Question

Bayes' Rule Problem

Consider the following simplified view of a manager and worker. The worker selects either high (H) or low (L) effort. Given effort, the firm's profit is either x1 or
x2, with x1 < x2.

Assume that the probability of x1, given that the worker selected H, is f(x1 | H) = 1/10 and the probability of x1, given that the worker selected L, is f(x1 | L) = 4/5 . Note
that the f functions are just a formal way of writing conditional probability functions.

As the only possible realizations of x are x1 and x2, we have the probability of x2, given that the worker selected H, is f(x2 | H) = 9/10 and the probability of x2 , given that the worker selected L, is f(x2 | L) = 1/5 .

Assume that the manager's prior probability that the worker selected H is 1/2 .

Assume that, given the profit realization, the manager forms an updated belief about the worker's effort using Bayes' rule.

(a) Suppose the manager observes that x1 is realized. What is the manager's updated belief that the worker selected H?
(b) Suppose the manager observes that x2 is realized. What is the manager's updated belief that the worker selected H?

Answer 1