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Archives of Business Research – Vol. 10, No. 8
Publication Date: August 25, 2022
DOI:10.14738/abr.108.12754.
Vogel, H. L. (2022). Playing With Power-Law Curves: A New Way to Analyze Market Structures and Sectors. Archives of Business
Research, 10(8). 158-174.
Services for Science and Education – United Kingdom
Playing With Power-Law Curves: A New Way to Analyze Market
Structures and Sectors
Harold L. Vogel
Vogel Capital Management
Powerlaw Analytics, LLC
ABSTRACT
This paper outlines a universal empirical method that illustrates how such curves
can be used to reveal changes in economic structures and relationships. The
method takes power law analysis from a static snapshot to something more akin to
a motion picture. The approach leads to an alternative to the classic Herfindahl- Hirschman sector concentration approach. It also provides a relatively stable
characteristic numerical description for each sector and market segment and
eliminates the biases caused by price inflation. Collection of such segment data can
lead to a characteristic number, a signature, for a region’s economy or a company’s
overall competitive position. The method and the valuable insights it can provide
is applicable in an unlimited number of fields. Examples include everything from
mergers and acquisition and portfolio management strategies to areas involving
studies in antitrust, banking, insurance, sports, medical science, education, crime- control, travel, weather, and entertainment.
Keywords: Power-Laws, Concentration, Antitrust, Movies, Travel, Banking, EconLit
Codes: A10, G18. G34, K24, L4, M38, Z3
INTRODUCTION
Power laws have been specified and explored for more than 100 years and are evident in
virtually all parts of the economy and human endeavors. It all began with the discovery by
Italian sociologist and economist Vilfredo de Pareto (1848-1925) whose studies found that in
general 80% a nation’s wealth was held by 20% of the population.
This 80/20 split roughly applies to a diverse set of circumstances, e.g., 20% of posts to
websites approximately generate 80% of the traffic. Similarly, 20% of a company’s products
might generate 80% of sales and/or profits even though in some industry sectors such as
music, 97% of profits might be generated by 3% of artists and the percentages need not
necessarily always add to 100%.1
A popular application of this was the discovery (by George K. Zipf, (1902-1950) that word
frequencies in languages are inversely proportional to rank in frequency tables. For example,
in English the word “the” occurs most frequently and by itself accounts for nearly 7% and “of”
for around 3.5% of all word occurrences.2
1 An example according to https://www.kevin-indig.com/power-laws-and-the-pareto-principle-powerful-ideas,
77% of Wikipedia articles are written by 1% of its editors. See also Elberse (2013), and Krueger (2019).
2 See Piantadosi, 2014, Moreno-Sanchez et al. (2016) and Clauset et al. (2009).
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Vogel, H. L. (2022). Playing With Power-Law Curves: A New Way to Analyze Market Structures and Sectors. Archives of Business Research, 10(8).
158-174.
URL: http://dx.doi.org/10.14738/abr.108.12754
Eliazar (2020 pp. 1-6), however, further explains that there is an assortment of different types
of power laws, with the two principal statistical forms being classified as distribution
functions and rank distributions.3
“A tail distribution function describes the proportion of the values that are greater than a given
threshold level...the threshold level is the input, and the corresponding proportion is the
output... The rank distribution orders the values decreasingly: the largest value is assigned
rank one, the second largest value is assigned rank two...etc. In this statistical form the rank is
the input and the corresponding value is the output.”
Pareto’s Law is an example of the relationship “between the input and output of a tail
distribution”, whereas Zipf’s Law represents “a power relation between the input and output
of a rank distribution.4 This is the feature explored in this paper.
An idealized version of a power law with reference to movie box office rankings appears in
Fig. 1 and would include thousands of film titles. This idealized version indeed substantially
resembles the empirically derived presentation in which Axtell (2001 ranked U.S. cities by
size of populations and found an almost perfectly straight downward-sloping (-1.03 versus a
perfect -1.0) power law distribution (also shown in Gabaix 2016).
