Consider the following version of the lemons problem. There is a continuum of buyers and sellers in the market; the total mass of each group is 1. Each seller has one car to sell and each buyer wishes to buy at most one car, but only sellers know the quality of their cars before trading. It is common knowledge however that the quality of cars, denoted s, is drawn from a uniform distribution on the interval [0, 1] (hence, the probability that a car’s quality is below some number x is equal to x if 0 ≤ x ≤ 1 and is equal to 1 if x ≥ 1). It is also common knowledge that a fraction α the sellers are of type 1 and have a payoff U₁ = p – s/8 if they sell their cars and 0 otherwise, and a fraction 1 – α of the sellers are of type 2 and their payoff is U₂ = p – s/4 if they sell their cars and 0 otherwise, where p is the price of the car (note that the two types of sellers differ only with respect to their payoffs but not with respect to the quality of cars they have to sell). There is a continuum of buyer types: the payoff of a type-buyer if he buys a car whose quality is s is U(θ) =θs - p, where θis distributed uniformly on the unit interval. If a buyer does not buy a car his payoff is 0. The buyers cannot observe the quality of cars before they buy nor can they observe the type of seller they face.

1. Compute the supply of cars by type 1 sellers, type 2 sellers, and the aggregate supply of cars (i.e., compute the fraction of cars that will be supplied at a given price by each type of sellers and then add the two to obtain the aggregate supply). Show your answer in a figure.

2. Let ŝ (p) denote the average quality of cars supplied on the market as a function of p. Using your answer to (1), compute ŝ (p). How does ŝ (p) vary with p and with  ? Explain the intuition for this.

3. Assume that buyers correctly anticipate ŝ (p) and compute the demand for cars (i.e., the fraction of buyers that will wish to buy a car at a given price) and show your answer in the figure you drew in Part (1). Explain the shape of the demand function.

4. Assume that the market is perfectly competitive and solve for the equilibrium price, p^*.