2. Consider an entrepreneur who has a business item. It requiresan initial investment of $10 million. The probability of thesuccess of the business is determined by the efforts that theentrepreneur puts in the business. For simplicity, let theprobability of success be ??=?? where 0????1 is her effort level.However, efforts generate a disutility (i.e., a negative utility).Her utility function is ??(??,??)=???25??2 where ?? is her earningsfrom the business in million dollars. She can earn $50 million ifthe business succeeds with probability ?? and earn nothing if thebusiness fails with probability 1???.

(a) Suppose that the entrepreneur has $10 million. She canimmediately start the business by investing her entire wealth intothe business. What is her expected utility?

(b) Under the condition of (a), which effort level should shechoose to maximize her expected utility?

(c) Suppose that the entrepreneur currently has nothing. Sheneeds to borrow $10 million from an “angel investor.” An angelinvestor is an investor who provides capital for a start-up, inexchange of high returns if the business succeeds. Let the returnthat the angel investor will receive be ?? (in million dollars).However, if the business fails, the angel investor will not requireanything. What is the entrepreneur’s expected utility?

(d) A rational investor will not invest unless the entrepreneurpromises enough returns. Say, the angel investor in (c) requiresthe half of the earnings, i.e., ??=25. Under the condition of (c),which effort level should the entrepreneur choose to maximize herexpected utility? (e) Compare the results in (b) and (d). What isthe implication of the difference in two results?