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Properties of Utility Functions

Econ 2100, Fall 2012

Lecture 4, 6 September

Outline

1 Structural Properties of Utility Functions

1 Monotonicity2 Convexity3 Quasi-linearity

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From Last Class

Utility Representation

Theorem (Debreu)

Suppose X Rn . The binary relation % on X is complete, transitive, andcontinuous if and only if there exists a continuous utility representation u : X ! R.

Choices are actions that maximize utility for a given budget.

Proposition

If% is a continuous preference relation and A Rn is nonempty and compact,then C%(A) is nonempty and compact:

C%(A) = arg maxx2A u(x):

Where the induced choice rule C% is dened byC%(A) = fx2 A : x% y for all y 2 Ag

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Structural Properties of Utility Functions

Aside from existence and continuity, further restrictions on the utility function are

often useful.

From now on, assume X = Rn .If xi yi for each i, we write x y.

The idea is to connect properties of preferences with properties of the utilityfunction that represents them.

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Monotonicity

Denition

A preference relation % is weakly monotone if x y implies x% y.A preference relation % is strictly monotone if x y and x6= y imply x y.

Monotonicity says more is better.

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Monotonicity: An Example

Example

Suppose % is the preference relation dened on R2 dened by x% y if and only if

x1 y1.This % is weakly monotone, because if x y, then x1 y1.It is not strictly monotone, because (1; 1) (1; 0) and (1; 1) 6= (1; 0), yetnot(1; 1) (1; 0) since (1; 0) % (1; 1).

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Strict Monotonicity: An Example

ExampleThe lexicographic preference on R2 is strictly monotone.Proof: Suppose x y and x6= y.

Then either (a) x1 > y1 and x2 y2 or (b) x1 y1 and x2 > y2 .If (a) holds, then

x% y because x1 y1,and noty% x because neither y1 > x1 (excluded by x1 > y1) nor y1 = x1 (alsoexcluded by x1 > y1).

If (b) holds, then

x% y because either x1 > y1 or x1 = y1 and x2 y2.and noty% x because noty1 > x1 (excluded by x1 y1) and noty2 x2(excluded by x2 > y2).

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Monotonicity and Utility Functions

Denition

A function f : Rn

! R isnondecreasing if x y implies f(x) f(y);strictly increasing if x y and x6= y imply f(x) > f(y).

Monotonicity is equivalent to nondecreasing corresponding utility function

being 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 5a. Problem Set 2, due next Tuesday.

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Local Non Satiation

Denition

A preference relation % is locally nonsatiated if for all x

2X 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 strictlypreferred to it.

DenitionA utility function u : X! R is locally nonsatiated if it represents a locallynonsatiated preference relation %, i.e. if for every x2 X and " > 0, there existssome y such that ky xk < " and u(y) > u(x).

Example

The lexicographic preference is locally nonsatiated.Proof:Fix (x1; x2) and " > 0.

Then (x1 +

"

2;

x2) satises k(x1 +"

2;

x2) (x1:x2)k

< "

and (x1 +

"

2;

x2) (x1;

x2).

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Local Non Satiation

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 x.Note that

jjy

x

jj=

vuut

n

Xi=1

"

n2

="

pn< ":

Thus % is locally nonsatiated.

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Shapes of Utility Functions

DenitionSuppose C is a convex subset of X. A function f : C

!R is:

concave if f(x + (1 )y) f(x) + (1 )f(y), for all 2 [0; 1] andx; y 2 C;strictly concave if f(x + (1 )y) > f(x) + (1 )f(y), for all 2 (0; 1)and x; y 2 X such that x6= y;convex if f(x + (1 )y) f(x) + (1 )f(y), for all 2 [0; 1] andx; y 2 C;strictly convex if f(x + (1 )y) < f(x) + (1 )f(y), for all 2 (0; 1)and x; y 2 C such that x6= y;ane if f is concave and convex, i.e.

f(x + (1 )y) = f(x) + (1 )f(y), for all 2 [0; 1] and x; y 2 C;quasiconcave if f(x) f(y) implies f(x + (1 )y) f(y), for all 2 [0; 1];strictly quasiconcave if f(x) f(y) and x6= y implyf(x + (1

)y) > f(y), for all

2(0; 1).

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Convex Preferences

DenitionA preference relation % is convex if

x% y ) x + (1 )y% y for all 2 (0; 1)

A preference relation % is strictly convex if

x% y and x6= y ) x + (1 )y 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 decisionmaker better o.

