# Spending multiplier/Tutorials

## The algebra of the spending multiplier identity

An injection of 100 currency units is assumed to be made into a circular flow of income model of the economy and that the marginal propensity to consume of its recipients is c (round 1).

Of the 100 units injected, an amount equal to 100 times c is spent (round 2)

The recipients of that amount spend the same proportion of it (round 3)

- and so on as below

expenditure saving round 1 100 round 2 100c 100(1 - c) round 3 100c ^{2}100(1 - c) ^{2}... round n 100c ^{n}100(1 - c) ^{n}

The total spending in the economy after n rounds is

- 100 + 100c + 100c
^{2}+ 100c^{3}....+ 100c^{n}

- 100 + 100c + 100c

- which is a geometric progression.

It can be proved that such a geometric progression converges to the value 100/(1 - c) as n approaches infinity.

The final outcome is therefore a total expenditure in the economy that is a multiple 1/(1 - c) of the initial injection, where 1 - c is definitionally equal to the marginal propensity to save.