Exercise 13.23. Consider the following model. Population at time t is L(t) and grows at the constant rate n (i.e., L˙ (t) = nL(t)). All agents have preferences given by Z ∞ 0 exp (−ρt) C(t)1−θ − 1 1 − θ dt, where C is consumption defined over the final good of the economy. This good is produced as Y (t) = •Z N 0 y (ν, t) β dν¸1/β , where y (ν, t) is the amount of intermediate good ν used in production at time t. The production function of each intermediate is y (ν, t) = l(ν, t) where l(ν, t) is labor allocated to this good at time t. New goods are produced by allocating workers to the R&D process, with the production function N˙ (t) = ηNφ (t)LR (t) where φ ≤ 1 and LR (t) is labor allocated to R&D at time t. So labor market clearing requires R N(t) 0 l(ν, t) dv + LR (t) = L(t). Risk-neutral firms hire workers for R&D. A firm who discovers a new good becomes the monopoly supplier, with a perfectly and indefinitely enforced patent. (1) Characterize the balanced growth path in the case where φ = 1 and n = 0, and show that there are no transitional dynamics. Why is this? Why does the long-run growth rate depend on θ? Why does the growth rate depend on L? Do you find this plausible? (2) Now suppose that φ = 1 and n > 0. What happens? Interpret. (3) Now characterize the balanced growth path when φ < 1 and n > 0. Does the growth rate depend on L? Does it depend on n? Why? Do you think that the configuration φ < 1 and n > 0 is more plausible than the one with φ = 1 and n = 0?