If the rate of inflation for the next 20 years is 2.5% per year, what annual income 20 years from now will have the same purchasing power as a $30000 annual income today?

Introduction / Context:
This problem applies compound interest ideas to inflation instead of investment returns. Inflation reduces the purchasing power of money over time, so a fixed nominal income in the future will buy fewer goods and services than the same nominal amount today. To preserve purchasing power, future income must grow at least as fast as inflation. The question asks how large a future annual income must be to match the purchasing power of 30000 today, given a constant annual inflation rate.

Given Data / Assumptions:

• Current annual income with desired purchasing power = 30000 units of currency
• Inflation rate i = 2.5% per year
• Time horizon t = 20 years
• Inflation compounds annually at a constant rate
• We must find the future income F that maintains the same real purchasing power

Concept / Approach:
Under constant inflation, the price level after t years is multiplied by a factor of (1 + i)^t. To have the same purchasing power, income must also be multiplied by this factor. Therefore, the required future income F is given by F = 30000 * (1 + i)^t, with i expressed in decimal form. This mirrors the compound interest formula, but instead of money growing in an account, we are inflating the nominal amount needed to buy the same basket of goods.

Step-by-Step Solution:
Present required income = 30000 Inflation rate i = 2.5% = 0.025 Time t = 20 years Future income F = 30000 * (1 + 0.025)^20 F = 30000 * 1.025^20 Using a calculator, 1.025^20 ≈ 1.6386 F ≈ 30000 * 1.6386 ≈ 49158 So the required future income is approximately 49158 units

Verification / Alternative check:
We can sanity check this result by noting that simple interest style inflation over 20 years at 2.5% would increase prices by about 50% (since 20 * 2.5% = 50%), leading to a required income of approximately 45000. Because of compounding, the actual price increase is slightly more than 50%, so the required income should be somewhat higher than 45000. Our calculated value of about 49158 fits this expectation, confirming that the result is reasonable.

Why Other Options Are Wrong:
39158 would correspond to a much lower inflation rate over 20 years and would not preserve purchasing power. 59158 and 69158 imply higher inflation or an unnecessarily large increase in income, going beyond the requirement of matching a 30000 income in today s terms. Only 49158 matches the correct application of the compound inflation formula at 2.5% for 20 years.

Common Pitfalls:
Many learners mistakenly multiply 30000 by (1 + 0.025 * 20) and ignore compounding, which underestimates the required future income. Another error is to reverse the direction and divide instead of multiply, treating inflation as if it were a discount rate. Carefully recognizing that inflation increases required nominal income over time is crucial for solving such questions correctly.

Final Answer:
An annual income of approximately 49158 will be needed in 20 years to have the same purchasing power as 30000 today when inflation is 2.5 percent per year.