# Central Bank and the Money Supply

> AP Macroeconomics · Unit 4: Financial Sector
> Source: https://www.owlsprep.com/study/ap-macroeconomics-u4-central-bank-and-the-money/

This module covers core functions of central banks, how fractional reserve banking expands the money supply, the money multiplier, and all major monetary policy tools that change the money supply for AP Macroeconomics.

**Prerequisites:** [Definitions of M1/M2 money supply](https://www.owlsprep.com/study/ap-macroeconomics-u4-money-definitions/); [Basics of fractional reserve banking](https://www.owlsprep.com/study/ap-macroeconomics-u4-fractional-reserve-banking/)

## Learning objectives

- Describe core functions of a central bank in a modern economy
- Calculate changes in the money supply using simple and adjusted money multipliers
- Analyze the impact of each core monetary policy tool on the money supply
- Avoid common directional and calculation mistakes tested on the AP exam

## Core Concepts: Central Bank Role and Monetary Base

A central bank is the official monetary authority of a sovereign nation, with core responsibilities including controlling the money supply, regulating commercial banks, stabilizing the financial system, and acting as a lender of last resort during banking panics.

**Monetary Base** — Also called high-powered money, it is the portion of the money supply directly controlled by the central bank.

*Notation:* MB

$$MB = C + R$$

Where $C$ = currency in circulation held by the public, and $R$ = total reserves held by commercial banks at the central bank. Total reserves are split into required reserves ($RR$, mandated by the central bank) and excess reserves ($ER$, reserves held beyond requirement): $R = RR + ER$.

**Required Reserve Ratio** — The percentage of total demand deposits commercial banks are legally required to hold as reserves, calculated as $rr = \frac{RR}{D}$ where $D$ = total demand deposits.

*Notation:* rr

**Worked example:** The public holds $1.2 trillion in currency, and commercial banks hold $0.3 trillion in total reserves at the central bank. Calculate the monetary base.

1. Identify the given values:
2. $$C = 1.2 \text{ trillion}, R = 0.3 \text{ trillion}$$
3. Use the monetary base formula:
4. $$MB = C + R = 1.2 + 0.3 = 1.5$$
5. The monetary base is $1.5 trillion.

## The Money Multiplier

In a fractional reserve banking system, banks lend out excess reserves, which creates new demand deposits and expands the total money supply. The simple money multiplier calculates the maximum possible change in the money supply from a change in the monetary base, assuming no excess reserves and no public currency holding.

**Simple Money Multiplier** — The maximum ratio of change in total money supply to change in monetary base, calculated as:

*Notation:* m

$$m = \frac{1}{rr}$$

The maximum change in total money supply is then $\Delta MS = m \times \Delta MB = \frac{1}{rr} \times \Delta MB$.

**Worked example:** The central bank adds $200 million in new reserves to the banking system. The required reserve ratio is 10%, banks hold no excess reserves, and the public holds no additional currency. What is the maximum total change in the money supply?

1. Identify given values: $\Delta MB = +$200$ million, $rr = 0.10$, no excess reserves or currency drain so the simple multiplier applies.
2. Calculate the simple money multiplier:
3. $$m = \frac{1}{rr} = \frac{1}{0.10} = 10$$
4. Calculate the maximum change in the money supply:
5. $$\Delta MS = m \times \Delta MB = 10 \times 200 = 2000$$
6. Adding new reserves expands the money supply, so the total change is a +$2 billion ($2000 million) increase.

If excess reserves or currency drain are given, use the adjusted money multiplier formula, where $er$ = excess reserve ratio and $c$ = currency drain ratio: $m = \frac{1 + c}{rr + er + c}$.

**Worked example:** A central bank injects $3 trillion in new reserves, $rr = 0.10$, banks hold $er = 0.14$, and public currency drain is $c = 0.01$. Calculate the total change in the money supply.

1. Plug values into the adjusted multiplier formula:
2. $$m = \frac{1 + 0.01}{0.10 + 0.14 + 0.01} = \frac{1.01}{0.25} = 4.04$$
3. Calculate change in money supply:
4. $$\Delta MS = 4.04 \times 3 = 12.12$$
5. The total change is a +$12.12 trillion increase in the money supply, far lower than the simple multiplier maximum of $30 trillion.

> **Exam tip:** If the AP exam does not mention excess reserves or currency drain, always use the simple $\frac{1}{rr}$ money multiplier.

## Open Market Operations

Open market operations (OMO) are the most frequently used monetary policy tool, involving the purchase or sale of government securities to change the monetary base and total money supply.

> **Direction Mnemonic**
>
> BITE: **B**uy = **I**ncrease money supply, **S**ell = **D**ecrease money supply

When the central bank buys bonds, it adds new reserves to the banking system, increasing the monetary base and expanding the money supply (expansionary policy). When it sells bonds, it removes reserves, decreasing the monetary base and contracting the money supply (contractionary policy).

**Worked example:** The central bank sells $50 million in government bonds to commercial banks. The required reserve ratio is 20%, banks hold no excess reserves, and the public holds no additional currency. What is the total change in the money supply?

1. Selling bonds removes $50 million in reserves, so $\Delta MB = -$50$ million.
2. Calculate the simple money multiplier:
3. $$m = \frac{1}{0.20} = 5$$
4. Calculate the total change in the money supply:
5. $$\Delta MS = 5 \times (-50) = -250$$
6. Selling bonds is contractionary, so the money supply decreases by $250 million.

## Other Core Monetary Policy Tools

Central banks use two additional core tools to adjust the money supply: the discount rate and interest on reserves.

**Discount Rate** — The interest rate commercial banks pay to borrow directly from the central bank to cover reserve shortfalls. Lowering the discount rate increases borrowed reserves and expands the money supply; raising it contracts the money supply.

