Exercise 1 (Job Market Signaling – 20 points). Suppose that low-ability workers have productivity of D, while high-ability workers have productivity of A, where A > D. Firms can not tell low-ability workers from high-ability workers ex ante, but can observe a worker’s education level e. Firms know that half of all workers are low-ability, and half are high-ability.

Any worker can acquire as much education as she wishes, but getting e units of edu- cation costs a low ability worker B · e, where B > 1, and costs a high-ability worker e. Assume the labor market is competitive, so that a worker earns her expected productivity.

1. Suppose A = 15, B = 4, and D = 1. Does there exist a pooling equilibrium in which both high-and low-ability workers get 1 unit of education? If so, describe a wage function and the belief system that support this equilibrium outcome. If not, explain why.
2. Suppose A = 15, B = 4, and D = 1. Does there exist a pooling equilibrium in which both high- and low-ability workers get 3 units of education? If so, describe a wage function and a belief system that support this equilibrium outcome. If not, explain why.
3. Suppose A = 15, B = 4, and D = 1. Solve for a separating equilibrium which does not satisfy the intuitive criterion. Describe a wage function and a belief system that support this outcome in an equilibrium. Explain why this equilibrium fails the intuitive criterion.
4. For general A, B, D, solve for the unique equilibrium which does satisfy the intuitive criterion as a function of A, B, D. How does the level of education obtained by the high types vary in D in this equilibrium? What is the intuition?

Exercise 2 (Signaling). Consider a market of second-hand car. The car is either “high” or “low” quality. The seller knows the quality of his car. The buyer only knows that the car is of high quality with probability 30% and is of low quality with probability 70%. The two parties’ valuations for the car are:

1

low high buyer 50 100

seller 40 70

The seller attempts to signal quality by offering a warranty for the duration of y months. The expected cost of the warranty is Ch(y) := 5y if the quality is high, and Cl(y) := 10y if the quality is low.

1. Does the market game have “separating equilibria” in which ph = 100, pl = 50, yl = 0? If yes, find the range of yh that can be supported as separating equilibria.

2. Does the market game have “pooling equilibria” in which ph = pl = 75, yh = yl = yp? If yes, find the range of yp that can be supported as pooling equilibria.