Zero marginal utility: total utility is maximal, and marginal utility is negative beyond

Proved

MarginalUtility.satiation_point_is_maximum

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

concavityeconomicsmarginal-utilitymicroeconomics

Throughout, a utility function of one quantified variable is a map $U:\mathbb R\to\mathbb R$ (all other variables held fixed, ceteris paribus); $\Delta U(s_1\to s_2)=U(s_2)-U(s_1)$ is the change in utility from state $s_1$ to state $s_2$ ; the marginal utility at $g$ is $\partial U/\partial g=U'(g)$ ; and $U$ has diminishing marginal utility on a set $D\subseteq\mathbb R$ when $U$ and $U'$ are differentiable on $D$ and $\partial^2U/\partial g^2<0$ at every point of $D$ . An open interval $D$ below means an open convex subset of $\mathbb R$ (possibly empty, possibly unbounded).

Let $D\subseteq\mathbb R$ be an open interval, let $U$ have diminishing marginal utility on $D$ , and suppose the marginal utility vanishes at some $x_0\in D$ . Then total utility is at its maximum at $x_0$ , i.e. $U(x)\le U(x_0)$ for all $x\in D$ , and beyond that point marginal utility is negative: $\partial U/\partial g\,(x)<0$ for every $x\in D$ with $x>x_0$ .

Preamble

import Mathlib
import Definitions.Def_MarginalUtility_Model

open Filter Topology Set

Formal statement

namespace MarginalUtility

theorem satiation_point_is_maximum (U : ℝ → ℝ) (D : Set ℝ)
    (hDo : IsOpen D) (hDc : Convex ℝ D) (hU : HasDiminishingMarginalUtility U D)
    (x₀ : ℝ) (hx₀ : x₀ ∈ D) (hzero : marginalUtility U x₀ = 0) :
    IsMaxOn U D x₀ ∧ ∀ x ∈ D, x₀ < x → marginalUtility U x < 0 := by sorry

end MarginalUtility

Source

Wikipedia, "Marginal utility" (PDF export supplied with the request); https://en.wikipedia.org/wiki/Marginal_utility

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 $U:\mathbb R\to\mathbb R$ and let $D\subseteq\mathbb R$ be open and convex. Assume $U$ has diminishing marginal utility on $D$ ( $U$ and $U'$ differentiable on $D$ , $(U')'<0$ on $D$ ). Let $x_0\in D$ with $\mathrm{MU}_U(x_0)=U'(x_0)=0$ . The conclusion has two parts:

$x_0$ is a maximum point of $U$ on $D$ : $U(x)\le U(x_0)$ for every $x\in D$ ;
for every $x\in D$ with $x>x_0$ , $U'(x)<0$ .

The statement does not assert that such an $x_0$ exists; it is a 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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