Chapter 3
PREFERENCES AND UTILITY
Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved.
1
Outline
• Preference
– Axioms of rational choice
• Completeness
• Transitivity
• Continuity
• Utility
– Marginal rate of substitution (MRS) and convexity
• Graph
• Mathematical derivation
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Axioms of Rational Choice
• Completeness
– if A and B are any two situations, an
individual can always specify exactly one of
these possibilities:
• A is preferred to B
• B is preferred to A
• A and B are equally attractive
3
Axioms of Rational Choice
• Transitivity
– if A is preferred to B, and B is preferred to
C, then A is preferred to C
– assumes that the individual’s choices are
internally consistent
4
Axioms of Rational Choice
• Continuity
– if A is preferred to B, then situations suitably
“close to” A must also be preferred to B
– used to analyze individuals’ responses to
relatively small changes in income and
prices
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Utility
• Given these assumptions, it is possible to show
that people are able to rank in order all possible
situations from least desirable to most
• Economists call this ranking utility (introduced
by Jeremy Bentham 19 century)
– if A is preferred to B, then the utility assigned to A
exceeds the utility assigned to B
U(A) > U(B)
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Utility
• Utility rankings are ordinal in nature
– they record the relative desirability of commodity
bundles
• Any set of numbers that accurately reflects a
person’s preference ordering will do.
• Because utility measures are not unique, it
makes no sense to consider how much more
utility is gained from A than from B
• It is also impossible to compare utilities
between people (may use very different scale)
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Utility
• Utility is affected by the consumption of
physical commodities, psychological
attitudes, peer group pressures, personal
experiences, and the general cultural
environment
• Economists generally devote attention to
quantifiable options while holding
constant the other things that affect utility
– ceteris paribus assumption
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Utility
• Assume that an individual must choose among
consumption goods x1, x2,…, xn
• The individual’s rankings can be shown by a
utility function of the form:
utility = U(x1, x2,…, xn; other things)
– x’s refer to the quantities of the goods that might be
chosen
– this function is unique up to an order-preserving
transformation
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utility
• Quite often to write as
utility = U(x1, x2,…, xn)
If only two goods
utility = U(x, y)
Everything else is being held constant
10
Arguments of utility functions
• The utility functionis used to represent
how an individual ranks certain bundles of
goods that might be purchased at one
point in time. On occasion we will use
other arguments in the utility function
– utility = U(w): wealth
– utility = U(c,h): consumption and nonwork time
– utility = U(c1, c2): consumption in period 1 and 2
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Economic Goods
In the utility function, the x’s are assumed to
be “goods”
– more is preferred to less
Quantity of y
Preferred to x*, y*
?
y*
Worse
than
x*, y*
?
Quantity of x
x*
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Trade and substitution
• Most economic activity involves trading
between individuals.
• When someone buys a loaf of bread, he
or she is voluntarily giving up one thing
(money)
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Indifference Curves
• An indifference curve shows a set of
consumption bundles among which the
individual is indifferent
Quantity of y
Combinations (x1, y1) and (x2, y2)
provide the same level of utility
y1
y2
U1
Quantity of x
x1
x2
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Indifference curve
• The slope of the indifference curve is
negative, showing that if the individual is
forced to give up some y, he or she
must be compensated by an additional
amount of x
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Marginal Rate of Substitution
• The negative of the slope of the
indifference curve at any point is called
the marginal rate of substitution (MRS)
Quantity of y
dy
MRS  
dx U U1
y1
y2
U1
Quantity of x
x1
x2
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Marginal Rate of Substitution
• MRS changes as x and y change
– reflects the individual’s willingness to trade y
for x
Quantity of y
At (x1, y1), the indifference curve is steeper.
The person would be willing to give up more
y to gain additional units of x
At (x2, y2), the indifference curve
is flatter. The person would be
willing to give up less y to gain
additional units of x
y1
y2
U1
Quantity of x
x1
x2
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MRS
• The slope of U1 and the MRS tell us
something about the trade this person
will voluntarily make.
• MRS diminishes between x1,y1 and x2,y2
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Indifference Curve Map
• Each point must have an indifference
curve through it
Quantity of y
Increasing utility
U3
U2
U1 < U2 < U3
U1
Quantity of x
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Indifference curve map
• Indifference curves are similar to
contour lines on a map in that they
represent lines of equal “altitude”
20
Transitivity
• Can any two of an individual’s indifference
curves intersect?
Quantity of y
The individual is indifferent between A and C.
The individual is indifferent between B and C.
Transitivity suggests that the individual
should be indifferent between A and B
C
B
A
U2
But B is preferred to A
because B contains more
x and y than A
U1
Quantity of x
21
Convexity of indifference curve
• An alternative way of stating the
principle of a diminishing marginal rate
of substitution (MRS) uses the
mathematical notion of a convex set
22
Convexity
• A set of points is convex if any two points
can be joined by a straight line that is
contained completely within the set
Quantity of y
The assumption of a diminishing MRS is
equivalent to the assumption that all
combinations of x and y which are
preferred to or x* and y* form a convex set
y*
U1
Quantity of x
x*
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Convexity
• If the indifference curve is convex, then
the combination (x1 + x2)/2, (y1 + y2)/2 will
be preferred to either (x1,y1) or (x2,y2)
Quantity of y
This implies that “well-balanced” bundles are preferred
to bundles that are heavily weighted toward one
commodity
y1
(y1 + y2)/2
y2
U1
Quantity of x
x1
(x1 + x2)/2
x2
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Convexity and balance in
consumption
• Individual prefer some balance in their
consumption
• Any proportional combination of two
indifferent bundles of goods will be preferred
to the initial
• The assumption of convexity and diminishing
MRS rule out the possibility of an indifference
curve being straight over any portion of its
length
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Utility and the MRS
• Suppose an individual’s preferences for
hamburgers (y) and soft drinks (x) can
be represented by
utility  10  x  y
• Solving for y, we get
y = 100/x
• Solving for MRS = -dy/dx:
MRS = -dy/dx = 100/x2
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Utility and the MRS
MRS = -dy/dx = 100/x2
• Note that as x rises, MRS falls
– when x = 5, MRS = 4 (the person is willing to give
up 4 hamburger for another soft drink)
– when x = 20, MRS = 0.25
– Point C is midway between points A and B and
has 12.5 hamburgers and 12.5 soft drinks.
Utility =12.5
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A mathematical derivation
• A mathematical derivation of the MRS
concept provided additional insights
about the shape of indifference curves
and the nature of preferences
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Marginal Utility
• Suppose that an individual has a utility
function of the form
utility = U(x,y)
• The total differential of U is
U
U
dU 
dx 
dy
x
y
• Along any indifference curve, utility is
constant (dU = 0)
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Deriving the MRS
• Therefore, we get:
dy
MRS  
dx
Uconstant
U
 x
U
y
• MRS is the ratio of the marginal utility of
x to the marginal utility of y
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Diminishing Marginal Utility
and the MRS
• Intuitively, it seems that the assumption
of decreasing marginal utility is related to
the concept of a diminishing MRS
– diminishing MRS requires that the utility
function be quasi-concave
• this is independent of how utility is measured
– diminishing marginal utility depends on how
utility is measured
• Thus, these two concepts are different
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Convexity of Indifference
Curves
• Suppose that the utility function is
utility  x  y
• We can simplify the algebra by taking the
logarithm of this function
U*(x,y) = ln[U(x,y)] = 0.5 ln x + 0.5 ln y
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Convexity of Indifference
Curves
• Thus,
U * 0.5
y

