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Indifference Curves An indifference curve shows a set of
consumption bundles among which the
individual is indifferent
Quantity of X
Quantity of Y
X1
Y1
Y2
X2
U1
Combinations (X1, Y1) and (X2, Y2)provide the same level of utility
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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 X
Quantity of Y
X1
Y1
Y2
X2
U1
1UU
dX
dYMRS
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Marginal Rate of Substitution MRS changes as Xand Ychange
reflects the individuals willingness to trade Y
forX
Quantity of X
Quantity of Y
X1
Y1
Y2
X2
U1
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 curveis flatter. The person would bewilling to give up less Y to gainadditional units of X
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Indifference Curve Map Each point must have an indifference
curve through it
Quantity of X
Quantity of Y
U1
U2U3 U1 < U2 < U3
Increasing utility
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Transitivity Can two of an individuals indifference
curves intersect?
Quantity of X
Quantity of Y
U1
U2
A
BC
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
But B is preferred to Abecause B contains more
Xand Y than A
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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 X
Quantity of Y
U1
The assumption of a diminishing MRS is
equivalent to the assumption that all
combinations ofXand Ywhich are
preferred to X* and Y* form a convex set
X*
Y*
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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 X
Quantity of Y
U1
X2
Y1
Y2
X1
This implies that well-balanced bundles are preferred
to bundles that are heavily weighted toward one
commodity
(X1 + X2)/2
(Y1 + Y2)/2
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Utility and the MRS
Suppose an individuals preferences for
hamburgers (Y) and soft drinks (X) can
be represented by
YX 10utility
Solving forY, 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 Xrises, MRSfalls
When X= 5, MRS= 4 When X= 20, MRS= 0.25
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Marginal Utility
Suppose that an individual has a utility
function of the form
utility = U(X1, X2,, Xn)
We can define the marginal utility of
good X1 by
marginal utility ofX1 = MUX1 = U/X1
The marginal utility is the extra utility
obtained from slightly more X1 (all else
constant)
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Marginal Utility
The total differential ofUis
n
n
dXX
UdX
X
UdX
X
UdU
...
2
2
1
1
nXXXdXMUdXMUdXMUdU
n
...21
21
The extra utility obtainable from slightly
more X1, X2,, Xn is the sum of the
additional utility provided by each of
these increments
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Deriving the MRS
Suppose we change Xand Ybut keep
utility constant (dU= 0)
dU= 0 = MUXdX+ MU
YdY
Rearranging, we get:
YU
XU
MU
MU
dX
dY
Y
X
/
/
constantU
MRSis the ratio of the marginal utility of
Xto the marginal utility ofY
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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 MRSrequires that the utilityfunction be quasi-concave
This is independent of how utility is measured
Diminishing marginal utility depends on howutility is measured
Thus, these two concepts are different
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Marginal Utility and the MRS Again, we will use the utility function
5050 ..utility YXYX
The marginal utility of a soft drink ismarginal utility = MUX= U/X= 0.5X
-
0.5Y0.5
The marginal utility of a hamburger is
marginal utility = MUY= U/Y= 0.5X0.5Y-
0.5X
Y
YX
YX
MU
MU
dX
dYMRS
Y
X
5050
5050
5
5
..
..
constantU .
.
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Examples of Utility Functions
Cobb-Douglas Utility
utility = U(X,Y) = XY
where and are positive constants The relative sizes of and indicate the
relative importance of the goods
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Examples of Utility Functions
Perfect Substitutes
utility = U(X,Y) = X+ Y
Quantity of X
Quantity of Y
U1U2
U3
The indifference curves will be linear.
The MRS will be constant along the
indifference curve.
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Examples of Utility Functions
Perfect Complements
utility = U(X,Y) = min (X, Y)
Quantity of X
Quantity of YThe indifference curves will be
L-shaped. Only by choosing more
of the two goods together can utility
be increased.
U1
U2
U3
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Examples of Utility Functions CES Utility (Constant elasticity of
substitution)
utility = U(X,Y) = X/ + Y/
when 0 and
utility = U(X,Y) = ln X+ ln Y
when = 0 Perfect substitutes = 1
Cobb-Douglas = 0
Perfect complements = -
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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 MRSdepends 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 MRSis the same at
every point
Perfect complements MRS= ifY/X>/, undefined ifY/X= /, and MRS= 0 if
Y/X< /
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Nonhomothetic Preferences Some utility functions do not exhibit
homothetic preferences
utility = U(X,Y) = X+ ln Y
MUY=U/Y= 1/Y
MUX= U/X= 1
MRS= MUX/ MUY= Y Because the MRSdepends on the
amount ofYconsumed, the utility function
is not homothetic
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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
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Important Points to Note:
The negative of the slope of the
indifference curve measures the marginal
rate of substitution (MRS)
This shows the rate at which an individual
would trade an amount of one good (Y) for one
more unit of another good (X)
MRSdecreases as Xis substituted forY This is consistent with the notion that
individuals prefer some balance in their
consumption choices
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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