A Dynamic Game of R and D: Patent Protection and Competitive Behavior

A theory of dynamic optimal resource allocation to R and D in an n-firm industry is developed using differential games. This technique represents a synthesis of the analytic methods previously applied to the problem: static game theory and optimal control. The use of particular functional forms allows the computation and detailed discussion of the Nash equilibrium in investment rules. THIS PAPER ADDRESSES THE PROBLEM of resource allocation to research and development. Among the important issues that a firm engaging in R and D must evaluate are: uncertainty regarding the feasibility and profitability of a particular innovation; the possibility of a protracted development period; the possibility that a rival firm may innovate first, capturing either a patent or a significant share of the new market; the possibility that a rival firm may imitate the innovation and appropriate some of the profits in the new market. In what follows, we will develop a theory of optimal resource allocation to research and development which incorporates the aspects of R and D enumerated above. We will use a dynamic game theoretic analysis, determining the Nash equilibrium strategies for n identical firms. The availability of perfect patent protection is shown to accelerate development of the innovation, and the effect of increasing rivalry is addressed. The impact of increasing rivalry on Nash equilibrium investment in R and D depends upon the degree of appropriability of rewards. If patent protection is perfect, then increasing the number of Nash rivals results in increased R and D effort. However, when imitation is rewarded, the opposite may be true. Finally, some notions of competitive and perfectly competitive equilibrium are examined in the context of the model developed below.