Introduction to Modern Economic Growth As before, a steady state is as an allocation in which the capital-labor ratio and consumption not depend on time, so again denoting this by *, we have the steady state capital-labor ratio as (6.38) β [f (k∗ ) + (1 − δ)] = 1, which is a remarkable result, because it shows that the steady state capital-labor ratio does not depend on preferences, but simply on technology, depreciation and the discount factor We will obtain an analogue of this result in the continuous-time neoclassical model as well Moreover, since f (·) is strictly concave, k∗ is uniquely defined Thus we have Proposition 6.3 In the neoclassical optimal growth model specified in (6.33) and (6.34) with Assumptions 1, and 3’, there exists a unique steady-state capitallabor ratio k∗ given by (6.38), and starting from any initial k (0) > 0, the economy monotonically converges to this unique steady state, i.e., if k (0) < k∗ , then the equilibrium capital stock sequence k (t) ↑ k∗ and if k (0) > k∗ , then the equilibrium capital stock sequence k (t) ↓ k∗ Proof Uniqueness and existence were established above To establish monotone convergence, we start with arbitrary initial capital stock k (0) and observe that k (t + 1) = s (k (t)) for all t ≥ 0, where s (·) was defined and shown to be nondecreasing in Proposition 6.1 It must be the case that either k (1) = s (k (0)) ≥ k (0) or k (1) = s (k (0)) < k (0) Consider the first case Since s (·) is nondecreasing and k (2) = s (k (1)), we must have k (2) ≥ k (1) By induction, h ki(t) = s (k (t − 1)) ≥ k (t − 1) = s (k (t − 2)) Moreover, by definition k (t) ∈ 0, k Therefore, in this case {k (t)}∞ t=0 is a non- decreasing sequence in a compact set starting with k (0) > 0, thus it necessarily converges to some limit k (∞) > 0, which by definition satisfies k (∞) = s (k (∞)) Since k∗ is the unique steady state (corresponding to positive capital-labor ratio), this implies that k (∞) = k ∗ , and thus k (t) → k∗ Moreover, since {k (t)}∞ t=0 is nondecreasing, it must be the case that k (t) ↑ k∗ , and thus this corresponds to the case where k (0) ≤ k∗ 295