Ordinal utility

In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. 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.

For example, suppose George tells us that "I prefer A to B and B to C". George's preferences can be represented by a function u such that:

:<math>u(A)=9, u(B)=8, u(C)=1</math>

But critics of cardinal utility claim the only meaningful message of this function is the order <math>u(A)>u(B)>u(C)</math>; the actual numbers are meaningless. Hence, George's preferences can also be represented by the following function v:

:<math>v(A)=9, v(B)=2, v(C)=1</math>

The functions u and v are ordinally equivalent – they represent George's preferences equally well.

Ordinal utility contrasts with cardinal utility theory: the latter assumes that the differences between preferences are also important. In u the difference between A and B is much smaller than between B and C, while in v the opposite is true. Hence, u and v are not cardinally equivalent.

The ordinal utility concept was first introduced by Pareto in 1906.

Notation

Suppose the set of all states of the world is <math>X</math> and an agent has a preference relation on <math>X</math>. It is common to mark the weak preference relation by <math>\preceq</math>, so that <math>A \preceq B</math> reads "the agent wants B at least as much as A".

The symbol <math>\sim</math> is used as a shorthand to the indifference relation: <math>A\sim B \iff (A\preceq B \land B\preceq A)</math>, which reads "The agent is indifferent between B and A".

The symbol <math>\prec</math> is used as a shorthand to the strong preference relation: <math>A\prec B \iff (A\preceq B \land B\not\preceq A)</math> if:

:<math>A \preceq B \iff u(A) \leq u(B)</math>

Indifference curve mappings

Instead of defining a numeric function, an agent's preference relation can be represented graphically by indifference curves. This is especially useful when there are two kinds of goods, x and y. Then, each indifference curve shows a set of points <math>(x,y)</math> such that, if <math>(x_1,y_1)</math> and <math>(x_2,y_2)</math> are on the same curve, then <math>(x_1,y_1) \sim (x_2,y_2)</math>.

An example indifference curve is shown below:

200px|indifference map

Each indifference curve is a set of points, each representing a combination of quantities of two goods or services, all of which combinations the consumer is equally satisfied with. The further a curve is from the origin, the greater is the level of utility.

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).

Revealed preference

Revealed preference theory addresses the problem of how to observe ordinal preference relations in the real world. 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.

Necessary conditions for existence of ordinal utility function

Some conditions on <math>\preceq</math> are necessary to guarantee the existence of a representing function:

Transitivity: if <math>A \preceq B</math> and <math>B \preceq C</math> then <math>A \preceq C</math>.
Completeness: for all bundles <math>A,B\in X</math>: either <math>A\preceq B</math> or <math>B\preceq A</math> or both.
Completeness also implies reflexivity: for every <math>A\in X</math>: <math>A \preceq A</math>.

When these conditions are met and the set <math>X</math> is finite, it is easy to create a function <math>u</math> which represents <math>\prec</math> by just assigning an appropriate number to each element of <math>X</math>, as exemplified in the opening paragraph. The same is true when X is countably infinite. Moreover, it is possible to inductively construct a representing utility function whose values are in the range <math>(-1,1)</math>.

When <math>X</math> is infinite, these conditions are insufficient. For example, lexicographic preferences are transitive and complete, but they cannot be represented by any utility function.

This definition of the MRS is based only on the ordinal preference relation – it does not depend on a numeric utility function. If the preference relation is represented by a utility function and the function is differentiable, then the MRS can be calculated from the derivatives of that function:

:<math>MRS = \frac{v'_x}{v'_y}.</math>

For example, if the preference relation is represented by <math>v(x,y)=x^a\cdot y^b</math> then <math>MRS = \frac{a\cdot x^{a-1}\cdot y^b}{b\cdot y^{b-1}\cdot x^a}=\frac{ay}{bx}</math>. The MRS is the same for the function <math>v(x,y)=a\cdot \log{x} + b\cdot \log{y}</math>. This is not a coincidence as these two functions represent the same preference relation – each one is an increasing monotone transformation of the other.

In general, the MRS may be different at different points <math>(x_0,y_0)</math>. For example, it is possible that at <math>(9,1)</math> the MRS is low because the person has a lot of x and only one y, but at <math>(9,9)</math> or <math>(1,1)</math> the MRS is higher. Some special cases are described below.

Linearity

When the MRS of a certain preference relation does not depend on the bundle, i.e., the MRS is the same for all <math>(x_0,y_0)</math>, the indifference curves are linear and of the form:

:<math>x+\lambda y = \text{const},</math>

and the preference relation can be represented by a linear function:

:<math>v(x,y)=x+\lambda y.</math>

(Of course, the same relation can be represented by many other non-linear functions, such as <math>\sqrt{x+\lambda y}</math> or <math>(x+\lambda y)^2</math>, but the linear function is simplest.)

Corresponding tradeoffs property

If the preferences are represented by an additive function, then a simple arithmetic calculation shows that

:<math>MRS(x_2,y_2)=\frac{MRS(x_1,y_2)\cdot MRS(x_2,y_1)}{MRS(x_1,y_1)}</math>

so this "corresponding tradeoffs" property is a necessary condition for additivity.

This condition is also sufficient.

enabled Andranik Tangian to develop methods for their construction from purely ordinal data.

In particular, additive and quadratic utility functions in <math>n</math> variables can be constructed from interviews of decision makers, where questions are aimed at tracing totally <math>n</math> 2D-indifference curves in <math>n - 1</math> coordinate planes without referring to cardinal utility estimates.

Comparison between ordinal and cardinal utility functions

The following table compares the two types of utility functions common in economics:

{| class="wikitable"

! !! Level of measurement !! Represents preferences on!! Unique up to !! Existence proved by !! Mostly used in

| Ordinal utility || Ordinal scale || Sure outcomes || Increasing monotone transformation || Debreu (1954) || Consumer theory under certainty

| Cardinal utility || Interval scale || Random outcomes (lotteries) || Increasing monotone linear transformation || Von Neumann-Morgenstern (1947) || Game theory, choice under uncertainty

References

External links

Lexicographic preference relation cannot be represented by a utility function. In Economics.SE
Recognizing linear orders embeddable in R2 ordered lexicographically. In Math.SE.
Murray N. Rothbard, "Towards a Reconstruction of Utility and Welfare Economics"