Question

Recall the job-market signaling game in Example 8.9 a. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education \((N E)\) and where the firm offers an uneducated worker a job. Be sure to specify beliefs as well as strategies. b. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education \((N E)\) and where the firm does not offer an uneducated worker a job. What is the lowest posterior belief that the worker is low-skilled conditional on obtaining an education consistent with this pooling equilibrium? Why is it more natural to think that a low-skilled worker would never deviate to \(E\) and thus an educated worker must be high-skilled? Cho and Kreps's intuitive criterion is one of a series of complicated refinements of perfect Bayesian equilibrium that rule out equilibria based on unreasonable posterior beliefs as identified in this part; see I. K. Cho and D. M. Kreps, "Signalling Games and Stable Equilibria," Quarterly Journal of Economics \(102(1987): 179-221\)

Short answer

Answer: The conditions for pooling equilibria are as follows: 1. For pooling equilibrium with No Education and Job Offer: a. The firm's belief about the proportion of high-skilled workers (π) in the population must satisfy: πV_H + (1-π)V_L > C b. Both high-skilled and low-skilled workers prefer not obtaining an education to obtaining an education, i.e., costs are lower for not obtaining education for both types of workers. 2. For pooling equilibrium with No Education and No Job Offer: a. The firm believes the proportion of high-skilled workers (π) is too low to justify offering a job to an uneducated worker, i.e., πV_H + (1-π)V_L ≤ C b. Both types of workers prefer not obtaining an education to obtaining an education even if they know they will not be offered a job. The cost of obtaining education is too high for low-skilled workers. c. The lowest posterior belief that the worker is low-skilled conditional on obtaining an education (π_E) is consistent with this pooling equilibrium if it upholds the following condition for the high-skilled worker: V_H - C_H(E) > π_E V_H + (1-π_E)V_L

Step-by-step solution

Part a: Pooling Equilibrium with No Education and Job Offer

Since both types of workers are not obtaining an education, the firm is unable to differentiate between them. Therefore, the firm's belief about the proportion of high-skilled workers (denote it as \(\pi\)) in the population is relevant. The firm offers a job to an uneducated worker if its expected payoff is greater than the cost of hiring a worker, which can be represented as: $$ \pi V_H + (1-\pi)V_L > C $$ Where \(V_H\) is the value of a high-skilled worker, \(V_L\) is the value of a low-skilled worker, and \(C\) is the cost of hiring. The pooling equilibrium occurs if both types of workers prefer not obtaining an education to obtaining an education. For high-skilled workers: $$ V_H - C_H(NE) \geq V_H - C_H(E) $$ For low-skilled workers: $$ V_L - C_L(NE) \geq V_L - C_L(E) $$ In both cases, the costs must be lower for not obtaining education for the equilibrium to hold.

Part b: Pooling Equilibrium with No Education and No Job Offer

In this case, the firm believes the proportion of high-skilled workers is too low to justify offering a job to an uneducated worker. The condition for this belief is: $$ \pi V_H + (1-\pi)V_L \leq C $$ The pooling equilibrium occurs if both types of workers prefer not obtaining an education to obtaining an education, even if they know that they will not be offered a job. The condition for a low-skilled worker to not deviate to E is: $$ 0 \geq V_L - C_L(E) $$ which means that the cost of obtaining education is too high. However, the lowest posterior belief that the worker is low-skilled conditional on obtaining an education (denote it as \(\pi_E\)) is consistent with this pooling equilibrium if it upholds the following condition for the high-skilled worker (since by obtaining education, the worker must believe it is possible to be offered a job): $$ V_H - C_H(E) > \pi_E V_H + (1-\pi_E)V_L $$ It is more natural to think that a low-skilled worker would never deviate to E and thus an educated worker must be high-skilled since the cost of obtaining education for low-skilled workers is generally higher compared to high-skilled workers, and the potential benefit of deviating to E is lower for a low-skilled worker (as they have lower value).

Key concepts

Pooling Equilibrium

In the context of job-market signaling, a pooling equilibrium occurs when different types of workers choose the same level of education, making it impossible for employers to distinguish between them based on educational attainment alone. In the scenario described, both high-skilled and low-skilled workers decide not to pursue further education. As a result, pooling happens because the firm cannot tell who is truly high-skilled based on education.

