Chapter 3 Preferences

Introduction The economic model of consumer behavior is very simple: people choose the best things they can afford. The last chapter was devoted to clarifying the meaning of “can afford”. This chapter will be devoted to clarifying the economic concept of “best things”. A consumer always chooses his most preferred one from his set of available alternatives. So to model consumers’ choices we must model their preferences.

Preference Relations p p f f ~ ~ Let x, y are consumption bundles. denotes strict preference: x y means that bundle x is strictly preferred 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. p p ~ f ~ f

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

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. Bundles are always comparable. Again, if both are true, then they are indifferent to the individual. ~ f ~ f

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

Assumptions about Preference Relations 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 y and y z x z. ~ f ~ f ~ f

Indifference Curves f ~ Take a reference bundle x’. The set of all bundles equally preferred to x’ is the indifference curve (set) containing x’; i.e., the set of all bundles {y: y ~ x’}. Weakly preferred set: bundles that are weakly preferred to x’. {y: y x’}. ~ f

Indifference Curves x2 x’ x’ ~ x” ~ x”’ x” x”’ x1

Indifference Curves p p x2 z x y x z y x1 If the consumer prefers more to less for each good, all bundles to the northeast of the indifference curve are strictly preferred to x, and all bundles to the southwest of the indifference curve are less preferred to x. z y x1

Indifference Curves I1 All bundles in I1 are strictly preferred to all in I2. x2 x z I2 All bundles in I2 are strictly preferred to all in I3. y I3 x1

Indifference Curves x2 x I(x’) I(x) x1 WP(x), the set of bundles weakly preferred to x. x I(x’) I(x) x1

Indifference Curves x2 x I(x) x1 WP(x), the set of bundles weakly preferred to x. x WP(x) includes I(x). I(x) x1

Indifference Curves x2 x I(x) x1 SP(x), the set of bundles strictly preferred to x, does not include I(x). x I(x) x1

Indifference Curves Cannot Intersect From I1, x ~ y. From I2, x ~ z. Therefore y ~ z. However, I1 and I2 represent different levels of preference. Contradiction! x2 I1 x y z x1

Goods 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. It is because when one has more of one good, one has to get less of another to make this bundle indifferent to the original one.

Slopes of Indifference Curves Good 2 Two goods a negatively sloped indifference curve. Better Worse Good 1

Bads If less of a commodity is always preferred then the commodity is a bad. e.g. rotten fruits; tobacco smoke (if you do not smoke) If one good is good and the other is bad, then the indifference curve would be upward sloping.

Slopes of Indifference Curves Good 2 One good and one bad a positively sloped indifference curve. Better Worse Bad 1

Neutrals If one just do not care about whether or how much to have a commodity, then the commodity is called a neutral good. e.g. goods that you don’t use and do not care about their existence. If one commodity is neutral, the other is good, the indifference curve would be vertical / horizontal.

Slopes of Indifference Curves

Perfect Substitutes If a consumer always regards units of commodities 1 and 2 as equivalent, then the commodities are perfect substitutes. Only the total amount (or a weighted sum) of the two commodities in bundles determines their preference rank-order. e.g. orange juice of two different brands.

Perfect Substitutes x2 Slopes are constant at - 1. 15 I2 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. 8 I1 x1 8 15

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. Only the number of pairs of units of the two commodities determines the preference rank-order of bundles. e.g. left shoes/right shoes.

Perfect Complements x2 45o 9 5 I1 x1 5 9 Each of (5,5), (5,9) and (9,5) contains 5 pairs so each is equally preferred. 9 5 I1 x1 5 9

Perfect Complements x2 45o 9 I2 5 I1 x1 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. 9 I2 5 I1 x1 5 9

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 preferences exhibiting satiation?

Indifference Curves Exhibiting Satiation Satiation (bliss) point x1

Indifference Curves Exhibiting Satiation Better Better Satiation (bliss) point Better x1

Indifference Curves Exhibiting Satiation Better Better Satiation (bliss) point Better x1

Discrete Commodities A commodity is infinitely divisible if it can be acquired in any quantity; e.g. water or cheese. A commodity is discrete if it comes in unit lumps of 1, 2, 3, … and so on; e.g. aircraft, ships and refrigerators.

Discrete Commodities 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. Aircraft 1 2 3 4

Well-Behaved Preferences A preference relation is “well-behaved” if it is Monotonic; and convex. Monotonicity: More of any commodity is always preferred (i.e. no satiation and every commodity is a good). Monotonicity implies that indifference curves are negatively sloped.

Well-Behaved Preferences Convexity: Mixtures of bundles are (at least weakly) preferred to the bundles themselves. For example, 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+y z is preferred to both x and y. x2+y2 z = 2 2 y y2 x1 x1+y1 y1 2

Well-Behaved Preferences -- Convexity z =(tx1+(1-t)y1, tx2+(1-t)y2) is preferred to x and y for all 0 < t < 1. y y2 x1 y1

Well-Behaved Preferences -- Convexity Preferences are strictly convex when all mixtures z are strictly preferred to their component bundles x and y. x x2 z y y2 x1 y1

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 Better The mixture z is less preferred than x or y. z y2 x1 y1

More Non-Convex Preferences Better The mixture z is less preferred than x or y. z y2 x1 y1

Slopes of Indifference Curves The slope of an indifference curve is its marginal rate-of-substitution (MRS). The MRS measures the rate at which the consumer is just willing to substitute one good for the other.

Marginal Rate of Substitution x2 MRS at x’ is the slope of the indifference curve at x’ x’ x1

Marginal Rate of Substitution x2 MRS at x’ is lim {Dx2/Dx1} Dx1 0 = dx2/dx1 at x’ Dx2 x’ Dx1 x1

Marginal Rate of Substitution x2 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’ dx2 dx1 x1

MRS & Ind. Curve Properties Good 2 Two goods a negatively sloped indifference curve Better MRS < 0. Worse Good 1

MRS & Ind. Curve Properties Good 2 One good and one bad a positively sloped indifference curve Better MRS > 0. Worse Bad 1

MRS & Ind. Curve Properties Good 2 MRS = - 5 MRS always increases (decreases in absolute value) with x1 (becomes less negative) if and only if preferences are strictly convex. MRS = - 0.5 Good 1 We call it a diminishing marginal rate of substitution.

MRS & Ind. Curve Properties x2 MRS decreases (becomes more negative) as x1 increases with nonconvex preferences. MRS = - 0.5 MRS = - 5 x1

MRS & Ind. Curve Properties x2 MRS is not always increasing as x1 increases with nonconvex preferences. MRS = - 1 MRS = - 0.5 MRS = - 2 x1