Introduction to Modern Economic Growth Exercise 13.2 Derive (13.9) and (13.10) from the profit maximization problem of a monopolist Exercise 13.3 Formulate the consumer optimization problem in terms of the current-value Hamiltonian and derive the necessary conditions Show that these are equivalent to (13.16) and (13.17) Exercise 13.4 Consider the expanding variety model of Section 13.1 and denote the BGP growth rates of consumption and total output by gC∗ and g ∗ (1) Show that gC∗ > g ∗ is not feasible (2) Show that gC∗ < g ∗ violates the transversality condition Exercise 13.5 This exercise asks you to construct and analyze the equivalent of the lab-equipment expanding variety model of Section 13.1 in discrete time Suppose that the economy admits a representative household with preferences at time given by ∞ X t=0 βt C (t)1−θ − , 1−θ with β ∈ (0, 1) and θ ≥ Production technology is the same as in the text and the innovation possibilities frontier of the economy is given by N (t + 1) − N (t) = ηZ (t) (1) Define an equilibrium (2) Characterize the balanced growth path equilibrium and compare the structure of the equilibrium to that in Section 13.1 (3) Show that there are no transitional dynamics, so that starting with any N (0) > 0, the economy grows at a constant rate Exercise 13.6 Complete the proof of Proposition 13.1, in particular, showing that condition (13.21) is sufficient for the transversality condition to be satisfied Exercise 13.7 Derive the consumption growth rate in the socially-planned economy, (13.22) Exercise 13.8 Consider the expanding input variety model of Section 13.1 Show that it is possible for the equilibrium allocation to satisfy the transversality condition, while the social planner’s solution may violate it Interpret this result Does it imply that the social planner’s allocation is less compelling? 602