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5 Choice - CHAPTER 5 CHOICE In this chapter we will put...

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Unformatted text preview: CHAPTER 5 CHOICE In this chapter we will put together the budget set and the theory of prefer- ences in order to examine the optimal choice of consumers. We said earlier that the economic model of consumer choice is that people choose the best bundle they can afford. We can now rephrase this in terms that sound more professional by saying that “consumers choose the most preferred bundle from their budget sets.” 5.1 Optimal Choice A typical case is illustrated in Figure 5.1. Here we have drawn the budget set and several of the consumer’s indifference curves on the same diagram. We want to find the bundle in the budget set that is on the highest indif- ference curve. Since preferences are well~behaved, so that more is preferred to less, we can restrict our attention to bundles of goods that lie on the budget line and not worry about those beneath the budget line. Now simply start at the right-hand corner of the budget line and move to the left. As we move along the budget line we note that we are moving to higher and higher indifference curves. We stop when we get to the highest 74 CHOICE (Ch. 5) indifference curve that just touches the budget line. In the diagram, the bundle of goods that is associated with the highest indifference curve that just touches the budget line is labeled (m’f,m§). The choice ($17323) is an optimal choice for the consumer. The set of bundles that she prefers to (sc’f,a:§)—the set of bundles above her indif— ference curveidoesn’t intersect the bundles she can afford—the bundles beneath her budget line. Thus the bundle (23339;) is the best bundle that the consumer can afford. Indifference curves ' Optimal choice. The'eptimfl censumptie'n position is where the indifference curve is tangent to the budget line, Note an important feature of this optimal bundle: at this choice, the indifference curve is tangent to the budget line. If you .think about it a moment you’ll see that this has to be the case: if the indifference curve weren’t tangent, it would cross the budget line7 and if it crossed the budget line, there would be some nearby point on the budget line that lies above the indifference curve—which means that we couldn’t have started at an optimal bundle. OPTIMAL CHOICE 75 Does this tangency condition really have to hold at an optimal choice? Well, it doesn’t hold in all cases, but it does hold for most interesting cases. What is always true is that at the optimal point the indifference curve can’t cross the budget line. So when does “not crossing” imply tangent? Let’s look at the exceptions first. First, the indifference curve might not have a tangent line, as in Fig— ure 5.2. Here the indifference curve has a kink at the optimal choice, and a tangent just isn’t defined, since the mathematical definition of a tangent requires that there be a unique tangent line at each point. This case doesn’t have much economic significancelit is more of a nuisance than anything else. Indifference curves xf - ' X1 Kinky tastes. Here is an optimal'consumption bundle where the indifference curve deesn’t have a tangent. ' The second exception is more interesting. Suppose that the optimal point occurs Where the consumption of some good is zero as in Figure 5.3. Then the slope of the indifference curve and the slope of the budget line are different, but the indifference curve still doesn’t cross the budget line. 76 CHOICE (Ch. 5) We say that Figure 5.3 represents a boundary optimum, while a case like Figure 5.1 represents an interior optimum. If we are willing to rule out “kinky tastes” we can forget about the example given in Figure 5.2.1 And if we are willing to restrict ourselves only to interior optima, we can rule out the other example. If we have an interior optimum with smooth indifference curves, the slope of the indifference curve and the slope of the budget line must be the same . . . because if they were different the indifference curve would cross the budget line, and we couldn’t be at the optimal point. indifference curves Xi“ 'I H " I X1 Boundary optimum. The? Optimal consumption involves con- suming Zero units of good 2.] The: indifference curveis not tan- gent to the budget3line.- ' ' ' We’ve found a necessary condition that the optimal choice must satisfy. If the optimal choice involves consuming some of both goodsflso that it is an interior optimumithen necessarily the indifference curve will be tangent to the budget line. But is the tangency condition a sufi‘icient condition for a bundle to be optimal? If we find a bundle where the indifference curve is tangent to the budget line, can we be sure we have an optimal choice? Look at Figure 5.4. Here we have three bundles where the tangency condition is satisfied, all of them interior, but only two of them are optimal. 