Understanding the Money Multiplier

How Banks Create Money: Fractional Reserve Banking

The modern banking system is a fractional reserve system. This means that banks are legally required to hold only a fraction of their deposits as reserves and can lend out the rest. The fraction of deposits that must be held in reserve is called the required reserve ratio (rr), which is set by the central bank.

When a bank makes a loan, it creates new money. The loan becomes a new deposit in the banking system, which then gets lent out again, creating a cascade effect. This process is often called the multiple deposit creation process.

To build an intuition for this process, let's watch how an initial injection of cash ripples through the banking system.

The Simple Money Multiplier

As you saw in the video, an initial deposit leads to a chain reaction of subsequent, smaller deposits. The total increase in the money supply is the sum of all these new deposits. We can represent this process mathematically.

If the required reserve ratio is $rr$, and there is an initial increase in bank reserves of $\Delta R$, the process unfolds as follows:

• Stage 1: The first bank receives $\Delta R$ and lends out $(1-rr)\times \Delta R$.
• Stage 2: This loan is deposited in a second bank, which then lends out $(1-rr)$ of that amount, which is $(1-rr{)}^2\times \Delta R$.
• Stage 3: This continues, with the third bank lending out $(1-rr{)}^3\times \Delta R$, and so on.

The total change in deposits ($\Delta D$) is the sum of this infinite geometric series:

Factoring out $\Delta R$:

The sum of an infinite geometric series $1+x+x^2+\dots$ is $1/(1-x)$. In our case, $x=(1-rr)$. Substituting this in, the term in the brackets becomes:

This gives us the final formula for the total change in deposits:

The term $1/rr$ is known as the simple money multiplier. It tells us the maximum amount the money supply (specifically, checkable deposits) can increase for every dollar increase in bank reserves.

The following video segments summarize this derivation and show how to apply the formula.

Applying the Money Multiplier: A Worked Example

Let's solve a problem similar to what you might see on your exam. This problem uses the concepts we've just discussed.

Let's break it down step-by-step:

1. Identify the given information:

   • Increase in bank reserves ($\Delta R$): $5
   • Desired reserve-deposit ratio ($rr$): 0.2
   • The problem also gives initial money supply ($1,000) and currency ($500), but for calculating the change, these are not needed. We only need the change in reserves and the multiplier.

2. Calculate the money multiplier:
   The formula for the simple money multiplier is $1/rr$.

   $$\text{Money Multiplier}=\frac{1}{0.2}=5$$

3. Calculate the increase in the money supply:
   The total increase in the money supply ($\Delta M$, which in this simple model equals the change in deposits $\Delta D$) is the increase in reserves multiplied by the money multiplier.

   $$\text{Increase in Money Supply}=\Delta R\times \text{Money Multiplier}$$ $$\text{Increase in Money Supply}=\$5\times 5=\$25$$

   This confirms the answer shown in the image. The initial $5 injection of reserves ultimately leads to a $25 expansion of the money supply.

An Important Distinction: New Reserves vs. Existing Currency

It is crucial to distinguish between two scenarios:

1. An injection of new reserves: When the central bank conducts an open market purchase, it buys bonds from banks and credits their reserve accounts. This is new money entering the banking system, and the full multiplier effect applies to this initial change in reserves. The example above illustrates this.
2. A deposit of existing currency: When an individual deposits cash (that was previously in circulation, e.g., under a mattress) into their bank account, the immediate effect on the money supply (M1 = Currency + Checkable Deposits) is zero. The currency component of M1 decreases, while the deposit component increases by the same amount. However, this deposit does increase the bank's reserves, which then get loaned out, starting the multiplier process.

The following video explains this critical distinction.

Beyond the Simple Multiplier: Leakages

The simple money multiplier $1/rr$ gives us the maximum possible expansion of the money supply. In reality, the actual multiplier is smaller due to two "leakages" from the deposit creation process:

1. Currency Drain: People and firms do not deposit all the money they receive; they choose to hold some of it as currency. Money held as currency cannot be used by banks to make new loans.
2. Excess Reserves: Banks may sometimes choose to hold more reserves than the legally required amount. These are called excess reserves. By holding excess reserves, banks are lending out less than they could, which dampens the multiplier effect.

A more sophisticated money multiplier accounts for these leakages. While a full derivation is beyond the scope of this lesson, it's useful to see the formula to appreciate the factors at play.

The decision by banks to hold significant excess reserves can dramatically shrink the money multiplier. This happened following the 2008 financial crisis, when heightened uncertainty led banks to hoard liquidity rather than lend it out. As a result, despite massive injections of reserves by the Federal Reserve, the money multiplier fell below 1, meaning the money supply expanded by less than the increase in reserves.

Conclusion

In this lesson, we have dissected the process of money creation in a fractional reserve banking system. We established that an initial change in reserves triggers a larger change in the money supply, a phenomenon captured by the money multiplier.

Key Takeaways:

• Banks create money by lending out a portion of the deposits they receive.
• The simple money multiplier provides an upper bound for this process and is calculated as $1/rr$, where $rr$ is the required reserve ratio.
• The total change in money supply is the initial change in bank reserves multiplied by the money multiplier.
• The real-world multiplier is smaller than the simple multiplier due to leakages, namely the public's holding of currency and banks' holding of excess reserves.

You are now equipped to calculate how central bank actions to change reserves will ripple through the economy to affect the overall money supply. In our next lesson, we will connect this concept to one of the oldest theories in macroeconomics: the Quantity Theory of Money. This will allow us to analyze how changes in the money supply ($M$) ultimately influence the aggregate price level ($P$) and real output ($Q$) in the long run.