# Money Supply and the Money Multiplier

Money, either in the form of currency or as bank reserves, is a liability of the central bank. The central bank controls the monetary base, expanding or contracting it at will, according to the needs of the economy. However, the actual money supply is a multiple of the monetary base, so what is the relationship between the supply of money and the **monetary base** (**MB**), which is the quantity of the individual units of money.

## Deposit Expansion Multiplier

Currency actually forms only a small part of the monetary base, since most money is stored electronically as account information. This electronic monetary base is multiplied through a process called **multiple deposit creation**, which results from the fact that the monetary base can be used in multiple financial transactions.

There is also a multiplier effect for currency. Imagine a group of 4 people who happened to have items for sale. Amy has $10, which she uses to buy Barbara's discount movie tickets. Barbara uses the $10 and pays Chris for a CD, who uses the $10 to buy LED Christmas lights from David. So, in this instance, the same $10 was used in 3 transactions for $30 worth of financial transactions; likewise, for bank reserves, except that a bank will keep a part of it as reserves to comply with the law and to carry out daily business.

To see in detail how bank deposits are multiplied, consider a series of banks as lenders and businesses as borrowers. We start this illustration with a number of assumptions:

- no bank holds excess reserves;
- the reserve requirement is 10%;
- the borrowed money is deposited into a checking account at another bank that is not any of the previous banks.

Bank 1 lends $1,000 to Borrower A, who then pays his supplier, Business B, the amount of the loan; Business B deposits the money in its own account at Bank B; Bank B lends out 90% of the deposit, or $900, to Business C, who pays its suppliers, Business D, the $900, and so on.

This leads to the following series of payments:

- $1,000
- $900
- $810
- $729
- ...

Because the banks keep some of each deposit as reserves, the amount of additional financial transactions that a particular deposit can generate is limited. However, if banks lent out all their deposits, there would be no limit to the number of financial transactions, just as currency can be used over and over again.

The formula for the **deposit expansion multiplier** is derived from the required reserves formula for deposits, where the **required reserves** (**RR**) are equal to the **required reserve ratio** (**r**) multiplied by **bank deposits** (**D**):

1. RR = r × D

Dividing both sides by RR, then transposing, yields:

2. D = RR / r

Hence, in the above example, if the money initially lent out by Bank A is continually re-deposited in different banks, the total quantity of money is: $1,000 / .1 = $10,000

Assuming that the reserve ratio remains constant, any change in reserves, whether positive or negative, causes a corresponding change in the potential deposit amount:

3. ΔD = ΔRR / r

Hence, if the reserve ratio is 10%, then increasing the reserves multiplies the increase in potential deposits by 10.

In the same way that increases in reserves expand deposits, decreases in reserves will cause a contraction by the same amount. So, if reserves increase by $10, then potential deposits increases to $100; if reserves decline by $10, then deposits contract by $100.

## Monetary Base And Money Supply

The monetary base is simply money, whether it is currency or reserves:

4. Monetary Base = Currency + Bank Reserves

However, the total **quantity of money** depends on how often each dollar is used in transactions. The** money multiplier** is the number of times that the monetary base is used in transactions:

5. Money Supply = Monetary Base × Money Multiplier

However, not all money is spent or lent out. That which is kept reduces the supply of money.

There are 2 factors that restrain the growth of the money supply when deposits expand:

- some banks keep
**excess reserves**(**ER**), which is the amount above what they are required to hold; - the public has a tendency to hold more cash as their income — and their income — rises.

When banks hold excess reserves, deposit multiplication is less. Indeed, although there is a legal distinction between required reserves and excess reserves, there is no economic distinction, because neither required reserves nor excess reserves is multiplied by the deposit multiplier. Nonetheless, banks tend to hold more excess reserves when their deposits increase, which is often expressed as an **excess reserves-to-deposit ratio** (**ER/D**). A bank's **total reserves** (**R**) can be expressed:

6. R = RR + ER

Substituting Equation 1:

1. RR = r × D

into Equation 6 and expressing excess reserves as a percentage of total deposits yields:

7. R = r × D + (ER/D) × D

Factoring out D yields:

8. R = (r + ER/D) × D

Likewise, the public tends to hold more **currency** (**C**) as their bank deposits increase, so that the amount of currency held by the public can be expressed as the **currency-to-deposit ratio** (**C/D**) times their deposits:

9. C = C/D × D

Hence, the monetary base can be expressed thus:

10. MB = C + R

This equation can be expressed as the currency held by the public being equal to a percentage of their deposits plus the total reserves held by the bank as expressed in Equation 8:

11. MB = (C/D) × D + (r + ER/D) × D

Factoring out D on the right hand side of the equation yields:

12. MB = (C/D + r + ER/D) × D

Dividing both sides by **C/D + r + ER/D** and transposing yields the amount of deposits as a multiple of the money base:

13. D | = | 1 C/D + r + ER/D | × | MB |

Since reserves are just deposits, then money (**M**) can be expressed as:

14. M = C + D

Substitute Equation 9:

9. C = C/D × D

into Equation 14, then factoring out D yields:

15. M = C/D × D + D

16. M = (C/D + 1) × D

Substituting Equation 13 into Equation 16 yields:

M | = | C/D + 1 C/D + r + ER/D | × | MB |

The 1^{st} term of the above equation is the **money multiplier** in terms of the currency-to-deposit ratio (C/D), the required reserve ratio (r), and the excess-reserves-to-deposit ratio (ER/D). Note that if banks decide to keep more excess reserves, the money supply will decline. Note also that even though the currency-to-deposit ratio is in both the numerator and denominator, an increase in the denominator will cause the ratio to decline more than a corresponding increase in the numerator will increase it. Hence, holding more currency tends to decrease the money supply.

How much currency is held by the public depends on costs and benefits. The opportunity cost of currency is the interest that it would earn as a deposit compared to the advantages of lower risk and greater liquidity as currency. Hence, the public will hold less currency if it can earn higher interest rates as a deposit. Likewise, the higher the interest rate difference between lent money and reserves, the less likely that banks will keep excess reserves.

The central bank controls the monetary base and usually controls the reserve requirement. Although banks decide how much excess reserves they will hold, the central bank can influence that decision by the amount of interest that it pays on the reserves.

What isn't under the central banks' control is the public's demand for currency, but it can be influenced by interest rates. Any increased demand for currency will probably cause the money supply to contract because withdrawing money as currency reduces reserves, which, because of the multiplier effect, will reduce the money supply by more than the amount withdrawn. When many banks failed during the Great Depression, many people withdrew most or all their money from the banks because they lost confidence in the banks, thereby worsening the Depression. Of course, there is a multiplier effect even with currency, if it is used in multiple transactions as currency, but, during hard times, such as the Great Depression or during the recent Great Recession, people and businesses hoard cash to protect themselves in an uncertain environment and future. Even in normal times, there isn't much of multiplier effect with currency because most people use currency to purchase goods or services from a business, who will then deposit the money in its checking account, putting it back into the banking system.