The algebra of the spending multiplier identity
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 100c2 100(1 - c)2 ... round n 100cn 100(1 - c)n
The total spending in the economy after n rounds is
- 100 + 100c + 100c2 + 100c3 ....+ 100cn
- 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.