An electric company must decide whether to install a scrubber to reduce sulfur dioxide emissions from its electric generating station. The plant currently has no such pollution control equipment, and must buy SO2 permits to cover its emissions at an annual cost of $500,000. The cost of installing the scrubber is $10 million; when installed, the need for buying 3 pollution permits will be eliminated. For every year that the company does not install the scrubber, it will have to purchase permits, and the demand for electricity-and therefore SO2 permits-is projected to grow at an annual rate of 5%. Assume that the plant managers base their decisions on a five-year capital budgeting cycle (i.e., time is t = 0, 1, 2, ..., 5, where t = 0 is now), and that changes in regulations mean that the company has to buy the scrubber within a 5-year grade period (i.e., by t = 5). Scrubbers are huge pieces of capital equipment that require time to build and bring on line, so if the scrubber is installed in year t, permits must be purchased for that year as well.
(a) If the company can get a 6 percent return on alternative investments, should the company install the scrubber now, or, if not, how long should it wait?
(b) What would the annual rate of return have to be for immediate installation of the scrubber to be cost-effective?
(c) Let P and S denote the (constant) costs of permits and the scrubber, and let the discount rate be r and the growth rate of the demand for permits be 0 < g < r. Assuming continuous time, set up the firm's problem and solve analytically for the optimal time t = T when the company would want to invest. Explain the impact of the parameters on your solution.