Emissions taxes and innovation

We consider a regulator who is tasked with finding an appropriate emissions tax level for a single power plant. It is common knowledge that the power plants marginal abatement costs are continuous and linear; and that without any emissions tax, the plant emits 1,000,000,000 Lbs CO2e per year. We believe that an emissions tax of $5 per 1,000 Lbs CO2e will cut emissions from the plant by 10% (i.e. down to 900,000,000 Lbs CO2e per year).

It is assumed throughout this problem that emissions are annual and that both the regulator and the power plant have infinite planning horizons.

Question 2.1 Find and illustrate the marginal abatement cost function for the power plant.

Question 2.2 Calculate the annual tax revenues and annual abatement costs for emissions tax levels of $5, $10, $15, and $20 (all per 1,000 Lbs CO2e).

Additional information for Questions 2.3 and onwards:

A new technology to cut abatement costs is available. This technology can reduce abatement costs by 60% relative to the current abatement cost function. You can find the new marginal abatement cost function by multiplying the function you found in Question 2.1 with a factor of 0.4. The technology requires an investment of $28 million today. The power plant applies a 5% discounting rate and considers any annual cost savings a perpetuity.

Question 2.3 Find the present values of the power plants cost savings for emissions tax levels of $5, $10, $15, and $20 (all per 1,000 Lbs CO2e).

Question 2.4 You consider three options:

A. An emissions tax of $5 per 1,000 Lbs CO2e and a subsidy (to the power plant) of $10 million should the power plant decide to invest in the new technology.

B. An emissions tax of $8 per 1,000 Lbs CO2e and a subsidy of $12 million should the power plant decide to invest in the new technology.

C. An emissions tax of $10 per 1,000 Lbs CO2e and a subsidy of $15 million should the power plant decide to invest in the new technology.

Which option would the power plant prefer, and which option would maximize tax revenues net of any subsidy to the power company?

Question 2.5 Assume that the marginal damage function is given by: MD(e) = (1/50) e. In 400 words or less explain how you could derive the efficient emissions level in context of potential technology shifts as the one above. You are free to use illustrations as well.