Utility (Economics) · aiwiki.page

In economics, utility represents how an individual ranks goods, activities, or outcomes. It is commonly expressed through a utility function that assigns higher numbers to preferred alternatives. In modern microeconomics, these numbers generally describe an ordering rather than measurable quantities of happiness. Utility provides a framework for analyzing consumer choice and evaluating alternatives, without requiring preferences to concern only money or material consumption. (ocw.mit.edu)

Preferences and utility functions

The starting point is a preference relation over alternatives, such as bundles containing different quantities of goods. Writing $x\succeq y$ means that bundle $x$ is considered at least as good as bundle $y$ . A utility function is a function representing this relation when

x\succeq y \quad\Longleftrightarrow\quad u(x)\geq u(y).

Strict preference corresponds to a higher utility number, while indifference corresponds to equal numbers. (ocw.mit.edu)

Standard models assume completeness, meaning that any two alternatives can be compared, and transitivity, meaning that rankings are internally consistent: if $x\succeq y$ and $y\succeq z$ , then $x\succeq z$ . On standard consumption domains, adding suitable continuity conditions permits representation by a continuous utility function. These assumptions describe a particular model of rational choice; they do not assert that every observed decision satisfies it. (ocw.mit.edu)

Ordinal utility preserves rankings but gives no independent meaning to numerical differences. A utility of 20 does not imply twice the satisfaction associated with 10. Any strictly increasing transformation $f(u)$ represents the same preferences. By contrast, cardinal utility attaches significance to additional numerical structure, such as ratios of utility differences. Cardinality does not, by itself, establish comparability between different people. (ocw.mit.edu)

Marginal utility and substitution

Marginal utility describes the change in utility associated with an additional quantity of a good. For a differentiable utility function, it is the partial derivative

MU_i=\frac{\partial u}{\partial x_i}.

Diminishing marginal utility means that this derivative decreases as consumption of that good increases, holding other quantities constant. It is a common modeling assumption, not a universal property of preferences. (ocw.mit.edu)

An indifference curve consists of bundles assigned the same utility. With two desirable goods, these curves typically slope downward: obtaining more of one good can compensate for giving up some of the other. The marginal rate of substitution measures this local trade-off. Using a positive convention,

MRS_{xy}=\frac{MU_x}{MU_y}, \qquad \frac{dy}{dx}\bigg|_{u=\mathrm{constant}}=-MRS_{xy}.

Unlike the magnitudes of marginal utilities, this ratio is unchanged by differentiable, strictly increasing transformations with positive derivatives. Thus substitution behavior, rather than numerical utility increments, is central to ordinal consumer theory. (ocw.mit.edu)

Perfect substitutes and perfect complements illustrate contrasting patterns. Linear utility $u(x,y)=ax+by$ , with positive coefficients, gives straight indifference curves. Utility $u(x,y)=\min\{x/a,y/b\}$ gives right-angled curves: extra quantities of one good alone cannot improve the bundle beyond the limit imposed by the other. (ocw.mit.edu)

Utility maximization and consumer demand

In consumer theory, choice is modeled as constrained optimization. With positive prices $p_i$ , income $m$ , and nonnegative quantities, the consumer solves

\max_{x\geq0}u(x) \quad\text{subject to}\quad \sum_i p_i x_i\leq m.

The budget constraint defines the affordable feasible set. The selected bundle is a most-preferred affordable alternative; different utility functions representing the same preferences yield the same maximizing choices. (live.ocw.mit.edu)

At a differentiable interior optimum with an exhausted budget, marginal utility per dollar is equal across purchased goods:

\frac{MU_x}{p_x}=\frac{MU_y}{p_y},

or equivalently $MRS_{xy}=p_x/p_y$ . This tangency condition is not universal: corner solutions and kinked indifference curves require different treatment. Under appropriate regularity and convexity assumptions, it characterizes optimal choice. Varying prices and income generates consumer demand. (live.ocw.mit.edu)

The indirect utility function records the maximum utility attainable at given prices and income. A complementary expenditure-minimization problem asks for the least spending needed to achieve a specified utility level. Together these tools allow price changes to be evaluated in monetary terms without treating utility itself as money. (live.ocw.mit.edu)

Expected utility and risk

In decision theory, uncertain alternatives may be represented as lotteries. Expected utility theory evaluates a lottery with outcomes $z_k$ and probabilities $p_k$ by

EU=\sum_k p_k u(z_k).

This is the expected value of utility, not necessarily the expected monetary payoff. Completeness, transitivity, continuity, and an independence axiom support the von Neumann–Morgenstern representation of lottery preferences. (ocw.mit.edu)

In this framework, outcome utilities are unique up to a positive affine transformation, $u'=au+b$ , where $a>0$ . Arbitrary increasing nonlinear transformations generally do not preserve expected-utility rankings. For monetary outcomes, an increasing concave utility function represents risk aversion: the decision-maker weakly prefers a certain amount to a lottery with the same expected amount. Linear utility represents risk neutrality, and increasing convex utility represents risk seeking. (ocw.mit.edu)

Welfare and empirical limitations

Welfare economics uses preferences to assess allocations. Pareto efficiency requires that no feasible change make someone better off without making another person worse off; it does not require interpersonal utility comparisons. Adding individuals’ utility numbers, however, requires additional assumptions about comparability and aggregation. Ordinal rankings alone cannot determine how one person’s gain should be weighed against another’s loss. (ocw.mit.edu)

Observed choices may also depart from standard utility models. Behavioral economics studies such departures, including sensitivity to framing and reference points. Prospect theory models risky choices using gains and losses relative to a reference point and nonlinear decision weights. These developments distinguish the usefulness of utility as a formal representation from the empirical adequacy of particular assumptions about choice. (nobelprize.org)