Revealed preference theory, pioneered by American economist Paul Samuelson, is a method by which it is possible to discern the best possible option on the basis of consumer behavior. Essentially, this means that the preferences of consumers can be revealed by their purchasing habits. Revealed preference theory came about because the theories of consumer demand were based on a diminishing marginal rate of substitution (MRS). This diminishing MRS is based on the assumption that consumers make consumption decisions based on their intent 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 creating a means to define utility functions by observing behavior.
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The revealed preference theory is trying to understand the preferences of a consumer among bundles of goods available to him, given his budget constraint. For instance, if the consumer buys the bundle of goods A over the bundle of goods B, where both the bundles of goods are affordable, it is revealed that A is directly preferred over B. It is assumed that the consumer's preferences are stable over the observed time period, i.e. the consumer will not reverse his relative preferences regarding A and B.
As a concrete example, if a person chooses the bundle {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.
This assumption implies that preferences are transitive. In other words, if we have bundles A, B, C, ..., Z, and A is revealed preferred to B, which is in turn revealed preferred to C and so on, then it follows that A is revealed preferred to C through Z. Under these hypotheses, economists are able to chart indifference curves which are employed in many models of consumer theory.
Let there be 2 bundles of goods (x1,x2) and (y1,y2) available at price (p1,p2),assuming that the consumer has an income 'm'. It is observed that the consumer buys (x1,x2) bundle of goods.To translate this arithmetically following equation is formulated p1y1+p2y2<m or p1y1+p2y2=m The above equation indicates that the bundle of goods (y1,y2) satisfies the budget constraint or in other words (y1,y2) is affordable by the consumer.But it is already stated that the consumer buys (x1,x2) bundle of goods,which implies p1x1+p2x2=m the above equation also satisfies the condition of the bundle of goods being affordable within the budget constraint and in this case it satisfies the condition with equality.Now, putting the above equations together, knowing that the bundle of goods (y1,y2) was affordable at the given budget constraint(p1,p2,m) the consumer bought (x1,x2) bundle of goods , thus the final equation of revealed preference is as stated below p1x1+p2x2>p1y1+p2y2 from the preceding equation we derive that the ,consumer prefers bundle of goods (x1,x2) over bundle of goods (y1,y2) or we can say that bundle of goods (x1,x2) is directly revealed preferred to (y1,y2). [1]
The weak axiom of revealed preference (WARP) is a characteristic on the choice behavior of an economic agent.The weak axiom of revealed preference states that if a consumer prefers bundle of good "A" over bundle of good "B" it will never happen so that in any situation where ,both "A" and "B" are present the consumer chooses bundle of good "B", we can also say that when good"A" is revealed preferred to good "B" good "B" will never be revealed preferred to good "A". For example, if an individual chooses orange out of a set of options including apple, they should never choose apple when faced with a choice of a different set of options which also includes orange and apple. More formally, if Apple is ever chosen when orange is available, then there can be no set containing both alternatives from which apple is chosen and orange is not. These two definitions however do not state the same necessary restrictions to satisfy WARP. The former prohibits ever choosing apple after orange was once chosen over apple. The latter (and weaker restriction) only requires to choose orange as well, if apple were to be chosen out of several choices.This characteristic can be stated as a characteristic of Walrasian demand functions as seen in the following example. Let pa be the price of apples and pb be the price of bananas, and let the amount of money available be m=5. If pa =1 and pb=1, and if the bundle (2,3) is chosen, it is said that the bundle (2,3) is revealed preferred to (3,2), as the latter bundle could have been chosen as well at the given prices. More formally, assume a consumer has a demand function x such that they choose bundles x(p,w) and x(p',w') when faced with price-wealth situations (p,w) and (p',w') respectively. If p·x(p',w') ≤ w then the consumer chooses x(p,w) even when x(p',w') was available under prices p at wealth w, so x(p,w) must be preferred to x(p',w').
The strong axiom of revealed preference (SARP) is an expansion of the concept of the weak axiom. A choice behavior that satisfies the weak axiom can form circles. That is if A is preferred to B and B to C then under the weak axiom it is possible that C is preferred to A. The strong axiom makes this behavior impossible, as it is the same as weak axiom plus the requirement that circles are not possible. (In two dimensions WARP=SARP).
Stanley Wong[2] argues that revealed preference theory is a failed research program. According to Wong, Samuelson's 1938 presented revealed preference theory as an alternative theory to utility theory, 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.[3] One of the critics of the revealed preference theory states that "Instead of replacing 'metaphysical' terms such as 'desire' and 'purpose'" they "used it to legitimize them by giving them operational definitions." Thus in psychology, as in economics, the initial, quite radical operationalist ideas eventually came to serve as little more than a "reassurance fetish" for mainstream methodological practice."[4]