Q.Consider an economy described by the following functions: $C = 20 + 0.80\,Y$, $I = 30$, $G = 50$, $TR = 100$.

The equilibrium condition in a simple Keynesian model is that aggregate demand equals income: $Y = AD$. Aggregate demand here is $AD = C + I + G$, where consumption depends on disposable income $Y_d = Y + TR - T$.

1. Equilibrium income and multiplier Disposable income is $Y_d = Y + 100 - 0 = Y + 100$ (no tax initially). Substituting into the consumption function:

   $C = 20 + 0.80(Y + 100) = 20 + 0.80Y + 80 = 100 + 0.80Y$

   Aggregate demand becomes:

   $AD = C + I + G = (100 + 0.80Y) + 30 + 50 = 180 + 0.80Y$

   At equilibrium, $Y = AD$:

   $Y = 180 + 0.80Y \implies 0.20Y = 180 \implies Y = 900$

   The autonomous expenditure multiplier is $\frac{1}{1 - c}$ where $c = 0.80$ is the marginal propensity to consume. Thus the multiplier is $\frac{1}{0.20} = 5$.
2. Impact of increased government expenditure If $G$ rises by 30, autonomous expenditure increases by 30. The change in equilibrium income is:

   $\Delta Y = \text{multiplier} \times \Delta G = 5 \times 30 = 150$

   Equilibrium income rises from 900 to 1050.
3. Impact of adding a lump-sum tax Now $T = 30$, so disposable income becomes $Y_d = Y + 100 - 30 = Y + 70$. Consumption is:

   $C = 20 + 0.80(Y + 70) = 20 + 0.80Y + 56 = 76 + 0.80Y$

   With $G = 80$ (the new level), aggregate demand is:

   $AD = (76 + 0.80Y) + 30 + 80 = 186 + 0.80Y$

   Equilibrium: $Y = 186 + 0.80Y \implies Y = 930$. …

Concept: aggregate-demand equilibrium

Equilibrium income is where output equals planned aggregate demand, $Y = C + I + G$. Consumption depends on disposable income $Y_d = Y - T + TR$, while $I$ and $G$ are autonomous. The multiplier $k = \tfrac{1}{1-c}$ magnifies any change in autonomous spending, because each injection becomes income that is partly re-spent.

(a) Equilibrium income and the multiplier

With no tax, $Y_d = Y + TR = Y + 100$, so

The autonomous-expenditure multiplier is …