**[Q:]**Suppose the required reserve ratio is 0.2.

If an extra $20 billion in reserves is injected into the banking system through an open market purchase of bonds, by how much can demand deposits increase?

Would your answer be different if the required reserve ratio was 0.1?

**[A:]** Great question! First we'll examine what the required reserve ratio is.

The reserve ratio is the percentage of depositors' bank balances that the banks have on hand. So if a bank has $10 million in deposits, and $1.5 million of those are currently in the bank, then the bank has a reserve ratio of 15%. In most countries banks are required to keep a minimum percentage of deposits on hand, known as the **required reserve ratio**. This required reserve ratio is put into place to ensure that banks do not run out of cash on hand to meet the demand for withdrawals.

What do the banks do with the money they don't keep on hand? They loan it out to other customers! Knowing this, we can figure out what happens when the money supply increases.

When the Federal Reserve buys bonds on the open market, it buys those bonds from investors, increasing the amount of cash those investors hold. They can now do one of two things with the money:

- Put it in the bank.
- Use it to make a purchase (such as a consumer good, or a financial investment like a stock or bond)

If every investor who sold a bond put her money in the bank, bank balances would initially increase by $20 billion dollars. It's likely that some of them will spend the money. When they spend the money, they're essentially transferring the money to someone else. That "someone else" will now either put the money in the bank or spend it. Eventually all of that 20 billion dollars will be put into the bank.

So bank balances rise by $20 billion. If the reserve ratio is 20%, then the banks are required to keep $4 billion on hand. The other $16 billion they can loan out.

What happens to that $16 billion the banks make in loans? Well, it is either put back into banks, or it is spent. But as before, eventually the money has to find its way back to a bank. So bank balances rise by an additional $16 billion. Since the reserve ratio is 20%, the bank must hold onto $3.2 billion (20% of $16 billion). That leaves $12.8 billion available to be loaned out. Note that the $12.8 billion is 80% of $16 billion, and $16 billion is 80% of $20 billion.

In the first period of the cycle, the bank could loan out 80% of $20 billion, in the second period of the cycle, the bank could loan out 80% of 80% of $20 billion, and so on. Thus the amount of money the bank can loan out in some period **n** of the cycle is given by:

$20 billion * (80%)^{n}

where **n** represents what period we are in.

To think of the problem more generally, we need to define a few variables:

### Variables

- Let
**A**be the amount of money injected into the system (in our case, $20 billion dollars) - Let
**r**be the required reserve ratio (in our case 20%). - Let
**T**be the total amount the bank loans out - As above,
**n**will represent the period we are in.

A*(1-r)^{n}

This implies that the total amount the bank loans out is:

T = A*(1-r)^{1} + A*(1-r)^{2} + A*(1-r)^{3} + ...

for every period to infinity. Obviously we cannot directly calculate the amount the bank loans out each period and sum them all together, as there are an infinite number of terms. However from mathematics we know the following relationship holds for an infinite series:

x^{1} + x^{2} + x^{3} + x^{4} + ... = x / (1-x)

Notice that in our equation each term is multiplied by A. If we pull that out as a common factor we have:

T = A[(1-r)^{1} + (1-r)^{2} + (1-r)^{3} + ...]

Notice that the terms in the square brackets is identical to our infinite series of x terms, with (1-r) replacing x. If we replace x with (1-r), then the series equals (1-r)/(1 - (1 - r)), which simplifies to 1/r - 1. So the total amount the bank loans out is:

T = A*(1/r - 1)

So if A = 20 billion and r = 20%, then the total amount the bank loans out is:

T = $20 billion * (1/0.2 - 1) = $80 billion.

Recall that all the money that is loaned out is eventually put back into the bank. If we want to know how much total deposits go up, we also need to include the original $20 billion that was deposited into the bank. So the total increase is $100 billion dollars. We can represent the total increase in deposits (D) by the formula:

D = A + T

But since T = A*(1/r - 1), we have after substitution:

D = A + A*(1/r - 1) = A*(1/r).

So after all this complexity, we are left with the simple formula **D = A*(1/r)**. If our required reserve ratio were instead 0.1, total deposits would go up by $200 billion (D = $20b * (1/0.1).

With the simple formula **D = A*(1/r)** we can quickly and easily determine what effect an open-market sale of bonds will have on the money supply.