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3 Prefernces - CHAPTER 3 PREFERENCES We saw in Chapter 2...

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Unformatted text preview: CHAPTER 3 PREFERENCES We saw in Chapter 2 that 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,” and this chapter will be devoted to clarifying the economic concept of “best things.” We call the objects of consumer choice consumption bundles. This is a complete list of the goods and services that are involved in the choice problem that we are investigating. The word “complete” deserves empha- sis: when you analyze a consumer’s choice problem, make sure that you include all of the appropriate goods in the definition of the consumption bundle. If we are analyzing consumer choice at the broadest level, we would want not only a complete list of the goods that a consumer might consume, but also a description of when, where, and under what circumstances they would become available. After all, people care about how much food they will have tomorrow as well as how much food they have today. A raft in the middle of the Atlantic Ocean is very different from a raft in the middle of the Sahara Desert. And an umbrella when it is raining is quite a difierent good from an umbrella on a sunny day. It is often useful to think of the 34 PREFERENCES (Ch. 3) “same” good available in different locations or circumstances as a different good, since the consumer may value the good differently in those situations. However, when we limit our attention to a simple choice problem, the relevant goods are usually pretty obvious. We’ll often adopt the idea de— scribed earlier of using just two goods and calling one of them “all other goods” so that we can focus on the tradeoff between one good and ev- erything else. In this way we can consider consumption choices involving many goods and still use two—dimensional diagrams. So let us take our consumption bundle to consist of two goods, and let x1 denote the amount of one good and .122 the amount of the other. The complete consumption bundle is therefore denoted by (331,172). As noted before, we will occasionally abbreviate this consumption bundle by X. 3.1 Consumer Preferences We will suppose that given any two consumption bundles, (171,332) and (y1,y2), the consumer can rank them as to their desirability. That is, the consumer can determine that one of the consumption bundles is strictly better than the other, or decide that she is indifferent between the two bundles. We will use the symbol > to mean that one bundle is strictly preferred to another, so that (331,332) >~ (y1,y2) should be interpreted as saying that the consumer strictly prefers ($1,332) to (gr/1,3,2), in the sense that she definitely wants the x—bundle rather than the y—bundle. This preference relation is meant to be an operational notion. If the consumer prefers one bundle to another, it means that he or she would choose one over the other, given the opportunity. Thus the idea of preference is based on the consumer’s behavior. In order to tell whether one bundle is preferred to another, we see how the consumer behaves in choice situations involving the two bundles. If she always chooses (£131,332) when (y1,y2) is available, then it is natural to say that this consumer prefers ($1,172) to (yhyg). If the consumer is indifferent between two bundles of goods, we use the symbol ~ and write (921,332) ~ (ghyg). Indifference means that the consumer would be just as satisfied, according to her own preferences, consuming the bundle (3:1, 2:2) as she would be consuming the other bundle, (3/1, yzl If the consumer prefers or is indifferent between the two bundles we say that she weakly prefers ($1, $2) to (y1,y2) and write (sham) t (91,312). These relations of strict preference, weak preference, and indifference are not independent concepts; the relations are themselves related! For example, if (931,;132) : (311,112) and (311,112) : (231,132) we can conclude that (271,332) ~ (yhyg). That is, if the consumer thinks that ($1,232) is at least as good as (311,312) and that (yl, 312) is at least as good as (331,132), then the consumer must be indifferent between the two bundles of goods. ASSUMPTIONS ABOUT PREFERENCES 35 Similarly, if ($1,332) t ($11,312) but We know that it is not the case that (£131,352) N (yhyg), we can conclude that we must have (331,302) > (yhyg). This just says that if the consumer thinks that (9:1,:r2) is at least as good as (y1,y2), and she is not indifferent between the two bundles, then it must be that she thinks that (5131,13) is strictly better than (y1,y2). 