Human Capital Growth Models
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Categories: CategoryModel CategoryMacro CategoryGrowth
Related Models: Solow Growth Model
Created: 2008-07-30
Suggested Citation: See license page.
Contents
Introduction
'Augmented' Solow models recognise the role of human capital in economic growth. The quality of labour varies across countries and labour productivity can be increased through education and training. The addition of human capital to the Solow model brings its predictions, for example the estimated capital share and the rate of convergence, much closer to the empirical evidence (for more on the empirics see 'Introduction to Economic Growth', C.I. Jones 2002).
Models
There are two specifications for human capital growth models: Mankiw, Romer and Weil introduce $$ H $$ as an accumulated factor in addition to $$ K $$ and $$ L $$, whereas Lucas presents investment in $$ H $$ as an alternative to labour.
Mankiw, Romer and Weil
Mankiw, Romer and Weil (mankiw_ea_1992) incude human capital $$ H $$ as an input into a Cobb-Douglas aggregate production function $$ Y = K^{\alpha} H^{\gamma} (AL)^{(1-\alpha-\gamma)} $$ where $$ \alpha+\gamma<1 $$ . There are now two factor accumulation equations, where $$ s_H $$ ($$ s_K $$) is the rate of investment rate into human (physical) capital and total investment is equal to the total fraction of output saved $$ (s_K + s_H)Y $$ . Both $$ K $$ and $$ H $$ are assumed to depreciate at the same rate $$ \delta $$:
- $$ \dot{K} = s_KY - \deltaK \Rightarrow \dot{\tilde{k}} = s_K\tilde{y} - (n+g+\delta)\tilde{k} $$
- $$ \dot{H} = s_HY - \deltaH \Rightarrow \dot{\tilde{h}} = s_H\tilde{y} - (n+g+\delta)\tilde{h} $$
At the steady-state equilibrium $$ \dot{\tilde{k}} = \dot{\tilde{h}} = 0 $$ which gives the solutions:
$$ \tilde{k*} = \left(\frac{s_K^{(1-\gamma)} s_H^{\gamma}}{(n+g+\delta)} \right)^{\frac{1}{(1-\alpha-\gamma)} $$
$$ \tilde{h*} = \left(\frac{s_K^{\alpha} s_H^{(1-\alpha)}}{(n+g+\delta)} \right)^{\frac{1}{(1-\alpha-\gamma)} $$
showing that physical $$ K $$ and human $$ H $$ capital accumulation are complimentary since an increase in $$ s_K $$ or $$ s_H $$ will increase both $$ \tilde{k*} $$ and $$ \tilde{h*} $$ .
As before, per-capita variables grow at rate $$ g $$ and aggregate variables at rate $$ (n+g) $$ . Neither $$ s_K $$ nor $$ s_H $$ has any effect on the long-run growth rate of output, only on its long-run level.
Steady-state output per effective worker is given by $$ \tilde{y*} = (\tilde{k*})^{\alpha}(\tilde{h*})^{\gamma} = \left(\frac{s_K^{\alpha} s_H^{\gamma}}{(n+g+\delta)^{\alpha+\gamma}} \right)^{\frac{1}{(1-\alpha-\gamma)} $$
thus steady state output per capita is $$ y* = A\left(\frac{s_K^{\alpha} s_H^{\gamma}}{(n+g+\delta)^{\alpha+\gamma}} \right)^{\frac{1}{(1-\alpha-\gamma)} $$ .
Lucas
Lucas (lucas_1988) also presents an augmented version of Solow, but in this model human capital is accumulated by an individual spending time in education or training rather than working. $$ H = e^{\psiu}L $$ : 'skilled labour' $$ H $$ is 'produced' from unskilled labour $$ L $$ where $$ u $$ is the fraction of time spent learning skills and is an exogenously-given constant. $$ \frac{dH}{du} = \psiH $$ so a unit change in $$ u $$ increases $$ H $$ by $$ \psi $$ .
The production function is given by $$ Y = K^{\alpha}(AH)^{(1-\alpha)} $$ , or in per-capita terms $$ y = k^{\alpha}(Ah)^{(1-\alpha)} $$ . $$ \frac{H}{L} = h = e^{\psiu} $$ which is a constant. Just as in the Solow model where variables were divided by $$ AL $$ to become per-effective-worker, the same can be done in this model by dividing by $$ Ah $$, giving $$ \tilde{y} = \tilde{k}^{\alpha} $$ . Physical capital accumulation is as before, so solving for the steady-state using $$ \dot{\tilde{y}} = \dot{\tilde{k}} = 0 $$ gives $$ \tilde{y*} = \left(\frac{s_K}{(n+g+\delta)} \right)^{\frac{\alpha}{(1-\alpha)} $$ which in per-capita terms becomes $$ y* = Ah \left(\frac{s_K}{(n+g+\delta)} \right)^{\frac{\alpha}{(1-\alpha)} $$ .
Economies with high investment in physical capital ($$ s_K $$), a high proportion of time spent accumulating skills ($$ u $$), low population growth rates ($$ n $$) and high levels of technology ($$ A $$) have a higher level of output per capita than those who do not. However, as with the original Solow model, it is only the growth rate of technology $$ g $$ which affects the growth rate of output per capita.
