Diminishing Marginal Utility

Diminishing Marginal Utility in Health Economics

Diminishing marginal utility is an economic principle about the shape of a utility function, under which the gain from one more unit of a good shrinks as a person already holds more of it. For health economists it is the standard explanation of why people insure against large medical bills, and of why the same sum of money is judged to matter more to a poor household than to a rich one. It sits behind risk aversion in expected utility models and behind the welfare weights, discount rate and health values in HM Treasury's Green Book. It also frames the argument over whether a QALY should count the same whoever gains it.

Marshall's law of satiable wants and the marginal utility of money

Alfred Marshall stated the idea as the law of satiable wants, or of diminishing utility. In his Principles of Economics he wrote that the total utility of a thing to anyone increases with every increase in their stock of it, but not as fast as the stock increases, and he called the utility of the part a person is only just induced to buy its marginal utility. He assumed that the person's tastes are not allowed time to change. The OpenStax Principles of Microeconomics text defines marginal utility as the additional utility from one more unit of consumption.

Marshall applied the law to money as well. The richer a person becomes, he wrote, the less is the marginal utility of money to them, so every increase in resources raises the price they are willing to pay for any given benefit. A willingness to pay figure mixes how much a person values a health gain with how much a pound means to them, so equal sums offered by a rich and a poor respondent do not show equal strength of preference.

In calculus terms the principle concerns the derivatives of a utility function $U(x)$ of a quantity $x$.

$$MU(x) = \frac{dU}{dx} > 0, \qquad \frac{d^{2}U}{dx^{2}} < 0$$

where $MU(x)$ is marginal utility at quantity $x$. The first condition says more is better and the second says each extra unit adds less than the last, which makes $U$ concave. A simple example is $U(y) = \ln y$ for income $y$, whose marginal utility is one divided by income, so a pound at an income of GBP 20,000 carries twice the marginal utility of a pound at GBP 40,000. OpenStax calls it an example of the general law of diminishing returns, but diminishing returns describes a production function, such as extra health from extra care, while diminishing marginal utility describes the utility function of a consumer or patient.

Why the principle needs more than a ranking of options

A falling marginal utility means something only if differences in utility can be compared. Osborne and Rubinstein show that any increasing function of a utility function represents the same preferences. Take $u(x) = \sqrt{x}$, which is concave, and the transformation $w(x) = u(x)^{4} = x^{2}$, which is convex. Both rank every quantity in the same order, yet one shows diminishing and the other increasing marginal utility, so a ranking alone cannot say which holds. In ordinal consumer theory the same work is done by a diminishing marginal rate of substitution along an indifference curve, which survives any such transformation.

The principle regains content in two settings. Under expected utility, Osborne and Rubinstein note that the Bernoulli function over prizes is unique up to a positive affine transformation, which never changes the sign of the second derivative, so concavity becomes a claim about attitudes to risk. In welfare comparisons it needs utility that is cardinal and comparable between people. Brouwer and colleagues describe the classical tradition of welfare economics, in which utilities were cardinally measured and added across individuals, and the later Paretian tradition, in which utility was usually ordinal and interpersonal comparisons were held to be impossible or meaningless. In the Paretian tradition the statement that a pound means more to a poorer person cannot be made, because it needs cardinal utility and an interpersonal comparison of utility as well.

Risk aversion and the demand for health insurance

Arrow's 1963 paper on medical care made the link with health insurance explicit. He treated medical costs as a random deduction from income, adopted the expected-utility hypothesis, which he traced to Daniel Bernoulli in 1738, and assumed that individuals are normally risk-averters, which in utility terms means a diminishing marginal utility of income. He accepted that gambling makes the assumption hard to apply in full.

