Utility Theory
• For consumer theory do not need to assign precise
numerical values to different indifference curves
Ordinal Utility
• Often convenient to do so especially if dealing with
more than 2 goods
ECON 370: Microeconomic Theory
• To do this, it is convenient to represent the
preference relation by a utility function U(x1,x2,…)
Summer 2004 – Rice University
Stanley Gilbert
Lecture 2
Econ 370 - Ordinal Utility
Conditions
Construction
• U(x) is a utility function representing the
preference relation ‘≿’:
• Construction of a utility function:
– Draw the indifference curves
– Drawing the diagonal (i.e. the line x1 = x2)
– Labeling each indifference curve with how far from the
origin the intersection of that indifference curve and the
diagonal it is.
– U(x) ≥ U(y) if and only if x ≿ y
– For preferences that satisfy A1 – A4
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
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Identical Representations
Equivalence
• Many different utility functions can describe same
set of preferences
• Our model of consumer behavior would be exactly
the same whichever of these utility functions were
assumed.
• Any monotonic transformation of a utility function
is still a utility function
• See that:
U a ( x ) = [U ( x )]2 + 42
• For Example: U ( x ) = x1 x2
U a ( x ) = 4 x12 x22 + 42
U b ( x ) = [U ( x )]
12
U b ( x ) = ( x1 x2 )1 2
U c ( x ) = ln[U ( x )]
U c ( x ) = ln x1 + ln x2
All three are monotonic transformations of U(x).
Econ 370 - Ordinal Utility
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Econ 370 - Ordinal Utility
Characteristics
Interpersonal Comparisons
• An associated issue is that of interpersonal
comparability:
• Consumer theory depends only on the shapes of
indifference curves and the direction in which
utility is increasing (normally northeast). Most
economists argue that precise numerical values
attached to individual indifference curves have no
economic meaning
– If Jack consumes cans of soda and pizzas and prefers
more of each good to less, we can conclude that he
would prefer 3 cans of soda and 2 pizzas per day to 1
can of soda and 1 pizza per day
– Suppose, however, that we have two consumers Jack
and Jill and Jack consumes 3 cans of soda and 2 pizzas
per day while Jill consumes 1 can of soda and 1 pizza
per day. Can we say Jack is happier?
• Because the ordering of indifference curves is all
that is important, this is described as an ordinal
theory of utility
• There is no objective (i.e. value-free) way to
compare the happiness of the two people
• That is, it can only be used to rank choices for a
single individual
Econ 370 - Ordinal Utility
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• In our analysis we treat utility functions as ordinal
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Econ 370 - Ordinal Utility
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Common Utility Functions
Indifference Curves
• Perfect Substitutes:
• If we have a utility function u = f(x1, x2):
– U(x) = ax1 + bx2
a, b > 0
• Indifference curves are represented by
• Perfect Complements:
– U(x) = min{ ax1, bx2}
– c = f(x1, x2)
– Where ‘c’ is some constant
a, b > 0
• Cobb-Douglas Preferences
– U(x) = x1αx21 - α
– U(x) = αln(x1) + (1 – α)ln(x2)
– (Why are these equivalent?)
0≤α≤1
0≤α≤1
• Quasilinear Preferences
– U(x) = x1 + f(x2)
Econ 370 - Ordinal Utility
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Marginal Utility
• The equation of an indifference curve is
U(x1,x2) ≡ k, a constant.
• MUi = change in U with small change in xi (holding constant
consumption of all other goods)
U(x1, x2) = x11/2x22
• Then,
MU(x1) = (½)x1-1/2x22
• And,
MU(x2) = 2x11/2x2
Econ 370 - Ordinal Utility
• Totally differentiating this identity gives
∂U
∂ xi
• For example, if
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Marginal Utilities and the MRS
• Marginal means “incremental”.
MU i =
Econ 370 - Ordinal Utility
∂U
∂U
dx1 +
dx2 = 0
∂x1
∂x2
MRS =
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Econ 370 - Ordinal Utility
or
dx2
∂U / ∂x1
MU ( x1 )
=−
=−
∂U / ∂x2
MU ( x2 )
dx1
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Example of Marginal Utilities and MRS
Monotonic Transformations and MRS
• Monotonic transformation of U does not
change underlying preference structure
• Suppose U(x1,x2) = x1x2. Then
• Example: U(x1, x2) = x1x2 with MRS = - x2/x1.
∂U
= (1)( x2 ) = x2
∂ x1
• Consider V = U2, or V(x1, x2) = x12x22:
MRS = −
∂U
= ( x1 )(1) = x1
∂ x2
so
Econ 370 - Ordinal Utility
MRS =
∂ V / ∂ x1
2x x2
x
= − 12 2 = − 2
∂ V / ∂ x2
x1
2 x1 x2
• This is the same as the MRS for U.
d x2
∂ U / ∂ x1
x
=−
=− 2
∂ U / ∂ x2
x1
d x1
• Monotonic transformation does not change MRS.
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