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 = 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 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) 217