Adverse Selection: Positive Selection of Worker Types (Mas-Collel)
In PO, you want all types with opportunity cost $r(\theta)\leq \theta$ to trade, because the firm gets more productivity than the worker has to give up on home productivity (&quot;opportunity cost&quot;).
By your assumptions $r(\theta_H) &lt; \theta_H$ and $r(\theta_L)&gt;\theta_L$ such that in PO, L shall stay at home and H shall work. However, you need to offer H a wage of at least $w\geq r(\theta_H)$.
The highest wage a firm would offer is the expected $\theta$ of a worker who is willing to accept this wage. Suppose that $\mathbb E[\theta]=\lambda \theta_H + (1-\lambda)\theta_L &gt;r(\theta_H)$ and, by definition,  $\theta_L&lt;\mathbb E[\theta]&lt;\theta_H$.
Now, it could be that $w=E[\theta] &gt; r(\theta_L)$ such that L would also want to accept this wage, but this is inefficient as
$$w &gt; r(\theta_L) &gt; \theta_L.$$
Low types should not work, but because there are so few of them ($\lambda$ is large), the firms would still profit if they also accepted $w$.
An uniformed third party could Pareto improve the allocation by changing the wage to $w' \in (r(\theta_H),r(\theta_L))$ such that L types would reject $w'$.