Introduction to Modern Economic Growth 9.7 Overlapping Generations with Perpetual Youth A key feature of the baseline overlapping generation model is that individuals have finite lives and know exactly when their lives will come to an end An alternative way of modeling finite lives is along the lines of the “Poisson death model” or the perpetual youth model introduced in Section 5.3 of Chapter Let us start with the discrete time version of that model Recall that in that model each individual is potentially infinitely lived, but faces a probability ν ∈ (0, 1) that his life will come to an end at every date (and these probabilities are independent) Recall from equation (5.9) that the expected utility of an individual with a “pure” discount factor β is given by ∞ X t=0 (β (1 − ν))t u (c (t)) , where u (·) is as standard instantaneous utility function, satisfying Assumption 3, with the additional normalization that u (0) = Since the probability of death is ν and is independent across periods, the expected lifetime of an individual in this model can be written as (see Exercise 9.15): < ∞ ν This equation captures the fact that with probability ν the individual will have a (9.31) Expected life = ν + (1 − ν) ν + (1 − ν)2 ν + = total life of length 1, with probability (1 − ν) ν, he will have a life of length 2, and so on This model is referred to as the perpetual youth model, since even though each individual has a finite expected life, all individuals who have survived up to a certain date have exactly the same expectation of further life Therefore, individuals who survive in this economy are “perpetually young”; their age has no effect on their future longevity and has no predictive power on how many more years they will live for Individual i’s flow budget constraint can be written as (9.32) (t + 1) = (1 + r (t + 1)) (t) − ci (t) + w (t) + zi (t) , which is similar to the standard flow budget constraint, for example (6.40) in Chapter Recall that the gross rate of return on savings is + r (t + 1), with the timing convention reflecting that assets at time t are rented out as capital at time t + 441