Osborne and Rubinstein give the formal result. A person whose preferences over lotteries follow expected utility is risk-averse if and only if the Bernoulli function is concave, and the proof rests on Jensen's inequality.

$$U\big(E[Y]\big) > E\big[U(Y)\big]$$

where $Y$ is uncertain income after medical costs, $E[Y]$ is its expected value and $U$ is a strictly concave utility function. The inequality says the certain mean is preferred to the gamble. Arrow used the same property to argue that a risk-averse person strictly prefers an actuarially fair health insurance policy to bearing the risk, and will still prefer a policy loaded for administrative costs, provided it is not too unfair.

Worked example with an illustrative household and a hospital bill

All numbers in this example are illustrative. A household has an income of GBP 40,000 and faces a 5% chance of an uninsured hospital bill of GBP 20,000, which would leave it with GBP 20,000. Its utility of income is $U(y) = \ln y$.

1. Expected income. 0.95 × 40,000 + 0.05 × 20,000 = 39,000, so the expected loss is GBP 1,000, which is the actuarially fair premium for full cover.

2. Expected utility without insurance. With $\ln 40{,}000 = 10.5966$ and $\ln 20{,}000 = 9.9035$, expected utility is $E[U] = 0.95 \times 10.5966 + 0.05 \times 9.9035 = 10.0668 + 0.4952 = 10.5620$.

3. Certainty equivalent. The sure income giving the same utility is $e^{10.5620}$, which equals 40,000 times 0.5 raised to the power 0.05, or about GBP 38,637.

4. Risk premium. The household would give up 39,000 − 38,637 = 363 pounds of expected income to shed the risk, so it would pay up to 40,000 − 38,637 = 1,363 pounds for full cover. Loaded premiums are worked through on the health insurance page.

5. Marginal utility in each state. Marginal utility is 1/20,000 = 0.00005 after the bill and 1/40,000 = 0.000025 without it. Insurance moves money into the state where each pound is worth twice as much to the household.

6. Interpretation. The GBP 363 premium comes entirely from the curvature of $U$. With linear utility the risk premium would be zero and the household would refuse any loaded policy.

Distributional weights and health values in the Green Book

HM Treasury's Green Book, the UK government's guidance on appraisal, rests several rules on the principle. It states that an additional GBP 1,000 of income is more valuable to a low-income household than to a high-income household and calls this the diminishing marginal utility of income. On that basis analysts may weight costs and benefits by the group that bears them, using the national and group median equivalised household incomes.

$$w_j = \left(\frac{\bar{y}}{y_j}\right)^{1.25}$$

where $w_j$ is the welfare weight for group $j$, $\bar{y}$ is the median equivalised income of the national population, $y_j$ is the median equivalised income of group $j$, and 1.25 is the Green Book's estimate of the elasticity of marginal utility of consumption. For an illustrative group whose median income is half the national median, $w_j = 2^{1.25} = 2.378$, so a GBP 1,000 benefit counts as about GBP 2,378. A group on one and a half times the national median gets $(2/3)^{1.25} = 0.602$. The Green Book requires weighted estimates to be presented alongside unweighted ones. A scheme whose cash benefits go mainly to low-income households therefore looks stronger in the weighted figures than in the unweighted ones.

The same elasticity enters the discount rate. The Green Book supplementary guidance on discounting, published in October 2026, derives the social time preference rate from the Ramsey formula, which includes a wealth effect arising because society gets richer over time.

$$\text{wealth effect} = \mu \times g = 1.25 \times 1.50\% = 1.875\%$$

where $\mu$ is the elasticity of marginal utility of consumption and $g$ is the expected annual growth rate of real per capita consumption. With pure time preference of 0.50% and a rare disaster risk term of minus 0.50%, the risk-free rate is 1.875%, which HM Treasury adjusts upwards to 2.00% after weighing other evidence. A 1.00% systematic risk premium then gives a discount rate of 3.00% in real terms for the first 30 years.

Health effects are handled differently. The Green Book states that social welfare from additional years of life, and from quality-adjusted years of life, is considered to rise over time as income rises, so the diminishing marginal utility of consumption does not apply, and future real health effects should be uprated by 2% a year. It gives a willingness-to-pay value of GBP 70,000 per QALY in 2020-21 prices. A Green Book appraisal of a vaccination programme therefore values a QALY gained in twenty years more highly in real terms than one gained today, before discounting.

