The first shop on the Street of Cunning Artificers was bought with money. The second was bought with the rent from the first. By the time the Merchants’ Guild owned half the street, it was buying workshops with money earned from workshops bought with money earned from workshops, and lending to the rest at interest. Nobody could remember which purchase had started it, but the accountants were delighted.
The cobblers still mended shoes and the bakers still baked. What changed was where the money went afterwards. More of what the city’s work earned now went to the people who owned the workshops, the warehouses, the trading companies and the loans, and less to the people who did the work. Wages rose when the city did. Fortunes rose faster, because a fortune earns on itself.
The Patrician could tax the fortunes, and had done so before. But the owners had become important people, and important people had opinions about taxation which they were generous enough to share over dinner at the Palace. The tenants had opinions too. They mostly shared them with one another, the Palace being a considerable walk away and experience suggesting the walk was wasted.
Most years the city prospered, and its rents were collected with admirable efficiency. Then one spring the Patrician noticed that the income from owning Ankh-Morpork had become large enough to buy what was left of it. He restored the old tax, and nothing much changed: the fortunes made in the meantime were still there, still earning. He restored the tax of ten years before. Still nothing. Preventing the city from being bought had been considerably cheaper than buying it back.
Outside the city that shape has a name, the fold, and the people who study lakes and climates have tools for asking how close a system is to one. The tools do not care what the system is made of.
Who sets the tax
Three things keep the city selling itself, and all three start from the same fact: returns buy more of the city. The more the few own, the more they earn, and the more they earn, the more of whatever comes up for sale next year they can afford.
In the first, the owners concluded long ago that they earned what they hold. The more they own, the firmer the belief, and the firmer the belief, the less tax they will stand for. What they will stand for is what the Patrician sets, because they have his ear and the tenants do not.
In the second, the tenants concluded that petitions go into a box in the Palace which is emptied into the river, which is true. The more the few own, the more certain the tenants are of it. The less they believe a petition is read, the fewer they write, so a larger share of the petitions reaching the Palace come from the owners. The Patrician reads the petitions he receives.
In the third, the more the few own, the more tax the tenants ask for, and what they ask for does raise what is set, by about the wages of one constable sent from Pseudopolis Yard to collect it.
The evidence for each of these, from card games to three decades of Dutch policy to a century of national accounts, is in The crown of creation, with the step it has not reached: no study has followed the winners of a rigged game into a lifetime of voting.
The bookkeeping
Each year the owners’ share changes by the gap between what their holdings return and what is taxed back, multiplied by how much they already hold, because a fortune earns on itself. Written down:
$$\Delta \text{share} = \bigl(\text{return} - \text{tax}(\text{share})\bigr) \times \text{share}$$
The return is the return on capital less the growth of everything else, Piketty’s r minus g, and the model holds it fixed. The tax is not fixed. The Patrician sets it, and the owners’ dinners and the tenants’ silence set the Patrician, so the tax depends on the share itself. The owners’ confidence and the tenants’ silence push it down as the share grows, the tenants’ demand pushes it up a little, and on the whole the more the few own, the less is taxed back.
A resting share is one at which the owners’ share neither grows nor shrinks, because what their holdings return equals what is taxed back. Some resting shares are stable: a bankruptcy or a windfall moves the share, and it drifts back. Others are not: a little more and it goes on rising, a little less and it falls away. With a strong tax the city has two stable shares, one spread out and one concentrated, and between them an unstable one it cannot hold.
The fold is the year the spread-out share disappears. Cut the tax a little each year and the spread-out share creeps up, still stable, while the unstable share above it creeps down to meet it. In the year they meet there is no spread-out share left to settle back to, and the city tips.
The way back has a fold of its own, much further along. Only when the tax is so heavy that the city is nearly evenly owned does the concentrated share give way and the city fall back to being spread out. The city’s records show it nearly evenly owned once, after the great fire, and it was regarded at the time as a disaster.
If the share alone told the Patrician where the city stood, he could read it off and act. It does not: the same share can belong to a city on its way in, still spread out, and to one on its way out, already tipped, and the two have different futures. A repair sized to the tax at which the city tipped falls short by the whole distance between the two folds, which is what it feels like from the Palace when the old tax is restored and nothing happens.
Where the shape comes from
The shape has been worked on. Bouchaud and Mézard, in 2000, wrote down the simplest economy that could be written: agents whose wealth grows with random returns and who exchange with one another at some rate. The tail of the wealth distribution came out as a power law, Pareto’s, without its being put in, and so did a transition. When exchange is limited, the distribution becomes very broad and the economy is dominated by a few individuals; when exchange is favoured, wealth spreads out. They called the first state wealth condensation, after the physics in which a finite fraction of a system’s particles fall into one state, and they noted, less surprisingly in their words, that raising taxes does the same job as favouring exchange. Concentration can be a place an economy comes to rest, in a model with no Guild, no tenants and no beliefs in it at all. The owners’ dinners and the tenants’ silence are the addition, a model of why the tax moves. The fold is not in any one of those papers. It appears when the compounding and the changing tax are coupled.
Gabaix, Lasry, Lions and Moll came at it from the other direction and asked how fast the standard random-growth models can move top income inequality, and found them an order of magnitude too slow to produce what the United States has seen since the 1970s. To match the data they had to add heterogeneity, some agents growing faster than others, which they read as superstar entrepreneurs and managers. A model that moves too slowly looks like a model missing something that feeds on itself. Their fix adds it at the level of individuals, some fortunes simply growing faster than others. The city adds it at the level of the Palace, where a growing fortune buys a lower tax. Neither is tested against the other.
