Assume that someone has inherited 2,000 bottles of wine from a rich uncle. He or
ID: 1205041 • Letter: A
Question
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 maximize 040 e–0.05tu(c(t))dt = 040 e–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). This person’s wine consumption decreases at a continuous rate of _________ percent per year. The number of bottles being consumed in the 30th year is approximately ________ . (NOTE: Write your answers in number format, rounding to tens (whole numbers, no decimals). Use a comma to separate groups of thousands). Show all steps.
Explanation / Answer
Ans. 15.6% and 200
Explanation-
Bottles consumed per year= 2000/40=50
1Year= 50 bottles
utility=0.5x50=25
Given, rate =0.05
Hence every year bottles added = 2000x 5/100 x 40 =4000
Given, rate= 0.05x0.5= .025
We have interest = 25x.025x25=15.6= rate
If rate =15.6, number of bottles consumed in 30th year
=2000x5/100x10=1000
or 4000/1000= 4 bottles
Calculating, bottles = 50 x 4=200