Question

The used car supply in Metropolis consists of 10,000 cars. The value of these cars ranges from \(\$ 5,000\) to \(\$ 15,000,\) with exactly one car being worth each dollar amount between these two figures. Used car owners are always willing to sell their cars for what they are worth. Demanders of used cars in Metropolis have no way of telling the value of a particular car. Their demand depends on the average value of cars in the market \((P)\) and on the price of the cars themselves \((P)\) according to the equation $$Q=1.5 P-P$$ a. If demanders base their estimate of \(P\) on the entire used car market, what will its value be and what will be the equilibrium price of used cars? b. In the equilibrium described in part (a), what will be the average value of used cars ac tually traded in the market? c. If demanders revise their estimate of \(P\) on the basis of the average value of cars actually traded, what will be the new equilibrium price of used cars? What is the average value of cars traded now? d. Is there a market equilibrium in this situation at which the actual value of \(P\) is consistent with supply-demand equilibrium at a positive price and quantity?

Short answer

In summary, we analyzed the used car market in Metropolis and, under different scenarios, sought to find the equilibrium price and average value of used cars traded. In the first scenario, the equilibrium price and average value of cars in the entire market were both \(\$10,000\). In the second scenario, we found the average value of cars actually traded to be \(\$7,500\), which led to a new equilibrium price of \(\$7,500\) and a revised average value of cars traded at \(\$6,250\). However, in the final scenario, we found that the market equilibrium at a positive price and quantity did not exist since the equilibrium price was higher than the average value of cars traded, making the supply-demand equilibrium inconsistent.

Step-by-step solution

Calculate the average value of cars in the market

Since the value of cars ranges from \(\$5,000\) to \(\$15,000\), we can find the average value by taking the middle value between these two figures, which is \(\frac{5000 + 15000}{2} = \$10,000\). Thus, the average value of used cars in Metropolis (P) is \(\$10,000\).

Use the demand equation to find the equilibrium price

Given the demand equation \(Q = 1.5P - P\), we can plug in the average value of cars (\(P = 10000\)) to find the equilibrium price. Plugging \(P = 10000\) into the equation, we find that: \(Q = 1.5 * 10000 - 10000 = 5000\). In this case, the equilibrium price of used cars (P) is also \(\$10,000\). #b. Calculate the average value of used cars actually traded#

Identify the cars traded in equilibrium

From part (a), we know that the equilibrium price is \(\$10,000\). Therefore, only cars worth less than or equal to \(\$10,000\) will be traded in the market because owners of cars worth more than \(\$10,000\) would not sell at that price.

Calculate the average value of cars traded

Since only cars worth less than or equal to \(\$10,000\) are being traded, we need to find the average value of the cars in this range. Using the middle value between \(\$5,000\) and \(\$10,000\), the average value of cars traded is \(\frac{5000 + 10000}{2} = \$7,500\). #c. Determine the new equilibrium price and average value of cars traded after demanders revise their estimate of P#

Update the demand equation with the revised P

Now demanders base their estimate on the average of the cars actually traded, which is \(\$7,500\). Plugging this new value into the demand equation, we find that: \(Q = 1.5 * 7500 - 7500 = 3750\). The new equilibrium price (P) is \(\$7500\), and the quantity traded (Q) has decreased to \(3,750\) cars.

Calculate the new average value of cars traded

Since the new equilibrium price is now \(\$7,500\), we know that only cars worth less than or equal to \(\$7,500\) will be traded in the market. Using the middle value between \(\$5,000\) and \(\$7,500\), the new average value of cars traded is now \(\frac{5000 + 7500}{2} = \$6,250\). #d. Check if there is a market equilibrium at a positive price and quantity#

Compare the equilibrium price and average value of cars traded

In the final equilibrium situation we found, the average value of cars traded was \(\$6,250\), and the equilibrium price was \(\$7,500\). Since these two values are not equal, there is a difference between what demanders are paying (the price) and what the cars are actually worth (average value). Since the equilibrium price is higher than the average value of the cars traded, we can conclude that there is no market equilibrium in this situation at a positive price and quantity where the actual value of P is consistent with supply-demand equilibrium.

Key concepts

Supply and Demand

Understanding the forces of supply and demand is critical for grasping how markets operate. In essence, the law of supply posits that, all else equal, the quantity of a good or service that is available for sale increases as the price increases. Conversely, the law of demand suggests that consumers will buy more of a good or service when its price is lower and less when it's higher.

When we apply these principles to the used car market in Metropolis, we see how these laws interact. The car owners (suppliers) are willing to sell each car at a price that reflects its value. Potential buyers (demanders) base their purchasing decisions on the average value of the cars, not knowing the individual worth of each vehicle. This interaction between what suppliers are willing to sell for and what demanders are willing to pay leads us to another crucial concept: the equilibrium price, which is where the quantity supplied equals the quantity demanded.

Equilibrium Price

The equilibrium price, also known as the market-clearing price, is the price at which the amount of goods producers want to sell is equal to the amount that consumers want to buy. It represents a balance in the market where there's no leftover supply or unmet demand at that price.

In our Metropolis example, car demanders are using the average value of all cars to determine how much they're willing to pay. When this estimated average value corresponds to the equilibrium price, transactions occur. However, as the exercise revealed, if assumptions are incorrect or if there's a disparity between the perceived average value and the actual average value, the equilibrium can fluctuate, leading to a potential mismatch between supply and demand. This can result in an imbalance where the actual trade may not occur at the presumed equilibrium price, necessitating a reevaluation of the market dynamics and, by extension, the equilibrium price.

Average Value Calculation

The average value calculation is a mathematical tool used to determine the central or 'typical' value of a data set. It's often used by economists and statisticians to simplify complex information about markets, particularly when dealing with diverse products like used cars with a range of values.

In our scenario with the Metropolis used car market, the average value calculation was used to estimate the mean value of the cars that potential buyers consider when evaluating their willingness to pay. Initially, the average was based on the full range of car values, but as only half of the supply was viable for trade, the average value calculation helped to adjust the demanders' perception of the market. This recalculation subsequently impacted the equilibrium price and the quantity of cars traded. This clearly shows how average value—which may seem to be a simple statistical measure—can have profound effects on market perception and real-world economic outcomes.