When you open a bank account, the teller doesn’t take your money and put it into a box labeled with your name. Instead, the bank holds a fraction of your account balance in reserve—hence the term fractional reserve banking—and it uses the rest of your money to make loans.

Banks earn profit on these loans. So do you. Competition among banks to attract your funds means that if the bank lends out your money and charges 5% interest, the bank must share some of that return with you. The bank will pay you, say, 2% for providing the money that they lend. The bank doesn’t just pay you interest, it also gives you useful services like check writing and check clearing, which are in part funded from bank profits on loans. Of course, you don’t get the full 5% return because the bank is bearing the risk on the loans, plus paying the costs of making the loans and monitoring the loan borrowers.

How much does the bank keep in reserve and how much does it lend? On one hand, banks need to keep some reserves around. In part, the law and the Federal Reserve require them to keep some reserves. More important, banks need those reserves to meet ordinary depositor demands for currency and payment services. Who would patronize a bank where the ATM machine was always empty? On the other hand, banks don’t want to hold too many reserves because money held in reserve isn’t being lent and lending is where banks earn most of their profits. Thus, there are opportunity costs to holding onto reserves. Banks balance these benefits and costs and thus they decide on the ratio between reserves and deposits. We define the reserve ratio, RR, as the ratio of reserves to deposits. If $1 in cash is held in reserve for every $10 of deposits, the reserve ratio is 1/10.

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The reserve ratio is determined primarily by how liquid banks wish to be. When banks are worried that depositors might want to withdraw their cash or when loans don’t seem so profitable anyway, they want a reserve ratio that is relatively high; when banks aren’t worried about depositors demanding cash and when loans are profitable, they want to have a relatively low reserve ratio.

It’s also useful to work with the inverse of the reserve ratio, called the money multiplier, MM. The money multiplier is the ratio of deposits to reserves, or in this case 10. Why is it called the money multiplier? Imagine that the Federal Reserve creates $1,000 of new money by crediting your bank account with an additional $1,000. Does that sound incredible? In fact, the Fed can create new money at will either by printing it or—the more modern method—by adding numbers to bank accounts held at the Fed. As we shall see, the Federal Reserve creates billions of dollars in just this way on a regular basis. So let’s imagine that your bank account has been credited with an additional $1,000. Your bank now has $1,000 in extra reserves, but remember that banks don’t want to keep all of their depositors’ money in reserve. Banks make a profit by lending so now that your bank has extra reserves, it will also feel comfortable making more loans. To restore its reserve ratio to 1/10, your bank will want to keep $100 in reserve and make additional loans of $900. So let’s say it lends $900 to Sam.

Now here’s where it gets tricky. You have an extra $1,000 in your account but in addition Sam now has an extra $900 in his bank account, which for convenience we will assume is held at another bank. Now Sam’s bank has an extra $900, but it too doesn’t want to hold all of its new money in reserves so it will keep $90 in reserve and make $810 in new loans—thus, the bank’s reserve ratio stays at 1/10. The process does not stop there as Sam’s bank now lends money to Tom and Tom’s bank lends money to Dick and … well you get the idea. This process keeps going through a ripple effect as one bank increases its loans, leading to an increase in deposits in another bank, which in turn increases its loans, which leads to an increase in deposits in another bank, which increases its loans … and so forth.

What is the end result of the ripple process? There are two ways to see the end result, the long way and the shortcut. We are going to save the long way for the appendix. Here’s the shortcut. If banks want a reserve ratio of 1/10, then when the Federal Reserve increases reserves by $1,000, deposits must ultimately increase by $10,000. Now remember that the money multiplier is the inverse of the reserve ratio, or 10. Did you notice that deposits eventually increase by the increase in reserves multiplied by the money multiplier? That’s why it’s called the money multiplier.

Let’s summarize: The money multiplier tells us how much deposits expand with each dollar increase in reserves. If the money multiplier is 10, for example, then an increase in reserves of $1,000 will lead to an increase in deposits of $10,000. Since checkable deposits are part of the money supply (M1 and M2), we can also say that an increase in reserves of $1,000 increases the money supply by $10,000. Thus, we have

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Change in money supply = Change in reserves × Money multiplier

or

ΔMS = ΔReserves × MM