Problem 5: Endogenous Innovationl (20 points) Answer the following questions for a Quality Improvement Model. Recall what you learn in the Variety Expansion Model. You may also use the textbook for reference. (3.) Consider the following production function of the competitive nal goods producer with quality q(v,t) ; 15 What is the optimal input variety demand as a function of quality-price ratio and labor demand? (2 points) Y(t) = [/0 q(v,t)m(v,t)1_'8dv L(t)5, (b) Suppose the marginal cost of producing one unit of $(U, t) input variety is (1 [3)q('u, t) units of nal goods. What is the prot of a monopolist that produces a variety? (2 points) (c) Suppose a successful new innovation improves the quality of a variety '0 E [0, l] by a factor of A > 1. Find the range of A (in terms of parameters) such that the monopolist of the improved variety can charge a monopoly price without worrying about the threat of the incumbent which does not have the improved quality. (4 points) (d) Suppose 2(1), 75) is the ow probability of innovation given R&D investment Z (t), t) on variety o E [0, l] and 2(1), t) = \"Z (out) / (1(1), 13). How does the aggregate quality improvement depends on aggregate R&D? (2 points) (e) How does the free entry condition relates patent value to quality? (2 points) (f) Consider constant 2(1), 13) = 2*, which is also the ow probability that the existing variety is replaced by an improved version. How does the value of a patent depend on its prot? (2 points) (g) Derive the steady state growth rate? (4 points) (h) Can competitive equilibrium induces too much growth? Briey discuss it. (2 points)