Equations of Motion for an Economy: Capital Deepening, Technology, and Firm Survival

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 ${\eta }^{\ast }\equiv (w/\kappa +1/\tau )/(1-f_p)$ sets the minimum viable capital productivity, where $\eta =Y/K$ [yr${}^{-1}$] is capital productivity, $\kappa =K/L$ is capital per worker, $w$ is the wage rate, $\tau$ is the capital lifetime, and $f_p$ is the production tax share. Four coupled relaxation equations govern $\kappa$, $\eta$, the frontier productivity ${\eta }_{\mathrm{new}}$ of new investment, and the labor share $q\equiv w/y$, with the sandwich constraint ${\eta }^{\ast }\le {\eta }_{\mathrm{new}}\le \eta$ maintained as an exact invariant. The frontier equation separates two physically distinct channels: a structural cheapening channel ($\mu$, always active, drives ${\eta }_{\mathrm{new}}$ downward) and a productivity channel ($\varphi$, historically zero). Calibration against BEA 2-digit NAICS sector data (1998--2023) confirms $\varphi =0$ for all identifiable sectors over 25 years; the 75-year postwar record extends this finding across four capital lifetimes. A step $\varphi =0.01$\,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 $\eta (t)$ 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.

Includes Supplemental Material for this article, with BEA/BDS/CBP data pipelines, derivations, and sector-by-sector calibration figures