Introduction to Modern Economic Growth A complete characterization of the equilibrium can now be obtained by looking at the dynamics of bequests It turns out that different types of bequests dynamics are possible along the transition path More can be said regarding the limiting distribution of wealth and bequests In particular, we know that k (t) → k∗ , so the ultimate bequest dynamics are given by steady-state factor prices Let these be denoted by w∗ = f (k∗ ) − k∗ f (k∗ ) and R∗ = f (k∗ ) Then once the economy is in the neighborhood of the steady-state capital-labor ratio, k∗ , individual bequest dynamics are given by bi (t) = β [w∗ + R∗ bi (t − 1)] 1+β When R∗ < (1 + β) /β, starting from any level bi (t) will converge to a unique bequest (wealth) level given by (9.30) b∗ = βw∗ + β (1 − R∗ ) Moreover, it can be verified that the steady-state equilibrium must involve R∗ < (1 + β) /β This follows from the fact that in steady state R∗ = f (k∗ ) f (k∗ ) < k∗ 1+β , = β where the second line exploits the strict concavity of f (·) and the last line uses the definition of the steady-state capital-labor ratio, k∗ , from (9.29) The following proposition summarizes this analysis: Proposition 9.9 Consider the overlapping generations economy with warm glow preferences described above In this economy, there exists a unique competitive equilibrium In this equilibrium the aggregate capital-labor ratio is given by (9.28) and monotonically converges to the unique steady-state capital-labor ratio k∗ given by (9.29) The distribution of bequests and wealth ultimately converges towards full equality, with each individual having a bequest (wealth) level of b∗ given by (9.30) with w∗ = f (k∗ ) − k∗ f (k∗ ) and R∗ = f (k ∗ ) 440