# Revealed preference theory (1938)

Pioneered by American economist Paul Samuelson (1915- ), revealed preference theory is a method by which it is possible to discern consumer behavior on the basis of variable prices and incomes.

A consumer with a given income will buy a mixture of products; as his income changes, the mixture of goods and services will also change. It is assumed that the consumer will never select a combination which is more expensive than that which was previously chosen.

Revealed preference theory deliberately ignores measures of utility and indifference. An empirical utility theory, it superseded cardinal utility in consumer theory.

Also see: social welfare function

Source:
P A Samuelson, ‘A Note on the Pure Theory of Consumers’ Behaviour’, Econometrica NS, 5 (1938), 353-54

## Motivation

Revealed preference theory tries to understand the preferences of a consumer among bundles of goods, given their budget constraint. For instance, if a consumer buys bundle of goods A over bundle of goods B, where both bundles of goods are affordable, it is revealed that they directly prefer 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.

## Definition and theory

If bundle b is revealed preferred over bundle a in budget set B, then the WARP says that bundle a can not be strictly revealed preferred over bundle b in any budget set B’. This would be equally true had a been located anywhere else in the pink area. The bundle c will not violate WARP even if it is chosen in budget set B’, because it is not in the pink area.

Let there be two bundles of goods, a and b, available in a budget set {\displaystyle 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 {\displaystyle B} is defined for two goods; {\displaystyle X=X_{1},X_{2}}, and determined by prices {\displaystyle p_{1},p_{2}} and income {\displaystyle m}, then let bundle a be {\displaystyle (x_{1},x_{2})\in B} and bundle b be {\displaystyle (y_{1},y_{2})\in B}. This situation would typically be represented arithmetically by the inequality {\displaystyle 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 {\displaystyle p_{1}x_{1}+p_{2}x_{2}=m} and {\displaystyle p_{1}y_{1}+p_{2}y_{2}=m} are satisfied. If, in this situation, it is observed that {\displaystyle (x_{1},x_{2})} is chosen over {\displaystyle (y_{1},y_{2})}, we conclude that {\displaystyle (x_{1},x_{2})} is (directly) revealed preferred to {\displaystyle (y_{1},y_{2})}, which can be summarized as the binary relation {\displaystyle (x_{1},x_{2})\succeq (y_{1},y_{2})} or equivalently as {\displaystyle \mathbf {a} \succeq \mathbf {b} }.[3]

### The 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 his 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 strictly 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, while prices remain constant. Formally:

{\displaystyle \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 {\displaystyle x} and {\displaystyle y} 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 }.

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

{\displaystyle p\cdot x(p’,m’)\leq m~\Rightarrow ~p’\cdot x(p,m)>m’~}.

Such that {\displaystyle x(p’,m’)\neq x(p,m)}.

### 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 consumer is not allowed to be indifferent between the two bundles that are compared. That is, if WARP concludes {\displaystyle \mathbf {a} \succeq \mathbf {b} }, SARP goes a step further and concludes {\displaystyle \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: {\displaystyle C(A,B)=A} , {\displaystyle C(B,C)=B} , {\displaystyle C(C,A)=C}, where {\displaystyle 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.

## Criticism

Several economists criticized the theory of revealed preferences for different reasons.

1. Stanley Wong claimed that revealed preference theory was a failed research program.[4] 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.
2. 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.[5]
3. The revealed preference theory assumes that the preference scale remains constant over time. If this were not the case all we can say 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 behavior (“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.[6]
4. The inability to define or measure preferences independently of ‘revealed-preferences’ leads some authors to see the concept as a tautological fallacy. See, inter alia, Amartya Sen’s critiques in a series of articles: “Behaviour and the concept of preference” (Sen 1973), “Rational Fools: A Critique of the Behavioural Foundations of Economic Theory” (Sen 1977), “Internal Consistency of Choice” (Sen 1993), “Maximization and the Act of Choice” (Sen 1997), and his book ‘Rationality and Freedom’ (Sen 2002).

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