Two interesting posts about bitcoin by JP Koning (post 1, post 2) got me thinking about the function of money. Usually we say that money serves three functions: unit of account, medium of exchange, and store of value. But what does it mean to be a "store of value"? More specifically, what does it mean for a form of money to be a "good store of value," i.e., performing this function well?

Suppose, for simplicity's sake, that an asset's value (defined in consumption terms) follows a geometric Brownian motion with constant percentage volatility and drift. So it satisfies:

Does "good store of value" mean that sigma, the volatility, is low? Or does it mean that mu, the drift, is high? Remember that in the short term, volatility dominates drift, while in the long term, drift dominates. Also remember that there should be a tradeoff between these two - assets with higher volatility will tend to have higher systematic risk, and thus will tend to have higher expected returns (drift). In other words, in general an asset can be either a good long-term store of value, or a good short-term store of value, but not both.
Stocks are a good example of an asset with high positive drift and high volatility. Their value bounces around a lot, but it tends to increase over time. If "store of value" means "value tends to rise over time", then stocks would be a very good candidate. Stocks are a good long-term store of value.
Fiat money with a 2%-inflation-targeting central bank is a good example of an asset with negative drift and low volatility. Over time, you can expect this currency to lose value, since there will tend to be about 2% inflation every year. But the value is highly predictable - it doesn't fluctuate very much at all from day to day. Fiat-money-with-2%-inflation-targeting is a good short-term store of value.
Looking out at the world, I see a whole lot of countries that use fiat money, with something like inflation targeting, as their medium of exchange (i.e., what they use to pay for stuff). And I see zero who use stocks as the medium of exchange, even though the technology now exists for us to make payments in stock shares quite easily (it's just the same as exchanging dollars electronically, really).
So I conclude that we want the medium of exchange - i.e., money - to be a good short-term store of value (i.e., to have low volatility), and that we don't need it to be a good long-term store of value (i.e., we don't care about its expected return).
Why is this the case?

It makes sense if you think about the way that we use money. People don't know exactly when they are going to need to spend money, or how much. If they keep their wealth in assets with high expected returns and high volatility - stocks, etc. - they run the risk of having to sell in a down market in order to pay for unexpected expenses. So it makes sense to keep some of their wealth in a low-volatility, low-expected-return asset like fiat-money-with-2%-inflation-targeting, in the expectation that they'll probably have to use it to pay for something. The low expected return - the fact that cash falls in value a little bit every year - doesn't matter so much, because you don't keep the cash around that long before you spend it.

(Note that this ignores correlations, but those won't end up mattering here.)

So this is why money should be a short-term store of value rather than a long-term store of value. This is why, as David Andolfatto pointed out, gold makes such a lousy form of money.

How about bitcoin? If it keeps experiencing high volatility, then it's not going to become the medium of exchange in the U.S. or other countries with inflation targets. But if volatility falls in consumption terms - in other words, if the bitcoin prices of goods and services become very stable - then bitcoin will have a good chance of becoming the medium of exchange.

One problem, though, is that there's a bit of a chicken-and-egg situation here. The more merchants use bitcoin, the less volatile its consumption value will probably be. But in order for merchants to use it, customers have to use it, and they'll only start using it if there's low volatility.

But if bitcoin eventually manages to solve this chicken-and-egg problem, its promoters hope that it will be able to offer about the same volatility as fiat money but with a higher expected return. That would make bitcoin dominate fiat money, and would kick fiat money right out of the universe of investible assets - or, more realistically, it would force central banks to adopt an inflation target lower than the rate at which bitcoin is mined. That, I think, is the hope of bitcoin enthusiasts who say that bitcoin will "compete with central banks."

So for bitcoin to become money, it has to figure out how to massively reduce the volatility of bitcoin prices of goods and services.

Update

Eli Dourado has a good response. I think we agree on the volatility thing. I glossed over other kinds of transaction costs, which Koning addresses somewhat; on those matters, I'm pretty ignorant, so I will let Eli and JP work it out...

Tyler Cowen thinks Bitcoin's volatility is a bad sign for its chances of future adoption, because it reflects a consensus that Bitcoin will never really catch on. I disagree with Tyler. Suppose, for simplicity's sake, that milk was the only good that people consumed. And suppose that in the future, bitcoin becomes the universal medium of exchange, and that at that time the bitcoin price of milk is about the same as it is today. In this case, there is no benefit to buying a lot of bitcoin today, even if you know for certain that it's going to become universally adopted. Because the price of bitcoin is already "right", in consumption terms. Hoarding a bunch of bitcoin right now doesn't actually improve your tradeoff between future milk and present milk. So the lack of bitcoin speculation doesn't necessarily mean that people have decided that bitcoin is doomed. It could even mean the exact opposite.