Consider an underground aquifer where the initial stock is X0=100,000 acre-feet. Andres bought the exclusive rights to pump water from the aquifer over a period of 20 time periods from the government. The revenues in a time period, t, is given by Rt = aWt – (b/2) W 2 where Wt is the amount of water pumped in acre-feet. The cost of pumping water is Ct = cWt/Xt. The stock of water in the aquifer in the future depends on the current stock minus the current amount of water pumped plus a constant recharge rate equal to D during each period. Q1: Write the objective function along with the constraints if Andres tries to maximize profit over his planning horizon. Assume that Andres does not value any leftover stock after his rights to pumping water expires and his discount rate is begin mathsize 12px style delta end style or, equivalently, his discount factor is ρ.
Q2: Assume that a=100, b=0.001, c=812,500, D = 35,000 and begin mathsize 12px style delta end style = 0.05. In an excel worksheet (sheet3), set up the problem above using an initial guess of Wt=100 for all relevant periods. Copy everything to sheet4 and use solver to derive the optimal water extraction path that maximizes profit. Graph extraction over time and stock over time in separate figures.