Question

Static patent race model. Assume n firms choose their R\&D investment and the probability of aning depends only on their R&D expenditure and their rival's R&D expenditure. Each firm oses its effort level x>0. The probability that a firm wins the race is equal to its effort level ided by the sum of *all* firms' effort levels. In other words, the probability of firm i winning the e (what we called the contest success function) is given by pi(success) =xi/Σn x n

. The cost of R\&D c, and assume that the patent is worth 10 (in other words the profits from winning the success given by π=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.)

Answer 1

In the static patent race model with two firms, solving for optimal R&D investment by setting up a profit maximization problem and invoking symmetry does not yield a positive effort level when costs are fixed and benefits depend only on relative effort due to the structure of the problem.

Step-by-step explanation:

To solve for the optimal level of R&D investment (x*) in the static patent race model where R&D expenditure directly influences the probability of winning the patent race, we begin by establishing the profit maximization problem for a generic firm. Given that the victory benefits are π=10 and the cost is c=5, the expected profit for firm i can be defined as:

πi = (probability of success) * (benefits) - (cost) * (effort) = (xi/Σnxn) * 10 - 5*xi

With 2 firms in the race (n=2), when we invoke symmetry (x1=x2=x*), the calculation simplifies because the denominator in the probability equation becomes 2x*. Hence, the expected profit for either firm would become:

πi = (x*/(2x*)) * 10 - 5*x* = (1/2)*10 - 5x* = 5 - 5x*

Maximizing this expected profit involves taking the derivative with respect to x* and setting it to zero:

dπi/dx* = 0 - 5 = -5

However, this derivative does not yield a maximum since it suggests a negative slope throughout. Therefore, the symmetrical effort level that maximizes expected profits for the firms would be x* = 0, which contradicts the requirement that each firm chooses x > 0. In this scenario, our base assumption or model may need reevaluation since in a setup with a fixed positive cost and benefits dependent only on relative effort, firms might lean towards minimal R&D effort due to the cost structure.

This demonstrates the intricacies of R&D investment decisions within a patent race, where spillover benefits and competitive dynamics play a significant role in determining corporate strategy.