Question

Consider a market for used cars. There are 100 sellers and 100 buyers. The sellers know the qualities of their cars, but the buyers only know that the quality θ of each used card is uniformly distributed over interval [0, 1]. The seller’s value of a car with quality θ is θ. (Thus, market supply is S(p) = 100p for 0 ≤ p ≤ 1 and S(p) = 100 for p > 1.) Each of 50 buyers values a car with expected quality θ at 2θ, but each of the other 50 buyers values such a car at 1.5θ. Assume (i) buyers and sellers are price-taking and (ii) buyers buy and sellers sell when they are indifferent.

a) Given price p ≥ 0, determine the expected quality of the cars supplied at price p.

b) Find market demand.

c) Find a competitive equilibrium. How many cars are traded in equilibrium? What is the expected quality?

Answer 1

Answer:

(i) As given in the question that the seller’s value of a car with quality θ is θ, Also the seller of the car having quality θ is willing to sell his car at price p if and only if

=> P ≥ θ

Therefore, the expected quality of the car supplied at price P will be

(b) Market demand :

θ to 2θ type A buyers are call buyers who value a car at the expected quality as mentioned initially.

And type 2 buyers are those other buyers.

here we need to observe that,

type A  buyer buys at price

and type B buyer buys at price,

Furthermore, the market demand function is

50 if 1.5θ(P)< p ≤ 2θ(p)

100

(c)Find a competitive equilibrium. How many cars are traded in equilibrium? Wha is the expected quality?

Let us assume that P1 >1,

Then S (P1) = 100 . But, as given in part A, θ (P1) = 1/2 and thus P1 >2θ (P1)

Subsequently,

D(P1) = 0.

AND D(P1)*6 = S (P1).

This implies that P1 >1 cannot be satisfied.

Also, and

Following the above outcome,

Similarly, ,D(P1) = 50

Since , S(1/2) = 50* P1 = 1/2 is going to satisfy D(P1) =S(P1).

That clearly means that,

P1 = 1/2 which is in a competitive equilibrium price.

At the price solved above,

the expected quality is,

θ(P1) =P1/2 = 1/4