Exercise 14.24. Consider the model of Section 14.3 and suppose that the R&D technology of the incumbents for innovation is such that if an incumbent with a machine of quality q spends an amount zq for incremental innovations, then the flow rate of innovation is φ (z) (and this innovation again increases the quality of the incumbent’s machine to λq). Assume that φ (z) is strictly increasing, strictly concave, continuously differentiable, and satisfies limz→0 φ0 (z) = ∞ and limz→∞ φ0 (z)=0. (1) Focus on steady-state equilibria and conjecture that V (q) = vq. Using this conjecture, show that incumbents will choose R&D intensity z∗ such that (λ − 1) v = φ0 (z∗). Combining this equation with the free entry condition for entrants and the equation for growth rate given by (14.55), show that there exists a unique BGP equilibrium (under the conjecture that V (q) is linear). (2) Is it possible for an equilibrium to involve different levels of z for incumbents with different quality machines? (3) In light of your answer to 2, what happens if we consider the “limiting case” of this model where φ (z)= constant? (4) Show that this equilibrium involves positive R&D both by incumbents and entrants. (5) Now introduce taxes on R&D by incumbents and entrants at the rates τ i and τ

e. Show that, in contrast to the results in Proposition 14.7, the effects of both taxes on growth are ambiguous. What happens if η (z) =constant?