The Effect of IP Protection on Radical and Incremental Innovation

Abstract

This paper examines the effects of risk in shaping and developing innovations at the country level by introducing the idea that strong patent protection can lead innovators to rest on their laurels. Specifically, in the context of a simple static framework, the model shows that there is a negative relationship between the ability of the economy to innovate and how strong patent protection should be.

Appendix

The winner will maximize its profits π _i subject to x. This maximization problem has the following FOC,

$$ x = {\left( {\frac{{{a^2}{A_i}}}{\kappa }} \right)^{{\frac{1}{{1 - a}}}}} $$

(5)

Since the firm must choose its s at the beginning of the patent race, before its innovation is materialized, firms solve the following problem, maximizing expected profits times the expected probability of winning the patent race, i.e. $\mathop{{\max }}\limits_{{{s_i}}} {p_i}{A_i}$. Using Eqs. 4–5, the FOC from this maximization is given by the following implicit function, $F\left( {{s_i}} \right) = \frac{{\partial {p_i}}}{{\partial {s_i}}}{\pi_i} + {p_i}\frac{{\partial {\pi_i}}}{{\partial {s_i}}} = 0$. Using the implicit function theorem on F(s _i ), accounting for Eq. 2 the following condition is derived,

$$ \frac{{\partial {s_i}}}{{\partial {A_j}}} = - \frac{{\left( {1 - 2a} \right)\frac{{\partial {p_i}}}{{\partial {A_j}}} + {{\left( {1 - a} \right)}^2}{A_i}\frac{{{\partial^2}{p_i}}}{{\partial {A_i}\partial {A_j}}}}}{{{M_i}\left( {2\left( {1 - 2a} \right)\frac{{\partial {p_i}}}{{\partial {A_i}}} + {{\left( {1 - a} \right)}^2}{A_i}\frac{{{\partial^2}{p_i}}}{{\partial A_i^2}}} \right)}} $$

(6)

In Eq. 6, if $a \in \left( {0.5,1} \right)$, then $\frac{{\partial {s_i}}}{{\partial {A_j}}}$ is always greater than zero if (a) M _i  > 0, and (b) $\frac{{\partial {p_i}}}{{\partial {A_j}}} > - \frac{{{{\left( {1 - a} \right)}^2}{A_i}}}{{\left( {1 - 2a} \right)}}\frac{{{\partial^2}{p_i}}}{{\partial {A_i}\partial {A_j}}}$. Furthermore, if $a \in \left( {0,0.5} \right)$, then $\frac{{\partial {s_i}}}{{\partial {A_j}}}$ is always greater than zero if a) M _i  > 0, and b) $\frac{{\partial {p_i}}}{{\partial {A_i}}} > - \frac{{{{\left( {1 - a} \right)}^2}{A_i}}}{{2\left( {1 - 2a} \right)}}\frac{{{\partial^2}{p_i}}}{{\partial A_i^2}}$.

Overall, the above discussion allows one to concentrate on the following question, how will the leader respond to an increase in A _j ? As it turns out, if μ _i  > γ and $\frac{{\partial {p_i}}}{{\partial {A_j}}} > - \frac{{{{\left( {1 - a} \right)}^2}{A_i}}}{{\left( {1 - 2a} \right)}}\frac{{{\partial^2}{p_i}}}{{\partial {A_i}\partial {A_j}}}$, $\frac{{\partial {p_i}}}{{\partial {A_i}}} > - \frac{{{{\left( {1 - a} \right)}^2}{A_i}}}{{2\left( {1 - 2a} \right)}}\frac{{{\partial^2}{p_i}}}{{\partial A_i^2}}$ any increase in A _j should ceteris paribus lead i to respond by adopting a greater s _i ; by assumption $\frac{{\partial {p_i}}}{{\partial {A_i}}} > \frac{{{\partial^2}{p_i}}}{{\partial A_i^2}}$ and $\frac{{\partial {p_i}}}{{\partial {A_j}}} > \frac{{{\partial^2}{p_i}}}{{\partial {A_i}\partial {A_j}}}$, thus the above inequalities always hold. In a similar fashion, if μ _j  > γ, when the follower j faces the leader’s higher technology it must also adopt a greater s _j in its innovation process; because $\frac{{\partial {s_j}}}{{\partial {A_i}}}$. The above intuition implies that if between two periods there is an increase in λ, which leads to more competition, then there will be an increase in s. As a result, considering that $\Delta {A_i} = \gamma + {s_i}\left( {{\mu_i} - \gamma } \right)$, if μ _j  > γ, increases in λ lead to a greater s, a greater expected innovation and a greater expected technology A _i .¹³