market for lemons with signals - equilibriums vs constraints

Question

I checked online for a question similar to this but was unable to find one.  I understand the concept of pooling and separating equilibrium and can apply it to labourers having different qualities of work but how can it be applied to the market for lemons with signalling through warranty?
Also, is there a connection between separating / pooling equilibrium and participation / self-selection constraint?  I know one is related to signalling and the other to incentives from reading Hal Varians Micro book.  But is there more?
For example lemons are worth 20 to consumer and 10 to producer and peaches are worth 60 to consumer and 50 to producer.  Suppose a 50-50 split.  This means the expected value to consumers is 40.  The pooling equilibrium means all cars would be sold at the same value and the consumer will purchase at a maximum of 40 so only lemons will be sold.  How would I find the separating equilibrium for a warranty priced at w?
Is it solved like this?
60-w > 20   peaches under warranty
20 > 60-w   lemons not under warranty
Is solving the pooling vs separating equilibrium different from solving the participating vs self selection constraint?
Thank you in advance for any help.

user28372 · Answer

The two questions you ask are related but distinct. First recall what a separating equilibrium is: its an equilibrium where distinct type takes distinct actions, supported by a system of consistent beliefs.
Now lets come to your questions one by one:

In the Lemons Market, a separating equilibrium would involve the seller setting different prices for the lemons and peaches. Can such an equilibrium exist? Think about what the buyer's beliefs would be and if the lemon can mimic the peaches?

I guess you mean Incentive Compatibility constraints instead of Participation Constraints. You must have encountered them in Principal Agent models. In these models, the agents play a (bayesian) game and the Principal can commit to design the rules of the game. The Participation Constraint ensure that the agents are willing to take part in the game designed by the Principal. The Incentive Compatibility Constraint ensures that the ensuing equilibrium of the game is a separating equilibrium - i.e. different types of agent will be choosing different actions in the game. The specific rules of the game designed by the Principal depends on the context.

market for lemons with signals - equilibriums vs constraints

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