Is the flat tax fair?

I came up with this scenario today in response to this post I came across on a blog at the Cato Institute.

Question about flat taxes:

Person 1 earns 25,000 a year. Person 1 pays 500 dollars a month for rent, 200 for food, 200 for utility bills (phone, electricity, heating), 100 for bus pass/train pass, for a total basic expenditure of 1000 per month. A flat tax on his total income of what, 15%, 10%, 12% would be another 257 each month. This leaves Person 1 with less than half of their total income per month, just about 825 dollars. Notice I did not include any costs for spouse, children, clothing, student loan payments, health insurance costs, car payments, car insurance, or savings. Let alone spending money for normal consumer expenditures like entertainment, Christmas shopping, Friday night movies, Monday night football with the guys, getting your nails or hair done with the girls.

Person 2 earns 250,000 a year. Person 2 spends 2250 for a studio apartment in lower Manhattan, however, still probably only spends 200 dollars on food, 200 for utility bills, 100 for the subway pass, apply that same flat income tax (averaged at 12.33%) and he will pay a much greater amount of 2568. This still leaves Person 2 with 15515.33 left over for the month. Again I am not factoring in any peripheral expenses.

So clearly, the 12.33% flat income tax has a much greater impact on Person 1 than Person 2. I think that is discriminatory.

Furthermore, since Person 2 has a greater disposable income, it is likely that Person 2 has a greater ability to use that income to make investments. These investments, if safe, are likely to bring a return. So on top of Person 2′s earned income of 250k you can assume that there are additional incomes collected. Are these additional incomes not earnings that would be taxed as interest, dividends, rental property income, etc? If either Person 1 or Person 2 suddenly earned a windfall inheritance does that count as income to be taxed?

Isn’t it more fair to say that it is discriminatory against Person 1 to charge him the same flat tax as Person 2?