Q.Suppose $C = 100 + 0.75\,Y_D$, $I = 500$, $G = 750$, taxes are 20 per cent of income, $X = 150$, $M = 100 + 0.2\,Y$. Calculate equilibrium income, the budget deficit or surplus and the trade deficit or surplus.

In the Keynesian framework, equilibrium income is achieved when aggregate demand equals aggregate supply. For an open economy with government, the aggregate demand ($AD$) is given by $AD = C + I + G + (X - M)$. At equilibrium, $Y = AD$.

First, we express consumption ($C$) and imports ($M$) in terms of income ($Y$).

Given $C = 100 + 0.75\,Y_D$ and taxes ($T$) are 20 per cent of income, so $T = 0.20\,Y$.

Disposable income ($Y_D$) is $Y - T = Y - 0.20\,Y = 0.80\,Y$.

Substituting $Y_D$ into the consumption function:

$C = 100 + 0.75(0.80\,Y) = 100 + 0.60\,Y$.

Imports are given as $M = 100 + 0.2\,Y$.

Now, we set $Y$ equal to aggregate demand:

$Y = C + I + G + (X - M)$

$Y = (100 + 0.60\,Y) + 500 + 750 + (150 - (100 + 0.2\,Y))$

$Y = 100 + 0.60\,Y + 500 + 750 + 150 - 100 - 0.2\,Y$

$Y = (100 + 500 + 750 + 150 - 100) + (0.60\,Y - 0.2\,Y)$

$Y = 1400 + 0.40\,Y$

$Y - 0.40\,Y = 1400$

$0.60\,Y = 1400$

$Y = \frac{1400}{0.60} = \frac{14000}{6} = \frac{7000}{3} \approx 2333.33$

Next, we calculate the budget deficit or surplus. This is given by $T - G$.

Taxes $T = 0.20\,Y = 0.20 \times \frac{7000}{3} = \frac{1}{5} \times \frac{7000}{3} = \frac{1400}{3} \approx 466.67$.

Government expenditure $G = 750$.

Budget balance $= T - G = \frac{1400}{3} - 750 = \frac{1400 - 2250}{3} = \frac{-850}{3} \approx -283.33$.

Since the value is negative, there is a budget deficit. …

In macroeconomics, the equilibrium level of income in the short run is determined by the interaction of aggregate demand and aggregate supply. The Keynesian Cross model, a fundamental tool for understanding this, posits that equilibrium occurs when the total amount of goods and services produced in an economy (aggregate supply, $Y$) is exactly equal to the total amount that people want to spend (aggregate expenditure, $AE$). When these two are equal, there is no unplanned inventory accumulation or depletion, and firms have no incentive to change their production levels.

In an open economy with government, aggregate expenditure ($AE$) comprises consumption ($C$), investment ($I$), government spending ($G$), and net exports ($X - M$). Each of these components can depend on various factors, including income, interest rates, and policy decisions. For instance, consumption typically increases with disposable income, while imports tend to rise with overall income. Taxes reduce disposable income, affecting consumption. The equilibrium condition is thus $Y = C + I + G + (X - M)$.

Let's calculate the equilibrium income, budget deficit/surplus, and trade deficit/surplus using the given information.

1. Derive the Aggregate Expenditure (AE) function:

   We are given the following equations:

   • Consumption function: $C = 100 + 0.75\,Y_D$
   • Investment: $I = 500$
   • Government spending: $G = 750$
   • Taxes: $T = 0.2Y$ (20 per cent of income)
   • Exports: $X = 150$
   • Imports: $M = 100 + 0.2\,Y$

   First, we need to express disposable income ($Y_D$) in terms of total income ($Y$):

   $Y_D = Y - T$

   Substitute the tax function:

   $Y_D = Y - 0.2Y$

   $Y_D = 0.8Y$

   Now, substitute $Y_D$ into the consumption function to express $C$ in terms of $Y$:

   $C = 100 + 0.75(0.8Y)$

   $C = 100 + 0.6Y$

   Next, we assemble the aggregate expenditure function:

   $AE = C + I + G + (X - M)$

   Substitute all the components into the $AE$ equation:

   $AE = (100 + 0.6Y) + 500 + 750 + (150 - (100 + 0.2Y))$

   $AE = 100 + 0.6Y + 500 + 750 + 150 - 100 - 0.2Y$

   Combine the autonomous components (constants) and the income-dependent components:

   $AE = (100 + 500 + 750 + 150 - 100) + (0.6Y - 0.2Y)$

   $AE = 1400 + 0.4Y$

2. Calculate Equilibrium Income ($Y$):

   Equilibrium occurs when aggregate output ($Y$) equals aggregate expenditure ($AE$).

   $Y = AE$

   $Y = 1400 + 0.4Y$

   Subtract $0.4Y$ from both sides:

   $Y - 0.4Y = 1400$

   $0.6Y = 1400$

   Divide by $0.6$:

   $Y = \frac{1400}{0.6}$

   $Y = \frac{14000}{6}$

   $Y = \frac{7000}{3}$

   $Y \approx 2333.33$

3. Calculate the Budget Deficit or Surplus:

   The budget balance is the difference between government tax revenue ($T$) and government spending ($G$).

   Budget Balance = $T - G$ …