Economic growth and economic development 653

Introduction to Modern Economic Growth where v−1 represents the value of any follower (irrespective of how many steps behind it is) The maximization problems involved in the value functions are straightforward and immediately yield the following profit-maximizing R&D decisions µ ả ắ ẵ [vn+1 ] 01 (14.56) ,0 zn = max G ả ắ ẵ [v0 − v−n ] ∗ 0−1 z−n = max G (14.57) ,0 ả ắ ẵ [v1 v0 ] ∗ 0−1 z0 = max G (14.58) ,0 , ω∗ where G0−1 (·) is the inverse of the derivative of the G function, and since G is twice continuously differentiable and strictly concave, G0−1 is continuously differentiable and strictly increasing These equations therefore imply that innovation rates, the zn∗ ’s, are increasing in the incremental value of moving to the next step and decreasing in the cost of R&D, as measured by the normalized wage rate, ω∗ Note also that since G0 (0) > 0, these R&D levels can be equal to zero, which is taken care of by the max operator The response of innovation rates, zn∗ , to the increments in values, vn+1 − , is the key economic force in this model For example, a policy that reduces the patent protection of leaders that are n + steps ahead (by increasing κ) will make being n + steps ahead less profitable, thus reduce vn+1 − and zn∗ This corresponds to the standard disincentive effect of relaxing IPR protection However, relaxing IPR protection may also create a beneficial composition effect; this is because, typically, ∗ ∗ {vn+1 − }∞ n=0 is a decreasing sequence, which implies that zn−1 is higher than zn for n ≥ (see Proposition 14.8 below) Weaker patent protection (in the form of shorter patent lengths) will shift more industries into the neck-and-neck state and potentially increase the equilibrium level of R&D in the economy Given the equilibrium R&D decisions, the steady-state distribution of industries across states µ∗ has to satisfy the following accounting identities: ¡ Â (14.59) zn+1 + z1 + àn+1 = zn∗ µ∗n for n ≥ 1, (14.60) (14.61) ¡ Â z1 + z1 + à1 = 2z0∗ µ∗0 , ∗ 2z0∗ µ∗0 = z−1 + κ 639