Chapter 16, Problem 1P

(a)

To determine

The expected value of the used car to the buyer.

Explanation of Solution

The problem of lemon arises when the quality of the used cars are not known to the buyers. This information asymmetry would culminate in the wrong selection of the cars from the used market. The sellers on the other hand would have more information about the used cars which help them to sell the lemons instead of selling the plums which are cars of good quality. Thus, the problem of lemon is the case where the asymmetric information makes the consumer to choose wrongly the low quality lemon goods from the market instead of the good quality plums.

Here in this case, the consumer knows that the chances for getting the lemons instead of the plums is 50 percent. This means that 1 out of 2 cars will be plums and lemons. However, the buyers are unable to predict which is lemon and plum. This makes the issue more serious. As a result of the uncertainty, the buyers will consider the values of both lemons and plums and they will be multiplied with their respective percentage chances of getting and added together to calculate the value of the used car. The value of the lemon is give to be $8,000, whereas the value of plum is $16,000. Thus, the expected value of the used car can be calculated as follows:

$\begin{array}{c}\text{Expected value of used car}=\left[\begin{array}{l}\left(\text{Price of lemon}\times \text{Percentage of getting}\right)+\\ \left(\text{Price of plum}\times \text{Percentage of getting}\right)\end{array}\right]\\ =\left(8,000\times 0.5\right)+\left(16,000\times 0.5\right)\\ =4,000+8,000\\ =12,000\end{array}$

Thus, the expected value of the used car will be $12,000 by the buyers when they are unable to identify the lemons and plums and their chances of getting lemons instead of plums is 50 percent.

Economics Concept Introduction

Asymmetric information:  The asymmetric information prevails when there are differences in the level of information that the seller and the buyer have. When any one party has more knowledge about the goods and services exchanged in the market, then it is imperfect or asymmetric information.

(b)

To determine

The quantity of lemons and plums offered at the expected value of the used car.

Explanation of Solution

When the buyers value the used car at $12,000, it is beneficial for the seller to sell more lemons and sell less plums. Thus, the quantity offered at the expected market price of the used cars will be ${\text{Q}}_{\text{L}}=\frac{1}{10}{\text{P}}_{\text{L}}-200{\text{ and Q}}_{\text{H}}=\frac{1}{20}{\text{P}}_{\text{H}}-200$, respectively. Thus, the quantities can be calculated by substituting the expected value instead of Ps in the equation as follows:

$\begin{array}{c}{\text{Q}}_{\text{L}}=\frac{1}{10}{\text{P}}_{\text{L}}-200\\ =\frac{1}{10}\times 12,000-200\\ =\frac{12,000}{10}-200\\ =1,200-200\\ =1,000\end{array}$

Thus, there will be 1000 lemons offered. The quantity of plums offered can be similarly calculated as follows:

$\begin{array}{c}{\text{Q}}_{\text{H}}=\frac{1}{20}{\text{P}}_{\text{H}}-200\\ =\frac{1}{20}\times 12,000-200\\ =\frac{12,000}{20}-200\\ =600-200\\ =400\end{array}$

Thus, there will be 400 plums of the high quality cars offered at the expected value of the used car.

(c)

To determine

The deadweight loss in the market for used car.

Explanation of Solution

When the market had perfect information, the quantity of low quality lemons was 600 at the price of $8,000 each. Similarly, the quantity of high quality plums was 600 at the price of $16,000 each. When the market became not a perfect information market, the quantities changed from 600 lemons to 1,000 lemons at a price of $12,000 each and plums from 600 to 400 at $12,000 each. Thus, the quantity of high quality cars sold in the market decreases which causes a deadweight loss in the market for high quality cars. It can be calculated as follows:

$\begin{array}{c}{\text{Deadweight loss}}_{\text{High quality}}=\frac{1}{2}\times \left[\begin{array}{l}\left({\text{Perfect information}}_{\text{quantity}}-{\text{Imperfect information}}_{\text{quantity}}\right)\\ \times \left({\text{perfect information}}_{\text{price}}-{\text{imperfect information}}_{\text{price}}\right)\end{array}\right]\\ =\frac{1}{2}\times \left(600-400\right)\times \left(16,000-12,000\right)\\ =\frac{1}{2}\times 200\times 4,000\\ =4,000\end{array}$

Thus, the deadweight loss in the market for high quality used cars is $4,000. Similarly, there will be deadweight loss in the market for low quality used cars because of the higher than the actual price at which it is sold. It can be calculated as follows:

$\begin{array}{c}{\text{Deadweight loss}}_{\text{Low quality}}=\frac{1}{2}\times \left[\begin{array}{l}\left({\text{Imperfect information}}_{\text{quantity}}-{\text{Perfect information}}_{\text{quantity}}\right)\\ \times \left({\text{Imperfect information}}_{\text{price}}-{\text{Perfect information}}_{\text{price}}\right)\end{array}\right]\\ =\frac{1}{2}\times \left(1,000-600\right)\times \left(12,000-8,000\right)\\ =\frac{1}{2}\times 400\times 4,000\\ =8,000\end{array}$

Thus, the deadweight loss in the market for low quality used cars is $8,000.