Mental Zooming as Variable Asset Integration in Inter-Temporal Choice

Introduction

Loewenstein and Prelec (1992) were the first to give a good overview of anomalies in inter-temporal choice. Anomalies are defined to be violations of the discounted utility (DU) model of Samuelson (1937). While Samuelson’s ambitions were very modest for this model, it gained widespread popularity as it represented rational inter-temporal choice equivalent to Expected Utility Theory (EUT) in risky decisions. To this day, it serves as a valuable benchmark. The anomalies in intertemporal choice include hyperbolic discounting (discount rates fall with the length of the time horizon), magnitude effects (small outcomes discounted more than large outcomes), the sign effect (gains are discounted more than losses), preference for improving sequences, and the delay-speedup asymmetry (Loewenstein and Prelec, 1992).

This paper aims to contribute to the literature on hyperbolic discounting and magnitude effects and their possible explanations. Hyperbolic discounting differs from exponential discounting (DU-model) in two ways. It puts higher weight on the present, and it involves a higher degree of patience for more distant prospects than the DU-model dictates. This paper refers to the latter as general hyperbolic discounting in contrast to the former, which is impatience in the form of present bias. The focus here is on general hyperbolic discounting. In particular, the study investigates to what extent the zooming theory, proposed by Holden and Quiggin (2017), is consistent with the empirical regularities found in incentivized lab-in-the-field experiments. The rationale behind the zooming theory and model is that larger and more distant amounts pave the way for a broader financial assessment compared to smaller near future amounts. In other words, the lower discount rates observed for larger and more distant amounts arises because the respondents take a more long-term and holistic view of their financial situation for such decisions. This means that they, to a larger extent, integrate the amount with other assets and consumption plans. In contrast, smaller and near future amounts call for narrower framing and thereby less asset integration. At a technical level, the zooming in the zooming models works through variable asset integration, where small near future amounts involve close to zero asset integration whereas larger amounts and longtime horizons call for considerable asset integration. Section 3 gives the algebraic formulation of the model and how it is applied to experimental data.

Different theories have been proposed to explain hyperbolic discounting. First, the most well-known and documented is the present bias associated with immediate pleasure, addiction, self-control problems and commitment devices, and liquidity constraints. Present bias is associated with quasi-hyperbolic discounting (Loewenstein and Prelec, 1992; Augenblick et al., 2015; Augenblick and Rabin, 2019; Balakrishnan et al., 2020) - also defined as the (β, δ)-formulation and originated from Phelps and Pollak (1968). Second, risk or uncertainty about future payments versus immediate payments is another potential reason for apparent time-inconsistent choices, and that has been studied (Halevy, 2008, 2015; Epper et al., 2011). To control for such differences in risk between immediate and future payments, some studies include delayed up-front points in time, such as introducing a one-week delay. A recent study in Kenya revealed that even very short delays in initial payment eliminated present bias (Balakrishnan et al., 2020). Other studies have provided guarantees related to future amounts. Grijalva et al. (2014) provided such guarantees and found diminishing impatience in a Multiple Choice List (MCL) experiment with time horizons of 5, 10, and up to 20 years into the future.¹ Moreover, a Convex Time Budget (CTB) experiment² with similar long time horizons and guaranteed future payments (Grijalva et al., 2018), also found diminishing impatience associated with longer time horizons. Their estimated discount rates were an order of magnitude lower than rates found with the CTB approach over much shorter time horizons of 5-14 weeks by Andreoni and Sprenger (2012), who found high discount rates and no present bias. The discount rate gap across these studies may be due