We study preferences over lotteries that pay a specific prize at uncertain future dates: time lotteries. The standard model of time preferences, Expected Discounted Utility (EDU), implies that individuals must be risk seeking in this case. As a motivation, we show in an incentivized experiment that most subjects exhibit the opposite behavior, i.e., they are risk averse over time lotteries (RATL). We then make two theoretical contributions. First, we show that RATL can be captured by a generalization of EDU that is obtained by keeping the postulates of Discounted Utility and Expected Utility. Second, we introduce a new property termed Stochastic Impatience, a risky counterpart of standard Impatience, and show that not only the model above, but also substantial generalizations that allow for non-Expected Utility and non exponential discounting, cannot jointly accommodate it and RATL, showing a fundamental tension between the two.
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