# Using Bayes’s Rule to Think About a Bitcoin Bubble

Is there a Bitcoin bubble? Jason Kuznicki thinks so and believes that he has conclusive proof. He blogs three graphs that show more or less that there is a lot of speculation in Bitcoin. But does speculation prove that there’s a bubble? Let’s use Bayes’s rule to think about this carefully.

Bayes’s rule is a mathematical tool for thinking about the incorporation of new evidence into subjective probabilities. Let’s suppose that there is some proposition A for which you have a prior belief. Somebody offers evidence B for or against A. How much should you change your belief in A based on evidence B?

Bayes’s rule boils the answer down to a simple mathematical form:

$P(A|B) = P(B|A)\dfrac{P(A)}{P(B)}$

In English, the probability of A given B equals the probability of B given A, times the probability of A and divided by the probability of B.

So to evaluate Jason’s argument and see how much we should change our estimate of a Bitcoin bubble based on the evidence that there’s speculation, we can simply assign the proposition and the evidence to A and B. In this case, A is the proposition that there’s a bubble, and B is the evidence that there’s speculation in Bitcoin. If we figure out our subjective probabilities for B|A and B, we can use those to determine how different P(A|B) should be from P(A).

So what is B|A? Since B is the evidence that there is speculation in Bitcoin and A is the proposition that there is a bubble, B|A simply states the proposition that given that there is a bubble, there is speculation. It seems pretty much impossible to have a bubble without speculation, so I’ll go with a subjective probability of 1. Picking a different value here will only work against Jason’s argument.

So what is the probability of B, the fact that there is speculation in Bitcoin? The Bitcoin ecosystem isn’t built out yet. Most of the protocol’s most exciting uses haven’t even seen the light of day yet. As I blogged last week, multisignature transactions are barely in use yet, but they form the foundation for a decentralized architecture of arbitration. Ed Felten at Princeton is working on decentralized prediction markets. Jerry Brito points to microtransactions, or even nearly-continuous transactions, as another exciting future use scenario.

Given that we don’t know whether this ecosystem will ever materialize, holders of bitcoin are necessarily speculating. If the ecosystem matures and is useful, bitcoins will be worth something. If none of these innovations come about, or if we decide they’re not that useful after all, then bitcoins will probably be worth nothing. There’s no way out of speculating, because we simply don’t know for sure if the ecosystem will come along. Almost the entire “fundamental value” of Bitcoin rests on future events.

So the probability of B, I think, is 1. When P(B|A) is 1, and P(B) is 1, what does Bayes’s rule reduce to?

$P(A|B) = P(A)$

B simply offers no information as to whether A is true.

A similar argument can be made when Bitcoin’s volatility is offered as evidence of a bubble. Bitcoin is a thinly-traded asset where supply does not adjust to accommodate demand. It is going to be volatile. So the fact that Bitcoin is volatile adds no new information to the question of whether it’s a bubble.

What does provide information? I think the most reliable evidence is on the maturation (or not) of the Bitcoin ecosystem. If Bitcoin seemed static right now, I would interpret that as evidence of a bubble. But it doesn’t. Every day, people are working to build businesses that leverage some of the unique features of Bitcoin’s protocol. As long as that continues, I think it’s most reasonable to be highly agnostic about the correct price of Bitcoin.

# Bitcoin and the No-Arbitrage Condition

One of my favorite aspects of the Bitcoin phenomenon is that it has people talking about monetary economics and finance. Just as recessions tend to produce advances in academic macroeconomics, Bitcoin is forcing a wide cast of characters, from libertarian nerds to economic journalists, to think more deeply than they otherwise would about money and monetary institutions.

Nevertheless, monetary economics can be difficult, and there is a lot of confusion out there. It seems that most of the confusion is due to two errors. The first is that Bitcoin appreciation is deflationary, and therefore, recessionary. A variant is that Bitcoin volatility will create massive booms and busts in the Bitcoin economy. I think that this point has been decisively refuted by Jerry Brito. The macroeconomic effects of a currency have to do with its unit-of-account status, not with its medium-of-exchange status. Consequently, unless people begin (foolishly) denominating their long-term contracts in Bitcoin, the cryptocurrency won’t have any macroeconomic drawbacks.

