Question

1. Consider a two-period model similar to the one discussed in the class. Assume that the inverse demand equations for a particular depletable resource (e.g., stock of oil) differs across the two periods, with MB. = 600 - 90 MB = 800 - 91 Where, 9. denotes the amount of oil consumed in period 0 and q, denotes the amount consumed in period 1. The higher demand for oil in period 1 might result, for example, from an increased population. On the other hand, suppose that the marginal cost equations for extraction remain fixed across the two periods with: MCo = 200 + 40 MC = 200 + 91 If there are a total of 500 barrels of oil available for the two, then a. How much oil would each time period wish to have, ignoring the other time period? That is, what is the statically efficient amount of oil to consume in each period? (Show it graphically and mathematically) (4 points/ Now, instead suppose that there are only 200 barrels of oil available for the two periods and the discount rate is 6.25% (i.e., r=0.0625): b. How much would be allocated to period 0 and how much would be allocated to period 1 in a dynamically efficient allocation? (Show it graphically and mathematically) 17 points) c. What would be the efficient prices in the two periods? (Show is graphically and mathematically) (4 points) d. Define marginal user's cost. [2 points) e. What would be the marginal user's cost in each period? (3 points

Answer 1

Solution a. Eauch period will want to extract oil as long as the marginal benefit from dang so is atleast as great as marginal cost of extraction .MB, MCI (t20,1) and they will stop the Extraction once MB = my (t=0,1)) after that point, any additional extrauction costs more Cat the margin) than it benefits users Cout the margin) For period o, then we have a statically efficent quantity of oil for period o. where 90 sholes a MBO - MCO (or) 600-agt 2 200 +mote Cor) 400 - 2a$ cor) qot - 200 for period I, we have the stastically efficent quantity of oil q solves MB1 - Mclcor 800 -q t - 206+q, or 600 = 29+ (or) aiz 300

Suppose that there are only 200 barrels of oil available for the two period - period o problem is to find the intersection of margind benefits and marginal cost Curves depicted graph below. emp MCO 00 MBO 200 100 200 300 90 > for period I as indicate below q, 2300 600 400- 200 0 100 200 300 I b Dynamic officency to maximize the present value of the total net benefit received from the use of oil. pr(MNBO) - PVC MNB17

when the net gains the each period wuld receive hom having a little mole oil are the same, it not the case we should shitt some oil to the higher valued time period and increase the total net benefit. MNBO = MBg-MCO -600-90 ( 200+90) = 400 - 090 i The marginal benefits occurs today is no discount necessary PVC NB] = MNBO 400-290 is for period I we know that these MNB I MB1 = MCL = 800 - 91 - C200 tu - 600-200 - In this case we have to write for a period to recieve tharese benefits we need to take the consumption The cost of writing using I discount PV[MNB 1] = MNBI -600-2001 1 0625 = 606 2 al 1.0605 1:0625 = 564.71- 1-8891

two caual yields 400-001-584-41-1889 We have 200 barrels of oil available for use in the two period ince both period want mole. thoon One oil is available 90 to 1:200 cor al = 200-aro_o i 200 - 54.25 = q 1 = 145.45 er substituing this situation 1 Substituting this solution in above 400 - 2010 -564-71 - 1188/200-90) 400-290-564.71-376.5+1.8890 YO54.55 800 Roo 600 ID +400 2007 200 200 00-54-55 000 NO 9 1 = 145.45

C) The price in each period needs to be high enough encourage the individuals in that period to consume exauctly the efficent quaintily an individual to purchase oll aos using long as the marginal benefits oil exceeds the price. Those will stop Consuming oil once their marginal benefits exactly equals the price Po = MBolao - 600-90 = 600 - 54.55 = 545.45 Similarly for period I, PIEMBilal = 800-91 = 600-145.45 2 654-55 poriod I is higher than period y MUCO = M N Bolao = 400-290 =400-2/5455) 290.9 for periodi MUCO: MNB 1/24 = 60o-dal =600-2 (145:45) = 30911 Marginal usen cost is higher in period,