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Utility

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Preferences - A Reminder
ux     y: x is preferred strictly to y.
ux ~ y: x and y are equally preferred.
ux     y: x is preferred at least as much as is y.
Textové pole: p
p
~
f

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Preferences - A Reminder
uCompleteness:  For any two bundles x and y it is always possible to state either that
                      x      y
or that
                      y      x.
~
f
~
f

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Preferences - A Reminder
uReflexivity:  Any bundle x is always at least as preferred as itself; i.e.
                         x      x.
~
f

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Preferences - A Reminder
uTransitivity:  If
x is at least as preferred as y, and
y is at least as preferred as z, then
x is at least as preferred as z; i.e.
  x      y and y      z          x      z.
~
f
~
f
~
f

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Utility Functions
uA preference relation that is complete, reflexive, transitive and continuous can be represented by
a continuous utility function.
uContinuity means that small changes to a consumption bundle cause only small changes to the
preference level.

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Utility Functions
uA utility function U(x) represents a preference relation      if and only if:
        x’     x”                U(x’) > U(x”)
        x’     x”                U(x’) < U(x”)
        x’ ~  x”                U(x’) = U(x”).
~
f
Textové pole: p
p
p

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Utility Functions
uUtility is an ordinal (i.e. ordering) concept.
uE.g. if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to  bundle y.  But x is not
preferred three times as much as is y.

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Utility Functions & Indiff. Curves
uConsider the bundles (4,1), (2,3) and (2,2).
uSuppose (2,3)      (4,1) ~ (2,2).
uAssign to these bundles any numbers that preserve the preference ordering;
e.g.  U(2,3) = 6 > U(4,1) = U(2,2) = 4.
uCall these numbers utility levels.
Textové pole: p
p

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Utility Functions & Indiff. Curves
uAn indifference curve contains  equally preferred bundles.
uEqual preference Þ same utility level.
uTherefore, all bundles in an indifference curve have the same utility level.

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Utility Functions & Indiff. Curves
uSo the bundles (4,1) and (2,2) are in the indiff. curve with utility level U º 4
uBut the bundle (2,3) is in the indiff. curve with utility level U º 6.
uOn an indifference curve diagram, this preference information looks as  follows:

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Utility Functions & Indiff. Curves
U º 6
U º 4
(2,3)      (2,2) ~ (4,1)
x1
x2
Textové pole: p
p

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Utility Functions & Indiff. Curves
uAnother way to visualize this same information is to plot the utility level on a vertical axis.

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U(2,3) = 6
U(2,2) = 4
U(4,1) = 4
Utility Functions & Indiff. Curves
3D plot of consumption & utility levels for 3 bundles
x1
x2
Utility

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Utility Functions & Indiff. Curves
uThis 3D visualization of preferences can be made more informative by adding into it the two
indifference curves.

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Utility Functions & Indiff. Curves
U º 4
U º 6
Higher indifference
curves contain
more preferred
bundles.
Utility
x2
x1

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Utility Functions & Indiff. Curves
uComparing more bundles will create a larger collection of all indifference curves and a better
description of the consumer’s preferences.

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Utility Functions & Indiff. Curves
U º 6
U º 4
U º 2
x1
x2

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Utility Functions & Indiff. Curves
uAs before, this can be visualized in 3D by plotting each indifference curve at the height of its
utility index.

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Utility Functions & Indiff. Curves
U º 6
U º 5
U º 4
U º 3
U º 2
U º 1
x1
x2
Utility

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Utility Functions & Indiff. Curves
uComparing all possible consumption bundles gives the complete collection of the consumer’s
indifference curves, each with its assigned utility level.
uThis complete collection of indifference curves completely represents the consumer’s preferences.

