Economic growth and economic development 137

Introduction to Modern Economic Growth Proposition 3.2 Suppose Assumption 1’ and 2’ are satisfied Then the unique steady-state equilibrium of the augmented Solow model with human capital, (k∗ , h∗ ), is globally stable in the sense that starting with any k (0) > and h (0), we have (k (t) , h (t)) → (k ∗ , h∗ ) A formal proof of this proposition is left to Exercise 3.6 Figure 3.2 gives a diagrammatic proof, by showing the law of motion of k and h depending on whether we are above or below the two curves representing the loci for k˙ = and h˙ = 0, respectively, (3.14) and (3.15) When we are to the right of the (3.14) curve, there is too much physical capital relative to the amount of labor and human capital, and consequently, k˙ < When we are to its left, we are in the converse situation and k˙ > Similarly, when we are above the (3.15) curve, there is too little human capital relative to the amount of labor and physical capital, and thus h˙ > When we are below it, h˙ < Given these arrows, the global stability of the dynamics follows We next characterize the equilibrium in greater detail when the production function (3.13) takes a Cobb-Douglas form Example 3.2 (Augmented Solow model with Cobb-Douglas production functions) Let us now work through a special case of the above model with CobbDouglas production function In particular, suppose that the aggregate production function is (3.18) Y (t) = K β (t) H α (t) (A (t) L (t))1−α−β , where < α < 1, < β < and α + β < Output per effective unit of labor can then be written as yˆ (t) = k β (t) hα (t) , with the same definition of yˆ (t), k (t) and h (t) as above Using this functional form, (3.14) and (3.15) give the unique steady-state equilibrium as ả1 ¶α ! 1−α−β sh sk (3.19) k∗ = n + g + δk n + g + δh õ ¶β µ ¶1−β ! 1−α−β sh sk h∗ = , n + g + δk n + g + δh 123