Revealed preference

Revealed preference theory, pioneered by American economist Paul Samuelson, is a method 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.

Theory

Revealed preference theory tries to understand the preferences of a consumer among bundles of goods, given her 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 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 her 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.

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.

Algebraic Analysis

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 This equation indicates that the bundle of goods (y1,y2) satisfies the budget constraint or in other words (y1,y2) is affordable by the consumer. The consumer's purchase of (x1,x2) instead 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. 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, 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).

The weak axiom

The weak axiom of revealed preference (WARP) is a characteristic on the choice behavior of an economic agent. WARP states that if a consumer prefers bundle "A" over bundle "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 an 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 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 that the consumer choose orange whenever apple is chosen. This characteristic can be stated as a characteristic of Walrasian demand functions as seen in the following example. Let p_a be the price of apples and p_b be the price of bananas, and let the amount of money available be m=5. If p_a =1 and p_b=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

The strong axiom of revealed preference (SARP) expands 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).

Criticism

Stanley Wong argued that revealed preference theory was a failed research program. According to Wong, in 1938 Samuelson presented revealed preference theory as an alternative 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. 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."

Counter Example: Given that I prefer the second cheapest flower in a set of flowers {x,y}, then C{x, y} = {x}. If a less expensive flower is added to the set then C{x,y,z} = {y}, which contradicts WARP. Sen 1993, p 501 lays out an explicit argument.

See also

• Conjoint analysis
• Hedonic regression
• Travel cost analysis
• Contingent valuation or stated preference methods

References

• Nicholson, W. (2005) Microeconomics, Thomson, Southwestern.
• Mas-Colell, A.; Whinston, M.D.; Green, J.R. (1995) "Microeconomic Theory", First Edition, New York: Oxford University Press, New York
• Samuelson, P. (1938). A Note on the Pure Theory of Consumers' Behaviour. Economica 5:61-71.
• Varian, H. (1992) Microeconomic Analysis, Third edition, New York: Norton, Section 8.7

External links

• Revealed Preference, review by Hal R. Varian, 2005, prepared for Samuelsonian Economics and the 21st Century.
• Lecture Notes in Microeconomic Theory, book by Ariel Rubinstein, 2005.