Exercise 11.4. Consider the following continuous time neoclassical growth model: U (0) = Z ∞ 0 exp (−ρt) (c (t))1−θ − 1 1 − θ , with production function Y (t) = A h L(t) σ−1 σ + K (t) σ−1 σ i σ σ−1 . (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. (3) Prove that the equilibrium is Pareto optimal in this case. (4) Show that if σ ≤ 1, sustained growth is not possible. (5) Show that if A and σ are sufficiently high, this model generates asymptotically sustained growth due to capital accumulation. Interpret this result. (6) Characterize the transitional dynamics of the equilibrium path. (7) What is happening to the share of capital in national income? Is this plausible? How would you modify the model to make sure that the share of capital in national income remains constant? (8) Now assume that returns from capital are taxed at the rate τ . Determine the asymptotic growth rate of consumption and output.