# Equilibria in a Signaling Model of Education

#### The Model

Suppose that there are two types of workers: type H and type L. Type H workers are more productive than type L workers and type H workers find it less costly to obtain a given level of education e. Specifically, the marginal products of labor for each type of worker is: $$m_H(e) = m_H e$$ $$m_L(e) = m_L e,$$ where $$m_L < m_H$$ are constants. And the costs to each type of obtaining a given level of education $$e$$ is: $$c_H(e) = c_H e^2$$ $$c_L(e) = c_L e^2,$$ where $$c_L > c_H$$. A worker's type is only observable by the worker but the amount of education a worker obtains is observed by a potential employer. So while a worker's wage cannot depend directly on that worker's type, the wage can be a function of that worker's chosen level of education. We consider two classes of equilibria: Pooling and separating.

#### Pooling Equilibria

A pooling equilibrium in ths model is a threshold value of education $$\bar{e}$$ such that:

1. If a worker obtains $$e<\bar{e}$$ units of education, then employers believe that worker is type L and pay that worker a wage $$w(e) = m_Le$$.
2. If a worker obtains $$e\geqslant\bar{e}$$ units of education, then employers believe that worker is type H with probability $$p$$ and pay that worker a wage $$w(\bar{e}) = \bar{m} \bar{e} = \left( pm_H + (1-p)m_L\right)\bar{e}$$.

#### Separating Equilibria

A separating equilibrium in ths model is a threshold value of education $$\bar{e}$$ such that:

1. If a worker obtains $$e<\bar{e}$$ units of education, then employers believe that worker is type L and pay that worker a wage $$w(e) = m_Le$$.
2. If a worker obtains $$e\geqslant\bar{e}$$ units of education, then employers believe that worker is type H and pay that worker a wage $$w(e) = m_H e$$.

#### Notes on the simulations

For the animated figures above, I used the following parameter values: $$m_L = 4$$, $$m_H = 8$$, $$c_L = 0.5$$, $$c_H = 0.25$$, and $$p = 0.25$$. For the pooling equilibrium animation, I computed equilibria for $$e \in[0,19]$$. And for the separating equilibrium animation, I computed equilibria for $$e \in[0,34]$$. The Python code that I used to create these animations is available in this Github repository.