Economic growth and economic development 529

Introduction to Modern Economic Growth µh : c (t)1−θ − H (a, h, c, ih , µa , µk ) = + µa (t) [(r (t) − n)a (t) + w (t) h (t) − c (t) − ih (t)] 1−θ +µh (t) [ih (t) − δ h h (t)] Now the necessary conditions of this optimization problem imply the following (see Exercise 11.8): (11.25) µa (t) = µh (t) = µ (t) for all t w (t) − δh = r (t) − n for all t c˙ (t) = (r (t) − ρ) for all t c (t) θ Combining these with (11.24), we obtain that f (k (t)) − δ k − n = f (k (t)) − k (t) f (k (t)) − δ h for all t Since the left-hand side is decreasing in k (t), while the right-hand side is increasing, this implies that the effective capital-labor ratio must satisfy k (t) = k∗ for all t We can then prove the following proposition: Proposition 11.3 Consider the above-described AK economy with physical and human capital, with a representative household with preferences given by (11.1), and the production technology given by (11.21) Let k∗ be given by (11.26) f (k∗ ) − δ k − n = f (k ∗ ) − k∗ f (k∗ ) − δ h Suppose that f (k∗ ) > ρ + δ k > (1 − θ) (f (k∗ ) − δ) + nθ + δ k Then, in this economy there exists a unique equilibrium path in which consumption, capital and output all grow at the same rate g ∗ ≡ (f (k ∗ ) − δ k − ρ)/θ > starting from any initial conditions, where k∗ is given by (11.26).The share of capital in national income is constant at all times Proof See Exercise 11.9 Ô The advantage of the economy studied here, especially as compared to the baseline AK model is that, it generates a stable factor distribution of income, with a 515