A visual explainer of Jensen's inequality and the Spirit Level critique.
In The Spirit Level (2009), the epidemiologists Richard Wilkinson and Kate Pickett argued that more equal societies almost always have better outcomes — better health, better education, less violence, less mental illness — across both rich and poor. Their proposed mechanism is psychosocial: inequality is corrosive to social trust, raises status anxiety, and damages everyone's wellbeing through chronic stress.
It's a powerful thesis. But there's a quieter, more uncomfortable possibility lurking inside the data — one Wilkinson and Pickett don't dwell on, and that the behavioural scientist Daniel Nettle has pressed on: the correlation they observe between inequality and worse average outcomes might be largely a mathematical inevitability, not an empirical discovery. It falls out of an old result called Jensen's inequality the moment you accept that money has diminishing returns to wellbeing.
This page tries to make that argument visible.
Almost every plausible function from income to outcome — health, life expectancy, years of education, self-reported happiness — is concave. The first £1,000 of income transforms a person's life. The hundred-thousandth £1,000 is statistical noise. Curves like √x and log(x) are the standard pedagogical stand-ins; the real curves are messier but qualitatively the same shape.
That curvature is the seed of the whole argument.
The Danish mathematician Johan Jensen proved in 1906 that for any concave function f and any random variable X:
In words: the outcome you'd get if everyone earned the average is at least as good as the average outcome when income is spread around that average. Equality only holds if either f is linear or X has zero variance — i.e., everyone earns exactly the same.
The two-point case captures the whole intuition. Take two people equidistant from the mean — one poor, one rich. Find their outcomes. Connect those two outcomes with a straight line (the chord). The midpoint of that chord is the average outcome. On a concave curve, the chord lies below the curve, so the average outcome lies below what you'd get if both people had earned the mean. The gap between the two is the cost inequality imposes on the average — what we'll call the Jensen gap.
Real countries don't have just two earners — they have millions, distributed across a whole range of incomes. But the two-earner thought experiment we used above is a useful microcosm: imagine zooming in on just two of those millions of people, one a bit poorer than average, one a bit richer. The wider the gap between them, the more unequal the society they live in.
The standard statistical measure of that "spread" is variance — the average squared distance of incomes from the mean. Variance has slightly odd units (£k², squared thousands of pounds) but it turns out to be the right quantity to track. A small Taylor-series approximation makes this precise: the inequality penalty is roughly −½ × f″(μ) × variance, where f″(μ) is the curvature of the outcome function at the mean. Doubling the variance roughly doubles the penalty.
For our two-earner case, the maths is especially clean: variance is simply the gap, squared. Two earners at ±£10k from a £30k mean give variance = 100 £k²; widen the gap to ±£20k and variance quadruples to 400 £k². The Jensen gap roughly quadruples too. Real income distributions extend the same idea to millions of earners and add a long right tail of high earners — making them skewed rather than symmetric — but the core result holds for any distribution at all.
Below, the same two earners appear in two charts, controlled by a single set of sliders. The top chart shows them as points on an income distribution, with red dotted lines marking their incomes on the x-axis. The bottom chart shows the same two incomes mapped through the outcome curve, where the Jensen gap appears as the vertical distance between f(μ) and the chord midpoint.
Drag the sliders and watch both charts update together. The two purple dots in the top chart spread apart as you increase inequality; the chord in the bottom chart sags further below the curve as that happens. Try the linear option to confirm the Jensen gap vanishes whenever the curve is straight.
So far the argument has had a deflationary feel — inequality damages average outcomes "automatically", as a property of arithmetic, without any social mechanism. But the same curvature that creates the Jensen gap also makes it cheap to close.
The key fact is that on a concave curve, the slope at low incomes is much steeper than the slope at high incomes. That means the same £1,000 of income buys a lot of outcome for a poor person and very little for a rich person. So if you transfer £1,000 from the rich earner to the poor earner — leaving the average income unchanged — the poor earner's outcome rises by much more than the rich earner's outcome falls. The transfer produces a net gain in the average.
