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

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: $$n_B = 4, n_Q = 2, n_L = 4$$ $$r_{B,Q} = 18, r_{B,L} = 6, r_{S,Q} = 8, r_{S,L} = 5$$ $$p_1 = 8, p_2 = 5$$

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.

• I agree with all that you say. However in markets with full information the concept of market equilibrium exists without such precise assumptions. The question is: Is the market in equilibrium with the $p_1$ price? It seems to me I was not clear on this. As you say games where this is a NEP and where this is not a NEP can both be constructed. – Giskard Apr 10 '16 at 17:21
• ok, sry, I misread. Yes, $p=5$ is a market equilibrium here: It equates demand and supply, $D(5)=n_B=4$ and $S(5)=n_Q=4$. The deviation argument you outlined above, while intuitively plausible, doesn't play a role here. On the one hand, the definition of market equilibrium doesn't call for it. On the other hand, without making these extra assumptions it is simply not possible to assess what it means "to deviate" for buyers and whether such a deviation would be profitable. – Bob Apr 10 '16 at 18:24
• There was probably no misreading, after your answer I went back and edited the question to be more clear. – Giskard Apr 10 '16 at 18:27
• btw, in the example you gave, it all comes down to what $n_B$ is. Had you chosen $n_B > 6$, the only equilibrium would have been $p=10$ (making buyers indifferent so that markets indeed equate at $Q=6$). – Bob Apr 10 '16 at 18:30
• I disagree (that it is the only equilibrium), I think there is another equilibrium at $p = 6$ when buyers are indifferent and demand is set-valued. – Giskard Apr 10 '16 at 19:14

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.