3 Prefernces

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 amouut 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 (£131,352) 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 (rhea) 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 (114:2) t (ybyg) 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 (331,562) > (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, xz) Weakly preferred set. The shaded area consists of all bun— dies 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 (1:1 + Aarl,:c2). 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 ($1 + 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 (331,1:2) 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. . " indifferEnce curves .L-jiig 5 E substitutes. _.The;.consumer.gn1ngmaaSmut the mtal ._ I 5ffinite?9ffiefififisginfit“About-theirpolarsq’fhusme mass-mg. - 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 proportiOnsi to each other. Thus the . indifference curves are Lshaped.. . .- -. - 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-likespeppemni- 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 cousumer 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 (351,552) 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. Suppose, for example, that the two goods are chocolate cake and ice cream. There might well be some optimal amount of chocolate cake and 44 PREFERENCES (Ch. 3) ice cream that you would want to eat per week. Any less than that amount would make you worse off, but any more than that amount would also make you worse off. If you think about it, most goods are like chocolate cake and ice cream in this respecteyou can have too much of nearly anything. But people would generally not voluntarily choose to have too much of the goods they consume. Why would you choose to have more than you want of something? Thus the interesting region from the viewpoint of economic choice is where you have less than you want of most goods. The choices that people actually care about are choices of this sort, and these are the choices with which we will be concerned. Discrete Goods Usually we think of measuring goods in units where fractional amounts make sensexyou might on average consume 12.43 gallons of milk a month even though you buy it a quart at a time. But sometimes we want to examine preferences over goods that naturally come in discrete units. For example, consider a consumer’s demand for automobiles. We could define the demand for automobiles in terms of the time spent using an automobile, so that we would have a continuous variable, but for many purposes it is the actual number of cars demanded that is of interest. There is no difficulty in using preferences to describe choice behavior for this kind of discrete good. Suppose that $2 is money to be spent on other goods and $1 is a discrete good that is only available in integer amounts. We have illustrated the appearance of indifference “curves” and a weakly preferred set for this kind of good in Figure 3.8. In this case the bundles indifferent to a given bundle will be a set of discrete points. The set of bundles at least as good as a particular bundle will be a set of line segments. The choice of whether to emphasize the discrete nature of a good or not will depend on our application. If the consumer chooses only one or two units of the good during the time period of our analysis, recognizing the discrete nature of the choice may be important. But if the consumer is choosing 30 or 40 units of the good, then it will probably be convenient to think of this as a continuous good. 3.5 Well-Behaved Preferences We’ve now seen some examples of indifference curves. As we’ve seen, many kinds of preferences, reasonable or unreasonable, can be described by these simple diagrams. But if we want to describe preferences in general, it will be convenient to focus on a few general shapes of indifference curves. In WELL—BEHAVEDPREFERENCES 45 Bundles weakly preferred to (1, x2) 1 _ 2 A Indifference "curves" 3 00,00 1 2 3 GOOD 1 B Weakly preferrred set A discrete good. Here good 1- is only available in integer ' amounts. In panel-A the. dashedlines connect together the bundles that are indifferent,'and in panel B the vertical lines represent bundles that are at least as good asthe indicated bundle. ' - ' this section we will describe some more general assumptions that we will typically make about preferences and the implications of these assumptions for the shapes of the associated indifference curves. These assumptions are not the only possible ones; in some situations you might want to use different assumptions. But we will take them as the defining features for well-behaved indifference curves. First we will typically assume that more is better, that is, that we are talking about goods, not bads. More precisely, if (mum) is a bundle of goods and (y1,y2) is a bundle of goods with at least as much of both goods and more of one, then (y1,y2) >— ($1,332). This assumption is sometimes called monotonicity of preferences. As we suggested in our discussion of satiation, more is better would probably only hold