Question

Individals are of two di §erent abilities: high (H) and low (L). An individual knows his or her own

ability but an employer does not. Employers believe that both types of individuals are equally likely

in the population, and therefore has an initial (prior) belief that an individual is high ability with

probability 1. Before an individual looks for a job, he or she has an opportunity to go to college. 2

Assume that attending college does not change an individualiÌs ability; this problem shows that even in

this circumstance, college can perform a useful social function. The e §ort cost of attending college is

di §erent for individuals of di §erent abilities. In order to attend college, high ability individuals must

expend e §ort 1, and low ability individuals must expend e §ort 1, where a < 3 (this means that a 3a

low ability individual has a higher cost of attending college). Not attending college costs 0. After

observing an individualiÌs education (that is, whether he or she attended college) an employer makes

the individual a job o §er. The wage that the individual receives is equal to the employeriÌs belief that

the individual is high ability, given the level of schooling that the individual has chosen to acquire. For

example, if the employer believes that a college graduate is high ability with probability 2 he will o §er 3

a college graduate a wage of 2 . If he believes that an individal with no college education is high ability 3

with probability 1 , he will o §er an individual without college a wage of 1 : An individualiÌs payo § is 44

equal to his wage, minus the cost of education..

First consider a separating equilibrium. In a separating equilibrium, high ability individuals attend college, but low ability individuals do not.

1. (a) If the employer observes that an individual did not attend college, what is his belief about the individualiÌs ability? What wage will he o §er individuals who do not attend college?

2. (b) If the employer observes that an individual did attend college, what is his belief about the indi- vidualiÌs ability? What wage will he o §er individuals who attend college?

3. (c) Given the wages paid to college graduates and those who do not attend, will the high-ability individual chooose to attend college? (Hint: compare the payo § of attending college for the high ability graduate to the payo § of not attending college)

4. (d) Given the wages paid to college graduates and those who do not attend, for which values of a will the low-ability individual chooose to attend college? (Hint: compare the payo § of attending college for the low ability graduate to the payo § of not attending college)

5. (e) In a separating equilibrium, a low ability individual does not attend college. For which values of a is this expectation consistent with the low ability individualiÌs strategy?

   Now, consider a pooling equilibrium in which all individuals attend college; that is, both high and low ability individuals attend college. If some individual were to choose not to attend college, the employer would believe that he or she must be low ability, and would o §er a wage of 0.

6. (f) If the employer observes that an individual attended college, what is his belief about the individ- ualiÌs ability? What wage will he o §er to college graduates? (Hint: The employer expects that all individuals attend college. This means that attending college conveys no information about the individualiÌs true ability).

7. (g) Given the wages paid to college graduates (and the wage that would be paid to non-grads) would the high-ability individual choose to attend college? For which values of a would the low ability individual choose to attend college?

8. (h) In a pooling equilibrium, both high and low ability individuals attend college. For which values of a is this expectation consistent with the individualsiÌstrategies?

   Finally, consider a sem-separating equilibrium in which high-ability individuals attend college (for sure), but low ability students attend college with probability 0 < s < 1:

9. (i) If the employer sees that an individual attends college what is his belief that the individual is high-ability? What is the wage to college graduates? (Hint: Remember, all high ability individuals attend, but low ability individuals attend with probability s. Divide the probability of an individual being high ability and attending college by the overall probability of an individual attending college).

10. (j) If the employer sees that an individual did not attend college, what are his beliefs about the individualiÌs ability? (Hint: all high ability individuals attend college)

11. (k) The low-ability individual is willing to randomize between attending college and not attending college. Use this to Önd the probability that the low-ability individual attends in equilibrium.

    Bonus: Comment on the purpose of college illustrated in this example. How is this purpose a §ected as college becomes easier for the low-ability type (i.e. as a increases)?