Revealed preference

If the budget set ${\displaystyle B}$ is defined for two goods; ${\displaystyle X,Y}$, and determined by prices ${\displaystyle p,q}$ and income ${\displaystyle m}$, then let bundle a be ${\displaystyle (x_{1},y_{1})\in B}$ and bundle b be ${\displaystyle (x_{2},y_{2})\in B}$. This situation would typically be represented arithmetically by the inequality ${\displaystyle pX+qY\leq m}$ 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 ${\displaystyle px_{1}+qy_{1}=m}$ and ${\displaystyle px_{2}+qy_{2}=m}$ are satisfied, need to be considered. If, in this situation, it is observed that ${\displaystyle (x_{1},y_{1})}$ is chosen over ${\displaystyle (x_{2},y_{2})}$, it is concluded that ${\displaystyle (x_{1},y_{1})}$ is (directly) revealed preferred to ${\displaystyle (x_{2},y_{2})}$, which can be summarized as the binary relation ${\displaystyle (x_{1},y_{1})\succeq (x_{2},y_{2})}$ or equivalently as ${\displaystyle \mathbf {a} \succeq \mathbf {b} }$.[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. 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:

${\displaystyle \left.{\begin{matrix}\mathbf {a} ,\mathbf {b} \in B\\\mathbf {a} \in C(B,\succeq )\\\mathbf {b} \in B'\\\mathbf {b} \in C(B',\succeq )\end{matrix}}\right\}~\Rightarrow ~\mathbf {a} \notin B'}$

where ${\displaystyle \mathbf {a} }$ and ${\displaystyle \mathbf {b} }$ are arbitrary bundles and ${\displaystyle C(B,\succeq )\subset B}$ is the set of bundles chosen in budget set ${\displaystyle B}$, given preference relation ${\displaystyle \succeq }$.

In other words, if a is chosen over b in budget set ${\displaystyle B}$ where both a and b are feasible bundles, but b is chosen when the consumer faces some other budget set ${\displaystyle B'}$, then a is not a feasible bundle in budget set ${\displaystyle B'}$.

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 if 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 preserved. 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.

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.

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 ${\displaystyle x^{i}Rx^{j}}$ implies not ${\displaystyle x^{j}P^{0}x^{i}}$.[5] This establishes that if consumption bundle ${\displaystyle x^{i}}$ is revealed preferred to ${\displaystyle x^{j}}$, then the expenditure necessary to acquire bundle ${\displaystyle x^{j}}$ given that prices remain constant, cannot be more than the expenditure necessary to acquire bundle ${\displaystyle x^{i}}$.[6]

To satisfy the generalised axiom of revealed preference a dataset must also not establish a preference cycle. Therefore, when considering the bundles {A,B,C}, the revealed preference bundle must be an acyclic order pair as such, If ${\displaystyle A\succeq B}$ and ${\displaystyle B\succeq C}$, then ${\displaystyle B\nsucceq A}$ and ${\displaystyle A\succeq C}$ thus ruling out “preference cycles” while still holding transitivity.[4]

As the generalised axiom is closely related to the strong axiom of revealed preference, it is very easy to demonstrate that each condition of SARP can imply the general axiom, however, the generalised axiom does not imply the strong axiom. 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]