lecture

Chapter 3: Preferences Remember: Rationality in Economics What does it mean to be rational? A person always chooses the most preferred alternative from their set of available alternatives. Budget constraint showed the “available alternatives.” What about preferred alternatives? So to model choice we must model preferences. Preference Relations Comparing two different consumption bundles, x and y: strict preference: x is more preferred than is y. indifference: x is exactly as preferred as is y. weak preference: x is as at least as preferred as is y. Preference Relations denotes strict preference x y means that bundle x is preferred strictly to bundle y. For two goods: (x1,x2) (y1,y2) means we prefer bundle (x1,x2) to bundle (y1,y2). x1 and y1 stand for different values of good 1, and x2 and y2 stand for different values of good 2. p p p Preference Relations denotes strict preference; x y means bundle x is preferred strictly to bundle y. ~ denotes indifference; x~ y means x and y are equally preferred. (3,2) ~ (2,3) , means I like 3 slices of pizza and 2 beers as much as 2 slices and 3 beers. Given the choice between the two, I don’t care which bundle I get, both make me equally happy p p Preference Relations denotes strict preference so x y means that bundle x is preferred strictly to bundle y. ~ denotes indifference; x ~ y means x and y are equally preferred. denotes weak preference; x y means the consumer is either indifferent between x and y or prefers x to y. A bit tricky but different from it means I prefer or am indifferent between two bundles. ~ f ~ f p p p Preference Relations 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. Weak Preference Relation Examples If (2,3) (3,2) And (3,2) (2,3) What can we say about a consumer’s preference for (3,2) vs. (2,3). Then, (2,3) ~ (3,2) ~ f ~ f Weak Preference Relation Examples If we know (4,5) (5,4) And it is not the case that (4,5) ~ (5,4) What can we say about a consumer’s preference for (4,5) vs. (5,4). ~ f The Axioms of Consumer Theory Preferences are complete Preferences are reflexive Preferences are transitive 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. Or x~ y Intuitively, completeness means that all soul searching has been done. You know who you are, and you know that you like one bundle better than the other. ~ 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 STARBURSTS Do star burst experiment here Use Clickers. Indifference Curves A convenient method of expressing preferences graphically. Shows all bundles that the consumer is indifferent between. Indifference Curves x2 x1 x” x”’ x’ ~ x” ~ x”’ x’ Indifference Curves x2 x1 z x y 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 the set of bundles weakly preferred to x. I1 Indifference Curves x2 x1 the set of bundles weakly preferred to x. includes x I1 I1 Indifference Curves x2 x1 the set of bundles strictly preferred to x, does not include x I1 I1 Indifference Curves Cannot Intersect x2 x1 x y z I1 I2 From I1, x ~ y. From I2, x ~ z. Therefore y ~ z. Thick Indifference curves can exist. x1 x y I1 z This type of preferences are uncommon, means indifferent to Having more of both goods x2 Slopes 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. Practice translating ICs Take verbal/written description of preferences and draw a typical indifference curve in 4 steps. Slopes of Indifference Curves Better Worse Good 2 Good 1 Two goods a negatively sloped indifference curve. Slopes of Indifference Curves If less of a commodity is always preferred then we call it a bad. Me and Coconut parfait I want to have less of this and not more. Practice translating ICs 1) Choose a consumption bundle (x1,x2) 2) Imagine you add a little x1 3) How much do you need to change x2 so that you are indifferent to adding a little x2? 4) Repeat and map out a curve. Slopes of Indifference Curves Worse Good 2 Good 1 good 1 is a bad a positively sloped indifference curve. Types of Preferences Three standard types (with associated indifference curves): Perfect substitutes Perfect complements Well Behaved Example 1 of Preferences; 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. Sometimes it is comparing two goods that have different amounts of some underlying commodity: 20 oz. mustard bottle vs. 10 oz. mustard bottle. For substitutes we might only care about consuming as much mustard as possible, in this sense these are interchangeable. Will still pay more for the 20 oz bottle, since it has more mustard. Example 1), perfect substitutes J. The smoker is addicted to nicotine. J. only cares about how much nicotine J. consumes. J. can get the same amount of nicotine from either 1 cigarette or 1 pinch of chewing tobacco. Now, given that good 1 is cigarettes and good 2 is chewing tobacco, draw a typical indifference curve. Perfect Substitutes Chart Like cigerettes and chewing tobacco, willing to exchange on a 1:1 ratio. Extreme Cases of Indifference Curves; Perfect Substitutes Cigarettes 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. Pinches of Tob. Policy Implication of Perfect Substitutes If we want to get people to quit being addicted to nicotine should we tax cigarettes? What if we want to reduce second hand smoke exposure? Example 2 of Preferences; 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. Ex: left shoes and right shoes Example 2, Perfect Complements Schools always purchase 1 computer with 1 copy of educational software. Hardware and software are perfect complements. Perfect Complements Chart Only buy one unit of hardware with one unit of software. 