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Random Intertemporal Choice Available online at www.sciencedirect.com ScienceDirect Journal of Economic Theory 177 (2018) 780–815 www.elsevier.com/locate/jet ✩ Random intertemporal choice ∗ Jay Lu a, , Kota Saito b a University of California, Los Angeles, United States b California Institute of Technology, United States Received 11 April 2017; final version received 27 May 2018; accepted 15 August 2018 Available online 17 August 2018 Abstract We provide a theory of random intertemporal choice. Agents exhibit stochastic choice over consump- tion due to preference shocks to discounting attitudes. We first demonstrate how the distribution of these preference shocks can be uniquely identified from random choice data. We then provide axiomatic char- acterizations of some common random discounting models, including exponential and quasi-hyperbolic discounting. In particular, we show how testing for exponential discounting under stochastic choice in- volves checking for both a stochastic version of stationarity and a novel axiom characterizing decreasing impatience. © 2018 Elsevier Inc. All rights reserved. JEL classification: D8; D9 Keywords: Stochastic choice; Intertemporal choice; Discounting; Stationarity 1. Introduction In many economic situations, it is useful to model intertemporal choices, i.e. decisions in- volving tradeoffs between earlier or later consumption, as stochastic or random. For instance, in typical models of random utility used in discrete choice estimation, this randomness is driven by ✩ We want to thank David Ahn, Jose Apesteguia, Miguel Ballester, Yoram Halevy, Yoichiro Higashi, Vijay Krishna, Tomasz Strzalecki, Charlie Sprenger and participants at D-Day at Duke, LA Theory Bash, UCSD, Barcelona GSE Stochastic Choice Workshop, D-TEA and the 16th SAET Conference at IMPA for their helpful comments. * Corresponding author. E-mail addresses: [email protected] (J. Lu), [email protected] (K. Saito). https://doi.org/10.1016/j.jet.2018.08.005 0022-0531/© 2018 Elsevier Inc. All rights reserved. J. Lu, K. Saito / Journal of Economic Theory 177 (2018) 780–815 781 unobserved heterogeneity where the econometrician is not privy to all the various determinants of discounting attitudes.1 Even when considering the behavior of a single individual, intertem- poral choices can still be stochastic.2 Decisions involving tradeoffs at different points in time are heavily influenced by visceral factors which are often of an uncertain nature even from the perspective of the decision-maker.3 In addition to being descriptively more accurate, a model of random intertemporal choice would also be useful for welfare analysis. An agent whose discounting is random but his utility is deterministic may behave as if his discounting is deterministic but his utility is random. However, the welfare analysis of an agent whose discounting is random would naturally be different from that of an agent with deterministic discounting. Given all these issues, any care- ful analysis and interpretation of behavioral patterns in intertemporal choice would require a probabilistic, or random model of discounting. In this paper, we provide a theoretical framework to study random intertemporal choice. We model random discounting as a distribution of preference shocks to discounting attitudes. Impor- tantly, we focus on random discounting as the sole source of stochastic choice. This allows us to precisely characterize the relationship between random discounting and stochastic choice data.4 In applications such as demand estimation where the relevant variable of economic interest is probabilistic choice, this is a useful and important exercise. Moreover, since random discounting is modeled as preference shocks on discounting attitudes, our theory yields robust comparative statics as demonstrated in recent work by Apesteguia and Ballester (2018). Our model is flexible enough to allow for random discounting to be interpreted in two ways. In the first interpretation, we consider stochastic choice as the aggregated choice frequencies made by agents in a group. Aggregated choices are random due to unobserved heterogeneity in the population from the perspective of an outside observer such as an econometrician. This is the case in most applications of discrete choice estimation or in typical intertemporal choice experiments. In the second interpretation, we consider stochastic choice as probabilistic choice from a single agent due to individual shocks to discounting attitudes. For instance, the agent is asked to choose from a menu of consumption streams repeatedly over a short interval of time. Under this interpretation, we can obtain stochastic data from experiments such as in Tversky (1969), Camerer (1989), Ballinger and Wilcox (1997), and more recently Regenwetter et al. (2011) and Agranov and Ortoleva (2017).5 In this case, final payoffs are randomized across the agent’s choices, so the agent considers each choice problem independently of the others.6 Random choice is then obtained from the frequency of the agent’s repeated choices. In both interpretations, we interpret stochastic choice as arising from ex-ante choices with commitment. By ex-ante, we mean that we observe choices before any consumption is realized. Indeed, this is the case in the experiments mentioned above. For example, in 1 For further discussions on random utility and discrete choice estimation, see McFadden (2001)and Train (2009). 