Revealed preference

Revealed preference theory, pioneered by American economist Paul Samuelson, is a method for comparing the influence of policies on consumer behavior. These models assume that the preferences of consumers can be revealed by their purchasing habits. Revealed preference theory came about because existing theories of consumer demand were based on a diminishing marginal rate of substitution (MRS). This diminishing MRS relied on the assumption that consumers make consumption decisions to maximize their utility. While utility maximization was not a controversial assumption, the underlying utility functions could not be measured with great certainty. Revealed preference theory was a means to reconcile demand theory by defining utility functions by observing behavior.

Definitions and theory

Let there be 2 bundles of goods (x1,x2) and (y1,y2) available at price (p1,p2),assuming that the consumer has an income 'm'. It is observed that the consumer buys (x1,x2) bundle of goods. To translate this arithmetically following equation is formulated p1y1+p2y2<m or p1y1+p2y2=m This equation indicates that the bundle of goods (y1,y2) satisfies the budget constraint or in other words (y1,y2) is affordable by the consumer. The consumer's purchase of (x1,x2) instead implies p1x1+p2x2=m the above equation also satisfies the condition of the bundle of goods being affordable within the budget constraint and in this case it satisfies the condition with equality. Putting the above equations together, knowing that the bundle of goods (y1,y2) was affordable at the given budget constraint(p1,p2,m) the consumer bought (x1,x2) bundle of goods, the final equation of revealed preference is as stated below p1x1+p2x2>p1y1+p2y2. From the preceding equation we derive that the consumer prefers bundle of goods (x1,x2) over bundle of goods (y1,y2) or we can say that bundle of goods (x1,x2) is directly revealed preferred to (y1,y2).

The weak axiom

Given any situation (budget sets) with options {A,B,C,D,...}, if a customer chooses A as one of his (possibly many) choices, then we say A is "(directly) revealed as good as" to B. If, in addition to choosing A, the customer does not choose B, then we say A is "(directly) revealed preferred" to B.

The weak axiom of revealed preference (WARP) states that if A is revealed as good as B, then it is never the case that B is revealed preferred to A.

In other words, if a consumer considers A as one of his best alternatives in a particular situation where B is also possible; then it cannot be that at another situation, where both A and B are possible, that the consumer considers B to be superior to A (in psychology, such behavior is known as a reversal of preference).

Completeness and Strong axiom

If A is directly revealed preferred to B, and B is directly revealed preferred to C, then we say A is indirectly revealed preferred to C. Note that it is possible for A and C to be (directly or indirectly) revealed preferable to each other at the same time, creating a "loop". In mathematical terminology, this says that transitivity is violated.

Consider the following choices: c(A,B)=A, c(B,C)=B, c(C,A)=C, where c is the choice function taking a set of options (budget set) to a choice. Then by our definition A is (indirectly) revealed preferred to C (by the first two choices) and C is (directly) revealed preferred to A (by the last choice).

It is often desirable in economic models to prevent such loops from happening, for example if we wish to model choices with utility functions (which have real-valued outputs and are thus transitive). One way to do so is to impose completeness on the revealed preference relation with regards to the situations, i.e. every possible situation must be taken into consideration by a consumer. This is useful because if we can consider {A,B,C} as a situation, we can directly tell which option is preferred to the other (or if they are the same). Using the weak axiom then prevents two choices from being preferred over each other at the same time; thus it would be impossible for "loops" to form.

Another way to solve this is to impose the strong axiom of revealed preference (SARP) which ensures transitivity. This is characterized by taking the transitive closure of direct revealed preferences and require that it is antisymmetric, i.e. if A is revealed preferred to B (directly or indirectly), then B is not revealed preferred to A (directly or indirectly).

Note that these are two different approaches to solving the issue; completeness is concerned with the input (domain) of the choice functions; while the strong axiom imposes conditions on the output.

Motivation

Revealed preference theory tries to understand the preferences of a consumer among bundles of goods, given her budget constraint. For instance, if the consumer buys bundle of goods A over bundle of goods B, where both bundles of goods are affordable, it is revealed that she directly prefers A over B. It is assumed that the consumer's preferences are stable over the observed time period, i.e. the consumer will not reverse her relative preferences regarding A and B.

As a concrete example, if a person chooses 2 apples/3 bananas over an affordable alternative 3 apples/2 bananas, then we say that the first bundle is revealed preferred to the second. It is assumed that the first bundle of goods is always preferred to the second, and that the consumer purchases the second bundle of goods only if the first bundle becomes unaffordable.

Criticism

Stanley Wong argued that revealed preference theory was a failed research program. According to Wong, in 1938 Samuelson presented revealed preference theory as an alternative to utility theory, while in 1950, Samuelson took the demonstrated equivalence of the two theories as a vindication for his position, rather than as a refutation.

If there exist only an apple and an orange, and an orange is picked, then one can definitely say that an orange is revealed preferred to an apple. In the real world, when it is observed that a consumer purchased an orange, it is impossible to say what good or set of goods or behavioral options were discarded in preference of purchasing an orange. In this sense, preference is not revealed at all in the sense of ordinal utility. One of the critics of the revealed preference theory states that "Instead of replacing 'metaphysical' terms such as 'desire' and 'purpose'" they "used it to legitimize them by giving them operational definitions." Thus in psychology, as in economics, the initial, quite radical operationalist ideas eventually came to serve as little more than a "reassurance fetish" for mainstream methodological practice."

Counter Example: Given that I prefer the second cheapest flower in a set of flowers {x,y}, then C{x, y} = {x}. If a less expensive flower is added to the set then C{x,y,z} = {y}, which contradicts WARP. Sen 1993, p 501 lays out an explicit argument.

See also

• Conjoint analysis
• Hedonic regression
• Travel cost analysis
• Contingent valuation or stated preference methods

References

• Nicholson, W. (2005) Microeconomics, Thomson, Southwestern.
• Mas-Colell, A.; Whinston, M.D.; Green, J.R. (1995) "Microeconomic Theory", First Edition, New York: Oxford University Press, New York
• Samuelson, P. (1938). A Note on the Pure Theory of Consumers' Behaviour. Economica 5:61-71.
• Varian, H. (1992) Microeconomic Analysis, Third edition, New York: Norton, Section 8.7

External links

• Revealed Preference, review by Hal R. Varian, 2005, prepared for Samuelsonian Economics and the 21st Century.
• Lecture Notes in Microeconomic Theory, book by Ariel Rubinstein, 2005.