Revealed preference

Revealed preference theory, pioneered by economist Paul Anthony Samuelson in 1938,[1][2] is a method of analyzing choices made by individuals, mostly used for comparing the influence of policies[further explanation needed] on consumer behavior.

Revealed preference theory arose 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 maximise their utility.

Revealed preference theory was a means to reconcile demand theory by defining utility functions by observing behaviour.

Taking economics to be an empirical subject, there is the issue that one cannot observe preferences.

If it is observed that a is chosen over b, then a is considered (directly) revealed preferred to b.

This situation would typically be represented arithmetically by the inequality

and graphically by a budget line in the positive real numbers.

[3] WARP is one of the criteria which needs to be satisfied in order to make sure that the consumer is consistent with their preferences.

where both a and b are feasible bundles, but b is chosen when the consumer faces some other budget set

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.

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 {A,B,C} 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 is a generalisation of the strong axiom of revealed preference.

It is the final criteria required so that constancy may be satisfied to ensure consumers preferences do not change.

This axiom accounts for conditions in which two or more consumption bundles satisfy equal levels of utility, given that the price level remains constant.

It covers circumstances in which utility maximisation is achieved by more than one consumption bundle.

[4] A set of data satisfies the general axiom of revealed preference if

given that prices remain constant, cannot be more than the expenditure necessary to acquire bundle

thus ruling out “preference cycles” while still holding transitivity.

This is a result of the condition in which the generalised axiom is compatible with multivalued demand functions, where as SARP is only compatible with single valued demand functions.

As such, the generalised axiom permits for flat sections within indifference curves, as stated by Hal R Varian (1982).

[5] If a set of preference data satisfies GARP, then there exists a strictly increasing and concave utility function that rationalizes the preferences (Afriat 1967).

[7]r Revealed preference theory has been used in numerous applications, including college rankings in the U.S.[8][9] Several economists criticised the theory of revealed preferences for different reasons.

If bundle a is revealed preferred over bundle b in budget set B, then the WARP says that bundle b cannot be revealed preferred over bundle a in any budget set B'. This would be equally true had b been located anywhere else on or below the blue line. The bundle c will not violate WARP even if it is chosen in budget set B', because it is not on or below the blue line of affordable choices at the time of the observed choice of a .
The dataset illustrates a budget constraint in which there are two consumption bundles a and b. Both bundles achieve utility maximisation, violating the SARP, but satisfying GARP.