Does the concept of Nash-equilibrium conflict with the concept of market equilibrium in the lemon market

Question

Consider a version of Akerlof's Lemon market with two types of sellers. One type sells Quality cars the other type sells Lemons. Buyers' reservation prices are $r_{B,Q}$ for a Quality car and $r_{B,L}$ for a Lemon. Sellers' reservation prices are $r_{S,Q}$ for a Quality car and $r_{S,L}$ for a Lemon. The buyers cannot differentiate between the types of the sellers but the sellers know their type. Given a market price $p$ sellers decide whether to sell or not by maximizing their surplus $p - r_{S,t}$. (Not selling yields a surplus of zero.) Buyers decide whether to buy or not by maximizing their expected surplus $E(r_{B,t}) - p$. (Not buying yields a surplus of zero.)

Given a number of buyers $n_B$ a number of sellers for both types $n_Q, n_L$ we can reason about the type of equilibrium. Depending on the parameters you can have total market collapse (if no purchase occurs), adverse selection (if only Lemons are bought and sold) and also markets where both types of cars are sold. For some parameter sets you have multiple market equilibria. That is you have a price $p_1$ at which adverse selection occurs and the corresponding demand at $p_1$ is equal to the supply at $p_1$. You also have a price $p_2$ at which adverse selection does not occur and the corresponding demand at $p_2$ is equal to the supply at $p_2$.

If the surplus of a single buyer is larger at $p_2$ and the surplus of sellers is not smaller at $p_2$, can I claim that $p_1$ is an equilibrium? Seems like any buyer would benefit by deviating from $p_1$ and unilateraly setting $p_2$ or $p_2 + \epsilon$. In competitive equilibrium it is assumed that market actors are price takers but if they are sufficiently small (have no bargaining power like a monopoly) then this coincides with their strategic interests. Here even if the buyer is insignificant, this does not hold (probably due to asymmetric information). So is the market in equilibrium with price $p_1$ or not?

Such parameter combinations exist, an example: $$ n_B = 4, n_Q = 2, n_L = 4 $$ $$ r_{B,Q} = 18, r_{B,L} = 6, r_{S,Q} = 8, r_{S,L} = 5 $$ $$ p_1 = 8, p_2 = 5 $$

Answer 1

So this seems to be a known issue. Quoting from the Wilson article of 1980, The Nature of Equilibrium in Markets with Adverse Selection:

Using a variant of Akerlof's model of the used car market, we examine the equlibrium of the model under three distinct conventions: (1) an auctioneer sets the price; (2) buyers set the price; (3) sellers set the price. Only in the case of the auctioneer is the equilibrium necessarily characterized by a single price which equates supply and demand.

Thus the answer to the question is a partial yes. Given some (most) variants of the lemon market the two equilibrium concepts do not result in identical outcomes.

Answer 2

In order to talk about whether the competitive equilibrium is also a Nash equilibrium, you first have to properly define a game. For instance, is the buyers setting the prices or the sellers or is there bargaining. How do they meet, are there search costs etc. And importantly, what is the order of events! Your reasoning implicitly assumed that buyers would first announce a price at which they are willing to buy, and then sellers could choose from all the offered prices if they want to sell to any buyer. Assume there are no search costs etc and that if multiple sellers are willing to accept, then the buyer selects a random one. In this case, with a configuration of parameters that permits multiple equilibria, the same set of equilibria is also a Nash equilibria here, but not necessarily a subgame perfect Nash equilibrium. That is, in the equilibrium where only bad quality cars are sold but where buyers are better off with the average quality at $p2$, it is indeed optimal for buyers to offer $p1$ if they believe that high quality sellers will not accept $p2$. If you want to rule that out, you need to restrict attention to equilibria that are sequentially rational/subgame perfect.