Questions: Signaling Games: Separating and Pooling Equilibria

Question 1

In a job market signaling model, all workers — regardless of ability — choose the same level of education. What type of equilibrium is this, and what does an employer learn by observing the education level?

• A. A separating equilibrium — the employer infers ability from how easily each worker obtained the degree
• B. A pooling equilibrium — the employer learns nothing about individual ability from observing education
• C. A separating equilibrium — different abilities choose the same signal but for different internal reasons
• D. A pooling equilibrium — the employer infers average productivity and pays accordingly, revealing type information

Answer: (B)

When all types choose the same action, the signal contains no information — it does not distinguish types. An employer observing a common education level cannot update their beliefs about any individual worker's ability beyond the prior. This is the defining feature of a pooling equilibrium: the signal is uninformative because it is not differentially costly. Note that option D is partially correct about wages (the employer pays expected average-productivity wages) but wrong that this reveals type — pooling means types are hidden, not revealed.

Question 2

In Spence's job market signaling model, education functions as a credible signal of worker ability even when education provides no direct productivity improvement. Why?

• A. Employers observe credentials and assume all degree holders are productive regardless of effort
• B. Education is differentially costly across ability types: high-ability workers bear lower costs, making it rational for them to signal while low-ability workers find mimicry too expensive
• C. High-ability workers intrinsically prefer education regardless of its cost
• D. Government regulations require credentialing for skilled positions, so education signals compliance rather than ability

Answer: (B)

The signaling logic depends entirely on cost differences, not on education's direct effect. If education were equally costly for all types, low-ability workers could mimic high-ability workers and the signal would be uninformative. The single-crossing property — that high-ability workers have lower marginal signaling cost — means there exists an education threshold high enough that low-ability workers rationally decline to reach it. High-ability workers signal not because they love education, but because for them the cost of the credential is less than the wage premium it commands.

Question 3

In a separating equilibrium, the employer's beliefs about a worker's type after observing their signal are the same as the employer's prior beliefs before any signal was sent.

• T. True
• F. False

Answer: False

In a separating equilibrium, different types choose different signals, so observing the signal fully reveals the worker's type. The employer's posterior beliefs after seeing the signal concentrate entirely on the type that would rationally choose that signal — a dramatic update from the prior. If prior beliefs were 50% high-ability and 50% low-ability, observing a separating signal of 'high education' shifts the posterior to nearly 100% high-ability. This is precisely the point of a separating equilibrium: information is fully transmitted.

Question 4

The Intuitive Criterion eliminates pooling equilibria that are sustained only by the belief that any worker who deviates from the pooling action must be the least productive type.

• T. True
• F. False

Answer: True

This is the core logic of the Intuitive Criterion. A pooling equilibrium can be sustained if the employer responds to any deviation with pessimistic beliefs (assuming the deviator is low-type). But if the deviation would never benefit a low-type worker regardless of employer response — it is 'equilibrium dominated' for low types — then it is unreasonable to attribute the deviation to them. The employer should infer the deviator is high-type, which makes the deviation profitable and unravels the pooling equilibrium. The criterion asks: are the supporting beliefs reasonable given who would plausibly benefit from deviating?

Question 5

What is the 'single-crossing property' in signaling games, and why is it a necessary condition for a separating equilibrium to exist?

Model answer: The single-crossing property means that the indifference curves of different types cross exactly once in the signal-wage space, with high types having flatter curves (lower marginal cost of the signal). This ensures that if a high-type worker is indifferent between a given signal level and no signal, a low-type worker strictly prefers no signal at that level. Without single-crossing, a low-type worker could always find it beneficial to mimic whatever signal a high-type sends, preventing any signal from being credibly informative. Single-crossing is what makes separation self-enforcing: each type's optimal choice differs because their cost-benefit tradeoffs differ fundamentally.

Single-crossing is the knife's edge that makes signaling work. If cost differences between types are too small (violating single-crossing), mimicry is always rational for low types, and only pooling equilibria exist. If cost differences are large enough (satisfying single-crossing), there exists a signal threshold that high types clear and low types rationally avoid.