Positive marginal utility raises utility; negative marginal utility lowers it

Proved

MarginalUtility.utility_monotone_of_marginal_sign

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 and let $U$ be differentiable on $D$ . (a) If the marginal utility is positive at every point of $D$ , then every increase of consumption within $D$ strictly increases utility: $U$ is strictly increasing on $D$ . (b) If the marginal utility is negative at every point of $D$ , then every increase of consumption within $D$ strictly decreases utility: $U$ is strictly decreasing on $D$ .

Preamble

import Mathlib
import Definitions.Def_MarginalUtility_Model

open Filter Topology Set

Formal statement

namespace MarginalUtility

theorem utility_monotone_of_marginal_sign (U : ℝ → ℝ) (D : Set ℝ)
    (hDo : IsOpen D) (hDc : Convex ℝ D) (hU : DifferentiableOn ℝ U D) :
    ((∀ g ∈ D, 0 < marginalUtility U g) → StrictMonoOn U D) ∧
      ((∀ g ∈ D, marginalUtility U g < 0) → StrictAntiOn U D) := 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 $D\subseteq\mathbb R$ , with $D$ open and convex (so $D$ is an open interval, possibly empty or unbounded), and $U$ differentiable on $D$ . Write $\mathrm{MU}_U(g)$ for the derivative of $U$ at $g$ . The conclusion is the conjunction of two implications:

if $\mathrm{MU}_U(g)>0$ for every $g\in D$ , then for all $a,b\in D$ with $a<b$ we have $U(a)<U(b)$ ;
if $\mathrm{MU}_U(g)<0$ for every $g\in D$ , then for all $a,b\in D$ with $a<b$ we have $U(a)>U(b)$ .

If $D$ is empty, both conclusions hold trivially. Zero marginal utility is not addressed.

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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