Asymmetric information assumption

This question isn't about asymmetric information per se but rather one of the assumptions. Consider a market for second hand cars; there are lemons (low-quality cars) and plums (high-quality cars).

For lemons, the valuations are: $V_{b} = 1500$ and $V_{s} = 1000$. The subscripts denote buyer (b) and seller (s).

For plums: $V_{b} = 4000$ and $V_{s} = 3000$

The buyer knows that there is an equal amount of each car (i.e., the probability of the car being high or low quality is 50%) but she doesn't observe the quality. This results in market failure because the buyer is willing to pay

$\frac{1}{2}(1500 + 4000) = 2750$

Therefore the sellers of plums leave the market because $2750 < 3000$ and only sellers of lemons remain. And now the buyer is only willing to pay 1500, therefore only lemons will be traded.

What is confusing me is that it seems like we are assuming that the buyer knows the sellers valuations. I.e., the buyer knows that because the seller values plums at 3000, and the buyers expected valuation is below this, those sellers will leave the market. Am I misunderstanding something or is this the actual assumption in this case?

In this setting, both buyers and sellers are usually assumed to be risk neutral. So adding an additional layer of uncertainty over seller's valuation is not really going to change the model's prediction in any "qualitative" way, in the sense that market unraveling (perhaps only partial in this case) is still an inevitable outcome, even though the exact equilibrium may be different.

Suppose the buyer is unsure of the seller's valuation, but the buyer believes that the seller's valuation follows a distribution with a mean of $3000$. Further, let's assume that under this distribution, a fraction $p\in[0,1)$ of the plum seller's value is below $2750$, the buyer's average value of the two types.

So, by offering to buy a car of unknown quality at $2750$, the buyer gets an expected payoff of $$\frac{p}{2}(4000-2750)+\frac12(1500-2750)=1250\left(\frac{p}2-\frac12\right)<0.$$ This is worse than the zero payoff the buyer gets by offering $1500$ for a lemon.

Since the mean of seller's valuation is $3000$, there is always a positive fraction of the sellers with values greater than $2750$, i.e. $1-p>0$. These sellers will leave the market as they don't anticipate fetching a price higher than their own values for the car. The exit will then trigger the usual cycle of downward revision of buyer's conditional expected value and more exits. Depending on the distribution of seller value, full market unraveling may or may not occur.

Nevertheless, what underpins the unraveling of (necessarily a fraction of) the plum market is still the same: the buyer's anticipation of a positive fraction $(1-p)$ of the plum sellers not willing to sell at $2750$.

• The word "expect" is abused here. If the seller's valuation had a distribution with mean 3000 like 99%: 2000, 1%:122000 then buying the car for ca. 2750 would be the right move. – Giskard May 16 '18 at 17:33
• @denesp: In your case, when the buyer offers 2750 for a car with unknown quality, he'd have a $0.99/2$ chance of getting a payoff of $(4000-2750)$ and a $1/2$ chance of getting $(1500-2750)$; the expected payoff is still negative, which is worse than just offering 1500 and getting a lemon with zero payoff. But I agree that the incentive structure is not quite the same as the original problem. – Herr K. May 16 '18 at 19:56
• Note that I wrote "ca. 2750" as in close to 2750. With these numbers an offer of 2700 would yield a higher payoff than one of 1500. Hence considering expected values is not enough to find the equilibrium. – Giskard May 17 '18 at 8:52
• @denesp I think my updated answer should have addressed that concern. – Herr K. May 17 '18 at 11:28
• @denesp: In the first paragraph I said adding uncertainty over seller's valuation is not going to change the model's prediction in "any qualitative way", meaning that adverse selection is still going to occur. The exact equilibrium will be different, as you rightly point out, but market unraveling (possibly only partial) is still inevitable. As I argue in my update, the fraction of plum sellers whose values are above 2750 is always positive, since the mean is at 3000. These sellers will always leave the market, putting a downward pressure on buyer's conditional expected value of the car. – Herr K. May 17 '18 at 14:14