microlower.jpg © 2010 W. W. Norton & Company, Inc. microtitle.jpg microedition.jpg varianname.jpg 3 Preferences microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Preference Relations ux y and y x imply x ~ y. ~ f ~ f microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Assumptions about Preference Relations uReflexivity: Any bundle x is always at least as preferred as itself; i.e. x x. ~ f microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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”. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Indifference Curves x2 x1 x” x”’ x’ ~ x” ~ x”’ x’ microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Indifference Curves x2 x1 z x y Textové pole: p p Textové pole: p p x y z microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Indifference Curves x2 x1 I(x’) x I(x) WP(x), the set of bundles weakly preferred to x. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Indifference Curves x2 x1 WP(x), the set of bundles weakly preferred to x. WP(x) includes I(x). x I(x) microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Indifference Curves x2 x1 SP(x), the set of bundles strictly preferred to x, does not include I(x). x I(x) microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Indifference Curves Cannot Intersect x2 x1 x y z I1 I2 From I1, x ~ y. From I2, x ~ z. Therefore y ~ z. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Slopes of Indifference Curves Good 2 Good 1 Two goods a negatively sloped indifference curve. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Slopes of Indifference Curves uIf less of a commodity is always preferred then the commodity is a bad. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Slopes of Indifference Curves Good 2 Bad 1 One good and one bad a positively sloped indifference curve. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Indifference Curves Exhibiting Satiation x2 x1 Satiation (bliss) point microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Indifference Curves Exhibiting Satiation x2 x1 Satiation (bliss) point microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Indifference Curves Exhibiting Satiation x2 x1 Satiation (bliss) point microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Indifference Curves With a Discrete Good Gas-oline Aircraft 0 1 2 3 4 Indifference “curves” are collections of discrete points. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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). microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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 microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Non-Convex Preferences x2 y2 x1 y1 z The mixture z is less preferred than x or y. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› More Non-Convex Preferences x2 y2 x1 y1 z The mixture z is less preferred than x or y. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Slopes of Indifference Curves uThe slope of an indifference curve is its marginal rate-of-substitution (MRS). uHow can a MRS be calculated? microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Marginal Rate of Substitution x2 x1 x’ MRS at x’ is the slope of the indifference curve at x’ microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› Marginal Rate of Substitution x2 x1 MRS at x’ is lim {Dx2/Dx1} Dx1 0 = dx2/dx1 at x’ Dx2 Dx1 x’ microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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’ microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› MRS & Ind. Curve Properties Good 2 Good 1 Two goods a negatively sloped indifference curve MRS < 0. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› MRS & Ind. Curve Properties Good 2 Bad 1 One good and one bad a positively sloped indifference curve MRS > 0. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› 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. microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› MRS & Ind. Curve Properties x1 x2 MRS = - 0.5 MRS = - 5 MRS decreases (becomes more negative) as x1 increases nonconvex preferences microlower.jpg © 2010 W. W. Norton & Company, Inc. ‹#› MRS & Ind. Curve Properties x2 x1 MRS = - 0.5 MRS = - 1 MRS = - 2 MRS is not always increasing as x1 increases nonconvex preferences.