Economic growth and economic development 459

Introduction to Modern Economic Growth
capital stock dynamics. Therefore, this model gives one example of a situation
in which continuous time methods turn out to be more appropriate than discrete
time methods (whereas the baseline overlapping generations model required discrete
time).
Recall that in the neoclassical model without technological progress, the consumer Euler equation admitted a simple solution because consumption had to be
equal across dates for the representative household. This is no longer the case in
the perpetual youth model, since different generations will have different levels of
assets and may satisfy equation (9.36) with different growth rates of consumption
depending on the form of the utility function u (·).
To simplify the analysis, let us now suppose that the utility function takes the
logarithmic form,
u (c) = log c.
In that case, (9.36) simplifies to
(9.37)

c (t + 1 | τ )
= β [(1 + r (t + 1)) (1 − ν) + ν] ,
c (t | τ )

and implies that the growth rate of consumption must be equal for all generations.
Using this observation, it is possible to characterize the behavior of the aggregate
capital stock, though this turns out to be much simpler in continuous time. For this
reason, we now turn to the continuous time version of this model (details on the
discrete time model are covered in Exercise 9.22).
9.8. Overlapping Generations in Continuous Time
9.8.1. Demographics, Technology and Preferences. We now turn to a
continuous time version of the perpetual youth model. Suppose that each individual faces a Poisson death rate of ν ∈ (0, ∞). Suppose also that individuals have

logarithmic preferences and a pure discount rate of ρ > 0. As demonstrated in

Exercise 5.7 in Chapter 5, this implies that individual i will maximize the objective
function
(9.38)

Z

∞

exp (− (ρ + ν) t) log ci (t) dt.

0

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