1 Otherwise, this book might get an R rating. OPTIMAL CHOlCE 77 So in general, the tangency condition is only a necessary condition for optimality, not a sufficient condition. indifference curves ~ Nonoptimai I bundle Budget line x1 More than one .tangency. Here there are three tangencies, but only two optimal points, so the tangency condition is nec» essary but not sufficient. - ' - . . However, there is one important case Where it is sufficient: the case of convex preferences. In the case of convex preferences, any point that satisfies the tangency condition must be an optimal point. This is clear geometrically: since convex indifference curves must curve away from the budget line, they can’t bend back to touch it again. Figure 5.4 also shows us that in general there may be more than one optimal bundle that satisfies the tangency condition. However, again con« vexity implies a restriction. If the indifference curves are strictly convex— they don’t have any fiat spotsrthen there will be only one optimal choice on each budget line. Although this can be shown mathematically, it is also quite plausible from looking at the figure. The condition that the MRS must equal the slope of the budget line at an interior optimum is obvious graphically, but what does it mean econom— ically? Recall that one of our interpretations of the MRS is that it is that rate of exchange at which the consumer is just willing to stay put. Well, the market is offering a rate of exchange to the consumer of —p1/p2#if 78 CHOICE (Ch. 5) you give up one unit of good 1, you can buy pl/pg units of good 2. If the consumer is at a consumption bundle where he or she is willing to stay put, it must be one where the MRS is equal to this rate of exchange: MRS: 31. Another way to think about this is to imagine what would happen if the MRS were different from the price ratio. Suppose, for example, that the MRS is A532 / A331 2 ~1/2 and the price ratio is 1 / 1. Then this means the consumer is just willing to give up 2 units of good 1 in order to get 1 unit of good Qfibut the market is willing to exchange them on a one-to—one basis. Thus the consumer would certainly be willing to give up some of good 1 in order to purchase a little more of good 2. Whenever the MRS is different from the price ratio, the consumer cannot be at his or her optimal choice. 5.2 Consumer Demand The optimal choice of goods 1 and 2 at some set of prices and income is called the consumer’s demanded bundle. In general when prices and income change, the consumer’s optimal choice will change. The demand function is the function that relates the optimal choich-the quantities demandedkto the different values of prices and incomes. We will write the demand functions as depending on both prices and income: m1(p1,p2, m) and 932(101, p2, m). For each different set of prices and income, there will be a different combination of goods that is the optimal choice of the consumer. Different preferences will lead to different demand functions; we’ll see Some examples shortly. Our major goal in the next few chapters is to study the behavior of these demand functions—how the optimal choices change as prices and income change. 5.3 Some Examples Let us apply the model of consumer choice we have developed to the exam- ples of preferences described in Chapter 3. The basic procedure will be the same for each example: plot the indifference curves and budget line and find the point where the highest indifference curve touches the budget line. Perfect Substitutes The case of perfect substitutes is illustrated in Figure 5.5. We have three possible cases. If p2 > 191, then the slope of the budget line is flatter than the slope of the indifference curves. In this case, the optimal bundle is SOME EXAMPLES 79 where the consumer spends all of his or her money on good 1. If p1 > p2, then the consumer purchases only good 2. Finally, if p1 : p2, there is a whole range of optimal choices—rany amount of goods 1 and 2 that satisfies the budget constraint is optimal in this case. Thus the demand function for good 1 will be m/p1 when 291 < 102; 3:1 2 any number between 0 and m/p1 when m 2 pg; 0 when p1 > 192. Are these results consistent with common sense? All they say is that if two goods are perfect substitutes, then a consumer will purchase the cheaper one. If both goods have the same price, then the consumer doesn’t care which one he or she purchases. Indifference curves Slope : —1 Budget line Optimal choice xi" = m/p1 X1 Optimal choice with perfect substitutes. If the goods are perfect substitutes, the optimal choice will usually be on the boundary. Perfect Complements The case of perfect complements is illustrated in Figure 5.6. Note that the optimal choice must always lie on the diagonal, Where the consumer is purchasing equal amounts of both goods, no matter what the prices are. 80 CHOICE (Ch. 5) In terms of our example, this says that people with two feet buy shoes in - 2 pa1rs. Let us solve for the optimal choice algebraically. We know that this consumer is purchasing the same amount of good 1 and good 2, no matter what the prices. Let this amount be denoted