The setback test
If a city is approaching a fold, it announces itself in how it recovers. A bankruptcy, a windfall or a bad harvest moves the owners’ share. Far from the fold the share drifts back within a season. Near the fold the pull back weakens, so each recovery takes longer than the last, the share comes to resemble last year’s share, and its swings widen. A person who has watched a friend take a little longer to get back up after each knock knows the signal before the name. Scheffer and colleagues set out these generic early-warning signals in Nature in 2009, and the year before, Dakos and colleagues had looked for them in eight ancient abrupt climate shifts and found each preceded by a slowing of the fluctuations, starting well before the shift. The signal can be generic because it depends on the shape of the fold and not on what the system is made of. In principle a lake, an ice sheet and a distribution of wealth can show the same signature.
The share alone does not give the warning. A city near the fold looks much like a city far from it, until it tips, and the same small cut in tax that one year is absorbed by the next spring can, another year, remove the resting place. What tells the two apart is the recovery.
Guttal, Raghavendra, Goel and Hoarau took three American and two European stock indices through the crashes of 1929, 1987, 2000 and 2008 and found no slowing before any of them. The swings grew before every crash, but the slowing did not come, and they argued that those crashes were better explained as jumps driven by growing noise, which can happen far from any fold, than as an approach to one. Markets are the noisiest series there are, and the result may say more about markets than about the method. It does say that a system can look alarming for the wrong reason.
Whether the top wealth share is taking longer to recover from its shocks has not, as far as searching finds, been tested. The series to test is in the World Inequality Database, and it has two candidate tipping points in it: the collapse of concentration between 1914 and 1945, and the rise since 1980. The likely difficulty is not the method but the data. The usual tests work best with long, evenly sampled, relatively high-resolution series, and wealth shares are annual, short, and smoothed by the way they are built. The result may be that the signal cannot be seen at that resolution, which would be a finding in its own right about what can be read at the speed of a tax code.
Quiet streets
Ankh-Morpork is not bought all at once. The Guild buys first where the rents are highest and last where the tenants are most stubborn. A city is a map as well as a series, and on the map an approach to a fold can show before it shows in the whole. Halleck Vega, Patuelli, van Voorn and Weinans did it with unemployment across the 96 departments of France from 1982 to 2019 and found neighbouring departments moving together more strongly before crises, most clearly before 2008, with the signal readable in the map when it was not readable in the national series, and with different regions leading before and after each shock, which they compare to response diversity in an ecosystem. They caution that the measure also trends upward overall.
The tenants who stop writing are the same idea from the other end. Vinković and Kirman showed that Schelling’s model, in which mild preferences about neighbours produce sharply separated neighbourhoods, is the physics of clustering, with the boundaries between groups behaving as a surface tension, and the separated state behaving either as liquid droplets that slowly merge or as frozen clusters that stay where they are. A city in which the tenants have stopped writing has gone quiet in patches, and the patches have edges. The quantity to measure is not whether the tenants are quiet in aggregate but whether quietness clusters: whether neighbouring districts come to resemble one another in turnout and in what they believe about being heard, and whether that clustering strengthens before a larger shift.
Buying it back
Near the fold the tax rate is no longer the useful question by itself. A rate one notch lower does nothing, and then everything, and an argument in the Palace held in rates, this year’s against last year’s, is an argument about the wrong quantity. The quantity is how fast the tax answers the share, and that is set by whose petitions are read, not by the rate in force.
Restoring the tax of the year before the city tipped does not untip it, because the fortunes made in between are still earning, and because the tenants who once wrote and the constable who once collected have partly gone. The city comes back only at the far fold, not by returning to the near one. So the lever that lasts is the tenants’ petitions, not the rate. A tax set from the Palace while the tenants stay quiet is bought back by the next year’s returns.
The thing to measure is the recovery, not the share. After a shock, a fire, a plague, a run on the banks, how long does the share take to settle, and where? If it goes back to concentrated within a season, concentration is a deep basin, the fold is behind rather than ahead, and the fold that counts is the one on the way out, which the city’s records put far past any rate now in force. If each recovery takes longer than the last, the fold is ahead and the signal is there to be read.
And the way out has a price that rises. Naomi Klein says a spiral can be flipped, and the fold says what flipping costs: more than not entering, and more with each year the tenants stay quiet. Her own example of tenants starting to write again is in Memphis, where she describes KeShaun and Justin Pearson organising against a data centre by rooting the fight in the city’s civil rights and environmental justice movements, back to the freed people who built Boxtown. That is tenants writing again where the few own most, and it is the kind of move that can shift the fold itself, because it changes how the tax answers the share, and that is what sets where the city tips.
What the model does not know
The physics supplies the compounding. Piketty supplies the pressure towards concentration. The owners’ dinners and the tenants’ silence supply a tax that changes with the share. Together they make a constructed system in which a fold can exist. The ecology supplies a way of asking, from the series alone, whether the fold is near. None of it shows that the fold exists in any real economy. It is the model’s hypothesis, not a historical law.
Nineteenth-century wealth concentration had already tipped, and it was not undone by taxes drifting back to where they had been; it took two wars, a depression and progressive taxation to push redistribution all the way to the far fold. That was the great fire, outside the city. The last independent shop on the Street of Cunning Artificers is still open. It sells what it always sold, to the people it always sold it to, and nobody has yet asked its owner to sell. He has noticed that the shops on either side now pay their rent to the same address.