Monetary budget identity: value of excess demand for goods plus excess demand for money is zero

Proved

SaysLaw.monetary_walras_law

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 also holds money: $m_i$ initially and $m'_i$ as planned.

If every agent satisfies the monetary budget constraint $\langle p,d_i\rangle+m'_i=\langle p,s_i\rangle+m_i$ , then

\langle p,z\rangle+\sum_{i\in\iota}(m'_i-m_i)=0 .

Thus the value of the aggregate excess demand for goods is exactly the negative of the aggregate excess demand for money: a deficiency of demand for goods in value terms is the same thing as an excess demand for money (hoarding), as in J. S. Mill's account of a general glut.

Preamble

import Mathlib
import Definitions.Def_SaysLaw_Model

Formal statement

namespace SaysLaw

theorem monetary_walras_law {ι G : Type*} [Fintype ι] [Fintype G]
    (p : G → ℝ) (supply demand : ι → G → ℝ) (m m' : ι → ℝ)
    (hbudget : MonetaryBudget p supply demand m m') :
    bundleValue p (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 $\iota,G$ be arbitrary finite types, $p:G\to\mathbb R$ any prices (no sign condition), $s,d:\iota\to G\to\mathbb R$ and $m,m':\iota\to\mathbb R$ arbitrary. Assume for every $i$ : $\sum_g p_g d_{i,g}+m'_i=\sum_g p_g s_{i,g}+m_i$ . Then, with $z_g=\sum_i(d_{i,g}-s_{i,g})$ ,

\sum_{g\in G}p_g z_g+\sum_{i\in\iota}(m'_i-m_i)=0 .

This is an exact equality of real numbers; it holds trivially (both terms $0$ ) when $\iota$ is empty.

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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