Persistence and Pressure: How capital lifetime determines the depletion rate of the resources that compose it: Evidence from 28 natural resources

If a product lasts half as long, twice as many must be manufactured to maintain the same functional stock. Each manufacturing cycle extracts materials from finite geological stocks. Therefore, the lifetime of capital has a direct, arithmetically necessary connection to the depletion rate of the resources that compose it. This paper formalizes and tests that connection. We derive the Φ–p identity: under stationary capital stock, the pressure on a resource's stock is inversely proportional to the lifetime of capital built with it (p ∝ 1/L). We test this prediction using 28 natural resources for which we compute p from primary sources (USGS, IEA, FAO, NASA GRACE). Results reveal a taxonomy of five Φ–p regimes. Combined with prior findings that GDP per capita correlates with the inverse of mean capital persistence and that declining product lifetimes impose a 51% hidden surcharge on household budgets, this result completes a three-level argument: richer economies rotate capital faster (macro), households pay the cost in working hours (micro), and the planet pays in depleted stocks (resource-level).