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+
N
this requirement. The household s endowments are denoted eh " and can be
+
traded.
In a production economy, the shareholdings of households in firms are denoted
F
ch " ,
N
A consumer s net demand is denoted zh a" xh - eh " .
Each consumer is assumed to have a preference relation or (weak) preference
ordering which is a binary relation on the consumption set Xh (?, Chapter 7).
Since each household will have different preferences, we should really denote
household h s preference relation , but the subscript will be omitted for the
h
time being while we consider a single household. Similarly, we will assume for
the time being that each household chooses from the same consumption set, X ,
although this is not essential.
Recall (see ?) that a binary relation R on X is just a subset R of X × X or a
collection of pairs (x, y) where x " X and y " X .
If (x, y) " R, we usually just write xRy.
Thus x y means that either x is preferred to y or the consumer is indifferent
between the two (i.e. that x is at least as good as y).
The following properties of a general relation R on a general set X are often of
interest:
Revised: December 2, 1998
CHAPTER 4. CHOICE UNDER CERTAINTY 65
1. A relation R is reflexive
Ð!Ò!
xRx "x " X
2. A relation R is symmetric
Ð!Ò!
xRy Ò! yRx
3. A relation R is transitive
Ð!Ò!
xRy, yRz =Ò! xRz
4. A relation R is complete
Ð!Ò!
"x, y " X either xRy or yRx (or both) (in other words a complete relation
orders the whole set)
An indifference relation,
from every preference relation:
1. x y means x y but not y x
2. x
The utility function u : X ’! represents the preference relation if
u(x) e" u(y) Ð!Ò! x y.
If f: ’! is a monotonic increasing function and u represents the preference
relation , then f æ% u also represents , since
f (u(x)) e" f (u(y)) Ð!Ò! u(x) e" u(y) Ð!Ò! x y.
If X is a countable set, then there exists a utility function representing any pref-
erence relation on X . To prove this, just write out the consumption plans in X in
order of preference, and assign numbers to them, assigning the same number to
any two or more consumption plans between which the consumer is indifferent.
If X is an uncountable set, then there may not exist a utility function representing
every preference relation on X .
Revised: December 2, 1998
66 4.3. AXIOMS
4.3 Axioms
We now consider six axioms which it are frequently assumed to be satisfied by
preference relations when considering consumer choice under certainty. (Note
that symmetry would not be a very sensible axiom!) Section 5.5.1 will consider
further axioms that are often added to simplify the analysis of consumer choice
under uncertainty. After the definition of each axiom, we will give a brief ratio-
nale for its use.
Axiom 1 (Completeness) A (weak) preference relation is complete.
Completeness means that the consumer is never agnostic.
Axiom 2 (Reflexivity) A (weak) preference relation is reflexive.
Reflexivity means that each bundle is at least as good as itself.
Axiom 3 (Transitivity) A (weak) preference relation is transitive.
Transitivity means that preferences are rational and consistent.
Axiom 4 (Continuity) The preference relation is continuous i.e. for all con-
sumption plans y " X the sets By a" {x " X : x y} and Wy = {x " X : y
x} are closed sets.
Consider the picture when N = 2:
We will see shortly that By, the set of consumption plans which are better than or
as good as y, and Wy, the set of consumption plans which are worse than or as
good as y, are just the upper contour sets and lower contour sets respectively of
utility functions, if such exist.
E.g. consider lexicographic preferences:
Lexicographic preferences violate the continuity axiom. A consumer with such
preference prefers more of commodity 1 regardless of the quantities of other com-
modities, more of commodity 2 if faced with a choice between two consumption
plans having the same amount of commodity 1, and so on.
"
In the picture, the consumption plan y lies in the lower contour set Wx but B (y)
"
never lies completely in Wx for any . Thus, lower contour sets are not open, and
upper contour sets are not closed.
Theorems on the existence of continuous utility functions have been proven by
Gerard Debreu, Nobel laureate, whose proof used Axioms 1 4 only (see ? or ?)
and by Hal Varian, whose proof was simpler by virtue of adding an additional
axiom (see ?).
Revised: December 2, 1998
CHAPTER 4. CHOICE UNDER CERTAINTY 67 [ Pobierz całość w formacie PDF ]