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Utility Functions in Decision Making

Learn about utility functions, preferences, and indifference curves in modeling choices. Explore examples of perfect substitutes and complements to grasp how preferences are represented. Dive into the concept of ordinality in utility functions.

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Utility Functions in Decision Making

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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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