Question 1:
Mary has a little cake with size W1 that is supposed to last indefinitely. However, her fridge is bad so that whatever amount of food stored in the fridge, half of it will go bad. That is, the transition equation governing next day's cake size and today's consumption is
Wt+1 = 1/2(Wt - ct). where ct is the amount of cake Mary decides to eat at day t and Wt is the amount of cake at the start of day t. She has flow utility u(c) = ln c and time-discount factor 0 < β < 1.
1. Write down Mary's infinite period constrained maximization problem.
2. Solve the problem you wrote down in 1. by the Lagrange multiplier method. What is the value function of the problem? If W1 = 10 and β = 0.8, what is the optimal consumption at day 2?
3. Rewrite the problem you wrote down in 1. as a dynamic programming problem.
4. Solve the problem you wrote down in 3. by making a guess of the value function and using the Method of Undetermined Coefficient discussed in class. Verify that your answer (optimal consumption sequences) coincide with the answer to 2.
Question 2:
Mary lives in a mysterious kingdom where cake is not private property. At the end of each day, she is required to turn in the remainder of the cake to the Central Cake Committee (CCC) that tries to reallocate the cake in a fair and equal way. Next morning, she will go to the CCC to claim the cake allocated to her at that day according to the amount of cake she turned in last evening. To be precise, for any amount of cake x surrendered to the CCC, the amount the person can take back in the next morning is x^0.5. For example, if 0.25 pieces of cake is turned in by the end of day, she will be entitled to take 0.250.5 = 0.5 pieces by next morning, but if 4 pieces of cake are turned in by the end of day, she will only be entitled to take 40.5 = 2 pieces the next morning. Therefore, the relevant transition equation is now Wt+1 = (Wt - ct)^0.5 where ct is the amount of cake Mary decides to eat at day t and Wt is the amount of cake she is entitled to at the start of day t. When Mary reaches adulthood, she is given W1 size of cake. She has flow utility u(c) = ln c and time-discount factor 0 < β < 1.
1. Write down Mary's infinite period constrained maximization problem.
2. Rewrite the problem you wrote down in 1. as a dynamic programming problem and solve it by making a guess and using the Method of Undetermined Coefficient discussed in class. If W1 = 10 and β = 0.8, what is the optimal consumption at day 2? (Hint: Try a two period problem first to see what the value function looks like. Also, there is another arithmetic law for log function that's useful: ln(a^b) = b ln a.)
3. Do your best to solve the problem by the Lagrange multiplier method. (Hint: enlarge the problem so that Wt, ct are all choice variables, except that they satisfy
the transition equation. i.e, consider max{Wt+1,ct ∞t=1
P∞
t=1 β
t-1
ln ct such that Wt+1 =
(Wt - ct)
0.5 and that 0 ≤ ct ≤ Wt
for t = 1, 2, ...)