3.2 Assumptions about Preferences Economists usually make some assumptions about the “consistency” of consumers’ preferences. For example, it seems unreasonable——not to say contradictory—to have a situation where (111,.CE2) > (311,312) and, at the same time, (311,312) >— ($1,113). For this would mean that the consumer strictly prefers the x—bundle to the y—bundle . . . and vice versa. So we usually make some assumptions about how the preference relations work. Some of the assumptions about preferences are so fundamental that we can refer to them as “axioms” of consumer theory. Here are three such axioms about consumer preference. Complete. We assume that any two bundles can be compared. That is, given any x—bundle and any y—bundle, we assume that (:51, 362) i (y1,y2), or (y1,y2) : ($1,372), or both, in which case the consumer is indifferent between the two bundles. Reflexive. We assume that any bundle is at least as good as itself: ($1,302) t ($1,562)- Transitive. If (231,:c2) i (yhyg) and (y1,y2) t (21,22), then we assume that (:51, 232) t (21, 22). In other words, if the consumer thinks that X is at least as good as Y and that Y is at least as good as Z, then the consumer thinks that X is at least as good as Z. The first axiom, completeness, is hardly objectionable, at least for the kinds of choices economists generally examine. To say that any two bundles can be compared is simply to say that the consumer is able to make a choice between any two given bundles. One might imagine extreme situations involving life or death choices where ranking the alternatives might be difficult, or even impossible, but these choices are, for the most part, outside the domain of economic analysis. The second axiom, reflexivity, is trivial. Any‘ bundle is certainly at least as good as an identical bundle. Parents of small children may occasionally observe behavior that violates this assumption, but it seems plausible for most adult behavior. The third axiom, transitivity, is more problematic. It isn’t clear that transitivity of preferences is necessarily a property that preferences would have to have. The assumption that preferences are transitive doesn’t seem 36 PREFERENCES (Ch. 3) compelling on grounds of pure logic alone. In fact it’s not. Transitivity is a hypothesis about people’s choice behavior, not a statement of pure logic. Whether it is a basic fact of logic or not isn’t the point: it is whether or not it is a reasonably accurate description of how peOple behave that matters. What would you think about a person who said that he preferred a bundle X to Y, and preferred Y to Z, but then also said that he preferred Z to X 7 This would certainly be taken as evidence of peculiar behavior. More importantly, how would this consumer behave if faced with choices among the three bundles X, Y, and Z? If we asked him to choose his most preferred bundle, he would have quite a problem, for whatever bundle he chose, there would always be one that was preferred to it. If we are to have a theory where people are making “best” choices, preferences must satisfy the transitivity axiom or something very much like it. If preferences were not transitive there could well be a set of bundles for which there is no best choice. 3.3 Indifference Curves It turns out that the whole theory of consumer Choice can be formulated in terms of preferences that satisfy the three axioms described above, plus a few more technical assumptions. However, we will find it convenient to describe preferences graphically by using a construction known as indif— ference curves. Consider Figure 3.1 where we have illustrated two axes representing a consumer’s consumption of goods 1 and 2. Let us pick a certain consump— tion bundle (3:1,1'2) and shade in all of the consumption bundles that are weakly preferred to ($1, 332). This is called the weakly preferred set. The bundles on the boundary of this set—the bundles for which the consumer is just indifferent to (x1,a:2)——form the indifference curve. We can draw an indifference curve through any consumption bundle we want. The indifference curve through a consumption bundle consists of all bundles of goods that leave the consumer indifferent to the given bundle. One problem with using indifference curves to describe preferences is that they only show you the bundles that the consumer perceives as being indifferent to each other—they don’t show you which bundles are better and which bundles are worse. It is sometimes useful to draw small arrows on the indifference curves to indicate the direction of the preferred bundles. We won’t do this in every case, but we will do it in a few of the examples where confusion might arise. If we make no further assumptions about preferences, indifference curves can take very peculiar shapes indeed. But even at this level of generality, we can state an important principle about indifference curves: indiflerence curves representing distinct levels of preference cannot cross. That is, the situation depicted in Figure 3.2 cannot occur. EXAMPLES OF PREFERENCES 37 Weakly preferred set: bundles weakly preferred to (x1, x2) Indifference curve: bundles indifferent “3 (x1, x2) Weakly preferred set. The shaded area consists of all bun— dles that are at least as good as the bundle ($1,332). In order to prove this, let us choose three bundles of goods, X, Y, and Z, such that X lies only on one indifference curve, Y