Assume a closed economy, no government spending and no taxes, and no depreciation. National income accounting states unambiguously:
C + I ≡ Y ≡ C + S
Where C, I, S are ex post. This is an identity. Sometimes it is simplified by putting “=” in place of “≡”, but this simplification often leads to confusion, because of this statement,
C + I = AD = Y
Which is a definition of aggregate demand as the sum of ex ante or “planned” consumption and planned investment. In equilibrium, AD = Y, that is production ramps up or down to match aggregate demand. That is why such a model is sometimes called “demand determined”.
When planned consumption plus planned investment equal ex post consumption and ex post investment, then AD = Y and planned saving equals planned investment.
When is there a “multiplier”? When planned consumption takes the linear form C = a + bY, investment is exogenous I*, then substituting in:
C + I = a + bY + I* = AD = Y
a + I* = Y – bY
a + I* = (1-b)Y
Y0 = (1/(1-b))(a + I*)
That is the level of equilibrium income is a multiple (since 1/(1-b) > 1 for 0 < b < 1) of the level spending that takes place irrespective of the level of income. a and I* qualify, and that is why they are called autonomous spending.
To find the multiplier, take the total differential of last equation:
ΔY = (1/(1-b))(Δa + ΔI*)
So the multiplier for a change in autonomous investment is found by holding the change in autonomous consumption at zero. Then:
ΔY = (1/(1-b))(ΔI*)
ΔY/ΔI* = (1/(1-b))
What are the key assumptions (in my mind)?
1. Consumption behavior is such that the marginal propensity to consume is less than unity, b < 1.
2. Output is demand is determined.
1. Don’t reason from identities.
2. Understand what are ex ante and ex post quantities before trying to reason out causality.