Why one bitcoin could well be worth one million dollars

Adrien FABRE
Dec 28, 2017 · 7 min read

Short version

As the exchange rate of bitcoin has increased ten-fold in a few months, many wonder: is bitcoin overvalued? Many economists say so; for example, the Nobel-winning Jean Tirole affirmed in the Financial Times that it has “no intrinsic value”. Yet, this is true for any currency, the value of which only comes from its long-term use by a large community. The bitcoin community could indeed prove sufficiently strong for bitcoin to become a common and stable currency, and that would translate into a much higher appreciation. For technical reasons, I am skeptical that this will happen; for political reasons, I don’t hope it will. But one cannot predict the future, and bitcoin already benefits from an impressive ecosystem promoting its development.

Long version

Is bitcoin overvalued?

96% of analysts think that bitcoin is overvalued and that its skyrocketing price is driven by a huge speculative bubble that will — like any bubble — eventually burst. They may be right, but for the wrong reasons. Indeed, speaking of bitcoin’s price falsely give an impression that it is an asset as any other, whereas bitcoin is indeed a currency, with an exchange rate rather than a price. In addition, bitcoin has been thought of as a money that will benefit a lot its creator(s), and more generally, all first-comers. Indeed, the total number of bitcoins that will ever be in circulation is 21 millions, and Satoshi Nakamoto, the unknown creator(s) of the bitcoin, holds 980,000 bitcoins, almost 5% of the total. While creating tremendous inequalities, capping the future supply of bitcoins produces scarcity. Now, let us see under which conditions the one bitcoin could be worth one million dollars through the effect of this scarcity.

The one-million-dollar equation

Wikipedia tells us that the transactions velocity of money is “the frequency with which the average unit of currency is used in any kind of transaction in which it changes possession”. Formally, the velocity is V=P.T/M, where M is the monetary mass (i.e. the total amount of currency in circulation), T the vector of all transactions of a given period denominated in this currency, and P the vector of corresponding prices. In the following, I use capital letters for dollar and small ones for bitcoin, ignoring other currencies for simplicity purpose.

For the moment, let us make the assumptions that in a given period, the exchange rate between bitcoin and dollar is constant and equal to E, and the monetary mass is also constant. Let us also assume that the law of one price is respected, that is: any good or service sold in dollar would have the same real price in bitcoin, and vice-versa. In other words, any transaction can be expressed indifferently in bitcoin or in dollar using the exchange rate: P=Ep. The velocity of bitcoin can then be written v=p.t/m=P.t/(mE), i.e. E=P.t/(mv). The law of one price is a standard and credible assumption, while taking m and E as constant is a good approximation as long as one reasons over a sufficiently short period of time, or in a stable phase as we often conceive the very long-term. I use a final assumption which is disputable because it concerns the behavior of agents: this belief characterizes the difference between bitcoin aficionados and bitcoin skeptics. It consists in assuming that the bitcoin will eventually replace other currencies, at least for some part, and that people will use the bitcoin exactly in the same manner as they now use the dollar or the euro: for transacting, accounting and storing value. In particular, the velocity of the bitcoin will be approximately the same as the velocity of the dollar: v≈V. Finally, we thus have: E≈P.t/(mV). Overall, the velocity of a currency does not vary substantially (V was in the range [3; 10] for the last 60 years). In the case of bitcoin, the monetary mass is bounded, and will converge slowly from 17 millions as of the beginning of 2018 to 21 millions in 2140. Then, the main determinant of the exchange rate bitcoin/dollar is the volume P.t of transactions set in bitcoins but expressed in dollars. According to the bitcoin believers, this volume could prove very substantial in the future, say in the order of magnitude of the US GDP ($20 trillions), i.e. around 1–7% of global transactions (the share in terms of transactions is lower than the share of the US in the Global World Product by a factor equal to the velocity of money). Taking a high estimate of 10 for the velocity V, we get E≈20*1⁰¹²/(20*1⁰⁶*10)≈100,000 $/bitcoin. If one further assumes that the bitcoin will eventually become the hegemonic currency and serve as the medium of most transactions globally, one may believe that one bitcoin could be worth one million dollars in a few decades.

The real math behind bitcoin’s value

For those who are confortable with basic equations and derivatives, I provide a more detailed formula: by relaxing some assumptions, I disentangle speculation from the size of the bitcoin community in the determination of the bitcoin/dollar exchange rate. Others can skip this part and jump directly to the technological and societal discussion over bitcoin.

