Microeconomics I #1: Several theorems in Choice Theory

Another technical post before I switch back to my unfinished work on Freefall over the weekend. This time is something micro.

Some preliminaries:

1. Preference is a binary relation ≿ between x,y∈X, the set of all possible choices such that x≿y if and only if "x is at least as good as y".
2. Preference is complete if ∀x,y, either x≿y or y≿x or both. Preference is transitive if x≿y and y≿z imply x≿z.
3. Rational choice theory assumes that preferences are both complete and transitive.
4. Now, suppose we have established a preference ≿, and B is the set of available choices, define the choice rule C(B;≿)={x∈B: x≿y for all y∈B}. Notice C(B;≿) may not be singleton. When B is finite, C(B;≿) is non-empty; however, when B is infinite, C(B;≿) can be empty.
5. For a given choice rule C(X), the Houthaker's Axiom of Revealed Preference (HARP) states if x,y∈A∩B with x∈C(A) and y∈C(B), then x∈C(B) and y∈C(A).

Some propositions:

1. If a predetermined preference ≿ is complete and transitive over the choice set X, then the choice rule C(X;≿) satisfies HARP.
2. Now suppose we only have revealed preferences in the forms of non-empty choice rule C, then there is a complete and transitive preference ≿ with C(A)=C(A;≿) for all subsets A of X if and only if C satisfies HARP.

It is easier if we can represent a preference relation by a utility function.

1. A utility function, u, is a mapping from the choice set X to the real numbers. Hence values of the utility function cannot be positive or negative infinity.
2. We say a utility function u represents a preference relation ≿ (or the preference relation ≿ affords the utility function u) if there is an equivalence between x≿y and u(x)≥u(y).

Some more propositions on utility representation of preferences:

• If the choice set X is finite, then every complete and transitive preference has a utility representation.

Proof: let n be the size of X, label elements in X with subscripts x_1, x_2,...,x_n, with x_n≿x_(n-1)≿...≿x_2≿x_1. Now let u(x_i)=i. Clearly u is a valid utility representation for the preference ≿. QED

• If the choice set X is countable, then every complete and transitive preference has a utility representation.

Proof: Let X={x_1,x_2,...}. Define the set "No Better Than" of some element x, NBT(x)={y∈X: x≿y}. By observation, we know ∀x,y∈X, either NBT(x)⊆NBT(y) or NBT(y)⊆NBT(x). Now define the utility function, u(x)=∑(0.5)^n, where n is the subscript of x_n for all x_n∈NBT(x). Then it is simple to verify (1) u(x) is well-define, in particular is convergent; (2) u is indeed a valid utility representation for the preference ≿. QED

• Note, the above proposition holds for any complete and transitive preferences over a countable choice set X, but not any arbitrary infinite set X. Consider for example X=[0,1]×[0,1], for any two points x=(a,b) and y=(c,d) in X, x≿y if either a>c or a=c but b≥d. This lexicographic preference has no proper utility representation even though it is complete and transitive.

Proof: By the definition of the preference, let α<β∈[0,1], then (β,1)≿(β,0)≿(α,1)≿(α,0). Suppose there is a utility representation u affording this preference, then u(β,1)≥u(β,0)≥u(α,1)≥u(α,0). In fact these inequalities hold strictly. Now consider the two disjoint open intervals, A=(u(α,0),u(α,1)) and B=(u(β,0),u(β,1)). Due to the density of the rational numbers, there exist rational numbers p,q such that p∈A and q∈B. But there are uncountable pairs of real numbers (α,β) with α<β∈[0,1], hence uncountable disjoint open intervals of those forms. This implies there are uncountable rational numbers in [0,1], but it contradicts with the fact that rational numbers are countable. Hence, by contradiction, such utility representation does not exist for this lexicographic preference defined on X. QED

• Sufficient and necessary condition: ≿ over X has a utility representation if and only if ∃X*⊆X with X* being countable such that ∀x,y∈X with x≻y, ∃x*∈X* with x≿x*≻y.