Mill: no excess of all other commodities together with an excess of money

Proved

SaysLaw.mill_no_glut_with_excess_money

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

economicsgeneral-equilibriummacroeconomicssays-law

Throughout, $\iota$ is a finite set of agents and $G$ a finite set of goods; $p=(p_g)_{g\in G}$ is a real price vector; $s_i=(s_{i,g})_g$ and $d_i=(d_{i,g})_g$ are real vectors giving the quantities agent $i$ brings to market (supplies) and plans to buy (demands); $\langle p,x\rangle=\sum_{g\in G}p_g x_g$ is the market value of a bundle $x$ ; and $z_g=\sum_{i\in\iota}(d_{i,g}-s_{i,g})$ is the aggregate excess demand for good $g$ . Each agent holds money $m_i$ initially and plans to hold $m'_i$ .

Assume there is at least one good, all prices are strictly positive, and every agent satisfies the monetary budget constraint $\langle p,d_i\rangle+m'_i=\langle p,s_i\rangle+m_i$ . Then it is impossible that simultaneously

every good is in excess supply ( $z_g<0$ for all $g\in G$ ), and
money is not in excess demand ( $\sum_i(m'_i-m_i)\le 0$ ).

This is Mill's rescue of the no-general-glut principle by counting money as one of the commodities.

Preamble

import Mathlib
import Definitions.Def_SaysLaw_Model

Formal statement

namespace SaysLaw

theorem mill_no_glut_with_excess_money {ι G : Type*} [Fintype ι] [Fintype G] [Nonempty G]
    (p : G → ℝ) (hp : ∀ g, 0 < p g) (supply demand : ι → G → ℝ) (m m' : ι → ℝ)
    (hbudget : MonetaryBudget p supply demand m m') :
    ¬ (GeneralGlut (excessDemand supply demand) ∧ excessMoneyDemand m m' ≤ 0) := by sorry

end SaysLaw

Source

Wikipedia, "Say's law" (PDF export supplied with the request); https://en.wikipedia.org/wiki/Say%27s_law

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.

Let $\iota$ be an arbitrary finite type, $G$ a finite type assumed nonempty, $p:G\to\mathbb R$ with $p_g>0$ for every $g$ , and $s,d:\iota\to G\to\mathbb R$ , $m,m':\iota\to\mathbb R$ arbitrary. Assume for every $i$ : $\sum_g p_g d_{i,g}+m'_i=\sum_g p_g s_{i,g}+m_i$ . Then it is not the case that both (a) $z_g<0$ for every $g\in G$ , where $z_g=\sum_i(d_{i,g}-s_{i,g})$ , and (b) $\sum_i(m'_i-m_i)\le 0$ . Equivalently: if every $z_g<0$ then $\sum_i(m'_i-m_i)>0$ . If $\iota$ is empty, (a) fails because $z\equiv 0$ .

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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