Revealed preference

Revealed preference theory, pioneered by American economist Paul Samuelson, is a method of analyzing choices made by individuals, mostly used 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.

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.

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 ${\displaystyle \mathbf {a} \succeq \mathbf {b} }$.

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