The math behind a beautiful world (September 2026)

The derivations behind A Beautiful World, and Nobody Owns It. Nothing here is new mathematics — it’s a production function, a limit, a profit identity and an accounting identity — but the post leans on these hard enough that they should be checkable rather than asserted.

Sections 1 and 2 are a stripped-down version of the model in Kording and Marinescu’s Brookings working paper on intelligence saturation, which nests both sectors properly and allocates labor endogenously. If you want the real thing, read theirs. This one is kept single-level on purpose, so that §3 and §4 — the industrial organization and the factor shares, which are what the post is actually about — sit on the smallest production function that carries them.

Four things get proved, in increasing order of how much they matter:

1. When the physical bottleneck binds — a closed-form condition in the factor shares, which turns out to be satisfied for plausible parameters.
2. Where the AI sector’s revenue goes as cognition gets free — the limit is either zero or everything, with nothing in between, and the switch is \(\sigma = 1\).
3. Why a genuine quality lead doesn’t imply a profit — four inequalities, one per horseman.
4. Where the surplus lands instead — factor shares, and what has to happen before workers lose in an economy six times richer.

1. The bottleneck, in closed form

Cobb-Douglas is the \(\sigma = 1\) special case of the model in §2, and normally I’d skip straight to the general version. It earns its place here because it’s the only case that gives a number for when the physical constraint starts binding.

Throughout, \(L\) and \(T\) are quantities of input — hours of physical work, units of cognitive work — not spending on them. Their prices are \(P_L\) and \(P_T\), and expenditure on each is \(E_L = P_L L\) and \(E_T = P_T T\). Every result below is about what happens to expenditure when a price falls and the quantity it buys rises in response, so the two must not be conflated.

Take output \(Q\) from physical labor \(L\) and cognitive input \(T\), with decreasing returns:

\[ Q = L^{\alpha} T^{\beta}, \qquad \alpha + \beta < 1. \]

Normalize the output price to 1, so profit is \(\Pi = L^{\alpha}T^{\beta} - P_L L - P_T T\). The first-order conditions are the familiar factor-share identities:

\[ \frac{\partial \Pi}{\partial L} = \alpha L^{\alpha-1}T^{\beta} - P_L = 0 \;\Longrightarrow\; P_L L = \alpha Q, \]

\[ \frac{\partial \Pi}{\partial T} = \beta L^{\alpha}T^{\beta-1} - P_T = 0 \;\Longrightarrow\; P_T T = \beta Q. \]

Substituting \(L = \alpha Q / P_L\) and \(T = \beta Q / P_T\) back into the production function and solving for \(Q\):

\[ Q^{*} = \left( \frac{\alpha^{\alpha}\beta^{\beta}}{P_L^{\alpha}P_T^{\beta}} \right)^{\frac{1}{1-\alpha-\beta}}. \]

Since \(L^{*} = \alpha Q^{*}/P_L\) and \(P_L\) is unchanged, a fall in the price of cognition from \(P_{T,0}\) to \(P_{T,1}\) pulls physical labor up by

\[ \frac{L_1}{L_0} = \left(\frac{P_{T,0}}{P_{T,1}}\right)^{\frac{\beta}{1-\alpha-\beta}}. \]

This is the demand for hardhats created by cheap thinking. Now cap it. Suppose the physical workforce can at most double — displaced knowledge workers retrain, but only so many and only so fast — so \(L_{\max} = 2L_0\). The cap binds when

\[ \left(\frac{P_{T,0}}{P_{T,1}}\right)^{\frac{\beta}{1-\alpha-\beta}} > 2. \]

Take a fourfold drop in the cost of cognition, \(P_{T,1} = P_{T,0}/4\). Then \(4^{\beta/(1-\alpha-\beta)} > 2\) reduces, pleasingly, to a condition on the factor shares alone:

\[ \boxed{\;\alpha + 3\beta > 1\;} \]

That is not a knife-edge. At a labor share of \(\alpha = 0.6\) and a cognitive share of \(\beta = 0.2\) it gives \(1.2 > 1\) — the bottleneck binds, comfortably, on parameters nobody would call aggressive. The physical ceiling in the complements case isn’t a corner case constructed to make a point. It’s the default.

