Question

4. (28 points) The inverse demand for a scarce depletable resource, same function for every period, is given by P = 20 – 0.29e, where t = 0,1 denotes the time. Period t = 0 is now and period t = 1 is the next period. The marginal cost of extraction, same function for every period, is given by MC = 2+0.196. (a) Assume that the available stock of this depletable resource is = 40 units and the discount rate is r = 5%. Find the per period efficient quantities, prices, and marginal user costs (MUCs). Round your answers to two decimal places. (8 points) (b) Repeat part (a) for Q = 40 units and r = 10%. What happens to your findings when the discount rate increases? Explain. (10 points) (c) Repeat part (a) for r = 5% and Q = 60 units. What happens to your findings when the available stock increases? Explain. (10 points)

Answer 1

20 -0.29t 4. (a) Pt Demand: 2+0'lat cost MC : MNB Pe-mce- 18 - 0:3 ve Equilibrium: MNB I PV (MNB2) 18-0.392 ari 18 - O: 3 1:05 a, 18. 9- o 3150 18-0.392 o o o av 0.392- 0.3159, cr, 0.3159,-0.392 av, 0.3159, 0-3(40-91) a on, 0:3159,- 12 +0:39,- 0.9 0.615 qi 12'9 20: 98 } 19:02 2 Pi = 20-0.3 (20.98) = 15.804 :. Þa 46 muca .. MUG III

- PV (MNB2) (6) IS Equilibrium: MNB 18-0.3 92 18- 0:39 0:39, a or, 1918 - 0:33q, 18-0.392 0:339, 0.392 l'8 = 1.8 ori 0:339,-0.3(40-91) 13. 8 ar, 0:339, +0:39L 0.63 ai ori 1 3.& 2191 :.qi Teioa 18:09 .. az As pi = increases present period extraction rises, 15:618, P2 16' 382 Present period price falls price falls and future price rises. muca = Muca ми с falls mise in = 11:427 n with

(c) Equilibrium: MNB, PV (MNB2) 18- 0.392 aro 18-0139 ('05 or, 0-315 q. o 3 az al 0.3159,-0.3(60-9.) 09 ar, 0.315 qui 18 + 0.39 = 0.9 o.615 a 18.9 qi 30:13 } az 29:27 2 with both period extraction rises. Pi 13' 85 P2 14'15 with the prices fall in both period muq = muca g'12 With a Muc falls