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In 1999, Ted O’Donoghue and Matthew Rabin published a paper in the American Economic Review that, 8 years later, would serve as the catalyst for my interest in Behavioural Economics. For the longest time, economists had recognised that people place a lower weight on things that happen later in the future than things that happen sooner. The problem was that the way in which this was represented mathematically, although simple and convenient, suggested that people would behave consistently about the decisions they made over time. In practice, this is not true. Many experiments have shown that people are quite impatient when rewards are right in front of them, and they are unwilling to wait for something better. However, push the same decision further into the future, and people suddenly decide to become more patient.

This makes sense from an evolutionary perspective. Stuff far into the future is abstract and has lots of risks of not being realised. Stuff right now, on the other hand, is concrete. Imagine we were around a thousand or so years ago. We had some food right in front of us, but were promised 1.5x as much if we waited until tomorrow. The implications of us waiting and then getting nothing tomorrow may have been a matter of life and death.

O’Donoghue and Rabin’s (ODR) paper made a simple change to the classical model that can explain a lot of interesting and more realistic behaviour. In what follows, I present a couple of the classic examples from that paper to help you understand the logic behind them.

#### TCs, Sophisticates, and Naifs

Classical economic theory assumed that people discounted the future in a constant way. This discount factor is represented by δ, where δ is between 0 and 1. Every time period you go into the future, you discount what you get by δ. Suppose δ=0.5, and a time period was a day. Then getting 10 today would be worth 10, but getting 10 tomorrow would be worth 5 to you right now. Getting 10 in two days time would be worth 10 x 0.5 x 0.5 = 2.5 to you now, and so on. Obviously, the closer δ is to 1, the more you care about the future. This process is known as exponential discounting, because every period into the future is discounted by δ raised to the power of however many periods in future we are talking about. Since these people are time consistent, they are referred to as TCs in the paper.

In order to model the fact that people are present-biased (i.e. we place a much greater value on stuff right now than at any time in the future), ODR change the original formulation very subtly. They introduce a new parameter, β, which is also between 0 and 1. Now, every period in the future is discounted by βδ. This is obviously smaller than just δ on its own if β<1. People who follow this type of discounting are known as β-δ discounters.

There are two types of these discounters. Sophisticates are people who know they are present biased, and know they will be in future. Naifs are the same, with the crucial difference being that they think they will be TCs in future. In other words, they are naive because they don’t think they will be subject to the same present bias in future as they are today.

Comparing these 3 ‘types’ of people leads to some very interesting explanations for why people might do things too late (procrastinate) or too early (preproperate).

#### Being Sophisticated helps you to avoid procrastination…

In their first scenario, imagine you have to do a report, and that you have 4 possible weekends to do it. The problem is that there is a film on every weekend at the cinema, and doing the report that weekend means you have to miss the film. There is an added issue: each week, the film showing is better than the previous week’s one. To be more specific, doing the report in the first week will cost you 3 units of pleasure, doing it in the second will cost you 5, the third will cost you 8, and the fourth will cost you 13.

Let’s assume δ=1 and β=0.5 for simplicity. What happens to each of the types of people? When will they choose to do the report?

##### The TC

Recall that TCs don’t have a β. Since their δ=1, it means they don’t discount any cost or reward in the future. It’s easiest to start from the last period and work backwards to see what they will do. In what follows, I’m going to use the letter D to mean ‘do the report’ and W to mean ‘wait until the next week’.

• Week 4: You have to do the report if you haven’t done it before now. D.
• Week 3: If you do it now, you get -8 from missing the film. If you wait, you will end up getting -13 in week 4. So you should do it now. D.
• Week 2: If you do it now, you get -5. If you wait, you will get -8 from doing it in week 3. D.
• Week 1: If you do it now, you get -3. If you wait, you get -5 next week. D.

So the TCs will do the report straight away in week 1. They always get the task done as soon as possible because they know they will always lose more by waiting. You might think of this as ‘ideal’ behaviour.

##### The Naif
• Week 4: You have to do the report if you haven’t done it before now. D.
• Week 3: If you do it now, you get -8 from missing the film. If you wait, you will end up getting -13 x 0.5 = -6.5 in week 4. So you wait, because the cost doesn’t seem as bad to you now as it will next week. W.
• Week 2: If you do it now, you get -5. If you wait, you think you will do it in week 3, like a good TC would. Hence, you expect to get -8 x 0.5 = -4 if you wait. W.
• Week 1: If you do it now, you get -3. If you wait, you think you will do it next week, which gives you -5 x 0.5 = -2.5. W.