There is a peculiar type of transaction in the vector t of all bitcoin transactions: exchanges of dollars against bitcoins. Indeed, at each period, a quantity “$” dollars are exchanged against $/E bitcoins. If in a given period, more dollars are sold (in bitcoin) relative to a previous period, it means that this quantity $ has increased. Now, let us note t° all other transactions, and P° their corresponding price (in dollars), so that P=(1 P°) and t=($ t°’)’.Over an infinitesimal period of time, the transactions velocity of bitcoin writes v=P.t/(Em). Differentiating the logarithm of this equation: dln(v)=dln(P.t/E)-dln(m). For the sake of simplicity, we keep the assumptions that the velocity V, the monetary mass m and prices P° are constant, so that: 0=Ed(P.t/E)/(P.t)=Ed($/E+P°.t°/E)/(P.t)=E(d$/E+P°.dt°/E-P.tdE/E²)/(P.t). I apologize that my equations are quite unreadable on medium, but here is what we finally get, re-arranging the previous relation:

dE/E = d$/(P.t) + P°.dt°/(P.t) = ad$/$ + (1-a)P°.dt°/(P°.t°) where a=$/(P.t) is the fraction of currency exchanges in all bitcoin transactions.

This can be interpreted as follows: the growth in the exchange rate dE/E (i.e. a bitcoin appreciation) is decomposed into two components: the growth of speculation d$/$ and the growth of transactions denominated in bitcoins P°.dt°/(P°.t°), weighted by the relative share of what is called the reservational and the transactional demands in all bitcoin transactions.

What will happen?

Although there is no data on the volume of transactional transactions denominated in bitcoin, this figure certainly evolves slowly and smoothly. Hence, the recent surge of bitcoin/dollar exchange rate E (and its even more recent correction) is clearly driven by speculation. However, these speculations about a future increase of E may well turn out to be right. Indeed, as a confidential, decentralized and widely known currency, bitcoin is vastly used for all clandestine purposes: drugs and arms markets that take place in the untraceable and anonymous dark web, money laundering and tax evasion. Recent research estimates that illegal activity is the reason behind 46% of bitcoin transactions (i.e. most non-speculative transactions), for a volume of $76 billion per year. The penetration of bitcoins in the drug market is substantial, as the total drug market is estimated at $150 billion. Besides, other purposes are also emerging: using bitcoins in place of the national currency, when the latter is not stable (which is more and more the case in Venezuela), or using bitcoins as a medium of rapid and almost free international transfers. Altogether, one can reasonably believe that bitcoin is on the verge to become THE medium of exchange for all clandestine purposes (and even some legal ones). This view is encouraged by the observation that the most downloaded app for iPhone is currently a bitcoin wallet, and that one billion dollars has already been invested in firms that develop the bitcoin ecosystem, outweighing all other cryptocurrencies.

Yet, bitcoin is the first blockchain ever created, and it does not favorably compare to most recent technologies, in any respect. Two technologies allow to build ethical financial systems: the free currency duniter is a fair and sustainable cryptocurrency, that allocate its new tokens to each member through a universal dividend, in a way that does not favour first-comers; while taler is a taxable and private payment system, meaning that the incomes of someone can be taxed without being disclosed. But the selfish nature of bitcoin is a feature, not a bug, so let us look at the characteristics sought by bitcoiners. For anonymity purpose, the cash-like monero is way better, and main drug markets already accept this other cryptocurrency. Iota — which does not rely on blockchain but on tangle, a new protocol without mining nor transaction fees — but also ripple or many other cryptocurrencies provide credible solutions to the technical and environmental problems of bitcoin: its limitation to 7 transactions per second (to be compared to the 56,000 of the VISA network) and its exorbitant energy consumption (1 or 2 thousands of all electricity consumed worldwide).

Admittedly, bitcoin proponents will argue that updates are being made on the bitcoin protocol, notably the establishment of ‘Segregated Witness’ last August, which allows to double the number of transactions per second, and the lightning network, under development, that will arguably solve the scalability issue. Hence, the bitcoin ecosystem may have reached the critical size allowing it to resist competitors and attract users through efficient apps and advertising. I am not expert enough to assess the relative features of the many cryptocurrencies, but in any case, the large adoption of one currency will likely depend on notoriety and political acceptability rather than on its intrinsic characteristics.

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