1. In this problem, we will explore how a firm can provide incentives for its executives using the firms stocks. Suppose WeRock is a public company in the market of online broadcasting. The CEO of WeRock has a personal cost of working C(H) = 50H2, where H is the number of days she works hard each quarter. Without any effort, the companys stock is expected to be worth $100, but for each day she works hard, there is a 0.5% probability WeRocks next earnings announcement will top theWall Street consensus estimate, in which case the company stock price will increase by $20. WeRock has 5,000,000 shares outstanding. Shareholders do not observe the CEOs effort but can observe the share price. Define a firms outside equity value as the share price times the number of shares held by outside investors. For example, if all WeRocks shares are held by outside investors and the CEO exerts no effort, WeRocks expected outside equity value is $100 5,000,000 = 500 million dollars. Suppose now WeRock offers 50,000 shares (known as inside equity) to the CEO. That is, the number of shares held by outside investors is 5,000,000 50,000 = 4.95 million

(a) How many days will the CEO work hard? (Hint: for every day of hard work, the expected marginal increase of stock price is 0.5% $20 per share. Use that and the fact that the CEO is offered 50,000 shares to calculate the CEOs expected marginal revenue for each day of hard work. Then, use MC = MR to find the answer, which should be between 0 and 365.)

(b) What is WeRocks expected stock price given your answer to part (a)? (Hint: if the CEO works hard for H days, WeRocks share price is $120 with probability 0.5% H, and $100 with the remaining probability.)

(c) What is WeRocks expected outside equity value (in million dollars) given your answer to part (b)? (Hint: the expected outside equity value is the expected share price times the numbers of shares held by outside investors)