Section 6 – Ec1818 Jeremy Barofsky March 10 th and 11 th, 2010

Section 6 Outline Power Laws – General defintion / scale invariance – Zipf’s Law – Pareto’s Law Net Logo Tutorial OFFICE HOURS – THURSDAY March 11, 10-11, CGIS N outside room 320.

Power Law Definition In general: y = S -a where y = frequency/ rank of event, S = event size, and a constant exponent >= 1. Distribution: Also can be represented as a complimentary CDF where if X is a random variable, then 1 – F(X) = Pr( X > x) = bx -α where α is the constant this time, x = event size and y in this case is the Pr(X > x). The tails of this distribution (the probability of rare events) increases as α / a falls. Since as the slope of the distribution becomes steeper rare events become less likely (the tails get thinner). When 0 < α < 2, var(X) = infinity, when α ≤ 1, then E(X) = infinity!!! (moments of the distribution don’t exist).

Power Laws in the Real World Previous to Axtell, 2001 firm sizes had been described as log- normally distributed. Axtell shows that the Pr( Firm S > s) = bs -a. And he finds a = 1.25 for the U.S. and close to 1 for other nations. Biological power laws – Kleiber’s law: metabolic needs of mammal increase = mass ¾. Geoffrey West (from Santa Fe Institute) explained from the transport system for energy to the body through an efficient branching circulatory network. Strogatz (NY Times blog 2009) – Find 3/4ths power laws in city size infrastructure too: “For instance, if one city is 10 times as populous as another one, does it need 10 times as many gas stations? No. Bigger cities have more gas stations than smaller ones (of course), but not nearly in direct proportion to their size. The number of gas stations grows only in proportion to the 0.77 power of population. The crucial thing is that 0.77 is less than 1. This implies that the bigger a city is, the fewer gas stations it has per person. Put simply, bigger cities enjoy economies of scale. In this sense, bigger is greener.”

Scale Invariance of Power Laws Random variables (R.V.) that are power law distributed exhibit scale invariance, meaning that we can multiply city size (our R.V.) by any units and still retain the distribution’s shape. Therefore, S is unit-free. Only rank matters. Take a rank- preserving transformation of distribution (take logs) and we have a simple test for Zipf’s law. Ln y = ln b – α ln(S). Shown to hold in U.S. 1890, 1940, 1990, and most other nations within modern times. India 1911, China 1850s (references from Gabaix). Deep intuition into processes of city growth / income distribution or just a statistical reality like the CLT that provides no insight?

Zipf’s Law Discovered by George Kingsley Zipf, 1949, in analysis of word rank and word usage (Human Behavior and the Principle of Least Effort). Specific type of power law where Pr( X > x) = bx - α and α = 1. So, the probability of seeing extreme events (large cities) falls at the rate of α = 1. For city sizes, S= city size: Pr( S > s) = Rank of city size = bs - α Meaning that the prob. of city being > size s = 1/s. The larger the city size the less likely we are to see a city of that size. Not a vacuous result: could have had normally or log-normally distributed city sizes. Implications? (Use board and STATA).

Pareto’s Law Another power law form, discovered to describe the tail-end of the income distribution around 1900 by Italian economist Vilifredo Pareto. Pr( S > s) = s - α If income is distributed Pareto, then this implies the rule, that 80% of the wealth is held by 20% of population – Pareto principle. Verified that income distributed this way around the world. Continuous version of Zipf where b = 1.