The term time inconsistency is used to describe a situation where one’s ‘optimal’ decision depends on ‘when’ he/she makes the decision and the preferred decision at each point in time is different to what you thought you would prefer in the past. In other words, there is (was or will be?) disagreement between the past-me, me, and future me. We don’t need to look around too hard to find a few examples of this happening in reality. My personal favourite is problems revolving around going to the gym.
By now you are probably starting to realise where I’m going with this. How many times have we paid a huge membership upfront after a rough calculation of the number of times we need to attend to make it worth the payment? And also, how many times have we deviated from this simple plan, only to waste a couple of hundreds of dollars?
Dellavigna and Malmendier (‘Paying not to go to the gym’, 2005) have written an interesting paper exploring this issue (among many others). In particular the paper describes how a certain type of discounting may help economists understand these time inconsistency issues.
This helpful type of discounting function is called hyperbolic discounting. This function simply discounts the near future much steeper than the far future. An implication of this type of discounting is that the agents exhibit a tendency to prefer immediate gratification over future gratification, and hence he is likely to be on a time-inconsistent decision path.
Now let’s return to the gym problem. The interesting question here arises from the fact that the consumers of this service have a choice of contracts they can choose from. They can choose to pay an upfront fee and for a fixed length of time or they can pay per visit. With the standard exponential discounting used in the majority of economic models, there is no scope for time inconsistency since the agent discounts utility at a constant rate, wherever he/she stands in time. With no time inconsistency, if an agent plans a certain number of visits to the gym, he will actually visit the gym that number of times. Consequently, the optimal contract he chooses is simply the one that he needs to pay less for, given his planned number of visits. What does the data say about gym contract choices in the real world?
Dellavigna and Malmendier find that people on contracts which require a flat monthly fee of $70 attend on average 4.8 times a month So what? It is important to note that these people have another contract choice, where they can pay $10 a visit. Doing some quick back-of-the-envelope calculations, it can easily be seen that the gym users are paying approximately $20 more each month than they actually need to. Does this mean that real people are irrational? Or does it simply mean that real people are rational, but time inconsistent.
The answer probably lies in between the two extremes, but I would personally say it is marginally closer on the side of the latter. If we assume hyperbolic discounting functions and some level of sophistication (knowledge about his/her own preference for immediate gratification) in the agent, we can see why he/she would be willing to pay slightly more than he/she needs to.
Consider for example, an agent who goes into the gym to consult which membership plan he/she should be on. He/She is aware that costs looming in the immediate future are perceived to be more intense than if they are in the far future (imagine planning to start a diet tomorrow, and actually not eating when the food is right in front of you). With this knowledge the agent knows that it is going to be harder to get himself/herself to actually go to the gym as many times as he/she wishes to now.
Now, if there exists some device that could sink a portion of the costs involved in traveling to the gym, he/she will demand it and use it, in the hope of being able to stick more closely to his/her originally planned number of visits. In other words, by paying a flat-rate fee for a given time period, an agent is using a commitment device to help his future selves stay committed to the path he/she originally planned.
This has been a grossly simplified explanation of how these intra-personal inter-temporal conflicts are modeled, but I hope it provided some insight into how behavioural theory can shed some light on issues that conventional approaches may conclude to be irrational.