You are planning to retire in $30$ years. You want to be able to spend $$40,000$ per year in retirement, adjusted for inflation $($so you will spend the equivalent in each year of $$40,000$ in today$$s dollars$).$ You will withdraw your retirement spending at the end of each year. So your first retirement withdrawal will be at the end of year $31.$ Assume you will live for $25$ years after retirement and the effective annual interest rate $($EAR$)$ is $8\%$ throughout.

A$.$ How much is the first retirement withdrawal at the end of year $31$ if the annual inflation rate is $2\%$ per year?

B$.$ What is the present value of all your retirement spending? Continue to assume inflation is $2\%$ per year and give the PV as of today $($year $0).$

C$.$ Before retirement, you are planning to contribute to your savings once a year for the next $30$ years, at the end of each year. For the first $10$ years, you will contribute the same amount. However, at the end of year $11,$ you will double the annual savings contribution amount but keep it fixed for the remaining $20$ years. $($So the annual savings contribution amount for years $11$ through $30$ is twice the amount for years $1$ through $10.)$ How much do you need to contribute this year to afford your retirement plans?