Concave utility: each equal gain adds less than the previous one

Proved

MarginalUtility.successive_gains_decrease_of_strictConcaveOn

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 $U$ be strictly concave on a convex set $D\subseteq\mathbb R$ (a concave relation between objective gains and subjective value). If $h>0$ and both $x$ and $x+2h$ lie in $D$ , then the second gain of size $h$ produces a strictly smaller increase in subjective value than the first:

U(x+2h)-U(x+h)<U(x+h)-U(x).

Preamble

import Mathlib
import Definitions.Def_MarginalUtility_Model

open Filter Topology Set

Formal statement

namespace MarginalUtility

theorem successive_gains_decrease_of_strictConcaveOn (U : ℝ → ℝ) (D : Set ℝ)
    (hU : StrictConcaveOn ℝ D U) (x h : ℝ) (hh : 0 < h)
    (hx : x ∈ D) (hx2 : x + 2 * h ∈ D) :
    utilityChange U (x + h) (x + 2 * h) < utilityChange U x (x + h) := 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 $D\subseteq\mathbb R$ , and assume $U$ is strictly concave on $D$ (this includes that $D$ is convex; no openness, differentiability or continuity is assumed). Let $x,h\in\mathbb R$ with $h>0$ , $x\in D$ and $x+2h\in D$ . The conclusion, with the defined utility change $\Delta U(s_1,s_2)=U(s_2)-U(s_1)$ , is

U(x+2h)-U(x+h)\;<\;U(x+h)-U(x).

The point $x+h$ is not assumed to be in $D$ separately. The step $h$ is an arbitrary positive real, not necessarily $1$ .

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