I'm thinking specifically of bitcoins.
What are the pros and cons of having a fixed number of coins, as opposed to more "normal" currencies? Would the currency have no inflation?
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FooBar is quite right that unless you expect GDP growth to stop, fixed nominal supply currencies will lead to deflation.
A moderate degree of currency inflation serves a number of useful functions in the economy. The most obvious are:
Without inflation you miss out on these benefits. The first benefit might not seem like a big deal if you think you can simply set the rate of inflation at zero percent. But it is very hard to hold inflation constant at some target level so attempting to hit zero inflation will almost certainly result in occasional lapses into deflation, with the attendant negative economic consequences.
It is a fallacy to conclude that a steady number of coins will give you no change on the monetary value (inflation/deflation). The classical quantity theory of money can be used as a first-order approximation here:
$MV = PY$
where $M$ is money, $V$ is the velocity of money, $P$ is the price level and $Y$ is the quantity of real goods. The equation says that the money stock must be sufficient to "buy" all produced goods in the economy, corrected for velocity of money.
Now take $M$ as constant (as in bitcoins). We immediately see that changes in $V$ or $Y$, unless they happen to cancel each other, will affect the price level $P$.
If bitcoins ever took over as the number one real currency, given that we see GDP increasing over time, c.p., they would have to increase in value (forever).
The other answers are correct in respect of what would happen if the money supply of a currency was kept constant, however there is nothing in bitcoins to ensure that money supply will stay fixed
This is such a common misconception so I'll repeat it. Bitcoins limit the money base, but that does not limit the money supply.
The money supply is the money base times the money multiplier, and the money multiplier comes from fractional reserve banking. There is nothing in bitcoins to prevent fractional reserve banking, and we are already starting to see website that allow you to lend or invest bitcoins.
Someone could, if they wished, set up a bitcoin bank. Savers would deposit bitcoins and earn interest, while borrowers could take out bitcoin loans. As soon as you do this the money supply (the $M$ in $MV=PY$) increases.
So, if bitcoins became a global currency, then you would almost certainly find that the money supply was much larger than the base limit - the money multiplier would expand as more banks entered the system. We don't have any concrete evidence to suggest what the limit on the money multiplier would be, but even in "normal" currencies the money multiplier can exceed 10x. The legal limit in the EU is 50x. With no regulatory capital or reserve requirements, who knows what the practical limit could be.
So what does fixed money base do?
Having a fixed money base means that there is no central bank to control the money supply. If the economy heats up, you would normally raise rate, which would reduce the money base and tighten the money supply.
In this case there would be nothing anyone could do - the world would simply rely on the "free banking sector" to do the right thing. This was actually not far off the way the US ran it's currency during the period when it had no central bank (between 1836 and 1913).
What you could do is control the money multiplier directly. This would mean passing laws and regulations forbidding fractional banking above a certain fraction. For bitcoins doing so would be almost impossible, not to mention against the whole "free" concept. You would have reintroduced a central controller for the currency.
The free banking era saw wild swings in money supply (e.g. more than doubling in a year) and hence huge inflation or deflation as banks grew and failed. It is anyone's guess what would be in store if bitcoins became a major currency, but there is no reason to believe they wouldn't have inflation.
The old monetary equation is MV=PT, where M is the money supply, V is the velocity of money, P is the price level, and T is the amount of transactions.
The fear in this case is that M will remain fixed. In a growing economy, T, the level of economic activity, will rise. Then, to make the equation hold, one of two things must happen: 1) V must rise to compensate for the rise in T, or 2) the price level must fall to compensate for the rise in T, to hold PT constant with M times V, both of which we're assuming remain constant.
A rise in V can be caused in improvements in money management; banking, ATM cards, smart phone payments etc. That, in fact, is a large part of what has driven economic growth in the past 50 years.
The other possibility, a fall in P, is what scares people. It's called deflation, and represents "sticker shock." Imagine a world in which prices were going down 3% every year, because the economy was growing 3% a year. Your wages would fall 3% a year. You shouldn't worry because your costs are also falling 3% a year, or more (but you will). There will be "reverse" millionaires," people who used to be millionaires, who aren't any longer, but who are better off because prices have fallen faster than their nominal wealth. It would be a strange, funky world, which is why policy makers avoid it, to the extent of making the "opposite" mistake of allowing inflation. Which may be one reason that the Bitcoin experiment is not viewed favorably by central banks.