5.A static patent race model. Assume n firms choose their R\&D investment and the probability of winning depends only on their R\&D expenditure and their rival's R\&D expenditure. Each firm chooses its effort level x>0. The probability that a firm wins the race is equal to its effort level divided by the sum of "all* firms' effort levels. In other words, the probability of firm i winning the race (what we called the contest success function) is given by pi (success) =nxnxi. The cost of R\&D is xc, and assume that the patent is worth 10 (in other words the profits from winning the success are given by =10 ) and that c=5. a. What is the optimal level of effort of each firm (i.e. x ) if n=2 ? (Hint: setup a profit maximization problem as we did in the "Competition and Innovation" lecture. As all firms face the same problem, solve for a generic firm's solution - once you take the derivative with respect to the firm's effort level, invoke symmetry that x1=x2=x. The resulting x will characterize the behavior of all of the firms in the race.) b. Solve for the optimal level of effort if n=4. c. How does the optimal level of effort exerted by each firm change as more firms effort? What does this tell us about the role of competition in incentivizing innovation