Asymmetric information assumption

Question

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?

Answer 1

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 \begin{equation} \frac{p}{2}(4000-2750)+\frac12(1500-2750)=1250\left(\frac{p}2-\frac12\right)<0. \end{equation} 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 $2750$.

Answer 2

This behavior happens over time. There is cultural knowledge about used car salesmen. A buyer will assume the car is of average quality, which makes it unprofitable to sell above average quality cars. This removes them from the market, which over time will be information that spreads through the market and perhaps a few years down the line, people will have an even poorer opinion of used car salesmen than they had before. This process continues until it reaches an equilibrium where the expected value of cars is high enough above the real value that no further attrition takes place.