<p>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 <a href="https://www.princeton.edu/~dixitak/Teaching/EconomicsOfUncertainty/Slides&amp;Notes/Slides15.pdf" rel="nofollow noreferrer">this set of slides</a> and in those from any second-year graduate course in Public Economics.</p>

<ul>
<li>In this paradigm, mixed strategy equilibria are ruled out. So when we say that "no equilibrium exists", we are actually just saying that no <em>pure strategy</em> equilibrium exists.</li>
<li>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.</li>
<li>There <em>may</em> or <em>may not</em> 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).</li>
<li><strong>EDIT</strong>: 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.</li>
</ul>

<p>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. </p>

<h2>Additional information</h2>

<p>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.</p>

CC BY-SA 3.0
2017-06-03T17:21:31.557Z
Thanks for your answer. What I do not understand about the no pooling equilibrium argument is that, when another firm enters and takes away the low-risk types and leaves the existing firm with losses, why won't there be some equilibrium in the future? If the existing firms closes its door then the high-risk types have nowhere to go since the other firm knows that there are high-risk. Wouldn't that leave the other firm intact?
Taufi
CC BY-SA 3.0
2017-06-03T17:38:02.817Z
Equilibrium requires every firm to make zero profits and for there to be no profitable unilateral deviations. Suppose, as you say, that a firm takes away the low risk types (making positive profits). The old firm is making a loss (not zero profits). That can't be an equilibrium. The old firm shuts down and competition gradually ensures that the new firm offers more and more generous insurance to the low risk types it attracted. But at some point, the high risk types see that contract as worthwhile! So they jump in and we have a pooling contract. But we know there is no pooling equilibrium!
saturno
CC BY-SA 3.0
2017-06-04T10:32:00.533Z
-1 "Every game has an equilibrium in mixed strategies." This is not true. You need additional conditions, such as quasiconcavity of the payoff functions and compactness of the strategy sets. An example: The 'whoever says the bigger number wins' game has no equilibrium, because the strategy sets are not compact. In the game defined by Rothschild and Stiglitz the strategy sets are also not compact. (Though you could introduce a 'set of relevant strategies' which would be.)
Giskard
CC BY-SA 3.0
2017-06-04T11:33:13.910Z
The payoff functions are quasiconcave in RS. The point about the compactness of strategy spaces is fair, though you resolve it yourself. In any case I agree that there are obviously conditions that should be attached to a statement like that. The key point is that RS made a deliberate decision not to consider mixed strategies and that this is an important caveat to the claim that "there are no equilibria". Feel free to edit if you think any of this discussion adds value in terms of answering Taufi's questions.
saturno
CC BY-SA 3.0
2017-06-04T14:11:32.633Z
@saturno I fail to see why you simply do not remove your one false claim yourself. Since you gave me permission I will do so.
Giskard
CC BY-SA 3.0
2017-06-04T22:27:38.207Z
@denesp regarding "You need additional conditions, such as quasiconcavity of the payoff functions and compactness of the strategy sets." I'm pretty sure that's not true. Nash's existence theorem only requires finiteness in the number of players and strategy space- not quasiconcavity of a utility function, nor any other quality.
AndrewC
CC BY-SA 3.0
2017-06-05T06:58:27.220Z
@AndrewC To be clear I am not claiming that these properties are necessary for equilibrium, but they are among the generally used sufficient conditions. The game class in Nash's theorem - mixed strategy expansions of finite games - does fulfill both conditions. Finite games do not necessarily have equilibria unless you allow for mixed strategies. I understand that you mean mixing strategies is obviously allowed, but I would say that is not obvious. It is the mixed strategy space of finite games that is compact. The expected payoff functions are linear in strategies and hence quasiconcave.
Giskard
CC BY-SA 3.0
2017-06-03T17:16:32.670Z
2017-06-04T14:12:05.457Z
2017-06-04T14:12:05.457Z
saturno
<p>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. </p>

