## Why is the Money Multiplier 1 over the Reserve Ratio?February 25, 2011

Posted by tomflesher in Macro, Teaching.
Tags: , , , , , , , ,

Or, How does the banking system create money?

The Introduction to Macroeconomics class I’m TAing just got to the Monetary System. This was a difficult chapter for the students last semester, so I want to place a clear explanation online and give a mathematical proof for some of the more motivated students who look grumpy when I handwave away the explanation. For those who want to play with this themselves, I’ve created a spreadsheet that shows the process of money creation. You can choose an initial deposit and a reserve ratio (which I set at $1000 and 1/10), then trace out the changes as people borrow the money. It boils down to what banks are for. The biggest mistake you, in your capacity as a consumer, can make is to think that the purpose of a bank is to give you a place to save money. Really, the purpose of a bank is to serve as a financial intermediary, which allows people who want to lend money (savers) to find people who want to borrow money (borrowers). When you put your money in a savings account, you’re putting money into the pool of loanable funds, and the bank will pay you a small amount of interest. Why don’t they let you do that yourself? Two reasons: 1. They have more information about who wants to borrow money (specifically, businesses and consumers ask them for loan), and 2. They offer a small return with certainty, whereas if you found a borrower yourself, you’d have to collect your return yourself. In order to provide this certainty, the banks have to hold a certain amount of each deposit in reserve, in order to make sure that people who want to take their money out can do so. This is the reserve ratio, and it’s set by the Board of Governors of the Federal Reserve. Adjusting the reserve ratio upward means that a bank needs to hold a larger portion of each deposit, so they have less to lend out. That means that all things being equal, money will be harder to get than with a smaller reserve ratio, and so interest rates rise and less money is created. Conversely, lowering the reserve ratio means that banks have more money they can lend out and more money will be created. When you deposit$1000 in the bank, if the reserve ratio is 1/10 or 10%, the bank has to hold $100 (or .1*$1000, or 1000/10) as reserves, but they can lend out $900. That means they’ve effectively created$900 out of thin air. They can loan out an additional 9/10 or 90% of the initial deposit. They’ll loan that money to a company which then uses it to, for example, pay an employee.

The employee will deposit $900 in the bank. The bank has to hold$90 ($900/10) as reserves, but they can lend out$810. Money is created, but the creation is smaller. As you can see, this cycle will repeat, with 10% being taken out every time, and if you iterate this enough times, you can see that the process approaches a final value because the new loanable funds that turn into new deposits are smaller every time.

In the case of a reserve ratio of .1 or 1/10, using my spreadsheet above, you can see that $1000 jumps to$6500 in deposits by the tenth person to be depositing the money, and the amount continues growing but at a slower rate. By the fiftieth person, the total deposits have reached about $9950, and by the one-hundredth depositor, it’s at$9999.73. The extra loanable funds here are only about 3 cents, so there’s going to be an almost imperceptible change. This is called convergence, and by playing with different reserve ratios, you can see that each reserve ratio converges. The total new deposits represent new money created. So, the change in the money supply is equal to the initial deposit divided by the reserve ratio, or, mathematically, $\Delta MS = (1/rr)*\Delta D$

Since 1/rr is a little inconvenient to write, we made up a new term: the Money Multiplier. $\Delta MS = MM*\Delta D$

Mathematically, this process is called a geometric series. Initially, the deposit is \$1000 and the new loanable funds are (9/10)*1000. The second deposit is (1/10)*1000 and the new loanable funds are (1/10)*(1/10)*1000. This process continues, and can be represented using the following notation: $\displaystyle\sum\limits_{i=0}^{\infty} r^i * 1000$

Rearranging slightly, $1000*\displaystyle\sum\limits_{i=0}^{\infty} r^i$

A geometric series with a ratio of less than one – which our ratio, 9/10, is – has the solution $1/(1-r) = 1/(1 - 9/10) = 1/(1/10) = 10$

Thus, the sum of all of these new deposits will be $10*1000$

or, more generally, $MM*Deposit$

That’s exactly what we wanted to show. So, if you’re mathematically inclined, remember that the Money Multiplier is the reciprocal of the reserve ratio because the process of borrowing and depositing money is a geometric series.