When Does Heterogeneity Matter? A Spectral Theory of Wealth Distribution and General Equilibrium

Heterogeneous agent models (Aiyagari, 1994; Bewley, 1986) have become the workhorse of quantitative macroeconomics, but their computational demands are severe: the state variable is the entire wealth distribution — an infinite-dimensional object. Krusell and Smith (1998) discovered that a few moments suffice empirically, but lacked a theoretical explanation. We provide one through the Latent Number \(\rho\) of the wealth distribution. When \(\rho\) is high, the distribution is spectrally concentrated: one or two moments capture most information, and the representative agent model is nearly exact. When \(\rho\) is low, the full distribution matters, and heterogeneity fundamentally changes equilibrium outcomes. Specifically: the Krusell-Smith \(N\)-moment approximation error decays as \(\rho^{-N}\), the Gini coefficient satisfies \(G \propto 1/\rho\), the fraction of borrowing-constrained agents scales as \(1/\rho\), and the representative agent model is the exact \(\rho \to \infty\) limit. We establish a bridge to asset pricing: heterogeneity amplifies SDF variance by a factor \((1 + \kappa)\) where \(\kappa \propto 1/\rho\), providing a distributional channel for the equity premium. 18 theorems, machine-verified, 0 axioms.