Consider the simplest version of the Spence signalling model. There are two types of worker, with either productivity $\theta_H$ or instead productivity $\theta_H < \theta_L$. The proportions of the workers are $p_H$ (share of high types) and $1 - p_H$ respectively. Risk-neutral and competitive firms pay wages that may depend on a worker's education level $e \in \{0, 1 \}$. Finally, getting education ($e = 1$) costs $c_H$ for the high types and $c_L > c_H$ for the low types.
My question is simple:
What are the possible wage schedules that sustain a pooling equilibrium? In particular, can a pooling equilibrium wage schedule specify a wage of zero following $e = 1$?
I will briefly elaborate on my understanding of the problem and what is unclear to me. In any pooling equilibrium, workers must be paid their expected productivity. So following $e = 0$, the specified wage must be $$ w(e = 0) = \mathbb{E}(\theta) = p_H \theta_H + (1 - p_H)\theta_L $$ Furthermore, to ensure that nobody wants to deviate (by getting education), we only need to ensure that the high types won't deviate. (If they won't want to deviate, then neither will the low types). This requires that $$ w(e = 1) - c_H \leq w(e = 0) = p_H \theta_H + (1 - p_H)\theta_L $$ So the wage following $e = 0$ is uniquely determined in the pooling equilibrium (the equation). Moreover, the wage following $e = 1$ must be sufficiently low to ensure that nobody gets education (the inequality). I am wondering if there are any additional constraints that the wage schedule in a pooling equilibrium must satisfy, or whether any wage schedule satisfying these constraints will sustain pooling (as a PBE).
To make the issue a bit more salient, suppose that we propose some wage $w(e = 1) < \theta_L$. For example, suppose you make the wage for those who get education zero. Clearly, nobody will choose education, i.e. the inequality holds. Moreover, wages don't need to be set optimally following $e = 1$ since this happens with zero probability in equilibrium. On the other hand, no possible type could have a productivity of zero, so this wage schedule does not follow from any possible firm beliefs. This suggests that the wage following zero education needs to also satisfy $$ w(e = 1) \in [\theta_L, \theta_H] $$ Is this indeed a further constraint (in any pooling equilibrium where both types choose $e = 0$)?