Question 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 In E {1,2}, with each agent in charge of an independent project. The principal puts in eEorts E1 and E2 into idea generation for each project In, while agent it puts in effort ck into idea generation for his own project. As before, this means that the principal produces an idea for project it. with probability E15, while agent 1:: produces an idea for project In: with probability ek. The realization of ideas is independent across players and projects. The principal's payo' function is sf + it]; %[E1 + EQJQ, and agent k's utility is a: %e%. As before, if the principal's idea is adopted for project k, then the principal's pays-E for that project is a: = B while agent k's payolf is :rrf,I = Eb, whereas if the agent's idea is adopted for project k, then the principal's payo' for that project is 7r: = orB while the agent's payoff is WE = 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 e'ort levels. 2. Ideas for each project are realized. 3. For each project, the principal makes 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 payo' in terms of the effort choices E1,E2,el, 62. Do the same for the expected payoff of each of the agents