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. Hardware Software L shape comes from being better off (higher IC), only when you increase both goods in fixed proportions 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. Policy Implications If we want to encourage families and/or schools to use technology for education does it make sense to only subsidize the purchase of computers? Example 3, Well behaved preferences The Diamond Water Paradox Adam Smith Pointed Out: We need water to live. Diamonds are a luxury. Yet, water is cheap and diamonds are expensive. The answer to the paradox is contained in well behaved preferences. There is a difference between value in exchange and value in use. Well-Behaved Preferences A preference relation is “well-behaved” if it is monotone and convex. Monotonicity: More of any commodity is always preferred (i.e. no “bads” and no “neutrals”). When monotonicty holds, preferences are described as monotone. Convexity: prefer mixtures of goods to extremes. Well-Behaved Preferences 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. x2 y2 x1 y1 X = (1,2) Y = (2,1) z = (1.5,1.5) Z is strictly preferred to both x and y. 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. 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 Example well behaved preferences: The Water Diamond Paradox Example well behaved preferences: Water vs. Diamonds Choose between consuming water and Diamonds. Diamonds are expensive and useless but rare Water is abundant but necessary for life. Why are we willing to give up a lot of water for only a little diamond? To understand this Need to learn how to quantify the rate of exchange. Interlude: Slopes of Indifference Curves What is the Significance of the slope of the IC? The slope of an indifference curve is its marginal rate-of-substitution (MRS). How 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 Dx2 Dx1 x’ Marginal Rate of Substitution x2 x1 MRS at x’ is the derivative of the equation for the line evaluated at point x’. Dx2 Dx1 x’ Marginal Rate of Substitution x2 x1 dx2 dx1 Willingness to pay interpretation MRS is the rate at which the consumer is only just willing to exchange commodity 2 for a small amount of commodity 1. x’ Example of MRS in the case of convex preferences. For convex indifference curves what happens to the slope as we increase x1? The slope decreases, the more X1 we have the less x2 we need to compensate for the loss of a little x1. The willingness to pay in terms of x1 decreases as x1 is less scarce. We call this having a diminishing marginal rate of substitution. This follows directly from the convexity assumption. MRS in the case of convex preferences Good 2 Good 1 MRS = - 5 MRS = - 0.5 The magnitude of the MRS always decreases with x1 (becomes less negative) if and only if preferences are strictly convex. Weirder Preferences Already went over bads: cases where you do not like one or more of the goods. Whether you like the commodity depends on how much you consume Only value commodities if consumed in discrete chunks. Preferences in the case of addiction (drug addicts, cigarette smokers, etc). Graph when good 1 is a bad Graph when good 2 is a bad Graph when both are bads. What if both goods are bads? Preferences Exhibiting Satiation Satiation: the idea of satiation is that there is a limit to how much of a good the consumer enjoys. Consumer is happier as they get closer to some fixed point If they consume beyond this point less happy. The point where they are “stuffed” or “satiated” is called a bliss point or a satiation point. Example 6: Satiation Jabba loves toads (good 1) and tobacco (good2). Before being satiated Jabba exhibits typical, well-behaved preferences. However, even Jabba can have his fill. If jabba is stuffed after 10 toads, after this point toads become a bad. Jabba starts coughing after 5 ounces of tobacco, after this point tobacco is a bad. Indifference Curves Exhibiting Satiation x2 x1 Satiation (bliss) point Indifference Curves Exhibiting Satiation x2 x1 Better Better Better Satiation (bliss) point Indifference Curves Exhibiting Satiation x2 x1 Better Better Better Satiation (bliss) point non-convex preferences Typical in cases of addiction Violates the assumption of convexity, want to consume at extremes. Usually think of in terms of one addictive good and one “everything else” good. Example 8: Ms. Heroin Imagine Ms. Heroin: Either consumes Heroin or other goods. If she consumes too much Heroin she quickly becomes addicted and faces a higher and higher marginal rate of substitution for Heroin. This leads to her to using the drug as much as possible. If she doesn’t take much Heroin, she has a higher and higher rate of substitution for “everything else.” Quickly only consumes all other goods. Non-Convex Preferences x2 y2 x1 y1 z Better The mixture z is less preferred than x or y. More Non-Convex Preferences x2 y2 x1 y1 z Better The mixture z is less preferred than x or y. MRS in the case of convex preferences 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 Extra Slides Weak Preference Relation a more detailed Example Two bundles A and B. Each bundle contains two goods A: (a1,a2) and B: (b1,b2). Now I assume I have the following preferences: I strictly prefer the bundle where there is more of the first good. If there are two bundles with equal amounts of the first good, I prefer the bundle that has more of the second good. If the bundles have the same amount of both goods, I am indifferent between them. Compare: (3,3) and (1,7) (3,4) and (4,5) (3,a2) and (3,b2), where it is always the case that a2>=b2. Example well behaved preferences: Consumers like water and like diamonds. If they have more than 6 tea spoons of water they will be willing to give 2 tea spoons of water for 1 very small diamond. If they have less than 6 tea spoons of water they will only give up a tea spoon for two more diamonds. An example problem: Talesha Another example problem: Quintin Cola Example:Nancy Learner Example of Bads: Pollution and Consumption Worse Everything else Pollution Example one good one neutral: Air and Food Air is in such abundance that I am indifferent between having a little more air (air is a neutral). Food is a good, and consuming more makes me happier.