Consider a used car market with 600 buyers and 500 sellers. 400 cars are of good quality (peaches) and 100 are of bad quality (lemons). A buyer's valuation is Rs. 100 for a peach a... Consider a used car market with 600 buyers and 500 sellers. 400 cars are of good quality (peaches) and 100 are of bad quality (lemons). A buyer's valuation is Rs. 100 for a peach and Rs. 10 for a lemon. A seller's valuation is Rs. 50 for a peach and Rs. 10 for a lemon. Sellers know the quality of their cars, but buyers do not. (a) What outcome maximizes the aggregate surplus of the economy? Explain. (b) (i) Derive the supply of used cars as a function of price and draw the supply curve. (ii) Derive the demand for used cars as a function of price and draw the demand curve. (iii) Find all possible competitive equilibria, mentioning which types of cars are bought and sold in each equilibrium. (c) Now suppose that buyers know the identity of all cars. Use a similar demand-supply analysis to solve for all possible competitive equilibria. Read more

Steps to Solve

1. Surplus-Maximizing Outcome Under Asymmetric Information

To maximize surplus, all cars should be sold since the lowest value of a car ($v_L = 1$) is greater than the highest value to a buyer ($b_L = 1.5$).

2. Supply Curve Under Asymmetric Information

The supply curve depends on the proportion of peaches ($\alpha$) and lemons ($1-\alpha$). A seller will offer a car if its value is less than or equal to the price $P$. Thus:

If $P < 1$, no cars are supplied. If $1 \le P < 2$, only lemons are supplied. The supply is $(1-\alpha)$. If $P \ge 2$, both lemons and peaches are supplied. The supply is $1$.

3. Demand Curve Under Asymmetric Information

Buyers are willing to pay their expected value for a car. Given the proportion of peaches and lemons, the expected value $E[V]$ is:

$E[V] = \alpha b_H + (1 - \alpha)b_L = \alpha(3) + (1 - \alpha)(1.5) = 3\alpha + 1.5 - 1.5\alpha = 1.5\alpha + 1.5$

So, the demand curve is a horizontal line at $P = 1.5\alpha + 1.5$. However, buyers will only buy if their expected value is greater than or equal to the price.

4. Competitive Equilibria Under Asymmetric Information

Equilibrium occurs where supply equals demand.

• If $1.5\alpha + 1.5 < 1$, no cars are sold. This requires $\alpha < -\frac{1}{3}$ which is impossible because $\alpha$ must be between 0 and 1. So, this case is impossible.
• If $1 \le 1.5\alpha + 1.5 < 2$, only lemons are supplied. The price would be $P = 1.5\alpha + 1.5$. The quantity sold would be $1-\alpha$. This requires $1 \le 1.5\alpha + 1.5 < 2$ or $-\frac{1}{3} \le \alpha < \frac{1}{3}$. Since $\alpha \ge 0$, we have $0 \le \alpha < \frac{1}{3}$.
• If $1.5\alpha + 1.5 \ge 2$, All cars are sold. The price is $P = 1.5\alpha + 1.5$. This requires $1.5\alpha + 1.5 \ge 2$ or $\alpha \ge \frac{1}{3}$.

Therefore:

• If $0 \le \alpha < \frac{1}{3}$, $P = 1.5\alpha + 1.5$ and quantity $Q = 1 - \alpha$.
• If $\frac{1}{3} \le \alpha \le 1$, $P = 1.5\alpha + 1.5$ and quantity $Q = 1$.

5. Surplus-Maximizing Outcome Under Perfect Information

Under perfect information, all cars should still be sold because the buyers' values are greater than the sellers' values i.e. $b_L > v_L$ and $b_H > v_H$.

6. Supply Curve Under Perfect Information

The supply curve is now different for lemons and peaches.

• For lemons: If $P \ge 1$, all lemons are supplied.
• For peaches: If $P \ge 2$, all peaches are supplied.

7. Demand Curve Under Perfect Information

The demand curve reflects the true values of the cars.

• For lemons: Buyers are willing to pay $1.5$.
• For peaches: Buyers are willing to pay $3$.

8. Competitive Equilibria Under Perfect Information

With perfect information, there are two separate markets: one for lemons and one for peaches.

• Lemons: Price $P = 1.5$, quantity $Q = 1 - \alpha$.
• Peaches: Price $P = 3$, quantity $Q = \alpha$.