I have this function

U(C0, C1,C2) = ln(C_0) + (?)*lnln(C_1) + (?^2)*(ln(C_2)

where? =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) + (?)*lnln(C_1) + (?^2)*(ln(C_2)

where? = 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

Price: $15