1. L03
Utility

2. 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.

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 ~ f

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 "mountain”
Utility function and Preferences
Utility function and Indifference curves
Utility function and MRS

6. Utility function: Geometryx2 z
All bundles in I1 are
strictly preferred to all in I2. x1

7. Utility function: Geometryx2 z
All bundles in I1 are
strictly preferred to all in I2. x1

8. Utility function: Geometryx2 z
All bundles in I1 are
strictly preferred to all in I2. x1

9. Utility function: Geometry5 x2 3
All bundles in I1 are
strictly preferred to all in I2. z x1

10. Utility function: Geometry
U(x1,x2)
Utility 5 x2 3
All bundles in I1 are
strictly preferred to all in I2. z 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 ~ f ~ f 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: Topographical 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 p U=6 U=4 U=4 U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
Define V = U2.
V(x1,x2) = x12x (2,3) (4,1) ~ (2,2).
V preserves the same order as U and so represents the same preferences. p
V= V= V=

17. Monotone Transformation
U(x1,x2) = x1x2
V= U2 x2 x1

18. 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

19. Three Examples Perfect Substitutes (Example: French and Dutch Cheese)
Perfect Complements (Right and Left shoe)
Well-behaved preferences (Ice cream and chocolate)

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)
U(x1,x2) = L

25. Perfect Complements (Proportions)
Coffee
1:2
U(x1,x2) =
Sugar