As an economics student, or if learning for fun, you’ll often run into signaling problems. I’ll break down a straightforward labor-economics signaling model with education and two types of workers. We’ll work out:
- What wage firms will set without directly observing ability.
- The threshhold share of high-productive workers above which education is not worthwhile.
- How much a worker gains by signaling themselves.
You’ll see how this model can generate both a pooling and a separating equilibria. Leave any questions in the comments, and let’s get into it:
Let’s say two types of workers exist in our (theoretical) world. There are high productive workers, and low productive workers. Each type of worker produces a different marginal productivity, which is value of output they produce for the firm they work for.
High productivity workers have a marginal productivity of 40,000, and low productivity workers have a marginal productivity of 20,000. Workers cannot change their own productivity, and firms cannot directly observe the productivity of workers.
Instead, firms must rely on signals about the worker. In our problem, workers can acquire education for a cost of 15,000, and education can be a signal to the firm about the quality of the worker.
The first question we’ll ask is what salary a firm will offer (remember, they can’t distinguish between low and high productivity workers). To solve this, we’ll need to find the expected value of marginal productivity to the firm; firms will pay this exact amount in salary.
Since expected marginal productivity depends on the number of high quality versus low quality workers, we’ll assign a variable theta (θ) to denote the percent of workers who are high-productivity. We could use another variable to denote low-productivity workers, but we know these proportions must sum to one, so therefore we can denote the proportion of low-productivity workers as 1 – θ.
With that in mind, let’s set up our expected value equation. We’ll multiply θ and 1 – θ by the marginal productivity of those worker types:
(40,000)(θ) + 20,000(1 – θ)
In this way, if 10% workers are high productive (θ = .1), then the expected marginal productivity to the firm is 4,000 + 18,000 = 22,000 and so firms will offer this amount as a salary. Let’s expand the equation:
(40,000)(θ) + 20,000(1 — θ)
40,000θ + 20,000 – 20,000θ
20,000θ + 20,000
So, given some percentage of high-value workers in the marketplace, we can easily calculate the salary paid by the firm.
Our next question concerns the values of the proportion θ of highly productive workers for which workers are not incentivized to acquire education.
This is really asking what salary has to be offered to make getting an education and paying 15,000 worthwhile to a high-productive worker. To solve this, we’ll set the expected values from getting education and not getting education equal to one another, and then solve for theta.
Let’s start with the expected value (salary) if a worker goes to school. We know that if a worker signals herself as a high productive worker she gets a salary of 40,000, and so if she must pay 15,000 to get this, then she will get 40,000 – 15,000 = 25,000. For the expected salary without going to school, we already calculated that equation above—the 20,000θ — 20,000. Let’s set those equal to eachother and solve:
20,000θ + 20,000 = 40,000 – 15,000
20,000θ = 25,000 — 20,000
20,000θ = 5,000
θ = 5,000/20,000
θ = 1/4 = 25%
To visualize what this means, let’s plug .2 and .3 into the equation:
θ = .2
20,000(.2) + 20,000 = 40,000–15,000
24,000 < 25,000
θ = .3
20,000(.3) + 20,000 = 40,000–15,000
26,000 > 25,000
If theta is under 1/4 such as in our θ = .2 setup, then high productive workers will get an education because the 25,000 they get from signaling themselves as a high value worker even after paying 15,000 beats the 24,000 earned from not acquiring an education.
However, if theta is over 1/4 like in our θ = .3 equation, then the cost of school isn’t justified for high-productive workers.
(Keep in mind that since education doesn’t change worker productivity, low productivity workers won’t pay for education because they will still get paid 20,000(.2) + 20,000.)
For the next question, let’s assume that θ = .1: what can a high productive worker gain by signaling herself as such? To solve this, let’s simply plug our theta value into the equation and find the salary spread:
20,000(.1) + 20,000 = 40,000–15,000
22,000 < 25,000
So, by signaling as high productive, the high-productive workers will gain 3,000.
That’s the problem! Note that an equilibrium with signaling is a separating equilibrium, since it separates out the workers by type, and an equilibrium without signaling is a pooling equilibrium.
Else, hope you found this helpful! Leave any questions in the comments! Check out my other articles on economics here (useful for studying, or just learning).