Spending multiplier

In economics, the spending multiplier effect describes a process by which an initial increase of one economic aggregate is amplified and provokes a larger increase than the initial raise. The idea is that the raise of a first agent income improves the situation of a second agent by the way of consumption, and so on.

The spending multiplier is a key concept in Keynesian economics for it explains how the government purchases can have a strong stimulating effect on the national output, depending on the marginal propensity to consume.

Consider a closed economy in which private agents consume in average 80% of their income. If the government increases its purchases by 100, then the national output will increase by 500.

Agent	Consumption	Saving
Government	100	0
Through consumption, the government increases by 100 the income of one of its suppliers (agent #2).
#2	80	20
Agent #2 saves 20% of its new wealth and spends the remaining money, increasing by 80 the income of agent #3.
#3	64	16
#4	51	13
#5	41	10
#6	33	8
#7	26	6
...	...	...
Total	500	125

In mathematics, this result is known as the sum of a convergent geometric serie.

An other demonstration relies on the following accountant relation in a closed economy :

Income = Consumption + Investment
Income = Private Consumption + Governmental Consumption + Saving - Taxes
Y = C + G + I - T
Since C = cY with c the propensity to consume, then
Y = cY + G + I - T
(1-c)Y = G + I - T
Y = (G + S - T)/(1-c)
Thus, an increase of G by 1 implies an increase of Y by 1/(1-c).