2 For some recent evidence, see Short Experiments 2 of Agranov and Ortoleva (2017). 3 Frederick et al. (2002) provides a detailed discussion on such visceral influences on intertemporal choice. 4 In Section 6, we generalize our model to allow for taste shocks as well. 5 See the introduction of Agranov and Ortoleva (2017)for more experiments on stochastic data. 6 Assuming the agent is an expected utility maximizer, this payment procedure means that he considers each choice problem separately. 782 J. Lu, K. Saito / Journal of Economic Theory 177 (2018) 780–815 Agranov and Ortoleva (2017), each subject is presented with the same choice problem repeat- edly and his choices are elicited before any payment.7 Our main contributions are twofold. First, we show that the distribution of random discounting can be uniquely identified from random choice. In other words, an outside observer such as an econometrician can recover the entire distribution of discount attitudes given sufficient stochas- tic choice data. Second, we provide axiomatic characterizations of our model including random exponential and quasi-hyperbolic discounting as special cases. As a result, we extend the charac- terizations of classic models of intertemporal choice to the domain of random choice with novel implications. ,In particular our characterization of random exponential discounting sheds new light on one of the benchmark properties of rational choice, stationarity. Originally proposed by Koopmans (1960), the classic stationarity axiom states that choices are not affected when all consumptions are delayed by the same amount of time and it is a defining property of exponential discounting. We propose a stochastic version of the stationarity axiom which we call Stochastic Stationarity. It states that choice probabilities are not affected when all consumptions are delayed by the same amount of time. Under stochastic choice, the relationship between Stochastic Stationarity and exponential dis- counting is weaker; one can find a random choice model that satisfies Stochastic Stationarity and all the standard properties but is not exponential discounting.8 In fact, testing for random expo- nential discounting involves checking not only for Stochastic Stationarity but a new axiom which we call Decreasing Impatience. By itself, Decreasing Impatience exactly characterizes decreas- ing discount ratios.9 Testing for quasi-hyperbolic discounting involves checking for a weaker version of Stochastic Stationarity and Decreasing Impatience. All these relationships are novel and unique to random intertemporal choice. In general, our main model is a random utility maximization model with discounted utilities. A discount functionD is a decreasing function over time that has value 1 initially and satisfies → →∞ the tail condition s>t D (s) 0as t . Random choice is characterized by a distribution μ on the set D of discount functions and a fixed taste utility u. More precisely, the probability that an infinite-period consumption stream f = (f(0), f (1),...) is chosen from a menu F of consumptions streams is the probability that f is ranked higher than every other consumption streams in F . In other words, if we let ρF (f ) denote this probability, then = ∈ D ≥ ∈ ρF (f ) μ D D (t) u (f (t)) D (t) u (g (t)) for all g F . t t We call this a random discounting model. One could interpret each realization of a discount function as corresponding to a particular agent in a population (as in most applied models of random utility) or to a particular realization of a preference shock to discounting for an individ- ual. 7 On the other hand, by ex-post, we mean that we observe choices only after all consumptions are realized. It may be difficult to take this ex-post interpretation literally as consumption steams in our paper have infinite length. However, with a slight modification of our axioms for finite-period consumption streams, we can interpret our model as ex-post as well. We consider infinite streams as it is a standard domain in the literature of intertemporal choice and allows for easy axiomatic comparisons. 8 See Proposition 2 for an explicit example. 9 If we
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