Preferences
What properties would we expect preferences to
exhibit?
Which of these properties allow us to derive an
indifference curve?
What will that indifference curve look like?
More importantly, what is not ruled out?
What else do we need to assume about properties
to generate the nice indifference curves required to
produce sensible demand curves?
Axioms of Consumer Theory
• This presentation covers the lectures on the
Axioms of Consumer preference. There are
some changes compared to the coverage in
the lectures. Continuity has been moved up
the order to a more appropriate and I have
simplified the coverage on Convexity.
Terminology
The bundle containing x1 and y1
( x1 , y1 )  ( x2 , y2 )
is strictly preferred to (better
than) the bundle containing x2
and y2.
( x1 , y1 )  ( x2 , y2 ) (x1, y1) is weakly preferred to (at
least as good as) (x2, y2) .
( x1 , y1 ) ~ ( x2 , y2 ) the individual is indifferent
between the two bundles.
Assumptions about Preferences
1. Complete - either
( x2 , y2 )  ( x1 , y1 )
or
or both, i.e
2. Reflexive -
( x1 , y1 )  ( x2 , y2 )
( x1 , y1 ) ~ ( x2 , y2 )
( x1 , y1 )  ( x1 , y1 )
3. Transitive -
if
( x1 , y1 )  ( x2 , y2 )
and ( x2 , y2 )  ( x3 , y3 )
then transitivity
=> ( x1 , y1 )  ( x3 , y3 )
Fundamental Axioms of
Consumer Theory
1.Completeness
2. Reflexivity
3. Transitivity
Known as the Three Fundamental
Axioms of consumer theory.
These allow consumers to arrange
bundles in order of preference.
If we have the three axioms above we can rank all
bundles in the x,y space we have drawn below.
For example, if we take any bundle (x1, y1) then we can
establish the bundles which satisfy the two
relationships:
( x1 , y1 )  ( x2 , y2 )
( x1 , y1 )  ( x2 , y2 )
y1
x1
If we have the three axioms above we can rank all
bundles in the x,y space we have drawn below.
For example, if we take any bundle (x1, y1) then we can
establish the bundles which satisfy the two
relationships:
( x1 , y1 )  ( x2 , y2 )
( x1 , y1 )  ( x2 , y2 )
The boundary of
the set is the
indifference curve
y1
x1
Could indifference curves ever
cross?
Units of good Y
30
20
10
u1
0
0
10
Units of good X
20
30
u2
Units of good Y
Could we get this?
Ans: No
20
Proof by
Contradiction
10
u1
0
0
10
Units of good X
20
30
Units of good Y
Consider points a,b
and c
And assume a is
preferred to c
20
a
b
10
c
u1
0
0
10
Units of good X
20
30
Claim: a is preferred to c
Units of good Y
u2
20
a
b
Indifference curve u1 implies
that Point a is indifferent to b
And u2=> point b is
indifferent to c
=> a is indifferent to c
10
Contradicts Claim
c
u1
0
0
10
Units of good X
20
u2
Units of good Y
30
Crossing Indifference
curves lead to a logical
contradiction, so
indifference curves can
never cross
20
a
10
b
c
u1
0
0
10
Units of good X
20
So what other properties must
Indifference curves have?
• Well thus far - none.
• Almost anything goes.
y
y
x
x
(a)
(b)
y
y
(c)
x
(d)
x
Or even these cases here:
y
y
x
(e)
x
(f)
What does case (a) mean?
y
u2
u1
u2
u0
u0
(a)
u1
x
Continuity
• For any given bundle, the set of bundles
which are weakly preferred to it, and the set
of bundles to which it is weakly preferred,
are closed sets (that is, they contain their
own boundary).
• Closed set: Football Pitch, Tennis Court
• Open set: Rugby Pitch, Cricket
Effectively means that points on the indifference
curve are ‘close’ to one another and that there are no
gaps.
y1
x1
4. Continuity
• For any given bundle the set of bundles
which are weakly preferred to it, and the set
of bundles to which it is weakly preferred,
are closed sets (that is, they contain their
own boundary).