This isn't a moral or political claim. It's just the geometry of concavity, made vivid below. To make the asymmetry as obvious as possible, the chart fixes a poor earner at £8k and a rich earner at £70k — a much wider gap than the chart above — so the steep and flat regions of the curve are both clearly in play. The two coloured arcs show how each earner's outcome changes when money moves from one to the other:
Drag the transfer-size slider. As you move money from rich to poor, both points slide along the curve toward the middle, but the poor earner climbs much further up the steep part of the curve than the rich earner slides down the flat part. The "net change" stat-card shows the gain in average outcome — always positive, as long as the curve is concave and the transfer doesn't reverse the income ordering.
Three observations:
The asymmetry is the whole point. If the curve were linear, the two arrows would be the same length and the transfer would be a wash. It's the curvature — diminishing returns — that makes redistribution productive.
The argument is local, not global. You don't need to redistribute all the way to perfect equality to see the benefit. Even a small transfer produces a small net gain. Each pound of redistribution buys some Jensen gap back; you can stop wherever the political or economic costs of further transfer outweigh the marginal gain.
This is the same maths, read in two directions. The Jensen gap (above) and the redistribution gain (here) are mirror images. Spreading incomes apart costs average outcome; pushing them back together recovers it. Both effects exist precisely because the outcome function curves.
If outcomes are a concave function of income, then a country with a wider income spread around the same mean will automatically show a worse average outcome — without any psychology, status anxiety, social mistrust, or stress hormones in the model at all. The correlation Wilkinson and Pickett document is, in this view, partly (or perhaps largely) a property of arithmetic that holds independent of any sociological mechanism.
Daniel Nettle made this point most forcefully in his 2017 essay Why inequality is bad, later collected in Hanging on to the Edges. He ran a 17-line simulation in which incomes were drawn from a distribution and passed through a concave function. The output reproduced the Spirit Level pattern reliably:
The plots could have come straight out of the pages of The Spirit Level… there is no delicate psychology of shame and anxiety; no response of the individual to their psychosocial milieu; no representation of the society's Gini coefficient in the head of any individual; and yet we see The Spirit Level's central result every time. That's mathematics for you. — Daniel Nettle, Why inequality is bad (2017)
Nettle is careful — he is not saying Wilkinson and Pickett are wrong, only that both the psychosocial mechanism and the Jensen-inequality mechanism plausibly contribute, and that it is striking how little space the latter is given in the book despite being widely discussed in the technical literature.
The critique is sometimes called "tautological" — a slightly loose use of the word, since the result is deductive rather than circular. But the political payload is interesting either way:
The deflationary reading. If inequality's harms can be derived from arithmetic alone, then the rich and the comfortable middle have no special reason to care: there's no toxic miasma of inequality affecting them. The "everyone benefits from equality" framing — the moral core of The Spirit Level — gets weaker.
The reinforcing reading. But as the redistribution chart above showed, the same arithmetic that demystifies the inequality–outcome correlation also argues for closing it. Moving a pound from a rich person to a poor person is net-positive for average outcome, precisely because of the diminishing returns. Jensen's inequality doesn't undermine the case for redistribution — it gives it a different, arguably more rigorous, foundation than the psychosocial story.
The two views diverge on the question of mechanism, and mechanism matters for policy. If inequality is toxic in itself, you need to compress the distribution. If it's just diminishing returns, then the level of the floor matters more than the shape of the distribution — and policies like a universal basic income, a higher minimum wage, or simply lifting people out of deep poverty become the priority. Most likely, both stories are true at once and to varying degrees, which is roughly where Nettle lands.
A few honest qualifications worth noting:
Real income distributions aren't symmetric. The chart above uses two symmetric points for visual clarity. Real income distributions are heavily right-skewed (a long tail of high earners), which complicates the simple two-point picture but does not change the basic Jensen-gap result.
The two mechanisms are observationally similar but not identical. A Jensen-only model predicts that average outcome depends on the full shape of the income distribution. A psychosocial model predicts effects driven specifically by perceived status differences — which can in principle be teased apart with the right data, particularly natural experiments where one varies while the other doesn't.
Concavity is empirical, not assumed. The whole argument rests on the income → outcome relationship genuinely being concave. For some outcomes (health, life expectancy, wellbeing) the evidence is strong; for others it's contested. If a function is roughly linear over the relevant range, the Jensen mechanism vanishes and the psychosocial story has to do all the work.