Microeconomics
In microeconomics, a mathematical model for preferences of consumers can be as follows. For a discussion on the validity of this model, see also Indifference curve.
Let S be the set of all "packages" of goods and services.
For each consumer there is assumed to be binary relation <=, called a preference relation, on S.
a<=b means: b is at least as preferable as a.
Assumed properties:
- The relation is reflexive: a<=a (this is logically true)
- The relation is transitive: a<=b and b<=c then a<=c (this is logically true, but in practice consumers may not always be that consistent)
- For all a and b in S we have a<=b or b<=a or both:
- a<=b and not b<=a; this is also written a
- the converse
- a<=b and b<=a; this is also written a~b: the consumer prefers a and b equally, he or she is indifferent to the choice.
In practice, S is a very large set and the consumer is not conscious of all preferences. For example, one does not have to make up one's mind about whether one prefers to go on holiday by plane or by train if one does not have enough money to go on holiday anyway (although it can be nice to dream about what one would do if one would win the lottery). Also, one does not have to choose between two cheap things if one can can easily afford a third more attractive version.
However, in practice the consumer makes lots of choices, and if a is chosen while b also could have been chosen (say, they cost the same), it is reasonable to assume that apparently b<=a.
The indifference relation ~ is easily shown to be an equivalence relation. Thus we have a quotient set S/~ of equivalence classes of S, which forms a partition of S.
Each equivalence class is a set of packages that is equally preferred.
Based on the preference relation on S we have a preference relation on S/~. As opposed to the former, the latter is antisymmetric and a total order.
In the case of only two products the equivalence classes can be graphically represented as indifference curves.
For a given preference relation on S we may construct a utility function U on S, with U(a)<=U(b) if and only if a<=b. It is not unique, it is determined up to a strictly monotonically increasing function.
Conversely, from a utility function follows a preference relation.
All the above is independent of the prices of the goods and services and independent of the budget of the consumer. These determine the feasible packages (those he or she can afford). In principal the consumer chooses a package within his or her budget such that no other feasible package is preferred over it; the utility is maximized.
See also
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