# Hyperbolic and exponential discounting

By Murray Bourne, 08 Apr 2015

I came across this article recently: Six Advantages of Hyperbolic Discounting…And What The Heck Is It Anyway?

I'd never heard of hyperbolic discounting and it looked mathematically interesting, so I followed the link. The article is a case of someone using a mathematical term but not being quite sure what it means.

The author says early in the article:

Since the phrase hyperbolic discounting is despicable jargon, let me explain it in terms that even I can understand.

Well, I don't know that it's "despicable" jargon (this is the kind of statement that discourages people from being comfortable with mathematics), and it worried me that a mathematical term was being maligned, so I read on.

## Immediate rewards

Question: If I offered you $100 today, or$105 in one month from now, which would you choose?

Most people choose the immediate reward, because waiting for a month for a bit more doesn't seem to make sense.

In plain English, that's all hyperbolic discounting really means. Most people would choose to claim a reward right now, rather than a larger reward some time in the future. It's what marketers use all the time to encourage you to buy now.

### Back to the article

Leaving the marketing behind for now, let's look at the mathematics used in the article.

The author correctly identifies an exponential curve and uses the following graph as an example (without saying what its relevance is, or indicating what the axes represent): [Fig 1: Image source]

Then he immediately states the following, suggesting some relationship between the exponential curve above, and hyperbolic curves. He simply says:

This is a hyperbolic curve: [Fig 2: Image source]

That's when it started to get interesting for me. The subject matter of the graph suggested a physics application, rather than some behavioural finance model. I began to suspect the author didn't really know what was going on in this topic (or certainly didn't understand the example graph).

### Exponential decay (increasing form) graphs

At first glance, this graph looked like an exponential decay (increasing form) to me. An exponential decay (increasing form) curve describes situations where a quantity grows quickly at first, then levels out to some limiting value. It is the mirror image of an exponentially decaying quantity. An example of such a curve arises in terminal velocity problems. A skydiver's velocity starts at zero, builds rapidly to around half of their final velocity, and as air resistance builds up, the velocity more gradually builds to some terminal value (usually around 200 km/h.) [Figure 3: Velocity of a skydiver V = 225(1 − et)]

Another case is the charge in a circuit containing a capacitor (see Example 1 on that page). Here's the graph of the charge from that example, given by q = 2(1 − e−10t). [Fig 4: Charge in a capacitor]

The curve rises fairly quickly then flattens out to some limiting value (which in this case is 2 coulombs).

It's called an exponential "decay" curve because the power of e is negative. We've just flipped the exponential growth curve (that is, reflected it in a horizontal axis) from its usual shape (steeply descending at first, then levelling off).

The Vmax term in the hyperbolic curve above (Figure 2) was what made me think of these exponential curves.

You can see the hyperbolic curve is similar to our "flipped" exponential decay ones, but not identical. The hyperbolic curve appears to have infinite slope at S = 0, whereas the exponential ones do not.

### Hyperbolic curves

I didn't accept that Figure 2 was a hyperbolic curve at first because such curves usually have 2 arms, something like this: [Fig 5: Rectangular hyperbola, xy = 1]

I felt the need to investigate further. Was Figure 2 hyperbolic or not?

### Enzyme Studies

The curve given in Figure 2 is not about behavioral economics at all (nor is it about physics, as I suspected). It actually represents the Michaelis-Menten equation, which arises in the study of enzymes. Here's what the original page (where the Figure 2 graph came from) said:

The Michaelis-Menten equation is a quantitative description of the relationship among the rate of an enzyme-catalyzed reaction [v1], the concentration of substrate [S] and two constants, Vmax (maximum reaction rate) and Km (a constant).

Here is the formula for V: The "velocity" refers to the speed of the reaction. For your convenience, here is the curve for Vmax (Figure 2) we saw earlier: They go on to say:

The Michaelis-Menten equation has the same form as the equation for a rectangular hyperbola; graphical analysis of reaction rate (V) versus substrate concentration [S] produces a hyperbolic rate plot.

Let's try it out with some simple constant values. I chose Vmax = 6, and Km = 1.7.

The formula becomes: Here is the graph of the hyperbola, showing the two arms I talked about before. But if we restrict our values of S such that S > 0, we obtain a curve that looks a lot like Figure 2. The slope at S = 0 is not infinite, but it is quite steep. So the graph referred to in the marketing article is indeed hyperbolic. The actual choice for their example was unfortunate, because the variables seemed to have nothing to do with economics (they don't), and I was no wiser about what was going on.

### Comparison with exponential decay curves

Let's now see how close the hyperbolic curve is to a similar exponential one. I graphed V = 6(1 − e−0.2S), (chosen so it has the same maximum value) and here's the result: Note the upper limit is the same as my hyperbolic curve, (Vmax = 6). The shape of the two curves is similar, but not exactly the same. Here are the 2 graphs on the same set of axes: ### Explanation

At the beginning of this post, I asked if you wanted $100 now or$105 in a month. In most countries currently, the interest and inflation rates mean it would be much better for you to wait for the month, as you would make 5% on your money. At that monthly rate, you could make around 80% per year!

Most people intuitively "discount" future money. That is, they know that if they wait, they should get some reward for waiting. Now mathematically, that reward grows exponentially (not as steep at the beginning), because interest on money grows exponentially. But in our minds, we want it now. That is, it's more like the hyperbolic case, where we are impatient and want a higher reward for waiting even a short while. Our impatience can be represented by the hyperbolic curve.

