Consider a world where people live for up to three periods. In the first period each person earns 10. In the second period, each person earns 20. Earnings in the third period are 0. Assume that utility in each period i is given by $U_i=\sqrt C_i$. Assume no restrictions on borrowing- only that lifetime earnings is equal to lifetime expenditure. Assume there is utility in bequeathing funds not spent. In particular, assume that if a person does not spend b during her lifetime, she gains half the utility he or she would have gained by spending b: $U_i=\frac12 \sqrt b$. How much would the representative consumer consume in each period if the consumer knows that he or she will live for only two periods in order to maximize utility?
I understand that to maximize anything, I need to take a first order condition of the function, however, I don't really understand the question itself. My understanding is that, utility in period one is $U_1=\sqrt C_1$ + $\frac12 \sqrt b$ where $b=10-C_1$ and then given that, we can just maximize it for the most optimal $C_1$ However, the professor did somehting in the class that is very different from how I'm thinking about it so I'm not really sure where to take it from there? Any help would me much appreciated.
I would have thought you had total utility $\sqrt{C_1}+\sqrt{C_2}+\sqrt{C_3}+\frac12\sqrt{b}$ where $C_1+C_2 +C_3+b=10+20+0$ as total earnings is equal to total expenditure and where $C_3=0$ as the consumer apparently knows the third period does not count.
As a shortcut, the concave nature of the square root function and symmetry in the equation means this total equation will be maximised when $C_1 = C_2$ and $b=30-C_1-C_2$ so you can just maximise $2\sqrt{C_1}+\frac12 \sqrt{30-2C_1}$ with $0 \le C_1 \le 15$. Simple calculus should help you find this is maximised when $C_1=\frac{40}{3}$ and so when $C_2=\frac{40}{3}$ and $b=\frac{10}{3}$ as well as $C_3=0$