From the Conversable Economist:
My long-ago memory is that basic classroom experiments in physics and
chemistry would produce a graph of data that was pretty close to a
straight line, or a smooth curve. In social science, it's more common to
see a rougher pattern, with the points scattered around. As a result,
the social science research on "power laws" is full of graphs that raise
my eyebrows, because there are lots of graphs where the points fall
very close to a straight line. The fit looks too good! Xavier Gabaix provides a readable overview in "Power Laws in Economics: An Introduction," appearing in the Winter 2016 issue of the Journal of Economic Perspectives, which pushed my (Full disclosure: I've worked as Managing Editor of JEP for 30 years now. All JEP article back to the first issue are freely available on-line compliments of the publisher, the American Economic Association.)
Here are some of Gabaix's examples from the paper, but he also refers to
power law results in a wide array of other papers. (For some readers,
it may be useful to add a few words on what a "power law" is. On a
typical linear graph, each equal distance on the graph represents a
change of the same absolute amount--say, 1, 2, 3, 4, ...--although the
units may be expressed in millions or years or percentage points or
dollars or whatever is useful. In a "power law," each equal distance on
the graph represents a rise in an exponential power--say, 101 , 102, 103, 104
... As a result, what appears to be an equal visual distance on the
graph now represents not an absolute change, but a proportional change:
for example, each equal visual distance in the powers-of-10 example
represents a 10-fold increase.)
Consider a graph based on the population of US cities, which in this
data is all cities with population above more 250,000. The horizontal
axis is population, expressed as powers-of-10. The vertical axis is the
rank of the city size--that is cities are ranked by population from #1
New York to #2 Los Angeles and so on. Again, the vertical axis is
expressed as powers-of-10. The result is very close to a straight line
with a slope of -1.
As Gabaix writes: "A slope of approximately
1 has been found repeatedly using data spanning many cities and countries
(at least after the Middle Ages, when progress in agriculture and transport could
make large densities viable, see Dittmar 2011). There is no obvious reason to expect
a power law relationship here, and even less for the slope to be 1."
Now here's an example looking at the distribution of the size of US
firms, measured by the number of employees on the horizontal axis, and
the number of firms of this size, measured on the vertical axis. Again,
both axes are measured in powers-of-10. Again, the slope is very close
to -1. But why should the ranks of cities as measured by population look
similar to the frequency of firms as measured by number of employees?
(As I said, these are the sorts of graphs that make your eyebrows go
up,)
Or here's an example about the distribution of daily stock market market
returns. You can read the details of the calculations in the Gabaix
article, but again, the axes are expressed as powers-of-10, and a linear
relationship seems to emerge....MORE
