Assume that someone has inherited 2,000 bottles of wine from a rich uncle. He or she intends to drink these bottles over the next 40 years. Suppose that this person’s utility function for wine is given by u(c(t)) = (c(t))0.5, where c(t) is each instant t consumption of bottles. Assume also this person discounts future consumption at the rate δ = 0.05. Hence this person’s goal is to maximize ∫ 0 to 40 of [e^(–0.05t) u(c(t))dt] = ∫ 0 to 40 of [e^(–0.05t) u(c(t))^0.5 dt] . Let x(t) represent the number of bottle of wine remaining at time t, constrained by x(0) = 2,000, x(40) = 0 and dx(t)/dt = – c(t): the stock of remaining bottles at each instant t is decreased by the consumption of bottles at instant t. The current value Hamiltonian expression yields: H = e^(–0.05t) (c(t))0.5 + λ(– c(t)) + x(t)(dλ/dt).

What is this person’s continuous decreasing rate of wine consumption?

What is the approximate number of bottles being consumed in the 30th year?