Quest ion 1 [n this question, we'll again use the model of authority to consider the implications of increasing span of control, in the sense of the principal being in charge of multiple projects, Suppose that there is one principal and two identical agents ls: E {1,2}, with each agent in charge of an independent project. The principal puts in eEorts E1 and E; into idea generation for each project it, while agent l: puts in effort ek into idea generation for his own project. As before, this means that the principal produces an idea for project l: with probability E11,, while agent k produces an idea for project I: with probability 31;. The realization of ideas is independent across players and projects. The principal's payo function is of + WE %{E1 + E212, and agent k's utility is arr}: %e%. As before, if the principal's idea is adopted for project k, then the principal's payoE for that project is in": = B while agent k's payoE is a: = Eh, whereas if the agent's idea is adopted for project k, then the principal's payed for that project is an": 2 QB while the agent's payoff is w: = h. To simplify the analysis, we consider only the case of centralization, i.e. the principal has decisionmaking authority over both projects. The timing of the game is as follows: 1. The principal and agent choose eEort levels. 2. Ideas for each project are realized. 3. For each project, the principal malies the decision over which idea [if there are two successful ideas] to adopt. 4. Payos are realized. We'll go through the problem stepbystep. a] Write down the principal's expected total payoff in terms of the effort choices E1,E2, 31, 2. Do the same for the expected payoff of each of the agents. b] Find the principal's and agent's bestresponse functions [i-e. what eEort choices does each player malce, given the other players' eEort choices?)