Mine ore extraction. Suppose we are granted operating a mine for 10 years, with initial available ore equal to 1000. We want to choose the optimal ore extraction u(t) such that we maximize the present value of cash flows from the mine over the 10 years. The model can be therefore formalized as follows, where the cash flows are modeled as a power function of the extraction (0< a < 1):

max fug XT t ¼ 0 bt ½uðtÞa s:t: yðt þ 1Þ ¼ yðtÞ  uðtÞ  0 for t ¼ 0; 1; 2; /; 9 and initial condition yð0Þ ¼ 1; 000 b ¼ 1 ð1 þ 0:08Þ is the discount factor and parameter a ¼ 1/2. Solve for the optimal ore extraction u*(t).