• Closed set: Football Pitch, Tennis Court
• Open set: Rugby Pitch, Cricket
5. Monotonicity
• The next assumption we need is called the
assumption of monotonicity or nonsatiation. It says that if
x2  x1 and y2  y1
and either
x2  x1 or y2  y1
then ( x2 , y2 )  ( x1 , y1 )
What does this mean?
y2  y1
y1
x2  x1
x1
y1  y2
Shaded area is the set
of points where
motonocity holds and
( x2 , y2 )  ( x1 , y1 )
y1
x2  x1
x1
y2  y1
y1
So up here is
definitely better
than
x2  x1
Down
Here
x1
So up here is
definitely better
than
y1
Down
Here
So the indifference
curve cannot go
through either of
these areas
x1
So up here is
definitely better
than
y1
u1
Down
Here
So it must
slope down
x1
Assumption 5.
• Monotonocity
• Gives us downward sloping indifference
curves
• Are we out of the woods yet
• NO!
All these satisfy properties 1-5
y
y
x
y
x
y
x
x
Lets take the third case first
y1
u1
x1
y1
u3
u1
u0
Why is this case a problem?
u2
y1
u3
u1
u2
u0
Let’s draw in a budget constraint
y1
u3
u1
u2
u0
So now we have two equilibria or two demands for x at
this set of prices
px
The Demand
Curve will look
like this
x
px
Why would that
be a problem?
x
px
Why would that
be a problem?
Suppose that we
observed this
data set
x
px
If we assumed it
was a straight
line demand
curve we would
get
x
px
But the true
demand curve is:
And if you told
the boss it’s the
red one, your
fired
x
px
S0
S1
Since
determining
equilibrium is a
problem and
supply curve
shifts even more
so
x
y1
u1
So how do we rule this case out?
This is known as a
convex curve
y1
We want the indifference curve to look like this
Concavity and (Quasi-)
Convexity
• In the lectures I tried to simplify this topic
compared with previous years and feel in
the circumstances I may have made things
worse. I have confined the material here to
the essential concept of what is called quasiconvexity. More detailed coverage of the
topic is contained in a separate presentation
Concavity and Quasi-convexity
y
If we look at a map of the utility
mountain then a nice, well
behaved indifference curve
should look like this.
We say it is Quasi-convex
because the cross-sections look
convex looking from the x,y
origin
x
What does Quasi-convex mean?
• Suppose we take a weighted average of two
bundles on the same indifference curve and
compare the utility we get from this new
bundle compared with the utility we got
from the originals.
• If it is higher we say that the function is
quasi-convex.
Formerly we can state this as: A
function is quasi-convex iff
U(x 3 , y3 )  U(0.5x 10.5x 2 ,0.5y10.5y 2 )
1
1
 U(x 1, y1 )  U(x 2 , y 2 )  U(x 1, y1 )
2
2
Where U(x1,y1) = U(x2,y2)
y
U(x1,y1) = U(x2,y2)
y1
U(x2,y2)
y2
x1
x2
x
Consider a new
bundle: (x3, y3) where
y
U(x1,y1)
y1
x3= half of x1 and x2
and
y3= half of y1 and y2
y3
U(x2,y2)
y2
x1
x3
x2
x
What is the utility
associated with this
U(x1,y1) new bundle?
