Rothschild-Stiglitz (RS) Separating Equilibrium In RS model, where there are only two risk types (high and low risks), there is no pooling equilibrium but a separating one (when there are sufficiently large number of high risks.)

Additionally, they state that if there are continuum of risk types, there can never be an equilibrium.

I have 2 questions:


I am trying to understand the analytical proof of the continuum case. I have the technical report of their paper and will provide that section here. However, I think the proof is not well-organised and I appreciate a clear explanation to the proof.**
In the proof, there is a section (highlighted) stating "necessary condition for existence is that.." and giving an inequality afterwards. To my understanding, this inequality gives how large the high risk population should be in order to guarantee the separating equilibrium. (Please let me know if I am wrong at this point). However, they take the derivative of the incentive compatibility constraint to derive this inequality, i.e. they take the derivative of an inequality to reach another inequality. Don't you think this approach is wrong as taking the derivative of an inequality does not have to yield another inequality?


** In the continuum case, I understand their reasoning that it is enough to show the inexistence of an equilibrium by only showing the inexistence for 2 close risk types, and they start from the maximisation problem of low risks. However, rest is foggy.
I recommend reading 'Informational Equilibrium' by John Riley (1979). It discusses this setting more generally. How the "insurance model" of Rotschild and Stiglitz is a special case of this is presented on pg. 335, and then Theorem 3 on pg. 343 is what you need. The proof is not trivial, but - in my opinion - well presented.