Cost-Effectiveness, Insurance Solvency, and Health Capital: A Joint Incompatibility Theorem and Dynamic Research Programme

This paper examines the mathematical architecture of health economics through four analytical traditions—Arrow’s welfare economics of medical care, Grossman’s health capital model, Garber’s cost-effectiveness analysis, and Zweifel’s actuarial insurance theory—and derives three original results. C1 (Joint Incompatibility Theorem): under empirically standard regularity conditions, no health system simultaneously satisfies the Garber–Phelps WTP-based cost-effectiveness criterion, Zweifel’s actuarial solvency condition under community rating, and the Pauly–Zweifel optimal coinsurance rule; the proof proceeds via three lemmas, each isolating a separate incompatibility. C2 (Dynamic Cost-Effectiveness Threshold): the cost-effectiveness threshold λ*(t) is derived as an endogenous co-state variable of a unified dynamic optimisation, and shown to equal the present value of future marginal health benefits discounted at the user cost of health capital ρ + δ(t). C3 (Demographic Suboptimality of Uniform Thresholds): as a corollary, uniform thresholds are dynamically inconsistent with any population whose age-specific depreciation rate is increasing, with the direction of the depreciation channel being theoretically determinate. Together, C1 converts a collection of separately noted tensions into a single structural result, while C2 and C3 convert that result into a constructive research programme for dynamic health economic evaluation.