What I find most fascinating about this is that our conscious minds are not very good at thinking about exponential growth. My favorite example is that of the twin brothers who invest with annual gains of 12% each: one twin saves $2,000 from age 19 to 26 (a total of $16,000) and then stops saving; the other twin doesn't start saving until age 27 but then puts away $2,000 every year until age 65 (a total of $78,000). Do the math, and you will see that the savings of the second twin, growing at 12% every year, never catch up with those of the first one.
Our perception might often be logarithmic, and we might understand the math of exponentiation, but our brains are not very good at 'seeing' exponential growth!
The Rule of 72 http://en.wikipedia.org/wiki/Rule_of_72 is a really handy tool the layperson can use to get a grip on how long it takes for compounding increases (which is most increases we deal with day to day--taxes, population, traffic) to result in doubling. It makes a much greater impact to say "this town's population will double in 10 years" than it does to talk about 7% growth rates.
This exact thing caught me out a few weeks ago, it still boggles my mind. I started a pension recently and I was given examples of how much I would be worth if I started as soon as possible (my 21st birthday) or if I waited until I was 22, 23, 24... if my total monthly contributions are $2,000 and I start on my 21st birthday I will be worth hundreds of thousands of dollars more than if I started at 23, or 25, but my brain thinks that is insane and even though I know it's true, I still can't get over it. The difference between 21 and 23 is $160,000!
going even more on a tangent, it's really a great shame this sort of thing isn't taught in schools. It seems obvious now after being taught that savings can vary so much this way with so little time, but until I was explained it I had no idea. I figured waiting until 25 or 30 to start a pension made most sense!
This is one of my favorite visualizations ever: http://www.nytimes.com/interactive/2011/01/02/business/20110... Be sure to carefully read what it is actually visualizing; I've posted this a couple times and people often knee-jerk a reaction to it based on a complete misreading of the chart.
In addition some stocks pay dividends; reinvesting these can also improve rate of return beyond what would be expected from just looking at the stock price.
Strange, at one point it says "High inlfation led to negative returns". When money is worth less, then you get more money when you sell something. In this case your share of a company. So why would the returns of investing in the S&P be affected by inflation?
Now the 18 years have speed buy and you go grab your very old stack of $10k from under the bed frame to celebrate your kid leaving the house. When you get to the car dealership, you find the equivalent car you could have purchased now costs $14,282. Your money has lost almost a third of it's value just by inflation.
Say you buy one share at $10 of a company with a billion outstanding shares. That company is worth, on the market, $10 billion.
Let's say that next year, the company's share price is still $10. Let's also say that inflation in that period was 4%. The company is still worth $10 billion, but each dollar is worth 4% less. The company's total value has dropped.
Inflation didn't cause the total value to drop. The company's value dropped for other reasons (bad sales, company president went crazy, market jitters, whatever causes these things) and inflation simply moved the numbers a bit in the opposite direction.
In short: inflation doesn't affect your returns, but it affects the numbers you use to measure them.
People encouraging you to save often give unrealistically high rates of return. 12% is unrealistic for any investment vehicle I know of (though people sometimes say it for the stock market -- but they're crazy optimistic).
Here's a spreadsheet to show you the impact of both rate of return and time:
You can find silly faults with individual examples all you want, but studies indicate that people have a tendency to think percentage-based about savings rather than absolute terms.
Which sounds like the smart way to go about it.
It seems a little illogical in the chocolate vs laptop $.50 off voucher, but it too makes sense. The laptop voucher provides almost no value to you, considering you are about to spent $700 dollars on the thing anyway. $.50 could be just the gas to go to the store to buy it.
$.50 for the chocolate on the other hand, will be taken only by cash strapped people (or coupon maniacs).
I doubt people with $700 at the ready to buy a laptop would at the same time clip a $.50 coupon for a chocolate bar.
Which also makes sense. $100,000 to me is serious money. To Bill Gates it's small change. Money is a relative - percentage thing.
A $20,000 car is a purchase you make once in 5-10 years. Saving $100 means nothing in that context --you're already parting with 200 times the amount, so if the $100 amount meant something for you, you'd have gone for a cheaper, say $15000, car in the first place...
Savings of $10 (20%) on small items, like food or garments, on the other hand, pile up.
