Say's law: there can never be a general glut

Proved

SaysLaw.no_general_glut

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 $g$ .

Assume there is at least one good, all prices are strictly positive ( $p_g>0$ ), and every agent obeys Say's budget principle — products are paid for with products, so each agent spends the full value of what he sells on other products: $\langle p,d_i\rangle=\langle p,s_i\rangle$ . Then there is no general glut:

\neg\big(\forall g\in G,\ z_g<0\big),

that is, some good $g$ has $z_g\ge 0$ .

This is the modern formulation of Say's law quoted in the article: "there can never be a general glut".

Preamble

import Mathlib
import Definitions.Def_SaysLaw_Model

Formal statement

namespace SaysLaw

theorem no_general_glut {ι G : Type*} [Fintype ι] [Fintype G] [Nonempty G]
    (p : G → ℝ) (hp : ∀ g, 0 < p g) (supply demand : ι → G → ℝ)
    (hsay : SaysBudget p supply demand) :
    ¬ GeneralGlut (excessDemand supply demand) := 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 of agents (possibly empty), $G$ a finite type of goods assumed nonempty, $p:G\to\mathbb R$ with $p_g>0$ for every $g$ , and $s,d:\iota\to G\to\mathbb R$ arbitrary real arrays. Assume that for every agent $i$ ,

\sum_{g\in G}p_g\,d_{i,g}=\sum_{g\in G}p_g\,s_{i,g}.

Then it is not the case that $\sum_{i\in\iota}(d_{i,g}-s_{i,g})<0$ for every $g\in G$ ; equivalently, there is at least one good $g$ with $\sum_i(d_{i,g}-s_{i,g})\ge 0$ . Without the nonemptiness of $G$ the conclusion would be false (the condition "every $z_g<0$ " holds vacuously for empty $G$ ); this case is excluded by the hypothesis.

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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