Example of Inconsistent Choices
The following example describes a pair of choices that are inconsistent with regularity properties P.1 - P.4 for preferences.
Figure 5: Choices that are inconsistent with regular preferences.
Intuitively, the problem with this pattern of choices is that, when the price of CDs is low (pCD = 8) and the price of DVDs is high (pDVD = 12), the consumption of DVDs exceeds the consumption of CDs, and when the price of CDs is high, and the price of DVDs is low, the consumption of CDs is high, and the consumption of DVDs is low.
Analysis of Inconsistency
The inconsistency in the choices xa and xb in figure 5 with the properties of preferences is demonstrated with a simple argument.
Role of the Properties of Preferences
One of the motivations for the properties of preferences is that, for a consumer whose preferences exhibit the regularity properties P.1 through P.4, and one additional property called continuity, the consumer's preferences can be represented by a utility function, and utility functions are useful because it is possible to evaluate changes in a consumer's choice patterns when the consumer's choice set varies. One important example of a consumer's response to changes in the choice set is a demand function. In the next section, an argument is developed that shows that if a consumer's preferences satisfy P.1 through P.4 and contnuity, then her preferences can be represented by a demand function u(x) such that u(x) > u(y) if and only if x y.
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