3 Utility and Marginal Utility

Utility represents preferences

In consumer theory, Preferences describe how a person ranks possible consumption bundles. A utility function assigns a number to each bundle to represent that ranking. Formally, a utility function $u$ represents preferences when

where $x \succeq y$ means that bundle $x$ is at least as preferred as bundle $y$. Equal utility values indicate indifference, while a higher value indicates a preferred bundle. Utility is therefore a numerical representation of preferences, not necessarily a measurable quantity of happiness.

For example, let $u(x,y)=xy$. Bundles $(2,6)$ and $(3,4)$ both have utility $12$, so they are indifferent in this representation. Bundle $(4,4)$, with utility $16$, is preferred to either, while bundle $(1,5)$, with utility $5$, is less preferred.

Ordinal utility and transformations

In standard consumer theory, utility is usually ordinal: the ranking of bundles matters, not the numbers themselves. A strictly increasing transformation preserves that ranking, so it can change utility values without changing which bundles are preferred.

For positive $x$ and $y$, the functions

and

represent the same preferences because the natural logarithm is strictly increasing. The utility numbers change, but the ranking of bundles does not. Consequently, an ordinal utility function is not unique: many functions can represent the same preferences.

Marginal utility

Marginal utility is the change in represented utility associated with a small increase in the quantity of a good, holding other goods fixed. For a differentiable utility function $u(x,y)$, the marginal utilities are

For a discrete change, marginal utility can instead be calculated as the difference between utility at two bundles, such as $u(x+1,y)-u(x,y)$.

Diminishing marginal utility

Consider the one-good utility function $U(q)=10q-q^2$, for $0\leq q\leq5$. Its marginal utility is

At $q=1$, marginal utility is $8$; at $q=3$, it is $4$. Utility still rises over this range, but each additional unit adds less than the previous one. This is diminishing marginal utility: marginal utility falls as consumption increases. It is a common pattern, not a universal rule.

Marginal utilities and trade-offs

Marginal utility helps describe the local trade-offs implied by a utility representation. With two goods, the slope of an indifference curve, or marginal rate of substitution, is

The ratio $MU_x/MU_y$ describes how much of good $y$ a person would give up for a small increase in good $x$, while remaining indifferent. For $u(x,y)=xy$, the marginal utilities are $MU_x=y$ and $MU_y=x$, so the indifference-curve slope is $-y/x$.

What changes with the utility representation

A strictly increasing transformation can change the marginal-utility numbers while preserving the preference ranking. Where derivatives exist and marginal utilities are nonzero, it also preserves the marginal rate of substitution. If $v=g(u)$ with $g'(u)>0$, then

The common factor cancels in the ratio of marginal utilities. Thus, marginal utility levels depend on the chosen utility representation, whereas the marginal rate of substitution is unchanged by such a transformation. Diminishing marginal utility in a particular representation should not, by itself, be treated as a representation-independent fact about preferences.