The Pareto distribution and r > g

The clear winner for the most cited mathematical formula of 2014 is Thomas Piketty’s famous inequality: r > g. The relationship concisely summarizes the argument at the heart of his “Capital in the Twenty-First Century”— the difference between the return on capital and the growth rate of the overall economy is a powerful force for economic divergence. In the months since the book was published in English, economists and others have fought about the Paris School of Economics professor’s relationship.

One of the reasons for the intensity of this debate is that Piketty’s argument doesn’t seem to mesh with widely cited models of economic growth. A new National Bureau of Economic Research working paper argues that the relationship between r and g can be best understood in the context of the Pareto distribution.The distribution is named after Vilfredo Pareto, an Italian economist who wrote about the unequal distribution of land.

The Pareto distribution follows a so-called power law: the portion of the distribution above a given cutoff is equal to the cutoff raised to some (constant) power. For instance, if the top 1 percent owns 40 percent of the wealth, then the top 0.01 percent owns 40 percent of the wealth of the top 1 percent, or 16 percent of the overall wealth. In that case, the constant power is approximately 0.8.

It’s in this context that we should think about r > g, according to Stanford University’s Charles I. Jones, the author of the new paper. Jones shows in the paper that many of the observations Piketty makes in “Capital in the 21st Century,” particularly the importance of r > g, also arise when thinking about income and wealth distribution in Pareto terms.

Jones explains that a Pareto distribution is the result of “exponential growth that occurs over an exponentially-distributed amount of time.” Jones has written on the sources of top-end income inequality before with Jihee Kim of the Korea Advanced Institute of Science and Technology.

Jones finds that the difference between r and g is at the heart of determining the top-end wealth distribution. But other factors come into play as well, among them the age distribution of the population and the tax rate. Piketty talks about these issues in the book, but Jones’ paper shows the importance of these underlying factors and the assumptions about them to the utility of the simple r > g inequality. If some of these assumptions don’t hold up, then r > g might not lead to the world Piketty predicts.

The new paper by Jones shouldn’t be thought of as supporting or attacking “Capital in the 21st Century,” but rather presenting the ideas of the book in a different manner. At the same time, the new paper helps explain the assumptions that Piketty makes and the forces that academics and policymakers alike need to look at in these various calculations in order to understand the future of economic inequality.

December 16, 2014

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