Descriptive Models of Intertemporal Choice 1 Running Head : DESCRIPTIVE MODELS OF INTERTEMPORAL CHOICE Descriptive Models of Intertemporal Choice

Two comparative models of intertemporal choice have recently been proposed as an alternative to the standard hyperbolic discounting model. One, the interval discounting model, retains the notion that intertemporal choice is governed by discounting, but proposes that discounting involves direct comparisons between the options along the time attribute. The other, the tradeoff model, discards the notion of discounting, and proposes that intertemporal choice is governed by direct comparisons along the time attribute and the money attribute. These comparative models have a broader coverage than the standard hyperbolic discounting model, but it is yet to be shown how much more accurate they are in quantitative analyses, and whether one is more accurate than the other. We parameterize all three models, and apply them to both primary and secondary data. The practical applicability of the two discounting models is limited, and their application is problematic. Moreover, the hyperbolic discounting model performs poorly, and the interval discounting model performs worse than the tradeoff model. We conclude that, both in terms of practical applicability and in terms of predictive accuracy, the tradeoff model is the best tool for empirical analyses of intertemporal choice. While the specific parameterizations of the respective models limit the generality of this result, they were the only ones that we could justify theoretically and estimate empirically. Descriptive Models of Intertemporal Choice 3 Descriptive Models of Intertemporal Choice Part 1: Accuracy and Applicability Intertemporal choices involve tradeoffs between costs and benefits that occur at different points in time. These include choices such as taking a job now or getting an education and having a chance at a better job later, and spending money now or saving it and having more to spend later. Theories of intertemporal choice have been tested on elementary choices between smaller-sooner and larger-later amounts of money, such as receiving $100 now or $150 in 3 months. Economics has provided a normative standard for these choices, and psychology has discovered preference patterns that are anomalous to that standard. This paper is about descriptive models of intertemporal choice, and the accuracy with which they predict anomalies in choices between smaller-sooner and larger-later amounts of money. The normative standard is Samuelson’s (1937) discounted utility model. In this model, each outcome is first integrated with the baseline consumption level, a utility is assigned to the consumption level resulting from this asset integration, the utility is discounted at a constant rate as a function of the delay to the outcome, and the outcome with the highest discounted utility is chosen. The anomalies to this model have come in three waves. The first wave, initiated by Thaler (1981), showed that the rate at which an outcome is discounted is not constant, but varies with the delay to an outcome, the magnitude and sign of the outcome, and changes in its timing (i.e., delay or speedup). These anomalies were accommodated by Loewenstein and Prelec’s (1992) hyperbolic discounting model (after Ainslie, 1975, who coined the term “hyperbolic” discounting). In this model, a value is assigned to each outcome, the value is discounted at a decreasing rate as a function of the delay to the outcome, and the outcome with the highest discounted value is chosen. Although a major development, the hyperbolic discounting model agrees with the discounted utility model that choice is Descriptive Models of Intertemporal Choice 4 alternative-based: The available options are independently assigned an overall value, these values are compared, and the option with the highest value is chosen. The second wave of anomalies showed that discount rates vary not only with the delay to the outcomes, but also with the interval between them (Read, 2001). A model that accommodates these anomalies is the interval discounting model (Scholten & Read, 2006). In this model, the time horizon is partitioned into a series of intervals. The first interval is the time to the first outcome on the horizon; the second is the time between the first outcome and the second; and so on. The value of an outcome is then discounted as a function of all intervals that precede it. The interval discounting model therefore incorporates attribute-based choice, because the options are directly compared along the time attribute before a value is assigned. However, it also incorporates alternative-based choice, because the options are assigned an overall value, even if, in contrast with the discounted utility model and the hyperbolic discounting model, these values are not independently assigned. An outcome at a given delay can therefore be worth more or less, depending on the delays to the other outcomes. The third wave of anomalies, initiated by Rubinstein (2001), showed that discount rates are affected by comparisons between outcomes as well as delays. These “similarity” effects have been treated by Rubinstein (2001, 2003) and Leland (2002), but their proposals do not also address the first two waves of anomalies. A proposal that does this is the tradeoff model (Scholten and Read 2010a). In this model, choice is purely attribute-based: The options are directly compared along both attributes (i.e., time and money), and the option favored by these comparisons is chosen. The tradeoff model has been shown to offer a qualitative account of the evidence that discounting models can and cannot address (Scholten & Read, 2010a), but it is yet to be shown to what extent it can also offer a quantitative account of that evidence. In this paper, we Descriptive Models of Intertemporal Choice 5 examine whether the tradeoff model at least matches the interval discounting model in accommodating the anomalies covered by both models (i.e., all anomalies except the similarity effects). We also examine to what extent both models improve on the hyperbolic discounting model. We are interested in whether the tradeoff model “at least” matches the interval discounting model, because there is an argument that favors the tradeoff model over the interval discounting model, all else equal. It is the argument of practical applicability. Problems in the application of discounting models arise when discount rates are inversely related to outcome magnitude, a robust anomaly called the ‘absolute magnitude effect. ’ In the hyperbolic discounting model and the interval discounting model, this anomaly is captured by a complex parametric specification of the value function (see al-Nowaihi & Dhami, 2009, and the specification proposed in this paper). In the tradeoff model, on the other hand, the absolute magnitude effect is captured by a simple model equation. The bottom line is that the tradeoff model can be estimated following standard procedures, whereas discounting models often force the user to resort to complex estimation techniques that may actually prevent these models from being estimated. We proceed as follows. First, we present the benchmark model, according to which outcomes are discounted at a constant rate. Then we discuss how the three descriptive models under consideration account for variation in discount rates. After this theoretical discussion, we deal with practical issues in the application of descriptive models. Then we apply the three models to both primary and secondary data. The hyperbolic discounting model performs poorly, and the interval discounting model performs worse than the tradeoff model. We conclude with methodological considerations in the formal analysis of intertemporal choice. Descriptive Models of Intertemporal Choice 6 The Benchmark Model We focus on intertemporal choices between pairs of single dated outcomes, one smaller-but-sooner (SS), the other larger-but-later (LL). An example is the choice between $100 in 1 month and $150 in 4 months. The outcomes are designated as xS and xL ($100 and $150), and their respective delays as tS and tL (1 and 4 months). Given such a choice, a person may prefer either SS or LL, or be indifferent between them. If indifferent, the delays and outcomes can be transformed into an empirical measure of discounting. Commonly, the model used for conducting this transformation assumes exponential (or ‘constant’) discounting of outcomes. According to this model, which we will call the benchmark model, the person is indifferent between SS and LL when L t S t x x L S δ δ = , (1) where δ is the discounting over any unit interval t → t + 1. A lower δ indicates more discounting: A lower proportion of x remains if x is delayed by one unit of time. An alternative measure of discounting used in the literature is ρ = (1 δ) / δ, a lower value of which indicates less discounting. Solving Equation 1 for δ, we get