week2b - Utility representation Main results ECON 404 Behavioral Economics 1 Econ 404 Brian W Rogers 2015 by Brian W Rogers Logistics Enrollment

# week2b - Utility representation Main results ECON 404...

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© 2015 by Brian W. Rogers ECON 404 Behavioral Economics Utility representation: Main results

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Econ 404 – Brian W. Rogers 2 Logistics • Enrollment finalized • Groups for class project Identify a behavioral research question Propose methodology to answer the question I expect about 12 pages written final project • Clickers Start bringing them Tuesday • HW 5 assignments throughout the semester Written, graded, individual
Econ 404 – Brian W. Rogers 3 Roadmap •Finish properties on choice data •Main result: choice data satisfies PROPERTIES If and only if There exists a utility function on alternatives such that maximizing utility coincides with choices •Survey of ways in which people often fail to be utility maximizers Leading eventually to a range of behavioral models

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Econ 404 – Brian W. Rogers 4 Recap of Tuesday •Goal: characterize which choice data is consistent with a model of utility maximization •Perspective: “as if” interpretation Rational/internally consistent behavior •The main characterization invokes Properties A (from Tuesday) and B (coming now)
Econ 404 – Brian W. Rogers 5 Property B (of choices) • Take menu A and a subset B A • If x and y are both chosen from B [x,y c(B)] • And if y is chosen from A [y c(A)] • Then x must also be chosen from A [x c(A)] • Intuition: Since x and y are chosen together they are equally good Since y is chosen from A, it is at least as good as anything in A So x, being equally as good as y, as also at least as good as anything from A • Note: If the DM is never indifferent, then Property B holds automatically

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Econ 404 – Brian W. Rogers 6 Big picture •Observable data: choices from menus •Unobservable traits: preferences •Question: When can we find a utility function that: Represents the DM’s preferences Explains the choices as utility-maximization? •This is probably the most canonical setting in which economics frames the question of rationality
Econ 404 – Brian W. Rogers 7 Preference relation • Finite set of alternatives X • A binary relation R on a set X is a set of ordered pairs of elements of X If (x,y) R, we write xRy Interpret as “x is at least as good as y” • A binary relation is a preference relation if it is: Transitive : if xRy and yRz then xRz Reflexive : for all x in X, xRx • A preference relation is complete if: for all x,y in X, either xRy or yRx (possibly both)

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Econ 404 – Brian W. Rogers 8 Preference relation and utility •Interpretation: Here, rationality = complete pref. rel. •Definition: A utility function u represents a binary relation R if: For all x,y LQ ;± X²[³ X²\³ LI DQG RQO\ LI xRy •Theorem 1: X finite. A binary relation R on X has a utility representation if and only if it is a complete preference relation I.e., it is complete, transitive and reflexive
Econ 404 – Brian W. Rogers 9 Preference relation and choices •A choice correspondence c is rationalized by a preference relation R if for every subset A of X, we have: c(A) = {x A | xRy for all y A}

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