Chapter 4 Utility

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Course: Microeconomics
Text: Varian’s Intermediate
Microeconomics
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

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.
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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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

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.
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

Utility is an ordinal (i.e. ordering)
concept.
The number assigned only matters about
ranking, but the sizes of numerical
differences are not meaningful.
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Consider only three bundles A, B, C.
 The following three are all valid utility
function of the preference.

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


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).
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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

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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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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.

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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.

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


An indifference curve contains equally
preferred bundles.
Equal preference  same utility level.
Therefore, all bundles in an indifference
curve have the same utility level.
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x2
U6
U4
U2
x1
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Utility
U6
U5
U4
U3
U2
x2
U1
x1
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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.

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Utility
Utility
function
Units of
water are
goods
Units of
water are
bads
x’
Water
Around x’ units, a little extra water is a neutral.
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
Consider
V(x1,x2) = x1 + x2.
What does the indifference curve look like?
 What relation does this function represent
for these goods?

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x2
x 1 + x2 = 5
13
x1 + x2 = 9
9
x1 + x2 = 13
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V(x1,x2) = x1 + x2.
5
9
13
x1
These goods are perfect substitutes.
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
Consider
W(x1,x2) = min{x1,x2}.
 What does the indifference curve look like?
 What relation does this function represent
for these goods?
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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
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

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.
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
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.
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x2
Each curve is a vertically shifted copy of the others.
x1
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
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)
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x2
All curves are hyperbolic,
asymptoting to, but never
touching any axis.
x1
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

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 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
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
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
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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.
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

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
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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
.
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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
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
Therefore,
MRS 

 U /  x1
U /  x2

MU 1
MU 2
The Marginal Rate of Substitution is the
ratio of marginal utilities.
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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.

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
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
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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
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
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.
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

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?
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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
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

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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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.
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
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.
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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”.
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

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.
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
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?
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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.

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


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
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