by 3:. Then we have to satisfy the budget constraint p12: + 39255 = m. Solving for :U gives us the optimal choices of goods 1 and 2: m 1014—192. $1I$2=$= The demand function for the optimal choice here is quite intuitive. Since the two goods are always consumed together, it is just as if the consumer were spending all of her money on a single good that had a price of p1 +192. Indifference curves Optimal Choice Budget line X1* X1 Optimal choice with perfect complements. If the goods are perfect complements, the quantities demanded will always lie on the diagonal since the optimal choice occurs where 931 equals 932. .) . . . _ . * l)()ll l V\‘()I‘l‘_\-‘. we 11 got some more (‘xmtmg results later on. SOME EXAMPLES 81 Optimal choice Budget line \ /I( Budget line \ \ ““0 Optimal choice t y‘ ‘1 i 2 3 x1 1 2 3 X1 A Zero units demanded B 1 unit demanded Discrete goods. In panel A the demand for good 1 is zero, while in panel B one unit will be demanded. W Neutrals and Bads In the case of a neutral good the consumer spends all of her money on the good she likes and doesn’t purchase any of the neutral good. The same thing happens if one commodity is a bad. Thus, if commodity 1 is a good and commodity 2 is a bad, then the demand functions will be m 1-1:; :01 232:0. Discrete Goods Suppose that good 1 is a discrete good that is available only in integer units, While good 2 is money to be spent on everything else. If the con~ sumer chooses 1,2,3, - .. units of good 1, she will implicitly choose the consumption bundles (1, m mp1), (2, m — 2131), (3, m — 3191), and so on. We can simply compare the utility of each of these bundles to see which has the highest utility. Alternatively, we can use the indifference~curve analysis in Figure 5.7. As usual, the optimal bundle is the one on the highest indifference “curve.” If the price of good 1 is very high, then the consumer will choose zero units of consumption; as the price decreases the consumer will find it optimal to consume 1 unit of the good. Typically, as the price decreases further the consumer will choose to consume more units of good 1. Figure l" O. 7 82 CHOICE (Ch. 5) Concave Preferences Consider the situation illustrated in Figure 5.8. Is X the optimal choice? No! The optimal choice for these preferences is always going to be a bound— ary choice, like bundle Z. Think of what nonconvex preferences mean. If you have money to purchase ice cream and olives, and you don’t like to consume them together, you’ll spend all of your money on one or the other. Indifference curves Nonoptimal Choice '\x Optimal choice Optimal choice with concave preferences. The optimal choice is the boundary point, Z, not the interior 'tangency point, X, because Z lies on a higher indifference curve. Cobb—Douglas Preferences Suppose that the utility function is of the Cobb-Douglas form, u(:1:1,:c2) 2 flag. In the Appendix to this chapter we use calculus to derive the optimal ESTIMATING UTILITY FUNCTIONS 83 choices for this utility function. They turn out to be _ c m x1_c+dp; _ d m m2_c+dp-2_' These demand functions are often useful in algebraic examples, so you should probably memorize them. The Cobb-Douglas preferences have a convenient property. Consider the fraction of his income that a Cobb-Douglas consumer spends on good 1. If he consumes 51:1 units of good 1, this costs him p131, so this represents a fraction p123; /m of total income. Substituting the demand function for $1 wehave p1331_1_9£ c m_ c m mc+de1—c+d' Similarly the fraction of his income that the consumer spends on good 2 is d/ (c + d). Thus the Cobb-Douglas consumer always spends a fixed fraction of his income on each good. The size of the fraction is determined by the exponent in the Cobb-Douglas function. This is Why it is often convenient to choose a representation of the Cobb- Douglas utility function in which the exponents sum to 1. If u($1,m2) = $31552”, then we can immediately interpret a as the fraction of income spent on good 1. For this reason we will usually write Cobb—Douglas preferences in this form. 5.4 Estimating Utility Functions We’ve now seen several different forms for preferences and utility functions and have examined the kinds of demand behavior generated by these pref— erences. But in real life we usually have to work the other way around: we observe demand behavior, but our problem is to determine what kind of preferences generated the observed behavior. For example, suppose that we observe a consumer’s choices at several different prices and income levels. An example is depicted in Table 5.1. This is a table of the demand for two goods at the different levels of prices and incomes that prevailed in different years. We have also computed the share of income spent on each good in each year using the formulas 51 2 plan/m and 52 : pgzrg/m. For these data, the expenditure shares are relatively constant. There are small variations from observation to observation, but they probably aren’t large enough to worry about. The average expenditure share for good 1 is about 1/4, and