Revealed preference theory, pioneered by economist Paul Anthony Samuelson in 1938, is a method of analyzing choices made by individuals, mostly used for comparing the influence of policies on consumer behavior. Revealed preference models assume that the preferences of consumers can be revealed by their purchasing habits.
Revealed preference theory arose 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 maximise their utility. While utility maximisation 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 behaviour.
Therefore, revealed preference is a way to infer the preferences of individuals given the observed choices. It contrasts with attempts to directly measure preferences or utility, for example through stated preferences. Taking economics to be an empirical subject, there is the issue that one cannot observe preferences.
B
If the budget set
B
X,Y
p,q
m
(x1,y1)\inB
(x2,y2)\inB
pX+qY\leqm
px1+qy1=m
px2+qy2=m
(x1,y1)
(x2,y2)
(x1,y1)
(x2,y2)
(x1,y1)\succeq(x2,y2)
a\succeqb
WARP is one of the criteria which needs to be satisfied in order to make sure that the consumer is consistent with their 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 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 when both are affordable, even as prices vary. Formally:
\left.\begin{matrix} a,b\inB\\ a\inC(B,\succeq)\\ b\inB'\\ b\inC(B',\succeq) \end{matrix}\right\} ~ ⇒ ~a\notinB'
where
a
b
C(B,\succeq)\subsetB
B
\succeq
In other words, if a is chosen over b in budget set
B
B'
B'
The strong axiom of revealed preferences (SARP) is equivalent to the weak axiom of revealed preferences, except that the choices A and B are not allowed to be either directly or indirectly revealed preferable to each other at the same time. Here A is considered indirectly revealed preferred to B if C exists such that A is directly revealed preferred to C, and C is directly revealed preferred to B. In mathematical terminology, this says that transitivity is preserved. Transitivity is useful as it can reveal additional information by comparing two separate bundles from budget constraints.
It is often desirable in economic models to prevent such "loops" from happening, for example in order 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 choices at large, i.e. without any price considerations or affordability constraints. This is useful because when evaluating as standalone options, it is directly obvious which is preferred or indifferent to which other. 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 characterised 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.
Generalised axiom of revealed preference is a generalisation of the strong axiom of revealed preference. It is the final criteria required so that constancy may be satisfied to ensure consumers preferences do not change.
This axiom accounts for conditions in which two or more consumption bundles satisfy equal levels of utility, given that the price level remains constant. It covers circumstances in which utility maximisation is achieved by more than one consumption bundle.[1]
A set of data satisfies the general axiom of revealed preference if
xiRxj
xjP0xi
xi
xj
xj
xi
To satisfy the generalised axiom of revealed preference a dataset must also not establish a preference cycle. Therefore, when considering the bundles, the revealed preference bundle must be an acyclic order pair as such, If
A\succeqB
B\succeqC
B\nsucceqA
A\succeqC
As the generalised axiom is closely related to the strong axiom of revealed preference, it is very easy to demonstrate that each condition of SARP can imply the general axiom, however, the generalised axiom does not imply the strong axiom. This is a result of the condition in which the generalised axiom is compatible with multivalued demand functions, where as SARP is only compatible with single valued demand functions. As such, the generalised axiom permits for flat sections within indifference curves, as stated by Hal R Varian (1982).
If a set of preference data satisfies GARP, then there exists a strictly increasing and concave utility function that rationalizes the preferences (Afriat 1967).
Revealed preference theory has been used in numerous applications, including college rankings in the U.S.[4] [5]
Several economists criticised the theory of revealed preferences for different reasons.