The number Anthropic's economists left as a word

Abstract

The Anthropic Institute has published a model of how AI might reshape the US economy by 2030. The figures that will travel are a labour share falling from 60 to 45 percent and knowledge-worker unemployment reaching 17.9 percent. I rebuilt the model independently, from the printed equations, and this is what I think a reader of those headlines should know. First, the model does not say wages fall: in every scenario, including the most extreme, the average wage rises. Second, that conclusion rests on a single parameter with a threshold below which it reverses, and the paper says the wage falls if capital supply is "sufficiently small" without saying how small. It is 2.08, against their assumption of 3.0. Third, the GDP figures may not be measuring the thing people will argue about, because work that leaves the market disappears from the accounts while making people better off. Past roughly half of AI use going off-market, measured GDP falls below its no-AI path while welfare rises.

What the paper actually says

The alarming numbers are real, and they are about distribution rather than about everyone getting poorer. Here is the model's own answer, from its results table:

The no-AI baseline is 3.8 percent unemployment and a 60 percent labour share.

If you read "AI transforms the economy" and heard "we all get poorer", this model says the opposite. Average wages rise in every scenario. What happens instead is a transfer: away from people who do cognitive work for a living, towards everyone else, whose wages rise by a third in the extreme case because the economy around them became far more productive while their own work did not get automated.

The second transfer is from labour to capital. The labour share falling from 60 to 45 percent is the most dramatic number in the paper, and the authors note that the four-point fall in their middle scenario is "only slightly less than the entire decline in the U.S. labor share over the four decades after 1980", compressed into four years.

So the honest summary is this. On this model, AI makes the country richer, makes most workers better off, makes knowledge workers worse off, and moves a large slice of national income from wages to capital.

The number that decides it

Now the part I think is missing, and the reason I bothered.

That average wages rise is less robust than it looks. It depends on how easily the economy can add capital, which in the model is one number. The intuition needs no mathematics. AI makes tasks cheaper, and somebody captures the saving. If capital is easy to add, its price does not rise much and the saving flows to wages. If capital is scarce, its price rises and the saving is absorbed by whoever owns it. Push that far enough and workers end up worse off than with no AI at all: they have lost the tasks, and must now work alongside capital that is expensive.

The paper defines the threshold where this flips, states that at their assumed value the wage rises everywhere, and says it falls if the elasticity is "sufficiently small". It does not say how small.

It is a short computation away:

A margin of about 0.9 in the scenario everyone will quote.

Whether that is comfortable depends on how firmly 3.0 is pinned, and here the paper's own reasoning is informative. The value comes from a study whose long-run estimate implies roughly 6. The authors halve it, explaining that the original describes an economy taking decades to reach its resting point while theirs runs only to 2030. That is the right direction of judgement. It also moves the parameter two thirds of the way from the literature value to the threshold. Halving again, on the same reasoning applied to a shorter horizon, would cross it.

There is a second reason not to relax. This one number stands in for the supply of compute, and over five years compute does not behave like capital in general. Fabrication capacity, grid connections and power availability bind hard for two or three years and then stop binding. That is a supply curve which is steep and then flat, and a constant elasticity is the one shape that cannot represent a bottleneck which resolves. The paper leans its choice upward precisely because compute is "financed in a world market and built in a year or two", which is an argument about the far end of that curve applied to all of it.

I am not claiming the wage falls. At their assumption it rises, exactly as they say. I am claiming that "sufficiently small" is doing work that a number would do better, and that the number is 2.08.

Why the GDP figure may not mean what you think

The second thing worth knowing is a measurement problem, and it starts from the paper's own caveats:

We do not assign a value to AI's use in household production, which lies largely outside GDP.

Take that seriously and it has a sharp consequence.

In the model, AI can do a few things to a task. Capital can take it over. A worker can do it faster. New tasks can appear. Each of those leaves the work inside the market, where somebody pays somebody, so each is counted.

There is a possibility the model has no room for. The work can leave the market altogether. You stop paying an accountant and do your return with AI. You stop paying for tutoring. A small firm stops buying a paralegal's hours. The work still happens, the household is better off, and the transaction vanishes, taking with it the expenditure GDP is built from.

flowchart TD
    T["<b>A task AI can now do</b>"]
    T -->|"capital takes it over"| K["done by <b>machines</b>, in a firm"]
    T -->|"worker does it faster"| W["done by a <b>worker</b>, in a firm"]
    T -->|"new work appears"| N["a <b>new job</b> in the market"]
    T -->|"<b>you just do it yourself</b>"| H["done <b>at home</b>, unpaid"]
    K --> C["<b>counted in GDP</b>"]
    W --> C
    N --> C
    H --> X["<b>not counted</b><br/>you are better off,<br/>the statistic falls"]
    classDef inside fill:#dbeafe,stroke:#2563eb,stroke-width:1px,color:#1a1a1a,rx:6,ry:6
    classDef outside fill:#fee2e2,stroke:#dc2626,stroke-width:2px,color:#1a1a1a,rx:6,ry:6
    classDef counted fill:#dcfce7,stroke:#16a34a,stroke-width:2px,color:#1a1a1a,rx:6,ry:6
    class K,W,N inside
    class H,X outside
    class C counted

Every route the model has ends in the green box. The one it does not model ends in the red one, and it moves GDP the opposite way to everything else in the paper.

