Traditional vs Roth 401k

Let p be amount invested over N years. r = rate of return. t = tax rate.

Roth 401k
You pay taxes upfront. That reduces p to (1-t)p. After N years this becomes (1+r)^N(1-t)p which you can withdraw tax free.

Traditional 401k
You don’t pay any taxes on p. After N years the amount becomes (1+r)^Np. Now when you withdraw it, you pay tax and so the net amount becomes (1+r)^Np(1-t) which is same as earlier.

Of course, the catch is the assumption that t remains same in both cases.

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