the average income share for good 2 is about 3 / 4. It appears 84 CHOICE (Ch. 5) Some data describing consumption behavior. .25 .75 25 150 100 75 24 304 i a that a utility function of the form u(:c1,:1:2) = mf :63 seems to fit these data pretty well. That is, a utility function of this form would generate choice behavior that is pretty close to the observed choice behavior. For convenience we have calculated the utility associated with each observation using this estimated Cobb-Douglas utility function. As far as we can tell from the observed behavior it appears as though the consumer is maximizing the function u(a:1, $2) = 2317‘ $21 . It may well be that further observations on the consumer’s behavior would lead us to reject this hypothesis. But based on the data we have, the fit to the optimizing model is pretty good. This has very important implications, since we can now use this “fitted” utility function to evaluate the impact of proposed policy changes. Suppose, for example, that the government was contemplating imposing a system of taxes that would result in this consumer facing prices (2, 3) and having an income of 200. According to our estimates, the demanded bundle at these prices would be 1200 =——=25 x1 4 2 3200 =——:50. 3” 4 3 The estimated utility of this bundle is 1 3 u(2:1,:c2) = 25350Z z 42. This means that the new tax policy would make the consumer better off than he was in year 2, but worse off than he was in year 3. Thus we can use the observed choice behavior to value the implications of proposed policy changes on this consumer. Since this is such an important idea in economics, let us review the logic one more time. Given some observations on choice behavior, we try to determine what, if anything, is being maximized. Once we have an estimate of what it is that is being maximized, we can use this both to lMPLlCATlONS OF THE MRS CONDlTlON 85 predict choice behavior in new situations and to evaluate proposed changes in the economic environment. Of course we have described a very simple situation. In reality, we nor- mally don’t have detailed data on individual consumption choices. But we often have data on groups of individuals~teenagers, middle«class house— holds, elderly people, and so on. These groups may have different prefer— ences for different goods that are reflected in their patterns of consumption expenditure. We can estimate a utility function that describes their con— sumption patterns and then use this estimated utility function to forecast demand and evaluate policy proposals. In the simple example described above, it was apparent that income shares were relatively constant so that the Cobb—Douglas utility function would give us a pretty good fit. In other cases, a more complicated form for the utility function would be appropriate. The calculations may then become messier, and we may need to use a computer for the estimation, but the essential idea of the procedure is the same. 5.5 Implications of the MRS Condition In the last section we examined the important idea that observation of de— mand behavior tells us important things about the underlying preferences of the consumers that generated that behavior. Given sufficient observa— tions on consumer choices it will often be possible to estimate the utility function that generated those choices. But even observing one consumer choice at one set of prices will allow us to make some kinds of useful inferences about how consumer utility will change when consumption changes. Let us see how this works. In well-organized markets, it is typical that everyone faces roughly the same prices for goods. Take, for example, two goods like butter and milk. If everyone faces the same prices for butter and milk, and everyone is optimizing, and everyone is at an interior solution . . . then everyone must have the same marginal rate of substitution for butter and milk. This follows directly from the analysis given above. The market is offer- ing everyone the same rate of exchange for butter and milk, and everyone is adjusting their consumption of the goods until their own “internal” mar- ginal valuation of the two goods equals the market’s “external” valuation of the two goods. Now the interesting thing about this statement is that it is independent of income and tastes. People may value their total consumption of the two goods very differently. Some people may be consuming a lot of butter and a little milk, and some may be doing the reverse. Some wealthy people may be consuming a lot of milk and a lot of butter while other people may be consuming just a little of each good. But everyone who is consuming the two goods must have the same marginal rate of substitution. Everyone 86 CHOICE (Ch. 5) who is consuming the goods must agree on how much one is worth in terms of the other: how much of one they would be willing to sacrifice to get some more of the other. The fact that price ratios measure marginal rates of substitution is very important, for it m...
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