Structural Properties of Utility Functions
Econ 2100
Fall 2015
Lecture 4, September 14
Problem Set 2 due now.
Outline
1
Monotonicity
2
Convexity
3
Quasi-linearity
4
Walrasian Demand
From Last Class
De…nition
The utility function u : X ! R represents the binary relation % on X if
x % y , u(x)
u(y ):
Theorem (Debreu)
Suppose X Rn . A binary relation % on X is complete, transitive, and continuous
if and only if it admits a continuous utility representation u : X ! R.
Today we study connections between a utility function and the underlying
preference relation it represents.
Structural Properties of Utility Functions
The main idea is to understand the connection between properties of
preferences and characteristics of the utility function that represents them.
NOTATION:
From now on, assume X = R n .
If xi
yi for each i, we write x
y.
Monotonicity
Monotonicity says more is better.
De…nitions
A preference relation % is weakly monotone if x
y implies x % y .
A preference relation % is strictly monotone if x
y and x 6= y imply x
In our notation, x
y and x 6= y imply xi > yi for some i.
Question
What does monotonicity imply for the utility function representing %?
y.
Monotonicity: An Example
Example
Suppose % is the preference relation on R2 de…ned by
x %y
if and only if x1
% is weakly monotone, because if x
y , then x1
y1
y1 .
It is not strictly monotone, because
(1; 1)
but
(1; 0) and (1; 1) 6= (1; 0)
not [(1; 1)
since (1; 0) % (1; 1).
(1; 0)]
Strict Monotonicity: An Example
The lexicographic preference on R2 is strictly monotone
Proof: Suppose x y and x 6= y .
Then there are two possibilities:
(a) : x1 > y1
and x2
y2
or (b) : x1
y1
and x2 > y2 :
If (a) holds, then
x % y because x1 y1 , and
not (y % x ) because neither y1 > x1 (excluded by x1 > y1 ) nor y1 = x1 (also
excluded by x1 > y1 ).
If (b) holds, then
x % y because either x1 > y1 or x1 = y1 and x2 y2 , and
not (y % x ) because not (y1 > x1 ) (excluded by x1 y1 ) and not (y2
(excluded by x2 > y2 ).
In both cases, x % y and not (y % x) thus x
y.
x2 )
Monotonicity and Utility Functions
De…nitions
A function f : Rn ! R is
nondecreasing if x
strictly increasing if x
y implies f (x)
f (y );
y and x 6= y imply f (x) > f (y ).
Monotonicity is equivalent to the corresponding utility function being
nondecreasing or increasing.
Proposition
If u represents %, then:
1
% is weakly monotone if and only if u is nondecreasing;
2
% is strictly monotone if and only if u is strictly increasing.
Proof.
Question 1a. Problem Set 3, due next Monday.
Local Non Satiation
De…nition
A preference relation % is locally nonsatiated if for all x 2 X and " > 0, there exists
some y such that ky xk < " and y x.
For any consumption bundle, there is always a nearby bundle that is strictly
preferred to it.
De…nition
A utility function u : X ! R is locally nonsatiated if it represents a locally
nonsatiated preference relation %; that is, if for every x 2 X and " > 0, there exists
some y such that ky xk < " and u(y ) > u(x).
Example: The lexicographic preference on R2 is locally nonsatiated
Fix (x1 ; x2 ) and " > 0.
Then (x1 + "2 ; x2 ) satis…es k(x1 + "2 ; x2 )
and (x1 +
"
2 ; x2 )
(x1 ; x2 ).
(x1 :x2 )k < "
Local Non Satiation and Strict Monotonicity
Proposition
If % is strictly monotone, then it is locally nonsatiated.
Proof.
Let x be given, and let y = x + n" e, where e = (1; :::; 1).
Then we have yi > xi for each i.
Strict monotonicity implies that y
Note that
jjy
x.
v
u n
uX "
xjj = t
n
Thus % is locally nonsatiated.
i =1
2
"
= p < ":
n
Shapes of Functions
De…nitions
Suppose C is a convex subset of X . A function f : C ! R is:
concave if
f ( x + (1
for all
)y )
f (x) + (1
)f (y )
)y ) > f (x) + (1
)f (y )
2 [0; 1] and x; y 2 C ;
strictly concave if
f ( x + (1
for all
2 (0; 1) and x; y 2 X such that x 6= y ;
quasiconcave if
f (x)
for all
2 [0; 1];
f (y ) ) f ( x + (1
)y )
f (y )
strictly quasiconcave if
f (x)
for all
2 (0; 1).
f (y ) and x 6= y ) f ( x + (1
)y ) > f (y )
Convex Preferences
De…nitions
A preference relation % is
convex if
x %y
strictly convex if
)
x % y and x 6= y
x + (1
)
)y % y for all
x + (1
)y
2 (0; 1)
y for all
2 (0; 1)
Convexity says that taking convex combinations cannot make the decision
maker worse o¤.
