Introduction to Modern Economic Growth common that it may deserve to be called the canonical overlapping generations model and will be analyzed separately in the next section In the current somewhat more general context, equation (9.8) implies s(t) (1 + n) w(t) = , (1 + n)ψ (t + 1) (9.13) k (t + 1) = or more explicitly, (9.14) k (t + 1) = f (k (t)) − k (t) f (k (t)) (1 + n) [1 + β −1/θ f (k(t + 1))−(1−θ)/θ ] The steady state then involves a solution to the following implicit equation: k∗ = f (k∗ ) − k∗ f (k∗ ) (1 + n) [1 + β −1/θ f (k∗ )−(1−θ)/θ ] Now using the Cobb-Douglas formula, we have that the steady state is the solution to the equation (9.15) h ¡ ¢(θ−1)/θ i (1 + n) + β −1/θ α(k∗ )α−1 = (1 − α)(k∗ )α−1 For simplicity, define R∗ ≡ α(k∗ )α−1 as the marginal product of capital in steadystate, in which case, equation (9.15) can be rewritten as i 1−α h −1/θ ∗ (θ−1)/θ R∗ = (R ) (9.16) (1 + n) + β α ∗ ∗ The steady-state value of R , and thus k , can now be determined from equation (9.16), which always has a unique solution We can next investigate the stability of this steady state To this, substitute for the Cobb-Douglas production function in (9.14) to obtain (9.17) k (t + 1) = (1 − α) k (t)α (1 + n) [1 + β −1/θ (αk(t + 1)α−1 )−(1−θ)/θ ] Using (9.17), we can establish the following proposition can be proved:1 Proposition 9.4 In the overlapping-generations model with two-period lived households, Cobb-Douglas technology and CRRA preferences, there exists a unique 1In this proposition and throughout the rest of this chapter, we again ignore the trivial steady state with k = 426