x
x
MRS 


U * 0.5 x
y
y
• MRS is diminishing as x increases and y
decreases. The indifference curves are
convex
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Convexity of Indifference
Curves
• If the utility function is
U(x,y) = x + xy + y
• There is no advantage to transforming
this utility function, so
U
1 y

x
MRS 

U 1  x
y
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Convexity of Indifference
Curves
• Suppose that the utility function is
utility 
x2  y2
• For this example, it is easier to use the
transformation (order preserving)
U*(x,y) = [U(x,y)]2 = x2 + y2
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Convexity of Indifference
Curves
• Thus,
U *
2x x

x
MRS 


U * 2y y
y
• as x increases and y increases, the MRS increases.
The curve are concave, not convex, and clearly not
a quasi-concave function
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Utility functions for specific
preferences
• Individuals’ rankings of commodity bundles and the
utility functions implied by this rankings are
unobservable.
• All we can learn about people’s preferences must
come from the behavior we observe when they
respond to changes in income, prices, and other
factors
• Examine a few of the forms of utility functions is
useful , both because such an examination may offer
some insights into observed behavior and understand
the properties of such functions can be of some help
in solving problems.
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Examples of Utility Functions
• Cobb-Douglas Utility
utility = U(x,y) = xy
where  and  are positive constants
– The relative sizes of  and  indicate the relative
importance of the goods
– Since utility is unique only up to a monotonic
transformation, it is often convenient to normalize
these parameters so that α+β=1
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Examples of Utility Functions
• Perfect Substitutes
utility = U(x,y) = x + y
Quantity of y
The indifference curves will be linear.
The MRS will be constant along the
indifference curve.
U3
U1
U2
Quantity of x
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Perfect substitutions
• utility = U(x,y) = x + y
dy
MRS  
dx |U cons tan t
U