Once the cap binds, \(L\) is stuck at \(2L_0\) and the firm optimizes \(T\) conditionally:

\[ T^{*} = \left(\frac{\beta (2L_0)^{\alpha}}{P_{T,1}} \right)^{\frac{1}{1-\beta}}, \qquad w = MP_L = \alpha (2L_0)^{\alpha-1}(T^{*})^{\beta}. \]

The wage equation is where the blue-collar spike comes from. \(T^{*}\) rises without bound as \(P_{T,1}\) falls, and \(w\) rises with \((T^{*})^{\beta}\). Cheap cognition bids up the price of hands, because hands are what’s left.

2. The switch at \(\sigma = 1\)

Now the general case. Write output as a CES aggregate with elasticity of substitution \(\sigma\) and returns to scale \(\gamma\):

\[ Q = A\left(\alpha L^{\rho} + \beta T^{\rho} \right)^{\gamma/\rho}, \qquad \rho = \frac{\sigma-1}{\sigma}. \]

As \(\sigma \to 1\) we have \(\rho \to 0\), and L’Hôpital on the indeterminate \(\tfrac{1}{\rho}\ln(\alpha L^{\rho} + \beta T^{\rho})\) gives

\[ \lim_{\rho \to 0} \frac{\alpha L^{\rho}\ln L + \beta T^{\rho}\ln T} {\alpha L^{\rho} + \beta T^{\rho}} = \frac{\alpha \ln L + \beta \ln T}{\alpha + \beta}, \]

which exponentiates back to the Cobb-Douglas form of §1. So the two models are one model, and \(\sigma\) is the dial.

Here is the result the post rests on. Take the ratio of the two first-order conditions. The common factor \(A(\gamma/\rho)(\cdot)^{\gamma/\rho - 1}\rho\) cancels, leaving \(P_T / P_L = (\beta T^{\rho-1})/(\alpha L^{\rho-1})\), and therefore the ratio of what the firm spends on each factor is

\[ \frac{E_T}{E_L} = \frac{P_T T}{P_L L} = \frac{\beta}{\alpha}\left(\frac{T}{L}\right)^{\rho}, \qquad \rho = \frac{\sigma - 1}{\sigma}. \]

Everything now turns on the sign of one exponent. Hold \(L\) at its physical cap and let cognition become free, \(P_T \to 0\), so \(T \to \infty\):

• Complements, \(\sigma < 1\). Then \(\rho < 0\), so \((T/L)^{\rho} \to 0\) and \(E_T \to 0\). The AI sector’s revenue goes to zero. Not “grows slowly” — goes to zero, as a share of spending, precisely because the thing it sells became abundant while its complement did not. This is Baumol’s cost disease in its purest form.
• Substitutes, \(\sigma > 1\). Then \(\rho > 0\), so \((T/L)^{\rho} \to \infty\) and \(E_T/E_L \to \infty\). The AI sector absorbs essentially all expenditure. This is the Jevons paradox: making a resource cheaper increases total spending on it, because consumption rises faster than price falls.

Two limits, zero and infinity, separated by \(\sigma = 1\). There is no gentle interpolation between these outcomes, which is why the post treats them as two futures rather than as a spectrum. And note what the result does not depend on: not on how good the models get, not on how much compute gets bought, not on anyone’s AGI timeline. Only on whether thinking substitutes for doing.

3. Four inequalities, one per horseman

Assume we landed in \(\sigma > 1\), so the sector is enormous. Does the frontier lab get paid?

Let \(P_O\) be the price of open-weight intelligence, driven to the marginal cost of compute; \(\lambda > 1\) the closed model’s quality multiplier, so one unit of \(T_{\text{closed}}\) does the work of \(\lambda\) units of \(T_{\text{open}}\); \(MC_C\) the marginal cost of serving a closed unit; \(F_{\text{train}}\) the fixed cost of the generation; and \(Q_C\) the volume served.

