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1. Suppose that workers have 4 levels of productivity in a labor market. High productivity is 10, medium productivity is 6, low productivity is 2, and trump productivity is negative 10. The distribution of workers is 1/8 high, 1/4 medium, 1/2 low, and 1/8 trump. The value of time in leisure is half of labor market productivity. Find the equilibrium wage. Is there adverse selection?
2. Let U=w-e where U is utility, w is wage, and e denotes effort. Profits are q-w, where q is output. Effort is zero or 10. If e=10, q=100 with probability 0.8 and q=0 with probability 0.2. If e=0, q=100 with probability 0.4 and q=0 with probability 0.6. The principle has an outside option of 20 and the agent has an outside option of 30. What is the equilibrium contract?
3. Suppose that high productivity is 10, medium productivity is 6, and low productivity is 2. Each type constitutes 1/3 of the labor market. The education signal can be zero or one. The cost of signaling is 20e divided by productivity. What is the equilibrium? Is it separating, semi-separating, or pooling?