Falling extraction costs due to new technology

Technological progress in the exhaustible resource industry has often been very rapid. This problem asks you to examine how anticipated progress affects the allocation of a resource. Consider an exhaustible resource that is to be allocated across two periods. You are given the following information:

• Demand in period 1: P1 = 100 - Q1
• Demand in period 2: P2 = 100 - Q2
• Marginal extraction cost in period 1: $40
• Marginal extraction cost in period 2: $20 (known in advance to be lower than MEC1)
• Total quantity of the resource is 50 units
• The interest rate is 100% (suppose the two periods are a generation apart)

Notice that extraction costs are known from the beginning to be lower in period 2 than in period 1 and that the cost reduction does not depend on how much has been extracted.

1. Find the efficient allocation of the resource between the two periods. Determine the price, royalty, quantity extracted and marginal extraction cost for both periods. Show all your work.
2. Now suppose that there's a backstop technology available in either period at a price of $60. Find the new efficient allocation, including all of the following variables for each period: the price, marginal extraction cost, royalty, quantity of raw extraction and quantity of backstop production. Explain in non-technical language how the backstop affects the problem. You may assume that everyone knows about the backstop at the beginning of period 1.