題組內容

V.A father-son pair use the following income streams to finance their living in 2021 and 2022. The father earns $100 in 2021, but plans to retire early and so will only earn $20 in 2022. The son is undergoing a training program and earns only $20 in 2021, but will finish the training and earn $100 in 2022. By consuming $c1$ in 2021 and $c2$ in 2022, the father obtains utility $UF = \min\{c1, c2\}$, and the son obtains $US = \ln c1 + 0.8 \ln c2$. Both $c1$ and $c2$ are continuous.

(b)(6%) Suppose that they have no access to the financial market, but can make the following arrangement. The father gives the son an amount of $R in the red envelope in 2021, and the son gives the father some pocket money $P in 2022, where R and P ≥ 0 (a negative amount means the opposite flow of money). Describe all the Pareto optimal pairs (R, P), if any, that both would agree, and illustrate in a figure. (There is no need to compute the exact number.)