Assignment 1- Assume that a high-productivity worker has cost function CH(q) = q2 and marginal cost function MCH(q) = 2q, and a low-productivity worker has cost function CL(q) = 3q2 and marginal cost function MCL(q) = 6q. The two types of workers are equally likely, and both types of workers have a best alternative option value of $0. The firm's net revenue is $60 for each unit of output produced.
1. Find the optimal contract for the two types of workers in the full information setting.
2. Now consider the hidden information setting. Write the low-productivity worker's participation constraint and the high-productivity worker's incentive compatibility constraint. Use the notations presented in the lecture.
3. Write the profit the firm receives from both types of workers.
4. Use the low-productivity worker's participation constraint to write πL as a function of qL.
5. Use the low-productivity worker's participation constraint and the high-productivity worker's incentive compatibility constraint to write πH as a function of qH and qL.
6. Write the firm's expected profit as a function of qH and qL.
7. Find the optimal output levels for the two types of workers in the hidden information setting.
8. Find the optimal piece rates for the two types of workers in the hidden information setting.
Assignment 2-
1. There are two types of workers. High-productivity workers can produce 25 units per hour, and low-productivity workers can only produce 20 units per hour. High-productivity workers can get jobs elsewhere with a wage of $18 per hour, and low-productivity workers can get jobs elsewhere with a wage of $15 per hour. What range of piece rates, if any, will lead workers to self-select and solve the adverse selection problem? Be sure to show your work.
2. a) A probationary contract specifies that a worker is paid $2,110 per month for the first m months. After m months the worker is either fired or gets a raise to $2,500 per month for (10-m) months. A low-productivity worker has a 90% chance of being fired at the end of his probationary period. If low-productivity workers can earn $2,200 per month somewhere else, how many months must the probationary period last to keep them from applying?
b) If the chance of being fired at the end of the probationary period falls to 80%, how does your answer to part a) change, if at all? Explain.
3. a) Workers come in two types, high-productivity and low-productivity. High-productivity workers generate net revenue of $100,000 per year to their employers, and low-productivity workers generate net revenue of $50,000 a year. Individuals work for 4 years. Before they start working, workers have the opportunity to take a licensing exam. To pass the exam, a high-productivity worker would have to study nights, and the implicit cost would be $20,000. A low-productivity workers would have to study nights and weekends, and take a special test-taking course, for a total cost of $40,000. Assuming that workers are paid $50,000 per year in an alternative industry, what range of salaries can a firm offer to licensed workers to guarantee that only high-productivity workers seek licenses? (Assume that future payments are not discounted.)
b) Is the above result an example of a separating equilibrium or a pooling equilibrium? Explain.