As Bitcoin skeptics have come to terms with Jerry’s point, they have resorted to a second error, that Bitcoin’s long-run fixed supply would generate persistent, long-run deflation, which will cause hoarding of the currency. This would make it unsuitable as a medium of exchange, because no one would be willing to exchange it. Transactional demand for Bitcoin would be zero. Matt Yglesias and Matt O’Brien make versions of this argument.

I think that Tim Lee has done a good job of refuting this line of thinking both on Twitter and in two posts at Forbes. But his arguments haven’t satisfied everyone. O’Brien in particular seems to be doubling down on Twitter.

The problem is that in Yglesias’s and O’Brien’s posts on Bitcoin, I have not come across the word “arbitrage.” This is a pretty good sign that their claims about the long-run equilibrium are not rigorous. The long-run equilibrium must be defined by a “no arbitrage” condition—if arbitrage between currencies is possible, then we are not in equilibrium.

Let’s try to write down an equation that describes a first approximation of this condition:

$1 + i_\ = \dfrac{E_t(S_{t+k})}{S_t} - \pi_\sigma - \pi_l$

At a high level, this equation says that the expected return to holding dollars has to equal the risk- and liquidity-adjusted expected return to holding bitcoins. On the left side of the equation is the return to holding dollars, which is given by the nominal interest rate on dollars, $i_\$. On the right side of the equation, $\frac{E_t(S_{t+k})}{S_t}$ represents the expected appreciation in bitcoins from time t to time t + k, while $\pi_\sigma$ is the risk premium and $\pi_l$ is the liquidity premium. I am assuming for now that it is not possible to have a bitcoin-denominated loan, and therefore no interest rate on bitcoins, although I could imagine that there might be an overnight rate of some sort. But for now, assume that the equalization of returns has to happen via appreciation.

What this equation makes clear is that there is no free lunch from hoarding bitcoins. Bitcoin hoarders will be compensated for the risk they are bearing, for the illiquidity of the Bitcoin market, and for the opportunity cost of holding bitcoins, but for nothing else.

The fact that bitcoins are expected to appreciate in value does not increase the incentive to hoard bitcoins at the margin. Instead, all of the change in the value of bitcoins happens in the spot rate, $S_t$. The price of bitcoins adjusts now to accommodate any future expected increase in the value of bitcoins, and there are no further gains from hoarding bitcoins. There is therefore no disincentive to transactional use of bitcoins.

I expect further that in the future, forward contracts on the Bitcoin-dollar exchange rate will reduce or eliminate the risk premium, and more sophisticated entrants (read: hedge funds) into the Bitcoin market will supply additional liquidity, making even the nominal return to holding bitcoins about the same as that of holding other currencies. The model above is not meant to be a complete account of the market for bitcoins. But I think it serves as a good baseline for thinking about what claims about an incentive to hoard entail.

Finally, I’ll note that even if almost all of the eventual 21 million bitcoins are “hoarded,” a mere 1000 bitcoins would be more than adequate to supply the transactional needs of an economy as large as the United States. Each bitcoin is divisible into 10^8 “satoshis,” and there are only around 10^9 dollars, or 10^11 cents, in circulation. One thousand bitcoins would be 10^11 satoshis. If each satoshi equalled one cent, the market capitalization of bitcoin would have to rise to 2.1 quadrillion dollars. When the market capitalization exceeds that figure, I will concede that bitcoins have been over-hoarded.

# Copyright Reform and the Incentive to Create

Mercatus has a new book out on copyright, edited by Jerry Brito, called Copyright Unbalanced: From Incentive to Excess. I am pleased to be one of an otherwise-illustrious group of contributors.

I expect that the book will create some controversy in policy circles. In this post, I want to address what is likely to be a knee-jerk response from our critics, that copyright reform will substantially decrease the incentive to produce creative works.

Content creators anticipate that their products will generate some amount of revenue each year after they are released. The expectation is generally that the creative work will generate the highest revenue in the first year, and less revenue in each subsequent year. To model this revenue stream, I’m going to assume exponential decay. Exponential decay lets us pick a half-life, $h$, and assume that $h$ years after the work was released, it will generate revenue at half the initial rate. After $2h$ years, it will generate revenue at one-fourth the rate, and so on.

In year $t$, the revenue that the content creator will receive if there is copyright is $e^{\frac{-t \ln2}{h}}$ times the initial revenue. Consequently, the total revenue that a copyright holder will receive over the life of a 95-year copyright term is

$\sum\limits_{t=0}^{94} e^{\frac{-t \ln 2}{h}}$

times the initial revenue.