Strict convexity says that taking convex combinations makes the decision
maker better o¤.
Question
What does convexity imply for the utility function representing %?
Convex Preferences: An Example
Let % on R2 be de…ned as x % y if and only if x1 + x2
Proof: Suppose x % y , i.e. x1 + x2
y1 + y2 , and …x
y1 + y2 is convex
2 (0; 1).
Then
x + (1
)y = ( x1 + (1
)y1 ; x2 + (1
)y2 )
So,
[ x1 + (1
)y1 ] + [ x2 + (1
)y2 ]
=
[x1 + x2 ] + (1
| {z }
)[y1 + y2 ]
y1 +y2
[y1 + y2 ] + (1
)[y1 + y2 ]
= y1 + y2 ;
proving x + (1
)y % y .
This is not strictly convex, because (1; 0) % (0; 1) and (1; 0) 6= (0; 1) but
1
1
1 1
(1; 0) + (0; 1) = ( ; ) % (0; 1):
2
2
2 2
Convexity and Quasiconcave Utility Functions
Convexity is equivalent to quasi concavity of the corresponding utility function.
Proposition
If u represents %, then:
1
% is convex if and only if u is quasiconcave;
2
% is strictly convex if and only if u is strictly quasiconcave.
Convexity of % implies that any utility representation is quasiconcave, but not
necessarily concave.
Proof.
Question 1b. Problem Set 3, due next Monday.
Quasiconcave Utility and Convex Upper Contours
Proposition
Let % be a preference relation on X represened by u : X ! R. Then, the upper
contour set is a convex subset of X if and only if u is quasiconcave.
Proof.
Suppose that u is quasiconcave.
Fix z 2 X , and take any x ; y 2% (z).
Wlog, assume u(x ) u(y ), so that u(x )
u(y )
u(z), and let
By quasiconcavity of u,
u(z) u(y ) u( x + (1
so x + (1
)y % z.
Hence x + (1
)y belongs to % (z), proving it is convex.
Now suppose that % (z) is convex for all z 2 X .
2 [0; 1].
)y );
Let x ; y 2 X and 2 [0; 1], and suppose u(x ) u(y ).
Then x % y and y % y , and so x and y are both in % (y ).
Since % (y ) is convex (by assumption), then x + (1
)y % y .
Since u represents %,
Thus u is quasiconcave.
u( x + (1
)y )
u(y )
Convexity and Induced Choices
Proposition
If % is convex, then C% (A) is convex for all convex A.
If % is strictly convex, then C% (A) has at most one element for any convex A.
Proof.
Let A be convex and x; y 2 C% (A).
By de…nition of C% (A), x % y .
Since A is convex: x + (1
)y 2 A for any 2 [0; 1].
Convexity of % implies x + (1
)y % y .
By de…nition of C% , y % z for all z 2 A.
Using transitivity, x + (1
)y % y % z for all z 2 A.
Hence, x + (1
)y 2 C% (A) by de…nition of induced choice rule.
Therefore, C% (A) is convex for any convex A.
Now suppose there exists a convex A for which C% (A)
2.
Then there exist x ; y 2 C% (A) with x 6= y .
Since x % y and x 6= y , strict convexity implies x + (1
)y y for all
2 (0; 1).
Since A is convex, x + (1
)y 2 A, but this contradicts the fact that
y 2 C% (A).
Quasi-linear Utility
De…nition
The function u : Rn ! R is quasi-linear if there exists a function v : Rn
such that u(x; m) = v (x) + m.
1
!R
We usually think of the n-th good as money (the numeraire).
Proposition
The preference relation % on Rn admits a quasi-linear representation if and only
1
2
3
1
2
3
(x; m) % (x; m0 ) if and only if m
m0 , for all x 2 Rn
1
and all m; m0 2 R;
(x; m) % (x ; m ) if and only if (x; m + m ) % (x ; m + m00 ), for all x 2 Rn
and m; m0 ; m00 2 R;
0
0
for all x; x 0 2 Rn
00
1
0
0
, there exist m; m0 2 R such that (x; m)
1
(x 0 ; m0 ).
Given two bundles with identical goods, the consumer always prefers the one
with more money.
Adding (or subtracting) the same monetary amount does not chage rankings.
Monetary transfers can always be used to achieve indi¤erence.
Proof.
Question 1c. Problem Set 3, due next Monday.