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Utility Functions & Indiff. Curves
x1
x2

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Utility Functions & Indiff. Curves
x1
x2

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Utility Functions & Indiff. Curves
x1
x2

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Utility Functions & Indiff. Curves
x1
x2

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Utility Functions & Indiff. Curves
x1
x2

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Utility Functions & Indiff. Curves
x1
x2

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Utility Functions & Indiff. Curves
x1

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Utility Functions & Indiff. Curves
x1

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Utility Functions & Indiff. Curves
x1

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Utility Functions & Indiff. Curves
x1

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Utility Functions & Indiff. Curves
x1

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Utility Functions & Indiff. Curves
x1

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Utility Functions & Indiff. Curves
x1

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Utility Functions & Indiff. Curves
x1

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Utility Functions & Indiff. Curves
x1

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Utility Functions & Indiff. Curves
x1

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Utility Functions & Indiff. Curves
uThe collection of all indifference curves for a given preference relation is an indifference map.
uAn indifference map is equivalent to a utility function; each is the other.

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Utility Functions
uThere is no unique utility function representation of a preference relation.
uSuppose U(x1,x2) = x1x2 represents a preference relation.
uAgain consider the bundles (4,1),
(2,3) and (2,2).

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Utility Functions
uU(x1,x2) = x1x2, so
U(2,3) = 6 > U(4,1) = U(2,2) = 4;
that is, (2,3)     (4,1) ~ (2,2).
Textové pole: p
p

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Utility Functions
uU(x1,x2) = x1x2           (2,3)     (4,1) ~ (2,2).
uDefine V = U2.
Textové pole: p
p

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Utility Functions
uU(x1,x2) = x1x2           (2,3)     (4,1) ~ (2,2).
uDefine V = U2.
uThen V(x1,x2) = x12x22  and
V(2,3) = 36 > V(4,1) = V(2,2) = 16
so again
(2,3)     (4,1) ~ (2,2).
uV preserves the same order as U and so represents the same preferences.
Textové pole: p
p
Textové pole: p
p

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Utility Functions
uU(x1,x2) = x1x2           (2,3)     (4,1) ~ (2,2).
uDefine W = 2U + 10.
u
Textové pole: p
p

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Utility Functions
uU(x1,x2) = x1x2           (2,3)     (4,1) ~ (2,2).
uDefine W = 2U + 10.
uThen W(x1,x2) = 2x1x2+10 so
W(2,3) = 22 >  W(4,1) = W(2,2) = 18.  Again,
(2,3)     (4,1) ~ (2,2).
uW preserves the same order as U and V and so represents the same preferences.
Textové pole: p
p
Textové pole: p
p

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Utility Functions
uIf
–U is a utility function that represents a preference relation    and
– f is a strictly increasing function,
u then V = f(U) is also a utility function
representing     .
~
f
~
f

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Goods, Bads and Neutrals
uA good is a commodity unit which increases utility (gives a more preferred bundle).
uA bad is a commodity unit which decreases utility (gives a less preferred bundle).
uA neutral is a commodity unit which does not change utility (gives an equally preferred bundle).

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Goods, Bads and Neutrals
Utility
Water
x’
Units of
water are
goods
Units of
water are
bads
Around x’ units, a little extra water is a neutral.
Utility
function

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Some Other Utility Functions and Their Indifference Curves
uInstead of U(x1,x2) = x1x2 consider
              V(x1,x2) = x1 + x2.
What do the indifference curves for this “perfect substitution” utility function look like?

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Perfect Substitution Indifference Curves
5
5
9
9
13
13
x1
x2
x1 + x2 = 5
x1 + x2 = 9
x1 + x2 = 13
V(x1,x2) = x1 + x2.

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Perfect Substitution Indifference Curves
5
5
9
9
13
13
x1
x2
x1 + x2 = 5
x1 + x2 = 9
x1 + x2 = 13
All are linear and parallel.
V(x1,x2) = x1 + x2.

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Some Other Utility Functions and Their Indifference Curves
uInstead of U(x1,x2) = x1x2 or
V(x1,x2) = x1 + x2, consider
             W(x1,x2) = min{x1,x2}.
What do the indifference curves for this “perfect complementarity” utility function look like?

