Here’s a result so fresh that the paper doesn’t even exist yet; Cho (yes, the Cho from Cho-Kreps) presented a version of the following at a seminar here recently and I wanted to jot some notes down while it’s fresh in my mind.
Take the standard Akerlof problem. There is a unit mass of buyers, a unit mass of high quality sellers, and a unit mass of low quality sellers. Quality is known only to the individual seller. All buyers value high quality goods at H and low quality goods at L. Low quality sellers have a reservation value of 0 and high quality sellers a reservation value of C, where H-C>L. Thus, efficiency is maximized by each high quality seller selling at some price to a buyer. Car quality is unobservable, so assume there is a pooling equilibrium where high and low quality cars are sold at the same price. Then consumers value these cars at (H+L)/2. So if (H+L)/2 is less than C, high quality sellers will be unwilling to sell, only low quality sellers will remain in the market, and we say the market has collapsed. It is a pretty robust result in these types of asymmetric information models that the lowest quality types always remain in the market, and the high quality types are often pushed completely out.
What if this market, however, were dynamic? What we mean here is that, in period one, buyers and sellers match randomly with each other; there are twice as many sellers as buyers, so only half the sellers match. In any given match, a price is offered for the transaction. If either party rejects the price, no match is made, payoffs are zero this period, and both buyer and seller rejoin the unmatched pool in the next period. If it is accepted, the good is sold, its value is realized, and the seller receives payoff of the price minus his cost, while the buyer receives either H or L minus the price, depending on what quality the good ended up being. This relationship is maintained, with precisely the same payoffs, every period into the future, except that with probability (1-delta) the relationship ends and both buyer and seller rejoin the unmatched pool. We then move to the next period and everyone who is currently unmatched is randomly matched again. We are interested in the existence of price(s) that form an undominated stationary equilibrium as the period length (and hence discounting between periods) goes to zero.
Will the static logic, that all buyers match with low quality types at a price somewhere between 0 and L, maintain? It will not. If that were an equilibrium, then every seller unmatched will be a high quality seller. So an individual buyer who rejoins the unmatched pool will match almost certainly with a high quality seller, and if the price is C+epsilon, the seller will accept the offer and the buyer will improve his payoff. In any equilibrium, then, there must be at least two prices being offered, and at least some high quality sellers must match.
Note, now, the incentive for low quality sellers. By pretending to be a high quality seller (only accepting the price C+epsilon, and not the price at which only low types would accept), the low quality sellers improve their payoff. So, again, in equilibrium, low quality sellers will mix between accepting any offer and only accepting the high offer in which they pretend to be high quality types. Cho shows that this deception leads to a couple worrying results: first, this deception means that, in the limit, buyer’s expected payoffs go to zero in any equilibrium, and second, that some buyers do not match for arbitrarily long numbers of periods. The second result obtains because buyers are getting arbitrarily close to zero payoff, so the harm from waiting is very low, and if they accept a low quality seller masquerading as a high quality seller, they will receive a negative payoff.
Note that all of the intuition above appears (at least to me) robust to allowing a continuum of seller types or to allowing sellers and buyers to break existing relationships the period after they begin. What remains to be seen, though, is what economic problem looks like Mortensen-Pissarides plus Akerlof lemons. I’d be surprised, given the importance of relational contracts in the world of finance, if we couldn’t find something suitable in that venue.
http://www.kellogg.northwestern.edu/research/math/seminars/2011-12/In-Koo_Cho_Abstract.pdf (Extended abstract – I’m fairly certain no draft of this paper is out yet)
A Fine Theorem 
Benjamin Lester of the Philly Fed has a similar working paper
“Trading Dynamics in Decentralized Markets with Adverse Selection”
RE: your last observation, there is “something” but I think it’s still an active area of research. Did you ever read Briana Chang’s JMP?
Although the nature of quesitons is a bit different (and… economically a bit more meaning and insightful), these papers are also interesting concerning this topic
Igal Hendel & Alessandro Lizzeri, 1999. “Adverse Selection in Durable Goods Markets,” vol. 89(5), pages 1097-1115
Maarten C. W. Janssen & Santanu Roy, 2002. “Dynamic Trading in a Durable Good Market with Asymmetric Information,” International Economic Review, vol. 43(1), pages 257-282.
Especially, since in the second we can see that all qualities will be traded, so the adverse selection problem is, basically, dealt with.
I mean… if we assume that the Akerlof problem is really about a durable good (like car), and not some other stuff that has low and high quality with asymmetric valuations on both sides of the market.
Then these are nice papers, especially, since the market for used goods emerges endogenously
If it is purely an intellectual exercise, then Cho’s paper shall be a nice one when it gets done.