Consider the following continuous-time neoclassical growth model:
(a) Define and characterize a competitive equilibrium for this economy.
(b) Prove that the equilibrium is Pareto optimal.
(c) Show that if σ ≤ 1, sustained growth is not possible.
(d) Show that if A and σ are sufficiently large, this model generates asymptotically sustained growth because of capital accumulation. Interpret this result.
(e) Characterize the transitional dynamics of the equilibrium path.
(f ) What is happening to the share of capital in national income? Is this behavior plausible? How would you modify the model to ensure that the share of capital in national income remains constant?
(g) Now assume that returns from capital are taxed at the rate τ . Determine the asymptotic growth rate of consumption and output.