# The mathematics of the multiplier process

This was written by PRIME economists for the second report of the Green New Deal group, of which we  (i.e. Douglas Coe and Ann Pettifor at PRIME ) are co-authors:

The theory of the multiplier process can be illustrated by considering an increase in government expenditure of £1billion, with leakages to saving only, and the assumption that households will spend 80 per cent of any increase in income.

The direct effect of the increase in public expenditure is £1billion. The first repercussion of the increase in expenditure is that households spend 80 per cent of the additional income, equivalent to £800 million. The next repercussion is the expenditure of 80 per cent of the additional income of £800 million, and so on. The process is encountered in schools as the mathematics of a geometric progression: 1 + 0.8 + 0.82 + 0.8 3 + ... etc. The total of this process can be simply expressed as equal to 1 / (1 – 0.8) = 5, so that the multiplier is 5, and the aggregate impact of an increase in government expenditure of £1billion is £5billion. Assuming that none of the increase goes to prices or imports, the total change in employment will be five times the direct increase in employment.

The whole of this £5billion accrues to households, with £4billion spent (80 per cent) and £1billion saved, meaning that the amount saved is equal to the original amount of the expenditure. In this way Keynes also demonstrated that the original outlay in government expenditure matched the new saving of households, proving the critics that said government expenditure would divert funds from other productive uses in the private sector wrong. (He might also have added that in a recession almost by definition private companies don’t demand funds.)

This example is entirely hypothetical and not realistic. In practice, any employment gain will be a far lower multiple of the original outlay, but the basic equality of the saving and new expenditure is true no matter what share of new income is spent by households.

Today, estimates of the multiplier can be determined from the National Accounts.

The multiplier can be obtained as 1 / ( 1 – c + m )

where:

c is the share of an increase in aggregate income that goes to household consumption, or the marginal propensity to consume (mpc) (this approach should crudely account for any leakages to profits and other costs), and

m is the share that goes to imports, or the marginal propensity to import (mpi).

In the UK the mpc is about 2/3 and the mpi 1/3, so that the multiplier is 1.5. In the US the tendency is to consume more and import less, so that the multiplier is higher and closer to 2.  There are concerns about the extent of leakages to price, but at times of high unemployment this is both unlikely and of limited consequence, given that a small rise in prices would also help company revenues.