Exponential inter-temporal utility function question:
I have this function:
U(C0, C1,C2) = ln(C_0) + (δ)*lnln(C_1) + (δ^2)*(ln(C_2)
where δ =0.8
Suppose I have $60 in period 0 (C_0). How much should 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.)
HELPFUL INFO:
Constraint:
C_0 + C_1 + C_2 = 60
A SOLUTION OF A SIMILAR PROBLEM:
Suppose you 7 hours of leisure spend over 3 periods (days).
U(L_0, L_2,L_2) = ln(C_0) + (δ)*lnln(C_1) + (δ^2)*(ln(C_2)
where δ = 1/2
MU_0 = (1/2) MU_1
(1/2) MU_1=(1/4) MU_1
Constraint:
L_0 + L_1 + L_2 = 7
MU_n= 1/L_n = (du/dL)
(1/L_0)= (1/2)(1/L_1)
and
(1/2)(1/L_1)=(1/4)(1/L_2)
s.t.
L_0 + L_1 + L_2 = 7
So, this simplifies to:
L_0=2*L_1
and
L_1=2*L_1
So our lifetime consumption plan is:
L_0= 4
L_1= 2
L_2= 1
NOTES:
I'm having trouble figuring out where the 4 came from in the example problem. I'm not sure how to translate that to the problem where I have 60, instead of 7. The delta is also different and I'm not sure how to divide the 60 into three periods based on the utility function.