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Lotteries over time

Georg Weizsäcker, HU Berlin

On March 4, the film studios MGM and Universal announced the postponement of the new James Bond film’s release (“No Time To Die”) from April to November 2020. At the time of the announcement, Chinese movie theaters had already closed, but little else was known about the SARS-CoV-2 restrictions worldwide. Pushing the date so far back seemed a rather cautious move – but the film producers faced a difficult choice: had they re-scheduled the release to an earlier date, e.g. in July 2020, they would stand to gain an earlier stream of revenues if the shutdown is over in July. If it is not over, however, another re-scheduling would become necessary. In this case, it would likely be too late for a release in November – release date scheduling is complex – and the producers would have to delay until an even later date. In essence, they made a choice between a safe-but-late option and an early-but-risky option.

The reader will recognize this trade-off, as many economic agents now face similar problems during the shutdown. The timing of access to markets is uncertain, which impedes all scheduling. When can a conference, or trade fair, take place? When does a training program start? When are delivery chains and product demand strong enough for production facilities to plan a ramp-up? In each case, planning for a too-early re-invigoration runs the risk of having to postpone even more than what would be possible with more prudent planning.

This highlights the importance of the agents’ risk attitudes with respect to time. Do they like or dislike uncertainty in the timing of things, and what is their willingness to pay for avoiding it? The bulk of the literature on choice under uncertainty takes timing as given and discusses randomness in the size of earnings, or other outcomes. In the present discussion, we hold the size of the outcome constant, and ask about the willingness to accept risks in the time of obtaining it. The literature on this issue is very small, but two recent papers by DeJarnette et al. (2019) and Ebert (2019) give a very good introduction, and make a clear theoretical prediction.

Perhaps the reader wants to pause here, and make a guess?

The theoretical prediction is that under expected discounted utility, which is the standard model of economic decision making over time, economic agents are risk seeking: they prefer adding uncertainty about the time of receiving a good. For many of us, this is a surprising and unintuitive result. Yet, as DeJarnette et al. (2019) and Ebert (2019) show, it is a very general result and holds under a wide variety of preference assumptions. The logic is simple: if discounting is convex over time – e.g. a constant discount factor is multiplied for every period of delay – then a time-risky reward has a larger expectation of the reward’s discounted value than if the same reward were to arrive at the average time for sure. This is the math – to get an intuition, think of someone with a nagging impatience who desperately wants to receive a good now: she is willing to take risks in arrival time if they satisfy her impatience with sufficient likelihood. If receiving the good now is all that counts, then even a decent probability of receiving it now is better than receiving the good a little later for sure. This logic extends to the much more general result.

What economic insights follow from this discussion, for the current shutdown? First, we need to be aware that the theoretical result may be empirically false, as is also suggested by the first controlled experiments: real people appear to be risk averse with respect to time (see DeJarnette et al, 2019). Additional empirical assessments would be important here. Second, both risk seeking and risk averse behaviors will likely have negative externalities for the economy. Taking large scheduling risks now may induce repeated re-scheduling and thereby destroy more economic value. (For instance, movie theater owners may rely on the release date.) Conversely, delaying things too far would also slow the economy down. From a societal perspective, taking the externalities into account, it seems preferable to avoid both effects and reduce the uncertainty about the shutdown’s duration as far as is reasonably possible.

DeJarnette, P., D. Dillenberger, D. Gottlieb and P. Ortoleva, „Time Lotteries and Stochastic Impatience”, Econometrica, forthcoming. 

Ebert, S., “Decision-Making When Things are Only a Matter of Time”, Operations Research, forthcoming. 


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