# Preference

Preference (or "taste") is a concept, used in the social sciences, particularly economics. It assumes a real or imagined choice between alternatives and the possibility of rank ordering of these alternatives. More generally, it can be seen as a source of motivation.

For example, happiness is generally preferred to suffering, sadness, or grief. Also, more consumption of a normal good is generally (but not always) assumed to be preferred to less consumption.

## Microeconomics

In microeconomics, preferences of consumers and other entities are modelled with preference relations.

Let S be the set of all "packages" of goods and services (or more generally "possible worlds"). Then ≤ is a preference relation on S if it is a binary relation on S such that a ≤ b if and only if b is at least as preferable as a. It is conventional to say "b is weakly preferred to a", or just "b is preferred to a". If a ≤ b but not b ≤ a, then the consumer strictly prefers b to a, which is written a < b. If a ≤ b and b ≤ a then the consumer is indifferent between a and b.

• The relation is transitive: a ≤ b and b ≤ c then a ≤ c.
• The relation is complete: for all a and b in S we have a ≤ b or b ≤ a or both. This means the consumer is able to form an opinion about the relative merit of any pair of bundles.
• If S is a topological space, then the relation is continuous if for every pair of convergent sequences [itex]x_n \rightarrow x[itex] and [itex]y_n \rightarrow y[itex] with [itex]x_n \leq y_n[itex] for all n has x ≤ y. This is automatically satisfied if S is finite.

If ≤ is both transitive and complete, then it is a rational preference relation.

If a consumer has a preference relation that violates transitivity, then an inscrupulous person can milk them as follows. Suppose the consumer has an apple, and prefers apples to oranges, oranges to bananas, and bananas to apples. Then, the consumer would be prepared to pay, say, one cent to trade their apple for a banana, because they prefer bananas to apples. After that, they would pay once cent to trade their banana for an orange, and again the orange for an apple, and so on.

Completeness is more philosophically questionable. In most applications, S is an infinite 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). However, preference can be interpreted as a hypothetical choice that could be made rather than a conscious state of mind. In this case, completeness amounts to an assumption that the consumer can always make up their mind whether they are indifferent or prefer one option when presented with any pair of options.

Behavioral economics investigates the circumstances when human behavior is consistent and inconsistent with these assumptions.

The indifference relation ~ is 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. If there are only two commodities, the equivalence classes can be graphically represented as indifference curves. 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.

It is usually more convenient to describe a preference relation on S with a utility function [itex]u : S \rightarrow \textbf R[itex], such that u(a) ≤ u(b) if and only if a ≤ b. A continuous utility function always exists if ≤ is a continuous rational preference relation on [itex]R^n[itex]. For any such preference relation, there are many continuous utility functions that represent it. Conversely, every utility function can be used to construct a unique 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 principle the consumer chooses a package within his or her budget such that no other feasible package is preferred over it; the utility is maximized.

## References

Kreps, David (1990). A Course in Microeconomic Theory. New Jersey: Princeton University Press. ISBN 0691042640

Mas-Colell, Andreu; Whinston, Michael; & Green, Jerry (1995). Microeconomic Theory. Oxford: Oxford University Press. ISBN 0195073401

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