Two equivalent forms of comparative advantage in cloth

Proved

ComparativeAdvantage.comparative_advantage_iff

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

comparative-advantageeconomicsinternational-trade

Let Home have unit labour requirements $a_{LC},a_{LW}>0$ and Foreign $a'_{LC},a'_{LW}>0$ . Home is relatively more productive than Foreign in cloth versus wine if and only if Home has a lower opportunity cost of cloth in terms of wine:

\frac{a_{LC}}{a'_{LC}}<\frac{a_{LW}}{a'_{LW}}\iff\frac{a_{LC}}{a_{LW}}<\frac{a'_{LC}}{a'_{LW}}.

This lets the comparative-advantage assumption be used in whichever form is convenient.

Preamble

import Mathlib
import Definitions.Def_ComparativeAdvantage_Model

Formal statement

namespace ComparativeAdvantage

theorem comparative_advantage_iff (h f : Country) :
    h.aLC / f.aLC < h.aLW / f.aLW ↔ HasComparativeAdvantageInCloth h f := 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 pair of countries $h=(L,a_{LC},a_{LW})$ and $f=(L',a'_{LC},a'_{LW})$ , the statement asserts the equivalence

\frac{a_{LC}}{a'_{LC}}<\frac{a_{LW}}{a'_{LW}}\iff\frac{a_{LC}}{a_{LW}}<\frac{a'_{LC}}{a'_{LW}}.

The labour forces $L,L'$ play no role. There are no further hypotheses.

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).

View graph