Consider a used car market with asymmetric information. The owners of used cars know what their vehicles are worth but have no way of credibly demonstrating those values to potential buyers. Thus, potential buyers must always worry that the used car they are being offered may be a low quality "lemon."

a. Suppose that there are equal numbers of good and bad used cars in the market and that good used cars are worth $13,000 while bad used cars are worth $5,000. What is the average value of a used car?
b. By how much does the average value exceed the value of a bad used car? By how much does the value of a good used car exceed the average value?
c. Would a potential seller of a good used car be willing to accept the average value as payment for her vehicle?
d. If a buyer negotiates with a seller to purchase the seller’s used car for a price equal to the average value, is the car more likely to be good or bad?
e. Will the used-car market come to feature mostly—if not exclusively—lemons? How much will used cars end up costing if all the good cars are withdrawn?

a. $9,000

b. $4,000 & $4,000

c. The seller is probably not willing to accept the average value as payment for her vehicle.

d. The car is more likely to be bad.

e. The used car market will still have both good and bad cars. Used cars end up costing the same if all the good cars are withdrawn. In the long run, prices of used cars may fall.

Explanation:

a. Because there are equal numbers of good and bad used cars in the market, the probability that a car is a good car is equal to the probability that a car is a bad car (=0.5). So the average value of a used car is calculated by taking the probability that a car is a good car (0.5), multiplying that by the value of a good car ($13,000), and adding to that the probability that a car is a bad car (0.5), multiplying that by the value of a bad car ($5,000). To put in an equation, we have

Average value of used car = Average value of good used car + Average value of bad used car

= (0.5 x $13,000) + (0.5 x $5,000)

= $9,000

b. The average value exceeds the value of a bad used car by:

Average value of used car - Average value of bad used car

= $9,000 - $5,000

= $4,000

The value of a good used car exceeds the average value by:

Average value of good used car - Average value of used car

= $13,000 - $9,000

= $4,000

c. A potential seller of a good used car would know that her car was worth $13,000, more than the average value ($9,000). Therefore, it would not make sense to sell her vehicle for $9,000. The seller would probably not be willing to accept the average value as payment for her vehicle.

d. If a buyer negotiates with a seller to purchase the seller's used car for a price equal to the average value, the car is more likely to be bad. As in c, a seller of a good used car would probably not be willing to accept the average value as payment for her vehicle.

e. The used car market will always have good and bad used cars because the average price does not apply in the market. If buyers know that they are negotiating the price of a car down to a much lower price, it is likely that the car they are buying are lemons. Likewise, buyers who want good cars may be willing to pay a bit more to ensure that they are not getting bad cars.

If all the good cars are withdrawn, the price of used cars will stay unchanged as buyers do not know whether cars are good or bad (asymmetric information). However, in the long run, buyers will know that and get more information than they had before, so prices will probably fall.

you have to put the parentheses like this:

in fact, you have to perform the operation inside parentheses first, and then proceed in order:

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