Marginal utility as the limit of $\Delta U/\Delta g$

Proved

MarginalUtility.marginalUtility_eq_limit_of_ratio

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 $U:\mathbb R\to\mathbb R$ be differentiable at $g$ . Then the ratio of the change in utility to the size of the change in $g$ converges to the marginal utility:

\lim_{\Delta g\to 0,\ \Delta g\neq 0}\frac{U(g+\Delta g)-U(g)}{\Delta g}=\frac{\partial U}{\partial g}(g).

Preamble

import Mathlib
import Definitions.Def_MarginalUtility_Model

open Filter Topology Set

Formal statement

namespace MarginalUtility

theorem marginalUtility_eq_limit_of_ratio (U : ℝ → ℝ) (g : ℝ)
    (hU : DifferentiableAt ℝ U g) :
    Tendsto (fun Δg : ℝ => utilityChange U g (g + Δg) / Δg) (𝓝[≠] 0)
      (𝓝 (marginalUtility U g)) := 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 $U:\mathbb R\to\mathbb R$ be any function and $g\in\mathbb R$ any point, and assume $U$ is differentiable at $g$ . The claim is that the function

\Delta g\;\longmapsto\;\frac{U(g+\Delta g)-U(g)}{\Delta g}

tends to $\mathrm{MU}_U(g)$ (defined as the derivative of $U$ at $g$ ) as $\Delta g$ tends to $0$ through nonzero values (the punctured neighbourhood filter of $0$ ). The numerator is the defined utility change from state $g$ to state $g+\Delta g$ . The value at $\Delta g=0$ itself is excluded from the limit. Only differentiability at the single point $g$ is assumed; no domain restriction appears.

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).

View graph

Marginal utility as the limit of ΔU/Δg\Delta U/\Delta gΔU/Δg

Marginal utility as the limit of $\Delta U/\Delta g$