Gravity, Confinement, and the Swap That Doesn’t Exist
I have spent the past year establishing when a wealth tax doesn’t do things. A uniform tax on all assets at market value doesn’t bend portfolio choices, doesn’t change what a buyer will pay for a company, doesn’t favour the foreign investor over the domestic one — it behaves like a silent partner taking the same slice of every venture, and the slice divides out of every decision. Norway’s tax commission recently arrived, by its own route, at a reform proposal remarkably close to that design — uniform rates on market values.
But “doesn’t distort” is only half a theory of taxation. The other half is the question politicians actually ask: what does the tax do — to the distribution of wealth, to the top shares, to who holds what a generation from now? Five new papers, now on the project page, take up that half. They form a single arc, and the arc has a shape: it starts with a paradox, passes through a phase diagram and a hidden political choice, arrives at an impossibility result that speaks directly to the loudest proposal in the Scandinavian debate, and closes by assembling the whole system into one working model of Norway.
Gravity
Start with the paradox, because it is the result the wealth tax’s friends will like least — and it comes out of the very same mathematics as the neutrality results. (The tax’s opponents, I have learned, are no fonder of the first half: tell a committed critic that the tax distorts nothing, and they hear an argument for keeping it. The mathematics does not care whose argument it inconveniences.)
A proportional wealth tax pulls every fortune down at the same rate — one percent a year off the top of everything, big or small. Physically, that is a uniform gravitational field: every particle feels the same acceleration, so while everything falls, nothing changes shape. The pecking order of fortunes, the ratios between them, the share held by the top one percent — all of it is preserved, not approximately but exactly, at every point in time. The first paper of the arc proves this as a theorem: the tax that leaves your investment choices alone also leaves relative inequality alone. Whatever compression eventually arrives comes on the timescale at which generations replace one another — decades to centuries, far beyond any electoral horizon.
So the wealth tax’s most-advertised virtue and its quietest failure are one and the same property. A tax that distorts nothing redistributes nothing — at least not through what it does to market outcomes. (It still redistributes through what the revenue buys; that fiscal channel is real and the papers model it. But the tax itself, acting on the distribution, is gravity: it lowers the whole landscape without flattening it.)
If you want the tax itself to compress the distribution, you need a different force — one that pulls harder the higher you sit. Physicists call that confinement. Economists call it progressivity. And the moment you introduce it, you have deliberately broken the symmetry that made the tax neutral. That is the deal on the table, stated honestly: neutrality and redistribution-through-the-market are not both available. Tax design is the art of choosing how to break a symmetry you liked.
There is one immediate reward for breaking it. Under the proportional tax, growth and tax take the same percentage bite at every size, so the race between them looks the same whether a fortune is small or vast: no size is special, and the distribution has no anchor. A progressive schedule creates one. Low in the schedule, growth outruns the tax and a fortune climbs; high in the schedule, the tax outruns growth and a fortune shrinks — so every fortune is pushed, from both directions, toward the level where the two balance. That two-sided push acts as a restoring force: the bowl under the ball. The randomness of returns keeps scattering fortunes away from the anchor, the restoring force keeps pulling them back, and the standoff between the two is a distribution with a definite shape, reached at a definite speed — on realistic numbers, within something like a political lifetime rather than a generation. Speed, not just shape, is what progressivity buys.
Confinement
Two papers then do the engineering. The first asks the coolest-headed version of the design question: if you must raise a given revenue from capital, what combination of profit taxes and wealth taxes rearranges the economy least? The answer turns out to be a phase diagram. One dimensionless ratio — essentially the economy’s growth stacked against its volatility — decides everything: below a threshold the gentlest design is all wealth tax, above a second threshold it is all profit tax, and in between lies a narrow band where the honest answer is a precise mixture of both. On Norwegian numbers, the economy sits inside that narrow band. Nearly everywhere else, “a bit of both” would be a politician’s evasion; for Norway, it is exactly what the mathematics prescribes. And even this cool-headed question hides a warm-blooded one beneath it: two perfectly reasonable ways of scoring “how much a tax rearranges the economy” disagree about the verdict — measure disruption one way and the mixture wins; measure it another and profit taxes always win. Even minimal-distortion is a value judgement wearing a lab coat.