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Convex Preferences: An Example

Example

% on R2

dened by x% y if and only if x1 + x2 y1 + y2 is convex.Proof: Suppose x% y, i.e. x1 + x2 y1 + y2, and x 2 (0; 1).Then

x + (1 )y = (x1 + (1 )y1; x2 + (1 )y2)So,

[x1 + (1 )y1] + [x2 + (1 )y2] = [x1 + x2| {z }y1+y2

] + (1 )[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

2

(1; 0) +1

2

(0; 1) = (1

2

;1

2

) % (0; 1):

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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 notnecessarily concave.

Proof.Question 5b. Problem Set 2, due next Tuesday.

Q i U ili d C U C

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Quasiconcave Utility and Convex Upper Contours

Proposition

Let% be a preference relation on X represened by u : X! R. Then, the upper contour set of zis a convex subset of X if and only if u is quasiconcave.

Proof.Suppose that u is quasiconcave.

Fix z2 X, and take any x; y2% (z).Wlog, assume u(x)

u(y), so that u(x)

u(y)

u(z), and let

2[0; 1].

By quasiconcavity of u,u(z) u(y) u(x + (1 )y);

so x + (1 )y% z.Hence x + (1 )y belongs to % (z), proving it is convex.

Now suppose that % (z) is convex for all z2 X.Let x; y2 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 %,

u(x + (1 )y) u(y):Thus u is quasiconcave.

C i d I d d Ch i

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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 denition ofC%(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 denition ofC%, y% z for all z2 A.Using transitivity, x + (1 )y% y% z for all z2 A.Hence, x + (1

)y

2 C%(A) by denition 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 x6= y.Since x% y and x6= y, strict convexity implies x+ (1)y y for all 2 (0; 1).Since A is convex, x+ (1)y2 A, but this contradicts the fact that y2 C%(A).

Q i li Utilit

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Quasi-linear Utility

Denition

The function u : Rn

!R is quasi-linear if there exists a function v : Rn1

!R

such that u(x; m) = v(x) + m.

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 (x; m) % (x; m0) if and only if m m0, for all x2 Rn1 and all m; m0 2 R;2 (x; m) % (x0; m0) if and only if(x; m + m00) % (x0; m0 + m00), for all x2 Rn1

and m; m0; m00 2 R;3 for all x; x0

2R

n1, there exist m; m0

2R such that (x; m)

(x0; m0).

1 Given two bundles with identical goods, the consumer always prefers withmore money.

2 Rules out wealth eects: adding (or subtracting) the same monetary amountto (or from) each bundle does not aect their ordering.

3 Monetary transfers can be used to achieve indierence between any twobundles.

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Proof.

Fix some x0 2 Rn1.For every x2 Rn1, there exist n; n0 2 R such that (x; n) (x0; n0), by (3).

Subtract n0

from both sides and apply (2), so that (x; n n0

) (x0 ; 0).Dene v(x) = n0 n, and thus (x;v(x)) (x0 ; 0).Now add m + v(x) to both sides and apply (2) again:

(x; m) (x0; m + v(x)) (A)

for any m 2 R.Then

(x; m) % (x0; m0) , (x; m v(x) v(x0))| {z }(x0;mv(x0))

% (x0; m0 v(x) v(x0))| {z }(x0;mv(x))

; by (2)

, (x0; m v(x0)) % (x0; m0 v(x)); by (A), m v(x0) m0 v(x); by (2), v(x) + m v(x0) + m0:

Q i li P f d Utilit

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Quasi-linear Preferences and Utility

Quasi-linear utility representations are unique up to a constant.

Proposition

Suppose that the preference relation % on Rn admits two quasi-linearrepresentations: v(x) + m, and v0(x) + m, where v; v0 : Rn1 ! R. Then thereexists c2 R such that v0(x) = v(x) c for all x2 Rn1.

Proof.

Let v(x) + m and v0(x) + m both represent %.

Given any x0 2 Rn1, for all x2 Rn1 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) = v(x0) + nx and v0

(x) = v0

(x0) + nx

Thus nx = v0(x) v0(x0). Therefore

v(x) = v(x0) + v0(x) v0(x0)| {z }

=nxHence, for all x

2Rn1,

v(x) v0(x) =a constant c

z }| {v(x0) v0(x0) ) v0(x) = v(x) c:

Homothetic Preferences and Utility

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Homothetic Preferences and Utility

Homothetic preferences are also useful in many applications, in particular foraggregation problems.

Denition (Homothetic preferences)

The preference relation % on X is homothetic if for all x; y2 X,

x y) x y for each > 0

Proposition

The continuous preference relation % on Rn is homothetic if and only if it isrepresented by a utility function that is homogeneous of degree 1.

A function is homogeneous of degree r if f(x) = rf(x) for any x and > 0.

Proof.Question 5c. Problem Set 2, due next Tuesday.

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