Chapter ThreeRationality in EconomicsPreference RelationsSlide 4Slide 5Slide 6Slide 7Slide 8Slide 9Assumptions about Preference RelationsSlide 11Slide 12Indifference CurvesSlide 14Slide 15Slide 16Slide 17Slide 18Slide 19Indifference Curves Cannot IntersectSlide 21Slopes of Indifference CurvesSlide 23Slide 24Slide 25Extreme Cases of Indifference Curves; Perfect SubstitutesSlide 27Extreme Cases of Indifference Curves; Perfect ComplementsSlide 29Slide 30Preferences Exhibiting SatiationIndifference Curves Exhibiting SatiationSlide 33Slide 34Indifference Curves for Discrete CommoditiesSlide 36Indifference Curves With a Discrete GoodWell-Behaved PreferencesSlide 39Well-Behaved Preferences -- Convexity.Slide 41Slide 42Well-Behaved Preferences -- Weak Convexity.Non-Convex PreferencesMore Non-Convex PreferencesSlide 46Marginal Rate of SubstitutionSlide 48Slide 49MRS & Ind. Curve PropertiesSlide 51Slide 52Slide 53Slide 54Chapter ThreePreferencesRationality in Economics Behavioral Postulate:A decisionmaker always chooses its most preferred alternative from its set of available alternatives.So to model choice we must model decisionmakers’ preferences.Preference RelationsComparing two different consumption bundles, x and y: –strict preference: x is more preferred than is y.–weak preference: x is as at least as preferred as is y.–indifference: x is exactly as preferred as is y.Preference RelationsStrict preference, weak preference and indifference are all preference relations.Particularly, they are ordinal relations; i.e. they state only the order in which bundles are preferred.Preference Relations denotes strict preference; x y means that bundle x is preferred strictly to bundle y.Preference Relations denotes strict preference; x y means bundle x is preferred strictly to bundle y.denotes indifference; x  y means x and y are equally preferred.Preference Relations denotes strict preference so x y means that bundle x is preferred strictly to bundle y.denotes indifference; x  y means x and y are equally preferred. denotes weak preference;x y means x is preferred at least as much as is y.~~Preference Relationsx y and y x imply x  y.~~Preference Relationsx y and y x imply x  y.x y and (not y x) imply x y.~~~~Assumptions about Preference RelationsCompleteness: For any two bundles x and y it is always possible to make the statement that either x y or y x.~~Assumptions about Preference RelationsReflexivity: Any bundle x is always at least as preferred as itself; i.e. x x.~Assumptions about Preference RelationsTransitivity: Ifx is at least as preferred as y, andy is at least as preferred as z, thenx is at least as preferred as z; i.e. x y and y z x z.~~~Indifference CurvesTake a reference bundle x’. The set of all bundles equally preferred to x’ is the indifference curve containing x’; the set of all bundles y  x’.Since an indifference “curve” is not always a curve a better name might be an indifference “set”.Indifference Curvesxx22xx11x”x”x”’x”’x’ x’  x” x”  x”’ x”’x’Indifference Curvesxx22xx11zz xx yyxyzIndifference Curvesx2x1xAll bundles in I1 arestrictly preferred to all in I2.yzAll bundles in I2 are strictly preferred to all in I3.I1I2I3Indifference Curvesx2x1I(x’)xI(x)WP(x), the set of bundles weakly preferred to x.Indifference Curvesx2x1WP(x), the set of bundles weakly preferred to x. WP(x) includes I(x).xI(x)Indifference Curvesx2x1SP(x), the set of bundles strictly preferred to x, does not include I(x).xI(x)Indifference Curves Cannot Intersectxx22xx11xxyyzzII11I2From IFrom I11, x , x  y. From I y. From I22, x , x  z. z.Therefore y Therefore y  z. z.Indifference Curves Cannot Intersectxx22xx11xxyyzzII11I2From IFrom I11, x , x  y. From I y. From I22, x , x  z. z.Therefore y Therefore y  z. But from I z. But from I11 and Iand I22 we see y z, a we see y z, a contradiction. contradiction.Slopes of Indifference CurvesWhen more of a commodity is always preferred, the commodity is a good.If every commodity is a good then indifference curves are negatively sloped.Slopes of Indifference CurvesBetterBetterWorseWorseGood 2Good 2Good 1Good 1Two goodsTwo goodsa negatively sloped a negatively sloped indifference curve.indifference curve.Slopes of Indifference CurvesIf less of a commodity is always preferred then the commodity is a bad.Slopes of Indifference CurvesBetterBetterWorseWorseGood 2Good 2Bad 1Bad 1One good and oneOne good and onebad a bad a positively sloped positively sloped indifference curve.indifference curve.Extreme Cases of Indifference Curves; Perfect SubstitutesIf a consumer always regards units of commodities 1 and 2 as equivalent, then the commodities are perfect substitutes and only the total amount of the two commodities in bundles determines their preference rank-order.Extreme Cases of Indifference Curves; Perfect Substitutesxx22xx11888815151515Slopes are constant at - 1.Slopes are constant at - 1.I2I1Bundles in IBundles in I22 all have a total all have a totalof 15 units and are strictlyof 15 units and are strictlypreferred to all bundles inpreferred to all bundles in I I11, which have a total of, which have a total of only 8 units in them. only 8 units in them.Extreme Cases of Indifference Curves; Perfect ComplementsIf a consumer always consumes commodities 1 and 2 in fixed proportion (e.g. one-to-one), then the commodities are perfect complements and only the number of pairs of units of the two commodities determines the preference rank-order of bundles.Extreme Cases of Indifference Curves; Perfect Complementsxx22xx11I14545oo55995599Each of (5,5), (5,9) and (9,5) contains5 pairs so each is equally preferred.Extreme Cases of Indifference Curves; Perfect Complementsxx22xx11I2I14545oo55995599Since each of (5,5), (5,9) and (9,5) contains 5 pairs, each is less preferred than the bundle (9,9) which contains 9 pairs.Preferences Exhibiting SatiationA bundle strictly preferred to any other is a satiation point or a bliss point.What do indifference curves look like for