How much would it take to matter? Running that margin through the model, measured GDP falls below its no-AI path once more than about 54 percent of AI use in the middle scenario has left the market. That is a lot, and I am not predicting it. But the nearest thing anyone has measured is open-source software, where one study puts the value to users at roughly two thousand times the cost of supplying it, with essentially none of the gap appearing in national accounts. Free digital goods behave the same way. The honest position is that nobody is measuring this for AI, and we now know roughly what it would have to reach before the headline statistic starts pointing the wrong way.

There is a sting in the tail. The same effect raises the measured labour share, by up to eight points, because work that leaves the market is disproportionately the work machines would otherwise have done. A world with heavy off-market AI use would print a labour share of 52.6 percent where the model says 44.6, and would look, statistically, like a world where workers held their ground. They would not have. The number would be flattering for the wrong reason.

What I checked, and what survived

That is the substance. Here is the basis for it, briefly, because none of the above should be taken on trust.

I rebuilt the model from the printed equations, with no access to the authors' code, which they do not publish. The reproduction matches their published results to within 0.35 percentage points on output and 0.6 points on unemployment. Parts of it are formalised in Lean 4, a proof assistant, so a machine has checked the reasoning rather than me: that the model has exactly one solution, that the quantity being solved for behaves as claimed, and that the measurement argument above is arithmetically sound. One figure is computed with certified error bounds, carrying a proof of its own accuracy rather than my assurance.

flowchart LR
    subgraph P ["checked by machine"]
        direction TB
        P1["the model has<br/><b>exactly one</b> solution"]
        P2["the measurement<br/>argument is <b>sound</b>"]
    end
    subgraph C ["checked by reproduction"]
        direction TB
        C1["the <b>numbers</b>:<br/>2.08, 54%, 52.6%"]
    end
    P --> G["reasoning<br/><b>guaranteed</b>"]
    C --> H["arithmetic<br/><b>matched</b> to 0.35pp"]
    classDef proved fill:#dcfce7,stroke:#16a34a,stroke-width:2px,color:#1a1a1a,rx:6,ry:6
    classDef comp fill:#fef3c7,stroke:#d97706,stroke-width:2px,color:#1a1a1a,rx:6,ry:6
    class P1,P2,G proved
    class C1,H comp

I also went looking for a weakness and failed to find the one I expected. The model turns on how readily different kinds of work substitute for each other, a parameter borrowed from a study of broad occupation groups and applied to something much finer. I thought that was the soft spot, and that the authors should have tested it. They did not, and it turns out not to matter: across the whole plausible range the labour-share result moves by under a point. Two effects inside the equation cancel almost exactly, which is visible in the exact algebra and invisible in the simplified version printed for intuition. I had reasoned from the simplified version.

That is worth stating plainly because it cuts against everything else here. The part of the paper most likely to frighten people is the part that held up best under attack.

I made two further mistakes of the same shape, comparing a readable approximation the paper prints against an exact figure it reports and treating the difference as a finding. Neither survived the exact equations. The pattern is worth naming for anyone doing this sort of work: papers contain both exact results and readable approximations of them, and it is easy to pick up the readable one, because it is the one you understood.

Move the parameters yourself

Everything above is a claim about what happens when you change numbers the paper holds fixed. Rather than ask you to take that on trust, the model runs in your browser: their five sliders plus the three they fix, with the wage threshold marked and welfare shown against measured output. It recomputes their published scenarios on load and shows the comparison, so you can see whether it still agrees with the thing it reproduces.

What I would want to know next

The quantity nobody is measuring is how much AI use leaves the market. It decides whether the GDP figures in this debate mean what people think they mean, and it is measurable in principle: time-use surveys already ask how people spend their hours, and willingness-to-pay studies already exist for free digital goods. Someone should point them at this.

The other is compute supply, treated here as one constant when it is the parameter the wage result turns on. Separate it out, give it a shape that can bend, and "do wages rise" would have a far better-grounded answer than a single elasticity halved by judgement.

Neither is a criticism of a careful paper whose caveats are unusually candid about exactly these limits. It is a note about which of its numbers will be quoted, and which of its assumptions those quotes will be resting on.

The formalisation, the solvers and the calibration with per-parameter sources are in a private repository; happy to share with anyone who wants to check the arithmetic. Figures other than the certified one are ordinary floating point, reproduced independently rather than proved.

About the author: Eduardo Aguilar Pelaez is CTO and co-founder at Legal Engine Ltd. He writes on formal methods, AI agents, and the discipline of building systems that survive being walked away from.