Question 1 (8 marks)
Consider an owner of a mine who plans to mine over 3 periods. There are R0 = 8 units of the stock available. The total cost of extracting the resource is TC=4q. The inverse demand function is p=32-q. The market is perfectly competitive.
(a) Write down the maximisation problem the miner must solve and the associated Lagrangian if the discount rate is δ. (1 mark)
(b) Write down the equations derived from the first order conditions. (1 mark)
(c) What are the values of q0 , q1 , q2 that maximize the discounted sum of profit for the mine owner if the discount rate is 10% (4 marks)
(d) Now suppose there are R0 = 100 units of the stock available. Do the answers to part (c) change What is your intuition as to why and how the answers change What is the most the miner would mine in any given period (2 marks)
Question 2 (8 marks)
Consider a competitive market and mining takes place over two periods. The initial resource stock is R0. The interest rate is delta. The product mined incurs a tax on it in period 0 and 1 of τt so that the price paid by a consumer is pt+τ. Show that the mining path is unaffected if τ1=τ0(1+δ) for a linear demand function pt+τ=a-bqt.
What is the condition on the second period tax compared to the first period tax for mining to be higher in the second period (this is sometimes known as the green
paradox – see question 3 below as well).
Question 3. (17 marks)
Suppose the price of oil is given by pt=a-bqt, where qt is the rate of oil extraction in period t, a=2.4 and b=0.1. The initial oil reserve (R0) equals 220, the remaining oil reserves change according to Rt+1=Rt-qt, and there are no extraction costs. Further, assume a 10% discount rate.
(a)Suppose the oil industry is competitive, and each firm is interested in maximizing the present value of net revenue over the horizon t=0, 1, ….., T. Using Hotelling’s rule , plot the optimal time path of extraction (qt) and price (pt) for the life-cycle of the oil reserve using Excel and ‘Solver’.
Start off by finding the value of T. Also include a copy of your Excell Spreadsheet results. (Either just cut or paste into word or use the print screen option on your open file (Hold the Alt key down and print screen to make a copy, then paste into word.) (4 marks)
(b) The government is considering implementing a carbon tax τ=0.4 which would increases the price of oil by τ so that pt+τ=a-bqt, (pt is the price paid to the producer) For each of the questions below paste your Excell file into your answer as well as your plots. [Hint the expression which determines “T” is different to that used in part (a). The tax will reduce the choke price which is no longer equal to “a”.]