**Interest on Reserves (IOR)** — The interest rate the central bank pays commercial banks for reserves held at the central bank. Lowering IOR reduces the incentive to hold excess reserves, increasing lending and expanding the money supply; raising IOR contracts the money supply by increasing reserve holdings.

**Worked example:** The monetary base is $200 billion, no public currency holding, required reserve ratio is 10%, and banks initially hold 0% excess reserves. After the central bank raises interest on reserves, banks hold 5% of all deposits as excess reserves. Calculate the change in total money supply.

1. Calculate initial money supply with $rr = 0.1$, $er = 0$:
2. $$m_1 = \frac{1}{rr + er} = \frac{1}{0.1} = 10, MS_1 = 10 \times 200 = 2000$$
3. Calculate new money multiplier after IOR increase, $er = 0.05$:
4. $$m_2 = \frac{1}{0.1 + 0.05} \approx 6.67, MS_2 = 6.67 \times 200 \approx 1334$$
5. Calculate the change:
6. $$\Delta MS = 1334 - 2000 = -666$$
7. Raising IOR reduces the money supply by approximately $666 billion, holding the monetary base constant.

**Check your understanding**

Test your understanding of policy tool impacts

1. Which of the following actions by the central bank will lead to an increase in the money supply, all else equal?

   - A) The central bank sells $100 million in government Treasury securities
   - B) The central bank raises the required reserve ratio from 10% to 15%
   - C) The central bank lowers the interest rate it pays on reserves
   - D) The central bank raises the discount rate

   *Why:* Lowering interest on reserves reduces the incentive for banks to hold excess reserves, so banks lend more, increasing the effective money multiplier and total money supply. Selling bonds removes reserves, raising RR reduces the multiplier, and raising the discount rate reduces borrowed reserves, all of which decrease the money supply.

> **Exam tip:** When the problem gives an excess reserve ratio, always add it to $rr$ in the denominator of the money multiplier.

## Common pitfalls

- **Wrong:** Confusing the direction of open market operations, claiming selling government bonds increases the money supply
  - Why it fails: Students mix up who is transacting; when the central bank sells, banks pay the central bank, removing reserves from circulation
  - Correct: Use the mnemonic BITE: Buy = Increase, Sell = Decrease to confirm direction every time
- **Wrong:** Using $\frac{1}{rr}$ as the money multiplier when the problem explicitly states banks hold excess reserves
  - Why it fails: The simple multiplier assumes no excess reserves, which only applies when this is stated or implied
  - Correct: Scan the problem first for excess reserve or currency drain values; add them to $rr$ in the denominator if given
- **Wrong:** Claiming raising the discount rate increases the money supply
  - Why it fails: Students confuse the discount rate with market interest rates and assume higher rates encourage lending
  - Correct: Remember the discount rate is what banks pay to borrow; higher = more expensive = fewer reserves = lower money supply
- **Wrong:** Calculating $\Delta MS$ as $\Delta MB \times rr$ instead of $\Delta MB \times \frac{1}{rr}$
  - Why it fails: Students invert the multiplier because they confuse the required reserve ratio with the multiplier itself
  - Correct: Always write the full formula down before plugging in numbers
- **Wrong:** Assuming changing the required reserve ratio changes the monetary base
  - Why it fails: Students mix up how different tools affect the monetary base vs. the money multiplier
  - Correct: Changing the required reserve ratio changes the money multiplier, not the monetary base; lower $rr$ = higher multiplier = higher money supply for a fixed MB
- **Wrong:** Claiming raising interest on reserves increases the money supply
  - Why it fails: Students assume higher interest rates mean more lending, but IOR is paid to banks for holding reserves, not for lending
  - Correct: Higher IOR = more incentive to hold reserves = less lending = lower money supply

## Cheatsheet

| Category | Formula / Rule | Notes |
| --- | --- | --- |
| Monetary Base | $MB = C + R$ | Directly controlled by central bank |
| Required Reserve Ratio | $rr = \frac{RR}{D}$ | Set by central bank |
| Simple Money Multiplier | $m = \frac{1}{rr}$ | No excess reserves/currency drain |
| Adjusted Money Multiplier | $m = \frac{1 + c}{rr + er + c}$ | $er$ = excess reserve ratio, $c$ = currency drain |
| Change in Money Supply | $\Delta MS = m \times \Delta MB$ | +$\Delta MB$ = increase MS, - = decrease |
| Open Market Purchase | $\uparrow MB \rightarrow \uparrow MS$ | Expansionary policy |
| Open Market Sale | $\downarrow MB \rightarrow \downarrow MS$ | Contractionary policy |
| Discount Rate Change | Lower = $\uparrow MS$, Higher = $\downarrow MS$ | Changes borrowed reserves |
| Interest on Reserves Change | Lower = $\uparrow MS$, Higher = $\downarrow MS$ | Changes incentive to hold reserves |

## What's next

Mastering how central bank actions change the money supply is the foundational prerequisite for all subsequent study of monetary policy, the most heavily tested topic in AP Macroeconomics Unit 4. This topic links the structure of the financial sector to aggregate demand, macroeconomic stabilization, and inflation targeting. Without correctly identifying how each policy tool impacts the money supply, you cannot accurately analyze how monetary policy transmits to changes in interest rates, real GDP, and the price level — topics that make up a large share of both multiple-choice and free-response exam points. Next, you will explore how money supply changes impact nominal interest rates and move to full monetary policy analysis.

- [Monetary Policy Tools](https://www.owlsprep.com/study/ap-macroeconomics-u4-monetary-policy-tools/)
- [Quantity Theory of Money](https://www.owlsprep.com/study/ap-macroeconomics-u4-quantity-theory-of-money/)

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