1st Lecture, ECON 5200, Preferences
Kjell Arne Brekke
August 20, 2010
1
Introduction
Note that these handouts are not lecture notes. Proofs are lacking, they are often short
in discussing economic content. They are more meant to indicate what the lecture will
be about, and make it easier to prepare for class.
1.1
Problems
Note that there are a few problems given in the handout. My intention is that working
on these problem will be a good preparation for class.
2
Preferences
We here de…ne weak preferences by axioms and derive indi¤erence and strong preferences.
De…nition 1 A preference relation is rational if it is complete
8x; y : x
y or y
y and y
z implies x
x
and transitive
x
z
We de…ne strong preferences and indi¤erence
x
y means x
y but not y
x
y means x
y and y
1
x
x
We can now prove
Proposition 2 If
is rational then
1.
is both irre‡exive (never x
2.
is re‡exive (x
3. if x
3
y
x) and transitive (x
x) transitive and symmetric (x
z then x
y and y
y then y
z implies x
z)
x).
z:
Utility functions
We are used to get preferences from a utility function
x
y i¤ u(x)
u(y):
Proposition 3 A preference relation can be represented by a utility function only if it is
rational.
A note on logic terminology, let A and B be two claims
A ) B is the same as "B if A" or "A only if B"
A , B is the same as "A if and only if B" or for short "A i¤ B"
4
Choice rules
Why not starte with choices instead? Let B
X denote the feasible set within the larger
set X (E.g. the budget set within the set of all consumption bundles). A choice rule that
pick a subset
C(B)
B; C(B) 6= ;
e.g.
C(fx; y; zg) = fx; yg
That is if the set of alternatives are x; y and z the rule is to choose any of the elements
x and y.
2
A choice rule satis…es the weak axiom if the following property holds
x 2 C(B) with x; y 2 B =) x 2 C(B 0 ) if x; y 2 B 0 and y 2 C(B 0 )
We are particularly interested in the relationship between choices and preferences.
Problem 4 Consider the choice rule
C(fs; m; lg) = fmg
C(fs; mg) = fsg
C(fm; lg) = fmg
The story goes, you are hungry and want a large cake (large> medium> small), but do
not want to be unpolite and take the largest one. Show that the choice rule violates the
weak axiom.
4.1
Preferences and choice rules
We can derive preferences from choices. Revealed preference relations
x
y () there is a B 2 B such that x; y 2 B and x 2 C(B)
(B is the family of all possible choice set, a family and a choice rule is called structure
(B; C( )).) and we can get a choice rule from preferences
C (B; ) = fx 2 B : x
Proposition 5 If
y for all y 2 Bg
is rational then C (B; ) satis…es the weak axiom
Proposition 6 If the structure (B; C( )) satis…es (i) the weak axiom and (ii) B contains
all sets up to three elements and, then there is a unique preference that rationalizes C.
Problem 7 Consider the cake-choice above. Can you restate that choice to be consistent
with the axioms of rational preferences?
3
5
Consumer choice
In consumer theory typically
L
= f(x1 ; :::xL ) 2 RL : xl
X = R+
0 for all l = 1; :::; Lg
The budget set also re‡ects prices and income
L
Bp;w = fx 2 R+
:p x
wg
a choice for each budget set
x(p; w) = C(Bp;w )
Note here that x may be a correspondence: that is x(p; w) may be a set (of equally
good choices). If it is single-valued we call it a demand function. When we di¤erensiate,
we implicitly assume that it is a function.
Axiom 8 The Walrasian demand correspondence x(p; w) is homogenous of degree zero:
x( p; w) = x(p; w), for any p, w and
> 0.
Axiom 9 x(p; w) satis…es Walras law: for every p >> 0, w > 0 we have p x = w for all
x 2 x(p; w):
This is su¢ cient to derive some familiar results
Proposition 10 When walrasian deman is homogenous of degree zero
L
X
dxl pi
k=1
dxl w X
+
=
Elk xl + Elw xl = 0
dpi xl
dw xl
k=1
L
Proposition 11 When Walrasian demand satis…es Walras law
L
X
dxl
pl + xk = 0
dp
k
l=1
L
X
dxl
l=1
dw
4
pl = 1
5.1
The weak axiom for walrasian demand
De…nition 12 The walrasian demand function x(p; w) saties…es the weak axiom of revealed preferences if
p x(p0 ; w0 )
w and x 6= x0 then p0 x(p; w) > w
Problem 13 Is this de…nition equivalent to the one above?
5