Chapter 4

Utility
Reminder - Assumptions about
Preference Relations

2
Assumptions about Preference
Relations

3
Assumptions about Preference
Relations

4
 Utility Functions
A preference relation that is
complete, reflexive, transitive and
continuous can be represented by a
continuous utility function.
Continuity means that small changes
to a consumption bundle cause only
small changes to the preference level.
 Utility Functions
A utility function U(x) represents a
preference relation ≿ if and only if:

x’ ≿ x” U(x’) ≥ U(x”)

(x’ ~ x” U(x’) = U(x”)

x’ ≻ x” U(x’) > U(x”))
 Utility Functions
Utility is an ordinal (i.e. ordering)
concept.
E.g. if U(x) = 6 and U(y) = 2 then
bundle x is strictly preferred to
bundle y. But x is not preferred three
times as much as y.
Utility Functions & Indiff. Curves
Consider the bundles (4,1), (2,3) and
(2,2).
Suppose (2,3) ≻ (4,1) ~ (2,2).
Assign to these bundles any
numbers that preserve the
preference ordering;
e.g. U(2,3) = 6 > U(4,1) = U(2,2) = 4.
Call these numbers utility levels.
Utility Functions & Indiff. Curves
An indifference curve contains
equally preferred bundles.
Equal preference  same utility level.
Therefore, all bundles in an
indifference curve have the same
utility level.
Utility Functions & Indiff. Curves
So the bundles (4,1) and (2,2) are in
the indiff. curve with utility level U  
But the bundle (2,3) is in the indiff.
curve with utility level U  6.
On an indifference curve diagram,
this preference information looks as
follows:
Utility Functions & Indiff. Curves

x2 (2,3)
p (2,2) ~ (4,1)

U6
U4

x1
Utility Functions & Indiff. Curves
Another way to visualize this same
information is to plot the utility level
on a vertical axis.
 Utility Functions & Indiff. Curves
3D plot of consumption & utility levels for 3 bundles

Utility U(2,3) = 6

U(2,2) = 4
U(4,1) = 4

x2

x1
Utility Functions & Indiff. Curves
This 3D visualization of preferences
can be made more informative by
adding into it the two indifference
curves.
Utility Functions & Indiff. Curves

Utility
U

U
x2 Higher indifference
curves contain
more preferred
bundles.
x1
Utility Functions & Indiff. Curves
Comparing more bundles will create
a larger collection of all indifference
curves and a better description of
the consumer’s preferences.
Utility Functions & Indiff. Curves

x2

U6
U4
U2
x1
Utility Functions & Indiff. Curves
As before, this can be visualized in
3D by plotting each indifference
curve at the height of its utility index.
Utility Functions & Indiff. Curves

Utility
U6
U5
U4
x2 U3
U2
U1
x1
Utility Functions & Indiff. Curves
Comparing all possible consumption
bundles gives the complete collection
of the consumer’s indifference curves,
each with its assigned utility level.
This complete collection of
indifference curves completely
represents the consumer’s
preferences.
Utility Functions & Indiff. Curves

x2

x1
Utility Functions & Indiff. Curves

x2

x1
Utility Functions & Indiff. Curves

x2

x1
Utility Functions & Indiff. Curves

x2

x1
Utility Functions & Indiff. Curves

x2

x1
Utility Functions & Indiff. Curves

x2

x1
Utility Functions & Indiff. Curves

x1
Utility Functions & Indiff. Curves

x1
Utility Functions & Indiff. Curves

x1
Utility Functions & Indiff. Curves

x1
Utility Functions & Indiff. Curves

x1
Utility Functions & Indiff. Curves

x1
Utility Functions & Indiff. Curves

x1
Utility Functions & Indiff. Curves

x1
Utility Functions & Indiff. Curves

x1
Utility Functions & Indiff. Curves

x1
Utility Functions & Indiff. Curves
The collection of all indifference
curves for a given preference relation
is an indifference map.
An indifference map is equivalent to
a utility function; each is the other.
 Utility Functions
There is no unique utility function
representation of a preference
relation.
Suppose U(x1,x2) = x1x2 represents a
preference relation.
Again consider the bundles (4,1),
(2,3) and (2,2).
 Utility Functions
U(x1,x2) = x1x2, so

U(2,3) = 6 > U(4,1) = U(2,2) = 4;

that is, (2,3) ≻ (4,1) ~ (2,2).

