Minimum-Distortion Wealth Taxation, I: Information-Theoretic versus Transport-Geometric Optimality on the Proportional Class

We characterise minimum-distortion wealth taxation under two contrasting normative criteria within a Fokker-Planck framework on log-wealth: the JKO free-energy gap, an information-theoretic measure aligned with the Mirrleesian decision-distortion tradition, and the squared 2-Wasserstein distance from the no-tax distribution at horizon $T$, a transport-geometric measure aligned with the Saez-Zucman distributional-compression tradition. Restricting to the neutrality-preserving (C1)-(C3) schedule class of a companion paper (Froseth 2026), both optima admit closed forms in the two-dimensional design plane parametrised by the corporate-dividend retention $k = (1-τ_c)(1-τ_d)$ and the proportional wealth-tax rate $τ_w$. The JKO optimum partitions the regime axis into three phases as a function of the dimensionless ratio $ρ= Σ_0 m_0/σ^2$, with $m_0 = μ- σ^2/2$ the geometric mean log-return: a pure wealth-tax phase at low $ρ$, a mixed-instrument phase at intermediate $ρ$, and a pure flow-tax phase at high $ρ$. The $W_2$ optimum, by contrast, is degenerate in this calibration: it pins to the pure flow-tax corner across the whole regime axis. The criterion contrast admits an economically meaningful reading via a bluntness index $B(m_0) = b/(a m_0)$ that measures the wealth-tax channel's mean-displacement-per-revenue overshoot relative to the flow-tax channel; JKO weights $B$ linearly, $W_2$ weights it quadratically, and the two normative traditions correspond to this difference in weighting. Norwegian-flavoured calibrations sit inside the JKO mixed-instrument phase under stock-heavy portfolio volatility but move into the pure flow-tax phase under the realised effective volatility of typical real-estate-heavy households.