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Perfect Complementarity Indifference Curves
x2
x1
45o
min{x1,x2} = 8
3
5
8
3
5
8
min{x1,x2} = 5
min{x1,x2} = 3
W(x1,x2) = min{x1,x2}

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Perfect Complementarity Indifference Curves
x2
x1
45o
min{x1,x2} = 8
3
5
8
3
5
8
min{x1,x2} = 5
min{x1,x2} = 3
All are right-angled with vertices on a ray
from the origin.
W(x1,x2) = min{x1,x2}

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Some Other Utility Functions and Their Indifference Curves
uA utility function of the form
           U(x1,x2) = f(x1) + x2
is linear in just x2 and is called quasi-linear.
uE.g.     U(x1,x2) =  2x11/2 + x2.

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Quasi-linear Indifference Curves
x2
x1
Each curve is a vertically shifted copy of the others.

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Some Other Utility Functions and Their Indifference Curves
uAny utility function of the form
           U(x1,x2) = x1a x2b
with a > 0 and b > 0 is called a Cobb-Douglas utility function.
uE.g.    U(x1,x2) = x11/2 x21/2  (a = b = 1/2)
           V(x1,x2) = x1 x23           (a = 1, b = 3)

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Cobb-Douglas Indifference Curves
x2
x1
All curves are hyperbolic,
asymptoting to, but never
touching any axis.

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Marginal Utilities
uMarginal means “incremental”.
uThe marginal utility of commodity i is the rate-of-change of total utility as the quantity of
commodity i consumed changes; i.e.

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Marginal Utilities
uE.g. if U(x1,x2) = x11/2 x22 then

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Marginal Utilities
uE.g. if U(x1,x2) = x11/2 x22 then

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Marginal Utilities
uE.g. if U(x1,x2) = x11/2 x22 then

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Marginal Utilities
uE.g. if U(x1,x2) = x11/2 x22 then

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Marginal Utilities
uSo, if U(x1,x2) = x11/2 x22 then

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Marginal Utilities and Marginal Rates-of-Substitution
uThe general equation for an indifference curve is
       U(x1,x2) º k, a constant.
Totally differentiating this identity gives

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Marginal Utilities and Marginal Rates-of-Substitution
rearranged is

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Marginal Utilities and Marginal Rates-of-Substitution
rearranged is
And
This is the MRS.

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Marg. Utilities & Marg. Rates-of-Substitution; An example
uSuppose U(x1,x2) = x1x2.  Then
so

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Marg. Utilities & Marg. Rates-of-Substitution; An example
     MRS(1,8) = - 8/1 = -8
     MRS(6,6) = - 6/6 = -1.
x1
x2
8
6
1
6
U = 8
U = 36
U(x1,x2) = x1x2;

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Marg. Rates-of-Substitution for Quasi-linear Utility Functions
uA quasi-linear utility function is of the form  U(x1,x2) = f(x1) + x2.
so

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Marg. Rates-of-Substitution for Quasi-linear Utility Functions
uMRS = - f  (x1)  does not depend upon x2 so the slope of indifference curves for a quasi-linear
utility function is constant along any line for which x1 is constant.  What does that make the
indifference map for a quasi-linear utility function look like?
¢

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Marg. Rates-of-Substitution for Quasi-linear Utility Functions
x2
x1
Each curve is a vertically shifted copy of the others.
MRS is a constant
along any line for which x1 is
constant.
MRS =
- f(x1’)
MRS = -f(x1”)
x1’
x1”

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Monotonic Transformations & Marginal Rates-of-Substitution
uApplying a monotonic transformation to a utility function representing a preference relation
simply creates another utility function representing the same preference relation.
uWhat happens to marginal rates-of-substitution when a monotonic transformation is applied?

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Monotonic Transformations & Marginal Rates-of-Substitution
uFor U(x1,x2) = x1x2 the MRS = - x2/x1.
uCreate V = U2; i.e. V(x1,x2) = x12x22.  What is the MRS for V?
which is the same as the MRS for U.

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Monotonic Transformations & Marginal Rates-of-Substitution
uMore generally, if V = f(U) where f is a strictly increasing function, then
So MRS is unchanged by a positive
monotonic transformation.