Growth Theory II - 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:

Explain why perfect competition with constant returns to scale cannot endogenize technological progress, using Euler’s theorem
Derive Rebelo’s steady-state growth condition and classify the four cases in the taxonomy of growth possibilities
Solve the AK model and explain why it has no transitional dynamics and no convergence across countries
Distinguish firm-level CRS from aggregate IRS in the Romer (1986) model and derive the wedge between private and social growth rates
Describe the Lucas (1988) human capital model and explain how the externality parameter $\psi$ interacts with risk aversion $\rho$ to determine growth
Explain why finite horizons break Ricardian equivalence in the Blanchard (1985) model and how adjustment costs affect convergence speed in the q-Ramsey model

Euler’s theorem and the growth problem: with CRS and perfect competition, $Y = F_K K + F_L L$ exhausts all income on factor payments, leaving no resources for R&D
Rebelo’s taxonomy: the equation $0 = (\alpha - 1)\gamma + n(\alpha + \beta - 1)$ catalogs all possible steady-state growth outcomes for a general Cobb-Douglas economy
AK model: $Y = AK$ with constant marginal product of capital; growth rate $\dot{C}/C = \rho^{-1}(A - \delta_t)$ is constant and proportional to the saving rate
Knowledge externalities (Romer 1986): aggregate knowledge $\Xi = K$ creates IRS at the economy level while each firm faces CRS; the private growth rate $\rho^{-1}(\alpha - \delta_t)$ is below the social optimum $\rho^{-1}(1 - \delta_t)$
Broad capital (Lucas 1988): defining $\kappa = k^\alpha h^{1-\alpha}$ reduces the model to an AK structure; the engine of growth is human capital accumulation $\dot{h}/h = \phi(1-u)$
Blanchard perpetual youth (1985): finite-horizon consumers discount the future more heavily than an infinitely lived dynasty, reducing capital accumulation below the RCK level and breaking Ricardian equivalence
q-Ramsey model: quadratic adjustment costs slow convergence without changing the steady state; both investment smoothing (higher $\xi$ ) and consumption smoothing (higher $\rho$ ) have similar effects on convergence speed