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Introduction to Modern Economic Growth
been substituted in and there is no need to include an explicit expectations operator. Throughout, except in the stochastic growth analysis in Chapter 17, we do not
introduce expectations operators and directly specified the expected utility.
In Exercise 5.7, you are asked to derive a similar effective discount factor for an
individual facing a constant death rate in continuous time.
A second justification for the infinite planning horizon comes from intergenerational altruism or from the “bequest” motive. At the simplest level, imagine an
individual who lives for one period and has a single offspring (who will also live for
a single period and will beget a single offspring etc.). We may imagine that this individual not only derives utility from his consumption but also from the bequest he
leaves to his offspring. For example, we may imagine that the utility of an individual
living at time t is given by
u (c (t)) + U b (b (t)) ,
where c (t) is his consumption and b (t) denotes the bequest left to his offspring. For
concreteness, let us suppose that the individual has total income y (t), so that his
budget constraint is
c (t) + b (t) ≤ y (t) .

The function U b (·) contains information about how much the individual values
bequests left to his offspring. In general, there may be various reasons why individuals leave bequests (including accidental bequests like the individual facing random
death probability just discussed). Nevertheless, a natural benchmark might be one
in which the individual is “purely altruistic” so that he cares about the utility of
his offspring (with some discount factor).1 Let the discount factor apply between
generations be β. Also assume that the offspring will have an income of w without
the bequest. Then the utility of the individual can be written as
u (c (t)) + βV (b (t) + w) ,
where V (·) can now be interpreted as the continuation value, equal to the utility
that the offspring will obtain from receiving a bequest of b (t) (plus his own income
1The alternative to “purely altruistic” preferences are those in which a parent receives utility
from specific types of bequests or from a subcomponent of the utility of his or her offspring. Models
with such “impure altruism” are sometimes quite convenient and will be discussed in Chapters 9
and 22.

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