A levy cannot tax what its base cannot see

The base caps the region, the rate moves you within it: redistribution as a parameter space

Jason C. Braatz · preprint, posted 2026-08-29 · 20 pages · not peer reviewed, not submitted

Wealth that grows by multiplication concentrates on its own, without anyone steering it, so every distribution that is not fully concentrated is being pushed back on by something. This paper asks what those pushes actually have in common, and finds that the decisive thing is not how hard you push but how much of the gain you can see in the first place. A levy on income nobody has realised is, arithmetically, no levy at all.

Abstract

A multiplicative wealth process with a positive mean growth rate condenses: log-wealth variance grows without bound and the Gini approaches unity. Every distribution that is not condensed is therefore being opposed by something. This paper classifies the opposing mechanisms not by institutional origin but by four coordinates (base, rate, periodicity, threshold) plus the realisation share of the base, and asks which regions bound inequality below unity.

The base sets a ceiling the rate cannot cross. At a matched rate the two bases differ by roughly an order of magnitude in κ, the levy's compressive budget, for which the flow base admits a closed form, κ = r·E[η⁺]. But the budget is not the whole story, and the second half of it points the other way: per unit of budget the flow base compresses more, not less. Matched at κ ≈ 0.10 the two reach Gini 0.222 and 0.125, and the reason is visible in a statistic nobody reports: the variance of the per-period log multiplier is untouched by the stock levy (a change of 6 × 10⁻⁶) and falls by a third under the flow levy. A stock levy truncates the outcome; a flow levy damps the generator, and both register as a smaller Gini.

The stronger prediction this design was built to test, that a flow levy fails to oppose the multiplicative term regardless of rate, is false, and §3.1's rate sweep is what falsified it. The frontiers are nested, stock 0.000 against flow 0.125, not disjoint.

The surviving claim is narrower and better: the decisive quantity is realisation, the share of a period's gain the base can observe. At zero realisation a 100% levy on flow leaves the wealth vector exactly unchanged, agent by agent, because its base is the uniform wage and a uniform assessment with a uniform rebate is the identity. And the collapse from there is violently front-loaded: the first five per cent of the realisation axis covers 32% of the reachable range, and the first quarter covers 69%. A rate is an intensity; realisation is an observability, and the observability binds first.

Periodicity and threshold are trim rather than structure: they modulate the effective rate without opening or closing a region, and a threshold at a quarter of the mean is close to free. A methodological result of independent interest: a summary statistic with a hard ceiling cannot serve as a convergence criterion, because the Gini is capped at (N−1)/N, so a fully condensed economy also stops rising, and a drift test scores total condensation as bounded.

The claims are properties of a model class; no causal claim about any institution is made. All results reproduce from open code; 18 tests pin them.

Keywords: kinetic exchange models · wealth condensation · multiplicative growth · Gini coefficient · tax base · realisation · agent-based models · econophysics
JEL: D31, D63, H23, H24, C63

The other preprint: Two firms file the same numbers