You spend more than $20,000 on small items per year than you do on cars...
This isn't rational: they're clearly not buying the 100 gallons of gas for the gas-drive to make sense.
By necessity, really. If you've ever spent much time with electronic sensors, you may discover as I did that seemingly half of the work is making nonlinear input linear, because nonlinear input is a mess to compute with.
I thought about it too, and come up with idea that some equivalent of Weber-Fechner law  might be working at it this level. So I have made a small write-up about this idea that you might find interesting .
Seriously though, I thought we already knew that human perception is mostly logarithmic (for example, the eye performs well across 9 magnitudes of brightness – a linear system doesn't make sense). On the other hand, human perception of future value has generally found to be discounted hyperbolically, which is illogical (http://en.wikipedia.org/wiki/Hyperbolic_discounting).
I wonder is there is a correspondence with the harmonic mean but I can't get my head around it at the moment.
Of course, there are other criteria for rationality besides coherence/consistency (as the wikipedia article alludes to).
According to criteria that would label hyperbolic discounting irrational or illogical, debt collectors are equally confused for not accepting the promise of a $100,000 payoff 50 years from now rather than a settling of the debt today.
In the real world, you have to deal with the uncertain risk that the offer will be unfulfilled due to external factors, which always increases in time relative to present. You, the entity offering you the choice, or the entire environment both exist in, may not still exist in 60 months.
If the same offer is made 60 months from now, you should make a different choice, because the risk of non-existence between time of choice and time of reward has changed.
If the amount of hyperbolic discounting is genetically determined, then is likely that we are still adjusting to the massive drop in risk of death over the past several centuries.
Citation needed! I mean, if "half" is not understood as mathematically half, how can traditional societies even have an economy?
Without a source on this, it's a difficult article to take the article seriously. It offers no support for its most prominent claim.
"1000. No, wait..."
(I might actually say that)
I've also felt that teaching a number system that is linear means that it is harder to understand the difference between 100 and 1000000000. Even many adults find it difficult to understand the magnitude of numbers when you approach a billion or a trillion. I think this is largely because we are taught to think in a relatively small range of the number line and perceiving numbers outside that range, we try to relate it back to what we were taught in schools.
I'd love to see schools teach a logarithmical number line instead of the linear system kids are taught today.
The general form of the law is
ψ(I) = kI^a
where I is the magnitude of the physical stimulus, ψ(I) is the psychophysical function relating to the subjective magnitude of the sensation evoked by the stimulus, a is an exponent that depends on the type of stimulation and k is a proportionality constant that depends on the type of stimulation and the units used.
Also, there's the ambiguity of "inclusive or exclusive? And, which side?"
[1,9], (1,9], [1,9) or (1,9)?
If you notice how we perceive distances, it's logarithmic too. It's really hard to judge accurately the time to arrive over long distances. Like the child's example on the OP, people often perceive 1/4 of the distance as being "halfway there". It's some kind of psychological phenomenon.
I do believe this is a property of the universe and not just a psychological phenomenon. Our mind here is simply following the natural distribution and Benford's law is just the observed manifestation of this pattern.
This type of pattern is not odd if you are believer in the Bayesian interpretation of probability theory. The shape of Benford's distribution follows the shape of a maximum entropy, most uninformative distributions for a value of magnitude (as I mentioned, positions do not follow benford's law).
Take street lengths for example. Assuming that they follow a log prior simply means that for a length of street L, if you pick another random street, you are as likely to pick a street within the length range L/2 to L than L to 2*L. From the original street length, to get a street twice as short, you need to subtract much less than you would have to add to get a street twice as long. That is why this distribution is not linear or rather it is linear on the multiplication and division operation, not on additions or subtractions.
If instead you'd assume that a street x meter longer is as likely as a street x meter shorter you would end up with impossible probabilities. For example, for a street of 1 km, a 3 kilometer street would be as likely as a -1km street? Even if you'd assume probabilities were equal for all lengths between 0 and infinity that would mean you think there are as many streets measuring a tredecillion billion km long as there are street 5 km long. This is simply not how things are sized in the universe. Smaller things are in greater numbers. Log priors are one of these areas where the math predicts the universe logically and the universe is mirrored by the math beautifully iff you do your calculations properly (using Bayesianity). It seems evolution has made our psychology reflect this reality.