Pareto Exponent Research Articles

A test of the relationship between the Pareto exponent and sample size

This paper uses un-truncated city population data from three countries—the United States, Spain and Italy—to empirically test Proposition 1 put forth by Eeckhout (2004 American Economic Review, 94: 1429–1451). Eeckhout’s hypothesis was that the estimate of the Pareto exponent in a standard Zipf regression decreases with sample size, if the underlying city size distribution is lognormal. Using rolling sample regressions, we find that this proposition is only valid once we enter the lognormal body of the distribution; for the Pareto-distributed upper-tail, the estimated exponent does not vary with sample size.

Adjusted Pareto Scaling for Multivariate Calibration Models

ABSTRACTThe performance of multivariate calibration models ŷ = f(x) for the prediction of a numerical property y from a set of x‐variables depends on the type of scaling of the x‐variables. Common scaling methods are autoscaling (dividing the centered x by its standard deviation s) and Pareto scaling (dividing the centered x by sP with P = 0.5). The adjusted Pareto scaling presented here varies the exponent P between 0 (no scaling) and 1 (autoscaling) with the aim of obtaining an optimum prediction performance for ŷ. Related scaling methods based on the variable spread are range scaling and vast scaling; while level scaling is based on the location (central value) of the variable. These scaling methods and robust versions are compared for models created by partial least‐squares (PLS) regression. The applied strategy repeated double cross validation (rdCV) evaluates the model performance for test set objects and considers its variability. Results with three data sets from chemistry show: (a) the efficacy of the different scaling methods depends on the data structure; (b) optimization of the Pareto exponent P is recommended; (c) range scaling or vast scaling may be better than adjusted Pareto scaling; (d) in general a heuristic search for the best scaling method is advisable. Overall, the consideration of different variants of scaling allow for a flexible adjustment of the variable contributions to the calibration model.

More urbanization, more polarization: evidence from two decades of urban expansion in China

This article examines whether the patterns of urbanization in China over the past twenty years have resulted in a more balanced or increasingly polarized city size distribution. China is an important study area due to the immense scale of its urbanization and the substantial role of its planners in coordinating and striving for more balanced development patterns. This study utilizes measures of the spatial and temporal characteristics of built-up area (BUA) expansion across 23 urban agglomerations in China, focusing on size dynamics through the urban primacy index and Pareto exponent. The BUA within these 23 UAs increased by 280.90% in the study period. Coastal urban agglomerations were larger in size at baseline and maintained relatively high annual growth rates compared to inland agglomerations, indicating rising polarization. Changes within agglomerations were indicative of declining polarization, as evidenced by the reduction in the urban primacy index and the increase of Pareto exponent. Policies implemented to achieve enhanced urbanization coordination in China appear to be partially successful, but the growth of urban regions with existing large size continues to fuel uneven urban development and polarization.

Distribution of export size, Pareto exponent and financial crisis

This paper studies the impact of the 2008 financial crisis on the distribution of firms’ export sizes. Using Chinese customs data from 2003 to 2013, we first show that firms’ export sizes follow a Pareto distribution. By applying a difference-in-differences method, we find that the 2008 financial crisis induced significant changes in the values of the Pareto exponents. We argue that variations in the distribution of export size signal resource reallocations in the export sector at the macro level and are associated with exporters’ adjustments to their exporting behavior at the micro level. The findings suggest that the Pareto exponent is an informative indicator and offers policymakers a better understanding of firm-level adjustments to exporting behavior in the presence of external negative shocks.

Differential entropy estimation with a Paretian kernel: Tail heaviness and smoothing

Differential entropy extends the concept of entropy to continuous probability distributions, measuring the uncertainty associated with a continuous random variable. In financial data analysis, accurately estimating differential entropy is pivotal for understanding market dynamics and assessing risk. Traditional methods often fall short when dealing with the heavy-tailed distributions characteristic of financial returns. This paper introduces a novel approach to differential entropy estimation employing a Paretian kernel function adept at handling tail heaviness’s intricacies. By incorporating an additional smoothing parameter, the Pareto exponent, our method offers flexibility in adjusting to light and heavy-tailed distributions. We compare our approach against established estimators through a comprehensive Monte Carlo simulation, demonstrating its superior performance in various scenarios. Applying our method to foreign exchange market data further illustrates its practical utility in identifying stochastic regimes and enhancing financial analysis. Our findings advocate for integrating the Paretian kernel estimator into the toolkit of financial analysts and researchers for a more nuanced understanding of market behavior.

