Statistical Dynamics of Wealth Inequality in Stochastic Models of Growth

Abstract

Understanding the statistical dynamics of growth and inequality is a fundamental challenge to ecology and society. Recent analyses of wealth and income dynamics in contemporary societies show that economic inequality is very dynamic and that individuals experience substantially different growth rates over time. However, despite a fast growing body of evidence for the importance of fluctuations, we still lack a general statistical theory for understanding the dynamical effects of heterogeneneous growth across a population. Here we derive the statistical dynamics of correlated growth rates in heterogeneous populations. We show that correlations between growth rate fluctuations at the individual level influence aggregate population growth, while only driving inequality on short time scales. We also find that growth rate fluctuations are a much stronger driver of longterm inequality than earnings volatility. Our findings show that the dynamical effects of statistical fluctuations in growth rates are critical for understanding the emergence of inequality over time and motivate a greater focus on the properties and endogenous origins of growth rates in stochastic environments.

The Bayesian Origins of Growth Rates in Stochastic Environments

“…This result shows that learning a shared environment reduces growth rate variance on the same time scale as the dynamical effects introduced by growth rate variance [12]. It suggests that fast learning (by a sufficiently low k) equalizes information access, and is a suitable mechanism for reversing the long-term effects of heterogeneous growth on inequality.…”

“…We furthermore measure the relative increase in variance to resources using the coefficient of variation c v = σ r / r , to assess whether the increase in inequality outpaces the overall resource growth. More on this analysis is given in [12].…”

“…In much of this work, agents representing individuals or households (often with life cycles), grow or lose wealth through a multiplicative (geometric) stochastic process. This modeling choice is well supported empirically and introduces a number of key parameters as an agent's resources (or wealth), r, evolve exponentially with mean growth rate (over time), γ, fluctuate with standard deviation (volatility), σ [1,10,11] and vary across individuals of a population with standard deviation σ γ [12,13].…”

“…Particularly important are the statistics of heterogeneous growth rates within the population, which generally result in growing inequality over long times. In such contexts, agents with higher average growth rates [12] amass larger and larger amounts of relative wealth, thus reducing social mobility across the population. This phenomenon has been well known to economists, who have studied its emergence in models of elastic agent decision-making for goods exchanges [14][15][16], and its aggregate impacts via heterogeneous growth through firm innovation [17] and natural resource abundance [18].…”

“…To model information dynamics, we show that learning in the joint space of environmental states and agents' signals is developed optimally in terms of Bayesian inference, translating a maximization of predictability of environmental states into that of resource allocations and growth. We finish by exploring the consequences of learning a shared environment on the statistics of information and wealth, and discuss the consequences of these results for the role of general education and training on population dynamics and its potential to reverse long-term inequality [12].…”