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L03

L03. Utility. 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. f. ~. Preferences: A Reminder.

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L03

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  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. f ~ 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 "mountain” • Utility function and Preferences • Utility function and Indifference curves • Utility function and MRS (next class)

  6. z Utility function: Geometry x2 x1

  7. z Utility function: Geometry x2 x1

  8. z Utility function: Geometry x2 x1

  9. z Utility function: Geometry Utility 5 x2 3 x1

  10. z Utility function: Geometry U(x1,x2) Utility 5 x2 3 x1

  11. f f ~ ~ 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 U=6 U=4 U=4 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=

  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) R U(x1,x2) = L

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

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