The naif is the master procrastinator. Because the costs of missing a good film don’t seem that bad now compared with when that day actually arrives, naifs will always put activities off until later. In this case, naifs wait all the way until the final week to do the report.

##### The Sophisticate
• Week 4: You have to do the report if you haven’t done it before now. D.
• Week 3: If you do it now, you get -8 from missing the film. If you wait, you will end up getting -13 x 0.5 = -6.5 in week 4. So you wait, because the cost doesn’t seem as bad to you now as it will next week. W.
• Week 2: If you do it now, you get -5. If you wait, you are smart and know that you will put it off in week 3 as well. So you know that in week 4 you would end up getting -6.5. D.
• Week 1: If you do it now, you get -3. If you wait, you know you will do it next week, which gives you -5 x 0.5 = -2.5 . W.

The sophisticate procrastinates as well, just like the naif. But since they are correctly aware about their future proneness to present bias, they don’t procrastinate as much as the naif. They only put the report off one week, and get it done the following week.

#### … but Sophistication can make you do things too early

The second scenario is very similar to the first. There is no report to do any more, but there are still films running every week. The value of seeing the films are still (3,5,8,13) for each of the 4 weeks respectively. Now, however, you can only afford to go to one film this month. This time, I use the letter Y for watching the film, and N for not watching in a given week.

##### The TC
• Week 4: Watch the film. Y.
• Week 3: If you watch this week’s film, you get 8, but if you wait you get 13 next week. N.
• Week 2: If you watch this week’s film, you get 5, but if you wait you get 13 in week 4. N.
• Week 1: If you watch this week’s film, you get 3, but if you wait you get 13 in week 4. N.

Obviously, the best film is in the final week. TCs realise this, and so are fine with waiting until week 4.

##### The Naif
• Week 4: Watch the film. Y.
• Week 3: If you watch this week’s film, you get 8, but if you wait you get 13 x 0.5 = 6.5 next week. Y.
• Week 2: If you watch this week’s film, you get 5, but if you wait you think you will wait all the way until week 4, and get 6.5. N.
• Week 1: If you watch this week’s film, you get 3, but if you wait you think you will wait all the way until week 4, and get 6.5. N.

As before, the naive individual thinks they will do what a TC does In this case, they think they would wait until the final week to watch the film. This has the effect of making them wait longer than they otherwise would, and they end up watching the film shown in week 3.

##### The Sophisticate
• Week 4: Watch the film. Y.
• Week 3: If you watch this week’s film, you get 8, but if you wait you get 13 x 0.5 = 6.5 next week. Y.
• Week 2: If you watch this week’s film, you get 5, but if you wait you realise you will watch the film in week 3, and get 8 x 0.5 = 4. Y.
• Week 1: If you watch this week’s film, you get 3, but if you wait you realise you will watch the film in week 2, and get 5 x 0.5 = 2.5. Y.

Being sophisticated when there are immediate rewards to be had means that you will preproperate. Since you are aware of your present bias, you figure you might as well watch the film now. You know you are going to fall into that same trap whatever week it is.

#### Implications

What this shows us is that people who are present biased, i.e. those that give disproportionately more value to things right now relative to the future, will put off things when there are immediate costs today. In contrast, present biased people will get drawn in by immediate rewards and grab things too early. This is much more reflective of the decisions people make in real life than time consistency suggests, as much as it might pain us to admit.

The second thing ODR’s paper highlights is that how extreme procrastination or preproperation is depends on how aware you are of your biases. Naive people that think they are time consistent even when they are not suffer most when there is something unpleasant right now. They will keep putting tasks off because they expect to be more sensible in future, despite the fact that this never happens. Sophisticated people who realise their present bias suffer most when there is a nice juicy reward right in front of them. They realise they are flawed, so why delay the inevitable?

Sophisticated and naive behaviour can explain things like having formal, contractual commitments to do things at a certain date (so that they are unable to do things too early or leave them too late). It also may explain why people might decide to take addictive or harmful substances early on in life, despite being aware of future costs.

As with most economic models, however, it is designed in an ‘as-is’ way. In other words, it’s unlikely that someone is completely sophisticated or completely naive. These are hypothetical ‘rules’ for making decisions over time. People often behave as-if they are thinking like one or both of these types of individual.