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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.