In a pooling equilibrium where both uneducated workers are considered for jobs, the firm's decision depends on its beliefs about the proportion of high-skilled workers in the labor pool. The firm offers a job if the expected value of hiring an uneducated worker exceeds the cost of hiring. In mathematical terms, this is when:

• \[ \pi V_H + (1-\pi)V_L > C \]

where \(V_H\) represents the value of a high-skilled worker, \(V_L\) for low-skilled, and \(C\) is the hiring cost.An opposite result occurs when no job offers are made due to the belief that there are too few high-skilled workers to justify employment. Here, the belief condition reverses:

• \[ \pi V_H + (1-\pi)V_L \leq C \]

Posterior Belief

Posterior belief in economic models refers to the revised probability that a firm assigns to a worker being high-skilled after observing their decision of whether or not to obtain an education. In situations where signaling is used, firms update their beliefs about a worker's skill level based on observed traits such as educational attainment.

For instance, if a worker opts out of education, firms may infer certain characteristics about them, adjusting their beliefs. When both high-skilled and low-skilled workers choose not to be educated, posterior beliefs are particularly critical because these beliefs will affect firms' decisions whether to hire and at what wage.

The posterior belief can also affect wage differentials. If the firm's posterior view is that uneducated workers are mostly low-skilled, this can potentially justify offering lower wages due to assumed lower productivity. Conversely, if workers obtaining education are generally assumed to be high-skilled, updating beliefs toward higher skill levels can lead to higher salary offers.

Perfect Bayesian Equilibrium

Perfect Bayesian Equilibrium (PBE) is a refinement of Nash Equilibrium used in games with incomplete information, which sets out both the strategies and beliefs of players. In job-market signaling, PBE helps firms decide their hiring strategies based on observed characteristics and inferred abilities of workers.

The concept of PBE necessitates firms to form beliefs about the types of workers based on their educational choices, then select hiring policies that maximize expected returns based on these beliefs. Workers, on their part, choose education levels knowing what these beliefs may trigger in terms of job offers.
Here’s how it works:

• Employers form beliefs about a worker's type upon seeing an education choice.
• Workers choose their education strategy considering how it influences employer beliefs and subsequent job offers.
• The combined assessment of both parties produces an equilibrium that balances beliefs and strategies optimally given the available information and constraints.

In the economy described, PBE ensures that a low-skilled worker's incentive not to obtain costly education aligns with the firm's reduced posterior belief that such a worker is high-skilled without education.

High-skilled vs Low-skilled Workers

In the labor market, the distinction between high-skilled and low-skilled workers is critical due to the differentiated impact they have on productivity and wage offers. High-skilled workers generally possess characteristics such as advanced education or experience, leading to higher contributions to a company compared with low-skilled workers.

This distinction is crucial in job-market signaling games. Employers seek to correctly identify high-skilled workers to maximize productivity — for instance, by offering better positions or higher pay. Conversely, they may offer lower wages or different contracts to perceived low-skilled workers.
Employers thus try to distinguish between these two groups using signals such as education. However, not all education level differences perfectly indicate skill levels, especially in pooling equilibriums where both worker types may choose similar education strategies. This sometimes leads to:

• Misallocation of employment resources.
• Wage disparities not based on actual skill differences.

Cost of Education

The cost of education plays a pivotal role in job-market signaling, particularly in determining whether workers choose to enhance their skills through further education. This cost includes not only direct fees paid to educational institutions but also opportunity costs like foregone earnings during study time.

For high-skilled workers, the cost of education may be justified by higher future earnings and job opportunities, making it a worthwhile investment. But for low-skilled workers, the potential returns may not sufficiently outweigh the costs, discouraging them from pursuing further education.
In signaling games, differences in education costs emphasize why high-skilled workers are more likely to invest in education. Firms recognize this, adjusting their beliefs and offers accordingly. The decision to invest in education reflects not only personal financial calculations but also the perceived market position following educational attainment. This realization leads to selective educational strategies by workers and strategic decision-making by employers based on the education levels observed.