Question I Had
When we go from the “full formula” () to the “change formula” (), the constant disappears. Why? And why does this keep happening in every chapter?
Answer
In any linear function, only the slope survives into the change formula. The constant (autonomous part) cancels because it’s the same value before and after the change.
f(x) &= a + bx \\[6pt] f(x + \Delta x) - f(x) &= [\cancel{a} + b(x + \Delta x)] - [\cancel{a} + bx] \\[4pt] &= b \cdot \Delta x \end{aligned}$$ The constant $a$ appears in both the "before" and "after" — so when you subtract, it's gone. What remains is the slope ($b$) times the size of the change ($\Delta x$). **That's what autonomous means mathematically:** a term that doesn't respond to the variable, so it vanishes when you ask "what changed?" ## Why This Matters in Economics Every model in this course is built on linear functions where the constant is "autonomous" (doesn't depend on income) and the slope is "induced" (does depend on income). This pattern appears everywhere: | Full formula | Change formula | The slope that survives | |---|---|---| | $C = a + b(1-t)Y$ | $\Delta C = b(1-t) \cdot \Delta Y$ | MPC out of national income | | $AE = A + zY$ | $\Delta AE = z \cdot \Delta Y$ | Marginal propensity to spend | | $Y = \frac{A}{1-z}$ | $\Delta Y = \frac{1}{1-z} \cdot \Delta A$ | The simple multiplier | | $NX = X_0 - mY$ | $\Delta NX = -m \cdot \Delta Y$ | (Negative of) marginal propensity to import | The **simple multiplier** itself is a consequence of this principle: if $Y = \frac{A}{1-z}$, then $\Delta Y = \frac{\Delta A}{1-z}$ because $\frac{1}{1-z}$ is the slope of the equilibrium equation (it plays the role of $b$, and $A$ is the "variable" being changed while $z$ is the structural parameter). ## The Trap **This only works for changes to autonomous (constant) terms — parallel shifts of the function.** When the **slope itself** changes (e.g., the net tax rate $t$ changes, which changes $z$), the constant doesn't simply cancel. The whole function rotates, and you must solve for the new equilibrium directly. You cannot use the multiplier shortcut. | What changes | Type of AE movement | Use $\frac{\Delta A}{1-z}$? | |---|---|---| | $G$, $I$, $X$, or $a$ (autonomous terms) | Parallel shift | ✅ Yes — constant cancels | | $t$ or $m$ (slope components) | Rotation | ❌ No — slope changed, solve fresh | ## Example $C = 40 + 0.6Y$. National income rises from $Y = 500$ to $Y = 600$. | | Before ($Y = 500$) | After ($Y = 600$) | Change | |---|---|---|---| | Autonomous part | 40 | 40 | 0 (cancelled) | | Induced part | $0.6 \times 500 = 300$ | $0.6 \times 600 = 360$ | +60 | | **Total C** | **340** | **400** | **+60** | $\Delta C = 0.6 \times 100 = 60$ ✓ — only the slope mattered. --- <font style="color:999999"><b>North:</b></font><font style="color:666666"> <i>Where this comes from</i></font> - [[Slope Formula - Point Order and Negative Placement]] — the slope as rate of change - Properties of linear functions (algebra fundamentals) - The distinction between autonomous and induced expenditure in Keynesian macro models <font style="color:999999"><b>East:</b></font><font style="color:666666"> <i>What's the opposite?</i></font> - Nonlinear functions — where the "slope" itself changes with $x$, so this clean separation doesn't hold (leads to derivatives in calculus) - Slope changes (rotation) — when $z$ changes, the constant does NOT simply cancel, and the multiplier shortcut fails <font style="color:999999"><b>South:</b></font><font style="color:666666"> <i>Where this leads</i></font> - [[ECON-1221 Chapter 7 - Notes from the Textbook_backup]] — the simple multiplier formula $\Delta Y = \frac{\Delta A}{1-z}$ - The shift-vs-rotate distinction in fiscal policy (why $\Delta G$ uses the multiplier but $\Delta t$ does not) - Derivatives in calculus — the generalized version of "only the slope survives" for any function <font style="color:999999"><b>West:</b></font><font style="color:666666"> <i>What's similar?</i></font> - Marginal analysis in microeconomics — marginal cost, marginal revenue are all "change formulas" where fixed costs cancel - The concept of a derivative: $f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ — same subtraction, same cancellation of constants - Percentage change formulas — the base cancels in the same way when comparing ratios