The Economic Singularity: Core Mathematical Model

Анотація

This paper develops a dynamical systems model of an economy with recursively self-improving artificial intelligence and financialization. The model features quadratic self-amplification in both AI capability ($\lambda A^2$) and financial capital ($\gamma_F K_f^2$), coupled through investment flows. It is shown that under mild conditions, the system exhibits a finite-time singularity where AI capability, AI capital, and financial capital diverge. Near the singularity, the wealth ratio between capital owners and workers diverges super-exponentially, with financialization amplifying the exponent by a factor $\gamma_F/\eta$. Introducing taxes on AI returns ($\tau_{ai}$) and financial gains ($\tau_f$) yields three distinct long-run regimes: low-tax (extreme inequality), moderate-tax (stable mixed economy), and high-tax (post-scarcity with universal basic income), separated by critical thresholds derived from workers' budget constraint. Finally, a policy irreversibility result is established: a critical time exists before the singularity after which redistribution becomes politically impossible, as wealth concentration renders feasible tax rates vanishingly small. The results highlight the urgency of early intervention in AI-driven economies.

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