Spence's Signaling Model: Separating Equilibrium Conditions

Separating Equilibrium: Beliefs and Wages

In a separating equilibrium, each type of worker chooses a distinct level of education. Let's denote the high-type's choice as $e_H$ and the low-type's as $e_L$, where $e_H\ne e_L$.

On the equilibrium path, firms observe these education levels and update their beliefs perfectly.

• If a firm sees education level $e_L$, it believes with 100% certainty that the worker is low-ability.
• If a firm sees education level $e_H$, it believes with 100% certainty that the worker is high-ability.

Given that firms are in competition, they will offer a wage equal to the worker's expected productivity. In this case, the beliefs are certain, so the wages are simply:

• $w(e_L)={\theta }_L$
• $w(e_H)={\theta }_H$

Now, what should $e_L$ be? For the low-ability worker, any education is costly. If they are going to be correctly identified as a low type and paid ${\theta }_L$ anyway, their best strategy is to incur the minimum possible cost. Since we assume the cost of no education is zero ($c(0|\theta )=0$), the low-type worker will choose $e_L=0$. Any choice $e_L>0$ would yield the same wage ${\theta }_L$ but a lower utility because of the education cost.

So, for the rest of our analysis, we will work with $e_L=0$ and some positive education level for the high type, $e_H>0$.

The Incentive Compatibility (IC) Constraints

For this separating outcome $(e_L=0,e_H)$ to be a stable equilibrium, neither type of worker should have an incentive to deviate. This gives us two crucial conditions, known as incentive compatibility constraints.

1. The High-Type's IC Constraint (IC-H): The high-ability worker must prefer choosing $e_H$ and getting the high wage ${\theta }_H$ over deviating to choose $e_L=0$ and getting the low wage ${\theta }_L$.

   • Payoff from equilibrium path: $U({\theta }_H,e_H|{\theta }_H)={\theta }_H-c(e_H|{\theta }_H)$
   • Payoff from deviating: $U({\theta }_L,0|{\theta }_H)={\theta }_L-c(0|{\theta }_H)={\theta }_L$
   • The constraint is: ${\theta }_H-c(e_H|{\theta }_H)\ge {\theta }_L$

2. The Low-Type's IC Constraint (IC-L): The low-ability worker must prefer choosing $e_L=0$ and getting the low wage ${\theta }_L$ over trying to mimic the high type by choosing $e_H$ to get the high wage ${\theta }_H$.

   • Payoff from equilibrium path: $U({\theta }_L,0|{\theta }_L)={\theta }_L-c(0|{\theta }_L)={\theta }_L$
   • Payoff from deviating: $U({\theta }_H,e_H|{\theta }_L)={\theta }_H-c(e_H|{\theta }_L)$
   • The constraint is: ${\theta }_L\ge {\theta }_H-c(e_H|{\theta }_L)$

These two inequalities are the core conditions for any separating equilibrium.

The image below gives a graphical representation of what we are looking for. The separating equilibrium requires finding an education level $e^{\ast }(H)$ and a wage $w^{\ast }(H)$ that the high-ability worker prefers over the low-type's outcome $(e^{\ast }(L),w^{\ast }(L))$, while the low-ability worker finds that same high-type outcome to be worse than their own.

Worked Example: Deriving the Conditions

Let's apply these abstract conditions to a concrete example. The process involves using the two IC constraints to find the range of education levels $e_H$ that can support a separating equilibrium.

As the video demonstrates, the two constraints give us a lower and upper bound on the education level $e_H$:

• IC-H (${\theta }_H-c(e_H|{\theta }_H)\ge {\theta }_L$) puts an upper bound on $e_H$. The education level cannot be so costly that even the high-ability worker finds it not worth the wage premium.
• IC-L (${\theta }_L\ge {\theta }_H-c(e_H|{\theta }_L)$) puts a lower bound on $e_H$. The education level must be costly enough to deter the low-ability worker from mimicking.

The separating equilibrium is possible for any $e_H$ that lies between these two bounds.

Advanced Application: A Three-Type Model

Now, let's tackle a more complex problem that will test your understanding of this method. We'll analyze a model with three worker types: low, medium, and high productivity. The logic remains the same: for each type, we must ensure they prefer their equilibrium choice over mimicking any other type. This will generate a set of simultaneous inequalities that define the equilibrium education levels.

The key takeaway from this detailed example is the systematic application of the no-deviation principle. For a strategy profile $(e_L,e_M,e_H)$ to be a separating equilibrium, it must be that for every type, their designated education choice provides a higher payoff than choosing the education level of any other type.

Conclusion

In this lesson, you have moved from setting up the Spence model to actively solving for its most famous outcome: the separating equilibrium. You have learned to derive the core mathematical conditions that make such an equilibrium possible.

Key Takeaways:

• A separating equilibrium is sustained by a set of incentive compatibility (IC) constraints, which ensure no player type has an incentive to deviate and mimic another.
• The high-type's IC constraint (IC-H) ensures they find it worthwhile to invest in education to signal their type. This sets an upper bound on the signal's cost.
• The low-type's IC constraint (IC-L) ensures they find it too costly to mimic the high type. This sets a lower bound on the signal's cost.
• The single-crossing property is what guarantees that a range of education levels exists that can satisfy both constraints simultaneously, allowing for successful signaling.
• This logic can be systematically extended from a simple two-type model to more complex scenarios with multiple types.

In our last lesson on this topic, we will turn to a different kind of information asymmetry: moral hazard, or hidden action. While signaling addresses hidden information (adverse selection), the Principal-Agent model will give us tools to analyze situations where one party's actions are unobservable to the other.