Demand for engineers is perfectly competitive. Employers cannot distinguish the ability of any single engineer from another. An employer's total profit increase by 12 (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 UH=W0.5T2, where W is wage (in thousand dollars) and T is number of tasks performed. Meanwhile, low-ability engineers have utility function UL=WT2.

(a) (10 points) Suppose the employers design the following contract: "if you perform 5 tasks, you will be paid 5 (thousand dollars) per task; if you perform X number of tasks (X > 5), you will be paid 12 (thousand dollars)per task". Find X_min, the minimal number of task X such that this contract achieves a separating equilibrium? (Leave your answeras it is. No rounding of X_minrequired)

(b) (5 points) Suppose the employers revises the contract as follows: "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 12 (thousand dollars)per task". Find X_max, the maximal number of task X such that this contract still achieves a separating equilibrium?