Consider the market for medical services. Suppose there are three types of doctors:

i Quacks (Q), who are unqualified and likely to hurt a patient. These doctors are willing to offer their services for $100. The fraction of doctors that are quacks is q.

ii Regular doctors (R), who are reasonably skilled and can treat many diseases. These doctors are willing to offer their services for $1,000. The fraction of doctors that are regular is r.

iii Super doctors (S), who are very skilled and can treat most diseases. These doctors are willing to offer their services for $4,000. The fraction of doctors that are super is 1 q r.

Suppose consumers value the medical services of a quack at -$100 (i.e. they would pay NOT to see a quack), they value the services of a regular doctor at $2,000, and they value the services of a super doctor at $4,500. Consumers cannot tell whether a specific doctor is a quack, a regular, or a super doctor, but they do know the fractions of each type of doctor (q, r) and how much money each type is willing to work for. There are many potential customers and a limited supply of doctors, so that the prevailing price will be the maximum price consumers are willing to pay.

(a) Suppose that q = .25 and r = .5. What price will prevail for medical services? What types of doctors will offer their services?

(b) Again let q = .25 and r = .5. Suppose the government is able to identify quacks and stop them from practicing medicine (perhaps they put the quacks in jail). What price will prevail for medical services? What types of doctors will offer their services?

(c) Now suppose q = .1 and r is unknown. For what values of r does an equilibrium exist where all types of doctors offer their services?