Calculating Equilibrium Income

To find the equilibrium level of income ($Y$), we set aggregate demand (AD) equal to national income ($Y$). In this model, AD is the sum of consumption ($C$), investment ($I$), and government spending ($G$).

Aggregate Demand Components

The components are given as:

• Consumption: $C = 100 + 0.75 Y_d$
• Investment: $I = 60$
• Government Expenditure: $G = 40$
• Tax Rate: $t = 0.2$

Disposable Income Calculation

Disposable income ($Y_d$) is income after taxes. Taxes ($T$) are calculated as a percentage of income: $T = t \times Y$. Therefore,

$Y_d = Y - T = Y - (t \times Y) = Y(1-t)$

Substituting the tax rate ($t=0.2$):

$Y_d = Y(1 - 0.2) = 0.8Y$

Consumption Function Adjustment

Substitute the expression for $Y_d$ into the consumption function:

$C = 100 + 0.75 (0.8Y)$

$C = 100 + 0.6Y$

Equilibrium Condition

Equilibrium occurs when $Y = AD$. Aggregate Demand is $AD = C + I + G$. Substituting the values and the adjusted consumption function:

$Y = (100 + 0.6Y) + 60 + 40$

Solving for Equilibrium Income

Combine the terms and solve for $Y$:

$Y = 100 + 0.6Y + 100$

$Y = 200 + 0.6Y$

Subtract $0.6Y$ from both sides:

$Y - 0.6Y = 200$

$0.4Y = 200$

Divide by 0.4:

$Y = \frac{200}{0.4}$

$Y = 500$

The equilibrium level of income is 500.