Introduction to Modern Economic Growth Proposition 9.10 In the continuous time perpetual youth model, there exists a unique steady state (k∗ , c∗ ) given by (9.50) and (9.51) The level of capital-labor ratio is less than the level of capital-labor ratio that satisfies the modified golden rule, kmgr Starting with any k (0) > 0, equilibrium dynamics monotonically converge to this unique steady state Perhaps the most interesting feature of this equilibrium is that, despite finite lives and overlapping generations, there is no overaccumulation The reason for this is that individuals have constant stream of labor income throughout their lives and thus not need to save excessively in order to ensure smooth consumption Is it possible to obtain overaccumulation in the continuous time perpetual youth model? The answer is yes and is demonstrated by Blanchard (1985) He assumes that each individual starts life with one unit of effective labor and then his effective labor units decline at some positive exponential rate ζ > throughout his life, so that the labor earnings of an individual on generation τ at time t is exp (−ζ (t − τ )) w (t), where w (t) is the market wage per unit of effective labor at time t Consequently, individual consumption function changes from (9.46) to c (t | τ ) = (ρ + ν) [a (t | τ ) + ω (t | τ )] , where now ω (t | τ ) is the human wealth of an individual of generation τ at time t, given by (see Exercise 9.28): Z ∞ exp (− (¯ r (t − s) + ν)) exp (−ζ (s − τ )) w (s) ds, (9.52) ω (t | τ ) = t where exp (−ζ (s − τ )) is the correction factor taking into account the decline in effective labor units Repeating the same steps as before with this new expression for human wealth, we obtain (9.53) c˙ (t) k (t) = f (k (t)) − δ − ρ − ζ − (ρ + ν) (n + ζ) , c (t) c (t) while the behavior of capital-labor ratio is still given by (9.43) It can now be shown that for ζ sufficiently large, the steady-state capital-labor ratio k ∗ can exceed both the modified golden rule level kmgr and the golden rule level, kgold (see Exercise 9.28) 452