Question 3: A manufacturing company is looking to hire an agent to manage a new project. The agent can exert a high effort (𝑒𝑒𝐻𝐻)or low effort (𝑒𝑒𝐿𝐿). The high effort increases the probability of project success, but it is more costly for the agent. The company wants to design a contract that ensures the agent will choose to exert high effort and will also be willing to accept the job. If the agent exerts high effort (𝑒𝑒𝐻𝐻), the probability of success is 0.8. If the agent exerts low effort ( 𝑒𝑒𝐿𝐿), the probability of success is 0.4. The agent's cost of exerting high effort is 𝐶𝐶𝐻𝐻= 10, and the cost of exerting low effort is 𝐶𝐶𝐿𝐿 =2. The company's payoff is 100 if the project is successful and 0 if it is not. The agent's utility is given by: 𝑈𝑈 = 𝑊𝑊−𝐶𝐶, i.e. their wage 𝑊𝑊 minus their effort cost. The agent has an outside option with a utility of 5. (a) Design a contract (𝑊𝑊𝑆𝑆, 𝑊𝑊𝐹𝐹) where 𝑊𝑊𝑆𝑆 is the wage if the project is successful and 𝑊𝑊𝐹𝐹 is the wage if the project fails. Write down the incentive compatibility and participation condition that ensures the agent will accept the contract and choose to exert high effort. (8 Marks) (b) Solve the conditions from (a) and find the optimal contract with two sets of payments such that the agent will accept the contract and choose to exert high effort. (8 Marks) (c) What is the company’s expected profits under such contract? (2 Marks)