đź‘ŤDetermination of Pareto Exponents in Economic Models Driven by Markov Multiplicative Processes

Published in Econometrica, 2022

To obtain the double power law result in my 2014 JET paper, I assumed some conditional independence (which rules out persistent heterogeneity) and took the continuous-time limit, which was not entirely satisfactory. In those days, all power law results in economics relied on the IID assumption, which is clearly unrealistic. Around Footnote 13, Gabaix (1999) states

Let each city grow at an arbitrary mean rate, say 2 percent (it does not matter if this mean rate is time varying) […] This is a conjecture that we firmly believe to be true. […] However, we could not find any argument in the mathematical literature—here we deal with Markov chains with time-varying transition matrices—to help us establish this rigorously.

Ever since reading this quote, I was determined to establish the robustness of power laws, that is, proving the emergence of power laws and characterizing Pareto exponents under weak distributional assumptions.

In May 2016, while reading Nirei & Aoki (2016), I learned the characterization of the Pareto exponent \[ \upsilon \psi(z)=1, \] where \(\upsilon \in (0,1)\) is the survival probability, \(\psi(z)\) is the moment generating function of log growth rates, and the solution \(z=\zeta\) is the Pareto exponent. This equation appears in Proposition 5 of Nirei & Aoki (2016), who attribute it to Equation (10) in Manrubia & Zanette (1999). However, as Manrubia & Zanette (1999) is a physics paper, the derivation is based on a heuristic argument that lacks mathematical rigor.

So I tackled the problem, and after some work, I proved the formula using Fourier inversion and the residue theorem. You don’t know what happens in life. I studied complex and Fourier analysis purely out of curiosity in the late 1990s and early 2000s but never used again for more than 15 years. I showed my notes to Brendan and asked him to write a paper together. He found my proof unnatural because it required the random variable to have a finite support with evenly-spaced points, so instead he found the important paper Nakagawa (2007), which at that time was cited less than 10 times, and obtained a more general proof based on the Laplace transform. Nakagawa’s Tauberian theorem as well as the technique in Graham & Vaaler (1981) was crucial for making the results sharper and more rigorous. We then generalized the results to the Markov case instead of IID, \[ \rho(\Pi\odot \Upsilon \odot \Psi(z)) = 1,\] where \(\Pi\) is the transition probability matrix of the Markov chain, \(\Upsilon\) is the matrix of conditional survival probabilities, \(\Psi(z)\) is the matrix of conditional moment generating functions of log growth rates, \(\odot\) denotes the Hadamard (entry-wise) product, and \(\rho\) denotes the spectral radius of the matrix. Along the way learned many things about Laplace transform, Tauberian theory, Perron-Frobenius theory, matrix pencils, and log-convex functions.

It took us some time to clean up all the fine details and posted the working paper to arXiv in December 2017 and submitted it to Econometrica. We got two negative reports in July 2018 and the paper was rejected, but the editor seemed to be quite interested in the results and gave us a reject-and-resubmit, although the path to a successful revision was unclear. As the initial reviews failed to recognize the marginal mathematical contribution and criticized the two applications, we spent quite a bit of time explaining the marginal contribution and overhauling the applications.

We resubmitted the paper in December 2019. We got three reports in June 2020, one positive (likely written by an expert on power laws), one uninformative (likely written by a physicist), and one negative (likely written by a quantitative macroeconomist). Luckily the editor sided with the positive referee and gave us a revise-and-resubmit. Due to the difference in opinions (we put value in theorem-proving; some people don’t), we were never able to convince the quantitative macroeconomist. I thank the editor for understanding true values.

I am proud of this paper as it resolved the 20 year old robustness problem of power laws, an important problem that very few people recognize its importance.

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