Transcript L3

L03
Utility
Quiz
How much do you like microeconomics
A: I love it (unconditionally)
B: I cannot live without it
C: I would die for it
D: All of the above
Big picture
Behavioral Postulate:
A decisionmaker chooses its most
preferred alternative from the set of
affordable alternatives.
 Budget set = affordable alternatives
 To model choice we must have
decisionmaker’s preferences.

Preferences: A Reminder
 Rational
agents rank consumption bundles
from the best to the worst
 We
call such ranking preferences
 Preferences
f
~
satisfy Axioms: completeness and
transitivity
 Geometric
representation: Indifference Curves
 Analytical
Representation: Utility Function
Indifference Curves
x2
x1
Utility Functions
 Preferences
satisfying Axioms (+) can be
represented by a utility function.
 Utility function: formula that assigns a
number (utility) for any bundle.
 Today:
1. Geometric interpretation
2. Utility function and Preferences
3. Utility and Indifference curves
4. Important examples
Utility function: Geometry
x2
z
x1
Utility function: Geometry
x2
z
x1
Utility function: Geometry
x2
z
x1
Utility function: Geometry
Utility
5
x2
3
z
x1
Utility function: Geometry
U(x1,x2)
Utility
5
x2
3
z
x1
Utility Functions and Preferences
A
utility function U(x) represents
preferences f
if
and
only
if:
~
x
fy
~
x
p
y
x ~ y
U(x) ≥ U(y)
Usefulness of Utility Function
 Utility
function U(x1,x2) = x1x2
 What can we say about preferences
(2,3), (4,1), (2,2)
 Quiz
 A:
 B:
 C:
 D:
1:
Utility Functions & Indiff. Curves
 An
indifference curve contains equally
preferred bundles.
 Indifference
= the same utility level.
 Indifference
curve
 Hikers:
lines
Topographic map with contour
Indifference Curves
 U(x1,x2)
x2
= x1x2
x1
Ordinality of a Utility Function
Utilitarians: utility = happiness = Problem!
(cardinal utility)


Nowadays: utility is ordinal (i.e. ordering)
concept

Utility function matters up to the
preferences (indifference map) it induces

Q: Are preferences represented by a
unique utility function?
Utility Functions
= x1x2
 Define V = 5U.
 V(x1,x2)
V
= 5x1x2
p
 U(x1,x2)
U=6
(2,3)
U=4 U=4
(4,1) ~ (2,2).
V=
V=
V=
(2,3)
(4,1) ~ (2,2).
preserves the same order as U and
so represents the same preferences.
Monotone Transformation
 U(x1,x2)
x2
 V=
5U
x1
= x1x2
Theorem
(Monotonic Transformation)
 T:
(1)
(2)
Suppose that
U is a utility function that represents some
preferences
f(U) is a strictly increasing function
then V = f(U) represents the same preferences
Preference representations
 Utility
 Quiz
U(x1,x2) = x1x2
2: U(x1,x2) = x1 +x2
 A: V = ln(x1 +x2)+5
 B: V=5x1 +7x2
 C: V=-2(x1 +x2)
 D: All of the above
Three Examples
 Cobb-Douglas
preferences (most
goods)
 Perfect
Substitutes (Pepsi and Coke)
 Perfect
Complements (Shoes)
Example: Perfect substitutes
 Two
goods that are substituted at the
constant rate
 Example:
Pepsi and Coke
(I like soda but I cannot distinguish
between the two kinds)
Perfect Substitutes (Soda)
Pepsi
U(x1,x2) =
Coke
Perfect Substitutes (Proportions)
x2 (1 can)
U(x1,x2) =
x1 (6 pack)
Perfect complements
 Two
goods always consumed in the
same proportion
 Example:
 We
Right and Left Shoes
like to have more of them but
always in pairs
Perfect Complements (Shoes)
R
U(x1,x2) =
L
Perfect Complements (Proportions)
Coffee
2:1
U(x1,x2) =
Sugar