Economic growth and economic development 231

Introduction to Modern Economic Growth
Finally, let us introduce a third assumption and suppose that households discount the future “exponentially”–or “proportionally”. In discrete time, and ignoring uncertainty, this implies that household preferences at time t = 0 can be
represented as

(5.1)

∞
X

β ti ui (ci (t)) ,

t=0

where β i ∈ (0, 1) is the discount factor of household i. This functional form implies

that the weight given to tomorrow’s utility is a fraction β i of today’s utility, and
the weight given to the utility the day after tomorrow is a fraction β 2i of today’s
utility, and so on. Exponential discounting and time separability are convenient for
us because they naturally ensure “time-consistent” behavior.
We call a solution {x (t)}Tt=0 (possibly with T = ∞) to a dynamic optimization

problem time-consistent if the following is true: whenever {x (t)}Tt=0 is an optimal

solution starting at time t = 0, {x (t)}Tt=t0 is an optimal solution to the continuation

dynamic optimization problem starting from time t = t0 ∈ [0, T ]. If a problem is
not time-consistent, we refer to it as time-inconsistent. Time-consistent problems

are much more straightforward to work with and satisfy all of the standard axioms
of rational decision-making. Although time-inconsistent preferences may be useful
in the modeling of certain behaviors we observe in practice, such as problems of

addiction or self-control, time-consistent preferences are ideal for the focus in this
book, since they are tractable, relatively flexible and provide a good approximation
to reality in the context of aggregative models. It is also worth noting that many
classes of preferences that do not feature exponential and time separable discounting
nonetheless lead to time-consistent behavior. Exercise 5.1 discusses issues of time
consistency further and shows how certain other types of utility formulations lead to
time-inconsistent behavior, while Exercise 5.2 introduces some common non-timeseparable preferences that lead to time-consistent behavior.
There is a natural analogue to (5.1) in continuous time, again incorporating
exponential discounting, which is introduced and discussed below (see Section 5.9
and Chapter 7).
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