A Note on Consistent Naive Intertemporal Decision Making and an Application to the Case of Uncertain Lifetime

Abstract

Samuelson [4] indicated the existence of the problem concerning consistent intertemporal behaviour and Strotz [5] commenced the formal analysis in this area. Subsequent contributions were made by Pollak [3], Peleg and Yaari [2] and Blackorby, Nissen, Primont and Russell [1]. This note extends Strotz's result concerning the consistency of naive intertemporal decision making and then employs this theorem to indicate the extent to which naive decision making will be consistent in the context of uncertain lifetime. Assuming that interest rates are zero the consumer's problem is to maximize the (subjectively) discounted sum of all future utilities subject to a stock constraint. Notationally K is the initial and total stock of wealth for the individual, c(t) is the time path of consumption, a(t, r) is his discount function where t is time and r is the planning date and a is continuously differentiable in t and r, and u[c(t), t] is his period utility function, assumed to be twice continuously differentiable, and monotonic increasing and concave in c. The tools of optimal control theory are employed and it is assumed that replanning may occur at dates 1, '2~ ..' P Our theorem is a generalization of Strotz's result stating that naive intertemporal decision making is consistent if and only if the consumer's subjective discount function is exponential. Strotz restricted his analysis to the case where the discount function was time distance dependent. We require only that it be dependent on both the time of consumption and the planning date. Subsequently it is shown that necessary and sufficient conditions for consistent naive intertemporal decision making are multiplicative separability of a in time (t) and the planning date (X).