INTMIC9.jpg Chapter 3 Preferences Rationality in Economics u Behavioral Postulate: A decisionmaker always chooses its most preferred alternative from its set of available alternatives. uSo to model choice we must model decisionmakers’ preferences. Preference Relations uComparing 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 uStrict preference, weak preference and indifference are all preference relations. uParticularly, they are ordinal relations; i.e. they state only the order in which bundles are preferred. Preference Relations u denotes strict preference; x y means that bundle x is preferred strictly to bundle y. Textové pole: p p Textové pole: p p Preference Relations u denotes strict preference; x y means bundle x is preferred strictly to bundle y. u~ denotes indifference; x ~ y means x and y are equally preferred. Textové pole: p p Textové pole: p p Preference Relations u denotes strict preference so x y means that bundle x is preferred strictly to bundle y. u~ denotes indifference; x ~ y means x and y are equally preferred. u denotes weak preference; x y means x is preferred at least as much as is y. ~ f ~ f Textové pole: p p Textové pole: p p Preference Relations ux y and y x imply x ~ y. ~ f ~ f Preference Relations ux y and y x imply x ~ y. ux y and (not y x) imply x y. ~ f ~ f ~ f ~ f Textové pole: p p Assumptions about Preference Relations uCompleteness: For any two bundles x and y it is always possible to make the statement that either x y or y x. ~ f ~ f Assumptions about Preference Relations uReflexivity: Any bundle x is always at least as preferred as itself; i.e. x x. ~ f Assumptions about Preference Relations uTransitivity: 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. ~ f ~ f ~ f Indifference Curves uTake 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’. uSince an indifference “curve” is not always a curve a better name might be an indifference “set”. Indifference Curves x2 x1 x” x”’ x’ ~ x” ~ x”’ x’ Indifference Curves x2 x1 z x y Textové pole: p p Textové pole: p p x y z Indifference Curves x2 x1 x All bundles in I1 are strictly preferred to all in I2. y z All bundles in I2 are strictly preferred to all in I3. I1 I2 I3 Indifference Curves x2 x1 I(x’) x I(x) WP(x), the set of bundles weakly preferred to x. Indifference Curves x2 x1 WP(x), the set of bundles weakly preferred to x. WP(x) includes I(x). x I(x) Indifference Curves x2 x1 SP(x), the set of bundles strictly preferred to x, does not include I(x). x I(x) Indifference Curves Cannot Intersect x2 x1 x y z I1 I2 From I1, x ~ y. From I2, x ~ z. Therefore y ~ z. Indifference Curves Cannot Intersect x2 x1 x y z I1 I2 From I1, x ~ y. From I2, x ~ z. Therefore y ~ z. But from I1 and I2 we see y z, a contradiction. Textové pole: p p Slopes of Indifference Curves uWhen more of a commodity is always preferred, the commodity is a good. uIf every commodity is a good then indifference curves are negatively sloped. Slopes of Indifference Curves Good 2 Good 1 Two goods a negatively sloped indifference curve. Slopes of Indifference Curves uIf less of a commodity is always preferred then the commodity is a bad. Slopes of Indifference Curves Good 2 Bad 1 One good and one bad a positively sloped indifference curve. Extreme Cases of Indifference Curves; Perfect Substitutes uIf 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 Substitutes x2 x1 8 8 15 15 Slopes are constant at - 1. I2 I1 Bundles in I2 all have a total of 15 units and are strictly preferred to all bundles in I1, which have a total of only 8 units in them. Extreme Cases of Indifference Curves; Perfect Complements uIf 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 Complements x2 x1 I1 45o 5 9 5 9 Each of (5,5), (5,9) and (9,5) contains 5 pairs so each is equally preferred. Extreme Cases of Indifference Curves; Perfect Complements x2 x1 I2 I1 45o 5 9 5 9 Since 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 uA bundle strictly preferred to any other is a satiation point or a bliss point. uWhat do indifference curves look like for preferences exhibiting satiation? Indifference Curves Exhibiting Satiation x2 x1 Satiation (bliss) point Indifference Curves Exhibiting Satiation x2 x1 Satiation (bliss) point Indifference Curves Exhibiting Satiation x2 x1 Satiation (bliss) point Indifference Curves for Discrete Commodities uA commodity is infinitely divisible if it can be acquired in any quantity; e.g. water or cheese. uA commodity is discrete if it comes in unit lumps of 1, 2, 3, … and so on; e.g. aircraft, ships and refrigerators. Indifference Curves for Discrete Commodities uSuppose commodity 2 is an infinitely divisible good (gasoline) while commodity 1 is a discrete good (aircraft). What do indifference “curves” look like? Indifference Curves With a Discrete Good Gas-oline Aircraft 0 1 2 3 4 Indifference “curves” are collections of discrete points. Well-Behaved Preferences uA preference relation is “well-behaved” if it is –monotonic and convex. uMonotonicity: More of any commodity is always preferred (i.e. no satiation and every commodity is a good). Well-Behaved Preferences uConvexity: Mixtures of bundles are (at least weakly) preferred to the bundles themselves. E.g., the 50-50 mixture of the bundles x and y is z = (0.5)x + (0.5)y. z is at least as preferred as x or y. Well-Behaved Preferences -- Convexity. x2 y2 x2+y2 2 x1 y1 x1+y1 2 x y z = x+y 2 is strictly preferred to both x and y. Well-Behaved Preferences -- Convexity. x2 y2 x1 y1 x y z =(tx1+(1-t)y1, tx2+(1-t)y2) is preferred to x and y for all 0 < t < 1. Well-Behaved Preferences -- Convexity. x2 y2 x1 y1 x y Preferences are strictly convex when all mixtures z are strictly preferred to their component bundles x and y. z Well-Behaved Preferences -- Weak Convexity. x’ y’ z’ Preferences are weakly convex if at least one mixture z is equally preferred to a component bundle. x z y Non-Convex Preferences x2 y2 x1 y1 z The mixture z is less preferred than x or y. More Non-Convex Preferences x2 y2 x1 y1 z The mixture z is less preferred than x or y. Slopes of Indifference Curves uThe slope of an indifference curve is its marginal rate-of-substitution (MRS). uHow can a MRS be calculated? Marginal Rate of Substitution x2 x1 x’ MRS at x’ is the slope of the indifference curve at x’ Marginal Rate of Substitution x2 x1 MRS at x’ is lim {Dx2/Dx1} Dx1 0 = dx2/dx1 at x’ Dx2 Dx1 x’ Marginal Rate of Substitution x2 x1 dx2 dx1 dx2 = MRS ´ dx1 so, at x’, MRS is the rate at which the consumer is only just willing to exchange commodity 2 for a small amount of commodity 1. x’ MRS & Ind. Curve Properties Good 2 Good 1 Two goods a negatively sloped indifference curve MRS < 0. MRS & Ind. Curve Properties Good 2 Bad 1 One good and one bad a positively sloped indifference curve MRS > 0. MRS & Ind. Curve Properties Good 2 Good 1 MRS = - 5 MRS = - 0.5 MRS always increases with x1 (becomes less negative) if and only if preferences are strictly convex. MRS & Ind. Curve Properties x1 x2 MRS = - 0.5 MRS = - 5 MRS decreases (becomes more negative) as x1 increases nonconvex preferences MRS & Ind. Curve Properties x2 x1 MRS = - 0.5 MRS = - 1 MRS = - 2 MRS is not always increasing as x1 increases nonconvex preferences.