How do I mathematically prove the following statement is false?

Question

In terms of purchasing power, a fifty percent tax on the price of goods X and Y is equivalent to a fifty percent reduction in income

Answer 1

With an income of x, a person has x purchasing power when the tax is 0.

 

With 50 percent tax they can buy 2x/3 dollars worth  (I think this is called purchasing power) with an income of x.

 

This is greater purchasing power than they have with x/2 income.

 

Answer 2

To get the final cost of a product with 50% tax added, multiply the cost times 1.5. For example, a $10 item would cost 10 X 1.5, or $15. However,  a 50% reduction in income would cut your purchasing power in half. For example, a salary of $1000, reduced by 50%, results in a new salary of $500. Mathematically, the per cent change in cost of the $10 product has increased by $5/$15, or 1/3 of the original cost, whereas the income example results in a 1/2 decrease. We know that 1/2 is greater than 1/3, so the statement indicating their equivalence is false.