Question

Suppose 100 cars will be offered on the used-car market. Let 50 of them be good cars, cach worth \(\$ 10,000\) to a buyer, and let 50 be lemons, each worth only \(\$ 2,000\). a. Compute a buyer's maximum willingness to pay for a car if he or she cannot observe the car's quality. b. Suppose that there are enough buyers relative to sellers that competition among them leads cars to be sold at their maximum willingness to pay. What would the market equilibrium be if sellers value good cars at \(\$ 8,000 ?\) At \(\$ 6,000 ?\)

Short answer

Answer: The buyer's maximum willingness to pay for a car without knowing its quality is $6,000. If sellers value good cars at $8,000, only lemons will be sold in the market. If sellers value good cars at $6,000, both types of cars will be sold at the maximum willingness to pay price of $6,000.

Step-by-step solution

Find the probability of getting a good car or a lemon

We have a total of 100 cars, with 50 good cars and 50 lemons. So, the probability of getting a good car is \(P(G) = \frac{50}{100} = 0.5\), and the probability of getting a lemon is \(P(L) = \frac{50}{100} = 0.5\).

Compute the buyer's maximum willingness to pay for a car

To find the maximum willingness to pay for a car, we will multiply the probabilities of getting each type of car by their worth to the buyer and sum the results: \(WTP = P(G)\times V(G)+P(L)\times V(L) = 0.5\times \$ 10,000 + 0.5\times \$ 2,000 = \$ 5,000 + \$ 1,000 = \$ 6,000.\) So, the buyer's maximum willingness to pay for a car without knowing its quality is \( \$ 6,000\).

Analyze the market equilibrium when sellers value good cars at \( \$ 8,000\)

In this case, the sellers value the good cars at \( \$ 8,000\). Since the buyers' maximum willingness to pay for a car (\( \$ 6,000\)) is less than the sellers' valuation for good cars, good cars will not be sold in the market. Only lemons will be sold at the buyers' maximum willingness to pay (\( \$ 6,000\)), which is higher than their valuation for lemons (\( \$ 2,000\)).

Analyze the market equilibrium when sellers value good cars at \( \$ 6,000\)

Now, the sellers value the good cars at \( \$ 6,000\), which is equal to the buyers' maximum willingness to pay for a car. In this case, both the good cars and lemons will be sold at the buyers' maximum willingness to pay (\( \$ 6,000\)), since it is higher than the valuation for both types of cars (good cars: \( \$ 10,000\), lemons: \( \$ 2,000\)). In conclusion, the market equilibrium will depend on the sellers' valuation for good cars. If sellers value good cars at \( \$ 8,000\), only lemons will be sold in the market. If sellers value good cars at \( \$ 6,000\), both types of cars will be sold at the maximum willingness to pay price of \( \$ 6,000\).

Key concepts

Willingness to Pay

In economics, willingness to pay (WTP) refers to the maximum amount an individual is prepared to spend to acquire a good or service. It is a crucial concept that reflects the consumer's subjective valuation of a product.
In the case of the used-car market example, calculating the WTP for a car involved considering that there are two types of cars with different valuations (good cars and lemons) but with no distinct way to identify them beforehand. Thus, the buyer's WTP represented an average, based on the probability of ending up with either type.
Understanding WTP is vital for businesses as it helps set prices that consumers are willing to pay, while also ensuring they do not exceed this threshold and drive consumers away. In competitive markets, WTP can also indicate the market-clearing price, where goods are sold at a price point that buyers are willing to pay and that sellers are willing to accept.

Adverse Selection

The concept of adverse selection arises when there is an asymmetry of information between buyers and sellers, leading to negative outcomes in transactions. Specifically, those who have more information about the true quality or nature of the product are able to exploit their knowledge at the expense of those who are less informed.
For the used car scenario, if sellers know the quality of the cars (good or lemons) and buyers do not, sellers may only offer lemons at the price of good cars, driving good cars out of the market. This is known as the 'market for lemons' problem, famously discussed by economist George Akerlof. Adverse selection in such markets results in only the least desirable goods being traded, as seen in our exercise where the market equilibrium ends up being at the price of lemons if the sellers value good cars higher than the buyers' WTP.

Probability in Economics

The application of probability in economics is crucial as it allows economists to understand and predict economic behaviors under uncertainty. Probabilities enable the calculation of expected values which inform decision-making in risk-laden situations, like investments or market transactions.
In the used-car market example, probabilities were used to determine the expected WTP by weighing the value of good cars and lemons against their likelihood of occurrence (a 50-50 chance for each type). Use of probability conveys the idea that economics isn't just about certainties; it's about managing and anticipating the unpredictable, thereby influencing market dynamics, pricing, and consumer behavior.