lies only on the other indifference curve, and Z lies at the intersection of the indifference curves. By assumption the indifference curves represent distinct levels of prefer- ence, so one of the bundles, say X, is strictly preferred to the other bundle, Y. We know that X N Z and Z N Y, and the axiom of transitivity there— fore implies that X N Y. But this contradicts the assumption that X > Y. This contradiction establishes the result~—indifference curves representing distinct levels of preference cannot cross. What other properties do indifference curves have? In the abstract, the answer is: not many. Indifference curves are a way to describe preferences. Nearly any “reasonable” preferences that you can think of can be depicted by indifference curves. The trick is to learn what kinds of preferences give rise to what shapes of indifference curves. 3.4 Examples of Preferences Let us try to relate preferences to indifference curves through some exam— ples. We’ll describe some preferences and then see what the indifference curves that represent them look like. 38 PREFERENCES (Ch. 3} . Alle. ed indi erence curves X: Indifference curves cannot cross. If they did, X, Y, and Z would all have to be indifferent to each other and thus could _ not lie on distinct .indifierence curves. There is a general procedure for constructing indifference curves given a “verbal” description of the preferences. First plop your pencil down on the graph at some consumption bundle (1:1, 132). Now think about giving a little more of good 1, A331, to the consumer, moving him to (:31 + A331,:L'2). Now ask yourself how would you have to change the consumption of 2:2 to make the consumer indifferent to the original consumption point? Call this change A502. Ask yourself the question “For a given change in good 1, how does good 2 have to change to make the consumer just indifferent between (331 + A$1,$2 + A332) and ($1,552)?” Once you have determined this movement at one consumption bundle you have drawn a piece of the indifference curve. Now try it at another bundle, and so on, until you develop a clear picture of the overall shape of the indifference curves. Perfect Substitutes Two goods are perfect substitutes if the consumer is Willing to substitute one good for the other at a constant rate. The simplest case of perfect substitutes occurs when the consumer is willing to substitute the goods on a one-to-one basis. Suppose, for example, that we are considering a choice between red pen- cils and blue pencils, and the consumer involved likes pencils, but doesn’t care about color at all. Pick a consumption bundle, say (10,10). Then for this consumer, any other consumption bundle that has 20 pencils in it is EXAMPLES OF PREFERENCES 39 just as good as (10,10). Mathematically speaking, any consumption bun- dle (331,132) such that $1 + x2 _—_ 20 will be on this consumer’s indifference curve through (10, 10). Thus the indifference curves for this consumer are all parallel straight lines with a slope of —1, as depicted in Figure 3.3. Bundles with more total pencils are preferred to bundles with fewer total pencils, so the direction of increasing preference is up and to the right, as illustrated in Figure 3.3. How does this work in terms of general procedure for drawing indifference curves? If we are at (10, 10), and we increase the amount of the first good by one unit to 11, how much do we have to change the second good to get back to the original indifference curve? The answer is clearly that we have to decrease the second good by 1 unit. Thus the indifference curve through (10,10) has a slope of —1. The same procedure can be carried out at any bundle of goods with the same resultskin this case all the indifference curves have a constant slope of —1. ;._ . 3 j _ " indifferEnce curves . . . .. . . ' 9r,” ' Perfect substitutes The consumer only cares about the total number of pencfis not about their colors. This the indifference curves are straight lines with a siope (cf —-1 - - - The important fact about perfect substitutes is that the indifference curves have a constant slope. Suppose, for example, that we graphed blue pencils on the vertical axis and pairs of red pencils on the horizontal axis. The indifference curves for these two goods would have a slope of —2, since the consumer would be willing to give up two blue pencils to get one more pair of red pencils. 