[Paper Review] Wealth distribution: To be or not to be a Gamma?

Mehdi Lallouache, Aymen Jedidi+1 more|arXiv (Cornell University)|Apr 28, 2010

Complex Systems and Time Series Analysis18 references3 citations

TL;DR

This paper challenges the widely cited assumption that wealth distribution in a kinetic exchange model with global saving propensity follows a Gamma distribution. Using moment analysis and integral equation techniques, it proves the distribution cannot be Gamma and derives an upper bound for low-wealth behavior, showing a non-Gamma, non-power-law decay near zero wealth.

ABSTRACT

We review some aspects, especially those we can tackle analytically, of a minimal model of closed economy analogous to the kinetic theory model of ideal gases where the agents exchange wealth amongst themselves such that the total wealth is conserved, and each individual agent saves a fraction (0 < lambda < 1) of wealth before transaction. We are interested in the special case where the fraction lambda is constant for all the agents (global saving propensity) in the closed system. We show by moment calculations that the resulting wealth distribution cannot be the Gamma distribution that was conjectured in Phys. Rev. E 70, 016104 (2004). We also derive a form for the distribution at low wealth, which is a new result.

Motivation & Objective

• To rigorously test the conjecture that the steady-state wealth distribution in a closed economy with global saving propensity follows a Gamma distribution.
• To investigate whether analytical moment calculations support the Gamma distribution hypothesis proposed in prior work.
• To derive an analytical upper bound for the wealth distribution at low wealth values, where the Gamma distribution is expected to differ.
• To clarify the functional form of the distribution near zero wealth, challenging the robustness of the Gamma fit in the low-wealth regime.

Proposed method

• Performs moment calculations on the steady-state wealth distribution to test consistency with the Gamma distribution.
• Derives the Fokker-Planck-type integral equation for the distribution function f(x) based on the trading rule with constant saving propensity λ.
• Simplifies the integral equation by integrating over the random fraction ε, reducing it to a convolution involving f(x) and 1/(x_i + x_j).
• Applies constraints from the δ-function and Heaviside step function to define the integration region for x_i and x_j.
• Uses iterative bounding techniques on the resulting inequality to derive an upper bound for f(x) as x → 0.
• Employs asymptotic analysis and variable rescaling to show f(x) decays as O(x^α exp[−β(log x)^2]) near zero, with β > 0, contradicting Gamma decay.

Experimental results

Research questions

• RQ1 Is the steady-state wealth distribution in the global saving propensity model exactly Gamma-distributed?
• RQ2 What is the analytical form of the wealth distribution at low wealth values, and how does it differ from the Gamma distribution?
• RQ3 Can moment-based analysis rule out the Gamma distribution as the exact solution for the model?
• RQ4 What functional form provides a valid upper bound for f(x) as x approaches zero?
• RQ5 How does the decay behavior near zero wealth compare to the power-law or exponential decay of the Gamma distribution?

Key findings

• The moment calculations show that the third and fourth moments of the distribution do not match those of a Gamma distribution, thus disproving the conjecture.
• The wealth distribution cannot be Gamma-distributed because the Gamma distribution decays slower than the derived upper bound at low wealth.
• An analytical upper bound for f(x) as x → 0 is derived as f(x) = O(x^α exp[−β(log x)^2]), with β > 0, indicating a faster decay than any power law.
• The derived functional form at low wealth is incompatible with the Gamma distribution, which decays as x^{n-1}exp(−nx/⟨x⟩), exhibiting slower decay.
• The result confirms that the distribution is not Gamma, even though it may appear similar in numerical fits for moderate wealth values.
• The closed-form solution to the integral equation remains an open problem, despite the derivation of a valid upper bound.