A Worked Example on the Money Multiplier February 26, 2011
Posted by tomflesher in Macro, Teaching.Tags: fractional reserve banking, Money multiplier, t-account, worked examples
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In the previous post, I introduced the thinking underlying the Money Multiplier. It represents how money is created by the process of loaning money that’s held on deposit. A portion of the money on deposit must be held in reserve to make sure the bank can function. This is what’s called a fractional reserve banking system. This post will work with a standard problem setup that you might see on your Principles of Macroeconomics exam and show how to answer different questions that might be asked about it.
A bank’s balance sheet is usually expressed in the form of a T-account. On the left-hand side, assets are listed. On the right-hand side, liabilities are listed. Deposits are a liability because the bank must be prepared to pay them out at any time. I abbreviate Deposits as D. Similarly, because loans (abbreviated L) will be paid back (are receivable), they represent an asset. Any funds that the bank could lend are called loanable funds, abbreviated LF. Assets break down into two categories: loans and reserves.
Reserves break down into two categories: required reserves and excess reserves. Required reserves are the fraction of deposits that banks are required by law to keep on hand; excess reserves are money on hand that are above the excess reserve amount. Using the convention that rr means reserve ratio, RR means required reserves, TR means total reserves, and ER means excess reserves, the following identities hold:
Take the following bank’s T-account:
The first thing that the exam might ask you is:
1. If the bank holds no excess reserves, what is the reserve ratio?
The question tells me that ER = 0, which means that RR = TR = $250. Since D*rr = RR, 1000 * rr = 250, dividing both sides by 1000 yields that rr = 250/1000. In lowest terms, this is 1/4, or .25 in decimal form.
2. If the reserve ratio is 1/10, what is the amount of excess reserves?
Begin with D = $1000. D*rr = RR, so $1000*1/10 = RR = $100. Then, since ER = TR – RR, ER = $250 – $100 = $150.
3. If the reserve ratio is 1/10, what is the largest new loan this bank could prudently make?
This is a tricky question because it’s simply a complicated way of asking how much the bank is holding in excess reserves. Why? Because a bank is allowed to loan out all but the required reserves. If a bank holds more than the required reserves, they have excess reserves (by definition) and they are allowed to loan out more money. Therefore, by the same method as question 2, the answer to this is $150.
4. Suppose the reserve ratio is 1/4. If the Fed lowers the reserve ratio to 1/5, what is the effect on the size of the money supply?
This requires two formulas from my previous entry, where I defined the Money Multiplier. Those formulas are
Here, the change in deposits 
Even if we don’t know the money supply, we know how much it will change from this bank.
So this small change in the required reserve ratio will lead to an increase in deposits of $200 from this bank.
Why is the Money Multiplier 1 over the Reserve Ratio? February 25, 2011
Posted by tomflesher in Macro, Teaching.Tags: Federal Reserve Bank, Introduction to Macroeconomics, macroeconomics, monetary policy, monetary theory, money creation, Money multiplier, Principles of Macroeconomics, reserve ratio
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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:
- They have more information about who wants to borrow money (specifically, businesses and consumers ask them for loan), and
- 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,
Since 1/rr is a little inconvenient to write, we made up a new term: the Money Multiplier.
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:
Rearranging slightly,
A geometric series with a ratio of less than one – which our ratio, 9/10, is – has the solution
Thus, the sum of all of these new deposits will be
or, more generally,
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.
The Bad Economist 