Rising extraction costs, recycling and externalities

Suppose that a particular good can be produced from either raw materials or scrap. The marginal cost of raw production depends on the total amount extracted to date and is given by the equation: MEC = $200 + $1*QT, where QT is the total amount produced to date. Producing the good by recycling scrap has a marginal cost of $1000. The good is to be produced for 5 periods, each of which has a willingness to pay given by: Pi = 2000 – 2.5*Qi, where Pi and Qi are the price and quantity of the good in period i. To keep the algebra managable you may assume the interest rate is 0.

1. Find the efficient pattern of production and use of this good. Solve for the price, quantity, marginal extraction cost and royalty in each period. How much is produced from raw materials? How much by recycling? In what period does recycling start?
2. Now suppose that recycling produces a lot of pollution and that the damages associated with it are $200 per unit recycled. What is the new efficient pattern of production and use of the good across the periods? What is the efficient amount of raw production? What is the efficient amount of recycling?
3. Suppose the economy is initially at the market equilibrium and a group of citizens becomes concerned about the $200 externality from recycling. They argue that since raw production doesn’t create the externality, it should be encouraged. In fact, they argue, it should receive a $200 subsidy per unit from the government. Evaluate this proposal and calculate any relevant numbers. Does it solve the problem? Discuss in detail. You can assume this debate occurs before any production has actually taken place.