# Utility by huanghengdong

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```									Utility
UTILITY FUNCTIONS
 A preference relation that is complete,
reflexive, transitive and continuous can be
represented by a continuous utility
function (as an alternative, or as a
complement, to the indifference “map” of
the previous lecture).
 Continuity means that small changes to a
consumption bundle cause only small
changes to the preference (utility) level.
UTILITY FUNCTIONS
A utility function U(x) represents a
preference relation if and only if:
p
x0       x1     U(x0) > U(x1)

x0 p x1         U(x0) < U(x1)

x0 ~ x1         U(x0) = U(x1)
UTILITY FUNCTIONS
is an ordinal (i.e. ordering or
 Utility
ranking) concept.
 For example, if U(x) = 6 and U(y) = 2
then bundle x is strictly preferred to
bundle y. However, x is not
necessarily “three times better” than
y.
UTILITY FUNCTIONS
and INDIFFERENCE CURVES

 Consider   the bundles (4,1), (2,3) and
(2,2).
 Suppose (2,3) (4,1) ~ (2,2).
p
 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
and INDIFFERENCE CURVES

 Anindifference curve contains
equally preferred bundles.
 Equal   preference  same utility level.
 Therefore, all bundles on an
indifference curve have the same
utility level.
UTILITY FUNCTIONS
and INDIFFERENCE CURVES

 So the bundles (4,1) and (2,2) are on
the indifference curve with utility
level U  
 But the bundle (2,3) is on the
indifference curve with utility level U
6
UTILITY FUNCTIONS
and INDIFFERENCE CURVES

x2          (2,3) > (2,2) = (4,1)
x2

U6
U4

x1
xx11
UTILITY FUNCTIONS
and INDIFFERENCE CURVES

 Comparing   more bundles will create
a larger collection of all indifference
curves and a better description of the
consumer’s preferences.
UTILITY FUNCTIONS
and INDIFFERENCE CURVES

x2x2

U6
U4
U2
xx1
1
UTILITY FUNCTIONS
and INDIFFERENCE 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
and INDIFFERENCE 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
   If
(i) U is a utility function that represents a
preference relation; and (ii) f is a strictly
increasing function,
then
V = f(U) is also a utility function
representing the original preference
function.
Example? V = 2.U
A  good is a commodity unit which
increases utility (gives a more
preferred bundle).
 A bad is a commodity unit which
decreases utility (gives a less
preferred bundle).
 A neutral is a commodity unit which
does not change utility (gives an
equally preferred bundle).

Utility
Utility
function
Units of      Units of
water are     water are

x’            Water
Around x’ units, a little extra water is a neutral.
UTILITY FUNCTIONS
Cobb-Douglas Utility Function

U x1 , x2   x x (a  0, b  0)
a b
1 2
Perfect Substitutes Utility Function

U  x1 , x2   ax1  bx2      Note: MRS = (-)a/b

Perfect Complements Utility Function
U  x1 , x2   minx1, x2      Note: MRS = ?
UTILITY
Preferences can be represented by a
utility function if the functional form has
certain “nice” properties

Example: Consider U(x1,x2)= x1.x2

1.   u/x1>0      and     u/x2>0
2.   Along a particular indifference curve
x1.x2 = constant  x2=c/x1
As      x1          x 2
i.e. downward sloping indifference curve
UTILITY
Example U(x1,x2)= x1.x2 =16
X1       X2             MRS
1        16
2        8              (-) 8
3        5.3            (-) 2.7
4        4              (-) 1.3
5        3.2            (-) 0.8
3. As X1  MRS  (in absolute terms),
i.e convex preferences
COBB DOUGLAS UTILITY FUNCTION

 Any   utility function of the form

U(x1,x2) = x1a x2b

with a > 0 and b > 0 is called a Cobb-
Douglas utility function.
 Examples
U(x1,x2) = x11/2 x21/2 (a = b = 1/2)
V(x1,x2) = x1 x23      (a = 1, b = 3)
COBB DOUBLAS INDIFFERENCE CURVES

x2
All curves are “hyperbolic”,
asymptoting to, but never
touching any axis.

x1
PERFECT SUBSITITUTES
 Instead   of U(x1,x2) = x1x2 consider

V(x1,x2) = x1 + x2.
PERFECT SUBSITITUTES
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 COMPLEMENTS
 Instead of U(x1,x2) = x1x2 or
V(x1,x2) = x1 + x2, consider

W(x1,x2) = min{x1,x2}.
PERFECT COMPLEMENTS
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/corners
on a ray from the origin.
MARGINAL UTILITY

 Marginal  means “incremental”.
 The marginal utility of product i is the
rate-of-change of total utility as the
quantity of product i consumed
changes by one unit; i.e.
U
MU i 
 xi
MARGINAL UTILITY
U=(x1,x2)
MU1=U/x1  U=MU1.x1
MU2=U/x2  U=MU2.x2

Along a particular indifference curve

U = 0 = MU1(x1) + MU2(x2)

 x2/x1 {= MRS} = (-)MU1/MU2
MARGINAL UTILITY

 E.g.   if U(x1,x2) = x11/2 x22 then

 U 1 1/ 2 2
MU 1       x1 x2
 x1 2
MARGINAL UTILITY
 E.g.   if U(x1,x2) = x11/2 x22 then

 U 1 1/ 2 2
MU1       x1 x2
 x1 2
MARGINAL UTILITY
 E.g.   if U(x1,x2) = x11/2 x22 then

U        1/ 2
MU 2        2 x1 x2
 x2
MARGINAL UTILITY
 E.g.   if U(x1,x2) = x11/2 x22   then

U        1/ 2
MU 2        2 x1 x2
 x2
MARGINAL UTILITY
 So,   if U(x1,x2) = x11/2 x22 then

   U 1 1 / 2 2
MU 1         x1 x 2
   x1 2
   U
MU 2          2 x1 / 2 x 2
1

   x2
MARGINAL UTLITIES AND MARGINAL
RATE OF SUBISITUTION

 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 UTLITIES AND MARGINAL
RATE OF SUBISITUTION

U         U
dx1       dx2  0
 x1        x2
rearranging
U           U
dx2        dx1
 x2          x1
MARGINAL UTLITIES AND MARGINAL
RATE OF SUBISITUTION

U           U
dx2        dx1
 x2          x1
rearranging
d x2     U /  x1

d x1     U /  x2
This is the MRS.
MU’s and MRS: 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
MU’s and MRS: 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
MONOTONIC TRANSFORMATIONS AND MRS

 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?
(Hopefully, nothing)
MONOTONIC TRANSFORMATIONS AND MRS

   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 AND MRS

 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.

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