Bitcoin and the no-arbitrage condition
Apr 15, 2013

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 10⁸ “satoshis,” and there are only around 10⁹ dollars, or 10¹¹ cents, in circulation. One thousand bitcoins would be 10¹¹ 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.