 Utility Functions
U(x1,x2) = x1x2 (2,3) ≻ (4,1) ~ (2,2).
Define V = U2.
 Utility Functions
U(x1,x2) = x1x2 (2,3) ≻ (4,1) ~ (2,2).
Define V = U2.
Then V(x1,x2) = x12x22 and
V(2,3) = 36 > V(4,1) = V(2,2) = 16
so again
(2,3) ≻ (4,1) ~ (2,2).
V preserves the same order as U and
so represents the same preferences.
 Utility Functions
U(x1,x2) = x1x2 (2,3) ≻ (4,1) ~ (2,2).
Define W = 2U + 10.
 Utility Functions
u U(x1,x2) = x1x2 (2,3) ≻ (4,1) ~ (2,2).
u Define W = 2U + 10.
u Then W(x1,x2) = 2x1x2+10 so
W(2,3) = 22 > W(4,1) = W(2,2) = 18.
Again,
(2,3) ≻ (4,1) ~ (2,2).
u W preserves the same order as U and V
and so represents the same preferences.
 Utility Functions
If
– U is a utility function that
represents a preference relation ≿
and
– f is a strictly increasing function,
then V = f(U) is also a utility function
representing ≿ .
 Goods, Bads and Neutrals
A good is a commodity unit which
increases utility as it is more
consumed
A bad is a commodity unit which
decreases utility as it is more
consumed.
A neutral is a commodity unit which
does not change utility as it is more
or less consumed.
 Goods, Bads and Neutrals
Utility
Utility
function
Units of Units of
water are water are
goods bads

x’ Water
Around x’ units, a little extra water is a
neutral.
Some Other Utility Functions and
Their Indifference Curves
Instead of U(x1,x2) = x1x2 consider

V(x1,x2) = x1 + x2.

What do the indifference curves for

this “perfect substitution” utility
function look like?
Perfect Substitution Indifference
Curves
x2
x1 + x2 = 5
13
x1 + x2 = 9
9
x1 + x2 = 13
5
V(x1,x2) = x1 + x2.

5 9 13 x1
Perfect Substitution Indifference
Curves
x2
x1 + x2 = 5
13
x1 + x2 = 9
9
x1 + x2 = 13
5
V(x1,x2) = x1 + x2.

5 9 13 x1
All are linear and parallel.
Perfect Substitution Indifference
Curves
What if not one-to-one substitution?
In general, preferences for perfect
substitutes can be represented by a
Utility function of the form
𝑼 𝒙𝟏 , 𝒙𝟐 = 𝒂𝒙𝟏 + 𝒃𝒙𝟐
where 𝒂 and 𝒃 are some positive numbers

Note that the slope of the indifference

curve is given by −𝒂/𝒃
Some Other Utility Functions and
Their Indifference Curves
Instead of U(x1,x2) = x1x2 or
V(x1,x2) = x1 + x2, consider

W(x1,x2) = min{x1,x2}.

What do the indifference curves for

this “perfect complementarity” utility
function look like?
 Perfect Complementarity
Indifference Curves
x2
45o
W(x1,x2) = min{x1,x2}

8 min{x1,x2} = 8
5 min{x1,x2} = 5
3 min{x1,x2} = 3

3 5 8 x1
 Perfect Complementarity
Indifference Curves
x2
45o
W(x1,x2) = min{x1,x2}

8 min{x1,x2} = 8
5 min{x1,x2} = 5
3 min{x1,x2} = 3

3 5 8 x1
All are right-angled with vertices on a ray
from the origin.
Some Other Utility Functions and
Their Indifference Curves
What if the consumer want to
consume the goods in some
proportion other than one-to-one?
In general, a utility function that
describe perfect-complementary
preference: 𝑼 𝒙𝟏 , 𝒙𝟐 = 𝒎𝒊𝒏 𝒂𝒙𝟏 , 𝒃𝒙𝟐
where 𝒂, 𝒃 > 𝟎 that indicate proportions
in which the goods are consumed.

