MINIMUM
VIABLE SCALE

Javier Milei / Demian Reidel

Minimum
Viable Scale.

Bellman Barriers and the Economics of Survival

An economy can replace its capital.
And still choose to run it down.

Three results on which capital is worth preserving, its value at the threshold, and how to prove survival.

Research paper Plain language paper
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Research paper: PDF · 58 pages · English
Preserving capital or running it down From S, two plans give the highest welfare: one runs capital down and the other preserves a productive base. S lies above the physical minimum κ. The curves show these paths schematically. Preserving capital does not require it to settle at a fixed level. Time moves to the right. The drawing is not to scale. Survival Extinction Kt t S κ Schematic paths
S Economic thresholdκ Physical replacement threshold
Working paper / October 2026Economic dynamics · Increasing returnsScroll to explore ↓
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01 · Minimum viable scaleTheorem 1 · p. 6 ↗

Being able to preserve capital
is not enough.

Maintaining equipment means giving up consumption and leisure. That sacrifice must be worthwhile. In this model, with sufficient productivity and patience, we prove that a minimum viable scale, S, separates extinction from survival.

The model. With increasing returns, expanding capital and labor by the same proportion raises output more than proportionally. A planner chooses consumption, work and saving to maximize welfare from consumption and leisure.

Three futures for capitalChoose a starting point
Capital KTime Sκ Extinction Survival Productive region
K₀ < S

Capital runs down.

Every optimal plan drives capital and consumption toward zero.

K₀ = S

Two futures. The same welfare.

Depleting capital or preserving a productive base: both outcomes are optimal.

K₀ > S

Production survives.

Every optimal plan reaches a productive region and stays there, with capital and consumption bounded away from zero.

κ < S

Between κ and S, replacing equipment is feasible. Yet depleting it is optimal. The economic minimum exceeds the lower physical maintenance threshold.

The curves illustrate outcomes; they are not computed paths. The result classifies every optimal plan when work and saving are chosen together. It does not require capital to converge to a constant stock. Conditions and proof ↗

02 · The value of capitalTheorem 2 · pp. 7–8 ↗

Welfare does not jump.
Its slope does.

V(K) is the greatest welfare the economy can achieve over time when it starts with capital K. At S, preserving capital and depleting it deliver the same welfare. But additional capital is worth more on the survival side.

Crossing S makes extra capital more valuable Welfare does not jump at S. Just to the right, each extra unit of capital adds more welfare than just to the left. The lines show the slopes on each side of S and meet at V(S). This is a comparison near the threshold, not a plot of the whole value function. ΔV− ΔV+ ΔK ΔK V(K) V(S) S K Slopes at S
The same extra capital. A different welfare gain. The lines show local slopes, not the entire function.
V′+(S) > V′−(S)

Welfare is continuous, but not differentiable at S. Equal welfare does not by itself imply this kink. We prove that productive plans value additional capital more, even when either outcome has several optimal plans. Productive choices save more and work more.

03 · The new methodTheorems 5–6 · pp. 15–17 ↗

A new
mathematical tool

Our method rules out paths that cannot be optimal, without solving the full model. We call it First Passage Bellman Barriers.

For each group of paths, we calculate a welfare ceiling: none can deliver more. If a feasible plan from the same starting point guarantees more than that ceiling, the group is ruled out. One example is plans that always keep little capital. The comparison covers their entire future, including recovery.

We use this comparison to prove that every optimal plan reaches a productive region and stays there. We combine first passage analysis from physics with methods for bounding option values in finance.

Particle trajectories and first exit from a region An artistic composition, without experimental data. One trajectory starts at X(0), inside the open observation region Omega, and first crosses its boundary at the marked point. Tau of Omega is that first exit time. The force formula shows only its magnetic component. The curves are illustrative, not a calculated physical solution or an application of the paper to particles. First exit τΩ = inf {t ≥ 0 : X(t) ∉ Ω} Ω ∂Ω τΩ X(0) FB = q(v × B) Magnetic force
An artistic illustration of a particle collision in an accelerator. The first crossing of a region’s boundary inspires the economic comparison: assess a departure together with everything that follows.

The rest is in the paper. The full model, capital losses, extinction laws and barrier construction, with their assumptions and proofs.

The model · p. 3 ↗Losses · p. 10 ↗Extinction · p. 12 ↗
The full argumentWorking paper · October 2026
First page of the PDF: title, authors and abstract of Minimum Viable Scale.
First page of the document

Minimum
Viable Scale.

Bellman Barriers and the Economics of Survival

Javier Milei · Demian Reidel

Read and download the PDFPDF · 58 pages · English

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Milei, Javier and Demian Reidel (2026). Minimum Viable Scale: Bellman Barriers and the Economics of Survival. Working paper, October.

Assumptions, proofs and references in the full paper.

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