# Money supply model. Money multiplier

The supply of money (Ms) includes cash (C) outside the banking system and deposits (D), which economic agents can use for transactions if necessary (in fact, this is the Ml aggregate)

Ms = C+D

The modern banking system is a system with partial reserve coverage: only a part of their deposits banks keep in the form of reserves, and the rest are used to issue loans and other active operations.

Unlike other financial institutions, banks have the ability to increase the supply of money (“create money”). Credit multiplication is the process of issuing means of payment within the system of commercial banks.

Suppose bank 1’s deposits are up 1,000. 20% remains in the reserve, that is, 200%, and the rest are loaned (the reserve ratio – the ratio of reserves to deposits – in this case is 20% or 0.2). Thus, Bank 1 increased the money supply by 800, and now it is equal to 800 + 1000 = 1800. Depositors still have deposits of 1,000 units of currency, but borrowers also hold 800 units in their hands, that is, a banking system with partial reserve coverage is able to increase the supply of money.

Further, if these 800 units again fall into the bank, the process resumes: 20%, that is, 160 units bank 2 leaves in reserves, and the remaining 640 uses to issue loans, increasing the supply of money by another 640 units. The third bank, where this money can go, will add another 512 and so on.

If the process lasts until the last monetary unit is used, then the amount of money in the system can be determined as follows:

Initial deposit = 1000

1st bank loan (additional money supply) = (1-0.2)×1000 = 800

2nd bank loan = (1-0,2)[(1 – 0,2) ×1000] =(1-0,2)2×1000=640

3rd bank loan = (1-0,2)[(1 – 0,2)2× l000] =(1-0,2)3×1000=512

The total money supply is:

1000 ×[1 + (1-0,2)+ (1-0,2)2+(1-0,2)3 +…..] =

(in square brackets we have the sum of the members of the geometric progression with a denominator (1-0.2), i.e. less than one. By definition, this sum is equal to ).

In general, the additional supply of money resulting from the appearance of a new deposit is:

,

where rr is the rate of bank reserves; D – initial contribution.

The ratio is called a bank multiplier, or deposit multiplier.

A more general model of money supply is built not only taking into account the role of the Central (National) Bank, but also taking into account the possible outflow of part of the money from the deposits of the banking system to cash. It includes a number of new variables.

The monetary base (high-capacity money, reserve money) is cash outside the banking system, as well as the reserves of commercial banks stored in the Central (National) Bank. Cash is a direct part of the money supply, whereas bank reserves affect banks’ ability to create new deposits, increasing the money supply. Let’s denote the monetary base through MB, bank reserves through R, then:

MB = C+R,

where MB is the monetary base; C = cash; R  – reserves.

MS= C + D,

where MS is the offer of money;

C – cash;

D – demand deposits.

The money multiplier (m) is the ratio of the supply of money to the monetary base:

The cash multiplier can be represented through the ratio of cash-deposits cr (deposit ratio) and reserve-deposits rr (reserve ratio):

Divide the numerator and denominator of the right side of the equation by D (deposits) and get:

Where is

The value of cr is determined mainly by the behaviour of the population, which decides in what proportion cash and deposits will be. The ratio of rr depends on the required reserve ratio set by the Central Bank and on the amount of excess reserves that commercial banks intend to hold in excess of the required amount.

Now the money supply can be determined by the formula:

Thus, the supply of money directly depends on the size of the monetary base and the monetary multiplier (or multiplier of the monetary base). The money multiplier shows how the money supply changes as the monetary base increases by one. An increase in the deposit ratio and the reserve ratio reduces the cash multiplier.

The Central (National) Bank can control the supply of money primarily by influencing the monetary base. A change in the monetary base, in turn, has a multiplicative effect on the supply of money. Thus, the process of changing the volume of money supply can be divided into two stages:

initial modification of the monetary base by changing the obligations of the Central (National) Bank to the population and the banking system (impact on the amount of cash and reserves); the subsequent change in the supply of money through the process of “multiplication” in the system of commercial banks.

Monetary policy instruments adjust the value of the money supply by affecting either the monetary base or the multiplier.

There are three main instruments of monetary policy, with the help of which the Central (National) Bank carries out indirect regulation of the monetary sphere:

change in the discount rate (or refinancing rate), that is, the rate at which the Central (National) Bank lends to commercial banks; change in the rate of required reserves, that is, the minimum share of deposits that commercial banks must keep in the form of reserves (interest-free deposits) in the Central Bank; open market operations: purchase or sale by the Central Bank of government securities (used in countries with a developed stock market).

These operations are associated with a change in the amount of bank reserves, and, consequently, the monetary base.

However, the Central Bank cannot fully control the supply of money because, for example:

commercial banks themselves determine the amount of excess reserves (they are part of R), which affects the ratio of rr and, accordingly, the multiplier; The Central (National) Bank cannot accurately foresee the amount of loans to be issued to commercial banks; the value of cr is determined by the behavior of the population and other reasons that are not always related to the actions of the Central (National) Bank.