PREFERENCE RELATIONS

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F 5. _{The sequence definition of a boundary point}

2. Preference relations

2.1. Relations and equivalence classes. Our aim is to consider re- lations on the goods spaceRℓ

+. However, we begin with three examples from

outside preference theory.

E $ _{IV.8. For any two inhabitants from Leipzig, we ask whether} • one is the father of the other or

• they are of the same sex.

E $ _{IV.9. For the set of integers Z (the numbers ..., −2, −1, 0,} 1, 2, ...) , we consider the difference and examine whether this difference is an even number (i.e., from ..., −2, 0, 2, 4, ...).

All three examples define relations, the first two on the set of the in- habitants from Leipzig, the last on the set of integers. Often, relations are expressed by the symbol ∼ . To take up the last example on the set of inte- gers, we have 5 ∼ −3 (the difference 5 − (−3) = 8 is even) and 5 ≁ 0 (the difference 5− 0 = 5 is odd).

D _{IV.12 (relation). A relation on a set M is a subset of M×} M. If a tuple (a, b) ∈ M × M is an element of this subset, we often write a∼ b.

Relations have, or have not, specific properties:

D _{IV.13 (properties of relations). A relation∼ on a set M is} called

• reflexive if a ∼ a holds for all a ∈ M,

• transitive if a ∼ b and b ∼ c imply a ∼ c for all a, b, c ∈ M, • symmetric if a ∼ b implies b ∼ a for all a, b ∈ M,

• antisymmetric if a ∼ b and b ∼ a imply a = b for all a, b ∈ M, and • complete if a ∼ b or b ∼ a holds for all a, b ∈ M.

L _{IV.6. On the set of integers Z, the relation ∼ defined by} a∼ b :⇔ a − b is an even number

is reflexive, transitive, and symmetric, but neither antisymmetric nor com- plete.

“:⇔” means that the expression left of the colon is defined by the ex- pression to the right of the equivalence sign.

P . _{We have a− a = 0 for all a ∈ Z and hence a ∼ a; therefore,} ∼ is reflexive. For transitivity, consider any three integers a, b, c that obey a∼ b and b ∼ c. Since the sum of two even numbers is even, we find that

(a− b) + (b − c) = a− c

is also even. This proves a ∼ c and concludes the proof of transitivity. Symmetry follows from the fact that a number is even if and only if its negative is even.

∼ is not complete which can be seen from 0 ≁ 1 and 1 ≁ 0. Finally, ∼ is not antisymmetric. Just consider the numbers 0 and 2.

E _{IV.7. Which properties do the relations “is the father of” and} “is of the same sex as” have? Fill in “yes” or “no”:

property is the father of is of the same sex as reflexive

transitive symmetric antisymmetric complete

D _{IV.14 (equivalence relation). Let∼ be a relation on a set M} which obeys reflexivity, transitivity and symmetry. Then, any two elements a, b ∈ M with a ∼ b are called equivalent and ∼ is called an equivalence relation. By an equivalence class of a∈ M, we mean the set

[a] :={b ∈ M : b ∼ a} .

Our relation on the set of integers (even difference) is an equivalence relation. We have two equivalence classes:

[0] = {b ∈ M : b ∼ 0} = {..., −2, 0, 2, 4, ...} and [1] = {b ∈ M : b ∼ 1} = {..., −3, −1, 1, 3, ...}

E _{IV.8. Continuing the above example, find the equivalence class-} es [17] , [−23] , and [100]. Reconsider the relation “is of the same sex as”. Can you describe its equivalence classes?

Generalizing the above example, a∼ b implies [a] = [b] for every equiv- alence relation. Here comes the proof. Consider any a′ ∈ [a] . We need to show a′ ∈ [b]. Now, a′ _{∈ [a] means a}′_{∼ a. Together with a ∼ b, transitivity}

2. PREFERENCE RELATIONS 61

implies a′ ∼ b and hence a′ ∈ [b] . We have shown [a] ⊆ [b] . The converse, [b]⊆ [a], can be shown similarly.

The following lemma uses the above result and the observation a∈ [a] which is true by reflexivity.

L _{IV.7. Let ∼ be an equivalence relation on a set M. Then, we} have

a_∈M

[a] = M and

[a] = [b]⇒ [a] ∩ [b] = ∅.

Thus, equivalence classes form a partition of the underlying set.

The other direction holds also: Once we have a partition, we can define an equivalence relation whose equivalence classes are equal to the compo- nents of the partition. Just say that two elements are related if they belong to the same component.

2.2. Preference relations and indifference curves. We now as- sume that every household i has weak preferences (a weak preference rela- tion) on the goods spaceRℓ

+, denoted by i. x i y means that household

i finds y at least as good as x. If there is no doubt about the household we are talking about, we omit the index.

D _{IV.15 (preference relation). A (weak) preference relation} is a relation on Rℓ

+ that is complete, transitive and reflexive. Given a

preference relation , the indifference relation is defined by x∼i y :⇔ x iy and y ix

and the strict preference by

x≺i y :⇔ x i y and not y ix.

While it is hard to imagine preferences without reflexivity, completeness and transitivity are not as innocent as they seem. Completeness means that households can always make up their mind. However, “real” households will sometimes have a hard time to find out what they “really” want. Also, if confronted with many good bundles, people will often violate transitivity. We discuss the money-pump argument against the violation of transitivity in chapter III, pp. 36.

E _{IV.9. Is the indifference relation a preference relation or an} equivalence relation? How about the strict preference relation? Fill in:

property indifference strict preference reflexive

transitive symmetric complete

2 x 1 x 3 6 2 x 1 x 6 3

F 6. _{Numbers associated with indifference curves}

D _{IV.16 (better set, indifference set). Let be a preference} relation on Rℓ

+. The better set By of y is given by

By := x∈ Rℓ₊ : x y .

The worse set Wy of y is

Wy := x∈ Rℓ+: x y .

y’s indifference set Iy is the intersection of its better and worse set:

Iy := By ∩ Wy= x∈ Rℓ+: x∼ y

The geometric locus of an indifference set is called an indifference curve.

The set of indifference sets partition the goods space. This means that every bundle belongs to one and only one indifference curve. You know from intermediate microeconomics that indifference curves cannot intersect. This follows from the fact that indifference relations are equivalence relations and, in particular, from [a] = [b]⇒ [a] ∩ [b] = ∅ in lemma IV.7.

When we draw indifference curves, we often associate them with numbers where a higher number indicates strict preference. Consider fig. 6. The left- hand graph stands for preferences of so-called goods where the consumer would like to have more of both goods. The right-hand graph represents so-called bads, i.e., the consumer wants as small an amount of both goods as possible. Think of dirt and noise.

E _{IV.10. Sketch indifference curves for a goods space with just} 2 goods and, alternatively,

• good 2 is a bad (the consumer would like to have less of that good), • good 1 represents red matches and good 2 blue matches,

• good 1 stands for right shoes and good 2 for left shoes.

Lexicographic preferences lex are very interesting preferences. In the

two-good case they are defined by

3. AXIOMS: CONVEXITY, MONOTONICITY, AND CONTINUITY 63