<p>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. </p>

<p>Therefore, my <strong>questions</strong> are: </p>

<p>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? </p>

<p>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? </p>

CC BY-SA 3.0
2017-06-02T14:40:42.903Z
It is impossible to answer this question without you spefifying the exact model you are studying. I am guessing you will find adequate answers in Eric Rasmussen's book 'Games and Information', Chapter 8.
Giskard
CC BY-SA 3.0
2017-06-02T14:53:57.660Z
Thank you for the reference. I wasn't aware that there are distinct models for this.
Taufi
<microeconomics><competitive-equilibrium><contracts>
No competitive equilibrium for pooling contracts
Taufi
# No competitive equilibrium for pooling contracts

(Asked by: username_0 on 2017-06-02)

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?

**Tags:** microeconomics, competitive-equilibrium, contracts

> **Comments:**
> **username_1**: It is impossible to answer this question without you spefifying the exact model you are studying. I am guessing you will find adequate answers in Eric Rasmussen's book 'Games and Information', Chapter 8.  
> **username_0**: Thank you for the reference. I wasn't aware that there are distinct models for this.  

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## 1 answers

### Answer by username_2 on 2017-06-03. Score: 3 | Accepted answer

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](https://www.princeton.edu/~dixitak/Teaching/EconomicsOfUncertainty/Slides&Notes/Slides15.pdf) 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.

> **Comments:**
> **username_0**: Thanks for your answer. What I do not understand about the no pooling equilibrium argument is that, when another firm enters and takes away the low-risk types and leaves the existing firm with losses, why won't there be some equilibrium in the future? If the existing firms closes its door then the high-risk types have nowhere to go since the other firm knows that there are high-risk. Wouldn't that leave the other firm intact?  
> **username_2**: Equilibrium requires every firm to make zero profits and for there to be no profitable unilateral deviations. Suppose, as you say, that a firm takes away the low risk types (making positive profits). The old firm is making a loss (not zero profits). That can't be an equilibrium. The old firm shuts down and competition gradually ensures that the new firm offers more and more generous insurance to the low risk types it attracted. But at some point, the high risk types see that contract as worthwhile! So they jump in and we have a pooling contract. But we know there is no pooling equilibrium!  
> **username_1**: -1 "Every game has an equilibrium in mixed strategies." This is not true. You need additional conditions, such as quasiconcavity of the payoff functions and compactness of the strategy sets. An example: The 'whoever says the bigger number wins' game has no equilibrium, because the strategy sets are not compact. In the game defined by Rothschild and Stiglitz the strategy sets are also not compact. (Though you could introduce a 'set of relevant strategies' which would be.)  
> **username_2**: The payoff functions are quasiconcave in RS. The point about the compactness of strategy spaces is fair, though you resolve it yourself. In any case I agree that there are obviously conditions that should be attached to a statement like that. The key point is that RS made a deliberate decision not to consider mixed strategies and that this is an important caveat to the claim that "there are no equilibria". Feel free to edit if you think any of this discussion adds value in terms of answering username_0's questions.  
> **username_1**: @username_2 I fail to see why you simply do not remove your one false claim yourself. Since you gave me permission I will do so.  
> **username_3**: @denesp regarding "You need additional conditions, such as quasiconcavity of the payoff functions and compactness of the strategy sets." I'm pretty sure that's not true. Nash's existence theorem only requires finiteness in the number of players and strategy space- not quasiconcavity of a utility function, nor any other quality.  
> **username_1**: @username_3 To be clear I am not claiming that these properties are necessary for equilibrium, but they are among the generally used sufficient conditions. The game class in Nash's theorem - mixed strategy expansions of finite games - does fulfill both conditions. Finite games do not necessarily have equilibria unless you allow for mixed strategies. I understand that you mean mixing strategies is obviously allowed, but I would say that is not obvious. It is the mixed strategy space of finite games that is compact. The expected payoff functions are linear in strategies and hence quasiconcave.  

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