Consider the following model in discrete time. There exists a continuum of agents. At the beginning of each period, each agent becomes a worker or an employer. The probability of becoming a worker and the probability of becoming an employer are both 0.5 for each agent. Then, workers and employers are paired with each other (i.e., they are matched to form pairs of a worker and an employer). All agents have matches. Each pair of a worker and an employer can produce an amount y (> 0) of goods in the period if they agree to work together.

If a pair of a worker and an employer fail to agree to work together, the worker can produce an amount b (∈ (0, y)) of goods in the period, while the employer can produce no goods. Each agent’s instantaneous utility in each period equals the amount of goods received as income in the period. Assume each agent is risk-neutral and discount her future income by a gross discount rate 1 + r (i.e., the time discount factor is 1/(1 + r)), where r > 0. For all questions, consider a stationary equilibrium, in which the life-time utilities for a worker and an employer in each period do not vary across periods.

(a) Write down a worker’s Bellman equation.

(b) Write down an employer’s Bellman equation.

(c) Suppose that each pair of a worker and an employer determine the wage through Nash bargaining with equal bargaining power for each. Compute the wage for each worker.