# Revealed Preference

# Revealed Preference

The theory of revealed preference was originally developed by Paul Samuelson using choice as a primitive to derive some fundamental principles of demand theory. In contrast, in the traditional framework of utility theory approach, the weak preference ordering of a chooser is the primitive. The motivation was that an observer does not know or observe a chooser’s preferences, but a bundle of goods a consumer buys is observable. Put differently, Samuelson’s objective was to derive the laws of demand from assumptions regarding price-quantity data without getting into any traces of preferences. The basic approach is based on the notion that in a given price-income situation, a chooser has an affordable set of consumption bundles, the *budget set*, and if a consumer chooses a particular bundle over all alternative bundles in the budget set, then this choice is revealed to be the chooser’s underlying preference.

This revealed preference is not necessarily the true preference of the chooser. Rather, it is the chooser’s “as if” preference. Suppose for a price-income situation (*p,M* ), *B* (*p,M* ) is the budget set. A demand function is a rule that specifies exactly one element of *B* (*p*,*M* ) for every budget set *B* (*p,M* ). If both *x* and *y* are in *B* (*p*,*M* ), and *x* is chosen, then *x* is said to be (directly) strictly *revealed preferred* to *y*. It is *directly* revealed preferred because both *x* and *y* are compared in a budget set *B* (*p*,*M* ); it is *strictly* revealed preferred since *x* is chosen from a budget set while *y* is rejected, since demand function is single-valued.

Samuelson introduces a condition, known as the *weak axiom of revealed preference* (WARP), which says that if *x* is (directly) strictly revealed preferred to *y*, then there is no other price-income situation (i.e., budget set) in which *y* will be chosen while *x* is available. In other words, WARP requires that if *x* is (directly) strictly revealed preferred to *y*, then *y* cannot be (directly) strictly revealed preferred to *x*. It can be viewed as a two-term consistency. If the demand function satisfies the WARP, then it must satisfy the property that a demand function is homogeneous of degree zero in income and prices; it exhibits the nonpositivity of the own compensated price effect (i.e., the substitution effect); and for every normal good, the price effect must be nonpositive. There are additional implications of the utility theory approach to consumer behavior (namely, symmetry of the cross-substitution term) that are not implied by the WARP.

The rationalizability of the demand function in terms of weak preference orderings implies that the consumer behaves as if he or she has a weak preference ordering and, given the budget set, he or she optimizes on the basis of a realized preference ordering. Rationalizability in terms of weak preference orderings implies, but is not implied by, the WARP. So the WARP needs to be strengthened. Suppose *x* and *z* are two alternatives that never appear together in any budget set. In addition, suppose *x* is (directly) strictly revealed preferred to *y*, *y* is (directly) strictly revealed preferred to *u*, *u* is (directly) strictly revealed preferred to *w*, and *w* is (directly) strictly revealed preferred to *z.* Then we say *x* is *indirectly strictly revealed preferred* to *z*. Strengthening the WARP from a two-term consistency requirement to a multiterm consistency— known as the *strong axiom of revealed preference* (SARP), which says that if *x* is (directly) strictly revealed preferred to *y*, then *y* cannot be directly or indirectly strictly revealed preferred to *x* —it can be demonstrated that this stronger requirement is a necessary and sufficient condition for the rationalizability of a demand function in terms of weak preference orderings (Houthakker). Furthermore, the SARP, together with a continuity condition, implies the existence of a utility function that rationalizes the consumer’s revealed preferences. This establishes the logical equivalence of the revealed preference approach and the utility theory approach.

Following Samuelson’s pioneering work, the revealed preference approach was developed further in an abstract choice theoretic framework. For any given universal set of alternatives *X*, consider all possible nonempty subsets [*X* ] of *X*, which is the power set of the universal set minus the empty set. Consider a well-specified family *S* of nonempty subsets of *X* ; i.e., *S* is a subset of [*X* ]. A choice function is a rule, *C* (•), which for any element *A* of *S* specifies a nonempty subset of *A*, that is, *C* (*A* ) is a subset of *A*. Note that *C* (*A* ) is not necessarily single-valued. Clearly, demand function is a special case of a choice function. Now for any *A* in *S*, if *x* is in *C* (*A* ) and *y* is in *A*, then *x* is said to be (directly) revealed preferred to *y*. If *x* and *z* together are not in any *S*, and if *x* is (directly) revealed preferred to *y*, *y* is (directly) revealed preferred to *u*, *u* is (directly) revealed preferred to *w*, and *w* is (directly) revealed preferred to *z*, then *x* is said to be indirectly revealed preferred to *z*.