The second design paper contains the finding I would nominate as the arc’s quiet scandal. Take progressive wealth-tax schedules of the kind Norway actually uses — a threshold, a higher rate above it — and hold measured inequality fixed: consider only schedules that deliver exactly the same Gini coefficient, the number that dominates public debate. Across those “equally equal” designs, the share of wealth held by the top one percent is anything but equal: one schedule leaves them with about five percent of all wealth, another with about forty. Eight-fold. Two tax systems can be indistinguishable on the headline equality index while one of them leaves the very top with eight times more. Which yardstick you optimise — the broad index or the top share — is not a technicality delegated to economists; it is a political decision about whose inequality counts, hiding inside what looks like a measurement convention.
The same paper also writes down, without flinching, the entry in this ledger that cuts against the wealth tax: on calibrated Norwegian numbers, the wealth-tax channel does about one-sixth of the structural compression of the wealth distribution — flow taxes on profits and dividends do the other five-sixths — and under textbook assumptions the wealth tax’s entire structural contribution could be replicated by raising dividend taxation by about seven percentage points. Read that again: our own framework prices the wealth tax’s redistributive job, under standard assumptions, as cheaply replaceable.
That sounds like the opening of an argument for abolition. It is. Auditing that argument is the third act.
The swap that doesn’t exist
The Scandinavian debate has a famous number, which regular readers will recognise from the elephant post: at today’s interest rates, a Norwegian business owner would accept a corporate tax rate north of forty percent to be rid of the wealth tax. The number is real, and it measures a real thing — the burden the wealth tax places on an owner. And it invites what seems an obviously humane trade: abolish the wealth tax, raise profit taxes to compensate, collect the same revenue from the same people with less pain. Combine that with the seven-percentage-point result above and the swap looks not just humane but cheap.
That post caught the debate clutching a phantom harm — a valuation effect the mathematics rules out. Here, the same debate reaches for a phantom remedy. The elephant, it turns out, has more than one part that isn’t there.
The third-act paper takes the swap seriously enough to price it properly. The textbook price — seven points on dividends — rests on an assumption the data reject: that everyone earns the same return on wealth. Norwegian registry data show persistent, large differences in returns across investors (the use-it-or-lose-it post is about exactly this). And an eighty-year-old distinction in public finance explains why that matters here. Since Domar and Musgrave in 1944, economists have known that a profit tax is a partner in your returns — it takes a share of what your wealth earns. A wealth tax, as Stiglitz formalised in 1969, is a partner in your position — it takes a share of what your wealth is. The two coincide only when wealth reliably produces taxable income.
At the top of the Norwegian distribution, much of it doesn’t. Fortunes there sit in closely held companies whose gains compound inside holding structures — under Norway’s participation exemption, dividends and gains retained within them generate no personally taxable flow at all — alongside appreciating assets that pay no dividend to anyone. Wealth of that kind is invisible to a profit tax at any rate, the way a silent engine is invisible to a microphone. The argument here asks less than it may seem to: it does not require the rich to earn better returns than anyone else — only that their returns arrive in forms that produce no taxable flow. And where persistent return gaps do exist, they need not signal skill: scale, access to markets closed to ordinary savers, and the market’s own long memory — measured in a companion paper — can hold a gap open without anyone outsmarting anyone. The impossibility ahead rests on the composition of top wealth, not on a story about superior investors.
So the paper puts the swap on a ladder of increasing realism, and the price climbs: seven percentage points in the textbook world; roughly eighteen once returns differ across investors; roughly forty once the holding-company channel is accounted for. And then the ladder simply ends. There is a threshold — when less than roughly half of top-end wealth produces taxable income flow — below which no dividend-tax rate, however high, replicates what the wealth tax does. Not expensive: unavailable.
That finding stacks two claims with different standings, and they deserve to be kept apart. That the threshold exists is a theorem. That the actual Norwegian top sits below it is a calibration — built on the best available panel of the country’s largest fortunes, and stated in the paper with the word “plausibly” doing honest work. Calibrations can be argued with; the theorem cannot.
