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L03

L03. Utility. Class Quiz. Q: How much do you like economics I love it I cannot live without it I would die for it All of the above. REEF Polling: iclicker Laptop/smartphone/ iclicker. Big picture.

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L03

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  1. L03 Utility

  2. Class Quiz • Q: How much do you like economics • I love it • I cannot live without it • I would die for it • All of the above REEF Polling: iclicker Laptop/smartphone/iclicker

  3. 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.

  4. 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

  5. Indifference Curves x2 x1

  6. 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

  7. z Utility function: Geometry x2 x1

  8. z Utility function: Geometry x2 x1

  9. z Utility function: Geometry x2 x1

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

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

  12. f f ~ ~ Utility Functions and Preferences • A utility function U(x) represents preferences if x y U(x) ≥ U(y) x y x ~ y p

  13. 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:

  14. Utility Functions & Indiff. Curves • An indifference curve contains equally preferred bundles. • Indifference = the same utility level. • Indifference curve • Hikers: Topographic map with contour lines

  15. Indifference Curves • U(x1,x2) = x1x2 x2 x1

  16. 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?

  17. Utility Functions U=6 U=4 U=4 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=

  18. Monotone Transformation • U(x1,x2) = x1x2 • V= 5U x2 x1

  19. 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

  20. 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

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

  22. 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)

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

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

  25. 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

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

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

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