Transcript Chapter Four

Chapter 4
Utility
Introduction
Last chapter we talk about preference,
describing the ordering of what a consumer
prefers.
 For a more convenient mathematical treatment,
we turn this ordering into a mathematical
function.

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Utility Functions
A utility function U: R+nR maps each
consumption bundle of n goods into a real
number that satisfies the following conditions:
x’ x”
U(x’) > U(x”)
p

x’p x”
U(x’) < U(x”)
x’ ~ x”
U(x’) = U(x”).
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Utility Functions

Not all theoretically possible preferences have a
utility function representation.

Technically, a preference relation that is
complete, transitive and continuous has a
corresponding continuous utility function.
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Utility Functions
Utility is an ordinal (i.e. ordering) concept.
 The number assigned only matters about
ranking, but the sizes of numerical differences
are not meaningful.
 For example, 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 is y.

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An Example
Consider only three bundles A, B, C.
 The following three are all valid utility
functions of the preference.

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Utility Functions
There is no unique utility function
representation of a preference relation.
 Suppose U(x1,x2) = x1x2 represents a preference
relation.
 Consider the bundles (4,1), (2,3) and (2,2).
U(2,3) = 6 > U(4,1) = U(2,2) = 4.
That is, (2,3) (4,1) ~ (2,2).

p
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Utility Functions
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.

p
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Utility Functions
Define W = 2U + 10.
 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).
 W preserves the same order as U and V and so
represents the same preferences.

p
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Utility Functions

If
U
is a utility function that represents a preference
relation f and
~
 f is a strictly increasing function,
then V = f(U) is also a utility function
representing f
.
~

Clearly, V(x)>V(y) if and only if
f(V(x)) > f(V(y)) by definition of increasing
function.
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Ordinal vs. Cardinal
As you will see, for our analysis of consumer
choices, an ordinal utility is enough.
 If the numerical differences are also meaningful,
we call it cardinal.
 For example, money, weight, height are all
cardinal.

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Utility Functions & Indiff. Curves



An indifference curve contains equally
preferred bundles.
Equal preference  same utility level.
Therefore, all bundles on the same indifference
curve must have the same utility level.
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Utility Functions & Indiff. Curves
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Utility Functions & Indiff. Curves
U=6
U=5
U=4
U=3
U=2
U=1
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Goods, Bads and Neutrals
A good is a commodity which increases your
utility (gives a more preferred bundle) when
you have more of it.
 A bad is a commodity which decreases your
utility (gives a less preferred bundle) when
you have more of it.
 A neutral is a commodity which does not
change your utility (gives an equally preferred
bundle) when you have more of it.

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Goods, Bads and Neutrals
Utility
Units of
water are
goods
Units of
water are
bads
x’
Utility
function
Water
Around x’ units, a little extra water is a neutral.
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Some Other Utility Functions and
Their Indifference Curves

Consider
V(x1,x2) = x1 + x2.
What do the indifference curves look like?
 What relation does this function represent for
these two goods?

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Perfect Substitutes
x2
x1 + x2 = 5
13
x 1 + x2 = 9
9
x1 + x2 = 13
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V(x1,x2) = x1 + x2.
5
9
13
x1
These two goods are perfect substitutes for this consumer.
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Some Other Utility Functions and
Their Indifference Curves

Consider
W(x1,x2) = min{x1,x2}.
 What do the indifference curves look like?
 What relation does this function represent for
these two goods?
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Perfect Complements
x2
45o
W(x1,x2) = min{x1,x2}
min{x1,x2} = 8
8
min{x1,x2} = 5
min{x1,x2} = 3
5
3
3 5
8
x1
These two goods are perfect complements for this consumer.
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Perfect Substitutes and Perfect
Complements

In general, a utility function for perfect
substitutes can be expressed as
u (x, y) = ax + by;

And a utility function for perfect complements
can be expressed as:
u (x, y) = min{ ax , by }
for constants a and b.
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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.