Wealth Inequality in a Low Rate Environment

We study the effect of interest rates on wealth inequality. While lower rates decrease the growth rate of rentiers, they also increase the growth rate of entrepreneurs by making it cheaper to raise capital. To understand which effect dominates, we derive a sufficient statistic for the effect of interest rates on the Pareto exponent of the wealth distribution: it depends on the lifetime equity and debt issuance rate of individuals in the right tail of the wealth distribution. We estimate this sufficient statistic using new data on the trajectory of top fortunes in the U.S. Overall, we find that the secular decline in interest rates (or more generally of required rates of returns) can account for about 40% of the rise in Pareto inequality; that is, the degree to which the super rich pulled ahead relative to the rich.

Neoclassical production, scarcity of skills and Pareto distributions

The distribution of top earnings follows a “power-law”: the top tail of the distribution is thick relative to other distributions and is approximately Pareto-distributed. There are different mechanisms proposed in the literature that yield Pareto distributions and power-laws. This paper shows that a simple general equilibrium model where aggregate production is assumed of the constant-elasticity-of-substitution (CES) type over multiple labor inputs generates distributions in earnings and wages that follow power-laws. The finding is very general as the distribution of the skill-types is irrelevant for the shape parameter (the Pareto exponent) of the power-law. The model displays a tight relationship between the Pareto exponent and the elasticity of substitution between the labor inputs. Available empirical estimates of the elasticities of labor substitution and of the labor supply predict a Pareto exponent consistent with its empirical counterpart for the distribution of earnings in the US.

A Stochastically Optimized Two-Echelon Supply Chain Model: An Entropy Approach for Operational Risk Assessment.

Minimizing a company's operational risk by optimizing the performance of the manufacturing and distribution supply chain is a complex task that involves multiple elements, each with their own supply line constraints. Traditional approaches to optimization often assume determinism as the underlying principle. However, this paper, adopting an entropy approach, emphasizes the significance of subjective and objective uncertainty in achieving optimized decisions by incorporating stochastic fluctuations into the supply chain structure. Stochasticity, representing randomness, quantifies the level of uncertainty or risk involved. In this study, we focus on a processing production plant as a model for a chain of operations and supply chain actions. We consider the stochastically varying production and transportation costs from the site to the plant, as well as from the plant to the customer base. Through stochastic optimization, we demonstrate that the plant producer can benefit from improved financial outcomes by setting higher sale prices while simultaneously lowering optimized production costs. This can be accomplished by selectively choosing producers whose production cost probability density function follows a Pareto distribution. Notably, a lower Pareto exponent yields better supply chain cost optimization predictions. Alternatively, a Gaussian stochastic fluctuation may be proposed as a more suitable choice when trading off optimization and simplicity. Although this may result in slightly less optimal performance, it offers advantages in terms of ease of implementation and computational efficiency.

Tax progressivity and the Pareto tail of income distributions

In a continuous-time version of the Bewley–Huggett–Aiyagari model, this paper shows theoretically and numerically that the fatness of the Pareto upper tail of the income distribution depends on tax progressivity only through the general equilibrium effect on the interest rate. With confiscatory taxes (marginal income tax rate approaching 100% at the top), the Pareto exponent is independent of tax progressivity.

Stochastic Ordering of Stationary Distributions of Linear Recurrences: Further Results and Economic Applications

We investigate pairwise stochastic comparisons of stationary solutions to the linear recurrence Xt+1=AtXt+Bt, where At and Bt are non-negative random variables. We establish novel order-preserving properties, which enable us to obtain comparison theorems about well-known measures of conditional size, tail variability and skewness across probability distributions. While useful in studies of ergodic wealth accumulation processes and the persistence of inequality, our results can fruitfully be exploited to conduct comparative statics exercises in structural models entailing Kesten-type reduced-form representations. An application of our analysis to a dynamic asset accumulation model uncovers the qualitatively similar effects of capital income and earnings taxation on expected wealth concentration over higher quantiles as well as on conditional upper tail dispersion of wealth holdings, qualifying previous results that solely rely on the determination of Pareto exponents.

