### Chapter 4 Utility

```Course: Microeconomics
Text: Varian’s Intermediate
Microeconomics
1


Last chapter we talk about preference,
describing the ordering of what a
consumer likes.
For a more convenient mathematical
treatment, we turn this ordering into a
mathematical function.
2
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”).
3


Not all theoretically possible preference
have a utility function representation.
Technically, a preference relation that is
complete, transitive and continuous has a
corresponding continuous utility function.
4


Utility is an ordinal (i.e. ordering)
concept.
The number assigned only matters about
ranking, but the sizes of numerical
differences are not meaningful.
5
Consider only three bundles A, B, C.
 The following three are all valid utility
function of the preference.

6



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).
7
U(x1,x2) = x1x2, so
U(2,3) = 6 > U(4,1) = U(2,2) = 4;
that is, (2,3)
(4,1) ~ (2,2).
p

8
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
9
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
10

If
U is a utility function that represents a
preference relationf and
~
 f is a strictly increasing function,

then V = f(U) is also a utility function
representing f .
~ if and only if
 Clearly, V(x)>V(y)
f(V(x)) > f(V(y)) by definition of increasing
function.

11
12
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.
 e.g. money, weight, height are cardinal
 Cardinal utility can be useful in some areas,
such as preference under uncertainty.

13



An indifference curve contains equally
preferred bundles.
Equal preference  same utility level.
Therefore, all bundles in an indifference
curve have the same utility level.
14
x2
U6
U4
U2
x1
15
Utility
U6
U5
U4
U3
U2
x2
U1
x1
16
A good is a commodity which increases
utility (gives a more preferred bundle)
when you have more of it.
 A bad is a commodity which decreases
utility (gives a less preferred bundle)
when you have more of it.
 A neutral is a commodity which does
not change utility (gives an equally
preferred bundle) when you have more
of it.

17
Utility
Utility
function
Units of
water are
goods
Units of
water are
x’
Water
Around x’ units, a little extra water is a neutral.
18

Consider
V(x1,x2) = x1 + x2.
What does the indifference curve look like?
 What relation does this function represent
for these goods?

19
x2
x 1 + x2 = 5
13
x1 + x2 = 9
9
x1 + x2 = 13
5
V(x1,x2) = x1 + x2.
5
9
13
x1
These goods are perfect substitutes.
20

Consider
W(x1,x2) = min{x1,x2}.
 What does the indifference curve look like?
 What relation does this function represent
for these goods?
21
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
22


In general, utility function for perfect
substitutes can be expressed as
u(x , y) = ax + by
Utility function for perfect complement
can be expressed as:
u(x , y) = min{ ax , by }
for constants a and b.
23

A utility function of the form
U(x1,x2) = f(x1) + x2
is linear in x2 and is called quasi-linear.

E.g.
U(x1,x2) = 2x11/2 + x2.
24
x2
Each curve is a vertically shifted copy of the others.
x1
25

Any utility function of the form
U(x1,x2) = x1a x2b
with a > 0 and b > 0 is called a CobbDouglas utility function.
 E.g.
U(x1,x2) = x11/2 x21/2 (a = b = 1/2)
V(x1,x2) = x1 x23
(a = 1, b = 3)
26
x2
All curves are hyperbolic,
asymptoting to, but never
touching any axis.
x1
27


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.
28
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.

MU i 
U
 xi
29

E.g. if U(x1,x2) = x11/2 x22 then
MU 1 
MU 2 
 U
 x1
 U
 x2

1
2
1 / 2
x1
 2x
1/ 2
1
2
x2
x2
30
Marginal utility is positive if it is a good,
negative if it is a bad, zero if it is neutral.
 Its value changes under a monotonic
transformation: (Consider the
differentiable case)

MU i 

 f (U )
 xi
 f ' (U )
U
xi
So its value is not particularly meaningful.
31


The general equation for an indifference
curve is
U(x1,x2)  k, a constant.
