Question

Consumers know that some fraction x of all new cars produced and sold in the market are defective. The defective ones cannot be identified except by those who own them. Cars do not depreciate with use. Consumers are risk-neutral and value nondefective cars at $11,000 each. New cars sell for $7,000 and used ones for $6,000. What is the fraction x?

Answer 1

Fraction x = 0.8

Step-by-step explanation:

As we know that consumers value a non-defective car at $11,000, so we make assumption that only used cars for sale will be defective ones.

The used car price of $6000 is the value to consumers of a defective car.

For a risk-neutral buyer, the reservation price for a new car will be the expected value of a non-defective car( i.e., the value of a good car times the probability of getting a good car, plus the value of a bad car times the probability of getting a bad car.)

So to find x, we solve:

Expected value = (prob. of non-defective car) (value of non-defective car)

+ (prob. of defective car)(value of defective car)

$7000 = ( 1 − x )11,000 + ( x )6000

7000= 11000 - 11000x + 6000x

5000x = 4000

x = 4000/5000

x= 0.8