However, content creators prefer revenue now to revenue 90 years from now. In order to calculate the present value of this revenue stream, we need to apply a discount rate $r$. The ex ante value of the revenue stream generated by the 95-year copyright term is therefore

$\sum\limits_{t=0}^{94} \dfrac{e^{\frac{-t \ln 2}{h}}}{(1+r)^t}$

times the initial revenue.

And of course, this calculation generalizes to different copyright terms. If we returned to a 28-year term, as Tom Bell advocates in his chapter of our book, the ex ante revenue stream would be valued at

$\sum\limits_{t=0}^{27} \dfrac{e^{\frac{-t \ln 2}{h}}}{(1+r)^t}$

times the initial revenue.

We’re now at a point where we can start to run some numerical calculations based on plausible values for $h$ and $r$. What is a reasonable ex ante expectation about the half-life of the revenue stream of a new creative work? I expect that for our book, the half-life will be something like 1 year or less; we will probably sell less than half as many books in the second year the book is out as in the first. But let’s not use $h=1$. Let’s estimate that $h=10$ to be extremely conservative and generous to our critics.

What about $r$? Again, how about if we are conservative and give $r$ a low value, like $r=0.02$?

Now we can run some calculations. Using the values above, the ex ante present value of a 95-year copyright is around 11.726 times the initial revenue. The ex ante present value of a 28-year copyright is around 10.761 times the initial revenue. Consequently, shortening the copyright term from 95 years to 28 years (less than 30% of the current term!) retains about 91.8 percent of the incentive effect of the current copyright term.

It is unlikely that such a small decrease in the present-value of the revenue stream would reduce the amount of content production by much. To the extent that content producers cannot or do not substitute easily into other fields, they would simply take the 8.2 percent decline in compensation per project as a decrease in wages (not the end of the world), and there would be no decline in content production. To the extent that content producers can substitute into other fields, we would get less content, but we would also get more of other stuff—the welfare effects of less content are ambiguous, since there is a knowledge problem regarding the optimal amount of content.

If you want to do the calculation with different half-lives and interest rates, be my guest. I am confident that for all plausible values of $h$ and $r$, you will find that shortening the copyright term will have at most a modest effect on the incentive to create.

How about the value of the public domain? This is a little harder to model, because we care about the ex post value of works, not just the ex ante expectation that content creators have. In practice, there turn out to be works with much longer half-lives than others. This fact complicates any back-of-the-envelope calculation. We also don’t know exactly by how much content creation would fall.

But let’s abstract from this and model the value of the public domain as the revenue stream for a given project that otherwise would have gone to copyright holders above. One difference for the public domain is that it no longer makes sense to discount the stream of value—future generations aren’t sitting around, waiting to be born so that they can watch Star Wars for the first time. Therefore, normalized to our original, first-year revenue stream, an estimate of the value of the public domain under a 95-year term is

$\sum\limits_{t=95}^{\infty} e^{\frac{-t \ln 2}{h}}$.

Under a 28-year term, the value is

$\sum\limits_{t=28}^{\infty} e^{\frac{-t \ln 2}{h}}$.

Plugging in the value we selected earlier for $h$, 10, the former expression yields around 0.021 and the latter about 2.144. In other words, the value of the public domain would be around 100 times higher per creative work if we shortened the term to 28 years. Again, this value is highly dependent on our selection of $h$, but the reason I am doing these calculations is so that my critics can repeat them with values they find more plausible, if they so choose.

This analysis has been highly stylized, but it is also extremely conservative. The half-life of most creative works is probably much shorter than 10 years, and when valuing an uncertain revenue stream, most artists—and even content corporations—probably discount at a rate of higher than 2 percent. The value of the public domain has been understated in this analysis, because there are many works that turn out ex post to have longer half-lives (but it is still the ex ante estimate of value that matters for investment). I have also not factored in the gains from those derivative works that are impossible under the current regime due to transaction costs, or the savings in enforcement costs from having a shorter time during which enforcement is necessary, or indeed, many of the other issues discussed in our book.

I would be interested in reading further analyses like the one above from anyone who supports the current copyright term or a longer one. How do you justify such a long term? You don’t have to use my assumptions, just make your own explicit so that people can see what they are and quarrel with them. How many fewer works do you really think would be created if we shortened the term from 95 years to 28 years? Would we really be worse off? Please show your work.

