Question

Let the consumption function, tax function, and income identity be given by C = C₀+b(Y-T), T = T₀+tY, and Y=C+I₀+ G₀, respectively, where C₀, I₀, G₀, and T₀ are autonomous consumption, investment, government expenditure, and tax, respectively. If b = 0.75 and t = 0.1, then an increase in G₀ by Rs. 20 million will increase Y by Rs. ______ million (round off to 2 decimal places).

Answer 1

Correct Answer: 61.52

This is a problem in macroeconomic modeling, specifically calculating the change in equilibrium income ($Y$) resulting from a change in government expenditure ($G_0$), which involves finding the government expenditure multiplier.

$\text{1. Determine the Equilibrium Income Equation}$

The system of equations is:

Consumption Function: $C = C_0 + b(Y - T)$

Tax Function: $T = T_0 + tY$

Income Identity: $Y = C + I_0 + G_0$

Substitute the tax function (2) into the consumption function (1):

$$C = C_0 + b(Y - (T_0 + tY))$$

$$C = C_0 - bT_0 + bY - btY$$

$$C = (C_0 - bT_0) + b(1-t)Y$$

Now substitute the consumption function into the income identity (3):

$$Y = [(C_0 - bT_0) + b(1-t)Y] + I_0 + G_0$$

Group the autonomous terms and the income terms:

$$Y - b(1-t)Y = (C_0 - bT_0 + I_0 + G_0)$$

$$Y [1 - b(1-t)] = (C_0 - bT_0 + I_0 + G_0)$$

The equilibrium income ($Y$) is:

$$Y = \frac{1}{1 - b(1-t)} (C_0 - bT_0 + I_0 + G_0)$$

$\text{2. Determine the Government Expenditure Multiplier}$

The Government Expenditure Multiplier ($M_G$) is the coefficient of the autonomous government expenditure ($G_0$) in the equilibrium income equation:

$$M_G = \frac{\Delta Y}{\Delta G_0} = \frac{1}{1 - b(1-t)}$$

$\text{3. Calculate the Multiplier Value}$

Given values:

Marginal Propensity to Consume ($b$) $= 0.75$

Marginal Tax Rate ($t$) $= 0.1$

Substitute the values into the multiplier formula:

$$M_G = \frac{1}{1 - 0.75(1 - 0.1)}$$

$$M_G = \frac{1}{1 - 0.75(0.9)}$$

$$M_G = \frac{1}{1 - 0.675}$$

$$M_G = \frac{1}{0.325}$$

Now we calculate the numerical value of the multiplier:

$$M_G = \frac{1}{0.325} = 3.076923\dots$$

$\text{4. Calculate the Increase in Income}$

The increase in government expenditure ($\Delta G_0$) is $\text{Rs. } 20 \text{ million}$. The resulting increase in income ($\Delta Y$) is:

$$\Delta Y = M_G \times \Delta G_0$$

$$\Delta Y = \frac{1}{0.325} \times 20$$

$$\Delta Y \approx 3.076923 \times 20$$

$$\Delta Y \approx 61.53846$$

Rounding off to 2 decimal places:

$$\Delta Y \approx 61.54$$

$$\text{An increase in } G_0 \text{ by Rs. } 20 \text{ million will increase } Y \text{ by Rs. } \mathbf{61.54} \text{ million}$$