Economic growth and economic development 136

Introduction to Modern Economic Growth Next, we prove that (3.16) is steeper than (3.17) whenever (3.14) and (3.15) hold, so that can it most be one intersection First, observe that ¯ ¯ dh ¯¯ dh ¯¯ < dk ¯h=0 dk ¯k=0 ˙ ˙ m ∗ ∗ sh fk (k , h ) (δ k + g + n) − sk fk (k∗ , h∗ ) < (δ h + g + n) − sh fh (k∗ , h∗ ) sk fh (k∗ , h∗ ) m sk sh fk fh < sk sh fk fh + (δ h + g + n) (δ k + g + n) − (δ h + g + n) sk fk − (δ k + g + n) sh fh Now using (3.14) and (3.15) and substituting for (δ k + g + n) = sk f (k∗ , h∗ ) /k∗ and (δ h + g + n) = sh f (k ∗ , h∗ ) /h∗ , this is equivalent to f (k∗ , h∗ ) > fk (k∗ , h∗ ) k∗ + fh (k∗ , h∗ ) h∗ , which is satisfied by the fact that f (k ∗ , h∗ ) is a strictly concave function Finally, to establish existence note that Assumption 2’ implies that limh→0 f (k, h) /h = ∞, limk→0 f (k, h) /k = ∞, limh→∞ f (k, h) /h = and limk→∞ f (k, h) /k = 0, so that the curves look as in Figure 3.1, that is, (3.14) is below (3.15) as k → and h → ∞, but it is above (3.15) as k → ∞ and h → This implies that the two curves must intersect at least once Ô This proposition shows that a unique steady state exists when the Solow model is augmented with human capital The comparative statics are similar to the basic Solow model (see Exercise 3.7) Most importantly, both greater sk and greater sh will translate into higher normalized output per capita, yˆ∗ Now turning to cross-country behavior, consider two different countries that experience the same rate of labor-augmenting technological progress, g This implies that the country with greater propensity to invest in physical and human capital will be relatively richer This is the type of prediction can be investigated empirically to see whether the augmented Solow model gives us a useful way of looking at cross-country income differences Before doing this, however, we also need to check whether the unique steady state is globally stable The next proposition shows that this is the case 122