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 interpretation
Utility function and Preferences
Utility and Indifference curves
Important examples

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
x y U(x) ≥ U(y) x y x ~ y ~ f ~ f p

12. Usefulness of Utility Function
Utility function U(x1,x2) = x1x2
(2,3), (4,1), (2,2)
Quiz 1: U represents preferences A: B: C: D:

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 p U=6 U=4 U=4 U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
Define V = 5U.
V(x1,x2) = 5x1x (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= 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

19. Preference representations
Utility U(x1,x2) = x1x2
Quiz 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

20. Three Examples Cobb-Douglas preferences (most goods)
Perfect Substitutes (Pepsi and Coke)
Perfect Complements (Shoes)

21. 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)

22. Perfect Substitutes (Soda)
Pepsi
U(x1,x2) =
Coke

23. Perfect Substitutes (Proportions)
x2 (1 can)
U(x1,x2) =
x1 (6 pack)

24. 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

25. Perfect Complements (Shoes)
U(x1,x2) = L

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