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A degree can raise your wage even if nothing you studied is used on the job. That sounds like a waste, and in one famous model it is exactly a waste, on purpose. The diploma is worth paying for because it is harder for the wrong person to earn.

Signaling games

When you know something an employer cannot verify, saying “I am talented” is cheap and unbelievable. What works is doing something costly that a less talented person would not find worthwhile. Spence showed education can play that role, sorting workers by type without teaching a thing.

From earlier lessons

Two quick ones before we start.

A perfect Bayesian equilibrium pairs strategies with...

A signal is believed only when the belief it creates...

Predict, then play

Two worker types: Low productivity (worth 1) and High (worth 2), equally likely. Education costs $e/\theta$ , so a year of school is cheaper for the High type. Firms pay the productivity they infer. You are the High worker choosing how much education to get. How little is enough to prove you are not the Low type?

Why did that happen?

Take a guess, then reveal the reason

The signal works because it is differentially costly. Education does not make anyone more productive here, but it hurts the Low worker more per year than the High worker. So there is a level of schooling the High type will pay for and the Low type will not, and a firm seeing that level can safely infer High. The diploma is not proof of skill; it is proof that faking it was not worth the trouble.

The math

Low productivity is $\theta_L = 1$ , High is $\theta_H = 2$ , each with probability $\tfrac12$ . Education $e$ adds nothing to productivity but costs $c(e,\theta) = e/\theta$ . In a separating equilibrium the High type picks some $e^*$ and the Low type picks $0$ ; firms pay $w = 2$ for $e \ge e^*$ and $w = 1$ below.

Two incentive constraints must hold. The Low type must not want to mimic:

\underbrace{2 - e^*}_{\text{mimic}} \le \underbrace{1}_{\text{stay at }0} \quad\Longrightarrow\quad e^* \ge 1.

The High type must prefer to separate:

\underbrace{2 - \tfrac{e^*}{2}}_{\text{separate}} \ge \underbrace{1}_{\text{pool down}} \quad\Longrightarrow\quad e^* \le 2.

Any $e^*$ in $[1, 2]$ is a separating equilibrium, so there is a continuum. The least-cost one is $e^* = 1$ : wages are $w(0) = 1$ , $w(1) = 2$ , and the High worker nets $2 - \tfrac12 = 1.5$ .

Definition, single crossing and refinement

Single crossing means the two types rank increases in the signal differently: education is less costly at the margin for the High type, so a higher education level can attract High workers without attracting Low workers. That cost difference creates the interval of separating signals. The intuitive criterion of Cho and Kreps can then rule out equilibria supported by implausible off-path beliefs and select the least-cost separating signal in this example.

Work it out

Find the least-cost separating education

Write the two constraints. The Low type, if it buys education $e^*$ and is paid the High wage, nets $2 - e^*/1$ . It will not mimic as long as that is no better than staying at zero for the Low wage: $2 - e^* \le 1$ , so $e^* \ge 1$ .

The High type buys $e^*$ at cost $e^*/2$ and is paid $2$ , netting $2 - e^*/2$ . It prefers this to skipping school for the Low wage of $1$ as long as $2 - e^*/2 \ge 1$ , so $e^* \le 2$ .

Both hold on $[1, 2]$ . The cheapest signal that still separates is $e^* = 1$ : the Low type is exactly indifferent about mimicking. This separating equilibrium therefore uses the stated tie-break in favour of no education. The High type nets $2 - 0.5 = 1.5$ . Signalling any more than $e^* = 1$ only burns money.

Key idea

A costly action credibly reveals private information when it is cheaper for one type than another. Education can sort workers by talent without adding any, and the efficient signal is the smallest one the weaker type would not copy.

Check yourself

Check yourselfapply

Why is education a credible signal of productivity in this model, even though it teaches nothing job-relevant?

Check yourselfanalyse

There is a whole range of separating education levels, from 1 to 2. Why single out e = 1?

Check yourselfbrings back: perfect Bayesian equilibrium

What makes a separating profile here a perfect Bayesian equilibrium?

Sources

Spence, M. (1973). Job market signaling. Quarterly Journal of Economics, 87(3), 355-374.
Cho, I.-K., and Kreps, D. M. (1987). Signaling games and stable equilibria. Quarterly Journal of Economics, 102(2), 179-221.