David Friedman addresses the problem of inflation caused by private banks in the book "Future Imperfect". Here is the section:http://www.daviddfriedman.com/Future_Imperfect/Chapter6.html
"Some economists, in rejecting the idea of private money, have argued that such an institution is inherently inflationary. Since issuing money costs a bank nothing and gives it the interest on the assets it buys with the money, it is always in the bank’s interest to issue more. The rebuttal to this particular argument was published in 1776. When Adam Smith wrote The Wealth of Nations, the money of Scotland consisted largely of banknotes issued by private banks, redeemable in silver.5 As Smith pointed out, while a bank could print as many notes as it wished, it could not persuade other people to hold an unlimited number of its notes. A customer who holds $1,000 in virtual cash – or Scottish banknotes – when he only needs $100 is giving up the interest he could have been earning if he had held the other $900 in some interest-earning asset instead. That is a good reason to limit his cash holdings to the amount he actually needs for day-to-day transactions.
What happens if a bank tries to issue more of its money than people wish to hold? The excess comes back to be redeemed. The bank is wasting its resources printing money, trying to put it into circulation, only to have each extra banknote promptly returned for cash – in Smith’s case, silver. The obligation of the bank to redeem its money guarantees its value, and at that value there is a fixed amount of its money that people will choose to hold.
Let us suppose that all the paper of a particular bank, which the circulation of the country can easily absorb and employ, amounts exactly to forty thousand pounds; and that for answering occasional demands, this bank is obliged to keep at all times in its coffers ten thousand pounds in gold and silver. Should this bank attempt to circulate forty-four thousand pounds, the four thousand pounds which are over and above what the circulation can easily absorb and employ, will return upon it almost as fast as they are issued. (Wealth of Nations, Bk I, chapter 2)"