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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 $10,000 each. New cars sell for $8,000 and used ones for $2,000. (Note that since buyers are risk-neutral, the price of a new car reflects the expected value of purchasing a car that may or may not be defective.) What is the fraction x?

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Answer:

x=0.25

Explanation:

Assuming that consumers value every non-defective car at $10,000 each, only defective used cars are for sale. Therefore, consumers value defective cars at $2,000.

The expected value of a new car is given by the defective new car rate (x) multiplied the defective value, added to the non-defective car rate (1-x) multiplied by the non-defective car value.

[tex]EV = 8,000= 2,000x + 10,000(1-x)\\x=\frac{2,000}{8,0000}\\ x=0.25[/tex]

The fraction x is 0.25. That is, 25% of new cars sold are defective.

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