The Amount of Resource Extracted in Each Period
a. To determine the amount of the resource extracted in each period, we need to find the quantity that maximizes the net benefit, which is the difference between the demand function and the marginal cost.
The demand function is given by: P = 176 – 0.4Q
And the marginal cost is given as: MC = 44
To find the quantity that maximizes net benefit, we set the marginal cost equal to the marginal benefit (which is the slope of the demand function) and solve for Q:
44 = -0.4
Simplifying, we have:
0.4Q = 132
Q = 330
Therefore, the amount of the resource extracted in each period is 330 units.
To graphically represent this outcome, we can plot the demand function and the marginal cost on a graph with Q (quantity) on the x-axis and P (price) on the y-axis.
Graph
The point where the demand curve intersects with the marginal cost curve represents the quantity extracted in each period, which in this case is 330 units.
Resource Extraction with a Reduced Initial Supply
b. Now let’s consider a scenario where the initial resource endowment is reduced to 400 units and a discount rate of 8 percent is applied.
To determine the optimal extraction levels in each period, we need to equate the present value of the marginal benefit from the last unit in period 1 with the present value of the marginal net benefit in period 2.
The present value of a future benefit (B) or cost (C) can be calculated using the following formula:
PV = B / (1 + r)^t
Where PV is the present value, r is the discount rate, and t is the time period.
In this case, we want to solve for Q1 and Q2, which represent the quantities extracted in period 1 and period 2, respectively.
Let’s denote Q1 as the quantity extracted in period 1 and Q2 as the quantity extracted in period 2. The present value of the marginal benefit from the last unit in period 1 can be calculated as:
PV1 = (176 – 0.4Q1) / (1 + 0.08)
The present value of the marginal net benefit in period 2 can be calculated as:
PV2 = (176 – 0.4Q2 – MC) / (1 + 0.08)^2
To satisfy the condition that PV1 equals PV2, we set these two equations equal to each other and solve for Q1 and Q2:
(176 – 0.4Q1) / (1 + 0.08) = (176 – 0.4Q2 – MC) / (1 + 0.08)^2
Simplifying this equation, we have:
176 – 0.4Q1 = (176 – 0.4Q2 – MC) / (1 + 0.08)
Now let’s substitute the values for MC and solve for Q1 and Q2:
176 – 0.4Q1 = (176 – 0.4Q2 – 44) / (1 + 0.08)
Multiplying both sides by (1 + 0.08), we get:
(176 – 0.4Q1)(1 + 0.08) = 176 – 0.4Q2 – 44
Expanding and simplifying, we have:
193.6 – 0.448Q1 = 132 – 0.4Q2
Rearranging terms, we get:
0.448Q1 – 0.4Q2 = 61.6
To find Q1 and Q2, we need another equation. Since we are told that the demand is constant over both periods, we can set Q1 equal to Q2:
Q1 = Q2
Substituting this into our equation, we have:
0.448Q1 – 0.4Q1 = 61.6
Simplifying, we get:
0.048Q1 = 61.6
Q1 = 1283.33
Since Q1 cannot be a fractional value, we round down to the nearest whole number:
Q1 ≈ 1283
Since Q1 and Q2 are equal, this means that Q2 is also approximately equal to 1283 units.
Therefore, if we desire to satisfy the condition that the present value of the marginal benefit from the last unit in period 1 equals the present value of the marginal net benefit in period 2 with a reduced initial supply of 400 units and a discount rate of 8 percent, approximately 1283 units of the resource will be extracted in each period.
