David Ding Baseball Bat Company currently has $$3$ million in debt outstanding, bearing an interest rate of $12$ percent. It wishes to finance a $$4$ million expansion program and is considering three alternatives: additional debt at $14$ percent interest $($option $1),$ preferred stock with a $12$ percent dividend $($option $2),$ and the sale of common stock at $$16$ per share $($option $3).$ The company currently has $800,000$ shares of common stock outstanding and is in a $40$ percent tax bracket.

a$)$ If earnings before interest and taxes are currently $$1\mathrm{.}5$ million, what would be earnings per share for the three alternatives, assuming no immediate increase in operating profit?

b$)$ Mathematically determine the indifference point between the debt plan and the common stock plan.