up to a point. Thus the assumption of monotonicity is saying only that we are going to ex- amine situations before that point is reachedibefore any satiation sets inflwhile more still is better. Economics would not be a very interesting subject in a world where everyone was satiated in their consumption of every good. What does monotonicity imply about the shape of indifference curves? It implies that they have a negative slope. Consider Figure 3.9. If we start at a bundle (LE1, m2) and move anywhere up and to the right, we must be moving to a preferred position. If we move down and to the left we must be moving to a worse position. So if we are moving to an indificrcnt position, we must be moving either left and up or right and down: the indifference curve must have a negative slope. Figure 3.8 46 PREFERENCES (Ch. 3) Second, we are going to assume that averages are preferred to extremes. That is7 if we take two bundles of goods (321,332) and (y1, y2) on the same indifference curve and take a weighted average of the two bundles such as 1 + 1 1 + 1 _ — _CL’ _ 2331 291,2 2 21/2 5 then the average bundle will be at least as good as or strictly preferred to each of the two extreme bundles. This weighted-average bundle has the average amount of good 1 and the average amount of good 2 that is present in the two bundles It therefore lies halfway along the straight line connecting the x—bundle and the yibundle. _ I Better _ bundles . Xi Monotonic preferences. _'More Of both goods is a better bundle for this Consumer-glass of both goods represents a worse bundle. ' - _ _ ' Actually, we’re going to assume this for any weight it between 0 and 1, not just 1/2. Thus we are assuming that if (m1, m2) N (ghyg), then (“51 + (1 — tlylatb + (1 — 0.1/2) i (3017332) for any t such that 0 S t S 1. This weighted average of the two bundles gives a weight of t to the x-bundle and a weight of 1 ~ It to the y—bundle. Therefore, the distance from the X-bundle to the average bundle is just a fraction t of the distance from the x—bundle to the y-bundle, along the straight line connecting the two bundles. WELL-BEHAVED PREFERENCES 47 What does this assumption about preferences mean geometrically? It means that the set of bundles weakly preferred to (£131,502) is a convex set. For suppose that (y1,y2) and (2:1, 272) are indifferent bundles. Then, if aver- ages are preferred to extremes, all of the weighted averages of ($1,272) and (y1, y2) are weakly preferred to ($1,322) and (y1,y2). A convex set has the property that if you take any two points in the set and draw the line seg- ment connecting those two points, that line segment lies entirely in the set. Figure 3.10A depicts an example of convex preferences, while Figures 3.108 and 3.10C show two examples of nonconvex preferences. Figure 3.1OC presents preferences that are so nonconvex that we might want to call them “concave preferences.” (Y1: Y2) Averaged bundle A Convex B Nonconvex C Concave preferences preferences preferences Various kinds of preferences. - Panel A depicts convex pref— erences, panel B depicts nonconvex preferences, and panel C depicts “concave” preferences. Can you think of preferences that are not convex? One possibility might be something like my preferences for ice cream and olives. I like ice cream and I like olives . . . but I don’t like to have them together! In considering my consumption in the next hour, I might be indifferent between consuming 8 ounces of ice cream and 2 ounces of olives, or 8 ounces of olives and 2 ounces of ice cream. But either one of these bundles would be better than consuming 5 ounces of each! These are the kind of preferences depicted in Figure 3.100. Why do we want to assume that well-behaved preferences are convex? Because, for the most part, goods are consumed together. The kinds of preferences depicted in Figures 3.1013 and 3.100 imply that the con- 48 PREFERENCES (Ch. 3) sumer would prefer to specialize, at least to some degree, and to consume only one of the goods. However, the normal case is where the consumer would want to trade some of one good for the other and end up consuming some of each, rather than specializing in consuming only one of the two goods. In fact, if we look at my preferences for monthly consumption of ice cream and olives, rather than at my immediate consumption, they would tend to look much more like Figure 3.10A than Figure 3.10C. Each month I would prefer having some ice cream and some olivesialbeit at different times—to specializing in consuming either one for the entire month. Finally, one extension of the assumption of convexity is the assumption of strict convexity. This means that the weighted average of two in- different bundles is strictly preferred to the two extreme bundles. Convex preferences may have flat spots, while strictly convex preferences must have indifferences curves that are “rounded.” The preferences for two goods that are perfect substitutes are convex, but not strictly convex. 