Consider a market for 300 used cars where $1 / 3$ of all cars are good quality cars and the rest are bad quality cars. All these cars are owned by (potential) sellers to begin with and each seller owns only one car. Suppose the value of a bad car is 20 to the buyers and 10 to the seller The valuation of a good car, however, is 100 for a buyer and 50 for a seller. Assume the number of buyers on the market exceeds the number of sellers and as a result seller and the bargaining power(the price of any car in the market will be its value to the buyer)
Question: Suppose there exists a certification test costing $\mathbf{C}$ that can identify good cars with certainty. However, bad cars can also pass the test with probability $1 / 4$ (and fail with, remaining probability). Consider the possibility of a pooling equilibrium, in which both sellers of good cars and bad cars choose to undertake the test. Find out the range of certification test cost $\mathbf{C}$ for which such a pooling equilibrium exists.
My attempt: without certification buyers are willing to pay the average price i.e. is
$$
140 / 3=46.67
$$
The price of a good certified car in the pooling equilibrium will be
$$
200 / 3=66.67
$$
assuming that buyers know that test can certified lemon as a good car with probability 0.25 and that of lemon is 20 . Therefore,
$$
0<C<16.67
$$
i.e $\mathbf{C}$ can be anything between what buyers are willing to pay less valuation of good cars to seller
$$
(66.67-50)
$$
I am having doubts in my attempt. I am not sure if it is correct or not. It would be of great help if someone helps me with this. Thank you.
Edit: After certification, lemon will be sold for an average price of 40 .
Rationale of my attempt: Suppose $\mathrm{C}=17$, then the net benefit to a seller after certification is $23>20($ =value of a bad car to seller) for lemon and $49.67<50(=$ value of good car to the seller) for a good car. Therefore, the seller of good car will opt out of certification.
We are looking for a pooling equilibrium where both types choose to certify their cars.
The certification has two outcomes $\{$ success, failure $\}$.
The buyer faces three possible types of histories: (1) (type $h_{1}$ ) a successful certification, (2) (type $h_{2}$ ) an unsuccessful certification, (3) (type $h_{3}$ ) no certification. (each of these histories are followed by a standard lemons market probem with a price offer from the seller and a buy/not buy decision from the buyer). Then, by Bayes rule (wherever possible), the interim beliefs following history $h$ are:
1. $\operatorname{Pr}($ good $\mid$ success ful test $)=\frac{2}{3}$
2. $\operatorname{Pr}($ good $\mid$ unsuccess ful test $)=0$
3. Off path Belief: $\operatorname{Pr}($ good $\mid$ no test $)=0$.
Game following history $h_{1}:$
Buyer’s maximum willingness to pay is $\frac{2}{3} 100+\frac{1}{3} 20=\frac{220}{3}>50$. Hence the equilibrium price in this game $p\left(h_{1}\right)=\frac{220}{3}$ and both types sell their cars.
Gross Payoff for either type: $\frac{220}{3}-C$
Game following history $h_{2}$ :
Buyer’s maximum willingness to pay $=20<50$. In this case, the equilibrium price is $p\left(h_{2}\right)=20$ and hence only the lemons sell.
Gross payoff to lemons: 20-C Gross payoff to good cars: 50 – $\mathrm{C}$
Thus the expected payoff to Lemons if they choose to certify: $\frac{1}{4} \frac{220}{3}+\frac{3}{4} 20-C=\frac{400}{12}-C$.
The expected payoff to Good cars following certification: $\frac{220}{3}-C$.
Off-path choices (history $h_{3}$ ):
To sustain this equilibrium, neither type should deviate to no testing. Given off path beliefs, the buyer’s maximum willingness to pay is $20<50$. Hence only lemons are sold in this scenario, yielding payoffs 20 and 50 for lemons and good cars respectively.
Pooling can be sustained when:
1. $\frac{220}{3}-C>50 \Longrightarrow C<\frac{70}{3},$ and
2. $\frac{400}{12}-C>20 \Longrightarrow C<\frac{160}{12}$
Hence pooling is an equilibrium as long as $C<\frac{40}{3}$