Quest ion 1 In 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 E2 into idea generation for each project ls, while agent ls: puts in eEort ek into idea generation for his own project. As before, this means that the principal produces an idea for project l: with probability Ek, while agent In: produces an idea for project In with probability ek. The realization of ideas is independent across players and projects. The principal's payoE function is all: + WE %[E1 + Eg, and agent k's utility is 7r}; %e%. As before, if the principal's idea is adopted for project is, then the principalis payo for that project is 7r: 2 B while agent k's payoff is a}: 2 t, whereas if the agent's idea is adopted for project k, then the principalis payoff for that project is 7r: 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 effort 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 payoff in terms of the eEort choices E1,E2,e1, 82. Do the same for the expected payo' of each of the agents. b] Find the principal's and agent's bestresponse functions (Le. what effort choices does each player make, given the other players1 eEort choices?) c] It turns out that the principal will behave symmetrically, i.e. he will choose the same eort level E for both projects. Given this fact, rewrite the bestresponse functions for the principal and for each agent in terms of E instead of Eh. [Notice that this means that both agents will exert the same eort level as well.}