Introduction to Modern Economic Growth simpler: c2 (t + 1) = βR (t + 1) c1 (t) and implies that the saving rate should satisfy the equation β w (t) , (9.19) s (t) = 1+β which corresponds to a constant saving rate, equal to β/ (1 + β), out of labor income for each individual This constant saving rate makes this model very similar to the baseline Solow growth model of Chapter Now combining this with the capital accumulation equation (9.8), we obtain s(t) (1 + n) βw(t) = (1 + n) (1 + β) β (1 − α) [k(t)]α = , (1 + n) (1 + β) k (t + 1) = where the second line uses (9.19) and the last line uses the fact that, given competitive factor markets, the wage rate is equal to w (t) = (1 − α) [k(t)]α It is straightforward to verify that there exists a unique steady state with capital- labor ratio given by (9.20) ∙ β (1 − α) k = (1 + n) (1 + β) ∗ ¸ 1−α Moreover, starting with any k (0) > 0, equilibrium dynamics are identical to those of the basic Solow model and monotonically converge to k∗ This is illustrated in Figure 9.2 and stated in the next proposition: Proposition 9.5 In the canonical overlapping generations model with log preferences and Cobb-Douglas technology, there exists a unique steady state, with capitallabor ratio k∗ given by (9.20) Starting with any k (0) ∈ (0, k ∗ ), equilibrium dynam- ics are such that k (t) ↑ k∗ , and starting with any k (0) > k∗ , equilibrium dynamics involve k (t) ↓ k ∗ Exercise 9.6 asks you to introduce technological progress into this canonical model and to conduct a range of comparative static exercises Exercise 9.7, on 428