Utility - Cardinal and Ordinal Utility

Cardinal and Ordinal Utility

For more details on this topic, see cardinal utility.

Economists distinguish between cardinal utility and ordinal utility. When cardinal utility is used, the magnitude of utility differences is treated as an ethically or behaviorally significant quantity. On the other hand, ordinal utility captures only ranking and not strength of preferences.

Utility functions of both sorts assign a ranking to members of a choice set. For example, suppose a cup of orange juice has utility of 120 utils, a cup of tea has a utility of 80 utils, and a cup of water has a utility of 40 utils. When speaking of cardinal utility, it could be concluded that the cup of orange juice is better than the cup of tea by exactly the same amount by which the cup of tea is better than the cup of water. One is not entitled to conclude, however, that the cup of tea is two thirds as good as the cup of juice, because this conclusion would depend not only on magnitudes of utility differences, but also on the "zero" of utility.

It is tempting when dealing with cardinal utility to aggregate utilities across persons. The argument against this is that interpersonal comparisons of utility are meaningless because there is no good way to interpret how different people value consumption bundles.

When ordinal utilities are used, differences in utils are treated as ethically or behaviorally meaningless: the utility index encode a full behavioral ordering between members of a choice set, but tells nothing about the related strength of preferences. In the above example, it would only be possible to say that juice is preferred to tea to water, but no more.

Neoclassical economics has largely retreated from using cardinal utility functions as the basic objects of economic analysis, in favor of considering agent preferences over choice sets. However, preference relations can often be represented by utility functions satisfying several properties.

Ordinal utility functions are unique up to positive monotone transformations, while cardinal utilities are unique up to positive linear transformations.

Although preferences are the conventional foundation of microeconomics, it is often convenient to represent preferences with a utility function and analyze human behavior indirectly with utility functions. Let X be the consumption set, the set of all mutually-exclusive baskets the consumer could conceivably consume. The consumer's utility function ranks each package in the consumption set. If the consumer strictly prefers x to y or is indifferent between them, then .

For example, suppose a consumer's consumption set is X = {nothing, 1 apple,1 orange, 1 apple and 1 orange, 2 apples, 2 oranges}, and its utility function is u(nothing) = 0, u(1 apple) = 1, u(1 orange) = 2, u(1 apple and 1 orange) = 4, u(2 apples) = 2 and u(2 oranges) = 3. Then this consumer prefers 1 orange to 1 apple, but prefers one of each to 2 oranges.

In microeconomic models, there are usually a finite set of L commodities, and a consumer may consume an arbitrary amount of each commodity. This gives a consumption set of, and each package is a vector containing the amounts of each commodity. In the previous example, we might say there are two commodities: apples and oranges. If we say apples is the first commodity, and oranges the second, then the consumption set and u(0, 0) = 0, u(1, 0) = 1, u(0, 1) = 2, u(1, 1) = 4, u(2, 0) = 2, u(0, 2) = 3 as before. Note that for u to be a utility function on X, it must be defined for every package in X.

A utility function represents a preference relation on X iff for every, implies . If u represents, then this implies is complete and transitive, and hence rational.

In order to simplify calculations, various assumptions have been made of utility functions.

• CES (constant elasticity of substitution, or isoelastic) utility
• Exponential utility
• Quasilinear utility
• Homothetic preferences

Most utility functions used in modeling or theory are well-behaved. They are usually monotonic and quasi-concave. However, it is possible for preferences not to be representable by a utility function. An example is lexicographic preferences which are not continuous and cannot be represented by a continuous utility function.