Growth Theory I - Modeling Macroeconomics

Materials¶

The following computational notebook accompanies the source chapters in the Reading Assignment below:

By the end of this lecture, students will be able to:

Derive the Euler equation for consumption growth in the RCK model using the Hamiltonian and interpret it as the modified golden rule
Construct the phase diagram with the $\dot{k}=0$ and $\dot{c}/c=0$ loci and explain the saddle-path dynamics
Show that the decentralized competitive equilibrium replicates the social planner’s solution and state the conditions under which this result breaks down
Explain why Ricardian equivalence holds with lump-sum taxes and infinite horizons
Derive the effect of a capital income tax on the steady-state capital stock and explain why a lump-sum rebate does not restore efficiency
Determine the sign of the saving-growth relationship in the RCK model as a function of risk aversion

Modified golden rule (Keynes-Ramsey rule): the Euler equation $\dot{c}/c = \rho^{-1}[f'(k) - \delta - \delta_t - n - \rho g]$ that governs consumption growth along the transition path
Golden rule capital stock: the level of $k$ that maximizes steady-state consumption, where $f'(k^{GR}) = n + g + \delta$ ; the RCK steady state falls short of the golden rule when consumers are impatient
Saddle path: the unique trajectory in the $(k, c)$ plane that satisfies both the Euler equation and the transversality condition, converging to the steady state
First Welfare Theorem: in the RCK model without externalities or distortionary taxes, the decentralized equilibrium is Pareto optimal
Ricardian equivalence: with infinite horizons and lump-sum taxes, the timing of government taxation has no effect on consumption or capital accumulation
Capital income tax distortion: a tax on capital income reduces the after-tax return to saving and lowers the steady-state capital stock, even when revenue is rebated lump-sum