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\ufffd MUy(\ufffdy), which can be rewritten as MUy(\ufffdy) \ufffd MUx(\ufffdx). We can now solve for the slope of the indifference curve : Finally, since MRSx, y is the negative of the slope of the indifference curve, we observe that (3.5) ¢y ¢x 2 holding utility constant \ufffd MUx MUy \ufffd MRSx, y ¢y ¢x 2 holding utility constant \ufffd MUx MUy ¢y\ufffd¢x ¢U \ufffd MUx (¢x) \ufffd MUy(¢y) Food and Drug Administration to add graphic warning labels similar to those used in other coun- tries. Studies by economists have found that such warnings and advertising restrictions can have sig- nificant negative impacts on consumer demand for cigarettes. In June 2009 the Family Smoking Prevention and Tobacco Control Act was enacted in the United States. It bans promotions and advertising believed to be focused on youth. It also requires that the top half of cigarette packs, front and back, have stern health warnings. Within two years the law requires the 4You may recognize that this equation is an approximation of the change in utility that results from changing x and y by \ufffdx and \ufffdy, respectively. The approximation becomes more accurate when \ufffdx and \ufffdy are small because the marginal utilities will be approximately constant for small changes in x and y. c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 88 3.2 UTILITY FUNCTIONS 89 Diminishing Marginal Rate of Substitution For many (but not all) goods, MRSx, y diminishes as the amount of x increases along an indifference curve. To see why, refer to Figure 3.9. At basket A, to get 1 more hamburger, Eric would be willing to forgo as many as 5 glasses of lemonade. And this makes sense because at basket A Eric is drinking much lemonade and eating only a few hamburgers. So we might expect MRSx,y to be large. However, if Eric were to move to basket D, where he is consuming more hamburgers and less lemonade, he might not be willing to give up as many glasses of lemonade to get still another hamburger. Thus, his MRSx,y will be lower at D than at A. We have already shown that Eric\u2019s MRSx,y at basket D is 2, which is lower than his MRSx,y at basket A. In this case Eric\u2019s preferences exhibit a diminishing marginal rate of substitution of x for y. In other words, the marginal rate of substitu- tion of x for y declines as Eric increases his consumption of x along an indifference curve. What does a diminishing marginal rate of substitution of x for y imply about the shape of the indifference curves? Remember that the marginal rate of substitution of x for y is just the negative of the slope of the indifference curve on a graph with x on the horizontal axis and y on the vertical axis. If MRSx,y diminishes as the consumer in- creases x along an indifference curve, then the slope of the indifference curve must be getting flatter (less negative) as x increases. Therefore, indifference curves with di- minishing MRSx,y must be bowed in toward the origin, as in Figure 3.9. diminishing marginal rate of substitution A feature of consumer pref- erences for which the mar- ginal rate of substitution of one good for another good diminishes as the consump- tion of the first good in- creases along an indiffer- ence curve. when companies attempt to forecast the potential market for a new product. Nestor Arguea, Cheng Hsiao, and Grant Taylor (AHT) used data on prices in the U.S. automobile mar- ket to estimate what are known as hedonic prices for automobile attributes.5 A discussion of hedonic prices is the stuff of an advanced econometrics course, so we won\u2019t go into the details of AHT\u2019s methods here. Roughly speaking, a hedonic price is a measure of the marginal utility of a particular attribute. Given this, the ratio of hedonic prices for two different automo- bile attributes, such as horsepower and gas mileage, represents the marginal rate of substitution between these attributes for the typical automobile consumer. Based on AHT\u2019s estimates, the marginal rate of substitution of gas mileage for horsepower for a typ- ical U.S. auto consumer in 1969 was 3.79. This means that the typical consumer would be willing to forgo 3.79 horsepower to get an additional one mile per gallon in gas mileage. Between 1969 and 1986 the marginal rate of substitution of gas mileage for horse- power gradually fell, reaching 0.71 by 1986. We began this chapter by discussing one of the choices you would face as you decide whether to buy an auto- mobile, the level of fuel efficiency. But you will proba- bly also care about other attributes of the car you might buy. Should it be big or small? Should it have a big engine and lots of horsepower, or should it have a smaller engine and thus get better gas mileage? In other words, when you buy a car you are really buying a bundle of attributes. Just as we can build a theory of consumer choice among different goods by means of a utility function defined over those goods, we can also build a model of consumer choice among different varieties of the same good (such as automo- biles) by means of a utility function defined over the attributes of this good. For example, the satisfaction that consumers would derive from different brands of cars could be described by a utility function over horse- power, gas mileage, luggage space, and so forth. Market researchers often use this attribute-based approach A P P L I C A T I O N 3.2 How People Buy Cars: The Importance of Attributes 5N. M. Arguea, C. Hsiao, and G. A. Taylor, \u201cEstimating Consumer Preferences Using Market Data\u2014An Application to U.S. Automobile Demand,\u201d Journal of Applied Econometrics 9 (1994): 1\u201318. c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 89 90 CHAPTER 3 CONSUMER PREFERENCES AND THE CONCEPT OF UTIL ITY ding indifference curve) to another, and at these bun- dles the marginal rates of substitution may differ. The key point of this example is that marginal rate of substitution is more than just a theoretical concept. It can be estimated and used to help us un- derstand the trade-offs that consumers are willing to make between products and product attributes. This decline in the marginal rate of substitution of gas mileage for horsepower could reflect changes in consumer tastes, or it could also reflect simultane- ous changes in automobile prices, gasoline prices, and consumer incomes. As we will see in the next chapter, when changes in prices and income occur, consumers move from one consumption bundle (and correspon- Suppose a consumer has preferences between two goods that can be represented by the util- ity function U \ufffd xy. For this utility function, MUx \ufffd y and MUy \ufffd x.6 Problem (a) On a graph, draw the indifference curve associated with the utility level U1 \ufffd 128. Then answer the follow- ing questions: 1. Does the indifference curve intersect either axis? 2. Does the shape of the indifference curve indicate that MRSx, y is diminishing? Indifference Curves with Diminishing MRSx,y E S D L E A R N I N G - B Y- D O I N G E X E R C I S E 3 . 3 (b) On the same graph draw a second indifference curve, U2 \ufffd 200. Show how MRSx, y depends on x and y, and use this information to determine if MRSx, y is diminish- ing for this utility function. Solution (a) To draw the indifference curve U1 \ufffd 128 for the utility function U \ufffd xy, we plot points where xy \ufffd 128\u2014for example, point G (x \ufffd 8, y \ufffd 16), point H (x \ufffd 16, y \ufffd 8), and point I (x \ufffd 32, y \ufffd 4)\u2014and then connect these points with a smooth line. Figure 3.11 shows this indif- ference curve. 6To see how these marginal utilities can be derived from the utility function, you would use the calculus techniques illustrated in Learning-By-Doing Exercise A.7 in the Mathematical Appendix. FIGURE 3.11 Indifference Curves with Diminishing MRSx,y The indifference curves on this graph are for the utility function U \ufffd xy,