Running Head: INTERTEMPORAL TRADEOFFS The Psychology of Intertemporal Tradeoffs

It is commonly assumed that people make intertemporal choices by „discounting‟ the value of delayed outcomes, assigning discounted values independently to all options, and comparing the discounted values. We identify a class of anomalies to such alternative-based discounting, which collectively show that options are not treated independently but rather comparatively: The time difference, or interval, between the options sometimes counts more and sometimes counts less as a whole than if it were divided into shorter subintervals (superadditivity and subadditivity, respectively), and whether the interval counts more or less depends on the money difference, or compensation, involved (inseparability). We develop a model that replaces alternative-based discounting with attribute-based tradeoffs. In our model, people make intertemporal choices by weighing how much more they will receive or pay if they wait longer against how much longer the wait will be, or, conversely, how much less they will receive or pay if they do not wait longer against how much shorter the wait will be. This model, called the tradeoff model, accommodates, in a psychologically plausible way, all anomalies that the discounting approach can and cannot address. Intertemporal Tradeoffs 3 The Psychology of Intertemporal Tradeoffs Intertemporal choices involve tradeoffs between costs and benefits that occur at different points in time (Loewenstein & Elster, 1992, Loewenstein, Read, & Baumeister, 2003). 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. Most intensively investigated, however, are much more elementary choices between smaller-sooner and larger-later amounts of money, such as receiving $100 now or $150 in 4 months. It is commonly assumed that people make intertemporal choices by „discounting‟ the value of delayed outcomes, assigning discounted values to the available options, and then comparing these discounted values. For this choice, they would compare the value of $100 now with the discounted value of $150 in 4 months. We will argue that this does not accurately describe the psychology of intertemporal tradeoffs. Discounting models belong to the broad class of alternative-based choice models, in which the available options are independently assigned an overall value, these values are compared, and the option with the highest value is chosen. Such alternative-based choice models can be contrasted with attribute-based ones (Payne, Bettman & Johnson, 1988), in which the available options are directly compared along their attributes and the option favored by these comparisons is chosen. 1 Alternative-based discounting models can accommodate much of what we know about intertemporal choice, but there remains evidence, reviewed and strengthened in this paper, that can only be addressed by an attribute-based choice model. In this paper, we develop such a model, called the tradeoff model. Our model accommodates, in a psychologically plausible way, all anomalies that discounting models can and cannot address. We begin with a review of discounting models and the evidence covered by them. We then identify three anomalies to discounting theory. Given these anomalies, we develop the Intertemporal Tradeoffs 4 tradeoff model and show how it accommodates all evidence under consideration, whether consistent or inconsistent with discounting. After this largely qualitative development, we provide a fully parametric specification of the tradeoff model. We conclude by discussing how the tradeoff model deals with additional evidence and how it improves on related theories of choice. Discounting Theory and Anomalies in Intertemporal Choice 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). We first describe how an elementary choice like this is addressed by the discounted utility model, the normative standard for intertemporal choice, and then discuss how successive reformulations of discounting theory have addressed preference patterns anomalous to this normative standard. Delay discounting model 1: Samuelson’s (1937) discounted utility model In this model, intertemporal preferences are governed by the discounted utilities of the options. Discounted utilities are determined by first integrating the outcomes of choice with the baseline consumption level, assigning utilities to the consumption levels resulting from this asset integration, and then exponentially discounting these utilities as a function of the delays to the outcomes. Formally, the discounted utilities of SS and LL are given as ) ( ) ( c u x c u U L S t S t S      , ) ( ) ( L t t L x c u c u U L S      , where 0 <  < 1 is an exponential discounting parameter, c is a constant baseline consumption level, and u is a concave utility function over consumption levels. The decision maker will be indifferent between SS and LL when Intertemporal Tradeoffs 5 ) ( ) ( ) ( ) ( c u x c u c u x c u L S t t S L        . (1) For elementary choices between SS and LL, the normative status of the discounted utility model derives largely from two of its features (for a more complete discussion, see Frederick, Loewenstein, & O‟Donoghue, 2002): Exponential discounting and asset integration. Exponential discounting ensures consistency in the treatment of time, so that intertemporal preferences will not change merely because time has passed (Strotz, 19551956). The preference between two outcomes separated by a 3-months interval, for instance, should be the same regardless of whether the interval begins now, in 1 month, or in 9 months. Asset integration ensures consistency in the treatment of money. For instance, someone who prefers $150 in 4 months to $100 in 1 month should, in two subsidiary choices between $75 in 4 months and $50 in 1 month, consistently prefer the former over the latter (see Prelec & Loewenstein, 1991). Delay discounting model 2: Laibson’s (1997) quasi-hyperbolic discounting model In this model, indifference between SS and LL is given by Equation 1 when both outcomes are delayed, but not when SS is a smaller-immediate outcome. This is because a single „jolt‟ of discounting is added to the exponential discounting of delayed utilities: ) ( ) ( c u x c u U L t S S     , ) ( ) ( L t L x c u c u U L     , where 0 <  < 1 is a quasi-hyperbolic discounting parameter. Thus, the decision maker will be indifferent between SS and LL when ) ( ) ( ) ( ) ( c u x c u c u x c u L S tL       . (2) The implication of Equations 1 and 2 is the „immediacy effect‟ (Keren & Roelofsma, 1995; Read, Loewenstein, & Kalyanaraman, 1999; Weber & Chapman, 2005), also called „presentIntertemporal Tradeoffs 6 biased preferences‟ (O‟Donoghue & Rabin, 1999): There is more discounting over an interval that begins now than one that begins later. For instance, someone who is indifferent between $150 in 3 months and $100 now will prefer $150 in 4 months to $100 in 1 month. Delay discounting model 3: Loewenstein and Prelec’s (1992) hyperbolic discounting model This model drops the assumptions of exponential discounting and asset integration. In this model, preferences are governed by the discounted values of the options. Discounted values are determined by first assigning values to changes in consumption level, and then hyperbolically discounting these values as a function of the delay to the outcomes. Formally, the discounted values of SS and LL are then given as ) ( 1 S t i i S x v δ V S    , ) ( 1 L t i i L x v δ V L    , where v is a value function over changes in consumption level. For each time unit, i, there is a hyperbolic discounting parameter, 0 < i < 1, which increases with i. The hyperbolic discount function given by Loewenstein and Prelec (1992) describes how. At the indifference point, ) ( ) ( 1 L S t t i i x v x v δ L