The previous problem set defined optimal depletion trajectories for a competitive resource extraction firm. This problem set extends the analysis in several ways:

Monopoly

Suppose you have a monoply on the resource. Your marginal extraction costs are $4 + $0.05Q, but you are selling to a downward-sloping inverse demand schedule P = 12 - 0.05Q, so your total revenue is TR = PQ = $12Q = $0.05Q². The derivative of this is your marginal revenue function MR = $12 - $0.10Q. (Note that the slope of MR is twice the slope of the inverse demand schedule.)

There are two sources of rent here: the ordinary rent on the resource, plus a monopoly rent on the market that you control. The MRR function for this monopolist would be

Assuming the stock X = 500 and the discount rate r = 0.05 in either case, use a spreadsheet to solve and graphically compare the optimal depletion schedules for this monopoly versus the optimal trajectories of a competitive firm with MRR = $6 - $0.05Q.

Your solution should demonstrate how a monopolist maximizes profits by extracting less in the initial years (to maintain a higher market price) and prolongs the time to depletion. The monopolist sacrifices some resource rent in the process of extracting monopoly rent, and looks like a better "conservationist" than the competitive firm, but this excess conservation is economically harmful.

Stochastic demand

To this point you have been calculating smooth optimal trajectories to a stationary depletion target. In reality, future market conditions are uncertain, and current expectations about the future are constantly changing. That's why prices and extraction quantities in real resource markets are always fluctuating as resource extraction firms continually adjust their rent expectations and extraction schedules. Their optimal marginal rent and extraction trajectories keep shifting as the depletion target moves.

Try solving out an optimal resource depletion schedule with a stochastic (i.e., randomly-varying) price P = MR = $9 + 2*RAND() where RAND() is an Excel function generating a random values between 0 and 1 with a mean of 0.5. Assume MC = $4 + $0.05Q as before. This stochastic element carries through to the marginal rent function:

Set up your spreadsheet with terminal marginal rent "=5+2*RAND()" in the top cell of the RENT column, and calculate the optimal MRR's downward for about 20 preceding years based on that cell. Calculate the implied extraction quantities Q and stock levels X for each year.

Copy the values of MRR and Q for the year when X is closest to 500 into some empty columns on the right. Use "Paste Special" to copy cell values rather than formulas. If no single row approximates your current stock level closely enough, you can interpolate values between the closest rows.
In an adjacent column, calculate the stock remaining after you extract this first amount Q.

Each time you do any calculation in this spreadsheet, it generates a new value for RAND() and calculates new trajectories for MRR, Q and X based on the shifted MRR schedule. Identify the year when X is closest to the remaining stock size, and copy these values of MRR and Q into the next row on the right. Calculate the stock remaining.

Working forward in time, iterate this process until you have exhausted the stock. This should yield realistically irregular optimal trajectories of MRR, Q and X. Create XY-plots comparing the optimal MRR, Q and X trajectories under the stochastic MRR = $5 + 2*RAND() - $0.05Q schedule versus the trajectories under the base case non-stochastic MRR = $6 - $0.05Q. You should get something that looks like this (your numbers will be different).