y
y1
y3
U(x2,y2)
y2
x1
x3
x2
x
y
y1
U(x3,y3)
y3
y2
x1
x3
x2
x
y
If
y1
1
1
U(x 3 , y3 )  U(x 1, y1 )  U(x 2 , y 2 )
2
2
Then we say the indifference
curve is quasi-convex
y3
U(x3,y3)
y2
x1
x3
x2
x
U(x 3 , y 3 )
y
1
1
 U(x 1, y1 )  U(x 2 , y 2 )
2
2
 U(x 1, y1 )  U(x 2 , y 2 )
y1
y3
y2
x1
x3
x2
x
y
U(x 3 , y 3 )
 U(x 1, y1 )
y1
y3
y2
x1
x3
x2
x
Note The bundle need not be x3, y3,
but any point on the red line. That is,
we could use any fraction l instead of
1/2. If the indifference curve is quasiconvex the condition
y
y1
would still hold
y3
U(x 4 , y 4 )
 U(x 1, y1 )
y2
x1
x3
x2
x
y
But this
indifference curve is
convex, since
y1
y3
y2
x1
x3
x2
x
1
1
U(x 3 , y3 )  U(x 1, y1 )  U(x 2 , y 2 )
2
2
 U(x 1, y1 )
y
y1
U(x3,y3)
y3
But not Strictly
convex
y2
x1
x3
x2
x
Strict Convexity
• So we really need Strict convexity
• And it is STRICTLY convex if
U( lx1(1 - l )x 2 , ly1(1 - l )y 2 ) 
lU(x 1, y1 )  (1  l ) U(x 2 , y 2 )
 U(x 1, y1 )
Where l lies between 0 and 1
1
1
U(x 3 , y3 )  U(x 1, y1 )  U(x 2 , y 2 )
2
2
y
y1
Strictly Convex
y3
y2
x1
x3
x2
x
Strict Convexity rules out every case here except case (b)
y
y
x
x
(a)
(b)
y
y
(c)
x
(d)
x
Why is case (b) troublesome?
• Because it has a pointy bit where there can
be an equilibrium for more than one price:
y
u1
x
Differentiability
• To rule out case (b) we assume that the
indifference curve is differentiable
everywhere. That is, the function is smooth
and has no corners.
y
y
•
=>
x
x
Axioms of Consumer Theory
1.Completeness
2. Reflexivity
3. Transitivity
4. Continuity
5. Monotonicity
6. Convexity
7. Differentiability
• Conditions 1-5 allow us to write a utility
function: u = u(x,y).
• E.g.
u=x1/2y1/2
• Formally, if
( x2 , y2 )  ( x1 , y1 )
then u is a mathematical function such that
u( x2 , y2 )  u( x1 , y1 )
• Utility is ordinal, I.e.the function merely
orders bundles the actual number associated
with u is irrelevant.
• E.g. If u(x2,y2)=4 and u(x1,y1)=2, then the
x2, y2 bundle is preferred to the x1, y1
bundle, but we can’t say it is twice as good.
• That is, utility is not Cardinal
• Since utility is ordinal I can change the
function as long as it does not change the
ordering of the the bundle:
• So I can change u(x2, y2) to 2u(x2, y2) so
that
• u(x2, y2)=4 and u(x1, y1)=2 becomes
• 2u(x2, y2)=8 and 2u(x1, y1)=4
• But the initial condition still holds:
( x2 , y2 )  ( x1 , y1 )
We say that a utility function is unique up to
any positive monotonic transformation
PMT
Positive Monotonic
Transformation
• If u(x2,y2) > u(x1,y1), then any Positive
Monotonic Transformation of u, say f (u),
implies that
• f [u(x2,y2)] > f [u(x1,y1)]
Consider utility from each
bundle below
Bundle 1
Bundle 2
Bundle 3
Bundle 4
A
15
16
20
40
Eg of PMT : B = A +3
Bundle 1
Bundle 2
Bundle 3
Bundle 4
A
15
16
20
40
B
18
19
23
43
Eg2 of PMT: C=2.5A
Bundle 1
Bundle 2
Bundle 3
Bundle 4
A
15
16
20
40
B
18
19
23
43
C
37.5
40
50
100
Eg 3 of PMT: D=2 A+4
Bundle 1
Bundle 2
Bundle 3
Bundle 4
A
15
16
20
40
B
18
19
23
43
C
37.5
40
50
100
D
34
36
44
84
Eg of NOT a PMT: E=100/A
Bundle 1
Bundle 2
Bundle 3
Bundle 4
A
15
16
20
40
B
18
19
23
43
C
37.5
40
50
100
D
34
36
44
84
E
6.67
6.25
5
2.5