Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to ask how much better it is or how good it is.
All of the theory of consumer decision-making under conditions of certainty can be, and typically is, expressed in terms of ordinal utility.
if: Instead of defining a numeric function, an agent's preference relation can be represented graphically by indifference curves.
The slope of the curve (the negative of the marginal rate of substitution of X for Y) at any point shows the rate at which the individual is willing to trade off good X against good Y maintaining the same level of utility.
The curve is convex to the origin as shown assuming the consumer has a diminishing marginal rate of substitution.
It can be shown that consumer analysis with indifference curves (an ordinal approach) gives the same results as that based on cardinal utility theory — i.e., consumers will consume at the point where the marginal rate of substitution between any two goods equals the ratio of the prices of those goods (the equi-marginal principle).
The challenge of revealed preference theory lies in part in determining what goods bundles were foregone, on the basis of them being less liked, when individuals are observed choosing particular bundles of goods.
are necessary to guarantee the existence of a representing function: When these conditions are met and the set
Moreover, it is possible to inductively construct a representing utility function whose values are in the range
For example, lexicographic preferences are transitive and complete, but they cannot be represented by any utility function.
By the theorems of Debreu (1954), the opposite is also true: Note that the lexicographic preferences are not continuous.
This is in accordance with the fact, stated above, that these preferences cannot be represented by a utility function.
For every utility function v, there is a unique preference relation represented by v. However, the opposite is not true: a preference relation may be represented by many different utility functions.
The same preferences could be expressed as any utility function that is a monotonically increasing transformation of v. E.g., if where
This equivalence is succinctly described in the following way: In contrast, a cardinal utility function is unique up to increasing affine transformation.
that represents the amounts consumed from two products, e.g., apples and bananas.
Suppose the preference relation is monotonically increasing, which means that "more is always better": Then, both partial derivatives, if they exist, of v are positive.
[5]: 82 This definition of the MRS is based only on the ordinal preference relation – it does not depend on a numeric utility function.
This is not a coincidence as these two functions represent the same preference relation – each one is an increasing monotone transformation of the other.
, the preference relation can be represented by a quasilinear utility function, of the form where
:[6][5]: 87 In this case, all the indifference curves are parallel – they are horizontal transfers of each other.
For example, the preferences between bundles of apples and bananas are probably independent of the number of shoes and socks that an agent has, and vice versa.
By Debreu's theorem, if all subsets of commodities are preferentially independent of their complements, then the preference relation can be represented by an additive value function.
Here we provide an intuitive explanation of this result by showing how such an additive value function can be constructed.
Other points can be calculated in a similar way, and then continuity can be used to conclude that the functions are well-defined in their entire range.
An implicit assumption in point 1 of the above proof is that all three commodities are essential or preference relevant.
[7]: 9 In short, The mathematical foundations of most common types of utility functions — quadratic and additive — laid down by Gérard Debreu[9][10] enabled Andranik Tangian to develop methods for their construction from purely ordinal data.
variables can be constructed from interviews of decision makers, where questions are aimed at tracing totally
coordinate planes without referring to cardinal utility estimates.
[11][12] The following table compares the two types of utility functions common in economics: