A lottery ticket is available that costs $100 and has a probability p of yielding a prize of $10,000. A group of investors is thinking about buying a...

A lottery ticket is available that costs $100 and has a probability p of yielding a prize of $10,000. A group of investors is thinking about buying a ticket and sharing the proceeds if they win. The organizer offers the following deal: a person can buy a share " of the ticket (for $100"). If the ticket wins, he/she gets a share " of the prize. Assume each potential group member has income y in the absence of the lottery, and an expected utility function u(z), where u a) Derive an expression for the expected utility of a person who buys a share " of a ticket. b) Suppose a potential group member is risk neutral. Show she will join the group if p$0.01, but she will not join if p<0.01. c) Suppose a potential group member is risk averse, and p=0.02. Show that the person will want to join if " is relatively small. HINT: look at the first order condition at "=0. d) Suppose that potential group members have expected utility functions of the form where 0<d<1 Assuming that p=0.02, find the optimal share for an investor. Show that if d is higher, the investor will want a bigger share.