Question

1. You own 15 barrels of oil that you can sell this year or next year and is otherwise worthless. The price of abarrelofoilineachyearisp0 =21−1q0 andp0 =17−1q1 withadiscountfactorof.95andacostof

   1 dollars per barrel. Find the optimal extraction plan, the prices in each period and total discounted net revenue.

2. You own 15 barrels of oil that you can sell this year, next year, or the year after next year and is otherwise worthless. The price of a barrel of oil in each year is pi = 14 − 1 qi with a discount factor of .96 and no costs.

   2

   Find the optimal extraction plan, the prices in each period and total discounted net revenue.

3. You own 30 barrels of oil that you can sell this year or next year and is otherwise worthless. The price of a barrel of oil in each year is pi = 6 − qi with a discount factor of .92 and no costs. Find the optimal extraction plan, the prices in each period and total discounted net revenue.

Answer 1

Total quantity q = q0+ q1 = 15 barrels

P0 = 21- q0

p1 = 17 -q1

Total revenue in period 0: P0*q0 = (21-q0)*q0

Total revenue in period 1: P1*q1 = (17-q1)*q1

Total Discounted Revenue = P0*q0 + discount factor (P1 *q1) = (21-q0)*q0 + 0.95 [(17-q1)*q1]

Total cost = Cost per barrel * q = $1 *15 = $15

Profit = Total Discounted Revenue -Total Cost =

Optimal extraction plan is to maximize this profit subject to q0+ q1 = 15

Lagrange function: L =

differentiating L w.r.t. q0, we get

differentiating L w.r.t. q1, we get

substituting the value of lambda, we get,

now, q1 = 15-q0, substituting the value, we get,

Total discounted Net revenue = Total discounted revenue - total cost = (8.55*12.45) + 0.95 (10.55*6.45) - 15

Total discounted Net revenue = 106.45 + 64.65 -15 = 171.10 -15 = $156.10

Therefore, optimal extraction plan is q0* = 8.55 barrels, q1* = 6.45 barrels, Price of barrel per year is: p0 = $12.45 per barrel, p1 = $10.55 per barrel and.the total discounted net revenue is $156.10.