A buyer purchases the closed model only if it is no more expensive than buying the same effective work from the open one, which is the limit-pricing constraint:

\[ P_C \le \lambda P_O. \]

Profit over one model generation is \(\Pi_C = Q_C (P_C - MC_C) - F_{\text{train}}\), so the ceiling is

\[ \Pi_C \;\le\; Q_C\left(\lambda P_O - MC_C\right) - F_{\text{train}}. \]

Every horseman in the post is a way for the right-hand side to be non-positive:

1. Inference bloat. \(MC_C \ge \lambda P_O\). The gross margin is negative before fixed costs are considered at all: the model is too expensive to serve at the only price the market will bear. Note this failure is caused by the pursuit of \(\lambda\) — staying smarter is what made serving expensive.
2. The Red Queen. \(Q_C(\lambda P_O - MC_C) < F_{\text{train}}\). Gross margin is positive but the amortization window closes first, because \(\lambda \to 1\) on the open-source release cadence rather than on a schedule the lab controls.
3. The compute tax. Upstream suppliers observe \(Q_C\) and raise the price of silicon and power, which increases \(MC_C\) and \(F_{\text{train}}\) simultaneously. They can keep going until \(\Pi_C = 0\) and no further, which is exactly what a monopolist facing a derived demand curve does.
4. The router. Orchestration sends the easy mass of the task distribution to \(T_{\text{open}}\), which forces the effective \(\lambda \to 1\) over most of the workload and collapses the \(Q_C\) that reaches the frontier endpoint. The lab keeps its quality lead and loses the volume it needed to pay for it.

The four are not alternatives. Conditions 3 and 4 make 1 and 2 more likely by moving \(MC_C\) up and \(Q_C\) down, which is the sense in which the post calls them structural rather than merely possible.

4. Factor shares, and what a 6× economy does for wages

The results above are about the AI sector’s revenue. This one is about everyone else’s, and it is the reason the post can claim a good outcome for households while claiming a bad one for the labs.

Start from the identity that does most of the work. Output is income: with constant returns and competitive factor pricing, the value of output is exhausted by payments to factors,

\[ Q = \frac{\partial Q}{\partial L}L + \frac{\partial Q}{\partial T}T = P_L L + P_T T, \]

which is Euler’s theorem for a homogeneous-of-degree-one \(Q\). There is no residual. (In §1 we used \(\alpha + \beta < 1\), which leaves a pure profit of \((1-\alpha-\beta)Q\); that is a return to whatever fixed factor is causing the decreasing returns, and under free entry it is competed away.)

So the question is never whether the surplus exists, only how it splits. Labor’s share is

\[ s_L = \frac{P_L L}{Q} = \frac{\alpha L^{\rho}}{\alpha L^{\rho} + \beta T^{\rho}}. \]

Differentiate with respect to \(T\), or just read it off the expenditure ratio from §2.2. As cognition gets cheap and \(T\) grows:

• \(\sigma < 1\) (\(\rho<0\)): \(T^{\rho}\to 0\), so \(s_L \to 1\). Labor’s share rises toward everything.
• \(\sigma = 1\) (\(\rho = 0\)): \(s_L = \alpha/(\alpha+\beta)\), constant at any level of output. This is the Cobb-Douglas case, and it is Kaldor’s stylized fact.
• \(\sigma > 1\) (\(\rho>0\)): \(T^{\rho}\to\infty\), so \(s_L \to 0\). Labor’s share falls toward nothing.

Labor income is \(s_L Q\), a share times a level, so the two effects fight. Writing the change from an initial state,

\[ \frac{(s_L Q)_1}{(s_L Q)_0} = \underbrace{\frac{s_{L,1}}{s_{L,0}}}_{\text{share}} \cdot \underbrace{\frac{Q_1}{Q_0}}_{\text{level}}. \]

Workers are absolutely worse off only when the share falls by more than output rises. At \(Q_1/Q_0 = 6\) and \(s_{L,0} = 0.6\), that requires \(s_{L,1} < 0.1\). A halving of the share to \(0.3\) still leaves labor income at \(3\times\).

Two refinements matter for reading that number honestly.

Per-worker versus aggregate. The wage is \(w = \alpha Q/L\) under Cobb-Douglas, so \(w_1/w_0 = (Q_1/Q_0)/(L_1/L_0)\): output growth divided by growth in the physical workforce. The complements case assumes displaced cognitive workers retrain into physical work, which raises \(L\) and therefore dilutes the per-worker gain even as aggregate labor income rises.

Sectoral \(\sigma\), and a composition effect that compounds. If \(\sigma < 1\) in physical production and \(\sigma > 1\) in digital, the economy has both cases running at once. The within-sector effect raises \(s_L\) in the physical sector. Baumol’s result then adds a composition effect on top: with price-inelastic demand, the sector whose costs rise relative to the other absorbs a growing share of total expenditure — and that is the physical sector. Labor’s share of aggregate income is therefore pushed up twice, once within the sector and once by the sector’s growing weight.