Human Capital and Signaling — Part 1
The Basic Model of Labor Market Signaling
Human capital is viewed as an input in the production process. The leading alternative is to view education purely as a signal. Consider the following simple model to illustrate the issues.
There are two types of workers, high ability and low ability. The fraction of high-ability workers in the population is λ. Workers know their own abilities, but employers do not observe this directly. High-ability workers always produce yH, while low-ability workers produce yL. In addition, workers can obtain an education. The cost of obtaining an education is cH for high-ability workers and cL for low-ability workers.
The crucial assumption is that cL > cH, that is, education is more costly for low-ability workers. This is often referred to as the “single-crossing” assumption since it makes sure that in the space of education and wages, the indifference curves of high and low types intersect only once. For future reference, let us denote the decision to obtain education by e = 1.
For simplicity, we assume that education does not increase the productivity of either type of worker. Once workers obtain their education, there is competition among a large number of risk-neutral firms, so workers will be paid their expected productivity. More specifically, the timing of events is as follows:
- Each worker finds out their ability.
2. Each worker chooses education, e = 0 or e = 1.
3. A large number of firms observe the education decision of each worker (but not their ability) and compete a la Bertrand to hire these workers.
Clearly, this environment corresponds to a dynamic game of incomplete information, since individuals know their ability, but firms do not. In natural equilibrium concept, in this case, is the Perfect Bayesian Equilibrium. Recall that a Perfect Bayesian Equilibrium consists of a strategy profile σ (designating a strategy for each player) and a brief profile μ (designating the beliefs of each player at each information set) such that σ is sequentially rational for each player given μ (so that each player plays the best response in each information set given their beliefs) and μ is derived from σ using Bayes’s rule whenever possible. While Perfect Bayesian Equilibria are straightforward to characterize and often reasonable, in incomplete information games where players with private information move before those without this information, there may also exist Perfect Bayesian Equilibria with certain undesirable characteristics. We may therefore wish to strengthen this notion of equilibrium (see below).
In general, there can be two types of equilibria in this game.
(1) Separating, where high and low ability workers choose different levels of schooling, and as a result, in equilibrium, employers can infer worker ability from education (which is a straightforward application of Bayesian updating). (2) Pooling, where high and low ability workers choose the same level of education.
In addition, there can be semi-separating equilibria, where some education levels are chosen by more than one type.
A separating equilibrium
Let us start by characterizing a possible separating equilibrium, which illustrates how education can be valued, even though it has no directly productive role.
Suppose that we have
(2.1) yH − cH > yL > yH − cL
This is clearly possible since cH < cL. Then the following is an equilibrium: all high ability workers obtain an education, and all low ability workers choose no education. Wages (conditional on education) are:
w (e = 1) = yH and w (e = 0) = yL
Notice that these wages are conditioned on education, and not directly on ability since the ability is not observed by employers. Let us now check that all parties are playing the best responses. First, consider firms. Given the strategies of workers (to obtain education for high ability and not to obtain education for low ability), a worker with education has productivity yH while a worker with no education has productivity yL. So no firm can change its behavior and increase its profits.
What about workers? If a high ability worker deviates to no education, he will obtain w (e = 0) = yL, whereas he’s currently getting w (e = 1)−cH = yH −cH > yL. If a low-ability worker deviates from obtaining education, the market will perceive him as a high-ability worker, and pay him the higher wage w (e = 1) = yH. But from (2.1), we have that yH − cL < yL, so this deviation is not profitable for a low-ability worker, proving that the separating allocation is indeed an equilibrium.
In this equilibrium, education is valued simply because it is a signal about ability. Education can be a signal about ability because of the single-crossing property. This can be easily verified by considering the case in which cL ≤ cH . Then we could never have the condition (2.1) hold, so it would not be possible to convince high-ability workers to obtain an education while deterring low-ability workers from doing so.
Notice also that if the game was one of perfect information, that is, the worker type was publicly observed, there could never be education investments here. This is an extreme result, due to the assumption that education has no productivity benefits. But it illustrates the forces at work.
Pooling equilibria in signaling games
However, the separating equilibrium is not the only one. Consider the following allocation: both low and high ability workers do not obtain an education, and the wage structure is
w (e = 1) = (1 − λ) yL + λyH and w (e = 0) = (1 − λ) yL + λyH
It is straightforward to check that no worker has any incentive to obtain education (given that education is costly, and there are no rewards to obtaining it). Since all workers choose no education, the expected productivity of a worker with no education is (1 − λ) yL+λyH, so firms are playing best responses. (In Nash Equilibrium and Perfect Bayesian Equilibrium, what they do in response to a deviation by a worker who obtains education is not important, since this does not happen along the equilibrium path).
What is happening here is that the market does not view education as a good signal, so a worker who “deviates” and obtains education is viewed as an average ability worker, not as a high-ability worker. What we have just described is a Perfect Bayesian Equilibrium. But is it reasonable? The answer is no. This equilibrium is being supported by the belief that the worker who gets an education is no better than a worker who does not. But education is more costly for low-ability workers, so they should be less likely to deviate from obtaining an education. There are many refinements in game theory which basically try to restrict beliefs in information sets that are not reached along the equilibrium path, ensuring that “unreasonable” beliefs, such as those that think a deviation to obtaining an education is more likely from a low ability worker, are ruled out. Perhaps the simplest is The Intuitive Criterion introduced by Cho and Kreps.
The underlying idea is as follows. If there exists a type who will never benefit from taking a particular deviation, then the uninformed parties (here the firms) should deduce that this deviation is very unlikely to come from this type. This falls within the category of “forward induction” where rather than solving the game simply backward, we think about what type of inferences will others derive from a deviation
To illustrate the main idea, let us simplify the discussion by slightly strengthening condition (2.1) to
(2.2) yH − cH > (1 − λ) yL + λyH and yL > yH − cL.
Now take the pooling equilibrium above. Consider a deviation to e = 1. There is no circumstance under which the low type would benefit from this deviation, since by assumption (2.2) we have yL > yH − cL, and the most a worker could ever get is yH , and the low ability worker is now getting (1 − λ) yL+λyH . Therefore, firms can deduce that the deviation to e = 1 must be coming from the high type, and offer him a wage of yH . Then (2.2) also ensures that this deviation is profitable for the high types, breaking the pooling equilibrium.
The reason why this refinement is referred to as “The Intuitive Criterion” is that it can be supported by a relatively intuitive “speech” by the deviator along the following lines: “you have to deduce that I must be the high type deviating to e = 1 since low types would never ever consider such a deviation, whereas I would find it profitable if I could convince you that I am indeed the high type).” You should bear in mind that this speech is used simply as a loose and intuitive description of the reasoning underlying this equilibrium refinement. In practice, there are no such speeches, because the possibility of making such speeches has not been modeled as part of the game. Nevertheless, this heuristic device gives the basic idea.
The overall conclusion is that as long as the separating condition is satisfied, we expect the equilibrium of this economy to involve a separating allocation, where education is valued as a signal.
Generalizations
It is straightforward to generalize this equilibrium concept to a situation in which education has a productive role as well as a signaling role. Then the story would be one where education is valued for more than its productive effect because it is also associated with higher ability.
To be cond….. in part 2