3.6 The Marginal Rate of Substitution We will often find it useful to refer to the slope of an indifference curve at a particular point. This idea is so useful that it even has a name: the slope of an indifference curve is known as the marginal rate of substitution (MRS). The name comes from the fact that the MRS measures the rate at which the consumer is just Willing to substitute one good for the other. Suppose that we take a little of good 1, Am, away from the consumer. Then we give him A932, an amount that is just sufficient to put him back on his indifference curve, so that he is just as well off after this substitution of :62 for .131 as he was before. We think of the ratio Ana/A231 as being the rate at which the consumer is willing to substitute good 2 for good 1. Now think of A561 as being a very small chang%a marginal change. Then the rate A122 /A:v1 measures the marginal rate of substitution of good 2 for good 1. As Aml gets smaller, Awg / A331 approaches the slope of the indifference curve, as can be seen in Figure 3.11. When we write the ratio A332 /Aw1, we will always think of both the numerator and the denominator as being small numbers—as describing marginal changes from the original consumption bundle. Thus the ratio defining the MRS will always describe the slope of the indifference curve: the rate at which the consumer is just willing to substitute a little more consumption of good 2 for a little less consumption of good 1. One slightly confusing thing about the MRS is that it is typically a negative number. We’ve already seen that monotonic preferences imply that indifference curves must have a negative slope. Since the MRS is the numerical measure of the slope of an indifference curve, it will naturally be a negative number. THE MARCINAL RATE OF SUBSTITUTION 49 cup, 5i 5am aul'J, auuis "duuls' smywfiau 'e aA’eq 18min SSAIIK) souaieyrpur 21qu 1(1er SSDUSJSJQJd oiuoqouom seq: uses Apeen'e eA‘eM 'iequmu separatism e Knead/C1 st 1; rem s; SHIN sq; snoqe Birqu Buisnguoo Kuqfins 9110 '1 p003 JO uorqdumsuoo 3891 311ng e 10} 5 p003 Jo nondumsuoo QJOTII QT’I’HT 19. 9’! “VICan nn fi'lT'I'TTTM ’IQTIF Q1 TQITThQTIn" QTY! TT'u’rrlm rvn r\rvn-r firm Indifference - CUWB ' ' Ax . . Slope = —2 == marginal rate ' - M! of substitution XI The marginal rate of substitution (MRS). The marginal rate of substitution measures the slope of the indifference curve; I ' The marginal rate of substitution measures an interesting aspect of the consumer’s behavior. Suppose that the consumer has well-behaved prefer- ences, that is, preferences that are monotonic and convex, and that he is currently consuming some bundle (331422). We now will offer him a trade: he can exchange good 1 for 2, or good 2 for 1, in any amount at a “rate of exchange” of E. That is, if the consumer gives up A301 units of good 1, he can get EAsrl units of good 2 in exchange. Or, conversely, if he gives up A132 units of good 2, he can get Amg / E units of good 1. Geometrically, we are ofiering the consumer an opportunity to move to any point along a line with slope HE that passes through (331, 3:2), as depicted in Figure 3.12. Moving up and to the left from ($1,252) involves exchanging good 1 for good 2, and moving down and to the right involves exchanging good 2 for good 1. In either movement, the exchange rate is E. Since exchange always involves giving up one good in exchange for another, the exchange rate E corresponds to a slope of —E. We can now ask what would the rate of exchange have to be in order for the consumer to want to stay put at ($1,:v2)? To answer this question, we simply note that any time the exchange line crosses the indifference curve, there will be some points on that line that are preferred to ($1,:32)—that lie above the indifference curve. Thus, if there is to be no movement from Figure 3.12 50 PREFERENCES (Ch. 3) (:01, $2), the exchange line must be tangent to the indifference curve. That is, the slope of the exchange line, —E, must be the slope of the indifference curve at (331,532). At any other rate of exchange, the exchange line would cut the indifference curve and thus allow the consumer to move to a more preferred point. Indifference curves x1 X1 Trading at an exchange rate. Here we are allowing the con- sumer to trade the goods at an exchange rate E, which implies that the consumer can move along a line with slope —E. W Thus the slope of the indifference curve, the marginal rate of substitution, measures the rate at which the consumer is just on the margin of trading or not trading. At any rate of exchange other than the MRS, the consumer would want to trade one good for the other. But if the rate of exchange equals the MRS, the consumer wants to stay put. 3.7 Other Interpretations of the MRS We have said that the MRS measures the rate at which the consumer is just on the margin of being willing to substitute good 1 for good 2. We could also say that the consumer is just on the margin of being willing to “pay” some of good 1 in order to buy some more of good 2. So sometimes BEHAVIOR OF THE MRS 51 you hear people say that the slope of the indifference curve measures the marginal willingness to pay. If good 2 represents the consumption of “all other goods,” and it is measured in dollars that you can spend on other