- 1 Pre-endogeneous growth: new waves
- 1.1 Outlines
- 1.2 Romer (1986 JPE): general/knowledge capital
- 1.3 Lucas (1988 JME): human capital
- 1.4 Stokey (1988 JPE): learning-by-doing
- 1.5 Jones and Manuelli (1990)
- 1.6 Excercise: Physical and Human Capital Spillovers and Growth
- (a) Write down the Current-Value Hamiltonian \( H(c,u,k,h,\lambda) \) and derive the first-order conditions (for \( c, u \)) and the Euler equations (for \( k, h \))
- (b) Derive the Keynes-Ramsey equation
- (c) Determine the balanced growth rate \( \theta \). Under what condition would \( \theta \) be strictly positive?
- (d) Chracterize the balanced growth equilibrium with respect to \( (A, \phi) \). Explain your results intuitively.
- (e) What is an optimal subsidy policy that may fully eliminate the free-rider’s problem? Explain your results intuitively.
Nowadays when you hear of development economics it is probably associated to the works by Duflo and Banejee etc where they popularized the use of rct in study of the poor behavior but traditionally development economics has been a macroeconomics enterprise.
We start with with an aggregate production function i.e. \[ y = A f(k, h) \] where \( k \) is capital per-capita and \( h \) is human capital per-capita.
The philosophy of the World Bank back in 1960s was that poor countries don’t have enough \( k \) and are credit constrained. This comes from the paradigm of “development accounting”: To see how much if this simple macro production function explains differences in cross-country income level. For example, If most of it is driven by \( k \)then the answer to poverty is simple: Just give the developing world more \( k \) or \( h \)!
In reality it is a bit tricky: Differences are mostly driven by \( A \) and it’s much harder to know what to do Mankiw, Romer, and Weil (1992) measured that \( k \) and \( h \) contributed for about 80% of income differences between countires. But subsequent results is less optimistic: Klenow and Rodríguez-Clare (1997) use better human capital data and concluded that \( k \) and \( h \) together explain less than 50%. Similarly, Caselli (2005) argued that \( \frac{2}{3} \) of the variation is due to \( A \). . Moreover, in the original Ramsey model, the steady states implies \[ A \frac{\partial f}{\partial k} = \rho \] which means that returns to capital equals to the discount rate. If \( k \) response to \( A \) in equilibrium, the naive policy of giving \( k \) would not work. Furthermore, phase diagrams analysis shows that if \( k \) is above the steady state level, it would slowly move back to the steady state point.
So, instead the focus of the field has been on the residual, or tfp, or \( A \).
This notes follow the development of growth theory, from the macro-theoretical perspective.
1 Pre-endogeneous growth: new waves
1.1 Outlines
This literature mostly consists of old papers. Chapter 11 in Acemoglu (2009) is the last one in a sequence of chapters that discuss the neoclassical model and not endogeneous growth. Thus, even though these papers introduced in this section should rather be considereed as neoclassical model than endogeneous.
Remember that the neoclassical technology conditions to ensure the existence and uniqueness of the bgp are:
- \( F(K,L;A) \) is concave in \( K, L \) and crs in \( (K, L )\)
- Inada condition: \( \lim_{K, L\rightarrow0} F(\cdot, \cdot) = +\infty, \lim_{K, L\rightarrow\infty} F(\cdot, \cdot) = 0 \).
Back then, the struggle with neoclassical models is that they cannot accomodate for the difference in growth rate between countries. A natural next step is to try endogenize technological accumulation. There are basically two ways you go at it
- Introduce human capital as a second accumulating asset, typically done in ak models.
- This is not fruitful
- Introducing externality/irs, or learning-by-doing.
- However these models requires weird non-convexity of the production set or absence in production factor.
- Enter Jones and Manuelli (1990), where they introduced a convex model that is capable of explaining long run growth heterogeneity.
