# Question: Doctors Smith and Jones two rival ophthalmologists plan to enter

Doctors Smith and Jones, two rival ophthalmologists, plan to enter the market for laser corrective eye surgery in a medium-size community in Illinois. Smith and Jones both utilize the same laser technology and face constant long-run costs of \$1,000 per patient (LMC = LAC = \$1,000). For planning purposes, the doctors assume that all patients have both eyes corrected, so the \$1,000 cost is for correcting both eyes of each patient. The doctors know about each other’s plans to enter the market, and they both believe the (inverse) market demand for laser eye surgery is
P = 7,000 − 2Q
where Q = qs + qJ and qs and qJ are the number of patients Drs. Smith and Jones, respectively, treat annually. Each of the doctors will choose the size of their clinic, and thus the number of patients they intend to treat each year, without knowledge of their rival’s decision. Demand and cost conditions are common knowledge.
a. Using calculus, derive the equations for the best-response curves.
b. Sketch a graph of the two best-response curves. Be sure to label both axes and both response curves.
c. If Dr. Smith expects Dr. Jones to treat 500 patients annually, what is Dr. Smith’s best response? If Dr. Jones predicts Dr. Smith will treat 750 patients annually, what is Dr. Jones’s best response?
d. Find Nash equilibrium. How much profit does each doctor earn annually in Nash equilibrium?
e. If Drs. Smith and Jones agree to serve annually only 750 patients each (1,500 patients in total in one year), how much annual profit does each doctor earn? Why don’t they do this?
f. If Drs. Smith and Jones merge into a single company forming a monopoly in the community, how many patients will they serve each year? What price will they charge? How much profit will they earn?
g. Instead of the monopoly in part f, suppose perfect competition characterizes the market in the long run. How many patients will be treated? What price will they pay in a competitive market? How much profit will the doctors earn?

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