L03

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L03

• Utility

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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.

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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

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Indifference Curves
x2
x1
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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 "mountain
• Utility function and Preferences
• Utility function and Indifference curves
• Utility function and MRS (next class)

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Utility function Geometry
x2
x1
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Utility function Geometry
x2
x1
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Utility function Geometry
x2
x1
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Utility function Geometry
Utility
5
x2
3
x1
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Utility function Geometry
U(x1,x2)
Utility
5
x2
3
x1
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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
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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

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Utility Functions Indiff. Curves

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

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Indifference Curves

• U(x1,x2) x1x2

x2
x1
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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?

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Utility Functions
U6 U4 U4
p

• U(x1,x2) x1x2 (2,3) (4,1)
  (2,2).
• Define V U2.
• V(x1,x2) x12x22 (2,3) (4,1)
  (2,2).
• V preserves the same order as U and so represents
  the same preferences.

V V V
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Monotone Transformation

• U(x1,x2) x1x2
• V U2

x2
x1
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Theorem (Formal Claim)

• 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

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Three Examples

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

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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)

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

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