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In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. The 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.
Contents
- Notation
- Indifference curve mappings
- Revealed preference
- Necessary conditions for existence of ordinal utility function
- Continuity
- Uniqueness
- Monotonicity
- Marginal Rate of Substitution
- Linearity
- Quasi linearity
- Additivity with two goods
- Double cancellation property
- Corresponding tradeoffs property
- Additivity with three or more goods
- Uniqueness of additive representation
- Comparison between ordinal and cardinal utility functions
- References
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:
But the only meaningful message of this function is the order
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
The symbol
The symbol
A function
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
An example indifference curve is shown below:
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
When these conditions are met and the set
When
Continuity
A preference relation is called continuous if, whenever B is preferred to A, small deviations from B or A will not reverse the ordering between them. Formally, a preference relation on a set X is called continuous if it satisfies one of the following equivalent conditions:
- For every
A ∈ X , the set{ ( A , B ) | A ⪯ B } is topologically closed inX × X with the product topology (this definition requires X to be a topological space). - For every sequence
( A i , B i ) , if for all iA i ⪯ B i A i → A andB i → B , thenA ⪯ B . - For every
A , B ∈ X such thatA ≺ B , there exists a ball around A and a ball around B such that, for everya in the ball around A and everyb in the ball around b,a ≺ b (this definition requires X to be a metric space).
If a preference relation is represented by a continuous utility function, then it is clearly continuous. By the theorems of Debreu (1954), the opposite is also true:
Note that the Lexicographic preferences are not continuous. For example,
Uniqueness
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 only unique up to positive affine transformation. Every affine transformation is monotone; hence, if two functions are cardinally equivalent they are also ordinally equivalent, but not vice versa.
Monotonicity
Suppose, from now on, that the set
Then a preference relation
Suppose the preference relation is monotonically increasing, which means that "more is always better":
Then, both partial derivatives of v are positive. In short:
Marginal Rate of Substitution
Suppose a person has a bundle
Note that 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:
For example, if the preference relation is represented by
In general, the MRS may be different in different points
Linearity
When the MRS of a certain preference relation does not depend on the bundle, i.e, the MRS is the same for all
and the preference relation can be represented by a linear function:
(Of course, the same relation can be represented by many other non-linear functions, such as
Quasi-linearity
When the MRS depends on
Additivity with two goods
A more general type of utility function is an additive function:
There are several ways to check whether given preferences are representable by an additive utility function.
Double cancellation property
If the preferences are additive then a simple arithmetic calculation shows that:
so this "double-cancellation" property is a necessary condition for additivity.
Debreu (1960) showed that this property is also sufficient, i.e: if a preference relation satisfies the double-cancellation property then it can be represented by an additive utility function.
Corresponding tradeoffs property
If the preferences are represented by an additive function, then a simple arithmetic calculation shows that:
so this "corresponding tradeoffs" property is a necessary condition for additivity. This condition is also sufficient.
Additivity with three or more goods
When there are three or more commodities, the condition for the additivity of the utility function is surprisingly simpler than for two commodities. This is an outcome of Theorem 3 of Debreu (1960). The condition required for additivity is preferential-independence.
A subset A of commodities is said to be preferentially-independent of a subset B of commodities, if the preference relation in subset A, given constant values for subset B, is independent of these constant values. For example, suppose there are three commodities: x y and z. The subset {x,y} is preferentially-independent of the subset {z}, if for all
In this case, we can simply say that:
Preferential-independence makes sense in case of independent goods. For example, the preferences between bundles of apples and bananas are probably independent of the amount of shoes and socks that an agent has, and vice versa.
By Debreu's theorem, if all subsets of commodities are preferentially-independnet 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. The proof assumes three commodities: x, y, z. We show how to define three points for each of the three value functions
0 point: choose arbitrary
1 point: choose arbitrary
Choose
This indifference serves to scale the units of y and z to match those of x. The value in these three points should be 1, so we assign:
2 point: Now it's time to use the preferential-independence assumption. The relation between
This is good because it means that the function v can have the same value - 2 - in these three points. Select
and assign:
3 point: To show that our assignments so far are consistent, we must show that all points that receive a total value of 3 are indifference points. Here, again, the preferential-independence assumption is used, since the relation between
and similarly for the other pairs. Hence, the 3 point is defined consistently.
We can continue like this by induction and define the per-commodity functions in all integer points, then use continuity to define it in all real points.
An implicit assumption in point 1 of the above proof is that all three commodities are essential or preference-relevant. This means that there exists a bundle such that, if the amount of a certain commodity is increased, the new bundle is strictly better.
The proof for more than 3 commodities is similar. In fact, we don't have to check that all subsets of points are preferentially-independent; it is sufficient to check a linear number of pairs of commodities. E.g, if there are
Uniqueness of additive representation
An additive preference relation can be represented by many different additive utility functions. However, all these functions are similar: they are not only increasing-monotone-transformations of each other (as are all utility functions representing the same relation), they are increasing linear transformations of each other. Shortly:
Comparison between ordinal and cardinal utility functions
The following table compares the two types of utility functions common in economics: