Compost Science, Inc. $($CSI$),$ is in the business of converting Boston$$s sewage sludge into fertilizer. The business is not in itself very profitable. However, to induce CSI to remain in business, the Metropolitan District Commission $($MDC$)$ has agreed to pay whatever amount is necessary to yield CSI a $10\%$ book return on equity$($ROE$).$ At the end of the year CSI is expected to pay a $$4$ dividend. It has been reinvesting $40\%$ of earnings and growing at $4\%$ a year.

a$.$ Suppose CSI continues on this growth trend. What is the expected long$-$run rate of return from purchasing the stock at $$100?$

b$.$ Now the MDC announces a plan for CSI to treat Cambridge sewage. CSI$$s plant will, therefore, be expanded gradually over five years. This means that CSI will have to reinvest $80\%$ of its earnings for five years. Starting in year $6,$ however, it will again be able to pay out $60\%$ of earnings. What will be CSI$$s stock price once this announcement is made and its consequences for CSI are known?