x


U 
y
40
Perfect substitutes
• MRS is constant along the entire indifference
curve. A person would be willing to give up
the same amount of y to get one more x no
matter how much x was being consumed.
• A diminishing MRS do not apply
• Might describe the relationship between
different brands of the same product (e.g.
willing to give up 10 gallons of Shell in
exchange for 10 gallons of Exxon
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Examples of Utility Functions
• Perfect Complements
utility = U(x,y) = min (x, y)
Quantity of y
The indifference curves will be
L-shaped. Only by choosing more
of the two goods together can utility
be increased.
U3
U2
U1
Quantity of x
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Perfect complements
utility = U(x,y) = min (x, y)
Neither of the two goods will be in excess only if x =
y, that is y/x= / 
MRS=infinite
for y/x> / 
=undefined
for y/x= / 
=0
for y/x< / 
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Perfect complements
• These preferences would apply to
goods that “ go together”– Coffee and
cream, Peanut butter and jelly, and
cream cheese and lox
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Examples of Utility Functions
• CES Utility (Constant elasticity of
substitution)
utility = U(x,y) = x/ + y/
when   0 and  ≦ 1.
utility = U(x,y) = ln x + ln y
when  = 0
– Perfect substitutes   = 1
– Cobb-Douglas   = 0
– Perfect complements   = - (less
obvious-- using a limits argument)
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Examples of Utility Functions
• CES Utility (Constant elasticity of
substitution)
– The elasticity of substitution () is equal to
1/(1 - )
• Perfect substitutes   = 
• Fixed proportions   = 0
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Homothetic Preferences
• If the MRS depends only on the ratio of
the amounts of the two goods, not on
the quantities of the goods, the utility
function is homothetic
– Perfect substitutes  MRS is the same at
every point
– Perfect complements  MRS =  if y/x >
/, undefined if y/x = /, and MRS = 0 if
y/x < /
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Homothetic Preferences
• For the general Cobb-Douglas function,
the MRS can be found as
U
 1 

x
y
 y

x
MRS 

 
  1
U x y
 x
y
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Homothetic preference
• The importance of homothetic functions is
that one indifference curve is much like
another. Slopes of the curves depend only on
the ratio y/x, not on how far the curve is from
the origin.
• We can only look at one indifference curve or
at a few nearby curves without fearing that
our results would change dramatically at very
different levels of utility.
49
Nonhomothetic Preferences
• Some utility functions do not exhibit homothetic
preferences
utility = U(x,y) = x + ln y
U
1

x
MRS 
 y
U
1
y
y
• MRS diminishes as y decreases, but it is
independent of the quantity of x consumed.
50
Nonhomothetic preference
• Contrary to the homothetic case, a
doubling of both x and y doubles the
MRS in this case rather than leaving it
unchanged.
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The Many-Good Case
• Suppose utility is a function of n goods
given by
utility = U(x1, x2,…, xn)
• The total differential of U is
U
U
U
dU 
dx1 
dx2  ... 
dxn
x1
x2
xn
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The Many-Good Case
• We can find the MRS between any two
goods by setting dU = 0
U
U
dU  0 
dxi 
dx j
xi
x j
• Rearranging, we get
U
dx j
x i
MRS( x i for x j )  

U
dx i
x j
53
Multigood Indifference
Surfaces
• We will define an indifference surface
as being the set of points in n
dimensions that satisfy the equation
U(x1,x2,…xn) = k
where k is any preassigned constant
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Multigood Indifference
Surfaces
• If the utility function is quasi-concave,
the set of points for which U  k will be
convex
– all of the points on a line joining any two
points on the U = k indifference surface will
also have U  k
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Important Points to Note:
• If individuals obey certain behavioral
postulates, they will be able to rank all
commodity bundles
– the ranking can be represented by a utility
function
– in making choices, individuals will act as if
they were maximizing this function
• Utility functions for two goods can be
illustrated by an indifference curve map
56
Important Points to Note:
• The negative of the slope of the
indifference curve measures the marginal
rate of substitution (MRS)
– the rate at which an individual would trade
an amount of one good (y) for one more unit
of another good (x)
• MRS decreases as x is substituted for y
– individuals prefer some balance in their
consumption choices
57
Important Points to Note:
• A few simple functional forms can capture
important differences in individuals’
preferences for two (or more) goods
– Cobb-Douglas function
– linear function (perfect substitutes)
– fixed proportions function (perfect
complements)
– CES function
• includes the other three as special cases
58
Important Points to Note:
• It is a simple matter to generalize from
two-good examples to many goods
– studying peoples’ choices among many
goods can yield many insights
– the mathematics of many goods is not
especially intuitive, so we will rely on twogood cases to build intuition
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The end
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