SIMILARITY JUDGMENTS AND ANOMALIES IN INTERTEMPORAL CHOICE

I. INTRODUCTION Evidence accumulated over many years reveals the inadequacies of the Expected Utility Hypothesis as a descriptive model of choice under uncertainty. Over a much shorter period of time, evidence has accumulated revealing systematic violations of the standard model of choice over time, the Discounted Utility model. In Rubinstein (1988), Azipurua et al. (1993), and Leland (1994, 1998), choice anomalies under uncertainty occur because agents base their decisions on judgments regarding the similarity or dissimilarity of prizes and probabilities across alternatives. (1) This article specifies conditions under which such a procedure implies analogous violations of the Discounted Utility model in intertemporal settings. II. CHOICE ANOMALIES UNDER UNCERTAINTY AND OVER TIME Axioms assumed in models of choice place restrictions on what agents can choose across different pairs of alternatives. The independence axiom, for example, requires that for risky or riskless options [L.sub.1], [L.sub.2], and [L.sub.3], if [L.sub.1] is weakly preferred to [L.sub.2], then the lottery {[L.sub.1], p; [L.sub.3], 1 - p} must be weakly preferred to the lottery {[L.sub.2], p; [L.sub.3], 1 - p} for any p. One consequence of this requirement is that preferences between simple lotteries {$[x.sub.1], [p.sub.1]; $0, 1 - [p.sub.1]} and {$[x.sub.2], [p.sub.2]; $0, 1 - [p.sub.2]} must be invariant to changes in the values of [p.sub.1] and [p.sub.2], which leave their ratio undisturbed. In choices between S and R and between S' and R', for example, independence requires either the choice of S and S' or the choice of R and R'. S: {$3000, 0.90; $0, 0.10} R: {$6000, 0.45; $0, 0.55} S': {$3000, 0.02; $0, 0.98} R': {$6000, 0.01; $0, 0.99} The stationarity assumption of the Discounted Utility model of intertemporal choice places restrictions on how agents can choose between pairs of intertemporal prospects. Consider simple intertemporal prospects [T.sub.j] and [T.sub.k], shown below, where [T.sub.j] offers an increment to consumption [x.sub.j] in time period [t.sub.j] and [T.sub.k] offers an increment to consumption [x.sub.k] in time period [t.sub.k]. [T.sub.j]: {[x.sub.j], [t.sub.j]} [T.sub.k]: {[x.sub.k], [t.sub.k]} Assuming, for simplicity, a common baseline level of consumption per period, c, agents deciding between these options according to the Discounted Utility model will choose as follows where U(*) is a concave, ratio-scaled, utility function, [delta] is the one period discount factor, and > and ~ denote strict preference and indifference, respectively: (2) (1) [FORMULA NOT REPRODUCIBLE IN ASCII] Dividing through by [[delta].sub.j] and rearranging terms yields the following expression: (2) [FORMULA NOT REPRODUCIBLE IN ASCII] Expression 2 reveals that the only way discounting enters into the decision is through the absolute difference in the time periods. As such, agents given choices between [T.sub.1] and [T.sub.2] and between [T.sub.11] and [T.sub.12], shown below, must either select [T.sub.1] and [T.sub.11] or [T.sub.2] and [T.sub.12] since the absolute time interval in both choices is identically 1 period. [T.sub.1]: {$20, 1 month} [T.sub.2]: {$25, 2 months} [T.sub.11]: {$20, 11 months} [T.sub.12]: {$25, 12 months} Neither the restrictions implied by the independence axiom nor those following from stationarity hold empirically. Instead, regarding the independence axiom, Kahneman and Tversky (1979), among others, find that individuals choosing the safer option S over R nevertheless choose the riskier option R' over S' as the difference between probabilities declines, their ratio held constant. This phenomenon is referred to as the common ratio effect. They also find that for lotteries involving losses, the opposite pattern, RS', obtains. This phenomenon is referred to as the reflection effect. …