Totally differentiating this identity gives
U
 x1
dx1 
U
 x2
dx2  0
32
U
 x1
dx1 
U
 x2
dx2  0
We can rearrange this to
U
 x2
dx2  
U
 x1
dx1
Rearrange further:
d x2
d x1

 U /  x1
 U /  x2
.
33
d x2
d x1


 U /  x1
 U /  x2
.
Recall that the definition of MRS:
The negative of the slope of an
indifference curve is its marginal rate of
substitution.
MRS  
d x2
d x1
U
34

Therefore,
MRS 

 U /  x1
U /  x2

MU 1
MU 2
The Marginal Rate of Substitution is the
ratio of marginal utilities.
35
MRS also means how many quantities of
good 2 you are willing to sacrifice for one
more unit of good 1.
 One unit of good 1 is worth MU1.
 One unit of good 2 is worth MU2.
 Number of good 2 you are willing to
sacrifice for a unit of good 1 is thus
MU1 / MU2.

36

Suppose U(x1,x2) = x1x2. Then
U
 x1
U
 x2
so
MRS  
 ( 1)( x2 )  x2
 ( x1 )( 1)  x1
d x2
d x1

 U /  x1
 U /  x2

x2
.
x1
37
U(x1,x2) = x1x2;
x2
MRS 
x2
x1
8
MRS(1,8) = - 8/1 = -8
MRS(6,6) = - 6/6 = -1.
6
U = 36
U=8
1
6
x1
38

A quasi-linear utility function is of the
form U(x1,x2) = f(x1) + x2.
U
 x1
U
 f ( x1 )
MRS  
d x2
d x1
 x2

1
 U /  x1
 U /  x2
 f ( x1 ).
Thus MRS for a quasi-linear function only depends on x1.
39


MRS = f ’(x1) does not depend upon x2 so
the slope of indifference curves for a quasilinear 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?
40
x2
MRS =
f(x1’)
Each curve is a vertically shifted copy
of the others.
MRS = f(x1”)
x1’
x1”
MRS is a constant
along any line for
which x1 is
constant.
x1
41


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?
42
For U(x1,x2) = x1x2 the MRS = x2/x1.
 Create V = U2; i.e. V(x1,x2) = x12x22.
What is the MRS for V?
2
 V /  x1 2 x1 x2 x2

MRS 
 V /  x2

2
1 2
2x x

x1
which is the same as the MRS for U.
43

More generally, if V = f(U) where f is a
strictly increasing function, then
 V /  x1
f (U )   U /  x1
MRS 


 V /  x2
 U /  x1
 U /  x2
f ' (U )   U /  x2
.
Thus the MRS does not change with monotonic
transformation. So, the same preference with different
utility functions still show the same MRS.
44
Consider the goods are the attributes of
each mode of transportation
 TW=total walking time
 TT=total time of trip on the bus/car
 C= total monetary cost


Different means of transportation have
different values of the above “goods”,
forming different “bundles”.
45


E.g. walking all the way involves a high TW
and low C,
Traveling on a taxi has a high C, but low TW
and TT.
Taking public transport may be something in
between.
We can estimate (with suitable econometrics
methods) a utility function that represents
people’s preferences if we know their choices
and TW, TT and C.
46

Domenich and McFadden (1975) use the
linear form (recall what it represents?) and
have the following function:
U= -0.147TW – 0.0411TT – 2.24C


We can obtain the MRS. E.g. Commuters are
willing to substitute 3 minutes of walking for
1 minute of walking.
How much is one willing to pay to shorten
the trip (on vehicle) for one minute?
47
This chapter we introduce utility function
as a way to represent preference
numerically.
 One prefers a bundle of higher utility than
a bundle of lower utility.
 Utility function is ordinal, and is invariant
to increasing transformation.
 The ratio of marginal utility is also the
marginal rate of substitution.

48



Chapter 2: Budget constraint
-what is affordable/feasible
Chapter 3 / 4: Preference / Utility
-what one likes more
Chapter 5: Choice
-choose the one with highest
utility under budget constraint
49
```