Exercise 11.3. Consider the following continuous time neoclassical growth model: U (0) = Z ∞ 0 exp (−ρt) (c (t))1−θ − 1 1 − θ , with aggregate production function Y (t) = AK (t) + BL(t),

where A, B > 0. (1) Define a competitive equilibrium for this economy. (2) Set up the current-value Hamiltonian for an individual and characterize the necessary conditions for consumer maximization. Combine these with equilibrium factor market prices and derive the equilibrium path. Show that the equilibrium path displays non-trivial transitional dynamics. (3) Determine the evolution of the labor share of national income over time. (4) Analyze the impact of an unanticipated increase in B on the equilibrium path. (5) Prove that the equilibrium is Pareto optimal.