Two Math Problems, One Civilization: Institutional Arithmetic of Currency, Credits and Human Survival

Abstract: This paper presents a minimalist agent-based model of a closed economic system—10 individuals stratified into three income tiers, producing and consuming 10 units of daily bread with a fixed money supply of 90 units. Two distinct accounting rules are tested. In the first scenario, a single-currency system inevitably collapses on day 28, as the bottom tier exhausts its cash due to structural income-expenditure deficits. The upper tiers accumulate surplus money that ceases to circulate, causing aggregate demand failure despite unchanged production capacity. In the second scenario, a "mirror credit" mechanism is introduced: income receipts simultaneously debit an equivalent amount of complementary credits, while consumption expenditures credit them. Credits are redeemable for bread but not convertible to cash, and hybrid cash-credit payments are permitted. Under this rule, the bottom tier achieves a steady-state equilibrium (8.67 yuan cash + 0.33 credits per person) that perpetuates indefinitely—without money creation, debt issuance, or external expansion. The mathematical proof rests on a conservation law: for every individual, cash + credits ≡ initial endowment (9 yuan). The stability condition requires only that the bottom tier's cash expenditure rate matches its income rate, with credits covering the residual purchasing power gap. The paper then maps these two models onto real-world institutional choices: fiat inflation vs. credit hedging, debt-based stimulus vs. zero-sum accounting, territorial expansion vs. technological iteration. It argues that in a closed global system—constrained by nuclear deterrence and geographical limits—the single-currency framework is arithmetically unstable, while a dual-record accounting system offers a sustainable alternative without abolishing hierarchy, property rights, or competitive incentives. Keywords: monetary systems, complementary currencies, economic sustainability, distributional macroeconomics, agent-based modeling, institutional design