Sunday, 9 December 2012

The world of Pareto

After a week of absence Schoolonomic returns with a new post on a set of seemingly unrelated, yet similarily named phenomena. The Pareto efficiency wasn't the only contribution of Vilfred Pareto to the field of economics, but also an index, a chart, a law, a distribution and a principle. Today's post will explain each, to get a better grasp of the Italian economist's heritage.


The Pareto principle


While so many seemingly similarly named phenomena may sound necessarily complicated to some, the reasons for the naming scheme are that much of Pareto's work can be organised around the topic of inequality. The Pareto principle was the starting point of his observations on the issue; in 1906, the economist noticed that approximately 80% of land in contemporary Italy was owned by approximately 20%
of the country's population. Such an observation, though certainly intriguing, wouldn't have caused such a stir on it's own - but it was soon found that the 80-20 distribution ratio was also fairly accurate for a wide range of other phenomena - the claimed fields of application include advertisement, road traffic distribution, website traffic, profit distribution and many more. Hence, the principle became widely popular and is generally used as a rule of thumb - a non-proven, non-precise, but generally true claim - in business circles.


Pareto distribution and index


The 80-20 pair of number actually represents only one point of a simplified probability distribution function, the Pareto distribution - what X percentage of wealth is controlled by what Y percentage of people. (Generally, probability distributions are used to collectively depict the results of random experiments - for example, how often can you throw certain number sequences with a dice.)
If the minimum income is xm, and the income level we'd like to measure against the percentage of holders is
x, then the so-called Pareto distribution is given by
,
where the greek letter α (alpha) denotes the Pareto index. Since xm/x is smaller than 1, alpha has to be positive; but in order for the total income of a population to be finite, it has to be greater than 1. For the Pareto principle to be in effect, alpha has to equal approximately 1.16.
This is all really nice, you might say - but where does it connect to 80 and 20?
Imagine that the minimum income in country A (Xm) is  100$. Now if we'd like to measure the 80% of incomes in A, substitute 400$ for X; 400$+100$=500$, and 400/500 indeed equals 0.8.
Now what do we get when we substitute Xm/X to the Pareto index? 100$/400$ on a power of 1.16 - and it miraculously equals 0.2, or 20 percent.


Pareto chart


Finally, the Pareto chart - a nice simultaneous graphical representation of absolute and relative statistical values. For more clarity, consider the graph below, using hypothetical data.
 
 The bars represent the causes of "late arrivals" to a workplace in descending order with absolute values - the number of them caused by traffic, child care, and so on. The red curve represents the cumulative, relative values of causes - that is, the first dot presents the percentage of traffic-related late arrivals, the second represents the percentage of traffic AND child care-related late arrivals, and so on.

These various concepts beatifully represent the broadth even a single economist's research can successfully study during a lifetime. Post your comments, questions, etc. below.



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