Question

Show that, if the percentage rise in prices is equal to the percentage rise in nominal income, then one's real income does not change.

Short answer

The real income remains unchanged when the percentage rise in prices and the nominal income are the same. This is because the rise in price corresponds to the rise in nominal income, thus compensating for the increased cost of living and maintaining the same buying power or real income.

Step-by-step solution

Formulate the hypothesis

The task is to prove that if the percentage rise in prices (\( P \)) is equal to the percentage rise in nominal income (\( I \)), then the real income (\( R \)) does not change. It's known that real income is the ratio of nominal income to price level, or \( R = \frac{I}{P} \). This is the initial formula you'll work with.

Show the change in real income using the initial formula

Let’s assume the nominal income and the price level increased by \( x \% \). Then new nominal income will be \( I' = I * (1 + \frac{x}{100}) \) and new price level will be \( P' = P * (1 + \frac{x}{100}) \). Now, substitute these values into the Real income formula to compute the new real income. So, \( R' = \frac{I'}{P'} = \frac{I * (1 + \frac{x}{100})}{P * (1 + \frac{x}{100})} \).

Simplify the equation and prove the hypothesis.

The next part is simplification. When we simplify the equation above, we find \( R' = \frac{I}{P} = R \). Hence, it is proven that when the percentage rise in the prices is equal to the percentage rise in the nominal income, the real income remains unchanged.