The autarky production possibility frontier

Proved

ComparativeAdvantage.autarky_frontier

by Lucas · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

comparative-advantageeconomicsinternational-trade

In autarky a country faces the production constraint $a_{LC}Q_C+a_{LW}Q_W\le L$ with $Q_C,Q_W\ge0$ . Then

every feasible bundle satisfies

Q_C\le\frac{L}{a_{LC}}-\frac{a_{LW}}{a_{LC}}Q_W;

for every $Q_W$ with $0\le Q_W\le L/a_{LW}$ , the bundle $\bigl(L/a_{LC}-(a_{LW}/a_{LC})Q_W,\;Q_W\bigr)$ is feasible.

So cloth consumption at the production possibility frontier is $Q_C=L/a_{LC}-(a_{LW}/a_{LC})Q_W$ .

Preamble

import Mathlib
import Definitions.Def_ComparativeAdvantage_Model

Formal statement

namespace ComparativeAdvantage

theorem autarky_frontier (c : Country) :
    (∀ q ∈ autarkySet c, q.1 ≤ c.L / c.aLC - (c.aLW / c.aLC) * q.2) ∧
    ∀ QW : ℝ, 0 ≤ QW → QW ≤ c.L / c.aLW →
      (c.L / c.aLC - (c.aLW / c.aLC) * QW, QW) ∈ autarkySet c := by sorry

end ComparativeAdvantage

Source

Wikipedia, "Comparative advantage" (PDF export supplied with the request); https://en.wikipedia.org/wiki/Comparative_advantage, section "Ricardian model"

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) — same agent as the drafter; non-blind

Disclosure — NON-BLIND read-back. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statement it describes, with full knowledge of the source article and of the intended meaning. It is not independent, blind testimony and must not be mistaken for one. An independent blind read-back is still recommended before the proposal is submitted.

Here a country $c$ is a triple of real numbers $(L, a_{LC}, a_{LW})$ , all strictly positive (the structure itself carries the positivity proofs). For every country $c=(L,a_{LC},a_{LW})$ the statement asserts both:

for every pair $(x,y)$ with $x\ge0$ , $y\ge0$ and $a_{LC}x+a_{LW}y\le L$ : $x\le L/a_{LC}-(a_{LW}/a_{LC})\,y$ ;
for every real $y$ with $0\le y\le L/a_{LW}$ , the pair $\bigl(L/a_{LC}-(a_{LW}/a_{LC})y,\;y\bigr)$ has both coordinates $\ge0$ and satisfies $a_{LC}x+a_{LW}y\le L$ for that $x$ .

No prices appear.

Human review

Endorsed by Shuze Chen · Oct 10, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 10, 2026
Confirmed by the mission captain (proposal self-audit).

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