Chapter 4 UTILITY
Utility is seen only as a way to describe
preferences.
 A utility function

a way of assigning a number to every possible
consumption bundle such that more-preferred
bundles get assigned larger numbers than less
preferred bundles.
( x1 , x2 )
( y1 , y2 ) iff
u ( x1 , x2 )  u ( y1 , y2 )
Ordinal utility: rank matters, size does not.
 Cardinal utility: size matters, too.

Monotonic transformation

Transforming one set of numbers into another set of
numbers in a way that preserves the order of the
numbers.
Monotonic transformation

u(x1, x2) represents the preference:
u ( x1 , x2 )  u ( y1 , y2 )iff ( x1 , x2 )

( y1 , y2 )
f(u) is a monotonic transformation:
u ( x1 , x2 )  u ( y1 , y2 ) iff f (u( x1 , x2 ))  f (u( y1 , y2 ))


f (u( x1 , x2 ))  f (u( y1 , y2 )) iff ( x1 , x2 )
( y1 , y2 )
A monotonic transformation of a utility function is a
utility function that represents the same preference
as the original utility function.
4.1 Cardinal Utility

Cardinal utility:
 Both
order and size matter.
 Tells how much more the consumer likes one
bundle over another.
 Size is inconsequential to consumer behavior.
 Ordinal utility is enough.
4.2 Constructing a Utility Function




Suppose more distant
indifference curves
represents higher utility.
Draw a diagonal line.
Find the distance from the
origin to an indifference
curve along the diagonal
line.
Assign that distance to the
indifference curve.
4.3 Some Examples of Utility
Functions
u(x1, x2) =x1x2

A typical indifference
curve
{(x1, x2): k=x1x2}

Algebraic expression of
the indifference curve
x2=k/x1
4.3 Some Examples of Utility
Functions
Perfect Substitutes
 Preferences of perfect substitutes can be
represented by a utility function of the form
u(x1, x2)=ax1+bx2
 Here a and b measure the “value” of goods 1
and 2 to the consumer. The slope of a typical
indifference curve is given by -a/b.

4.3 Some Examples of Utility
Functions
Perfect Complements
 A utility function that describes a perfect
complement preference is given by
u(x1, x2)=min{ax1,bx2}
 where a and b are positive numbers that
indicate the proportions in which the goods are
consumed.

4.3 Some Examples of Utility
Functions

Quasilinear Preference
The utility function is
linear in good 2, but
nonlinear in good 1:
u(x1, x2)=v(x1)+x2

Cobb-Douglas Preference
u ( x1 , x2 )  x1c x2d
c and d are positive numbers that describe the preferences
of the consumer.
4.4 Marginal Utility

Marginal utility
consumer’s utility change as we give him or her a little
more of good 1.
u ( x1  x1 , x2 )  u ( x1 , x2 ) u ( x1 , x2 )
U
MU1  lim
 lim

x1 0 x
x1  0
x1
x1
1
The change in utility by a marginal consumption of good 1:
U  MU1x1
4.5 Marginal Utility and MRS



A utility function u(x1, x2) can be used to measure the
marginal rate of substitution (MRS).
Consider a change in the consumption bundle, (△x1,
△x2), that keeps utility constant
MU1△x1+MU2△x2=△U=0
Solving for the slope of the indifference curve we
have
MRS=△x2/△x1=－MU1/MU2