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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 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 and 22 228