Paper: arXiv 2604.22995

Authors: Robert T. Nachtrieb

Abstract

We derive equations of motion for capital deepening in a competitive economy directly from accounting identities, without assuming a production function. A profit imperative $η^* \equiv (w/κ+ 1/τ)/(1-f_p)$ sets the minimum viable capital productivity, where $η= Y/K$ [yr$^{-1}$] is capital productivity, $κ= K/L$ is capital per worker, $w$ is the wage rate, $τ$ is the capital lifetime, and $f_p$ is the production tax share. Four coupled relaxation equations govern $κ$, $η$, the frontier productivity $η_{\rm new}$ of new investment, and the labor share $q \equiv w/y$, with the sandwich constraint $η^* \leq η_{\rm new} \leq η$ maintained as an exact invariant. The frontier equation separates two physically distinct channels: a structural cheapening channel ($μ$, always active, drives $η_{\rm new}$ downward) and a productivity channel ($φ$, historically zero). Calibration against BEA 2-digit NAICS sector data (1998–2023) confirms $φ= 0$ for all identifiable sectors over 25 years; the 75-year postwar record extends this finding across four capital lifetimes. A step $φ= 0.01$,yr$^{-1}$ – a 1%/yr improvement in new-capital productivity, modest but historically unprecedented – nearly doubles the aggregate growth rate within one capital lifetime, a falsifiable prediction with a precise observable signature: upward-curving $η(t)$ in BEA sector data. Firms near the zero-profit threshold have a cash martingale, predicting establishment exit rate $\sim t^{-1/2}$; convolved with the Zipf firm-size distribution~\cite{WP}, this yields firm exit rate $\sim t^{-1/2}!\log t$ with apparent exponent $b = 0.295 \pm 0.03$, confirmed against BDS data with no free parameters.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper demonstrates high mathematical sophistication by deriving a four-equation dynamic system from accounting identities while simultaneously providing rigorous empirical validation using BEA national accounts and BDS firm-level data.

Research Flowchart

  flowchart TD
    A[<b>Research Goal:</b> Deriving Equations of Motion for Capital Deepening] --> B[<b>Method:</b> Accounting Identities & Relaxation Equations]
    B --> C[<b>Inputs:</b> BEA 2-digit NAICS & BDS Firm Data]
    C --> D[<b>Dynamics:</b> Sandwich Constraint & Profit Imperative $η^*$]
    D --> E[<b>Process:</b> Calibration of Structural ($μ$) vs Productivity ($φ$) Channels]
    E --> F([<b>Outcomes:</b> Historical $φ=0$ & Falsifiable $t^{-1/2} \log t$ Exit Law])
    
    style F fill:#f9f,stroke:#333,stroke-width:2px

Cited References (1)

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