Definition and theory
Two-dimensional example
If the budget set is defined for two goods; , and determined by prices and income , then let bundle a be and bundle b be . This situation would typically be represented arithmetically by the inequality and graphically by a budget line in the positive real numbers. Assuming strongly monotonic preferences, only bundles that are graphically located on the budget line, i.e. bundles where and are satisfied, need to be considered. If, in this situation, it is observed that is chosen over , it is concluded that is (directly) revealed preferred to , which can be summarized as theThe Weak Axiom of Revealed Preference (WARP)
WARP is one of the criteria which needs to be satisfied in order to make sure that the consumer is consistent with their preferences. If a bundle of goods a is chosen over another bundle b when both are affordable, then the consumer reveals that they prefer a over b. WARP says that when preferences remain the same, there are no circumstances ( budget set) where the consumer prefers b over a. By choosing a over b when both bundles are affordable, the consumer reveals that their preferences are such that they will never choose b over a when both are affordable, even as prices vary. Formally: : where and are arbitrary bundles and is the set of bundles chosen in budget set , given preference relation . In other words, if a is chosen over b in budget set where both a and b are feasible bundles, but b is chosen when the consumer faces some other budget set , then a is not a feasible bundle in budget set .Completeness: The Strong Axiom of Revealed Preferences (SARP)
The strong axiom of revealed preferences (SARP) is equivalent to the weak axiom of revealed preferences, except that the choices A and B are not allowed to be either directly or indirectly revealed preferable to each other at the same time. Here A is considered ''indirectly'' revealed preferred to B iff C exists such that A is directly revealed preferred to C, and C is directly revealed preferred to B. In mathematical terminology, this says that transitivity is violated. Transitivity is useful as it can reveal additional information by comparing two separate bundles from budget constraints. It is often desirable in economic models to prevent such "loops" from happening, for example in order 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 choices at large, i.e. without any price considerations or affordability constraints. This is useful because when evaluating as standalone options, it is ''directly'' obvious which is preferred or indifferent to which other. 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 characterised 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). 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.Generalised Axiom of Revealed Preference (GARP)
Criticism
Several economists criticised the theory of revealed preferences for different reasons. # Stanley Wong claimed that revealed preference theory was a failed research program. 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 behavioural options were discarded in preference of purchasing an orange. In this sense, preference is not revealed at all in the sense of ordinal utility. # The revealed preference theory assumes that the preference scale remains constant over time. Were this not the case all that can be stated is that an action, at a specific point of time, reveals part of a person's preference scale at that time. There is no warrant for assuming that it remains constant from one point of time to another. The "revealed preference" theorists assume constancy in addition to consistent behaviour ("rationality"). Consistency means that a person maintains a transitive order of rank on his preference scale (if A is preferred to B and B is preferred to C, then A is preferred to C). But the revealed preference procedure does not rest on this assumption so much as on an assumption of constancy—that an individual maintains the same value scale over time. While the former might be called irrational, there is certainly nothing irrational about someone's value scales changing through time. It is claimed that no valid theory can be built on a constancy assumption.See also
*Notes
References
* * Section 8.7External links