I’m impatient. My mom is even worse. For us having something now is of much higher value than having it later. Later is an awful word isn’t it?

I have been writing an awful lot about payment methods. I almost make it seem like this is all behavioural economists study. Do not fear, we study much more. One of the most interesting things we study, in my point of view of course, is temporal discounting.

I have already mentioned how __timing__ is important in how we decide to handle our money. I’ve talked about falling into the traps of temptation and other impulsive behaviours. This is what temporal discounting studies. The term may sound a bit heavy duty, but it is dirt simple: discounting means to make a lesser value of. We are making a lesser value of something we expect to happen in the future, to adjust it to its *present* value. This means that receiving €10 a week from now, is worth less than receiving €10 now. But how much less is it worth?

Some people are incredibly impatient, for them getting everything immediately carries immense value. These people have very steep discounting curves. Very patient people on the other hand have very shallow discounting curves. The steeper the curve, the more impatient, and the stronger the discounting factor is.

But how is discounting done? My favourite answer: it depends. There are currently three main models of discounting: exponential, hyperbolic and quasi-hyperbolic. We will discuss them in turn.

*Exponential Discounting*
This form of discounting was proposed by economists. It assumes that all discounting is equal. The discounting factor does not change. For example: I have a discounting factor of 10%. So to me, receiving €100 in a week is equal to receiving only €90 now. Receiving €90 in a week is equal to receiving only €81 now etc. The discount factor, 10%, is not dependent on how close or distant the future is. It remains stable. There is no difference between discounting from week 3 to week 2, and between discounting from week 23 to week 22.

Let’s run the maths. The formula as mentioned before is simple. When still assuming my 10% discounting rate: (1-10%)^delay, what is €100 worth:

With a delay of 23 weeks: Discount Factor = .0886 , Present Value = €8,86 With a delay of 22 weeks: Discount Factor = .0985, Present Value = €9,85 With a delay of 3 weeks: Discount Factor = .7290, Present Value = €72,90 With a delay of 2 weeks: Discount Factor = .8100 , Present Value = €81,00

*Hyperbolic Discounting*
As economists assume stability, psychologists do not. Stability of preferences is about the last thing psychologists believe in. As a result, a different model was proposed: hyperbolic discounting. This model assumes a changing discount factor, as it is important how close or distant the future is. For actual people, there is a tremendous difference between discounting from week 3 to week 2, and between discounting from week 23 to week 22. The latter is so far away it hardly matters, there’ll be little difference between the two values when discounted. The former matters a lot more. It is a third difference in waiting time.

Mathematically hyperbolic discounting assumes that the discount factor is derived from the following formulae: 1/(1 + discounting parameter x delay). So when discounting from week 23 to 22, my discount factor changes from 1/(1 + 23 x 10%) to 1/(1 + 22 x 10%). There is only an incredibly small change occurring. But there is a big difference between having a delay of only 3 weeks or 2. If we run the maths again, what is €100 worth?

With a delay of 23 weeks: Discount Factor = .3030, Present Value = €30,30 With a delay of 22 weeks: Discount Factor = .3125, Present Value = €31,25 With a delay of 3 weeks: Discount Factor = .7692, Present Value = €76,92 With a delay of 2 weeks: Discount Factor = .8333, Present Value = €83,33

*Quasi-hyperbolic Discounting*
The quasi-hyperbolic discount function was proposed by Laibson (1997). It is also known as the beta-delta model as those were the initial parameters used. Both β and δ are constants between 0 and 1. They signify the strength of discounting. This model combines the models we have seen so far. It assumes immense discounting moving from present to one delay away. This is known as the present bias. If the discounting does not happen between an immediate option and a delayed option, but between two delayed options, discounting is seemingly close to exponential again.
Mathematically, delta is first set to the power of the delay and then this number is multiplied by beta. No discounting occurs for immediate rewards when the delay is zero. Given that these numbers are different per person, it is hard to make an accurate estimation of what the formula might look like. An example can be shown of course, again working with our €100 for ease and assuming beta is .6 and delta is .9 (1-10%), (.6 x .9^delay):

With a delay of 23 weeks: Discount Factor = .0532 , Present Value = €5,32 With a delay of 22 weeks: Discount Factor = .0591, Present Value = €5,91 With a delay of 3 weeks: Discount Factor = .4374, Present Value = €43,74 With a delay of 2 weeks: Discount Factor = .4860 , Present Value = €48,60

We can see the difference in these models clearly now. The present bias incorporated in the quasi-hyperbolic model discounts everything severely in the beginning. Even with a mere two weeks delay, €100 has fallen to less than half its value and drops another 10% when delaying it yet another week. In hyperbolic discounting going from a two to three week delay is about as dramatic, but our starting values are much higher. Compared to exponential discounting, hyperbolic discounting starts slower but rather quickly catches up, having lower present values for longer delays. Compared to quasi-hyperbolic discounting, exponential discounting is slow, but ultimately ends up with higher values.

Overall, temporal discounting is a powerful thing. But it is not solely dependent on time alone. Our discounting curves become steeper as we perceive the risk of obtaining our rewards to be high. If someone promises me a choice between €100 now and €150 in a week, I’d need to have quite a high discounting factor (in all models) to want the current option more than the later option. However, what if I don’t trust the person? A lot can happen in a week. They could be trying to trick me in some way. And even if they were honest, they could die in a weeks’ time, and I’d have no money. Hell, I could die in a weeks’ time! There is a certain risk involved with waiting, and on average,__ ____people don’t like risk__** **that much.

In the next article I’m going to explain how the present bias, but also a thing we call the projection bias are influencing the way we see our current and our future preferences. And how this affects our spending and more importantly our savings. Especially when it comes to retirement.

*References*
Laibson, D. (1997). Golden eggs and hyperbolic discounting. *The Quarterly Journal of Economics*, *112*(2), 443-478.