Introduction to Modern Economic Growth in the economy This is a consequence of the specific utility function in (10.27), which ensures that there are no income effects in human capital decisions so that all agents choose the same “income-maximizing” level of human capital (as in Theorem 10.1) Next, note that since bequest decisions are linear as shown (10.32), we have Z bi (t) di K (t + 1) = Z mi (t) di = (1 − η) = (1 − η) f (κ (t)) h (t) , where the last line uses the fact that, since all individuals choose the same human capital level given by (10.35), H (t) = h (t), and thus Y (t) = f (κ (t)) h (t) Now combining this with (10.30), we obtain (1 − η) f (κ (t)) h (t) κ (t + 1) = h (t + 1) Using (10.35), this becomes (10.36) κ (t + 1) γ 0−1 [a (f (κ (t + 1)) − κ (t + 1) f (κ (t + 1)))] = (1 − η) f (κ (t)) γ 0−1 [af (κ (t)) − κ (t) f (κ (t))] A steady state, as usual, involves a constant effective capital-labor ratio, i.e., κ (t) = κ∗ for all t Substituting this into (10.36) yields (10.37) κ∗ = (1 − η) f (κ∗ ) , which defines the unique positive steady-state effective capital-labor ratio, κ∗ (since f (·) is strictly concave) Proposition 10.2 In the overlapping generations economy with physical and human capital described above, there exists a unique steady state with positive activity, and the physical to human capital ratio is κ∗ as given by (10.37) This steady-state equilibrium is also typically stable, but some additional conditions need to be imposed on the f (·) and γ (·) to ensure this (see Exercise 10.17) An interesting implication of this equilibrium is that, the capital-skill (k-h) complementarity in the production function F (·, ·) implies that a certain target level of 484