Consumption-savings Decisions with Quasi-geometric Discounting by per Krusell and Anthony

THE PURPOSE OF THIS PAPER is to study how an infinitely-lived consumer with "quasigeometric" discounting-thought of as represented by a sequence of "selves" with conflicting preferences-would make consumption and savings decisions. In light of experimental evidence suggesting that individuals do not have geometric discount functions (see, for example, Ainslie (1992) and Kirby and Herrnstein (1995)), it is important to understand how departures from geometric discounting affect an individual's consumption-saving decisions.2 We assume that time is discrete and that the consumer cannot commit to future actions. We assume that the consumer is rational in that he is able to forecast correctly his future actions: a solution to the decision problem is required to take the form of a subgameperfect equilibrium of a game where the players are the consumer and his future selves. We restrict attention to equilibria that are stationary: they are Markov in current wealth; that is, current savings cannot depend either on time or on any other history than that summarized by current wealth. The consumption-savings problem is of the simplest possible kind: there is no uncertainty, and current resources simply have to be divided into current consumption and savings. The period utility function is strictly concave, and the consumer operates a technology for saving that has (weakly) decreasing returns. A special case is that of an affine production function; this special case can be interpreted as one with a price-taking consumer who has a constant stream of labor income and can save at an exogenous interest rate. Our main finding is one of indeterminacy of Markov equilibrium savings rules: there is a continuum of such rules. These rules differ both in their stationary points and in their implied dynamics. First, there is a continuum of implied stationary points to which the consumer's asset holdings may converge over time. Second, associated with each stationary point is a continuum of savings rules, implying that there is a continuum of dynamic paths converging to each stationary point. We construct these equilibria explicitly-the savings rules are step functions. The discontinuities in the step functions are key: payoff functions with jumps can be optimal precisely because the different selves have conflicting