Page 242 - GHES-3-3

Global Health Economics and
Sustainability
Empirical resource allocation in healthcare

2. Theoretical framework and methods chips doubles each year) exemplifies this phenomenon.
These effects facilitate business growth but also contribute
In accordance with the law of the distribution of competitors, to the emergence of winner-takes-all markets.
there is a stable inverse statistical relationship between the
competitive “rank” of an economic entity in the market Equation I has several equivalent formulations,
hierarchy and the value of the resource it captures. It has including the Pareto-Zipf distribution (in rank form),
been demonstrated that elements of a large system that the Mandelbrot distribution (Manin, 2009), and Alfred
compete for a limited resource are ranked according to the Lotka’s Law (a hyperbolic distribution describing the
power law distribution (Equation I) (Arnold, 2015, p2). number of publications per scientist). “In most real-world
applications, the value of α falls in the range 2< α ≤3”
N = A m –a (I) (Newman, 2017).
Where N is the rank of the element, m represents In economic and mathematical models, where the
the share of the resource captured by the element, A is a dependencies of wide ranges of argument values (in our
constant, and α is the exponent of the power function” case, economic values) are estimated, a double logarithmic
(Arnold, 2015, p. 2).
scale is used. As a result, Equation I was transformed into
The first empirical approximation of economic the logarithmic form (Equation II).
indicators (such as income and wealth distribution) was ln N = A – a ln m (II)
proposed by Pareto. “Typically, the Pareto exponent is
around 1.5 for wealth and between 1.5 and 3 for income where N is the rank of the element, m represents the
(recall that a lower Pareto exponent means a higher degree share of the resource captured by the element, A is a
of inequality in a distribution)” (Gabaix, 2016). constant, and α is the exponent of the power function.
Modern empiricism has shown that Gaussian The graph of Equation II in a double logarithmic scale
distributions dominate when events are completely produces a straight line with a slope coefficient (−α)
independent of each other. Once the assumption of the showing the elasticity of the distribution function. Using
interdependence of events is introduced, a power law data for Metropolitan Statistical Areas provided in the
distribution – also known as the “long tail” – emerges. The Statistical Abstract of the United States (2012), the log
power law distribution is also called a Paretian distribution rank-log size relationship for U.S. cities with populations
since positive feedback tends to amplify small initial of 250,000 was obtained using Equation III.
changes. Chris Anderson’s “Long Tail” concept offers a ln (N) = 7.88 − 1.03 ln(m) (III)
modern example of a Pareto probability distribution,
where a few extreme events or “blockbusters” occupy The parameter N and parameter m were determined
the left side of the curve while a very long tail of much from Equation II.
less popular events extends to the right side of the curve This relationship is nearly linear, with a slope very close
(Anderson, 2008). to 1 (the standard deviation of the estimated slope is 0.01)

Gaussian and Pareto distributions are radically (Newman, 2017).
different. The Gaussian distribution can be completely Reinterpreting Pareto’s law in terms of resource
characterized by its mean and variance, whereas the Pareto distribution, we observe that for a fixed change in resource
distribution lacks a finite mean or variance. Thus, a power share (dm), the number of economic entities possessing a
law distribution has no mean that can be assumed to reflect share equal to or greater than m decreases proportionally.
the typical features of such a distribution and no finite That is, obtaining a higher rank is easier for entities already
standard deviations on which confidence intervals can ranked highly than for those lower in the hierarchy. The
be based. The term “Pareto principle” was introduced to effort required to move to the highest-ranked group is
the business world in 1940 by Joseph M. Juran, a quality inversely proportional to the existing rank.
engineer who applied the principle to business and started
the idea of the Six Sigma process (Godfrey et al., 2007). However, with the general recognition of this law of
The key component of the Pareto distribution and scaling uneven allocation of resources among competitors of
differing ranks, several aspects remain poorly understood:
effects is time: power laws strengthen over time and their
effects scale. Some distributions grow faster than others, but its dynamic characteristics, the heterogeneity of economic
they all exhibit growth. Digital technology, particularly the resources, and the specific influence of different resources
Internet, accelerates the effects of power law. In technology, on their allocation among competitors.
network effects and feedback loops amplify these patterns. A fair amount of research examining power law and
Moore’s Law (where the memory capacity of computer Pareto distributions across various domains found results

Volume 3 Issue 3 (2025) 234 https://doi.org/10.36922/ghes.8283