$5.$ Numerical problem II $(15$p$)$

Suppose the interest rate is $8\%,$ the income tax rate $40\%$ at the time of contribution, the

tax rate on investment income is $30\%,$ the expected income tax rate is $30\%$ at the time

of withdrawal and the investment horizon is $40$ years.

$($a$)$ What is the final payoff after tax if $$100$ pre$-$tax income is invested in a regular

savings account?

$($b$)$ What is the final payoff after tax if $$100$ pre$-$tax income is invested in a regular

$401($k$)$ account?

$($c$)$ What is the final payoff after tax if $$100$ pre$-$tax income is invested in a Roth $401($k$)$

account

$($d$)$ How big is the smoothing benefit?

$($e$)$ Which tax effect is bigger, GAP or SMOOTH?