No competitive equilibrium for pooling contracts In class we dealt with insurance economics and, specifically, adverse selection due to information asymmetry. As one possible solution we considered pooling contracts, i.e. the same contract for both high- and low-risk households for some average price. We showed and I understood why the high-risk households now would buy more insurance than the low-risk ones and thus everybody would act as if they were low-risk whether this is true or not. What I am not sure about is why there wouldn't be an equilibrium in a competitive market. 

It's clear that the low-risk individuals have reason to go for a lesser premium and, if some other insurer would offer this, then they would go there. 

Therefore, my questions are: 

1) So why won't there be an equilibrium? I would think that if a new insurer appears with a better deal than the pooling contract, then there would again be the problem with adverse selection. Hence, the new insurer wouldn't appear and offer a better deal in the first place? 

2) I also read that this is pareto-inefficient in that it disadvantages the low-risk households (obvious) but does not benefit the high-risk households. The latter point I do not understand since the high-risk households will have a higher demand since the pooling contract situation makes the insurance relatively cheap. Isn't that a benefit?
As one of the comments points out, there are many models of equilibria in insurance markets. But it sounds to me like you are referring to the Rothschild-Stiglitz (1976) paradigm. I will provide a basic summary of the key takeaways here, but there is a more complete explanation in this set of slides and in those from any second-year graduate course in Public Economics.


In this paradigm, mixed strategy equilibria are ruled out. So when we say that "no equilibrium exists", we are actually just saying that no pure strategy equilibrium exists.
There is never a pooling equilibrium. The basic reason for this is that a firm could profitably come in and offer a cheaper package with less insurance and poach away the low risk types. This leaves the existing firm with only the high risk types and it makes a loss.
There may or may not exist a separating equilibrium. If there does not, it is because - starting from a separating equilibrium - there exist pareto-improving cross-subsidies (from high to low risk types). This means that a firm can profitably come in and offer a package that attracts both types (i.e., a pooling contract).
EDIT: To answer your second question, the separating equilibrium - if one exists - is indeed not pareto efficient but it does benefit both high and low risk types. The pareto efficient separating contract would offer full insurance to both type, but the Rothschild-Stiglitz candidate cannot. To see why, suppose that it did. Then both receive full insurance but the low-risk types get it for less. Clearly the high risk types will pretend to be low types and the incumbent firm will make a loss.


In summary, this particular model offers a theory of pure strategy Nash Equilibria in insurance markets with adverse selection and multiple risk types. An equilibrium may or may not exist. If it does, then it is separating. 

Additional information

There are competing paradigms that solve this problem with assumptions about the timing of when offers are made and accepted (e.g., Miyazaki-Wilson-Spence). Another classic model to compare is Akerlof-style adverse selection. Neither alternative suffers from the non-existence problem.