Timing of adoption of clean technologies by regulated monopolies

We consider a monopolistic firm producing a good while polluting and using a fossil energy. This firm can adopt a clean technology by incurring an investment cost decreasing exponentially with the adoption date. This clean technology does not pollute and has a lower production cost because it uses a renewable energy. We determine the optimal adoption date for the firm in the case where it is not regulated at all, and in the case where it is regulated at each period of time i.e. the regulator looks for static social optimality. Interestingly, the regulated firm adopts the clean technology earlier than what is socially-optimal. However, the non-regulated firm adopts later than what is socially-optimal. The regulator can induce the firm to adopt at the socially-optimal date by a postpone adoption subsidy. Nevertheless, the regulator may be interested in the earlier adoption of the firm to encourage the diffusion of the use of clean technologies in other industries.