Maximizing Profit in Ore Mining
Maximizing Profit in Ore Mining
A mine owner has the right to extract ore for a time duration . Initially there is ore in the ground. As ore is extracted, the instantaneous stock of ore declines, and =-u(t) is the extraction rate. Assume that the ore sells at a constant price and the extraction cost is .
T
x(t=0)=
x
0
x(t)
dx
dt
p
2
(u(t))
x(t)
This Demonstration finds the optimal extraction rate that maximizes the profit (or performance measure), defined as .
u(t)
J=px(t)-dt
T
∫
0
2
(u(t))
x(t)
When you set the value of the parameters , , and , the Demonstration plots the rate of extraction , the instantaneous stock of ore , and the costate (also called shadow price) . Assume that the ore remaining in the ground at time has no value (i.e. there is no "scrap value", or ).
T
p
x
0
u(t)
x(t)
λ(t)
T
λ(T)=0
Two solution methods are adopted: (1) the discrete-time method (solutions indicated by the blue triangles), and (2) the continuous-time method (the red curves).
You can readily check that when the duration is decreased, the extraction rate is larger. Also for a fixed value of , if you increase the initial stock of ore in the mine, then both the profit and the extraction rate are larger. The values of for the discrete-time and continuous-time versions are indicated in blue and red, respectively. The two values show very close agreement. Finally, you can lower the ore price, which results in a lower profit.
T
T
J
J