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 = 0.05. Hence this person's goal is to maximize₀⁴⁰e^-0.05tu(c(t))dt =₀⁴⁰e^-0.05t(c(t))^0.5dt. 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+ (- c(t)) + x(t)(d/dt).

• Wine consumption decreases at a continuous rate ofwhat percent per year?
• What is the approximate number of bottles being consumed in the 30th year?

Please explain steps to solve.