Introduction to Modern Economic Growth following two equations: (3.14) sk f (k∗ , h∗ ) − (δ k + g + n) k∗ = 0, and (3.15) sh f (k∗ , h∗ ) − (δ h + g + n) h∗ = As in the basic Solow model, we focus on steady-state equilibria with k∗ > and h∗ > (if f (0, 0) = 0, then there exists a trivial steady state with k = h = 0, which we ignore as we did in the previous chapter) We can first prove that this steady-state equilibrium is in fact unique To see this heuristically, consider Figure 3.1, which is drawn in the (k, h) space The two curves represent the two equations (3.14) and (3.15) Both lines are upward sloping For example, in (3.14) a higher level of h∗ implies greater f (k∗ , h∗ ) from Assumption 1’, thus the level of k∗ and that will satisfy the equation is higher The same reasoning applies to (3.15) However, the proof of the next proposition shows that (3.15) is always shallower in the (k, h) space, so the two curves can only intersect once Proposition 3.1 Suppose Assumption 1’ and 2’ are satisfied Then in the augmented Solow model with human capital, there exists a unique steady-state equilibrium (k∗ , h∗ ) Proof First consider the slope of the curve (3.14), corresponding to the k˙ = locus, in the (k, h) space Using the implicit function theorem, we have ¯ dh ¯¯ (δ k + g + n) − sk fk (k∗ , h∗ ) (3.16) , = dk ¯k=0 sk fh (k∗ , h∗ ) ˙ where fk ≡ ∂f/∂k Rewriting (3.14), we have sk f (k∗ , h∗ ) /k∗ − (δ k + g + n) = Now recall that since f is strictly concave in k in view of Assumption 1’ and f (0, h∗ ) ≥ 0, we have f (k ∗ , h∗ ) > fk (k∗ , h∗ ) k∗ + f (0, h∗ ) > fk (k∗ , h∗ ) k∗ Therefore, (δ k + g + n) − sk fk (k ∗ , h∗ ) > 0, and (3.16) is strictly positive 120