Question

An economy saves 25% of its national income and invests a sum of Rs.1000 each year. The economy starts with Rs.10,000 as its initial national income. Assume that the consumption in any year depends on the income of the previous year. The national income of the economy after 10 years is Rs. ........... (rounded off to two decimal places).

Hint:

The formula for national income growth accounts for both initial savings and the impact of investment on the economy.

Answer 1

The formula for national income \( Y_t \) after \( t \) years with savings \( s \) and investment \( I \) is: \[ Y_t = Y_0 \times (1 + s)^t + \frac{I}{s} \times \left( (1 + s)^t - 1 \right) \] Where:
- \( Y_0 = 10,000 \) (initial income),
- \( s = 0.25 \) (savings rate),
- \( I = 1000 \) (annual investment),
- \( t = 10 \) (number of years).
Step 1: First calculate \( Y_0 \times (1 + s)^t \): \[ Y_0 \times (1 + s)^t = 10,000 \times (1.25)^{10} \approx 10,000 \times 9.313225746 = 93,132.25746 \] Step 2: Calculate the second part of the equation: \[ \frac{I}{s} \times \left( (1 + s)^t - 1 \right) = \frac{1000}{0.25} \times \left( (1.25)^{10} - 1 \right) = 4000 \times (9.313225746 - 1) = 4000 \times 8.313225746 \approx 33,252.90298 \] Step 3: Add both parts: \[ Y_t = 93,132.25746 + 33,252.90298 \approx 126,385.16 \] Thus, the national income of the economy after 10 years is approximately Rs.126,385.16. Final Answer: \[ \boxed{126385.16} \]