For Samuelson, a choice function must be single-valued, and the domain *S* must be the convex polyhedra (or the budget triangles in a two-commodity world) since the objective is to derive the laws of demand of a competitive consumer (i.e., a price taker) from observed data. For the domain *S* = [*X* ], the multiterm consistency requirement of the SARP is collapsed into a two-term consistency. Consequently, the WARP becomes equivalent to the SARP (Arrow).

Suppose a multivalued choice function is defined over any arbitrary domain *S*, a subset of [*X* ]. Marcel Richter (1966) introduces a condition, called *congruence axiom*, which says that if *x* is (directly) revealed preferred to *y*, then *y* cannot be directly or indirectly revealed preferred to *x*. This condition is necessary and sufficient for transitive rationalization of a multivalued choice function in a general domain framework. Clearly, using this general approach, the demand theory results can be derived as a special case. This generalized framework helps to analyze rational choice in a wider context, such as social choice theory, voting theory, and production analysis.

The revealed preference approach developed in two directions. Following the requirement of the preference-based approach that a weak preference relation is transitive, the revealed preference literature originally developed to focus on transitive rationalization of a choice function. For the existence of a maximum, transitivity is indeed very restrictive. A less restrictive requirement could be transitivity of strict preference relation (known as *quasi-transitivity* ) and the least restrictive requirement could be acyclicity of strict preference relation. The WARP requires that no revealed inferior alternative of a set *A* can be chosen in any other set in the presence of *any* revealed preferred alternative in *A*. This consistency condition can be weakened by simply requiring that no revealed inferior alternative of a set *A* can be chosen in any other set in the presence of *some* revealed preferred alternative in *A*. This weaker requirement turns out to be a necessary and sufficient condition for quasi-transitive rationalization of a choice function. Furthermore, if one requires that no revealed inferior alternative of a set *A* can be chosen in any other set in the presence of *some* alternative in *A*, then this condition turns out to be necessary and sufficient for (acyclic) rationalization of a choice function. These results are obtained for the domain *S* = [*X* ] using Samuelsonian revealed preference approach (Bandyopadhyay and Sengupta). Following Arrow’s approach, there is a large literature on alternative characterization of various weak rationalization for the domain *S* = [*X* ]. The results for the domain *S* = [*X* ] can also be obtained for the domain that consists with all pairs and triples in [*X* ]. However, there is no complete characterization of weaker rationalization result for any arbitrary *S* that is a subset of [*X* ].

The other line of development is in the context of stochastic demand theory. For every price-income situation (*p,M* ), a stochastic demand function (SDF) is a rule *d*, which, for a given budget set *B* (*p,M* ), specifies exactly one probability measure *q* over the class of all subsets of *B* (*p,M* ). Let *q* = *d* (*p,M* ), where *d* is an SDF, and let *A* be a subset of *B* (*p,M* ). Then *q* (*A* ) is the probability that the bundle chosen by the consumer from *B* (*p,M* ) will lie in the set *A*. Let *B* (*p,M* ) and *B* (*p*,M** ) be the budget sets for two price-income situations. A stochastic demand function, *d*, satisfies the *weak axiom of stochastic revealed preference* (WASRP) whenever for all pairs of price-income situations, and for all subsets of the intersection of the two budget sets, *A*, *q** (*A* ) - *q* (*A* ) ≤ *q* (*B* (*p,M* ) –*B* (*p*,M** )), where *q* = *d* (*p,M* ) and *q** = *d* (*p*,M** ). WASRP generalizes laws of demand from its deterministic environment to a stochastic environment (Bandyopadhyay, Dasgupta and Pattanaik, 1999). The rationalizability of a stochastic demand function in terms of stochastic orderings has also been established. Recently, these results are generalized further by considering a stochastic demand correspondence.

The revealed preference theory has been applied largely in the evaluation of economic index numbers and in empirical tests of consistency in consumer behavior. The WARP can be used to evaluate directly a given index number.

**SEE ALSO** *Demand; Preferences; Samuelson, Paul A.; Tastes; Utility Function*

## BIBLIOGRAPHY

Arrow, Kenneth J. 1959. Rational Choice Functions and Orderings. *Economica* 26: 121–127.

Bandyopadhyay, Taradas, and Kunal Sengupta. 1993. Characterization of Generalized Weak Orders and Revealed Preference. *Economic Theory* 3: 571–576.

Bandyopadhyay, Taradas, Indraneel Dasgupta, and Prasanta K. Pattanaik. 1999. Stochastic Revealed Preference and the Theory of Demand. *Journal of Economic Theory* 84: 95–110.

Houthakker, H. S. 1950. Revealed Preference and the Utility Function. *Economica* (N.S.) 17: 159–174.

Richter, Marcel K. 1966. Revealed Preference Theory. *Econometrica* 34: 635–645.

Samuelson, Paul. A. 1948. Consumption Theory in Terms of Revealed Preference. *Economica* 15: 243–253.

*Taradas Bandyopadhyay*

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