Each firm in a competitive research industry can undertake one project at a constant cost, f = $960 to try and develop a certain new product. The inverse demand curve for the new product is expected to be p = 100 - 2Q, and its marginal cost of production is expected to equal $20 (with no fixed production costs).
The interest rate r is 5%. The probability ρ(n) that at least one of n firms will succeed is given by the following table:
Using these rounded figures rather than the values of the function they approximate, and using ρ (n+1) - ρ (n) as an approximation for ρ'(n), calculate,
(a) The socially optimal number of firms in an R&D race, n*.
(b) The number of firms that would race to develop the product if the winner were granted a permanent patent (monopoly profits forever). Does it make sense to grant such patents?
(c) Calculate the government prize, P that would induce the optimal amount of research.