Fig. 1. Idealized power law applied to movie box office data.5
Gabaix (2008, 2016) provides a most comprehensive and important work on power laws and
shows how the underlying mathematics and empirical data are related. He illustrates with the
cumulative distribution of daily stock market returns for different capitalization sizes of
stocks. Differing growth patterns that emanate from initial circumstantial conditions
involving availability of capital, accrued expertise and knowledge, forecasting ability,
availability of skilled labor, and other economic factors, lead naturally to the development of
3 Other important power law forms include a Lorenz curves and Weibull’s Law, which characterizes a hazard
rate used to describe the lifespan of a system. Eliazar notes that Zipf was not the first to discover the law that
was named for him. There is, according to Eliazar, “a foundational statistical power-structure that underlies
all the aforementioned power statistics.”
4 In economics, the word “law” is much more loosely and informally applied than in the physical or biological
sciences. See Vogel (2017).
5 Many films have box-office grosses of under $50 million and only a few, more than $400 million. Avengers: Endgame, with the highest current dollar worldwide gross exceeding $2.8 billion would be at the far lower
right. In actuality, the representation is concave to the origin: it is depressed (i.e., sags) in the middle. Big winners have a high rank but a low frequency.
log (freq)
log (rank)
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Archives of Business Research (ABR) Vol. 10, Issue 8, August-2022
Services for Science and Education – United Kingdom
power law characteristics. Analysis of these characteristics provide interesting and significant
economic and socionomic insights.6
Power law distributions have been studied and applied in many different fields, for examples,
Covid-19 ( Jang, 2021 and and Neipel, )2020, Interned topology (Faloutsos, et al., 1999).
earthquakes (Meng et al,, 2019), trade (Eaton, Korum, and Kramarz, 2011), metabolic rates
(West, Brown, and Enquist, 2000), and wealth (Levy and Solomon, 1997). And Clauset et al
(2009) and Eliazar (2020) provide great depth in covering the statistical and empirical nature
of such distributions. Eliazar indeed notes that power laws are prevalent in the physical
sciences (e.g., Newton’s law of gravitation, Coulomb’s law of electrostatics, Kepler’s law of
planetary motions). It appears, however, that none of these have used the approach that is
here presented.
This paper aims to extend the existing framework by adding a dynamic aspect that provides a
new and universally applicable way to understand sequential changes in a wide variety of
sectors and product and service market share studies. As such it can also serve as an adjunct
to the classic Herfindahl-Hirschman Index that is frequently used to analyze market
concentration aspects and trends.
The study begins by describing how the classic power law relationship can be extended to
provide a more dynamic picture of what occurs to a sector’s share of market over time. To
make an analogy, the classic power law presents a snapshot; the method shown here is a bit
more like a movie.
The second section provides the basics of power law applications The third describes how the
“snapshot” can be turned into more dynamic representation. It conveys the underlying
thinking and methodology with a relatively long data series (41 years) that also includes the
pandemic year of 2020. The fourth section extends this methodology to a wider variety of
sectors and services and discusses comparative results. The fifth and last section summarizes.
POWER LAW BASICS
Power laws, sometimes called scaling laws, are generally described by the expression, Y = αX-B
where the variables are X and Y, B is the power law exponent, and α is constant. The sign
preceding the B is negative, which formulates the declining slope moving from left to right
along the lower axis. Data on the axis can be shown in terms of logs of rank and frequency
(Figure 1) or in ordinary numerical notation. Also, the rank and frequency axes can be
swapped and the characteristic downward slope will still appear.
Classic power law studies show ranks and frequencies as a snapshot, even though the periods
covered by the data can be extensive. Rankings, however, tend to change and such changes
can be at least partially captured by relatively simple comparisons related to different periods
of time. For instance, Fig. 2 shows changes in professional sports team values for the years
2017 and 2020. The entire valuation structure rose and over the intervening years shifted the
curve upward. This does not, however, preclude the possibility that there might be periods
when technological, economic, or demographic factors send the more recent curve below the
6 On socionomics and finance see Prechter (2016).