Microfoundations of Discounting

An important question in economics is how people choose between different payments in the future. The classical normative model predicts that a decision maker discounts a later payment relative to an earlier one by an exponential function of the time between them. Descriptive models use nonexponential functions to fit observed behavioral phenomena, such as preference reversal. Here we propose a model of discounting, consistent with standard axioms of choice, in which decision makers maximize the growth rate of their wealth. Four specifications of the model produce four forms of discounting—no discounting, exponential discounting, hyperbolic discounting, and a hybrid of exponential and hyperbolic discounting—two of which predict preference reversal. Our model requires no assumption of behavioral bias or payment risk.

Time-Consistent Investment and Consumption Strategies under a General Discount Function

In the present paper, we investigate the Merton portfolio management problem in the context of non-exponential discounting, a context that gives rise to time-inconsistency of the decision-maker. We consider equilibrium policies within the class of open-loop controls that are characterized, in our context, by means of a variational method which leads to a stochastic system that consists of a flow of forward-backward stochastic differential equations and an equilibrium condition. An explicit representation of the equilibrium policies is provided for the special cases of power, logarithmic and exponential utility functions.

Controlled Markov chains with non-exponential discounting and distribution-dependent costs

This paper deals with a controlled Markov chain in continuous time with a non-exponential discounting and distribution-dependent cost functional. A definition of closed-loop equilibrium is given and its existence and uniqueness are established. Due to the time-inconsistency brought by the non-exponential discounting and distribution dependence, it is proved that the equilibrium is locally optimal in some appropriate sense. Moreover, it is shown that our problem is equivalent to a mean-field game for infinite-many symmetric players with a non-exponential discounting cost.

Present Bias

Present bias is the inclination to prefer a smaller present reward to a larger later reward, but reversing this preference when both rewards are equally delayed. Such behavior violates stationarity of temporal choices, and hence exponential discounting. This paper provides a weakening of the stationarity axiom that can accommodate present‐biased choice reversals. We call this new behavioral postulate Weak Present Bias and characterize the general class of utility functions that is consistent with it. We show that present‐biased preferences can be represented as those of a decision maker who makes her choices according to conservative present‐equivalents, in the face of uncertainty about future tastes.

Exponential discounting in security games of timing

Strategic game models of defense against stealthy, targeted attacks that cannot be prevented but only mitigated are the subject of a significant body of recent research, often in the context of advanced persistent threats (APTs). In these game models, the timing of attack and defense moves plays a central role. A common assumption, in this literature, is that players are indifferent between costs and gains now and those in the distant future, which conflicts with the widely accepted treatment of intertemporal choice across economic contexts. This article investigates the significance of this assumption by studying changes in optimal player behavior when introducing time discounting. Specifically, we adapt a popular model in the games of timing literature, the FlipIt model, by allowing for exponential discounting of gains and costs over time. We investigate changes of best responses and the location of Nash equilibria through analysis of two well-known classes of player strategies: those where the time between players’ moves is constant, and a second class where the time between players’ moves is stochastic and exponentially distributed. By introducing time discounting in the framework of games of timing, we increase its level of realism as well as applicability to organizational security management, which is in dire need of sound theoretic work to respond to sophisticated, stealthy attack vectors.

Using the past to estimate sensory uncertainty

To form a more reliable percept of the environment, the brain needs to estimate its own sensory uncertainty. Current theories of perceptual inference assume that the brain computes sensory uncertainty instantaneously and independently for each stimulus. We evaluated this assumption in four psychophysical experiments, in which human observers localized auditory signals that were presented synchronously with spatially disparate visual signals. Critically, the visual noise changed dynamically over time continuously or with intermittent jumps. Our results show that observers integrate audiovisual inputs weighted by sensory uncertainty estimates that combine information from past and current signals consistent with an optimal Bayesian learner that can be approximated by exponential discounting. Our results challenge leading models of perceptual inference where sensory uncertainty estimates depend only on the current stimulus. They demonstrate that the brain capitalizes on the temporal dynamics of the external world and estimates sensory uncertainty by combining past experiences with new incoming sensory signals.

Time-inconsistent discounting and the Friedman rule: roles of non-unitary discounting

We examine the optimality of the Friedman rule by considering recent development of behavioral economics. We construct a simple macroeconomic model where agents discount consumption and leisure at different rates. We also consider a standard exponential discounting model and a hyperbolic discounting model, by assuming that the same discounting applies to both consumption and leisure. Money is introduced via a cash-in-advance constraint. Although the three models are observationally equivalent, they provide different policy implications. The Friedman rule is optimal in the latter two models, while it is not optimal in the first model if agents discount consumption is at a higher rate than leisure.