You are the manager of a reservoir that provides water to a small irrigation district. The reservoir is subject to stochastic a (random) inflow which makes the decision on the current amount of water to release a risky one. To keep things simple we will assume a two-period (t = 0, 1), two-state (s = 1, 2) problem. Specifically, let R0 denote the amount of water in the reservoir in t = 0 and W0 the amount of water released for irrigation, also in t = 0. Then, the amount of water in the reservoir in t = 1, s = 1 is R1,1 = R0 -W0 + S1,1 while the amount of water in the reservoir in t = 1, s = 2 is R1,2 = R0 - W0 + S1,2, where S1,2 > S1,1 > 0. The value of water released in t = 0 is given by the function U0 = αln(1 + W0), where α > 0 and ln(•) is the natural log operator. The value of water in the reservoir in t = 1, is U1,s = βln(R0 - W0 + S1,s), s = 1, 2, where β > α > 0 and ln(•) is once again the natural log operator. The probability of future state s = 1 is Pr(s = 1) = π while the probability of future state s = 2 is Pr(s = 2) = (1 - π), where 1 > π > 0. The value of water in the reservoir in t = 1, in either state, is discounted by the discount factor ρ = 1/(1 + δ), where δ > 0 is the discount rate.
(a) Let U denote the value of water released in t = 0 plus the discounted expected value of water in the reservoir in t = 1. What is the expression for U ?
(b) What is the first-order condition that might be solved for the optimal amount of water to be released in t = 0, i.e., W ∗?
(c) Suppose α = 5, β = 8, δ = 0.02, S1,1 = 10, S1,2 = 72.8, R0 = 500, and π = 0.9. Set up a
spreadsheet and use Solver to determine the value of W ∗ by (i) maximizing U and (ii) by driving
dU/dW0 = G(W0) to zero. Are the values for W ∗ using either approach the same?