Lecture 8: Preference, Choice, and Utility
Advanced Microeconomics I
Yosuke YASUDA
Osaka University, Department of Economics yasuda@econ.osaka-u.ac.jp
October 28, 2014
Preferences
To construct a model of individual choice, the notion of
preferences plays a central role in economic theory, which specifies the form of consistency or inconsistency in the person’s choices. We view preferences as the mental attitude of an individual toward alternatives independent of any actual choice.
We require only that the individual make binary comparisons, that is, that she only examine two choice alternatives at a time and make a decision regarding those two.
For each pair of alternatives in the choice set X, the description of preferences should provide an answer to the question of how the agent compares the two alternatives. We present two versions of question: questionnaires P and R. For each version we formulate the consistency requirements necessary to make the responses “preferences” and examine the connection between the two formulations.
Questionnaire P
P(x, y) for all distinct x and y in the set X:
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Q How do you compare x and y? Tick one and only one of the✆ following three options:
1 I prefer x to y (or, x is strictly preferred to y): x ≻ y
2 I prefer y to x (or, y is strictly preferred to x): y ≻ x
3 I am indifferent (or, x is indifferent to y): x ∼ y
Note that we implicitly assume that the elements in X are all comparable, and ignore the intensity of preferences.
A legal answer to the questionnaire P can be formulated as a function f which assigns to any pair (x, y) of distinct elements in X exactly one of the three values: x ≻ y, y ≻ x or x ∼ y. That is,
f(x, y) =
x≻ y y≻ x x∼ y
.
Preference P (1)
Preferences are characterized by axioms that are intended to give formal mathematical expression to fundamental aspects of choice behavior and attitudes toward the objects of choice.
The following basic axioms are (almost) always imposed. Definition 1
Preferences (P ) on a set X are a function f that assigns to any pair (x, y) of distinct elements in X exactly one of the three values: x ≻ y, y ≻ x or x ∼ y so that for any three different elements x, y and z in X, the following two properties hold:
1 No order effect: f (x, y) = f (y, x).
2 Transitivity:
1 if f (x, y) = x ≻ y and f (y, z) = y ≻ z, then f (x, z) = x ≻ z, and
2 if f (x, y) = x ∼ y and f (y, z) = y ∼ z, then f (x, z) = x ∼ z.
Preference P (2)
The first property requires the answer to P (x, y) being identical to the answer to P (y, x), and the second requires that the answer to P(x, y) and P (y, z) are consistent with the answer to P (x, z) in a particular way.
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Ex Non-preference relation✆
For any x, y ∈R, f(x, y)(= f(y, x)) = x ≻ y if x ≥ y + 1 and f(x, y) = x ∼ y if |x − y| < 1. This is not a preference relation since transitivity is violated. For instance, suppose
x= 1, y = 1.8, z = 2.6. Then,
f(x, y) = x ∼ y and f (y, z) = y ∼ z, but f (x, z) = z ≻ x, which violates transitivity (2-2).
Questionnaire R
R(x, y) for all x, y ∈ X, not necessarily distinct:
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Q Is x at least as preferred as y? Tick one and only one of the✆ following two options:
1 Yes (or, x is at least as good as y): x % y.
2 No (or, x is strictly worse than y): x y. Definition 2
Preferences (R) on a set X is a binary relation % on X satisfying the following two axioms.
Completeness (Axiom 1):
For any x, y ∈ X, x % y or y % x. Transitivity (Axiom 2):
For any x, y, z ∈ X, if x % y and y % z, then x % z.
Remarks on the Axioms
Completeness formalizes the notion that the individual can make comparisons, that is, that she has the ability to discriminate and the necessary knowledge to evaluate alternatives. It says the individual can examine any two distinct alternatives.
Transitivity gives a very particular form to the requirement that the individual choices be consistent. Although we require only that she be capable of comparing two alternatives at a time, the axiom of transitivity requires that those pairwise comparisons be linked together in a consistent way.
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Rm The money pump argument when transitivity is violated.✆ (see, for example, Rubinstein, lecture 3)
Equivalence of the Two Preferences
We can translate one formulation of preferences to another by the following mapping (bijection). Note that completeness guarantees
“x y and y x” never happen.
f(x, y) = x ≻ y ⇔ x % y and y x. f(x, y) = y ≻ x ⇔ y % x and x y. f(x, y) = x ∼ y ⇔ x % y and y % x.
In our lectures, we take the second definition, i.e., preference (R), and denote x ≻ y when both x % y and y x, and x ∼ y, when x% y and y % x.
Definition 3
A preference (R) is called a preference relation.
Utility Representation
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Rev Function U : X →✆ R represents the preference % if for all x and y ∈ X, x % y if and only if U (x) ≥ U (y). If the function U represents the preference relation %, we refer to it as a utility function and we say that % has a utility representation.
