Does a rising tide lift all boats? A wealth exchange model on a dynamic network with economic growth

Technical Summary: "Does a rising tide lift all boats? A wealth exchange model on a dynamic network with economic growth"

Problem Statement
Economic inequality has risen substantially since the early 21st century, with wealth and income distributions across diverse nations exhibiting a universal two-regime structure: a power-law tail for the top 5% and a Gamma distribution for the bulk. While agent-based models (ABMs) have been widely used to study these emergent properties, most existing literature relies on mean-field descriptions where agents interact randomly with all others. Furthermore, these models typically assume a strictly conservative economy, neglecting economic growth. Although some studies have introduced complex network topologies or stochastic growth separately, no existing work had analyzed the simultaneous influence of network topology and economic growth on wealth distribution dynamics.

Methodology
The authors propose a dynamic complex network agent-based model that integrates three alternating processes within a single Monte Carlo Step (MCS):

1. Network Topology Dynamics: Based on a model from Ref. [26], the network evolves via a preferential attachment mechanism where new agents connect to existing ones with probability proportional to their wealth. Connections are rewired in each step based on wealth differences, allowing the network structure to adapt dynamically.
2. Wealth Exchange: Agents trade wealth with their connected neighbors according to the "Yard-Sale" rule. The amount exchanged is determined by the minimum risk-weighted wealth of the two agents. Crucially, the direction of the flow is governed by a probability that favors the poorer agent, controlled by a social protection factor ($f$).
3. Stochastic Economic Growth: Unlike conservative models, each agent's wealth evolves independently via a stochastic process: $\omega_i(t+1) = \omega_i(t)(1 + \mu + \sigma dB_i) + \Delta\omega_i(t)$. Here, $\mu$ represents the drift (economic growth) benefiting all agents, while $\sigma$ represents the volatility (heterogeneity in productivity). $dB_i$ is a Brownian motion term.

The system is simulated for $N=10^3$ agents over $4 \times 10^4$ MCS. To manage computational constraints with growing wealth, agent wealth is rescaled at each step. The study analyzes stationary quantities (Gini index, assortativity) and dynamic evolution across varying parameters of social protection ($f$), economic growth ($\mu$), and production heterogeneity ($\sigma$).

Key Results

• Impact of Heterogeneity ($\sigma$): Increasing production heterogeneity ($\sigma$) consistently increases inequality (Gini index) and drives the system toward wealth condensation, where a single agent accumulates nearly all wealth. This leads to a highly disassortative network (rich agents connecting to poor ones) and, in extreme cases, a star-like topology where one agent holds all connections.
• Role of Social Protection ($f$): The social protection factor is the primary stabilizing force.
  • Low $f$ ($f \approx 0.01$): Wealth exchanges accelerate wealth condensation compared to a model with only stochastic growth. The system becomes highly unequal, and the top 1% holds over 95% of wealth even for moderate $\sigma$.
  • High $f$ ($f \geq 0.2$): Strong social protection stabilizes the network topology and wealth distribution early in the simulation. It prevents the rapid condensation seen at low $f$ and maintains a more egalitarian distribution.
• Effect of Economic Growth ($\mu$): Increasing economic growth ($\mu$) reduces inequality, but this effect is contingent on the presence of strong social protection.
  • When $f$ is low, growth does not prevent condensation.
  • When $f$ is high, growth significantly lowers the Gini index, particularly when $\sigma$ is large.
  • For the middle class (10–50% wealth share), economic growth only provides benefits when social protection is sufficiently strong ($f \geq 0.5$). Otherwise, the negative impact of high production volatility ($\sigma$) outweighs the benefits of growth.
• Dynamic Interplay: The relationship between exchange and production is non-trivial. In the intermediate range of social protection ($0.01 < f < 0.2$), the system exhibits complex dynamics where exchanges initially appear more unequal than pure production but eventually stabilize the distribution if heterogeneity is high enough. Conversely, at low $f$, exchanges accelerate the formation of hubs and wealth concentration.

Significance and Claims
The paper claims to be the first to simultaneously analyze the influence of network topology and economic growth within an agent-based framework. The authors assert that their work reveals a "very interesting phenomenology" regarding the temporal relationship between wealth exchanges and production.

The central finding challenges the notion that economic growth universally benefits all strata ("a rising tide lifts all boats"). The paper concludes that economic growth benefits the poorest agents and the middle class only when strong social protection is in place. Without such protection, production heterogeneity ($\sigma$) drives wealth condensation regardless of the growth rate ($\mu$). The study highlights that while the exchange mechanism (specifically the social protection factor $f$) acts as a stabilizer for the system, the production process inherently favors the growth of the upper classes when heterogeneity is present. Thus, effective policy to mitigate stratification requires not just economic growth, but robust mechanisms that favor poorer agents in transactions.