2 Utility and Its Representations

Utility represents preferences

A consumer’s preferences rank possible consumption bundles. A utility function assigns a number to each bundle while preserving that ranking: if bundle $A$ is preferred to bundle $B$, then $A$ must receive higher utility. For example, if a consumer prefers three apples and two oranges to two apples and two oranges, a representing utility function must assign the first bundle a higher value.

In standard consumer theory, utility is ordinal: it records the order of preference, not a measurable amount of happiness. Therefore, any strictly increasing transformation of a utility function represents the same preferences. If $U(x,y)$ represents preferences, then $2U(x,y)$ and $\ln U(x,y)$, where defined, rank bundles identically. The numerical utility levels themselves have no independent meaning.

Marginal utility and trade-offs

Marginal utility describes how a utility representation changes when consumption of one good increases slightly while consumption of other goods is held constant. For a differentiable utility function $U(x,y)$, the marginal utility of $x$ is $MU_x=\frac{\partial U}{\partial x}$, and the marginal utility of $y$ is $MU_y=\frac{\partial U}{\partial y}$.

For example, if $U(x,y)=xy$, then $MU_x=y$ and $MU_y=x$. At the bundle $x=2$, $y=5$, an additional small amount of $x$ has marginal utility $5$, while an additional small amount of $y$ has marginal utility $2$ in this particular representation. These marginal-utility values depend on the chosen representation and are not directly measurable units of satisfaction.

The ratio of marginal utilities gives the marginal rate of substitution: the consumer’s local willingness to trade one good for another while remaining equally preferred. Along an indifference curve, the marginal rate of substitution of $x$ for $y$ is $\frac{MU_x}{MU_y}$. This ratio is unchanged by a differentiable, strictly increasing transformation of utility.

Cobb–Douglas preferences

A common representation of Cobb–Douglas preferences is

where $\alpha$ and $\beta$ are positive constants. This function represents preferences for both goods and, in the usual case, a willingness to substitute between them.

The marginal utilities are $MU_x=\alpha x^{\alpha-1}y^{\beta}$ and $MU_y=\beta x^{\alpha}y^{\beta-1}$, giving

As the consumer has more $x$ relative to $y$, the amount of $y$ they are willing to give up for another unit of $x$ falls. For example, $U(x,y)=x^{1/2}y^{1/2}$ represents a preference for a mix of the two goods. For positive quantities, taking the logarithm gives $\frac{1}{2}\ln x+\frac{1}{2}\ln y$, an equivalent representation because the logarithm is strictly increasing.

Perfect substitutes and spending

Perfect substitutes are goods that a consumer is willing to exchange at a constant rate. A standard utility function is

where $a$ and $b$ are positive constants. Its indifference curves are straight lines, and its marginal rate of substitution is constant: $\frac{MU_x}{MU_y}=\frac{a}{b}$. For example, if $U(x,y)=x+2y$, one unit of $y$ contributes as much utility in this representation as two units of $x$.

With a budget constraint, the consumer compares the marginal utility per dollar of each good: $\frac{a}{p_x}$ for $x$ and $\frac{b}{p_y}$ for $y$. If $\frac{a}{p_x}>\frac{b}{p_y}$, buying $x$ gives more utility per dollar; if $\frac{b}{p_y}>\frac{a}{p_x}$, buying $y$ does. The consumer typically spends the budget on the better-value good. If the ratios are equal, every affordable bundle on the budget line provides the same utility.

How the representations differ

Utility functions give a convenient numerical representation of preference rankings, but their numbers are ordinal rather than direct measures of happiness. Marginal utility describes how the representation changes with consumption, while the ratio of marginal utilities gives the marginal rate of substitution. Cobb–Douglas functions represent goods with a changing trade-off; perfect-substitutes functions represent goods exchanged at a constant rate.