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Q Under what conditions do utility representations exist?✆ Theorem 4
If% is a preference relation on a finite set X, then % has a utility representation with values being natural numbers.
Proof.
There is a minimal (resp. maximal) element (an element a ∈ X is minimal (resp. maximal) if a - x (resp. a % x) for any x ∈ X) in any finite set A ⊂ X. We can construct a sequence of sets from the minimal to the maximal and can assign natural numbers according to their ordering.
Continuous Preferences
To guarantee the existence of a utility representation over consumption set, i.e., an infinite subset ofRn, we need some additional axiom.
Definition 5
A preference relation % on X is continuous (Axiom 3) if {xn} (a sequence of consumption bundles) with limit x satisfies the
following two conditions for all y ∈ X.
1 if x ≻ y, then for all n sufficiently large, xn≻ y, and
2 if y ≻ x, then for all n sufficiently large, y ≻ xn.
The equivalent definition of continuity is that the “at least as good as” and “no better than” sets for each point x ∈ X are closed. This axiom rules out certain discontinuous behavior and guarantees that sudden preference reversals do not occur.
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Fg Figures 1.2 and 1.3 (see JR, pp.9)✆
Continuous Utility
Given axioms 1-3, we can establish the existence of the (continuous) utility function.
Theorem 6
Assume that X is a convex subset ofRn. If % is a continuous preference relation on X, then% is represented by a continuous utility function.
Here are two remarks on continuity.
1 If % on X is represented by a continuous function U , then % must be continuous.
2 The lexicographic preferences are not continuous. Theorem 7
The lexicographic preference relation %L on[0, 1] × [0, 1], i.e., (a1, a2) %L(b1, b2) if a1 > b1 or both a1= b1 and a2 ≥ b2, does not have a utility representation.
Existence of Solutions to Consumer Problems
Theorem 8
If% is a continuous preference relation, then all consumer problems have a solution.
Proof.
Since the budget set is convex, we can apply the first theorem in the previous slide to establish that the preferences are represented by a continuous utility function.
Then, by the Weielstrass theorem, there exists a maximum (and minimum) value of continuous functions if the domain is a compact (that is, closed and bounded) set and a range isR. Since every budget set is compact and a utility function is continuous, there must exist a consumption bundle which gives a maximum utility value, a solution of the consumer problem.
Revealed Preferences
Important difference between choice (demand) and preferences or utility is that the former is observable while the latter cannot be. We may want to develop the theory which is based on the observable choice behaviors, not on preferences or utility.
We say that the preferences % (fully) rationalize the demand function x if for any (p, ω) the bundle x(p, ω) is the unique
%best bundle within B(p, ω).
We say that a is revealed to be better than b, if there is (p, ω) so that both a and b are in B(p, ω) and a = x(p, ω).
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Q What are general conditions guaranteeing that a demand✆ function x(p, ω) can be rationalized?
→ Present two axioms of revealed preferences.
Weak Axiom of Revealed Preferences
Definition 9 (Weak Axiom)
The weak axiom of revealed preferences (WA) is a property of choice function which says that it is impossible that a be revealed to be better than b and b be revealed to be better than a. That is,
if px(p′, ω′) ≤ ω and x(p, ω) 6= x(p′, ω′), then p′x(p, ω) > ω′.
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Fg Figure 2.3 (see JR, pp.92)✆
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Rm Note that any choice function rationalized by some✆ preference relation must satisfy WA.
Weak Axiom ⇒ Law of Demand
Theorem 10
Let x(p, ω) be a choice function satisfying Walras’s Law and WA. Then,
1 x(·) is homogeneous of degree zero, and
2 if ω′ = p′x(p, ω), then either x(p′, ω′) = x(p, ω) or (p′− p)(x(p′, ω′) − x(p, ω)) < 0.
The proof of 2.
Assume that x(p′, ω′) 6= x(p, ω). By Walras’s Law and the assumption that ω′ = p′x(p, ω):
(p′− p)(x(p′, ω′) − x(p, ω))
= p′x(p′, ω′) − p′x(p, ω) − px(p′, ω′) + px(p, ω)
= ω′− ω′− px(p′, ω′) + ω = ω − px(p′, ω′). By WA the right hand side is less than 0.
Strong Axiom of Revealed Preferences
The previous theorem implies that the compensated (Hicksian) demand function y(p′) = x(p′, p′x(p, ω)) satisfies the law of demand, that is, yk is decreasing in pk.
WA is not a sufficient condition for extending the binary relation % (defined from the choice function) into a complete and transitive relation. The following stronger condition than WA is known to be necessary and sufficient.
Definition 11 (Strong Axiom))
Choice function satisfies the strong axiom of revealed preferences (SA) if for every sequence of distinct bundles x0, x1, ..., xk, where x0 is revealed preferred to x1, and x1 is revealed preferred to x2, ..., and xk−1 is revealed preferred to xk, it is not the case that xk is revealed preferred to x0.