40 PREFERENCES (Ch. 3) In the textbook we’ll primarily consider the case where goods are perfect substitutes on a one-for—one basis, and leave the treatment of the general case for the workbook. Perfect Complements Perfect complements are goods that are always consumed together in fixed proportions. In some sense the goods “complement” each other. A nice example is that of right shoes and left shoes. The consumer likes shoes, but always wears right and left shoes together. Having only one out of a pair of shoes doesn’t do the consumer a bit of good. Let us draw the indifference curves for perfect complements. Suppose we pick the consumption bundle (10,10). Now add 1 more right shoe, so we have (11,10). By assumption this leaves the consumer indifferent to the original position: the extra shoe doesn’t do him any good. The same thing happens if we add one more left Shoe: the consumer is also indifferent between (10,11) and (10,10). Thus the indifference curves are L-shaped, with the vertex of the L oc- curring where the number of left shoes equals the number of right shoes as in Figure 3.4. LEFT SHOES . . indifference - curves RIGHT SHOES . Perfect complements. . The consumer always wants to con— same the goods. in fixed proportibnsi to each other. Thus the . indifference curves are LrShaPGd.- . .- -. - EXAMPLES OF PREFERENCES 41 Increasing both the number of left shoes and the number of right shoes at the same time will move the consumer to a more preferred position, so the direction of increasing preference is again up and to the right, as illustrated in the diagram. The important thing about perfect complements is that the consumer prefers to consume the goods in fixed proportions, not necessarily that the proportion is one-to-one. If a consumer always uses two teaspoons of sugar in her cup of tea, and doesn’t use sugar for anything else, then the indifference curves will still be L—shaped. In this case the corners of the L will occur at (2 teaspoons sugar, 1 cup tea), (4 teaspoons sugar, 2 cups tea) and so on, rather than at (1 right shoe, 1 left shoe), (2 right shoes, 2 left shoes), and so on. In the textbook we’ll primarily consider the case where the goods are conSumed in proportions of one—for—one and leave the treatment of the general case for the workbook. Bads A bad is a commodity that the consumer doesn’t like. For example, sup— pose that the commodities in question are now pepperoni and anchovies— and the consumer loves pepperoni but dislikes anchovies. But let us suppose there is some possible tradeoff between pepperoni and anchovies. That is, there would be some amount of pepperoni on a pizza that would compen- sate the consumer for having to consume a given amount of anchovies. How could we represent these preferences using indifference curves? Pick a bundle ($1,332) consisting of some pepperoni and some anchovies. If we give the consumer more anchovies, what do we have to do with the pepperoni to keep him on the same indifference curve? Clearly, we have to give him some extra pepperoni to compensate him for having to put up with the anchovies. Thus this consumer must have indifference curves that slope up and to the right as depicted in Figure 3.5. The direction of increasing preference is down and to the rightfithat is, toward the direction of decreased anchovy consumption and increased pepperoni consumption, just as the arrows in the diagram illustrate. Neutrals A good is a neutral good if the consumer doesn’t care about it one way or the other. What if a consumer is just neutral about anchovies?1 In this case his indifference curves will be vertical lines as depicted in Figure 3.6. 1 Is anybody neutral about anchovies? 42 PREFERENCES (Ch. 3) mm AN‘CHOVIES ' indifference . curves ' PEPPERONI Beds. Here anchovies are a “bad,” and pepperoni is a “good” for this consumer. Thus the indifference curves have a positive slope. ANCHOVIES Indifference curves ' PEPPERONI A neutral good. The consumer-likespepperoni- but 'isneutral about anchovies, so the indifference Curves are vertical lines. He only cares about the amount of pepperoni he has and doesn’t care at all about how many anchovies he has. The more pepperoni the better, but adding more anchovies doesn’t affect him one way or the other. EXAMPLES OF PREFERENCES 43 Satiation We sometimes want to consider a situation involving satiation, where there is some overall best bundle for the consumer, and the “closer” he is to that best bundle, the better off he is in terms of his own preferences. For example, suppose that the consumer has some most preferred bundle of goods (Eljg), and the farther away he is from that bundle, the worse off he is. In this case we say that ($1,252) is a satiation point, or a bliss point. The indifference curves for the c0nsumer look like those depicted in Figure 3.7. The best point is (5132) and points farther away from this bliss point lie on “lower” indifference curves. Indifference curves Satiation point 21 I xi Satiated preferences. The bundle (351,352) is the satiation point or bliss. point, and the indifierence curves surround this point. ' ' ' ' W.“- In this case the indifference curves have a negative slope when the con- sumer has “too little” or “too much” of both goods, and a positive slope when he has “too much” of one of the goods. When he has too much of one of the goods, it becomes a badfireducing the consumption of the bad good moves him closer to his “bliss point.” If he has too much of both goods, they both are bads, so reducing the consumption of each moves him closer to the bliss point. ...
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