An oil well manager extracts qt barrels of oil in every period. The profit, t, from extraction is given by ln(1 + qt). The goal of the manager is to maximize the present value of profits over his ten year horizon (t = 0, 1, 2, ...,9) subject to the standard equation of motion for non-renewable resources.

1. Write out the objective function and constraint for this problem. Clearly identify the control variables and the state variables.

2. Set up the Lagrangian for this problem.

3. Determine the first order conditions for profit maximization in this problem.

4. Provide an economic interpretation for each first order condition.