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Suppose n identical firms are racing to introduce the same innovation. The per-firm (fixed) cost of R&D is $50. R&D occurs at time t = 0. Assume for simplicity that if the innovation is developed at t = 0, then society benefits in all later periods; but if the innovation is not developed at t = 0, then research does not take place in subsequent periods. If one firm invests in R&D, it will succeed with 25% probability. Assume for simplicity that all successful firms have equal chance of obtaining the patent, and the patent never expires. The discount rate is 10%.

1. What is the expected present value of the patent when five firms invest in R&D? Will five or more firms invest in R&D? Explain.

2. How many firms will invest in R&D? Explain. Note: To support your answer, you can calculate the expected present value of the patent for n = 1, ...6 in Excel.

3. Suppose two firms invest in R&D. What is the expected discounted social value over lifetime? What is the net expected discounted social value over lifetime?

4. What is the socially optimal number of firms investing in R&D? Note: To support your answer, you can calculate net social value for n = 1, ...6 in Excel.

5. How does the equilibrium number of firms in the patent race compare to the socially optimal number? Illustrate your answer on a diagram showing (i) expected present value of the patent; (ii) expected discounted social value; (iii) social cost; (iv) the equilibrium number of investing firms, and (v) the socially optimal number of investing firms.

6. How can the government ensure that the equilibrium number of firms is equal to the socially optimal number?

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