$($Measuring growth$)$ Solarpower Systems earned $$20$ per share at the beginning of the year and paid out $$8$ in dividends to shareholders so$,D_0=$$$8)$ and retained $$12$ to invest in new projects with an expected return on equity of $19$ percent. In the future, Solarpower expects to retain the same dividend payout ratio, expects to earn a return of $19$ percent on its equity invested in new projects, and will not be changing the number of shares of common stock outstanding.

a$.$ Calculate the future growth rate for Solarpower's earnings

b$.$ If the investor's required rate of return for Solarpower's stock is $13$ percent, what would be the price of Solarpower's common stock?

c$.$ What would happen to the price of Solarpower's common stock if it raised its dividends to $$14$ and then continued with that same dividend payout ratio permanently?

Should Solarpower make this change? $($Assume that the investor's required rate of return remains at $13$ percent$)$

d$.$ What would happened to the price of Solarpower's common stock if it lowered its dividends to $$2$ and then continued with that same dividend payout ratio permanently? Does the constant dividend growth rate model work in this case? Why or why not? $($Assume that the investor's required rate of return remains at $13$ percent and that all future new projects will earn $19$ percent.$)$