Unified theory of growth

NOTE: The present notebook is coded in R. It relies heavily on the tidyverse ecosystem of packages. We load the tidyverse below as a prerequisite for the rest of the notebook - along with a few other libraries.

\(\rightarrow\) Don’t forget that code flows sequentially. A random chunk may not work if the previous have have not been executed.

library(tidyverse)   # Package for data wrangling
library(readxl)      # Package to import MS Excel files
library(latex2exp)   # Package for LaTeX expressions
library(quantmod)    # Package for stock data extraction
library(highcharter) # Package for reactive plots

The content of the notebook is heavily inspired from the book Advanced Macro-economics - An Easy Guide.

Context

In physics, scientists have tried for decades to build a theory of everything. There are two theories:
- quantum mechanics, which tries to explain what happens at very small scales (sub-atomic) - it is based on the strong nuclear, weak nuclear, and electromagnetic forces;
- general relativity, which seeks to explain what drives stellar bodies at very large scales (based on gravity mostly)

\(\rightarrow\) But reconciling the two is incredibly hard.

Here the aim is similar, but for economics. Indeed, we have sought to study models that could explain the post-industrial revolution growth.

At the same time, if a model wants to be exhaustive, it should also explain the centuries of stagnation that occurred before the 18th century (Malthusian stagnation).

The model

Production and utility

We start with the production function

\[Y_t=H_t^a(A_tX)^{1-a},\]

where \(H_t\) is the (total) efficiency of labor, \(A_t\) is productivity and \(X\) is the land used for production (assumed to be constant…). If we divide by labor \(L_t\), with \(x_t=A_tX/L_t\), \[y_t=h_t^ax_t^{1-a}\]

Utility is defined as \[u_t=c_t^{1-\gamma}(n_th_{t+1})^\gamma, \tag{1}\] with \(c_t\) being consumption, \(n_t\) the number of children and \(h_{t+1}\) their future efficiency (or “quality”). This idea comes from a family of models that consider overlapping generations.

The constraint of the model is built as follows. The consumption is bound by income, a wage \(w_t\) times quality \(h_t\). However, the total time dedicated to work is split between a unit of time, minus the number of children multiplied by the average time required to raise one chlid \(\tau\) plus a potential additional effort if the parent wants to increase \(h_{t+1}\) for the offspring; this effort is \(e_{t+1}\). The (saturated) constraint is then \[c_t=w_th_t(1-\underbrace{n_t(\tau+e_{t+1})}_{\text{time for kids}}). \tag{2}\] Basically, potential income \(w_th_t\) is split between consumption - or raising children.

Human capital is a function of two components: \[h_{t+1}=h(e_{t+1}, g_{t+1}), \tag{3}\] where \(g_{t+1}=A_{t+1}/A_t-1\) is the rate of technological progress and \(e_{t+1}\) is the effort put into education.

Solution

We plug the consumption value Equation 2 and human capital Equation 3 in Equation 1: \[u_t= (w_th_t(1-n_t(\tau+e_{t+1})))^{1-\gamma}(n_t h(e_{t+1}, g_{t+1}))^\gamma\]

Parents optimize their utility by chosing the number of children \(n_t\) and the intensity of education \(e_{t+1}\). We have \[\begin{align} \frac{\partial u_t}{\partial n_t}=-(1-\gamma)(\tau+e_{t+1}) w_th_t (w_th_t(1-n_t(\tau+e_{t+1})))^{-\gamma}(n_t h(e_{t+1}, g_{t+1}))^\gamma \\ +\gamma n_t^{\gamma-1}(w_th_t(1-n_t(\tau+e_{t+1})))^{1-\gamma} h(e_{t+1}, g_{t+1})^\gamma \end{align}\]

Hence the FOC is

\[(1-\gamma)(\tau+e_{t+1})w_th_t=\gamma n_t^{-1} w_th_t(1-n_t(\tau+e_{t+1}))\]

i.e., given the definition of \(c_t\) in Equation 2, \[\frac{1-\gamma}{\gamma}(\tau+e_{t+1})w_th_t=\frac{c_t}{n_t} \quad \Leftrightarrow \quad \frac{c_t}{w_th_t}=n_t \frac{1-\gamma}{\gamma}(\tau+e_{t+1}) \tag{4}\]

But from Equation 2, \(c_t/(w_th_t)=(1-n_t(\tau+e_{t+1}))\), hence

\[1-n_t(\tau+e_{t+1})=n_t \frac{1-\gamma}{\gamma}(\tau+e_{t+1}) \quad \Leftrightarrow \quad \gamma=n_t(\tau+e_{t+1}),\] i.e., the parents will dedicate a proportion \(\gamma\) to raising their child(ren).

Moreover, with respect to education, it holds that

\[\begin{align}\frac{\partial u_t}{\partial e_{t+1}}=-(1-\gamma)w_tn_th_t (w_th_t(1-n_t(\tau+e_{t+1})))^{-\gamma} (n_th(e_{t+1}, g_{t+1}))^\gamma \\ + \gamma n_t h_e(e_{t+1}, g_{t+1})(w_th_t(1-n_t(\tau+e_{t+1})))^{1-\gamma} (n_th(e_{t+1}, g_{t+1}))^{\gamma-1} \end{align}\]

where \(h_e\) is the derivative of h w.r.t. e. The FOC reduces to

\[(1-\gamma)w_tn_th_t =\gamma c_th_e(e_{t+1}, g_{t+1}) h(e_{t+1}, g_{t+1})^{-1}\]

i.e.,

\[\frac{1-\gamma}{\gamma h_e(e_{t+1}, g_{t+1})} n_t h(e_{t+1}, g_{t+1}) = \frac{c_t}{w_t h_t} \]

and plugging Equation 4, we get

\[\frac{1-\gamma}{\gamma h_e(e_{t+1}, g_{t+1})} n_t h(e_{t+1}, g_{t+1}) =n_t \frac{1-\gamma}{\gamma}(\tau+e_{t+1}) \] i.e., \[\frac{h(e_{t+1}, g_{t+1})}{ h_e(e_{t+1}, g_{t+1})} =\tau+e_{t+1}.\]

A natural assumption is that h increases with e : human capital benefits from education. However, it also makes sense to consider that \(h_g<0\) because technology increases knowledge obsolescence. Hence, parents will/should invest in education more when technology grows faster to limit the erosion of their child’s capabilities.

Closing the loop

The interesting feature of the model is the interplay between technology, population and human capital (investment). These are, in the model: \(n_t\) for population growth (via \(n_t=L_{t+1}/L_t\)), \(e_t\) for investment in education and \(g_t\) for technological growth. The latter has not formally been treated until now. We can make assumptions: \(g:=g(e,L)\), that is: technology progress is driven by how much we invest in education (well trained work-force or R&D) and by the population size.

From Equation 4, if the left-hand side is fixed, increasing quality via \(\tau+e_{t+1}\) imposes to reduce quantity, which is a feature observed in many developed countries since the beginning of the XXIst century. The transition from a Malthusian economy to sustainable growth is depicted below.