Inequality and Mobility in a Minimal Model for Evolving Income Distributions

TL;DR

This work analyzes the co-evolution of income inequality and economic mobility using a minimal framework that couples a population-scale PDE for income distributions with an agent-based SDE. Mobility is driven by two mechanisms: percentile-dependent growth $R(p)$ and year-to-year randomness $V$, with $R(p)$ shaping mean growth across percentiles and $V$ inducing re-ranking. Calibrated to US Census data from 1968–2021, the study demonstrates that increasing mobility via randomness can amplify inequality, while making growth across percentiles more egalitarian can improve mobility but does not fully counteract inequality accumulation. The framework provides Growth Incidence Curves and a $(P_1\to P_2)$-mobility metric to reveal non-equilibrium dynamics and the sensitivity of mobility and inequality to the underlying growth and randomness structure, offering insights for policy design under the veil of ignorance.

Abstract

In this paper we explore the dynamic relationship between income inequality and economic mobility through a pairing of a population-scale partial differential equation (PDE) model and an associated individual-based stochastic differential equation (SDE) model. We focus on two fundamental mechanisms of income growth: (1) that annual growth is percentile-dependent, and (2) that there is intrinsic variability from one individual to the next. Under these two assumptions, we show that increased economic mobility does not necessarily imply decreased income inequality. In fact, we show that the mechanism that directly enhances mobility, intrinsic variability, simultaneously increases inequality. Using Growth Incidence Curves, and other summary statistics like mean income and the Gini coefficient, we calibrate our model to US Census data (1968-2021) and show that there are multiple parameter settings that produce the same growth in inequality over time. Strikingly, these parameter settings produce dramatically different mobility outcomes. Naturally, the greater disparity there is between annual percentage growth in the upper and lower income levels, the less ability there is for individuals to climb the percentile ranks over a fixed period of time. However, more than this, the model shows that whatever mobility does exist, it decreases substantially over time. In other words, while it may remain true that opportunity to reach the upper ranks mathematically persists in the long run, that long run gets longer and longer every year.

Figures (8)

• Figure 1: Quantile upper limits in 2021 dollars. After adjusting for inflation we see how rapidly the incomes of the high-earners are growing compared to other groups.
• Figure 2: A sketch of the one-year income change behaviors that result from our four primary parameter regimes. At left, annual percentage growth has the same mean and variance across all percentiles. At right, the upper percentiles have a slight advantage. In order to calibrate the models so that there is similar growth in inequality, the curved $R(p)$ model requires a smaller degree of year-to-year randomness, leading to lower economic mobility.
• Figure 3: US Census data, 1968-2021: Log-quantiles and Growth Incidence Curve. Left. Using regression, we fit the average annual percentage growth rates for incomes at the 20th, 40th, 60th, 80th, and 95th percentiles. Right. The estimated growth rates are displayed (red circles) and a cubic fit of the form $\mathrm{GIC}(p) = \hat{\alpha}_0 + \hat{\delta} p^3$ was computed to estimate interpolated GIC values (black curve, $\hat{\alpha}_0 = 3.94, \hat{\delta} = 1.16$).
• Figure 4: Mobility/Inequality when there is egalitarianism among cohorts.(Top) Fifty-year mobility decreases over time (left) because the Gini coefficient increases over time (right). (Bottom) We display multiple mobility measures for the flat-$R(p)$ model ($R(p) = 4.49$), over a range of values for the randomness parameter $V$. Fifty-year mobility increased, but so did the 50th-year Gini coefficient.
• Figure 5: Numerical results under the four main models. An overlaid comparison of pdfs that arise from numerically solving the PDE model with the initial condition described in \ref{['sec:numerical:init']} and the $R(p)$ and $V$ selections described in \ref{['sec:numerical:dynamic']}. The distributions look similar, distinctions in the underlying mobility is laid out in Section \ref{['sec:numerical:mobility']}. Parameters of the initial distribution and PDE model are given in Tables \ref{['tab:lognormal-init']} and \ref{['tab:parameters-dynamic']}.

Theorems & Definitions (24)

• Definition 1.1: Growth Incidence Curve (GIC)
• Definition 1.2: ($P_1 \to P_2$)-mobility
• Definition 2.1: Growth and Variation
• Definition 2.2: Income distribution: PDE model
• Definition 2.3: Agent-based model
• Proposition 3.1: Flat $R(p)$ preserves log-normality
• proof
• Corollary 3.2
• proof
• Corollary 3.3