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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.
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