1.2 Romer (1986 jpe): general/knowledge capital
1.3 Lucas (1988 jme): human capital
1.4 Stokey (1988 jpe): learning-by-doing
1.5 Jones and Manuelli (1990)
1.6 Excercise: Physical and Human Capital Spillovers and Growth
Consider a modified model of Lucas (1988). Denote individual variables by lower cases – physical capital capital \( k \), human capital \( h \), working time, \( u \), and consumption \( c \) – and aggregate by upper cases – society’s physical capital \( K \) and society’s human capital \( H \). By normalizing total productive land as one, the fixed supplies of production land and school land are denoted as \( z \) and \( 1 - z \), respectively. Goods production is constant returns in both private inputs and social reproducile inputs (including society’s aggregate), so physical capital evolves according to:
\begin{equation} \label{k-dot} \dot{k} = A (k^{\beta}K^{1-\beta}z^{1-\beta})^{\alpha}(uh)^{1-\alpha} - c \end{equation}
where \( \alpha, \beta \in (0, 1) \) and \( A > 0 \). Education production takes the convex technology form (a la Jones and Manuelli 1990), featuring positive spillovers:
\begin{equation} \label{h-dot} \dot{h} = \phi [(1-u) h]^{1-\gamma} (1-z)^{\gamma} H^{\gamma} \end{equation}
where \( \gamma \in (0,1) \) and \( \phi > 0 \).
The lifetime utility is
\[ \int_{0}^{\infty} \frac{c^{1-\sigma^{-1}} - 1}{1-\sigma^{-1}} e^{-\rho t}~\mathrm{d}t \]
where \( \sigma > 0 \) and \( \rho > 0 \).
In equilibrium with population normalized to one, \( K = k \) and \( H = h \).
Let the co-state variable associated with
How does this production resembles the convex production in Jones and Manuelli (1990)? In their paper, the convex production that yields long-term growth is of the form: \[ y = A k + B k^{\gamma} \] where \( \gamma \in (0, 1) \). The rates of returns is then \( \frac{\partial y}{\partial k} = A + B \gamma k^{\gamma-1} \rightarrow A \) as \( k \rightarrow \infty \).
Here, in our case, since \( h \) can only accumulate but not de-accumulate, \( \dot{h} \) is the same as the production of \( h \) consider the rate of returns over human capital
\begin{align*} \frac{\partial \dot{h}}{\partial h} &= \phi (1-\gamma) (1-u)^{1-\gamma} h^{-\gamma} (1 - z)^{\gamma} H^{\gamma} \\ &= \phi (1-\gamma)(1-u)^{1-\gamma} (1-z)^{\gamma} \quad \because H = h \text{ in equilibrium} \end{align*}
which is constant. Thus ensured positive human capital growth in the bgp.
(a) Write down the Current-Value Hamiltonian \( H(c,u,k,h,\lambda) \) and derive the first-order conditions (for \( c, u \)) and the Euler equations (for \( k, h \))
The Current-Value Hamiltonian
\begin{align*} \mathcal{H} = \frac{c^{1-\sigma^{-1}} - 1}{1-\sigma^{-1}} &+ \lambda \left[ A \left( k^{\beta} K^{1-\beta} z^{1-\beta} \right)^{\alpha} \left( uh \right)^{1-\alpha} - c \right] \\ &+ \mu \phi [(1-u)h]^{1-\gamma} (1-z)^{\gamma} H^{\gamma} \end{align*}
The FoCs are as follow:
\begin{align*} c &:\qquad c^{- \frac{1}{\sigma}} = \lambda \\ u &:\qquad \lambda (1-\alpha) A \left( k^{\beta} K^{1-\beta} z^{1-\beta} \right)^{\alpha} h ^{1-\alpha} u^{-\alpha} = \mu (1-\gamma) \phi h^{1-\gamma}(1 - u)^{-\gamma} (1-z)^{\gamma} H^{\gamma} \end{align*}
In equilibrium, the FoCs for \( u \) is further reduced as:
\begin{gather*} \lambda (1-\alpha) A \left( k z^{1-\beta} \right)^{\alpha} h^{1-\alpha} u^{-\alpha} = \mu (1-\gamma) \phi h (1-u)^{-\gamma}(1-z)^{\gamma} \\ \Rightarrow \lambda (1-\alpha) A \left( k z^{1-\beta} \right)^{\alpha} h^{-\alpha} u^{-\alpha} = \mu (1-\gamma) \phi (1-u)^{-\gamma}(1-z)^{\gamma} \end{gather*}