### Mathematical/economic help requested, please

Driving to work today, I caught a moment of Terry Gross's show, *Fresh Air*. When I tuned in, someone was telling her that Bush's social security plan just won't work, no way. I'm very slow at math, and quite phobic about percentages/investments, etc., but what he said sounded like funny money to me. That is, I kept saying to myself, "Something is wrong here. That's just not right." I wasn't surprised, therefore, when Terry, pausing for station identification, identified her guest as Paul Krugman. I'm willing to accept that there are entirely valid arguments against the President's social security plan, but I'm unwilling to believe that any of those arguments could come from Krugman.
I was curious, though, about the benefits of a young worker keeping her money, as opposed to turning it over to the government. In essence, I imagined a scheme where the worker is forced to save a percentage of her income, rather than being allowed to spend it all. Using very simple numbers, because that's all I'm good at, and using a compound interest calculator, I did the following:
First, I envisioned my worker beginning at 20 years old with a $20,000 annual salary. I further envisioned a situation in which she never got a raise -- she just flatlined at that number, getting $20,000 every year of he working life. To continue with the easy numbers, I pretended she was self-employed, so she was required to fund the entire 15% of her forced savings, for an annual total of $3,000 (money that would normally go into Social Security's ever-hungry maw). Lastly, I had her retire at 65, so I had her gainfully employed at an annual salary of $20,000, with annual $3,000 contributions, for 45 years. Here's what I came up with:
If she had an average annual compound interest rate of only 2% for the 45 year period, she would end up with $227,305.26.
If she had an average annual compound interest rate at the slightly greater 3% for the 45 year period, she would end up with $297,849.16. The numbers really start taking off after that.
At an average compound interest rate of 4% per annum, she would have $395,135.23 to her name. And if she was able to achieve a 5% annual compound interest rate over the 45 year period, she'd end up with $530,010.51.
My sense is that, at the 2% rate, she'd be at parity with Social Security (and please correct me if I'm wrong). Any greater interest rate, she'd be ahead of Social Security.
Further, if one assumes that she actually increases her salary over her lifetime (whether because she ascends the employment ladder into a higher wage class or because of adjustments for inflation), she's got to end up with even higher numbers. But that's a complicated mathematical calculation I'm utterly incapable of doing.
Having set all this out, I'd appreciate it if any of my readers could explain to me where (and if) I'm right with this analysis and where (and if) I'm wrong. Please do it in words of one syllable, though, and with simple numbers, since otherwise I'll be like that female professor at Harvard who tried to faint or throw up when confronted with Larry Summer's intellectual challenges.

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