Beyond Reswitching: A Rank Obstruction to Scalar Capital Aggregation
Year: 2026
Type: article
Abstract: The Cambridge capital controversy is usually framed as a problem of reswitching, but this paper argues that the more primitive local obstruction to scalar capital aggregation is rank. Let z be the hidden heterogeneous-capital state. If an exact scalar aggregate K(z) supports an aggregate production relation Y = F(K, L) with competitive prices r = ∂F/∂K and w = ∂F/∂L, then for fixed labor the observable map z ↦ (Y, r, w) has Jacobian rank at most one; conversely, where that Jacobian has constant rank one, the constant-rank theorem yields a local scalar factorization. Exact scalar aggregation is therefore locally characterized, on regular sets, by one-dimensionality of the observable manifold rather than by wage-curve crossings, and a rank-two observable map cannot be repaired by renaming capital. The obstruction survives the usual escape routes, extending to locally Lipschitz aggregates almost everywhere, to Banach- and measure-valued capital via Fréchet differentiation, and to nonlinear accumulation through the observability codistribution. The paper constructs no-reswitching Sraffian economies whose audited bundle values are rank two, proving reswitching sufficient but not necessary; shows that a scalar capital stock must be a one-dimensional observable state, which unequal-depreciation Solow–Swan dynamics violate; and quantifies the failure by the second singular value of the observable Jacobian, or its invariant canonical-correlation form, which is the minimum first-order error of any scalar approximation and a target for measurement design. A positive value falsifies exact scalar capital, while a small one may still justify scalar approximation within explicit bounds, with conditional applications to growth accounting, production networks, and a two-capital climate model.
Author: Fathi, Kevin
Language: English
Topic: Economic Theory and Policy
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7138085682
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