# Can the War on Drugs Bootstrap Bitcoin?

A few weeks after my last post on Bitcoin, the cryptocurrency was featured on EconTalk. From there it captured the imaginations of libertarian geeks everywhere. When I last wrote about it on March 12, one Bitcoin was worth 88 cents; today one is worth \$17.14. There are now numerous startups dedicated to serving as Bitcoin exchanges, banks, e-wallets, etc. Notably, there are not yet any futures markets, which has led many commentators to quite reasonably cry bubble.

There is a sense in which all fiat currencies are based on bubbles. After all, by definition fiat currencies have no intrinsic value; they are valuable because they are liquid, and they are liquid because they are valuable. This circular reasoning is not far from the information cascades that economists discuss in the context of bubbles.

Let’s call the process by which a currency comes to circulate as a widely-accepted medium of exchange “bootstrapping.” For fiat currencies, bootstrapping typically involves some coercion: the government demands that taxes be paid in its fiat currency, which creates demand for the currency. If the currency has other properties that make it useful as money—it’s divisible, transportable, and a reasonably good store of value—then this coercion is enough to make the currency widely-accepted for non-tax payments as well.

Bitcoin does not have a sponsoring government that demands Bitcoin-denominated tax payments. But it has something close: the black market. This week Gawker ran a piece describing Silk Road, an online black market where you can buy everything from marijuana to heroin, plus drug lab supplies and small weapons (no WMDs—yet).

Silk Road is only accessible through the encrypted and decentralized Tor network, so I did what any anarcho-curious geek would do. I downloaded and configured Tor and merrily browsed the Silk Road website (link only works if you have Tor running). I can confirm that it is like a candy store for drug users. According to the merchant reviews, the drugs are shipped in vacuum-sealed packages that emit no odor to be detected by the drug-sniffing dogs. Furthermore, each merchant lists the country from which he ships; pick a merchant based in your country and your package won’t have to go through customs, further decreasing the likelihood of detection. Most customers seem very happy with the care taken in shipping as well as the quality of the products they have received.

Nearly all payments on Silk Road are made using Bitcoin. Bitcoin is an excellent fit for the black market because it is pseudonymous—every payment is made from and accepted at a public 33-character address, but users can generate as many addresses as they want to preserve anonymity.

The question remains of whether the quasi-anonymity of Bitcoin is enough to keep the Feds from being able to shut down Silk Road or to make it unsafe to use the site. As Jerry Brito points out, we are now observing a natural experiment on the anonymity of Bitcoin. The hacker group LulzSec has recently undertaken some activities that make it a prime FBI target and solicited donations at a publicly-listed Bitcoin address. Assuming that the address is a real one and not devised to throw the FBI off the scent, we’ll soon know whether the government is able to identify people on the basis of their Bitcoin transactions.

Assume that it turns out to be safe and convenient to use Bitcoin in the black market. This fact may then turn out to be enough to bootstrap Bitcoin. As I wrote above, bootstrapping a fiat currency involves coercion. In this case, that coercion is supplied by governments who enforce the illegality of black market activities. They are coercively creating demand for the currency that is most convenient to use in the black market. Once there is enough demand for Bitcoin for black market purposes, Bitcoin may become more widely-accepted for legitimate transactions, just as the demand for fiat currency that governments create through taxation spills over to non-tax payments.

In the short run, Bitcoin will likely become more widely associated with the black market and therefore demonized. If you want Bitcoin to succeed, you should be OK with this, since it’s pretty much inevitable. Read up on Agorism and counter-economics. The demonization of Bitcoin may make some legitimate users hesitant to adopt the currency. In my view, this is silly. I will happily accept your Bitcoin-denominated tips and donations at 1FMxbQLh2hEoWXgM4GggbSAMoR61iL7zdp.

I am by no means certain that Bitcoin will succeed. The rapid rise in value that it has experienced may in fact be because there is a Bitcoin bubble. This guy is evidence of that. But it’s hard a priori to differentiate between a bubble and the successful bootstrapping of a currency (this is one reason we need Bitcoin futures). However, I am confident that if Bitcoin succeeds, it will be because of the War on Drugs and other policies that increase demand for a quasi-anonymous, internet-transportable currency to engage in online black market activities.