Simple Money Multiplier Calculation

Introduction

Welcome to the next lesson in our module on Money and Banking. In our last session, we explored the fascinating process of how commercial banks create money through the fractional-reserve system. We saw that when a bank makes a loan, it effectively creates a new deposit, expanding the overall money supply. We concluded by introducing the concept of the simple money multiplier as a way to quantify the total potential effect of this process.

Today's lesson is dedicated entirely to mastering that concept. Our goal is to calculate the simple money multiplier given a required reserve ratio. This is a fundamental quantitative skill in macroeconomics. We will not only learn the formula but also understand its mathematical origins and, critically, how to apply it correctly in different scenarios. This will build a strong foundation for our next lesson, where we'll see how the central bank uses this multiplier effect to conduct monetary policy.

1. The Money Multiplier Formula and its Origin

In the previous lesson, you learned that the money creation process is a sequence of lending and redepositing. An initial deposit allows for a loan, which becomes a new deposit, which allows for another, smaller loan, and so on. This creates a cascade effect.

The total sum of this cascade can be calculated with a simple formula:

$$\text{Simple Money Multiplier}=\frac{1}{\text{Required Reserve Ratio (rrr)}}$$

Where rrr is expressed as a decimal (e.g., 10% is 0.10).

This formula isn't magic; it's the result of summing an infinite geometric series. Given your BTech background, you'll likely appreciate seeing the mathematical foundation for this economic tool. It demonstrates how an initial deposit, $D_0$, combined with a reserve ratio, $r$, generates a sequence of new deposits.

To see this derivation, let's look at a resource that connects the economic concept directly to the underlying mathematics.

This derivation shows that the total amount of deposits created is the sum of $(1-r{)}^0{D}_0+(1-r{)}^1{D}_0+(1-r{)}^2{D}_0+\dots$, which elegantly simplifies to $D_0/r$. The term $1/r$ is our money multiplier.

2. The Multiplier in Action: A Step-by-Step Example

Now that we've seen the theory, let's watch it in action. The following video provides a clear, intuitive walkthrough of how an initial injection of reserves gets multiplied through the banking system.

As the video shows, if the reserve ratio is 10% (or 0.1), the money multiplier is:

$$\text{Money Multiplier}=\frac{1}{0.10}=10$$

This means that every new dollar of reserves in the banking system can potentially support a total of $10 in deposits.

3. A Crucial Distinction: Calculating the Change in the Money Supply

Here is where we need to be precise. The way you calculate the total change in the money supply depends on the source of the initial funds. This is a common point of confusion, and mastering this distinction is key to correctly applying the multiplier.

There are two primary scenarios:

1. The Central Bank injects new reserves. (e.g., by buying a government bond from a bank). This money is entirely new to the banking system.
2. An individual deposits existing cash. This cash was already part of the money supply (M1) before it was deposited.

The following video does an excellent job of demonstrating the calculation for both scenarios.

Let's summarize the key distinction from the video:

• Scenario 1: New Reserves (e.g., Fed buys $100bn of bonds)
  The entire $100bn is new to the system.

  • Total Change in Money Supply = Initial Injection of Reserves × Multiplier
  • Calculation: $100bn × 10 = $1,000bn ($1 trillion)

• Scenario 2: Individual Deposits Existing Cash (e.g., Jill deposits $100 cash)
  The initial $100 was already in the money supply as currency. The banking system creates new money only through the loans made from this deposit.

  • Total Change in Deposits = Initial Deposit × Multiplier = $100 × 10 = $1,000
  • Total Change in Money Supply = Change in Deposits - Initial Cash Deposit = $1,000 - $100 = $900
  • Alternative way: The new money created is the total amount of new loans. The first loan is the initial deposit minus required reserves ($100 - $10 = $90 of excess reserves). Total new loans = Initial Excess Reserves × Multiplier = $90 × 10 = $900.

To reinforce this, the following reading breaks down the process using balance sheets.

4. Practice Problems

Now it's your turn to apply the formula. Solve the following problems.

Problem 1:
The central bank sets the required reserve ratio at 5% (0.05). What is the value of the simple money multiplier?

Problem 2:
Assume the required reserve ratio is 20% (0.20). The central bank decides to increase the money supply and purchases $50 million worth of government bonds from commercial banks. What is the maximum possible change in the money supply?

Problem 3:
Assume the required reserve ratio is again 20% (0.20). An individual finds $1,000 in cash that they had stored at home and deposits it into their checking account. What is the maximum possible change in the money supply?

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Solutions

Solution 1:

$$\text{Money Multiplier}=\frac{1}{\text{rrr}}=\frac{1}{0.05}=20$$

The simple money multiplier is 20.

Solution 2:
This is a Scenario 1 problem (new reserves from the central bank).

1. Calculate the multiplier: 1 / 0.20 = 5.
2. Calculate the total change in the money supply:
   Initial Injection × Multiplier = $50 million × 5 = $250 million.
   The maximum possible increase in the money supply is $250 million.

Solution 3:
This is a Scenario 2 problem (deposit of existing cash).

1. The multiplier is 1 / 0.20 = 5.
2. The initial $1,000 was already part of M1. The new money is created from the loans generated by this deposit.
3. Calculate initial excess reserves: The bank must hold 20% of $1,000 which is $200. The excess reserves are $1,000 - $200 = $800.
4. Calculate the change in money supply (total new loans):
   Initial Excess Reserves × Multiplier = $800 × 5 = $4,000.
   The maximum possible increase in the money supply is $4,000.

5. A Final Caveat: The "Simple" Multiplier

It is important to remember why we call this the simple money multiplier. It calculates the maximum potential change in the money supply. As we touched on in the last lesson, the real-world multiplier is often smaller due to two main "leakages":

1. Banks hold excess reserves: Banks may not lend out all of their excess reserves, especially during uncertain economic times.
2. Individuals hold cash: Not all loaned money is redeposited. If a borrower takes a loan and holds it as cash, that money leaves the banking system and cannot be used for the next round of lending.

The Marginal Revolution University video also briefly mentions this important point. You can re-watch the segment from 05:07 to 06:18 in "Fractional Reserve Banking and the Money Multiplier Made Simple" (LINK) for a quick refresher on these factors.

Conclusion

In this lesson, we focused on the essential skill of calculating the simple money multiplier and its effect on the money supply. We saw that while the formula is straightforward, its application requires careful attention to the source of the funds.

Key Takeaways:

• The simple money multiplier formula is 1 / rrr.
• The formula is derived from the sum of a geometric series, representing the successive rounds of lending in a fractional-reserve system.
• When calculating the change in the money supply, you must distinguish between an injection of new reserves from the central bank and a deposit of existing cash by an individual.
• The simple multiplier represents the maximum potential expansion of the money supply, which can be reduced by real-world leakages like banks holding excess reserves or individuals holding cash.

Preview of the Next Lesson:

Having mastered the money multiplier, you are now perfectly equipped to understand our next topic: the tools of monetary policy. We will explore how a central bank, like the U.S. Federal Reserve, uses instruments like open market operations, the discount rate, and reserve requirements to influence bank reserves. You will see that the money multiplier is the mechanism through which these seemingly small policy actions can have a large and powerful impact on the overall money supply and the economy.