Imperfect Information Problem Set

These problems demonstrate the basic issues of signaling, adverse selection, search, and matching. Fully-developed models provide insights into human resource policies and also provide insights into the problems of testing theories (as well as providing clues to solutions).

I) The Acquiring an Employee Problem

A worker comes to your firm's door. You have one opportunity to make a single wage offer to this worker. If your offer is greater than his earnings at home, he accepts the offer. If your offer is less than his earnings at home, he works at home.

You do not know the workers' earnings at home or his productivity with you, but you know that his earnings at home are uniformly distributed between 0 and $10.00 per hour (that is, it is equally likely to take on each penny value between $0.00 and $10.00). Whatever his earnings are at home is, he is 1.5 times as productive with you. For example, a worker who earns $3 at home will accept any offer over $3, and will produce $4.50 worth of output per hour at this firm. Thus, your offer should be between 0 (that is, make no offer) and $15.00, inclusive.

1. If you are interested in maximizing expected profits, what offer should you make? Why?

II) A Matching Model

Each period new graduates go to work at one of many companies. All workers and all companies initially appear indistinguishable. In fact, half of all new hires will flourish at any given company. At the same time, match quality (and, thus, productivity) at one firm is uncorrelated with match quality and productivity at the next employer. No worker or company can see match quality until the first period is over. Good matches have productivity = 10, while bad matches have productivity = 4. People live a long time and we assume no discounting of the future.

Assume that there are no barriers to entry, so that firms earn zero profits in the long run.

1. In equilibrium, what wage will firms pay new hires? (Hint: If it were below expected productivity, new firms would enter.)

2. After the end of the first period, what wage should the firm offer bad matches? (Hint: What is their marginal productivity?) Should such workers accept that offer or do something else? If not accepting, what should they do? (Hint: What is their expected wage and marginal productivity here and at another firm?)

Assume that good matches who remain at the firm receive their marginal product of 10. (In fact, a bargaining problem exists that we do not model here.)

3. What are the returns to initial tenure estimated in the cross-section (that is, the average of wages of employees with tenure of 1 / wages of employees with tenure of 2)? Does the observed result support human capital theory's predictions that wages rise when people learn more skills?

4. What are two examples of pieces of evidence that worker-employee matching on idiosyncratic tastes, skills, or job characteristics is sometimes important?

5. What does the theory of job matching predict about the relation between job satisfaction and tenure? Job satisfaction and age?

Note that what economists call the theory of “job matching” psychologists refer to as “person-job fit.” Also, note that the returns to tenure in a cross-sectional model may higher than longitudinal estimates if some firms pay persistently high wages for efficiency wage or rentsharing reasons.

III) Signaling

Over their lifetime, high-skill people produce $50 and low-skill people produce $20. Workers know their productivity, but workers appear identical to employers initially and even after working. Thirty percent of the labor force is highly skilled.

1. What wage will workers earn over their lifetime in a competitive market?

An entrepreneur invents "school." Schooling is costly for high-skilled workers, who must pay $10 per year to attend. Schooling is even more costly for low-skilled workers, who must pay $10 per year to attend. Unlike high-ability people, low-ability people also suffer an additional $15 in disutility from schooling.

Assume that after schools are invented, companies offer a lifetime wage of $39 to any worker who has attended at least one year of schooling and $20 to people with no schooling.

2. What is the private value to the high-skilled employee of attending school?

3. What is the private value to the low-skilled employee of attending school?

4. Who attends school, and why? What type of applicants apply for high-wage jobs?

5. What is the value to society (that is, change in productivity minus costs of schooling)?

6. What are two examples of behaviors that students undertake to signal their quality to universities?

7. What are two examples of behaviors that potential employees undertake to signal their quality to employers?

8. What are two examples of behaviors that organizations undertake to signal their quality to customers or shareholders?

The key results in a signaling model are:

Signals must be lower-cost investments (on average) for the group employers want to attract.
Signaling in this example has no social value. GDP declines as high-skill people invest in unproductive signals. In more general models, schooling also raises productivity or may help firms find workers well suited for a job. Nevertheless, in general some people will over-invest in the signaling activity.

Note that what economists call the theory of “signaling” sociologists refer to as “credentialism.” Much of the new institutionalist theory of sociology builds related concepts of legitimacy and status. Michael Spence, before teaching me microeconomics and becoming Dean at Stanford, wrote down this formal model. He recently won the Nobel prize for this contribution.

IV) Employer Learning about Employee Skill

Each period new graduates go to work at one of many companies. All workers initially appear indistinguishable. In fact, half of all new hires have productivity = 10 while half have productivity = 4. This productivity will be the same at all other firms. Only the employer sees each employee’s productivity.

What wage will workers earn in period 1?
What is the high-productivity workers’ market wage if the employer gives them a raise in period 2?
What is the high-productivity workers’ market wage if the employer does not give them a raise?
Is there any evidence that employers fear good treatment of an employee may raise their outside option?

In models where people have attributes that are hard to observe, organizations learn over time. What are two human resource policies that could accelerate such learning? What is a policy (other than promotions) that might use that learning?

Consider characteristics that are useful or costly to many firms, known to the employee, and not observed by the employee. What are two human resource policies that induce positive or adverse self-selection among applicants?

V) High search costs

Each period job searchers get one job offer. For half of applicants the offer is at a firm that is a good match, and the offer is $10; otherwise the offer is $4. At the same time, match quality (and, thus, productivity) at one firm is uncorrelated with match quality and productivity at the next employer. People live a long time and we assume no discounting of the future.

1. What is the first-period average wage?

2. What is the second-period average wage?

3. What are the returns to tenure estimated longitudinally (looking only at stayers)?

4. What are the returns to tenure estimated in the cross section = second-period average wage / first period average wage.

5. Why do the two estimate differ?

Overall notes: In models I and II the worker knows her productivity, but not the firm. In models II and V, neither knows the workers’ match-specific productivity, although the timing of learning differs. In model IV only this firm learns the workers’ productivity. In other models, other firms also learn workers’ productivity. Note how subtle changes in the information structure greatly affect the results. This flexibility means (i) many articles are published, (ii) any pattern of results can match at least one model, and (iii) it is important to create integrative models that can match many facts.