Diminishing marginal utility of health and the equal QALY

The QALY itself is usually built without diminishing marginal utility. Lipman, Brouwer and Attema note that the linear QALY model behind time trade-off and standard gamble weights ignores evidence of diminishing marginal utility of life years.

At the level of society the question becomes whether a health gain should be worth less to someone who already has a lot of health. Brouwer and colleagues note that applied cost-benefit analysis gives an equal weight to each money gain or loss, whoever receives it, with net utility expressed through willingness and ability to pay. Extra-welfarist cost-utility analysis uses something like need instead, which is unrelated to ability to pay, so welfare judgements are insulated from the income distribution. Where extra-welfarists give more weight to QALY gains for people in poor health or who have not had their fair innings, Brouwer and colleagues stress that this is done for reasons of equity rather than on a presumption of generalised diminishing marginal utility of health.

Distributional cost-effectiveness analysis makes this choice explicit. Asaria, Griffin and Cookson describe social welfare indices whose inequality aversion parameters set the trade-off between total health and health inequality. A social welfare function that is concave in each person's health behaves like diminishing marginal utility, because a QALY gained by someone with low lifetime health raises social welfare by more than the same QALY gained by someone with high lifetime health. Their tutorial reports the Atkinson index at several levels of relative inequality aversion. The concavity is a social value judgement about inequality, not a measured property of individual preferences, and an equity-weighted analysis should state which of the two it is applying.

Where the principle is misread

Most errors come from treating diminishing marginal utility as more general, or more measurable, than it is. The conditions attached to the law are often dropped in applied work.

• Marshall excluded changes in tastes over time and named habits, such as drunkenness, that grow on what they feed upon. Alcohol or tobacco use, where past consumption changes tastes, sits outside the law as he stated it.
• Risk aversion and diminishing marginal utility of income coincide only under expected utility. Lipman and colleagues note that expected utility has been shown to be descriptively invalid for health outcomes, so standard gamble answers may reflect loss aversion and probability weighting as well as curvature.
• OpenStax warns that utils cannot be compared between individuals. A conclusion that money or health should go to the worse off therefore needs an explicit distributional judgement, which Brouwer and colleagues locate in the choice of a Bergson-Samuelson social welfare function.

Sources

• Arrow KJ. Uncertainty and the welfare economics of medical care. American Economic Review. 1963;53(5):941-973. https://www.aeaweb.org/aer/top20/53.5.941-973.pdf
• Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19. https://doi.org/10.1177/0272989X15583266
• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. https://doi.org/10.1016/j.jhealeco.2007.07.003
• HM Treasury. The Green Book (2026). London: HM Treasury; updated 28 October 2026. https://www.gov.uk/government/publications/the-green-book--2/the-green-book-2026
• HM Treasury. Green Book supplementary guidance: discounting. London: HM Treasury; October 2026. https://www.gov.uk/government/publications/green-book-supplementary-guidance-discounting
• Lipman SA, Brouwer WBF, Attema AE. QALYs without bias? Nonparametric correction of time trade-off and standard gamble weights based on prospect theory. Health Economics. 2019;28(7):843-854. https://doi.org/10.1002/hec.3895
• Marshall A. Principles of Economics. 8th ed. London: Macmillan; 1920 (first published 1890). Book III, Chapter III. https://www.econlib.org/library/Marshall/marP.html?chapter_num=12
• Osborne MJ, Rubinstein A. Models in Microeconomic Theory. Expanded second edition. Cambridge: Open Book Publishers; 2023. Chapters 1 and 3. https://doi.org/10.11647/OBP.0362
• Shapiro D, MacDonald D, Greenlaw SA. Principles of Microeconomics. 3rd ed. Houston: OpenStax; 2022. Section 6.1. https://openstax.org/books/principles-microeconomics-3e/pages/6-1-consumption-choices