A more rigorous treatment of our model will help you understand why income changes by some multiple of any change in autonomous spending. (We use delta (D) to signify change. Thus, DY is read "change in income.") How much will total income change (DY) as a result of a given change in, say, autonomous investment spending (DI)? This ratio (DY/DI) is known as the autonomous spending multiplier.
We assume that consumption is related to income and that changes in income will cause consumption to change by a value equal to the mpc times the change in income. We know that
Y = C + I (1)
(output is either consumed or invested), so
DY = DC + DI (2)
(changes in output reflect changes in consumption and/or investment). If consumption spending is related to income by the mpc, then
DC = mpc·DY (3)
(this is the change in induced consumption). The change in consumption is equal to the change in income times the proportion of the change you intend to spend.
Substituting Equation (2) into Equation (3) yields
DY = mpc·DY + DI (4)
Now we need to move all income (DY) terms to one side by subtracting mpc·DY from each side of Equation (4), so
DY - mpc·DY = DI (5)
Factoring the DY terms on the left side of Equation (5) leaves
DY·(1 - mpc) = DI, (6)
and dividing both sides by (1 - mpc) yields
DY = DI·(1 - mpc), (7)
The term 1/(1 - mpc) is the autonomous spending multiplier. Since mpc + mps = 1, then mps = 1 - mpc. Thus, another way to write the autonomous spending multiplier is 1/mps.
We have used investment to show how hikes in autonomous spending yield increases in income via the multiplier. Mathematically identical effects occur if autonomous consumption, government spending, or exports are raised. Economists often use A to stand for all forms of autonomous spending when writing formulas for multipliers. Thus, the following are all equivalent ways to write the autonomous spending multiplier:
DY = 1 = __1___
D A mps 1 - mpc
If the marginal propensity to consume is 0.8, the autonomous spending multiplier will be 5, because 1/(1 - 0.8) = 1/0.2 = 5. Calculate multipliers for alternative values for mpc (e.g., 9/10, 6/7, 5/6, 4/5, 3/4, 2/3, and 3/5). Observe what happens to multipliers as mpc rises and mps falls. Naturally, reversed multiplier effects follow cuts in autonomous spending.