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 $\eta^* \equiv (w/\kappa + 1/\tau)/(1-f_p)$ 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 $\eta_{\rm new}$ of new investment, and the labor share $q \equiv w/y$, with the sandwich constraint $\eta^* \leq \eta_{\rm new} \leq \eta$ maintained as an exact invariant. The frontier equation separates two physically distinct channels: a structural cheapening channel ($\mu$, always active, drives $\eta_{\rm new}$ downward) and a productivity channel ($\phi$, historically zero). Calibration against BEA 2-digit NAICS sector data (1998--2023) confirms $\phi = 0$ for all identifiable sectors over 25 years; the 75-year postwar record extends this finding across four capital lifetimes. A step $\phi = 0.01$\,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.