The Bad Economist

A Worked Example on the Money Multiplier February 26, 2011

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 will be equal to the change in excess reserves .

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.