Pareto extrapolation: An analytical framework for studying tail inequality

We develop an analytical framework designed to solve and analyze heterogeneous‐agent models that endogenously generate fat‐tailed wealth distributions. We exploit the asymptotic linearity of policy functions and the analytical characterization of the Pareto exponent to augment the conventional solution algorithm with a theory of the tail. Our framework allows for a precise understanding of the very top of the wealth distribution (e.g., analytical expressions for top wealth shares, type distribution in the tail, and transition probabilities in and out of the tail) in addition to delivering improved accuracy and speed.

Determination of Pareto Exponents in Economic Models Driven by Markov Multiplicative Processes

This article contains new tools for studying the shape of the stationary distribution of sizes in a dynamic economic system in which units experience random multiplicative shocks and are occasionally reset. Each unit has a Markov‐switching type, which influences their growth rate and reset probability. We show that the size distribution has a Pareto upper tail, with exponent equal to the unique positive solution to an equation involving the spectral radius of a certain matrix‐valued function. Under a nonlattice condition on growth rates, an eigenvector associated with the Pareto exponent provides the distribution of types in the upper tail of the size distribution.

Capital and Labor Income Pareto Exponents Across Time and Space

We estimate capital and labor income Pareto exponents across 475 country‐year observations that span 52 countries over half a century (1967–2018). We document two stylized facts: (i) capital income is more unequally distributed than labor income in the tail; namely, the capital exponent (1–3, median 1.46) is smaller than labor (2–5, median 3.35), and (ii) capital and labor exponents are nearly uncorrelated. To explain these findings, we build an incomplete market model with job ladders and capital income risk that gives rise to a capital income Pareto exponent smaller than but nearly unrelated to the labor exponent. Our results suggest the importance of distinguishing income and wealth inequality.

The distribution of city sizes in Turkey: A failure of Zipf’s law due to concavity

AbstractThe linearity of the distribution of city sizes is often assumed in the existing literature. Although different functional forms were tried, almost all of them impose a certain functional shape. In this study, we investigate the urban hierarchy and Zipf’s law using data for 973 Turkish subprovincial cities in 2019. We contribute to the literature in several ways. We force no definite functional form to observe the natural shape and employ nonparametric and quadratic regressions. We incorporate formal procedures of spatial dependence in regression models. We demonstrate that the linear model overestimates the Pareto exponent for small cities and underestimates it for bigger cities. We show that city sizes are unevenly distributed in Turkey. The rank–size rule is not valid in Turkey, either above or below a certain city‐size truncation level. Thus, the Pareto exponent estimated from the linear model is not a reliable indicator as quadratic regressions perform much better.

Did Zipf's Law hold for Chinese cities and why? Evidence from multi-source data

The New Form of Urbanization was proposed in 2012, emphasizing the development of small and medium-sized cities and towns. Analyzing the city-size distribution is significant in various studies, such as urban sustainable development, urban planning, population movement dynamics, and policy making in urban issues. But there lacks systematic analysis and mechanism discussion covering all cities sample especially after 2012 because of statistical data missing. This study assessed the city-size distribution of 337 cities (administratively defined) across Chinese mainland from 2013 to 2018 using multi-source data. Four main procedures are involved: 1) the urban areas of 337 cities were extracted using Head/tail Division Approach from remotely sensed nighttime light imagery; 2) the urban patterns were explored using spatiotemporal analysis (Kernel Density Estimation); 3) ln(rank) and ln(rank-1/2) were used as dependent variable to estimate Pareto exponent respectively; and 4) the economical mechanisms were explored using robust estimates. The results indicated that:1) The second mean of pixel values of NPP/VIIRS data was recommended to be the cut-off for the identification of urban areas in Chinese mainland, comparing to 1% National Population Sample Survey of 2015 and master plans of typical cities. 2) Using the total nighttime light (TNL) of urban areas to measure the city size, Zipf's Law was mostly violated in Chinese mainland. Overall, urban system was unequally distributed in all cities sample, equally distributed in upper-tail cities sample, showed great heterogeneity in provincial cities sample, meanwhile small and medium-sized cities developed rapidly after 2012. 3) Provinces with the larger value of regional scale and industrial structure were characterized as the more evenness of the city-size distribution, but with the larger value of trading level were characterized by unevenly distributed urban systems. The findings of the study indicated that the discussion about the applicability of Zipf's Law in Chinese mainland using multi-source data was of great significance, and the change of urban system in China is greatly affected by policies.

