Chapter ThreePreferences 偏好Road Map• Describe preferences • Indifference curves (无差异曲线) • Well-behaved preferences • Marginal rate of substitution (边际替代率)Rationality in Economics• Behavioral Postulate: A decision-maker always chooses its most preferred alternative from its set of available alternatives. • So to model choice we must model decision-makers’ preferences.Preference Relations • Comparing two different consumption bundles, x and y: – strict preference (严格偏好): x is more preferred than is y (x y). – Indifference (无差异): x is exactly as preferred as is y (x~y). – weak preference (弱偏好): x is as at least as preferred as is y (x y).π π~φ φPreference Relations• Preference relations are ordinal relations; i.e. they state only the order in which bundles are preferred.Preference Relations• x y and y x imply x ∼ y.~φ φ~φ φPreference Relations• x y and y x imply x ∼ y.• x y and (not y x) imply x y.~φ φ~φ φ~φ φ~φπφπAssumptions about Preference Relations• Completeness (完备性): For any two bundles x and y it is always possible to make the statement that either x y or y x.~φ φ~φ φAssumptions about Preference Relations• Reflexivity (反身性): Any bundle x is always at least as preferred as itself; i.e.x x.~φ φAssumptions about Preference Relations• Transitivity (传递性): If x is at least as preferred as y, and y is at least as preferred as z, then x is at least as preferred as z; i.e.x y and y z x z.~φ φ~φ φ~φ φCounter Example• A, B, C,=> x, y, z, • A: x y z • B: y z x • C: z x y • Voting system: • x y, y z, z x.~φ φ~φ φ ~φ φ~φ φ~φ φ~φ φ~φ φ~φ φ~φ φIndifference 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 Curvesx x2 2x x” ”x x”’”’x x’ ’ ∼ ∼ ∼ ∼ x x” ” ∼ ∼ ∼ ∼ x x”’”’x x1 1x’Indifference Curvesx x2 2 z zx xy yx x1 1ππππxyzIndifference Curvesx2x1xAll bundles in I1are strictly preferred to all in I2.yzAll bundles in I2are strictly preferred to all in I3.I1I2I3x2I(x’)xI(x)x1WP(x), the set of bundles weakly preferred to x.Weakly Preferred Set (弱偏好集)Weakly Preferred Set (弱偏好集)x2x1WP(x), the set of bundles weakly preferred to x. WP(x) includes I(x).xI(x)Strictly Preferred Set (严格偏好 集)x2x1SP(x), the set of bundles strictly preferred to x, does not include I(x).xI(x)Indifference Curves Cannot Intersectx x2 2x x1 1x x y y z zI I1 1I2From IFrom I1 1, x , x ∼ ∼ ∼ ∼ y. From Iy. From I2 2, x , x ∼ ∼ ∼ ∼ z.z. Therefore y Therefore y ∼ ∼ ∼ ∼ z. But from Iz. But from I1 1and Iand I2 2we see y z, a we see y z, a contradiction.contradiction.π πSlopes 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 goods a 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 one bad a bad a positively sloped positively sloped indifference curve.indifference curve.Examples• Perfect substitutes (完全替代) • Perfect complements (完全互补) • Satiation (餍足) • Discrete goods (离散商品)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 Substitutesx x2 2x x1 18 88 815151515Slopes are constant at Slopes are constant at - - 1.1.I2I1Bundles in IBundles in I2 2all have a totalall have a total of 15 units and are strictlyof 15 units and are strictly preferred to all bundles inpreferred to all bundles inI I1 1, 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 Complementsx x2 2x x1 1I14545o o5 59 95 59 9Each of (5,5), (5,9) and (9,5) contains 5 pairs so each is equally preferred.Extreme Cases of Indifference Curves; Perfect Complementsx x2 2x x1 1I2I14545o o5 59 95 59 9Since 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。