Quasi-linear Preferences and Utility
Proposition
Suppose that the preference relation % on Rn admits two quasi-linear
representations: v (x) + m, and v 0 (x) + m, where v ; v 0 : Rn 1 ! R. Then there
exists c 2 R such that v 0 (x) = v (x) c for all x 2 Rn 1 .
Proof.
Let v (x) + m and v 0 (x) + m both represent %.
Given any x0 2 Rn 1 , for all x 2 Rn 1 there exists nx such that
(x; 0) (x0 ; nx ). (by (2) and (3) of the previous Proposition.)
Since the utility functions both represent % we have
v (x) + 0 = v (x0 ) + nx
Thus nx = v 0 (x)
and
v 0 (x) + 0 = v 0 (x0 ) + nx
v 0 (x0 ). Therefore
v (x) = v (x0 ) + v 0 (x) v 0 (x0 )
{z
}
|
=n x
Hence, for all x 2 Rn
v (x)
1
,
a constant c
z
}|
{
v (x) = v (x0 ) v 0 (x0 )
0
)
v 0 (x) = v (x)
c:
Homothetic Preferences and Utility
Homothetic preferences are also useful in many applications, in particular for
aggregation problems.
De…nition
The preference relation % on X is homothetic if for all x; y 2 X ,
x
y) x
y for each
>0
Proposition
The continuous preference relation % on Rn is homothetic if and only if it is
represented by a utility function that is homogeneous of degree 1.
A function is homogeneous of degree r if f ( x) =
Proof.
Question 1d. Problem Set 3, due next Monday.
r
f (x) for any x and
> 0.
Walrasian Demand
Main Idea
The consumer uses income to purchase goods (commodities) at the
exogenously given prices.
De…nition
The Budget Set Bp;w Rn at prices p and income w is the set of all a¤ordable
consumption bundles and is de…ned by
Bp;w = fx 2 Rn+ : p x
w g:
De…nition
Given a preference relation %, the Walrasian demand correspondence
x : Rn++ R+ ! Rn+ is de…ned by
x (p; w ) = fx 2 Bp;w : x % y for any y 2 Bp;w g:
By de…nition, for any x 2 x (p; w )
x %x
for any x 2 Bp;w :
Walrasian demand equals the induced choice rule given the preference relation
% and the available set Bp;w :
x (p; w ) = C% (Bp;w ):
Walrasian Demand With Utility
Although we do not need the utility function to exist to de…ne Walrasian
demand, if a utility function exists there is an equivalent de…nition.
De…nition
Given a utility function u : Rn+ ! R, the Walrasian demand correspondence
x : Rn++ R+ ! Rn+ is de…ned by
x (p; w ) = arg max u(x)
where
x 2B p;w
Bp;w = fx 2 Rn+ : p x
w g:
As before,
x (p; w ) = C% (Bp;w ):
and for any x 2 x (p; w )
x %x
for any x 2 Bp;w :
We can derive some properties of Walrasian demand directly from assumptions
on preferences and/or utility.
Walrasian Demand Is Homogeneous of Degree Zero
Proposition
Walrasian demand is homogeneous of degree zero: for any
>0
x ( p; w ) = x (p; w )
Proof.
For any
> 0,
B
because
p; w
= fx 2 Rn+ : p x
is a scalar
w g = fx 2 Rn+ : p x
w g = Bp;w
Since the constraints are the same, the utility maximizing choices must also be
the same.
The Consumer Spends All Her Income
This is sometimes known as Walras’Law for individuals
Proposition (Full Expenditure)
If % is locally nonsatiated , then
p x =w
for any x 2 x (p; w )
Proof.
Suppose not.
Then there exists an x 2 x (p; w ) with p x < w
Find some y such that
ky
xk < " with " > 0
and
p y
w
why does such a y exist?
By local non satiation, this implies y
x contradicing x 2 x (p; w ).
Problem Set 3: Partial List
Due Monday, 21 September, at the beginning of class
1
Prove that if u represents %, then:
1
2
3
% is weakly monotone if and only if u is nondecreasing;
% is strictly monotone if and only if u is strictly increasing.
% is convex if and only if u is quasiconcave;
% is strictly convex if and only if u is strictly quasiconcave.
u is quasi-linear if and only
1
2
3
4
(x ; m) % (x ; m 0 ) if and only if m m 0 , for all x 2 Rn 1 and all m; m 0 2 R;
(x ; m) % (x 0 ; m 0 ) if and only if (x ; m + m 00 ) % (x 0 ; m 0 + m 00 ), for all x 2 Rn
and m; m 0 ; m 00 2 R;
for all x ; x 0 2 Rn 1 , there exist m; m 0 2 R such that (x ; m) (x 0 ; m 0 ).
1
a continuous % is homothetic if and only if it is represented by a utility function
that is homogeneous of degree 1.