2. ( 25 points) Elon is a millionaire, and also a risk averter who tries to maximise the expected value of \( \ln (w) \), where \( w \) is his wealth. Elon has \( £ 400,000 \) in safe assets and he also owns a rocket worth \( £ 300,000 \). Elon is planning to launch his rocket into space. After launching, the rocket will explode with probability 0.5 , and will land back safely on earth with probability 0.5 .
(a) (5 points) Calculate Elon's expected utility if he doesn't buy rocket insurance.
(b) (5 points) Calculate the certainty equivalent of the lottery he faces if he doesn't buy rocket insurance.
(c) (5 points) Suppose that Elon can buy \( £ \mathrm{~K} \) worth of insurance at a cost of 0.6 K . How much insurance will Elon buy?
(d) (5 points) Elon has a competitor Johnny, who is also a millonaire and also has a rocket he is planning to launch into space. Johnny has utility function for consumption in the two states of nature \( u\left(c_{E}, c_{N E}\right)=c_{E}^{\pi} c_{N E}^{1-\pi} \), where \( c_{E} \) is consumption if an explosion occurs and \( c_{N E} \) is consumption in case of no explosion. The probability that the rocket will explode is also \( \pi=0.5 \). Johnny's wealth is exactly the same income as Elon and has access to the same insurance price as Elon does. What much insurance will Johnny purchase? (Hint: The problem you are trying to solve is \( \max u\left(c_{E}, c_{N E}\right)=c_{E}^{\pi} c_{N E}^{1-\pi} \) subject to the budget constraint).
(e) (5 points) In which ways are Elon and Johnny's preferences similar? In which ways are Elon and Johnny's preferences different?
