Assume you own a small natural gas well and live in a world with only two periods, the present (0) and next year (1). The stock (S) of natural gas in the ground under your well is 5000 Mcf (thousand cubic feet). The price of natural gas in period 0 is known to be $21.2 per Mcf feet and in period 1 to be $21.4 per Mcf. The cost function for extracting natural gas is 1000+1.2qi + .005q2. Your discount rate, r, is .07.
(a) Set up the problem of maximizing present value using a Lagrange multiplier framework.
(b) Derive each of the three first order conditions which must hold at the optimum.
(c) Solve numerically for optimal quantities of natural gas to be extracted in time periods 0 and 1 and solve for λ.
(d) Repeat parts (a)-(c) but now assume that you are a monopoly supplier. Price in both time periods is influenced by the number of Mcf of natural gas in the following manner: pi = 2800 − qi.
(e) IN THREE SENTENCES OR LESS Explain why the initial competitive market and the monopoly situation differ with respect to the amount of natural gas extracted in the first time period.