalized impulse-response functions representing the effect of a single standard deviation shock in mining costs (Bitcoin price) on Bitcoin price (mining costs) shown on the left (right). Black curve represents the estimated shock magnitude, the grey lines represent one standard deviation confidence intervals. On the bottom, the forecast error-variance decomposition (FEVD) is shown for Bitcoin price (left) and mining costs (right). The effect of shocks in mining onto Bitcoin price are tiny and perish quickly. However, price dynamics plays an important role in mining costs.

The impulse-response functions represent a reaction of one variable to a single standard deviation shock to the other variable. Here we see that in both directions, there is only a small reaction in the short-term. The overall effect of shocks to mining costs onto Bitcoin price are small even for a longer period and the effect is insignificant almost everywhere. From the other side, there is a small reaction of mining costs to shocks in prices in the short-term but after approximately three months, the mining costs start correcting for the shock and the correction stabilizes only after approximately one year. Forecast variance decomposition tells a very similar story as the mining costs play only a marginal role in the future dynamics of price and it forms only around 10% of the total price variance. The results for the mining costs reflect the dynamics of its impulse-response function – for three months, price forms only around 10% of the mining costs variance but it increases up to around 70% after 12 months. Bitcoin price thus plays a directing role in the dynamics of its mining costs and there is a strong tendency towards the common equilibrium.

5. Discussion and conclusions

Putting the results crudely, Bitcoin mining industry has been following its path towards a standard business. Even though the profits had been absurd in the retrospect, the last few months have shown that the standard economic forces are at work even at the crypto-markets. The 100%–200% margin that seems to be a lower floor for the business might look very high. However, we need to keep in mind that the mining costs we consider here are the marginal/operational ones, i.e. taking into consideration only the running costs, not the costs of purchasing the miners, power cords, other hardware and equipment at the mining sites and similar further costs that would be hard to quantify. As long as the mining components manufacturers keep their margins stable, the same can be expected for the minimal margin between the operational costs and Bitcoin price. The analysis we present here can be summarized in the following points.

Bitcoin price drives the mining costs and not (or only weakly) the other way around. This is a rather non-standard dynamics of the price formation as normally, the prices are formed with respect to their production costs. However, here we observe the reversed relationship which is tightly connected to an ongoing discussion about the fundamental price/value of Bitcoin or any other cryptocurrency or token. The fact that the mining profit margin was decreasing even during the bear market of 2014 and 2015 only shows that the mining costs catch up on the prices rather than the mining costs keeping the prices up. With the market stabilization in the second half of 2018, at least from the mining perspective, we might be able to see the relationship in a different phase of the market, i.e. a satiated one. In the retrospect, the unprecedented price hike of 2017 combined with the rather limited production capacities of the miners’ components rise a question whether the mining difficulty adjustment of 2016 blocks (approximately 2 weeks given 10 minutes per a block) is not too slow. A comparison of the proof-of-work cryptocurrencies with respect to their difficulty adjustment times and how it affects their ability to react to divergence from the long-term equilibrium is thus at hand.

It needs to be stressed that the mining business is far from being unprofitable. Even though the times when the Bitcoin mining was profitable even for home/fan miners are probably irreversibly gone, the electricity costs we consider ensure that the mining is profitable even for electricity prices above the minimal ones. Interestingly, we are getting to the point when the standard economic competition gets into play and the prime factor in the business becomes the electricity price. This has some quite unexpected implications. Mining will be pushed to places where the electricity prices are low. And when we get to the situation when one needs to get below $0.04/kWh, which forms a floor of our representative electricity costs, the renewable sources of energy become essential. The combination of these sources and cryptocurrency mining can help balancing the unstable systems of the renewables as mining is a non-stop business. The networks can thus become more stable as there could always be consumers for normally excessive capacities and overproduction of electricity. This direction is certainly worth a more detailed examination as the environmental impacts of cryptocurrency mining are far from solved when only Mora et al. (2018) provide some preliminary results that are, however, built mostly on the booming dynamics of the end of 2017. A deeper analysis taking our presented results into consideration, specifically the balancing forces between price and costs and thus also profits, will provide more realistic future scenarios. In addition, the discussion about advantages and disadvantages of fossil and renewable sources of energy can be enriched by inclusion of cryptocurrency mining into the equation. As most energy sources cannot be simply, cheaply and efficiently turned on and off repeatedly, a possible inclusion of the cryptocurrency miners as smoothing instruments for the overproduction of energy might be of interest. As our results suggest, the core of the cryptocurrency production connected to the energy issues behaves as a standard economic asset and it can be worked with and utilized with it in mind.

Interestingly, Bitcoin might become a victim of its own success. We see that the increasing price drives the mining costs up. These costs are mainly formed by the power consumption as the electricity prices as well as the mining efficiency remain rather stable. As it is hard to imagine much cheaper electricity prices, there would need to be a sustainable revolution in mining efficiency. Looking at the history, this is quite unlikely. Even if we consider the newly (November 2018) announced S15 and T15 miners that promise the consumption of 0.05 J/GH, it is only less than a four-times improvement over what was available at the end of 201410. The linear or slower increase in mining efficiency is not sufficient to keep the mining power consumption in check. A new rapid (speculation induced) bull run could get the mining costs and thus also associated power consumption to hardly sustainable proportions. And even though we have shown that the market adjusts towards equilibrium, the adjustment is rather slow and the shooting-up prices might threaten existence of the whole system. In a sense, the fact that the miners producers are rather slow at this moment (the waiting times are in a frame of several months) are friendly towards sustainability of the whole Bitcoin system. Should the miners be readily available in large amounts, a rocketing price would be quickly followed by a rocketing power consumption and a correction downwards would be problematic. However, the speculative bubbles are not the only threatening sources for sustainable mining and associated energy and environmental costs but also the interaction between the mining efficiency pace and the Bitcoin penetration and utilization development. As shown by Wheatley et al. (2019) and also used by Mora et al. (2018), the value of technology and in our specific case the price (or capitalization) of Bitcoin follows a power-law with respect to the utilization. As long as the mining efficiency grows at a higher pace than the penetration, which drives the price up, the system should remain stable in the sense of not exploding. Again, Bitcoin might fall victim to its own potential success. Nevertheless, the outlined dynamics and interactions certainly deserve a more detailed treatment as a follow-up to the results presented here.

Our results can be also seen as a contribution towards the ongoing discussion of Bitcoin and crypto-market in general being purely speculative. Even though we do not study the interactions between and price influences of speculative factors, herding, market microstructure, and market manipulation, we show that yet another fundamental factor is essential for understanding the complex dynamics of the cryptocurrency prices. We do not claim that the speculative factors are not in effect but we also deliver strong evidence that the fundamental (technical) factors play an important role as well.

To summarize, the Bitcoin market in general and its mining segment specifically still remain a fascinating field to examine. The second half of 2018 and the following months have seen a market stabilization and even though the speculators and fan-investors would rather see another price explosion through the whole crypto-market, it seems we are eventually getting to a stable market situation when the mining margins are well set, mining difficulty has stabilized and Bitcoin can look ahead towards its more standard times.

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