Group-Clicker Question (1 pt): Without using calculators, find df/dx where . 1. df/dx=2. df/dx=3. df/dx=4. df/dx=0%0%0%0%Preferences“No one else can ever make your choices for you. Your choices are yours alone. They are as much a part of you as every breath you will take, every moment of your life.”-Dr. Shad Helmstetter+=2() ln[(3)]xfx+22exp( 3)xx+213x+223xx2x2Individual ChoiceWhat you wantFoundationStructurePatternsWhat you can haveWhat you will chooseFirm Choice Game TheoryClass 4 – Foundations of Preferences3Class 4 – Foundations of PreferencesFoundationFramePatternsModeling Human Choice4Something behaves according to a functional specificationWhat is the best choice??Class 4 – Foundations of Preferences5How do people make a choice?Class 4 – Foundations of Preferences6Class 4 – Foundations of PreferencesGroup-Clicker Question (P): Which concert would yourather attend? 1. Justin Timberlake2. Eminem3. Keith Urban4. Avicii5. Deafheaven8Preference Relation:Given two bundles, a and b, define a preference relation, , as follows: !! if and only if (iff) is at least as good as . If this relationship holds, we sayFor an individual, "a is preferred to b"ababExample: UNC Duke!means “To me, I like UNC at least as much as I like Duke”Class 4 – Foundations of Preferences9Class 4 – Foundations of Preferences!Vacation Education10Baltimore Ravens Pittsburgh SteelersBaltimore Orioles New York YankeesAnyone else New York YankeesThe Wire Any TV Show Ever Made!!!!Define “a b” as “a is preferred to b”Class 4 – Foundations of Preferences11!!!!Baltimore Ravens San Francisco 49ersBaltimore Orioles New York YankeesAnyone else New York YankeesThe Wire Any TV Show Ever MadeDefine “a b” as “a is preferred to b”Class 4 – Foundations of Preferences12Preference Relation:≥!DOES NOT MEA N aabb!Example: 6 graded HW assignments 10 graded HW assignmentsClass 4 – Foundations of Preferences13Preference Relation:!Example: (Me) 10 graded HW assignments 6 graded HW assignments(You) 10 graded HW assignments 6 graded HW assignments!A preference relation is specific to an individual. Class 4 – Foundations of PreferencesGroup-Clicker Question (P): Which type of bundle does NOT make sense if 1 3!1. Punches to the head2. Grades of “F”3. Dollars4. Speeding ticketsGroup-Clicker Question (P): Which type of bundle does NOT make sense if 2 5!1. Days on vacation2. Dates with a person you really like3. Kicks to the shin4. Millions of dollars16Strict Preference Relation:Given two bundles, a and b, define a strictpreference relation, , as follows: ≻≻ if and only if (iff) is better than . If this relationship holds, we say "a is stFor an individual, rictly preferred to b"baabIndifference Relation:Given two bundles, a and b, define an indifference relation, , as follows: ~ if and only if (iff) is indifferent to . If this relationship holds, we say "For an individual,a is indifferent t o b"bbaa~Class 4 – Foundations of PreferencesGroup-Clicker Question (1 pt): If , what must be true?~ab1. a2. a3. a4. a≻!,bb a!!,bb a!,bb a!≻≺,bb a18Class 4 – Foundations of PreferencesFoundationFramePatternsModeling Human Choice19How to people make a choice?Class 4 – Structure of Preferencesa b!20Complete:A preference relation is complete if, for each x and y, either xyor y x or both.!Class 4 – Structure of Preferences!This does NOT mean “one is always better”. If just means you are able to make the decision, even if the decision is “I am indifferent”Group-Clicker Question (1 pt): A father brings his 3-year old daughter to Cold Stone Creamery. When he asks what she would like, she answers “I don’t care. I love all ice cream!”. Are her preferences over ice cream complete?1. Yes2. NoGroup-Clicker Question (P): A father brings his 3-year old daughter to Cold Stone Creamery. When he asks what she would like, her lower lip trembles, she starts to cry, and she runs out of the store. Are her preferences over ice cream likely complete?1. Yes2. NoSymmetric:A preference relation is symmetric if, for each x and y, if xythen y x.!!Group-Clicker Question (P): Which of the following best describes symmetric preferences?1. An individual is indifferent to identical bundles.2. An individual is indifferent to all bundles.3. An individual is indifferent to all bundles she can compare.4. If a is preferred to b then a bundle very similar to a will also be preferred to b.0%0%0%0%Symmetric:A preference relation is symmetric if, for each x and y, if xythen y x.!!25Symmetric:A preference relation is symmetric if, for each x and y, if xythen y x.!Class 4 – Structure of Preferences!26Transitive:A preference relation is transitive if, for every x, y, and z, if xyand y z then x z!Class 4 – Structure of Preferences!!!27Transitive:A preference relation is transitive if, for every x, y, and z, if xyand y z then x z!Class 4 – Structure of Preferences!!!28Transitive:A preference relation is transitive if, for every x, y, and z, if xyand y z then x z!Class 4 – Structure of Preferences!!!29Class 4 – Structure of PreferencesTo show preferences are NOT transitive, find an intransitive cycle. i.e. preferences of the form≻≻≻ab c aor ≻!!abca≻≻ab c c aor ~ ~ , but a or cTransitive:A preference relation is transitive if, for every x, y, and z, if xyand y z then x z!!!30Class 4 – Structure of PreferencesIf you can find a single intransitive cycle, preferences are not transitive.Transitive:A preference relation is transitive if, for every x, y, and z, if xyand y z then x z!!!Group-Clicker Question (P): On any given morning, you are faced with three options:1. Go to class2. Sleep in3. Go to the beachOn Monday, your car is in the shop and you have no way of getting to the beach. Of your remaining options, you decide to go to class. On Saturday, you get your car back. You and your roommate decide to sleep in instead of going to the beach. On Monday your roommate is extremely loud and wakes you up early (i.e. you cannot sleep in). After a brisk discussion, you decide to resolve your differences by skipping classes and going to the beach. Assume a particular choice implies a strict preference. Do these actions imply preferences that are transitive?1. Yes2. NoLocal Nonsatiation:A preference relation is locally