这是一篇来自新西兰的关于开采矿山相关问题的经济数学代写
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”.]
- Suppose that the carbon tax is imposed from the beginning . Write down the price path as seen by the producer with and without the tax. Write down the amount mined each time period with the tax and without the tax as a function of “T”. Now write down an equation for “T” with the tax [Hint the final price when demand is zero is pT=a- τ= 1.8.] (2 marks) Plot the optimal time path of extraction (qt) and the pre tax price (pt) for the life-cycle of the oil reserve. Include on your plots the paths for (qt) and the pre-tax price (pt) plotted in (a). (3 marks)
- Suppose that the government announces from the start that there will be a tax imposed at the start of period 8. Plot the time path of extraction (qt) and the pre-tax price (pt) for the life-cycle of the oil reserve. Include on your plots the paths for (qt) and the pre-tax price (pt) plotted in (a). [Hint vary the initial value of p0 until the final amount of reserves is zero just as the choke price of (a-0.4) is hit. (Remember that the demand function changes at the start of period 8)] (3 marks)
iii. Suppose that the government imposes the tax in period 8 and it is unanticipated (so the orginal extraction path is followed up to this point). Plot the time path of extraction (qt) and pre-tax price (pt) for the life-cycle of the oil reserve. Include on your plots the paths for (qt) and the pre-tax price (pt) plotted in (a). [Hint: Keep track of the reserves. At the start of period t=7 you will be left with R7. From then on the problem is very much like (i) but with a starting value for reserves of R7] (3 marks)
(c) Which policy is the best choice as far as reducing short term emmissions. Why is the results of policy (ii) sometimes referred to a “Green Paradox” (2 marks)Question 4. Constant Elasticity function. (16 marks)
Suppose there is perfect substitute for a resource when the price reaches p=ps. The interest rate is δ, and R0=80, The inverse demand function is pt aqt b .
(a) Suppose the market is perfectly competitive and there are no extraction costs. Write down the Lagrangian associated with this problem and derive the Hotelling rule. (2 marks)
(b) Write down an expression for qt as a function of T which is the time when the price hits the choke price. (3 marks)
(c) Suppose the market is perfectly competitive and there are no extraction costs. Find an expression for the optimal length of mining “T” and the initial price. Hint – you will need to use the formula (4 marks)
(c) Now suppose that there is a monopoly owner of the mine. Show that the mining path is the same (2 marks)
(d) Suppose that R0=90, δ=0.06, a=1.3, b=0.1, pS=1 . Using Excell solve for “T” and plot the price path and the mining path on the same graph (with different axes). (5 marks)
Question 5. Cost reducing technology (21 marks)
A primary response to resource scarcity is investment in cost-reducing technology. Suppose a firm is interested in maximizing profits from extracting a non-renewable resource over two periods (t=0,1), where their profits in time t are written as πt = Ptqt – c(qt)2 / Rt , where Pt is the price of the resource, qt is the amount extracted, c is a positive cost parameter, and R0 is the initial stock. Suppose the firm is interested in investing in research in period 0 that will potentially lower their cost parameter in t=1 from c to (c-αI0), where 0 < α < 1 and I0 is the level of investment in period 0. Suppose further, that they are uncertain of the outcome from investment, and they believe two states are possible in t=1. In state s=0, the research investment fails and the cost parameter stays at c. In state s=1, the research investment succeeds and the cost parameter in t=1 falls to (c – αI0). The probability of state 0 is ε, while the probability of state 1 is 1- ε.
(a) Derive an expression for the firm’s optimal extraction choice in t=1 for both possible states (s=0, 1). That is find expressions for each q*1,s ( 4 marks)
1011)(nk n k rr ar a(b) Using your results from part a, what expression determines the firm’s optimal expected profits in t=1? (5 marks)
(c) Suppose the investment level I0 in period 0 will cost the firm w0(I0 )2 , where w0 is a positive parameter. Using your results from part (a) and (b), what expression determines the firm’s optimal profits in t=0 as a function of q0 and I0? [Don’t try and find the optimal I0 and q0 ] (5 marks)
(d) Suppose the following parameter values: R0=100, P0=1, P1=1.1, c=2, w=5, α=0.1, ε=0.5,and a discount rate (δ) of 0.05. Using Solver, maximize your expression from part c by choosing q0 and I0. Use starting values of q0=20 and I0=0.04. What are the optimal values q*0 and I*0? Include a copy of your Excell Spreadsheet results. (4 marks)
(e) Using Solver, conduct comparative statics to see how investment changes as a result of a change in each of the eight parameters. (3 marks)