Question

A worker can have either high or low ability, each with probability 1/2. A worker knows his ability, but a firm which wants to hire the worker does not. The worker, whether a high or a low ability type, can choose between additional education or not. Choosing additional education does not enlarge the workerTMs productivity but may serve as a signal to the firm: a high ability worker can choose education without additional costs, whereas for a low ability worker the cost of education equals e > 0. The firm chooses either a high or a low wage, having observed whether the worker took additional education or not. The payoff to the firm equals the productivity of the worker minus thewage. The payoff to the worker equals the wage minus the cost of education; if, however this payoff is lower than the worker TMs reservation utility, he chooses not to work at all and to receive his reservation utility, leaving the firm with 0 payoff. Denote the productivities of the high and low ability worker by pH and p L, respectively, and denote the high and low wages by w h and w l Finally, let r H and r L denote the reservation utilities of both worker types. (All these numbers are fixed.)

(a) Determine the extensive form of this game.
Choose p
H = 10, p
L = 8, w
h = 6, w
l = 4, r
H = 3, r
L = 2, e = 3.

(b) Compute the strategic form of this game, and determine the pure strategy Nash equi-
libria. Also compute the perfect Bayesian equilibrium or equilibria in pure strategies,

determine whether they are separating or pooling and whether they satisfy the in-tuitive criterion.

Answer 1

The task is to construct and analyze game theoretic models, including extensive and strategic forms, to identify pure strategy Nash equilibria and the perfect Bayesian equilibrium, determining if the outcomes are separating or pooling equilibria and meet the intuitive criterion.

Step-by-step explanation:

The student's question involves the construction of game theoretic models, namely the extensive form of the game and the strategic form, as well as finding the Nash equilibria and the perfect Bayesian equilibrium for a given situation involving labor economics. With specific values provided for the worker's productivity, wages, reservation utilities, and the cost of additional education, the task is to analyze the choices of high-ability and low-ability workers in the context of signaling to a firm. The game explores how workers decide whether to pursue additional education as a signal of their ability and how firms respond to this signal by offering different wages. The pure strategy Nash equilibria and perfect Bayesian equilibrium will reveal whether the game results in separating or pooling equilibria and if these equilibria meet the intuitive criterion, where only reasonable outcomes that can be predicted based on the players' payoffs and actions are considered.