For example, U(x1,x2) = 2x11/2 + x2.
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Quasi-linear Indifference Curves
x2
Each curve is a vertically shifted copy of the
others.
x1
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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.
 For example, U(x1,x2) = x11/2 x21/2, (a = b = 1/2),
and V(x1,x2) = x1 x23 , (a = 1, b = 3).
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Cobb-Douglas Indifference Curves
x2
All curves are hyperbolic,
asymptoting to, but never
touching any axis.
x1
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Cobb-Douglas Utility Functions

By a monotonic transformation V=ln(U):
U( x, y) = xa y b implies
V( x, y) = a ln (x) + b ln(y).

Consider another transformation W=U1/(a+b)
W( x, y)= xa/(a+b) y b/(a+b) = xc y 1-c
so that the sum of the indices becomes 1.
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Marginal Utilities
Marginal means “incremental”.
 The marginal utility of commodity i is the rateof-change of total utility as the quantity of
commodity i consumed changes; i.e.

U
MU i 
 xi
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Marginal Utilities

For example, if U(x1,x2) = x11/2 x22, then

MU 1 


MU 2 

U 1 1/ 2 2
 x1 x2
x1 2
U
 2 x11/ 2 x2
x2
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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
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Marginal Utilities and Marginal
Rates of Substitution
U
U
dx1 
dx2  0
 x1
 x2
It can be rearranged to
U
U
dx2  
dx1
 x2
 x1
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Marginal Utilities and Marginal
Rates of Substitution
And
U
U
dx2  
dx1
 x2
 x1
can be furthermore rearranged to
d x2
 U /  x1

.
d x1
 U /  x2
This is the MRS (slope of the indifference
curve).
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A Note
In some texts, economists refer to the MRS by
its absolute value – that is, as a positive number.
 However, we will still follow our convention.
 Therefore, the MRS is

dx2
MRS 
dx1
U
 U /  x1
MU1


 U /  x2
MU 2
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MRS and MU
Recall that MRS measures how many units of
good 2 you’re willing to sacrifice for one more
unit of good 1 to remain the original utility
level.
 One unit of good 1 is worth MU1.
 One unit of good 2 is worth MU2.
 Number of units of good 2 you are willing to
sacrifice for one unit of good 1 is thus MU1 /
MU2.

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An Example

Suppose U(x1,x2) = x1x2. Then
U
 (1)( x2 ) 
 x1
U
 ( x1 )(1) 
 x2
so
x2
x1
dx2
 U /  x1
x2
MRS 

 .
dx1
 U /  x2
x1
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An Example
U(x1,x2) = x1x2;
x2
8
x2
MRS  
x1
MRS(1,8) = -8/1 = -8
MRS(6,6) = - 6/6 = -1.
6
U = 36
1
6
U=8
x1
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MRS for Quasi-linear Utility Functions

A quasi-linear utility function is of the form
U(x1,x2) = f(x1) + x2.
U
 f ( x1 )
 x1
U
1
 x2
Therefore,
dx2
 U /  x1
MRS 

  f ( x1 ).
dx1
 U /  x2
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MRS for Quasi-linear Utility Functions
MRS = - f′(x1) depends only on x1 but not on
x2. So the slopes of the indifference curves for a
quasi-linear utility function are constant along
any line for which x1 is constant.
 What does that make the indifference map for a
quasi-linear utility function look like?

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MRS for Quasi-linear Utility Functions
x2
MRS = -f(x1’)
Each curve is a vertically
shifted copy of the others.
MRS = -f(x1”)
MRS is a constant
along any line for
which x1 is
constant.
x1’
x1”
x1
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Monotonic Transformations &
Marginal Rates of Substitution
Applying a monotonic (increasing)
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?

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Monotonic Transformations &
Marginal Rates of Substitution
For U(x1,x2) = x1x2 , MRS = -x2/x1.
 Create V = U2; i.e. V(x1,x2) = x12x22. What is
the MRS for V?

 V /  x1
2x x
x2
MRS  


 V /  x2
2x x
x1
2
1 2
2
1 2
which is the same as the MRS for U.
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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
The MRS does not change with a monotonic transformation.
Thus, the same preference with different utility functions still
show the same MRS.
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