A Number Needs a Unit

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Aug/04/2007
A Number Needs a Unit

5 + 3 = 8, right? But does it always, and under every circumstance, equal eight? I will here try to come up with some over-precise thinking in order to understand the traps we encounter when dealing with prices and other sizes.

In the equation above, 5 means really '5 ones', where 'one' is some abstract definition. Is the number 'one' something of universal nature or is it just man made? From a very young age we are taught to distinguish between 'one' and 'more than one.' From there on mathematics can be derived.

In first grade my teacher liked to use 'apples' as the unit. 5 'apples' + 3 'apples' = 8 'apples,' right? After a while units were not necessary any more, so that all of a sudden 5 + 3 = 8 became an unquestioned truth.

Let's go a bit further. How much is the weight of 8 apples? If every apple weighs 100 gram the answer is 8 'apples' * 100 'grams / apple' = 800 'grams.' But in real life you know that every apple is a bit different, so you cannot truly say that 8 'apples' = 8 'apples', can you?

This leads to the problem:
Every number needs a unit, but when numbers are compared they must have the exact same unit.

Stock Market Indices
An index is measured in points. The points shall reflect the move in the underlying stocks, in which are measured in dollars, or any other currency. But then again; what is a dollar? I don't know, except that it tends to depreciate in something we call value over time, so that a dollar today is not equal to a dollar tomorrow. A more down to earth example is the move of the S&P since 2000. In dollars the index is at about the same level today, but what if you measure it in Euros?

Scientific Units
If you are dealing with the SI units (meter, kilogram, second, ampere, Kelvin, mole, and candela) you can be quite confident that a meter is always a meter and a second always a second (although relativity theory suggesting this not being the case, I will leave this for the more educated.) 5 'meters' + 3 'meters' = 8 'meters,' just as 5 'ones' + 3 'ones' = 8 'ones'. When dealing with sizes you just use mathematics, as your size is measured in a defined unit. Back to the apples and dollars; it's the same, except for the SI units being much, much more precise.

Everyday Economics
So to the real problem. You're buying a product; is the price good or not? Say a t-shirt costs $10. You have an idea of what $1 is worth to you, and you will buy the t-shirt if the value from the piece of clothing is worth giving up ten units of 'a dollar' for. But then some other mental unit is probably more usable. The t-shirt costs 1 'ten dollars,' or maybe 0.5 'twenty dollar' is better if your hourly wage is $20. The mental price is all of a sudden 0.5 'hours of work.' Comparing the price to similar products is maybe even better. If a cheap t-shirt costs $5; is this shirt worth 2 'cheap shirts'?

There are numberless similar units, and one the most important aspects in economic day-to-day thinking is to find the best unit.

Trading
When trading stocks you are faced with the same problem. The most common problem, I think, is to use the purchase price as the unit. If the present price is >1 'purchase price,' selling is a lot easier than if it's not. This is called anchoring, and other anchors are round numbers, and previous highs and lows. My point here is that although you see prices in 'dollars / share,' you should try to become aware of what other units you unconsciously are using.

Caravaggio:
Interesting philosophical post JP. Re the last point on trading prices and units, it may be interesting to compare the volatility of penny stocks and low nominal priced stocks to high priced stocks (eg: Buffet's stock). My bet is that there is a negative correlation, with lower price stocks generally being more volatile. I wonder what the rational is for this. Also, I remember reading about stock splits (purely an accounting change) and how they seem to have a beneficial effect on stocks...if it still holds true it could provide an example of irrational investors/mkt, perhaps?

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