Consider the BMI–ECN example of a patent race. Assume that demand for the new good is P = 100 − 2Q, and that each firm believes that it will be able to produce this good at a unit cost of c = $50. Assume further that the discount factor R is so small that each firm cares only about the one-period profit it will make. (Alternatively, assume that one period is of a very long duration, say, thirty years or more.) The probability that such a lab will be successful and actually produce a discovery is ρ = 0.8.
a. Show that if one firm is successful in introducing the product, it will have a monopoly price of $75, sell 12.5 units, and earn monopoly profits (before paying for the research) of M = $312.50. Show also that consumer surplus is $156.25.
b. Show that if each firm sets up a lab and if both labs are successful, then each firm will earn a profit (before paying for the research) of $138.89. Consumer surplus in this case will be $277.78
c. Now show that the expected profit to BMI (or ECN) if it is the only firm to establish an R&D division is $250 − K while the expected profit to each firm if they both establish R&D divisions is $138.89 − K. Use these results to construct the payoff matrix for this case, now including the cost, K, of establishing an R&D division.
d. Show that if K, the cost of setting up the research lab, is such that K > $250, neither firm will set up a lab, while if K < $138.89,=”” both=”” firms=”” will=”” set=”” up=”” a=”” lab.=””>
e. Show that expected social surplus ignoring research costs if one firm establishes a research lab is $375, and if two research labs are established, is $416.67. Hence, show that the second lab is socially desirable only if K <>