54
Some Other Utility Functions and
Their Indifference Curves
A utility function of the form

U(x1,x2) = f(x1) + x2

is linear in just x2 and is called quasi-

linear.
E.g. U(x1,x2) = 2x11/2 + x2.
 Quasi-linear Indifference Curves
x2 Each curve is a vertically shifted
copy of the others.

x1
Some Other Utility Functions and
Their Indifference Curves
Any utility function of the form

U(x1,x2) = x1a x2b

with a > 0 and b > 0 is called a Cobb-

Douglas utility function.
E.g. U(x1,x2) = x11/2 x21/2 (a = b = 1/2)
V(x1,x2) = x1 x23 (a = 1, b = 3)
 Cobb-Douglas Indifference
x2 Curves
All curves are hyperbolic,
asymptoting to, but never
touching any axis.

x1
 Marginal Utilities
Marginal means “incremental”.
The marginal utility of commodity i is
the rate-of-change of total utility as
the quantity of commodity i
consumed changes; i.e.
U
MU i =
 xi
 Marginal Utilities
E.g. if U(x1,x2) = x11/2 x22 then

 U 1 − 1/ 2 2
MU1 = = x1 x2
 x1 2
 Marginal Utilities
E.g. if U(x1,x2) = x11/2 x22 then

 U 1 − 1/ 2 2
MU1 = = x1 x2
 x1 2
 Marginal Utilities
E.g. if U(x1,x2) = x11/2 x22 then

U 1/ 2
MU 2 = = 2 x1 x2
 x2
 Marginal Utilities
E.g. if U(x1,x2) = x11/2 x22 then

U 1/ 2
MU 2 = = 2 x1 x2
 x2
 Marginal Utilities
So, if U(x1,x2) = x11/2 x22 then
 U 1 − 1/ 2 2
MU1 = = x1 x2
 x1 2
U 1/ 2
MU 2 = = 2 x1 x2
 x2
Marginal Utilities and Marginal
Rates-of-Substitution
The general equation for an
indifference curve is
U(x1,x2)  k, a constant.
Totally differentiating this identity gives
U U
dx1 + dx2 = 0
 x1  x2
Marginal Utilities and Marginal
Rates-of-Substitution
U U
dx1 + dx2 = 0
 x1  x2
rearranged is
U U
dx2 = − dx1
 x2  x1
Marginal Utilities and Marginal
Rates-of-Substitution
And U U
dx2 = − dx1
 x2  x1
rearranged is
d x2  U /  x1
=− .
d x1  U /  x2
This is the MRS.
Marg. Utilities & Marg. Rates-of-
Substitution; An example
Suppose U(x1,x2) = x1x2. Then
U
= (1)( x2 ) = x2
 x1
U
= ( x1 )(1) = x1
 x2
d x2  U /  x1 x2
so MRS = =− =− .
d x1  U /  x2 x1
Marg. Utilities & Marg. Rates-of-
Substitution; An example
x2
x2 U(x1,x2) = x1x2; MRS = −
x1
8 MRS(1,8) = - 8/1 = -8
6 MRS(6,6) = - 6/6 = -1.

U = 36
U=8
1 6 x1
 Marg. Rates-of-Substitution for
Quasi-linear Utility Functions
A quasi-linear utility function is of
the form U(x1,x2) = f(x1) + x2.
U U
= f ( x1 ) =1
 x1  x2
d x2  U /  x1
so MRS = =− = − f  ( x1 ).
d x1  U /  x2
Marg. Rates-of-Substitution for
Quasi-linear Utility Functions
MRS = - f  (x1) does not depend upon
x2 so the slope of indifference curves
for a quasi-linear utility function is
constant along any line for which x1
is constant. What does that make
the indifference map for a quasi-
linear utility function look like?
 Marg. Rates-of-Substitution for
x2
Quasi-linear Utility Functions
MRS = Each curve is a vertically
- f(x1’) shifted copy of the others.
MRS = -f(x1”) MRS is a
constant
along any line
for which x1 is
constant.

x1’ x1” x1
Monotonic Transformations &
Marginal Rates-of-Substitution
Applying a monotonic transformation
to a utility function representing a
preference relation simply creates
another utility function representing
the same preference relation.
What happens to marginal rates-of-
substitution when a monotonic
transformation is applied?
Monotonic Transformations &
Marginal Rates-of-Substitution
For U(x1,x2) = x1x2 the MRS = - x2/x1.
Create V = U2; i.e. V(x1,x2) = x12x22.
What is the MRS for V?
 V /  x1 2
2 x1 x2 x2
MRS = − =− =−
 V /  x2 2
2 x1 x2 x1
which is the same as the MRS for U.
 Monotonic Transformations &
Marginal Rates-of-Substitution
More generally, if V = f(U) where f is a
strictly increasing function, then
 V /  x1 f  (U )   U / x1
MRS = − =−
 V /  x2 f '(U )   U / x2
 U /  x1
=− .
 U /  x2
So MRS is unchanged by a positive
monotonic transformation.