Imagine a country where each person is either upper class, middle class or lower class,

and there is an equal proportion of each type. Regardless of one's class, being thought

to be upper class is worth 120; being thought to be middle class is worth 90; and

being thought to be lower class is worth 0. One way in which a person can try to

indicate their class to others is by choosing 'to signal'. This signal is a crude 'yes-or-no' choice; that is, each person can only choose whether to signal or not. The cost of

the signal is 200 for lower-class types; 10 for middle-class types; and 0 for upper-class

types. Whether or not a person chooses to signal, after he or she has made that

decision, he or she must take a compulsory 'high-society' test. The outcome of the

test is completely independent of whether or not a person has signaled. Upper-class

types always pass the test; lower-class types never pass; and middle-class types pass

with probability .5.

4.1 Suppose there were an equilibrium in which only the middle class signaled. In

such an equilibrium, what must society believe when it sees a person signaling

and passing the test; signaling and failing; not signaling and passing; or not

signaling and failing?

4.2 Show carefully that there is indeed an equilibrium in which only middle-class

types signal