If the conclusion has a familiar ring, it should: that the incomes of the very wealthiest are a poor handle on their wealth is the empirical heart of Gabriel Zucman’s case for taxing top fortunes directly, built from tax data across countries. The theorem adds a different kind of grounding — not an accounting of effective tax rates, but an impossibility derived inside the very framework that prices the swap and grants it its best case. And though the numbers are Norwegian, the mechanism is no Norwegian quirk: participation exemptions like the one doing the work here are the European norm, anchored in EU-wide practice. Any country whose top fortunes compound inside such structures faces the same threshold.
Two rejoinders usually arrive at this point. The first: surely that quiet wealth is taxed eventually, when the gains are finally realised? Eventually, yes — but for fortunes structured this way, “eventually” is measured in generations, and a tax deferred that long is worth a fraction of itself today. Deferral also creates a distortion of its own: the lock-in that rewards clinging to an asset you would otherwise sell, purely to avoid triggering the tax. Recent work by Ocampo, Schjelderup and Zoutman turns this observation into an efficiency argument for the wealth tax: a tax on the stock of wealth is precisely the instrument that unlocks the lock-in built by taxes on realised flows.
The second: then repeal the participation exemption — tax the holding companies directly. Notice what that concedes. It is no longer the swap; it is a different reform, one that would have to manufacture the very taxable flow the dividend tax needs. And the exemption exists for a reason — without it, the same profit is taxed again at every link of an ownership chain. Perhaps that price is worth paying; that is a debate worth having. But it is not the debate the swap’s advocates started, and it leaves the original claim — that today’s profit-tax instruments can substitute for the wealth tax — exactly as false as the theorem says.
The conclusion, phrased as carefully as I can: the popular replacement argument fails on its own terms. The wealth tax does one specific job — reaching wealth that produces no taxable income flow — that no profit tax can do at any rate. Emigration does not change this: the relocation that escapes a wealth tax escapes a dividend tax just as well, so exit — real, and modelled in an earlier paper as a tipping process rather than a smooth elasticity — is an argument about the tax’s costs, not about its replaceability. Whether the job is worth those costs remains a political question. But the swap being offered in its place does not exist.
One machine, one dial
The fifth paper closes the arc by assembling the whole Norwegian system — corporate tax, dividend tax, the wealth tax with its uneven valuations, the debt-deduction rules, the holding-company routing, the progressive brackets — into a single mathematical machine acting on everyone’s portfolios at once. Out of all that plumbing comes one strikingly clean result: the long-run concentration of wealth at the very top responds to exactly one dial in the entire apparatus — the top bracket rate. Everything else shapes the middle of the distribution, the incentives, the composition of portfolios. The far tail listens to one number.
Two disclosures about what the machine does not contain. The first is labour: nothing in the machine taxes wages — the whole arc studies how to tax capital, taking as given the prior and larger political choice of how much of the total burden capital should carry relative to labour. That choice — the territory Piketty made his own — is a redistributive lever in its own right, and it sits upstream of everything here.
The second is an assumption held fixed throughout: pre-tax returns are set by the world market — Norway is small and open — so the tax cannot push them up to compensate anyone. A distinguished counter-tradition, running from Pasinetti in the 1960s to Stiglitz in 2018, insists that it must: tax a dynastic capitalist, and the pre-tax return has to rise to keep his saving going. The next paper in this programme, nearly finished, audits that mechanism against the population that actually sits at the Norwegian top — founder-entrepreneurs holding illiquid, effort-dependent equity, not dynastic rentiers — and finds its structural preconditions failing one by one.
Step back and the arc reads in one line: a symmetry, a phase diagram, a political choice disguised as a measurement convention, an impossibility result, and a working model of Norway. And a framework can only be trusted to say the third-act thing if it also says the first-act thing. The same mathematics that shows the wealth tax cannot compress the distribution through the market is the mathematics that shows it reaches wealth nothing else can touch. Those are not contradictions. That is what it looks like when you take a tax apart honestly — and find that some of its parts do jobs no other tool can do, while others were never doing the job everyone assumed.
The five papers — on the redistribution paradox, minimum-distortion design, progressive schedules under Gini and top-share targets, the substitutability question under heterogeneous returns, and the combined Norwegian system — are on the wealth-tax project page.