## Where do Power Laws Come From?

Paper: Gabaix, Xavier. Power Laws in Economics: An Introduction, Journal of Economic Perspectives—Volume 30, Number 1—Winter 2016—Pages 185–206. https://pubs.aeaweb.org/doi/pdf/10.1257/jep.30.1.185.

Summary: This article begins with a brief background on power laws and their origins in economics. As early as 1969, on the hunt for a nontrivial and true law, economist Paul Samuelson seemed to only be able to come up with the law of demand (rather trivial) and comparative advantage (which requires rationality to hold true in practice). This paper, then, offers a modern answer to the very question Samuelson was charged with answering so many decades ago. To summarize, the answer Gabaix explores deeply in this paper is the concept of power laws. He begins with a simple definition (the same one learned in class): a power law is a scaling law that relates some variable Y to another variable X through the equation Y = a * X^b. The constant ‘a’ is generally unremarkable while the constant ‘b’ is of interest; when X is multiplied by a factor of 5, then Y is multiplied by a factor of 5^b.

Empirically, the paper points to the frequency of US cities with populations over 250,000, the frequency of firms with a given number of employees, and frequencies of people with given levels of income and wealth as prime examples that follow power laws. Interestingly, the data for the former two distributions have scaling factors very close to 1; this type of power law has a special name and is called Zipf’s law. The latter two distributions follow power laws with scaling factors ~3 and ~1.5 respectively, showing us that the distribution of wealth is generally more unequal than the distribution of income. These findings relating to wealth and income are considered to have been discovered by economist Pareto, and thus these particular power law scaling factors are often referred to as “Pareto exponents.”

Key Takeaways: It is first worth noting that a smaller Pareto exponent means a higher degree of inequality in a distribution; this rule is precisely how Pareto drew his conclusions over a century ago. Further, the empirical findings coupled with the theory described in the paper make a strong case for the efficacy of power laws. Theoretically, the author of this paper leaves us with two main mechanisms that can explain the origins of power laws. The first and foremost is proportional random growth, or the concept whereby an initial distribution grows and shrinks according to independent events. Assuming the same expected growth rate and standard deviation for the population (Gibrat’s law), we do not expect a steady state solution but rather a distribution that becomes closer to lognormal as the variance in the growth rate rises. One additional assumption, the assumption that individuals in the population cannot fall below a particular frequency level, brings us to a power law with some scaling factor that can quantify the end result of this proportional growth process.

Another way power laws manifest is described by the “economics of superstars” concept. This is the idea that there exist extremely high earners in particular distributions such as arts, sports, and business. As a result, the market for hiring the top talent can be considered such that each individual is assigned a value proportional to their expected growth rate (i.e. growth in value of art, exposure of a team, or profit margin). Then, the top earners will seek to hire these individuals in an efficient way where the top earner receives the top talent, the second highest earner receives the second highest talent, etc. The author presents a quantitive theory to solidify this model, but suffice it to say that we find adjacent individuals in the tail ends of this population will take on talent values varying according to a power law of their overall rank. The exponents for these distributions vary depending on the precise model, but any reasonable distribution described by the model in this paper can be shown to have some power law scaling factor. I found these concepts fascinating as many students in class were raising the question of where these power laws originate, and this paper gave excellent, mathematically sound and insightful arguments as to why we see so many power laws in modern economics and networks.