# 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