Assume the MPC of a hypothetical economy is 0.5. If a person spends $1 on a widget, then the person who sells it receives $1 of income, of which a portion is spent (MPC), this person then uses their $0.5 to purchase another good ad infinitum. Expressing this mathematically:
$1+MPC+MPC^{2}+\cdots$
As $MPC\leq 1$ then what results is the limiting sum of a geometric sequence.
i.e. $S_{\infty }=1+MPC+MPC^{2}+\cdots ={1 \over 1-MPC}$ (letting a=1, r=MPC)
The multiplier shows how one man's spending creates another man's income, through several time periods. In this case an initial new investment of $10 creates new income of $10, which is either spent or saved by those who have earned it. The proportion which is spent creates income for others in the second time period (shown by arrows from one period to the next). This is also spent or saved. The final new income which has flowed from the initial investment can be determined using the formula [$Y=k\times initial\ investment$] (i.e. $5\times 10=50$). Had the MPC been higher (say 0.9) the multiplier would be larger and more income would have been created.
.This diagram assumes no government and no trade, so AE = C + I.
Initial equilibrium occurred at the red dot on the upper panel of the diagram, where $300 was earned and spent by households and firms. S = I at $60 (red dot on the lower panel).
Then firms decided to increase I by $10 (shown on the diagram as I + ΔI). This created extra income in the economy, which gave consumers extra spending power. Because one man's spending is another man's income, there was a series of C, S and Y increases similar to those at the green dot on the upper panel of the diagram, where planned levels of expenditure and income had risen to $350. Once again, S = I at $70. The new equilibrium occurs at a higher level of income. A small change in investment has resulted in a large change in AY.