The AI Production Function

Why the classical model breaks

Industrial economics ties output Y to physical capital K and human labor L: $Y = A K^\alpha L^\beta$. AI-native production strains this: an agent can draft, review, and ship work with effectively zero human hours. That would read as infinite productivity — a sign the labor slot is mislabeled, not that output is unbounded.

The AI production function

The paper keeps capital in its classical role as a means of production but re-grounds the inputs for agentic systems: $Y = A C^\alpha X^\beta W^\gamma$. The factors are renamed and re-measured; the multiplicative form is deliberately a first-order approximation the data can reject.

Three factors

Code Capital (C) — reusable, executable software an agent can call. Unlike physical capital it is non-rival (near-zero marginal cost to replicate), but it decays through dependency rot as platforms and APIs move.

Context Capital (X) — accumulated knowledge the agent retrieves: prompt templates, retrieval corpora, post-mortems. The C/X split uses a consumption-mode partition: an artifact is C if it is executed, X if it is retrieved for its content.

Computational Work (W) — the flow of tokens and cycles that fills the labor slot. Treating compute as labor, not merely a billed intermediate, recognizes that it now performs the inference and execution once reserved for humans.

Accumulation and decay

Code accumulates through investment $I_c$ and decays at rate $\delta_c$ driven by substrate churn: $C_{t+1} = (1-\delta_c) C_t + I_c$.

Context accumulates through learning at efficiency $\eta$ and decays at $\delta_x$ driven by task-mix drift: $X_{t+1} = X_t + \eta D_t - \delta_x X_t$.

The asymmetry is managerial: code must be maintained to keep working (bug fixes); context must be curated to stay relevant (pruning stale knowledge). The two decay rates are driven by observably different covariates.

Returns to scale and where surplus accrues

The scale index $s = \alpha+\beta+\gamma$ governs whether doubling all inputs more than doubles output. Because C and X are digital, a firm can deploy one well-built agent "brain" across n tasks — a replication economy that opens a large gap between creation cost and deployed value.

Surplus accrues mainly to the owners of C and X. Compute (W) is essential but often rentable as a commodity; a firm's specific history and specialized tools are monopolistic assets rivals cannot easily copy.

Empirical pilot

A pilot on a self-instrumented AI software organization (repository-week data) takes the framework to data. Its findings are reported as identification diagnostics, not structural estimates.

The C/X partition earns its keep against a lumped "AI capital"; context leads output in this window (X outweighs C); and common-activity shocks inflate naive estimates — a busy week raises compute, code, and output together, making the true driver hard to isolate.

Implications

A specificity gradient shapes firm boundaries: context is specific and immobile, so firms keep it in-house and vertically integrate agent memory; compute is a commodity, so the future may pair deep specialized "brains" with "borrowed muscle" rented from compute markets.

Accounting standards under-recognize context as an asset, risking a productivity paradox where the most valuable investments are invisible on the balance sheet. As agents take over tasks, labor productivity Y/L becomes deceptive; output per unit of compute (Y/W) and per unit of accumulated capital (Y/C, Y/X) become the honest measures.