Rising demand

Suppose an exhaustible resource is to be allocated across three periods. The resource is subject to increasing marginal extraction costs and demand is increasing over time. In particular, suppose that the following is true:

• Demand in period 1: P1 = 100 - Q1
• Demand in period 2: P2 = 200 - Q2
  
• Demand in period 3: P3 = 300 - Q3
• Marginal extraction costs: MEC = QT/3

where QT is the total amount of the resource extracted to date. Throughout this problem you may assume the interest rate is zero.

1. Find the efficient allocation of the resource across the three periods. What will the price, quantity, marginal extraction cost and royalty be in each period? Be sure to show all your work.
   
2. Now suppose that a backstop resource is available at a marginal cost of $50. Calculate how the backstop would change the efficient use of the resource. Be sure to show what happens to prices, extraction costs, royalties and production in each period. In what period does the backstop begin to be used? Why? You should assume that everyone knows about the backstop at the beginning of period 1.