Chapter 3 Preferences Rationality in Economics Behavioral Postulate

Chapter 3 Preferences

Rationality in Economics Behavioral Postulate: A decisionmaker always chooses its most preferred alternative from its set of available alternatives. u So to model choice we must model decisionmakers’ preferences. u

Preference Relations u 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 u Strict preference, weak preference and indifference are all preference relations. u Particularly, they are ordinal relations; i. e. they state only the order in which bundles are preferred.

Preference Relations p p u denotes strict preference; x y means that bundle x is preferred strictly to bundle y.

Preference Relations p p 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. u

Preference Relations p p 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 f denotes weak preference; ~ x f y means x is preferred at least as ~ much as is y. u

Preference Relations ux f~ y and y f~ x imply x ~ y.

Preference Relations f~ y and y f~ x imply x ~ y. u x f y and (not y f x) imply x ~ ~ ux y. p

Assumptions about Preference Relations u Completeness: For any two bundles x and y it is always possible to make the statement that either x f y ~ or y f x. ~

Assumptions about Preference Relations u Reflexivity: Any bundle x is always at least as preferred as itself; i. e. x f~ x.

Assumptions about Preference Relations u Transitivity: 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 f~ y and y f~ z x f z. ~

Indifference Curves u 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’. u Since an indifference “curve” is not always a curve a better name might be an indifference “set”.

Indifference Curves x 2 x’ ~ x”’ x’ x” x”’ x 1

Indifference Curves p x z y x 1 p x 2 y

Indifference Curves I 1 x 2 x z I 2 y I 3 All bundles in I 1 are strictly preferred to all in I 2. All bundles in I 2 are strictly preferred to all in I 3. x 1

Indifference Curves x 2 WP(x), the set of x bundles weakly preferred to x. I(x) I(x’) x 1

Indifference Curves x 2 WP(x), the set of x bundles weakly preferred to x. WP(x) includes I(x). x 1

Indifference Curves x 2 SP(x), the set of x bundles strictly preferred to x, does not include I(x). x 1

Indifference Curves Cannot Intersect x 2 I 1 I 2 From I 1, x ~ y. From I 2, x ~ z. Therefore y ~ z. x y z x 1

Indifference Curves Cannot Intersect I 1 I 2 From I 1, x ~ y. From I 2, x ~ z. Therefore y ~ z. But from I 1 and I 2 we see y z, a contradiction. x y p x 2 z x 1

Slopes of Indifference Curves u When more of a commodity is always preferred, the commodity is a good. u If every commodity is a good then indifference curves are negatively sloped.

Slopes of Indifference Curves Good 2 Be tte r W or se Two goods a negatively sloped indifference curve. Good 1

Slopes of Indifference Curves u If less of a commodity is always preferred then the commodity is a bad.

Slopes of Indifference Curves Good 2 One good and one bad a r te t positively sloped e B indifference curve. e s or W Bad 1

Extreme Cases of Indifference Curves; Perfect Substitutes u 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 Substitutes x 2 15 I 2 8 I 1 Slopes are constant at - 1. Bundles in I 2 all have a total of 15 units and are strictly preferred to all bundles in I 1, which have a total of only 8 units in them. x 1 8 15

Extreme Cases of Indifference Curves; Perfect Complements u 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 Complements x 2 45 o 9 5 Each of (5, 5), (5, 9) and (9, 5) contains 5 pairs so each is equally preferred. I 1 5 9 x 1

Extreme Cases of Indifference Curves; Perfect Complements x 2 Since each of (5, 5), (5, 9) and (9, 5) contains 5 pairs, each is less I 2 preferred than the bundle (9, 9) which I 1 contains 9 pairs. 45 o 9 5 5 9 x 1

Preferences Exhibiting Satiation u. A bundle strictly preferred to any other is a satiation point or a bliss point. u What do indifference curves look like for preferences exhibiting satiation?

Indifference Curves Exhibiting Satiation x 2 Satiation (bliss) point x 1

Indifference Curves Exhibiting Satiation Be tte r r e t t e B Better x 2 x 1 Satiation (bliss) point

Indifference Curves Exhibiting Satiation Be tte r r e t t e B Better x 2 x 1 Satiation (bliss) point

Indifference Curves for Discrete Commodities u. A commodity is infinitely divisible if it can be acquired in any quantity; e. g. water or cheese. u A 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 u Suppose 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 Gasoline Indifference “curves” are collections of discrete points. 0 1 2 3 4 Aircraft

Well-Behaved Preferences u. A preference relation is “wellbehaved” if it is – monotonic and convex. u Monotonicity: More of any commodity is always preferred (i. e. no satiation and every commodity is a good).

Well-Behaved Preferences u Convexity: 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. x x 2 x+y is strictly preferred z= 2 to both x and y. x 2+y 2 2 y y 2 x 1+y 1 2 y 1

Well-Behaved Preferences -Convexity. x x 2 z =(tx 1+(1 -t)y 1, tx 2+(1 -t)y 2) is preferred to x and y for all 0 < t < 1. y y 2 x 1 y 1

Well-Behaved Preferences -Convexity. x x 2 y 2 x 1 Preferences are strictly convex when all mixtures z are strictly z preferred to their component bundles x and y. y y 1

Well-Behaved Preferences -Weak Convexity. Preferences are weakly convex if at least one mixture z is equally preferred to a component bundle. x’ z’ x z y y’

Non-Convex Preferences B x 2 r te et z y 2 x 1 y 1 The mixture z is less preferred than x or y.

More Non-Convex Preferences B r te et x 2 z y 2 x 1 y 1 The mixture z is less preferred than x or y.

Slopes of Indifference Curves u The slope of an indifference curve is its marginal rate-of-substitution (MRS). u How can a MRS be calculated?

Marginal Rate of Substitution x 2 MRS at x’ is the slope of the indifference curve at x’ x’ x 1

Marginal Rate of Substitution x 2 D x 2 x’ MRS at x’ is lim {Dx 2/Dx 1} D x 1 0 = dx 2/dx 1 at x’ D x 1

Marginal Rate of Substitution x 2 dx 2 x’ dx 1 dx 2 = MRS ´ dx 1 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 1

MRS & Ind. Curve Properties Good 2 r te et B Two goods a negatively sloped indifference curve MRS < 0. se or W Good 1

MRS & Ind. Curve Properties Good 2 One good and one bad a r te t positively sloped e B indifference curve e MRS > 0. s r o W Bad 1

MRS & Ind. Curve Properties Good 2 MRS = - 5 MRS always increases with x 1 (becomes less negative) if and only if preferences are strictly convex. MRS = - 0. 5 Good 1

MRS & Ind. Curve Properties x 2 MRS = - 0. 5 MRS decreases (becomes more negative) as x 1 increases nonconvex preferences MRS = - 5 x 1

MRS & Ind. Curve Properties x 2 MRS is not always increasing as x 1 increases nonconvex preferences. MRS = - 1 MRS = - 0. 5 MRS = - 2 x 1