Economic growth and economic development 241

Introduction to Modern Economic Growth
function of the age of the individual (a feature best captured by actuarial life tables,
which are of great importance to the insurance industry). Nevertheless, this is a
good starting point, since it is relatively tractable and also implies that individuals
have an expected lifespan of 1/ν < ∞ periods, which can be used to get a sense of
what the value of ν should be.

Suppose also that each individual has a standard instantaneous utility function
ˆ meaning that this is the
u : R+ → R, and a “true” or “pure” discount factor β,

discount factor that he would apply between consumption today and tomorrow if
he were sure to live between the two dates. Moreover, let us normalize u (0) =
0 to be the utility of death. Now consider an individual who plans to have a
consumption sequence {c (t)}∞
t=0 (conditional on living). Clearly, after the individual

dies, the future consumption plans do not matter. Standard arguments imply that
this individual would have an expected utility at time t = 0 given by

ˆ (0)
U (0) = u (c (0)) + βˆ (1 − ν) u (c (0)) + βνu

2
2
+βˆ (1 − ν)2 u (c (1)) + βˆ (1 − ν) νu (0) + ...
∞ ³
´t
X
βˆ (1 − ν) u (c (t))
=

t=0

(5.9)

=

∞
X

β t u (c (t)) ,

t=0

where the second line collects terms and uses u (0) = 0, while the third line defines β ≡ βˆ (1 − ν) as the “effective discount factor” of the individual. With this
formulation, the model with finite-lives and random death, would be isomorphic
to the model of infinitely-lived individuals, but naturally the reasonable values of
β may differ. Note also the emphasized adjective “expected” utility here. While
until now agents faced no uncertainty, the possibility of death implies that there
is a non-trivial (in fact quite important!) uncertainty in individuals’ lives. As a
result, instead of the standard ordinal utility theory, we have to use the expected
utility theory as developed by von Neumann and Morgenstern. In particular, equation (5.9) is already the expected utility of the individual, since probabilities have
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