2.2. (15 points) In the recorded lecture we reviewed the following screening problem: suppose a software engineer can be either high- or low-ability. An employer's total profit increase by 10 (thousand dollars) when a high-ability engineer completes a task, and by 5 (thousand dollars) when a low- ability one completes a task. High-ability engineers have utility function Ug = W -0.472, where W is yearly wage (in thousand dollars) and T' is number of tasks performed. Meanwhile, low-ability engineers have utility function U, = W 0.572. That is, both types of engineers prefer higher wage and fewer tasks. However, high-ability engineers have a lower marginal cost of performing an additional task. The employers cannot observe the type of each engineer. (a) (10 points) Suppose the employers design the following "screening" contract: if you perform 10 tasks, you will be paid 5 (thousand dollars) per task; if you perform X number of tasks (X > 10), you will be paid 10 (thousand dollars) per task. What is the minimal number of task X such that this screening contract achieves a separating equilibrium? Round your answer to the nearest (reasonable) integer. (Hint: follow the recorded lecture and set up the IC condition for both the low-ability and the high-ability engineers. To find the minimal X, the IC condition for low-ability engineers needs to be binding) (b) (5 points) In part (a), what is the maximal number of task X such that this screening contract still achieves a separating equilibrium? Round your answer to the nearest (reasonable) integer. (Hint: if X is too high, the high-ability engineer would prefer to choose the low-wage, few-task combination designed for low-ability engineers. Like how you solve problem 2.1(a), this means you need to set the IC condition for high-ability engineers to be binding. You may find the following result useful: 0.474 - 10TH + 10 = 0.4[(TH - 12.5)2 131.25].) 2.2. (15 points) In the recorded lecture we reviewed the following screening problem: suppose a software engineer can be either high- or low-ability. An employer's total profit increase by 10 (thousand dollars) when a high-ability engineer completes a task, and by 5 (thousand dollars) when a low- ability one completes a task. High-ability engineers have utility function Ug = W -0.472, where W is yearly wage (in thousand dollars) and T' is number of tasks performed. Meanwhile, low-ability engineers have utility function U, = W 0.572. That is, both types of engineers prefer higher wage and fewer tasks. However, high-ability engineers have a lower marginal cost of performing an additional task. The employers cannot observe the type of each engineer. (a) (10 points) Suppose the employers design the following "screening" contract: if you perform 10 tasks, you will be paid 5 (thousand dollars) per task; if you perform X number of tasks (X > 10), you will be paid 10 (thousand dollars) per task. What is the minimal number of task X such that this screening contract achieves a separating equilibrium? Round your answer to the nearest (reasonable) integer. (Hint: follow the recorded lecture and set up the IC condition for both the low-ability and the high-ability engineers. To find the minimal X, the IC condition for low-ability engineers needs to be binding) (b) (5 points) In part (a), what is the maximal number of task X such that this screening contract still achieves a separating equilibrium? Round your answer to the nearest (reasonable) integer. (Hint: if X is too high, the high-ability engineer would prefer to choose the low-wage, few-task combination designed for low-ability engineers. Like how you solve problem 2.1(a), this means you need to set the IC condition for high-ability engineers to be binding. You may find the following result useful: 0.474 - 10TH + 10 = 0.4[(TH - 12.5)2 131.25].)
