Q.Why must the equilibrium level of national income found using the Y=C+I approach always be identical to the equilibrium found using the S=I approach, for the same underlying consumption function and investment figure?

S=I is obtained by simply rearranging Y=C+I (since S=Y-C), so both conditions describe the exact same equilibrium algebraically, not two different equilibria.

The two apparent "approaches" to finding equilibrium income are not actually two DIFFERENT economic conditions — they are the SAME single condition, algebraically rearranged.

Starting from the equilibrium condition:

Subtract $C$ from both sides:

But by the basic income identity, $S\equiv Y-C$ (saving is, by definition, whatever part of income is not consumed). Substituting this definition directly into the left-hand side above:

This derivation shows that the $S=I$ condition is obtained from the $Y=C+I$ condition purely through simple algebraic rearrangement and the DEFINITION of saving — no new economic assumption has been introduced anywhere in this step. Consequently, ANY value of Y that satisfies $Y=C+I$ will, by this exact chain of algebra, automatically also satisfy $S=I$, and vice versa — the two conditions have EXACTLY the same solution set, which for a single linear equation in Y means exactly the same single equilibrium value. …

Students sometimes treat the C+I and S=I approaches as needing to be independently "verified against each other" as though they might legitimately disagree — algebraically they CANNOT disagree for the same input data, since one is a direct rearrangement of the other; if a computed answer using the tw …