Case Study 2 — Concavity in Behavioral Economics: Diminishing Marginal Utility

Field: Economics, behavioral economics, decision theory Calculus used: Concavity and the second derivative (§9.6), critical points and the second derivative test (§9.5–9.6), and the Mean Value Theorem (§9.7).

Why a billionaire and a student value a dollar differently

Offer \$100 to a college student living on ramen and you change their week. Offer the same \$100 to a billionaire and they may not bother bending to pick it up. The money is identical; the value of the money is not. Economists call the satisfaction a person derives from wealth their utility, written $U(w)$ as a function of wealth $w$, and the observation above is the single most important fact about it: the more you already have, the less an additional dollar is worth to you. This is the law of diminishing marginal utility, and it is, precisely and completely, a statement about concavity.

This case study shows how the curvature tools of §9.6 — the second derivative, concavity, and inflection — turn a vague intuition about human behavior into a sharp, testable mathematical claim, and how the Mean Value Theorem (§9.7) connects an individual's "marginal" feelings to their average experience of a windfall.

Marginal utility is the first derivative

"Marginal utility" is the economist's name for the derivative $U'(w)$ — the extra utility from one more unit of wealth. That this derivative is positive simply says more wealth is better: $U'(w) > 0$. Nearly every utility function shares this. The interesting claim is about the second derivative.

A standard model is the logarithmic utility function, proposed by Daniel Bernoulli in 1738 to resolve the St. Petersburg paradox:

$$U(w) = \ln w, \qquad w > 0.$$

Its marginal utility is

$$U'(w) = \frac{1}{w} > 0 \quad\text{for all } w > 0,$$

so more is always better — but notice how the marginal utility behaves. At $w = 100$, one more dollar adds $U'(100) = 0.01$ utils. At $w = 100{,}000$, one more dollar adds only $U'(100{,}000) = 0.00001$ utils — a thousand times less. The marginal value of a dollar shrinks as wealth grows. That shrinking is exactly the second derivative:

$$U''(w) = -\frac{1}{w^2} < 0 \quad\text{for all } w > 0.$$

The second derivative is negative everywhere, so $U$ is concave down on its entire domain (§9.6). Concavity is not a side feature of the model here — it is diminishing marginal utility. The economic law and the calculus statement are the same sentence in two languages: "each additional dollar is worth less than the last" $\iff$ "the slope $U'$ is decreasing" $\iff$ "$U'' < 0$" $\iff$ "$U$ is concave down."

The picture. A concave-down utility curve rises but bends over, like the upper-left quarter of a circle. It climbs steeply at low wealth (where dollars are precious) and flattens at high wealth (where dollars are trivial). The student and the billionaire are standing on the same curve — just at very different slopes.

Concavity explains risk aversion (the second derivative test in disguise)

Here is where concavity does real predictive work. Suppose you face a gamble: a 50/50 coin flip between ending with \$0 or \$200, versus a guaranteed \$100. The two options have the **same expected wealth**, \$100. A purely "expected-money" agent would be indifferent. But almost everyone prefers the sure \$100 — they are risk-averse — and concavity explains exactly why.

Compute the expected utility of the gamble against the utility of the certain amount, using $U(w) = \ln w$ (shift to $U(w) = \ln(1+w)$ to allow $w=0$):

$$\text{Gamble: } \tfrac12 U(0) + \tfrac12 U(200) = \tfrac12\ln(1) + \tfrac12\ln(201) = \tfrac12(0) + \tfrac12(5.303) = 2.652,$$ $$\text{Sure thing: } U(100) = \ln(101) = 4.615.$$

The certain \$100 delivers far more utility (4.615) than the gamble (2.652), even though both average \$100 in money. This is Jensen's inequality in action, and Jensen's inequality is nothing but concavity: for a concave-down $U$, the curve lies above every chord, so

$$U\!\left(\tfrac{a+b}{2}\right) > \tfrac12 U(a) + \tfrac12 U(b).$$

The utility of the average exceeds the average of the utilities — geometrically, the midpoint of the curve sits above the midpoint of the chord joining $(a, U(a))$ and $(b, U(b))$. The second-derivative sign $U'' < 0$ is the entire engine of risk aversion. Flip the sign — a convex $U$ with $U'' > 0$ — and you get a risk-seeking agent who prefers the gamble. The shape of one curve predicts whether a person buys insurance or lottery tickets.

