However, some things are worth noting. There is nothing intrinsically special about banks.
At least in the US, what's special about a bank (National Association Organization, ie, Chase Manahattan Bank, N.A.), is that it has a charter to literally create money by lending out more than it has on reserve and demanding the full sum in repayment, ie, fractional reserve practices.
If I borrow 1000฿ each from my pals Alice and Bob and promise to pay them back anytime they want then together we will consider ourselves to have more than 2000฿. That's exactly the same kind of inflation. And it's no big deal. It's no big deal because it quickly stabilizes and then there is no more inflation. The only way to have ongoing inflation is to increase the base money supply and Bitcoin prevents that.
This is not quite the same thing. I agree the two situations are deceptively similar, but here is where they differ.
Borrowing Example:
Alice has 1000BTC. Bill has 1000BTC. Christina needs 2000BTC, so she borrows 1000 from Bill and 1000 from Alice. Alice loses 1000, Bill loses 1000, and Christina gains 2000. Total in play before transaction: 2000. Total after transaction: 2000
Bank example:
BigBank has 3000 Bitcoins, but no one knows this except the BigBank CEO. They have a fancy office, though, so everyone assumes they have millions of bitcoins. Christina needs 2000 Bitcoins, so she goes to BigBank. Christina signs a pledge to return 2,100 bitcoins at this date next year (5% interest), and BigBank opens an two "accounts" for Christina. These accounts are nothing more than two lines on spreadsheets. The first:
Christina - Line of Credit - -2000BTC
The second:
Christina - Checking: 2000 BTC.
Before the transaction, BigBank had 3000 Bitcoins, and Christina had 0 Bitcoins, for a total of BTC3000. After, BigBank has 3000Bitcoins (they haven't actually given Christina any cryptographically-signed data - they just typed two numbers in a spreadsheet), and Christina "has" BTC2000. Now, if Christina demanded the BTC2000 in cash, BigBank would have to turn it over, but she's not going to do that. She's going to take the debit card that she got for her new checking account and leave. (Even if she doesn't do that in this instance, on average, most people will, because money in an account earns interest, so it "costs" them money to request it in cash. Also, maybe the debit card is easier to handle, etc. The entire fractional reserve system is based on the correct assumption that people will not demand cash an overwhelming percentage of the time).
Total Available to be Spent Before Transaction: 3000BTC (3000 BigBank + 0 Christina)
Total Available After Transaction: 500BTC (3000 BigBank + 2000 Christina).
The two thousand bitcoins that Christina got didn't exist before the transaction. They were created out of thin air. Of course, they really weren't minted. BigBank didn't hash them using a client, so in fact don't exist. But, through the "miracle" of fractional reserve banking, the following happens.
Christina goes to Dirty Dan's Dirtbike Dealership. Dirty Dan happens to except Bit-Visa. So, Christina swipes her debit card, and BigBank moves the 2000BTC number (which is in fact 0 real bitcoins) from her checking account into Dirty Dan's checking account. These 2000BTC don't exist, as we established above, but nonetheless, Christina just bought a Dirty Dan Dirtbike because both she and Dirty Dan both *think* they exist.
Later, a year rolls around, and Christina has to work two jobs to pay back the 2,100BTC she owes BigBank. What did those 2,100BTC cost BigBank? Nothing. They never even existed. Probably, they don't exist now, because Christina's employer, Ernie's Elastics, has an account with BigBank...and he took out a loan with BigBank to cover business expenses, and so her salary is made of bitcoins that don't exist...and on and on and on....
Even though the 2000 BC don't exist, the *purchasing power* of 2,000BTC has just entered the economy. This is the inflation that is problematic.
Thus, in the borrowing example, Alice and Bob lose money and Christina gets it. In the BigBank example, no one loses money, but everyone trusts BigBank, and so their pledge that Christina has 2,000BTC to spend is just as good as those 2,000BTC, even though their pledge is a lie, and those bitcoins do not exist.
That is the situation which causes inflation, and it is in no way related to the underlying supply of pieces of paper that make dollar bills, or hashed kilobytes that make real bitcoins. Its all, quite literally, imaginary. Thus, Bitcoin is subject to the same inflationary pressures as dollars, euros, or pounds.
The problem with many banks today is not fractional reserves. The problem is that their risks are guaranteed by governments. So they take huge risks and enjoy greats profits while it lasts. When it blows up taxpayers pay. To make it worse governments often pay with "free" freshly printed money. Again Bitcoin fixes the money printing issue but not the guarantee issue (or maybe a little if it makes it harder to collect taxes).
Governments do not pay with "freshly printed money." The money isn't printed, nor does it go through the government at all. The Federal Reserve, which is a private bank just like Chase or Wells Fargo or HSBC, merely writes an "account" for the troubled bank, and money pops magically into existence just like the example above. The government only has to print enough money to keep the banks from running out of *what the people ask for in cash*, which, in today's world of credit cards, PayPal, etc, is a very, very small fraction of the total money created. The rest of the money is blessed into existence whenever a loan is made, because to loan out five dollars of five trillion dollars, the loaning bank loses precisely zero dollars. The same is true with bitcoins - the accounts are fictitious numbers, and in no way represent the number of bitcoins, or dollars, or whatever else, that physically (or electronically) exist.
This sounds confused. In a bank run their problem is that they don't have enough real bitcoins and they can't create real bitcoins. What does the 21M limit have to do with anything?
The problem is not a bank run - the problem is inflation. Why would people demand real bitcoins? Why would they not just use their account balances. Heck, a website for a bank account could look *exactly* like the Bitcoin client, and users wouldn't know the difference. A run on the bank only occurs when people demand real currency (dollar bills, bitcoins, whatever). However, most of the time, even with paper money, people are willing to trust their bank balances and don't demand cold hard cash. If cold hard cash looked and felt and seemed exactly like the bank balances to most people (which they do with bitcoin - one computer screen or another, no one who's not a programmer knows the difference), then people, as with today, would not demand real bitcoins. They can tell the difference between a bank balance and a dollar bill, but as far as they know, real bitcoins and bank balances are *exactly* the same thing. So, a run on the bank is even *less* likely than with paper currency. In the meantime, inflation and a run on the bank are unrelated.