The Saving Function — APS, MPS and the Consumption-Saving Identity

Since a household's income can only be spent (consumed) or set aside (saved), income is always divided, by definition, into exactly these two uses:

Given a linear consumption function $C=a+bY$, the corresponding saving function is derived by substitution:

Notice the saving function has a NEGATIVE intercept, $-a$ — at zero income, since autonomous consumption $a$ still has to be spent (financed from past savings or borrowing), saving is actually NEGATIVE (dissaving) by exactly the amount $a$.

Exactly parallel to the consumption-side ratios:

Average Propensity to Save (APS) $=\dfrac{S}{Y}$ — the fraction of total income that is saved.

Marginal Propensity to Save (MPS) $=\dfrac{\Delta S}{\Delta Y}=1-b$ — the fraction of an additional rupee of income that is saved rather than consumed.

Because every rupee of income is, by definition, either consumed or saved, two identities always hold exactly:

$APC+APS=1 \qquad\text{and}\qquad MPC+MPS=1$ …