Introduction to Modern Economic Growth Assumption 2’: F satisfies the Inada conditions ∂F (K, H, AL) ∂F (K, H, AL) = ∞ and lim = for all H > and AL > 0, K→0 K→∞ ∂K ∂K ∂F (K, H, AL) ∂F (K, H, AL) = ∞ and lim = for all K > and AL > 0, lim H→0 H→∞ ∂H ∂H ∂F (K, H, AL) ∂F (K, H, AL) = ∞ and lim = for all K, H, A > lim L→0 L→∞ ∂L ∂L Moreover, we assume that investments in human capital take a similar form lim to investments in physical capital; households save a fraction sk of their income to invest in physical capital and a fraction sh to invest in human capital Human capital also depreciates in the same way as physical capital, and we denote the depreciation rates of physical and human capital by δ k and δ h , respectively We continue to assume that there is constant population growth and a constant rate of labor-augmenting technological progress, i.e., A˙ (t) L˙ (t) = n and = g L (t) A (t) Now defining effective human and physical capital ratios as k (t) ≡ H (t) K (t) and h (t) ≡ , A (t) L (t) A (t) L (t) and using the constant returns to scale feature in Assumption 1’, output per effective unit of labor can be written as Y (t) A (t) L (t) ả K (t) H (t) = F , ,1 A (t) L (t) A (t) L (t) ≡ f (k (t) , h (t)) yˆ (t) ≡ With the same steps as in Chapter 2, the law of motion of k (t) and h (t) can then be obtained as: k˙ (t) = sk f (k (t) , h (t)) − (δ k + g + n) k (t) , h˙ (t) = sh f (k (t) , h (t)) − (δ h + g + n) h (t) A steady-state equilibrium is now defined not only in terms of effective capital-labor ratio, but effective human and physical capital ratios, (k∗ , h∗ ), which satisfies the 119