Reference no: EM13223455

U(C0, C1, C2) = ln C0 + d ln C1 + d2 ln C2

where Ct is the amount of consumption, measured in dollars, that they get in period t, and d < 1 is the individual's discount factor (that is, how much they discount future consumption, relative to current time 0 consumption.)

d=0.9

a) Calculate the present value of $100 in year 0, in year 1, and in year 2.

b) Suppose this person has $300 in period 0. How much will they consume in each period? Show your math. (Set the discounted marginal utility of consumption between period 0 and 1 equal, and also the discounted marginal utility of consumption between period 1 and 2 equal. That gives you 2 equations and 3 unknowns. The third equation comes from the constraint. Recall that the derivative of ln x is 1/x.)

c) Check your answer to b), by comparing the present discounted value of utility of a person who follows the consumption plan you derived with the following suboptimal plan: C0 = 100, C1 = 100, C2 = 100.

d) Now consider this person at the beginning of time period 1, instead of 0. So, they have already made their optimal period 0 decision, and consumed that much stuff. Now they are deciding about what's left over. How much will they consume in period 1 and in period 2?