Introduction to Modern Economic Growth Exercise 13.18 Consider the model in Section 13.3 Suppose that there are two economies with identical preferences, technology and initial conditions, except country starts with population L1 (0) and country starts with L2 (0) > L1 (0) Show that income per capita is always higher in country than in country Exercise 13.19 Consider the lab equipment model of Section 13.1, but modify the innovation possibilities frontier to N˙ (t) = ηN (t)−φ Z (t) , where φ > (1) Define an equilibrium (2) Characterize the market clearing factor prices and determine the free entry condition (3) Show that without population growth, there will be no sustained growth in this economy (4) Now consider population growth at the exponential rate n, and show that this model generates sustained equilibrium growth as in the model analyzed in Section 13.3 Exercise 13.20 Consider the baseline endogenous technological change model with expanding machine varieties in Section 13.1 Suppose that x’s now denote machines that not immediately depreciate In contrast, once produced these machines depreciate as an exponential rate δ References and the rest of the production structure remain unchanged (1) Define an equilibrium in this economy (2) Formulate the maximization problem of producers of machines [Hint: it is easier to formulate the problem in terms of machine rentals rather than machine sales] (3) Characterize the equilibrium in this economy and show that all the results are identical to those in Section 13.1 Exercise 13.21 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 605