Home > Macro, Money > The money multiplier

## The money multiplier

If the interest rate is determined by the interaction between money supply and money demand, how can we model this in a simple way?

We can look at it through defining the concept of Central Bank money, or high-powered money (also known as the monetary base).

Firstly think in terms of assets and liabilities. Retail banks (the type individual consumers bank with) will hold as assets the loans they make (eg mortgages, personal loans), that need to be repaid back to them, they will hold bonds and shares etc, and they will keep a certain amount of liquid assets (like cash) as reserves in their account with the Central Bank. They need these reserves to have liquidity available to meet the daily needs of customers’ withdrawals, when customers want to withdraw money they want cash not bonds, so the bank needs to have a ready supply of it.

Retail banks effectively ‘bank’ with the Central Bank and they have their own accounts there, this is effectively what happens if you buy something with a debit card, say you bank with Barclays and the shop banks with NatWest, when you pay £30 for something on a debit card, then the electronic transaction deducts £30 from Barclays’ account at the Central Bank and credits NatWest’s account at the Central Bank with £30. At the same time, Barclays will deduct £30 from your current account balance, while NatWest will credit the shop’s balance by £30.

The liabilities the retail banks have are the customers’ deposits. When you use your debit card or withdraw cash from the ATM, the bank has to give you money (up to the sum of your current account balance) so that is a liability to it – something it owes.

Now the Central Bank will also have assets and liabilities. It holds foreign exchange reserves and bonds, shares, gold etc as its assets, and its liabilities are the retail banks balances (their reserves) in their accounts with the central bank, plus the money the Central Bank has ‘issued’ (ie coins and notes in circulation).

So high-powered money, or the monetary base, is equal to retail banks’ reserves plus currency in circulation.

With this in mind we can make a few definitions to make a simplifed model:

$M^d$, the total demand for money in the economy, is made up partly of currency $CU$(coins and notes floating around in circulation) and partly of current account deposits $D$. If we denote the proportion of total money made up of currency as $c$ then the demand for currency is $CU^d = cM^d$ and the demand for current account deposits is $D^d = (1-c)M^d$.

$H^d$, the demand for high-powered money (or the monetary base) is the equal to the demand for currency, $CU^d = cM^d$ plus demand for retail bank reserves, $R^d$. Consumers are depositing $D$ among of deposits in current accounts with the retail banks, but the banks won’t hold all of this as reserves, they operate a system of fractional reserve banking. This means that they know that (or are banking on the fact that) customers won’t suddenly all demand to withdraw their deposits at once, so they only need to keep a proportion of their total deposit liabilities to customers in reserve, to meet day to day withdrawals…and they will use the rest to lend out to other people in order to make interest on it. So if we denote the proportion as the ‘reserve ratio’, $\theta$, then the demand for retail bank reserves, will be $R^d=\theta D^d = \theta (1-c)M^d$

So putting that together, we have two expressions:
Total demand for money in the economy: $M^d = CU^d + D^d = cM^d + (1-c)M^d$
Demand for Central Bank money: $H^d = CU^d + R^d = cM^d + \theta (1-c)M^d$

From the second equation we can say that $H^d = M^d (c + \theta (1-c))$ So $M^d = \frac{1}{(c + \theta (1-c))}H^d$

This expression basically expresses how total demand for money relates to demand for high-powered money, ie demand for total money will always be higher than demand for high-powered money.

But remember that the total demand for money is determined by two things, the interest rate and overall incomes in the economy. If we assume that incomes are fixed in the short run then it will be the interest rate that determines the demand for money, when interest rates are high there will be a lower demand for both parts of high-powered money, currency and retail bank reserves (because customers have lower demand for current account deposits when interest rates are high, they are instead putting their money in illiquid forms of saving like bonds).

The money market will head into equilibrium, ie the demand and supply for money will come into equilibrium because the interest rate will adjust to get there, just like any other market comes into equilibrium due to adjustments in the price. When we get to the equilibrium we can say that $M^s = M^d =M$ and $H^s = H^d =H$ so $M = \frac{1}{(c + \theta (1-c))}H$

Here we have a money multiplier of $\frac{1}{(c + \theta (1-c))}$

The multiplier shows us how changes in high-powered money translate into changes in the overall amount of money in the economy. This is where the name high-powered comes from, the monetary base is a type of money that has magnified effects on the overall amount of money: if you increase the monetary base by £1, you get an increase in the overall amount of money in the economy of £$\frac{1}{(c + \theta (1-c))}$

As an example, suppose the total amount of high powered money is £1000000, the reserve ratio is 0.1 and the proportion of money which people hold as currency as being 0.2, then the total amount of money is $M = \frac{1}{(0.2 + 0.1(1-0.2))}1000000 = 3571428.57$. Our multiplier here is $\frac{1}{(0.2 + 0.1(1-0.2))} = 3.571429$

The Central Bank controls the amount of high-powered money in the economy. If it decides to increase the amount of high-powered money by £10000, then the multiplier implies that it will increase the total amount of money by $3.571(10000)=35714.29$. The new total amount of money in the economy is $M = \frac{1}{(0.2 + 0.1(1-0.2))}1010000 = 3607142.86$