The Depreciation‒Demurrage Equivalence: A User Cost and Self- Financing Theory of Health Capital and Money

This paper formalises, through the concept of user cost, the structural isomorphism between the depreciation rate δ of health capital in Grossman (1972) and the demurrage rate µ in the Gesell (1916) tradition. This isomorphism is termed the Depreciation–Demurrage Equivalence; the δ-µ matching principle is used as a derivative term denoting its normative content. The central result, Proposition 1, establishes that, under µ = δ, the canonical user-cost rate components r + δ (Grossman/Jorgenson type) and r + µ coincide to first order—and that, under a discrete-time gross-up normalization, substituting µ = δ yields the conditional identity (r + µ)/(1 − µ) = (r + δ)/(1 − δ) (see §5.1 of the manuscript). This delivers coherence in the time structure of the local arbitrage condition for health investment; it does not constitute a complete proof of dynamic general-equilibrium efficiency, since establishing the absence of allocative distortions in general equilibrium requires an explicit model of household problems, firm behaviour, exchange markets, the government budget, price determination, and income effects, an analysis that is deferred to a companion paper. Under µ = δ, the steady-state balance M* = p_I · H* and the self-financing condition µ · M* = F (an accounting equality between the demurrage flow and the new-issuance flow) hold (Theorem 5, Corollary 5.1). This paper formalises this structure as a six-axiom system together with associated propositions, theorems, and conjectures. The principle provides a theoretical foundation for the self-financing design of healthcare financing in the nominal accounting sense: a demurrage-currency arrangement, by aligning the duration of the benefit period of health-care provision (the depreciation rate of health capital) with that of the financing repayment horizon (the demurrage rate of currency), offers a formal accounting route for aligning the time structure of healthcare benefits and financing burdens, and a path for reconciling the temporal mismatch in intergenerational burden. Because demurrage currency carries an implicit-taxation function through its demurrage flow, however, the question of who ultimately bears the real burden of demurrage (including whether Ricardian equivalence holds in the relevant general-equilibrium fiscal analysis) lies beyond the partial-equilibrium framework adopted here and is deferred to a companion paper. The theoretical contribution of this study is concentrated in three points: (i) the structural isomorphism between health-capital theory and demurrage-currency theory (Depreciation–Demurrage Equivalence), the user cost coherence under µ = δ (Proposition 1), and the resulting derivation of the self-financing condition µ · M* = F; (ii) within the present δ-µ matching framework, the Friedman-rule benchmark µ* = 0 can be reinterpreted as the limiting case in which the depreciation-matching channel of Axiom IV is inactive (δ_j = 0 for all j) and externalities are absent; and (iii) under premises (Q-1)–(Q-6), a conditional welfare superiority hypothesis concerning the dual-currency regime that arises from the functional separation between consumption-purpose demurrage currency and accumulation-purpose conventional currency (Proposition 11; the complete welfare-comparison theorem is deferred to a companion paper). The Depreciation–Demurrage Equivalence also signals several directions for future research across multiple dimensions of policy design, including the normative design of demurrage-currency basic income (§7.2.2 of the manuscript) and possible applications to Piketty's r > g proposition (§7.2.3). Monetary-side implementation of negative natural rates of interest, reinterpretation within the Fiscal Theory of the Price Level (FTPL) valuation equation as debt with a self-decay term, two-dimensional optimisation of helicopter money over scale and recovery speed, integrated time-structure asset-liability management for the government and central bank, the multi-sector generalisation, and a general framework for currency with time structure are all deferred to a companion paper (Fujii, in preparation). This paper positions itself as an attempt at a theoretical bridge across four lineages: health economics, monetary economics, complementary-currency theory, and the universal-basic-income literature.