goods, then the marginal- willingness-to-pay interpretation is very natural. The marginal rate of sub- stitution of good 2 for good 1 is how many dollars you would just be willing to give up spending on other goods in order to consume a little bit more of good 1. Thus the MRS measures the marginal willingness to give up dollars in order to consume a small amount more of good 1. But giving up those dollars is just like paying dollars in order to consume a little more of good 1. If you use the marginal—willingness—to—pay interpretation of the MRS, you should be careful to emphasize both the “marginal” and the “willingness” aspects. The MRS measures the amount of good 2 that one is willing to pay for a marginal amount of extra consumption of good 1. How much you actually have to pay for some given amount of extra consumption may be different than the amount you are willing to pay How much you have to pay will depend on the price of the good in question. How much you are willing to pay doesn’t depend on the priceiit is determined by your preferences. Similarly, how much you may be willing to pay for a large change in consumption may be different from how much you are willing to pay for a marginal change. How much you actually end up buying of a good will depend on your preferences for that good and the prices that you face. How much you would be Willing to pay for a small amount extra of the good is a feature only of your preferences. 3.8 Behavior of the MRS It is sometimes useful to describe the shapes of indifference curves by de— scribing the behavior of the marginal rate of substitution. For example, the “perfect substitutes” indifference curves are characterized by the fact that the MRS is constant at —1. The “neutrals” case is characterized by the fact that the MRS is everywhere infinite. The preferences for “perfect complements” are characterized by the fact that the MRS is either zero or infinity, and nothing in between. We’ve already pointed out that the assumption of monotonicity implies that indifference curves must have a negative slope, so the MRS always involves reducing the consumption of one good in order to get more of another for monotonic preferences. The case of convex indifference curves exhibits yet another kind of be- havior for the MRS. For strictly convex indifference curves, the MRS—the slope of the indifference curveiedecreases (in absolute value) as we increase 911. Thus the indifference curves exhibit a diminishing marginal rate of 52 PREFERENCES (Ch. 3) substitution. This means that the amount of good 1 that the person is willing to give up for an additional amount of good 2 increases the amount of good 1 increases. Stated in this way, convexity of indifference curves seems very natural: it says that the more you have of one good, the more Willing you are to give some of it up in exchange for the other good. (But remember the ice cream and olives exampleror some pairs of goods this assumption might not hold!) Summary 1. Economists assume that a consumer can rank various consumption pos- sibilities. The way in which the consumer ranks the consumption bundles describes the consumer’s preferences. 2. Indifference curves can be used to depict different kinds of preferences. 3. Well-behaved preferences are monotonic (meaning more is better) and convex (meaning averages are preferred to extremes). 4. The marginal rate of substitution (MRS) measures the slope of the in- difference curve. This can be interpreted as how much the consumer is willing to give up of good 2 to acquire more of good 1. REVIEW QUESTIONS 1. If we observe a consumer choosing (9:1, 332) when (y1,y2) is available one time, are we justified in concluding that ($1,332) >— (y1,y2)? 2. Consider a group of people A, B, C and the relation “at least as tall as,” as in “A is at least as tall as B.” Is this relation transitive? Is it complete? 3. Take the same group of people and consider the relation “strictly taller than.” Is this relation transitive? Is it reflexive? Is it complete? 4. A college football coach says that given any two linemen A and B, he always prefers the one who is bigger and faster. Is this preference relation transitive? Is it complete? 5. Can an indifference curve cross itself? For example, could Figure 3.2 depict a single indifference curve? 6. Could Figure 3.2 be a single indifference curve if preferences are mono- tonic? REVIEW QUESTIONS 53 7. If both pepperoni and anchovies are bads, will the indifference curve have a positive or a negative slope? 8. Explain why convex preferences means that “averages are preferred to extremes.” 9. What is your marginal rate of substitution of $1 bills for $5 bills? 10. If good 1 is a “neutral,” what is its marginal rate of substitution for good 2? 11. Think of some other goods for Which your preferences might be concave. ...
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This note was uploaded on 09/06/2010 for the course FBE ECON2113 taught by Professor Franchsica during the Fall '09 term at HKU.

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

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