Suppose that in New Crankshaft, Pennsylvania, the quality distribution of the 6,000 used cars on the market is such that the number of used cars of value less than V is V/2.Original owners must sell their used cars.Original owners know what their cars are worth, but buyers can't determine a car's quality until they buy it.An owner can either take his car to an appraiser and pay the appraiser $400 to appraise the car (accurately and credibly) or sell the car unappraised.In equilibrium, car owners will have their cars appraised if and only if the car's value is at least

A) $1,200.
B) $3,000.
C) $400.
D) $800.
E) $1,600.

Let's assume that a car owner has a car of value V. If he decides to appraise his car, he needs to pay $400 for the appraisal. If the appraisal comes out to be less than V, then the owner will sell the car for less than its actual value, losing money. So, he will choose to sell the car unappraised if V < $400.
On the other hand, if he appraises the car and the value comes out to be greater than or equal to $400, he will sell the car at the appraised value to ensure that he is not losing out on any money. Therefore, the car owner will appraise his car only if V $\geq$ $400$.
Now, we need to find the value of V such that the number of cars of value less than V is equal to 6,000. We can set up the following equation:
$$\frac{V}{2}+\frac{V-1}{2}+\frac{V-2}{2}+...+\frac{1}{2}=6,000$$
Solving this equation, we get V = 800. Therefore, the car owner will have his car appraised if and only if the car's value is at least $800+400=$ $1,200$. Hence, the answer is (D) $800$.