A theory of the saving rate of the rich

Empirical evidence suggests that the rich have higher propensity to save than do the poor. While this observation may appear to contradict the homotheticity of preferences, we theoretically show that that is not the case. Specifically, we consider an income fluctuation problem with homothetic preferences and general shocks and prove that consumption functions are asymptotically linear, with an exact analytical characterization of asymptotic marginal propensities to consume (MPC). We provide necessary and sufficient conditions for the asymptotic MPCs to be zero. We calibrate a model with standard constant relative risk aversion utility and show that zero asymptotic MPCs are empirically plausible, implying that our mechanism has the potential to accommodate a large saving rate of the rich and high wealth inequality (small Pareto exponent) as observed in the data.

The Success of Top Professional Athletes Is Not Only Success in Sports

Professional athletes are a scalable occupation that potentially allows them to make money without an equivalent increase in labor and time because they become celebrity influencers. Earnings above medians of scalable occupations show considerable variation because the right tail of their distributions decays as a power law. This fact implies that professional athletes’ labor markets are of the winner-take-all type. We take data from Forbes magazine of the world’s top 100 highest-paid athletes from 2012 to 2020 and calculate Pareto exponents by ordinary least squares and maximum likelihood. We find we cannot dismiss their distribution of earnings following a power law. This result means we cannot explain the mega-earnings of the highest-paid athletes by merit in sports alone.

The impact of trade liberalization on productivity distribution under the presence of technology diffusion and innovation

We develop an international trade model in which firms' productivity is affected by technology diffusion and innovation, and the model generates an endogenous Pareto productivity distribution. Each new entrant to the market randomly searches for and meets incumbents and then adopts their existing technology, which is considered technology diffusion. The incumbents adopt the newly developed frontier technology, which is considered innovation. The endogenous response of the productivity distribution to trade liberalization differs depending on whether the incumbents possess a high or low frequency of innovation. In an economy with a low frequency of innovation, trade liberalization increases both the Pareto exponent and the exit-cutoff productivity level. This higher Pareto exponent shifts the weight of the productivity distribution from the upper-tail area to the area around the new lower productivity bound, which generates productivity losses. The dominant productivity gains through the increasing of exit-cutoff productivity increase average productivity across all firms. However, in an economy with a high frequency of innovation, trade liberalization increases moderately or decreases both the Pareto exponent and the exit-cutoff productivity level. The reduced Pareto exponent generates productivity gains by accumulating highly productive incumbents. In total, however, trade liberalization generates weaker productivity gains or productivity losses on average productivity across all firms due to the moderately increased or reduced exit-cutoff productivity. Consequently, the welfare gains from trade are smaller in economies with high frequency of innovation than in those with low frequency.

Efficient minimum distance estimation of Pareto exponent from top income shares

SummaryWe propose an efficient estimation method for the income Pareto exponent when only certain top income shares are observable. Our estimator is based on the asymptotic theory of weighted sums of order statistics and the efficient minimum distance estimator. Simulations show that our estimator has excellent finite‐sample properties. We apply our estimation method to US top income share data and find that the Pareto exponent has been ranging between 1.4 and 1.8 since 1985, suggesting that the rise in inequality during the last three decades is mainly driven by redistribution between the rich and poor, not among the rich.

On the emergence of a power law in the distribution of COVID-19 cases

The first confirmed case of Coronavirus Disease 2019 (COVID-19) in the US was reported on January 21, 2020. By the end of March, 2020, there were more than 180,000 confirmed cases in the US, distributed across more than 2000 counties. We find that the right tail of this distribution exhibits a power law, with Pareto exponent close to one. We investigate whether a simple model of the growth of COVID-19 cases involving Gibrat’s law can explain the emergence of this power law. The model is calibrated to match (i) the growth rates of confirmed cases, and (ii) the varying lengths of time during which COVID-19 had been present within each county. Thus calibrated, the model generates a power law with Pareto exponent nearly exactly equal to the exponent estimated directly from the distribution of confirmed cases across counties at the end of March.