Bernoulli's $\log$ and Cramer's $\sqrt{\cdot}$ have diminishing marginal utility

Proved

MarginalUtility.bernoulli_cramer_diminishing

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).

Both measures of the desirability of a sum of money $x>0$ proposed in the 18th century have diminishing marginal utility on $(0,\infty)$ : Bernoulli's natural logarithm $U(x)=\log x$ and Cramer's square root $U(x)=\sqrt x$ are twice differentiable on $(0,\infty)$ with $U''(x)<0$ for every $x>0$ .

Preamble

import Mathlib
import Definitions.Def_MarginalUtility_Model

open Filter Topology Set

Formal statement

namespace MarginalUtility

theorem bernoulli_cramer_diminishing :
    HasDiminishingMarginalUtility Real.log (Ioi 0) ∧
      HasDiminishingMarginalUtility Real.sqrt (Ioi 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.

The statement is the conjunction of two facts, both about the open half-line $(0,\infty)$ .

The real natural logarithm $\log$ (as a function $\mathbb R\to\mathbb R$ ; its values for $x\le 0$ are irrelevant here) has diminishing marginal utility on $(0,\infty)$ : $\log$ is differentiable on $(0,\infty)$ , its derivative function is differentiable on $(0,\infty)$ , and the second derivative is strictly negative at every $x>0$ .
The real square root $\sqrt{\cdot}$ (as a function $\mathbb R\to\mathbb R$ , equal to $0$ for $x\le 0$ ) has diminishing marginal utility on $(0,\infty)$ in the same sense.

The point $0$ is excluded from the domain in both cases.

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

Bernoulli's log⁡\loglog and Cramer's ⋅\sqrt{\cdot}⋅​ have diminishing marginal utility

Bernoulli's $\log$ and Cramer's $\sqrt{\cdot}$ have diminishing marginal utility