Economic growth and economic development 230

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.
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