L03

1
L03

• Utility

2
Big picture

• Behavioral PostulateA decisionmaker chooses
  its most preferred alternative from the set of
  affordable alternatives.
• Budget set affordable alternatives
• To model choice we must have decisionmakers
  preferences.

3
Preferences A Reminder

• Rational agents rank consumption bundles from the
  best to the worst
• We call such ranking preferences
• Preferences satisfy Axioms completeness and
  transitivity
• Geometric representation Indifference Curves
• Analytical Representation Utility Function

4
Indifference Curves
x2
x1
5
Utility Functions

• Preferences satisfying Axioms () can be
  represented by a utility function.
• Utility function formula that assigns a number
  (utility) for any bundle.
• Today
• Geometric representation-a mountain
• Utility function and Preferences
• Utility function and Indifference curves
• Utility function and MRS (next class)

6
Utility function Geometry
x2
x1
7
Utility function Geometry
x2
x1
8
Utility function Geometry
x2
x1
9
Utility function Geometry
Utility
5
x2
3
x1
10
Utility function Geometry
U(x1,x2)
Utility
5
x2
3
x1
11
Utility Functions and Preferences

• A utility function U(x) represents preferences
  if and only if
• x y U(x) U(y)
  x y x y

p
12
Usefulness of Utility Function

• Utility function U(x1,x2) x1x2
• What can we say about preferences
• (2,3), (4,1), (2,2), (1,1) , (8,8)
• Recover preferences

13
Utility Functions Indiff. Curves

• An indifference curve contains equally preferred
  bundles.
• Indifference the same utility level.
• Indifference curve
• Hikers Topographic map with contour lines

14
Indifference Curves

• U(x1,x2) x1x2

x2
x1
15
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?

16
Utility Functions
U6 U4 U4
p

• U(x1,x2) x1x2 (2,3) (4,1)
  (2,2).
• Define V 5U.
• V(x1,x2) 5x1x2 (2,3) (4,1) (2,2).
• V preserves the same order as U and so represents
  the same preferences.

V V V
17
Monotone Transformation

• U(x1,x2) x1x2
• V 5U

x2
x1
18
Theorem (Monotonic Transformation)

• T 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
• Examples U(x1,x2) x1x2

19
Three Examples

• Well-behaved preferences (Ice cream and
  chocolate)
• Perfect Substitutes (Example French and Dutch
  Cheese)
• Perfect Complements (Right and Left shoe)

20
Example Perfect substitutes

• Two goods that are substituted at the constant
  rate
• Example French and Dutch Cheese
• (I like cheese but I cannot distinguish between
  the two kinds)

21
Perfect Substitutes (Cheese)
Dutch
U(x1,x2)
French
22
Perfect Substitutes (Proportions)
x2 (1 Slice)
U(x1,x2)
x1 Pack (6 slices)
23
Perfect complements

• Two goods always consumed in the same proportion
• Example Right and Left Shoes
• We like to have more of them but always in pairs

24
Perfect Complements (Shoes)
R
U(x1,x2)
L
25
Perfect Complements (Proportions)
21
Coffee
U(x1,x2)
Sugar