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eco331-04-ExpectedUtilityHandout

# eco331-04-ExpectedUtilityHandout - Overview Preferences...

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Overview Preferences Choice under Risk and Uncertainty Expected Utility Certainty Equivalent Applications Eco331: Preferences and Expected Utility Marciano Siniscalchi October 5, 2009

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Overview Preferences Choice under Risk and Uncertainty Expected Utility Certainty Equivalent Applications Overview Preferences and Representation Certainty, Risk, and Uncertainty Preferences over random variables The St. Petersburg Paradox Expected Utility Certainty Equivalent Applications, including forecasts
Overview Preferences Choice under Risk and Uncertainty Expected Utility Certainty Equivalent Applications Preferences Definition Let F be a set of alternatives . A preference on F is a binary relation < ⊂ F × F . A preference < on F is: complete if, for all x , y ∈ F , either x < y or y < x ; transitive if, for all x , y , z ∈ F , x < y and y < z imply x < z . Your intermediate-micro preference Actually, also the most general definition. Will get more specific theories by choosing F . Two main tenets of economic analysis Preferences are assumed observable Any claim should be interpretable in terms of preferences.

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Overview Preferences Choice under Risk and Uncertainty Expected Utility Certainty Equivalent Applications Notation and Examples x < y x is as least as good as y weak preference x y x is strictly better than y strict preference x y x is as good as y indifference Examples: F = { ( x , y ) : x , y ∈ { 1 , . . . , 5 }} . Interpret ( x , y ) ∈ F as “ x apples and y bananas.” A preference statement: (3 , 2) < (2 , 3). Note: preference = binary relation = set. If < is complete, it contains at least 24 + 23 + 22 + . . . + 2 + 1 pairs!
Overview Preferences Choice under Risk and Uncertainty Expected Utility Certainty Equivalent Applications Representation This suggests that “listing” < is tedious. Definition A representation of a preference < on the set F is a function U : F → R such that x , x 0 ∈ F , x < x 0 if and only if U ( x ) U ( x 0 ) . Example: U (( x , u )) = 3 x - 5 y . 3 · 3 - 5 · 2 = - 1, 3 · 2 - 5 · 3 = - 9: (3 , 2) (2 , 3) Representations are invariant to strictly increasing transformations Example: V (( x , y )) = [ U (( x , y ))] 3 .

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Overview Preferences Choice under Risk and Uncertainty Expected Utility Certainty Equivalent Applications The Knightian Taxonomy X : set of prizes , or ultimate outcomes Can assume X = R : monetary outcomes . Certainty: F ⊂ X . Risk: Random variables as probability distributions over X . “Objective” probabilities: objects of choice Notation: F ⊂ Δ( X ); p , q ∈ F , traditionally called lotteries . Uncertainty: Random variables as functions from some S to X . “Subjective” probability: representation of preferences Notation: F ⊂ X S ; f , g ∈ F , traditionally called acts . In this course: Will distinguish between risk and uncertainty only if we must Deal jointly with both: preferences over random variables .
Overview Preferences Choice under Risk and Uncertainty Expected Utility Certainty Equivalent Applications Dominance Ideas Focus on F = random variables viewed as functions on some S .

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eco331-04-ExpectedUtilityHandout - Overview Preferences...

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