AI시대 경제윤리

Ethics in the Geometry of Capital–Labor

“Where Geometry Permits Freedom”

The equilibrium of an economic system has long been interpreted as a signal of resolution—a resting point of optimization, balance, or social efficiency. Yet our analysis reveals a different logic. In closed dynamical systems, especially those that model capital and labor under conditions of automation, equilibrium is not justice. It is a structural consequence.

The share-based dynamic system explored in this study, governed by:

\[ \frac{dc}{dt} = c(a_{2,2} - a_{2,4} c), \]

does not depict a market adjusting toward balance. It describes a phase-space curvature, a regime in which autonomous capital growth confronts internal saturation. When \(a_{2,1}\)—the coupling coefficient from labor to capital—is effectively zero, the system becomes structurally mutating: labor vanishes, and capital becomes decoupled, autopoietic, and unsustainable. Importantly, this is not a theoretical boundary case. It is the empirically dominant configuration across top 0.1%, 1%, and 10% U.S. wealth share trajectories (1989–2024).

Our simulations show that when labor ceases to contribute to capital formation—even under high automation growth rates (\(a_{2,2} > a_{2,3}\))—the entire system decays. Capital itself collapses. Dominance without anchoring becomes self-destructive. A capitalist system without labor is a closed loop of extraction without reproduction: it does not evolve—it consumes itself.

In contrast, calibrating \(a_{2,1}\) to a small but positive value (e.g., \(0.0007\)) transforms the trajectory. Capital shares stabilize at empirically plausible levels. The geometry of the system shifts. This demonstrates that structural persistence is governed not by equilibrium conditions, but by bifurcation thresholds in the feedback network. A minute coupling term becomes the difference between long-run stability and structural extinction.

This leads to a broader ethical insight: freedom in such systems emerges only where curvature permits deviation. Just as Milton Friedman once argued that economic freedom undergirds political liberty, we propose a parallel structural logic:
> Curvature is the precondition for meaningful intervention.

Where the system is flat, efforts dissipate. But where curvature is high—where local changes generate divergent long-term behavior—intervention becomes not only possible, but necessary.

Ethics in dynamical systems is not a matter of ex post redistribution. It is a matter of anticipatory geometry. If the system’s structure ensures the long-run disappearance of labor, then justice cannot be restored through compensatory policies. Once a group is eliminated from phase space, there is no feasible re-entry. Normative analysis must occur before the attractor forms.

This view casts doubt on the methodological neutrality of mainstream economic models. Formalism often equates stability with fairness. But as our system shows, stability may encode structural injustice. A model that predicts the disappearance of labor and yet interprets the result as an equilibrium has not merely erred—it has misunderstood the purpose of modeling.

The model does not fail because reality is messy.
The model fails because it permits no mess, and thus no future for those outside its logic.

A just system, in this framing, is not one that converges. It is one that preserves structural space for all agents to persist. It is not a system that assumes meritocratic outcomes, but one that resists the topological elimination of any group.

We began with a structural question:
> What happens when capital reproduces without labor?

We conclude with a geometric truth:
> A system that eliminates its agents is not efficient—it is pathologically optimized.

The role of mathematics is not to prescribe solutions.
Mathematics reveals, but it does not repair.

It is our ethics that prescribe; it is our courage that cures.
Mathematics illuminates which futures are geometrically open—and which are already foreclosed.

Let us not mistake convergence for harmony.
Let us read equilibrium not as justice, but as a possible injustice.

And let us recover, from the differential equations of capital and labor,
an ethics that is not imposed from above,
but one that must emerge from within the structure of the system itself.