Studies on peoples' (and animals') investing and savings habits have shown this to be the case. In one experiment, a group of subjects was offered $15 now, or they could wait and get more money later. The average responses were: One month later$20, one year later $50, and ten years later$100. These perceived "wait for reward" data follow the hyperbolic curve more closely than the exponential one. [Study by Thaler, R. (2005), Advances in Behavioral Economics, Russel Sage Foundation]

### Applications

Hyperbolic discounting has many implications in the areas of savings rates (small pain now for future gain), climate change (impact of energy policy now on future environmental conditions), attitudes to health screening (some discomfort now for future health), lifestyle choices (amount of exercise now for reducing obesity) and behavior due to weather predictions (how many crops to plant for future benefit).

### Conclusion

Next time some salesperson is pressuring you to buy something right now, ask him about the difference between exponential and hyperbolic discounting. It will give you some breathing time to work out whether it really is better to buy now, or wait.

### 8 Comments on “Hyperbolic and exponential discounting”

1. Obliczone.pl says:

The concept of hyperbolic discounting is really interesting, especially explanation about difference between the hyperbolic and exponential case ("where we are impatient and want a higher reward for waiting even a short while"). I'm curios if there exist some connection with behavioral finance (e.g. indifference curve)?

P.S. How do you produce such goodlooking plots?

2. Murray says:

@Obliczone: Thanks for your positive feedback. Yes, this topic is all about behavioral finance, as mentioned in the article.

The graphs were mostly made using GeoGebra or my graph plotter.

3. Obliczone.pl says:

4. Alan Cooper says:

Actually I tend to agree with the judgement that "hyperbolic discounting is despicable jargon"- though not to the credit of the 'kissmetrics' author.

One problem is that the term is often used in contexts where the specifically "hyperbolic" model is not what is relevant; in fact most of the "applications" in the 'kissmetrics' article fall into that category. It is also, in my opinion, inappropriate to name a model on the basis of a (relatively) esoteric aspect compared to the simplicity of its underlying assumption. Let me elaborate.

The basic idea of the 'kissmetrics' piece is that people often favour an immediate reward over a later reward which is worth more than the accepted reward would have been with accumulated interest at the prevailing rate, or equivalently that they prefer to postpone a payment now in favour of making a later payment which includes more than the interest they might earn in the meantime. But the mathematical model needed to describe this behaviour is not necessarily "hyperbolic"; it could just be a "spread" between the interest rate that the individual applies to transactions and what could be earned in the bank, or any of dozens of other equally likely models.

Indeed the distinction between exponential and "hyperbolic" discounting only really shows up in the nature of its "tail" behaviour for long periods and the fact that in the "hyperbolic" model the effective discount rate f(t)/f(t+1) is not constant (and in fact, though it may start high, it approaches 1 as t goes to infinity). Of course many other models also have this property in a way that is just as good at matching the data and to call all of them "hyperbolic" is incorrect.

But even when the hyperbolic model is being used, I do object a bit to the label because it plays on an aspect of the model which is much less "natural" than the fact that it is based on *linear* growth. Surely it would be more generally understandable and informative to call it the linear growth discounting model.

Some terminological mind games:

Although the function f(t)=1/(1+rt) has a hyperbolic graph it is not a hyperbolic function (because there are other functions which got that name first). But it is a *rational* function although many people think that using it for discounting is not rational!

Linear growth leads to hyperbolic discounting, but what leads to linear discounting? and is anything wrong with it? (Hint: Vernor Vinge and Ray Kurzweil may have the answer.)

5. Murray says:

@Alan: Thanks for your inputs, as always.

Actually, I doubt the premise that people consider actual interest rates or inflation or whatever when they are asked questions about how much reward they need for the pain of waiting. It's just a gut instinct, I suspect.

6. Alan Cooper says:

Yes, I agree. There's usually no conscious consideration of rates etc. And my gut often tells me that it's worth a small penalty to have one less thing that I must remember to deal with next month. I find it interesting that the effect reverses for longer deferrals though (which is the essence of the fat tail on the hyperbola compared to the exponential). Perhaps our gut is also telling us that exponential growth can't really go on forever! Thanks for bringing up an interesting topic.

7. Alex Yakovlev says:

I find this article very interesting and stimulating. I have been exploring similar differences in circuit theory, where you can have exponential if you discharge a cap with constant resistance. However if you discharge a cap with an exponentially increasing resistance, i.e. where the resistance is inversely proportional to the current voltage, the discharge is hyperbolic. The latter effect can be achieved if you connect a charged cap as a power supply to a ring oscillator (built of digital NOT gates). Inverters switch progressively slower as the power decreases for them. It is like humans who are less and less motivated to move fast when their energy supply goes down (I once tested my wife with a question about her agility to spend money at different times when a bank account is initially set with say £1000. She clearly said that her rate of spending will start to get slower as the amount of money on the account becomes smaller). Apparently, similar hyperbolic effects are in many areas in biology - for example, the levels of energy determine the luminescence decay rates.

I have a paper written about the math modelling of electronic circuits with hyperbolic decay of charge:
Below is a URL to our paper in the Wiley's Int. Circuit Theory and Applications journal (it is in open electronic access).
If you see that there is something interesting for you in this paper, please let us know.

http://onlinelibrary.wiley.com/doi/10.1002/cta.2010/pdf

8. Murray says:

@Alex: Thanks for sharing. The hyperbolic decay behaviour in your paper is very interesting. I'll have a closer look at it sometime before my next post on such things.

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