# Revealed preference

Revealed preference theory, pioneered by American economist Paul Samuelson,[1] is a method of analyzing choices made by individuals, mostly used 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.

## Definition and Theory

If bundle a is revealed preferred over bundle b in budget set B, then the WARP says that bundle b can not be strictly revealed preferred over bundle a in any budget set B'. This would be equally true had b been located anywhere else in the pink area. The bundle c never violates WARP because it is not in the pink area.

Let there be two bundles of goods; a and b available in a budget set $B$. If it is observed that a is chosen over b, we say that a is (directly) revealed preferred to b.

### Two-Dimensional Example

If the budget set $B$ is defined for two goods; $X = X_{1},X_{2}$, and determined by prices $p_{1},p_{2}$ and income $m$, then let bundle a be $(x_{1},x_{2}) \in X$ and bundle b be $(y_{1},y_{2}) \in X$. This situation would typically be represented arithmetically by the inequality $p_{1}X_{1} + p_{2}X_{2} \leq m$ and graphically by a budget line in the positive real numbers. Assuming strongly monotonic preferences, we only need to consider bundles that graphically are located on the budget line, i.e. bundles where $p_{1}x_{1} + p_{2}x_{2} = m$ and $p_{1}y_{1} + p_{2}y_{2} = m$ are satisfied. If, in this situation, it is observed that $(x_{1},x_{2})$ is chosen over $(y_{1},y_{2})$, we conclude that $(x_{1},x_{2})$ is (directly) revealed preferred to $(y_{1},y_{2})$, which can be summarized as the binary relation $(x_{1},x_{2}) \succeq (y_{1},y_{2})$ or equivalently as $\mathbf{a} \succeq \mathbf{b}$.[2]

### The Weak Axiom of Revealed Preference (WARP)

WARP is one of the criteria which need to be satisfied in order to make sure that the consumer is consistent with his preferences. If a bundle of goods a is chosen over another bundle b when both are affordable, then the consumer reveals that he prefers a over b. WARP says that when preferences remain the same, there are no circumstances (budget set) where the consumer strictly prefers b over a. By choosing a over b when both bundles are affordable, the consumer reveals that his preferences are such that he will never choose b over a, while prices remain constant. Formally,

$\left.\begin{matrix} x,y \in B\\ x \in C(B, \succeq) \\ x,y \in B' \\ y \in C(B', \succeq) \end{matrix}\right\} ~\Rightarrow~ x \in C(B', \succeq)$

where $x$ and $y$ are arbitrary bundles and $C (B, \succeq) \subset B$ is the set of bundles chosen in budget set $B$, given preference relation $\succeq$.

Alternatively, if a is chosen over b in budget set $B$ where both a and b are feasible bundles, but b is chosen over a when the consumer faces some other budget set $B'$, then a is not a feasible bundle in budget set $B'$. This equivalent statement of WARP can formally and more generally be expressed as

$p \cdot x(p',m') \leq m ~ \wedge ~ x(p',m') \neq x(p,m) ~\Rightarrow ~ p' \cdot x(p,m) > m'~$.

### Completeness and Strong axiom

The strong axiom of revealed preferences (SARP) is equivalent to the weak axiom of revealed preferences, except that the consumer is not allowed to be indifferent between the two bundles that are compared. That is, if WARP concludes $\mathbf{a} \succeq \mathbf{b}$, SARP goes a step further and concludes $\mathbf{a} \succ \mathbf{b}~$ .

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. 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).

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 their 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 he/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 their 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.[3] According to Wong, in 1938 Samuelson presented revealed preference theory as an alternative to utility theory,[1] 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.[4]