An optimum where utility meets cost (second derivative test)

Diminishing marginal utility sets up genuine optimization problems. Consider a worker deciding how many hours $h$ to work. Income buys utility $U(w)$, but each hour of work costs leisure, modeled as a quadratic disutility. Suppose total wealth from working $h$ hours at \$20/hour is $w = 20h$, and the worker's net satisfaction is

$$S(h) = 100\ln(1 + 20h) - h^2,$$

the first term the (concave) utility of income, the second the rising cost of fatigue. Find the satisfaction-maximizing number of hours.

$$S'(h) = \frac{100 \cdot 20}{1 + 20h} - 2h = \frac{2000}{1 + 20h} - 2h.$$

Set $S'(h) = 0$: $\frac{2000}{1+20h} = 2h$, so $2000 = 2h(1 + 20h) = 2h + 40h^2$, giving $40h^2 + 2h - 2000 = 0$, or $20h^2 + h - 1000 = 0$. The quadratic formula:

$$h = \frac{-1 + \sqrt{1 + 80000}}{40} = \frac{-1 + \sqrt{80001}}{40} \approx \frac{-1 + 282.8}{40} \approx 7.04\ \text{hours}.$$

Confirm it is a maximum with the second derivative test (§9.6):

$$S''(h) = -\frac{2000 \cdot 20}{(1 + 20h)^2} - 2 = -\frac{40000}{(1+20h)^2} - 2 < 0 \quad\text{for all } h,$$

which is negative everywhere — $S$ is concave down throughout, so the critical point at $h \approx 7.04$ is the global maximum. The worker's optimal effort is about seven hours; beyond that, fatigue's rising marginal cost overtakes income's falling marginal utility. The crossing point — where marginal benefit equals marginal cost — is the same "marginals meet" principle as the profit-maximization rule in §9.12, now driving a labor-supply decision.

One more connection ties the chapter's theory to this economics. A windfall raises a person's wealth from $a$ to $b$. Their total gain in utility is $U(b) - U(a)$; their average marginal utility over that range is $\frac{U(b) - U(a)}{b - a}$. The Mean Value Theorem (§9.7) guarantees there is some wealth level $c$ between $a$ and $b$ at which the instantaneous marginal utility equals that average:

$$U'(c) = \frac{U(b) - U(a)}{b - a}.$$

For $U(w) = \ln w$, this says $\frac{1}{c} = \frac{\ln b - \ln a}{b - a}$, so $c = \frac{b - a}{\ln b - \ln a}$ — the logarithmic mean of $a$ and $b$, which always lies strictly between them. Economically: somewhere inside any windfall there is a "representative" wealth level whose marginal utility captures the whole experience. Because $U$ is concave (marginal utility decreasing), this representative level $c$ sits below the midpoint $\frac{a+b}{2}$ — the early dollars of the windfall, received while still relatively poor, carry more than their share of the satisfaction. The MVT turns a fuzzy claim ("the first part of a raise feels like more") into an exact statement about where the average marginal utility is realized.

What the calculus delivered

A single property — $U'' < 0$ — generated the entire behavioral story. Negative second derivative is diminishing marginal utility; concavity is risk aversion (via Jensen's inequality); the second derivative test confirmed the labor-supply optimum is a true maximum; and the Mean Value Theorem located the representative wealth level inside a windfall. Behavioral economics often presents these as separate "effects" with separate names. The calculus of §9.6–9.7 reveals them as one geometric fact about a curve that bends downward — the same toolkit that sketches a polynomial, reading the architecture of human choice instead.

Discussion Questions

  1. A second worker has utility $U(w) = \sqrt{w}$ instead of $\ln w$. Compute $U'$ and $U''$ and confirm this worker is also risk-averse. Which worker is more risk-averse at $w = 100$? (Compare $|U''|/U'$, the Arrow–Pratt coefficient.)
  2. For the gamble in the risk-aversion section, the "certainty equivalent" is the guaranteed amount that gives the same utility as the gamble. Solve $\ln(1 + x) = 2.652$ for $x$. Why is $x$ less than \$100, and what does the gap represent?
  3. Re-examine the labor problem with wage \$30/hour instead of \$20. Set up the new $S'(h) = 0$ equation. Without solving, predict whether optimal hours rise or fall, and justify using the structure of the marginal-benefit term.
  4. The MVT gave a representative wealth $c = \frac{b-a}{\ln b - \ln a}$ below the midpoint. For a convex (risk-seeking) utility, would $c$ lie above or below the midpoint? Sketch the geometry.

Annotated Reading

  • Stewart, Calculus: Early Transcendentals (9th ed.), §4.3 ("How Derivatives Affect the Shape of a Graph"). Concavity, the second derivative test, and the chord-vs-curve picture underlying Jensen's inequality.
  • OpenStax Calculus Vol. 1, §4.5–4.6. Free coverage of concavity and curve analysis with applied optimization examples.
  • Varian, Intermediate Microeconomics (9th ed.), chapters on choice under uncertainty. Diminishing marginal utility, expected utility, and risk aversion in their economic home — the concavity-equals-risk-aversion argument made above, expanded.
  • Bernoulli, "Exposition of a New Theory on the Measurement of Risk" (1738, transl. Econometrica 1954). The original source of logarithmic utility and the first use of a concave function to model human valuation of money.