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Introduction to Modern Economic Growth finite For many models, this would also be acceptable, but as we will see below, models with overlapping generations require the set of households to be infinite We can either assume that households are truly “infinitely lived” or that they consist of overlapping generations with full (or partial) altruism linking generations within the household Throughout, we equate households with individuals, and thus ignore all possible sources of conflict or different preferences within the household In other words, we assume that households have well-defined preference orderings As in basic general equilibrium theory, we make enough assumptions on preference orderings (in particular, reflexivity, completeness and transitivity) so that these preference orderings can be represented by utility functions In particular, suppose that each household i has an instantaneous utility function given by ui (ci (t)) , where ui : R+ → R is increasing and concave and ci (t) is the consumption of household i Here and throughout, we take the domain of the utility function to be R+ rather than R, so that negative levels of consumption are not allowed Even though some well-known economic models allow negative consumption, this is not easy to interpret in general equilibrium or in growth theory, thus this restriction is sensible The instantaneous utility function captures the utility that an individual derives from consumption at time t It is therefore not the same as a utility function specifying a complete preference ordering over all commodities–here consumption levels in all dates For this reason, the instantaneous utility function is sometimes also referred to as the “felicity function” There are two major assumptions in writing an instantaneous utility function First, it imposes that the household does not derive any utility from the consumption of other households, so consumption externalities are ruled out Second, in writing the instantaneous utility function, we have already imposed that overall utility is time separable, that is, instantaneous utility at time t is independent of the consumption levels at past or future dates This second feature is important in enabling us to develop tractable models of dynamic optimization 216