4. Two agents are competing for a promotion. The winner gets S, the loser gets zero. The probability worker I wins the prize is given by: Prob(worker ] wins)=0.9 + 0.1 (E1-Ev), where agent I's effort is given by E and agent 2's by Er. Each agent's disutility of effort is given by E-/2. a) If both agents work equally hard, which one is more likely to win the promotion? b) Write down the formula for the expected utility of agent 1. Find his/her optimal effort as a function of the prize spread, S. c) Write down the formula for the expected utility of agent 2. Find his/her optimal effort as a function of the prize spread, S. Explain why, in this example, both agents work equally hard for any given prize spread. d) In the blank spreadsheet that is provided for this question, think of the various values of S as different possible prize spreads the firm is considering. In column 2, fill in the (common) effort level both agents will choose. e) Assume neither agent is paid to show up for work (a=0). Using the definition of utility (and the fact that both agents work equally hard) fill in both agents ' utility in columns 3 and 4. A) Suppose that, to get agent I to take the job, she must attain an expected utility level of 20. To get agent 2 to take the job, he must get an expected utility of 5. In columns 5 and 6, compute the level of a for each agent that just induces them to take the job (i.e. that gives them expected utilities of exactly 20 and 5 respectively). g) Suppose the firm chooses the levels of a given by part f. Let the expected value of each agent's output be given by 10E (where E is the agent's effort). Now compute the firm's output in column 7, and its profits in column 8. What is the profit-maximizing prize spread? Comment on the different levels of a for the two workers at this point