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In reality this assumption is not always true. Let\u2019s return to the example of con- suming hamburgers. Sarah may find that her total utility increases as she eats the first, second, and third hamburgers each week. For these hamburgers, her marginal utility B¿ ¢U\ufffd¢y \ufffd 0.25 ¢U\ufffd¢y ¢U\ufffd¢y principle of diminish- ing marginal utility The principle that after some point, as consump- tion of a good increases, the marginal utility of that good will begin to fall. c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 79 80 CHAPTER 3 CONSUMER PREFERENCES AND THE CONCEPT OF UTIL ITY is positive, even though it may be diminishing with each additional hamburger she eats. But presumably at some point she will find that an additional hamburger will bring her no more satisfaction. For example, she might find that the marginal utility of the seventh hamburger per week is zero, and the marginal utility of the eighth or ninth hamburgers might even be negative. Figure 3.3 depicts the total and marginal utility curves for this case. Initially (for values of y \ufffd 7 hamburgers), total utility rises as consumption increases, and the slope of the utility curve is positive (e.g., note that the segment RS, which is tangent to the utility curve at point A when Sarah is purchasing her second hamburger, has a posi- tive slope); thus, the marginal utility is positive (as depicted at point A\ufffd). However, the marginal utility is diminishing as consumption increases, and at a consumption level of seven hamburgers, Sarah has purchased so much of the good that the marginal util- ity is zero (point B\ufffd). Since the marginal utility is zero, the slope of the total utility curve is zero. (The segment MN, which is tangent to the utility curve at point B, has a slope of zero.) If Sarah were to buy more than seven hamburgers, her total satisfaction U (y) , to ta l u til ity o f h am bu rg er s y, weekly consumption of hamburgers 0 2 7 9 S U = ( y ) R A B M C N K L M U y, m a rg in al u tili ty o f h am bu rg er s y, weekly consumption of hamburgers 0 2 7 9 MUy A' B' C' (a) (b) FIGURE 3.3 Marginal Utility May Be Negative The utility curve U(y) is shown in panel (a), and the corresponding marginal utility curve is illustrated in panel (b). The slope of the utility curve in the top panel is positive at A; thus, the marginal utility is positive, as indicated at point A\ufffd in panel (b). At point B the slope of the utility curve is zero, meaning that the marginal utility is zero, as shown at point B\ufffd. At point C the slope of the utility function is negative; therefore, the marginal utility is negative, as indicated at point C \ufffd. c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 80 3.2 UTILITY FUNCTIONS 81 would decline (e.g., the slope of the total utility curve at point C is negative, and thus the marginal utility is negative, as indicated at point C\ufffd). Although more may not always be better, it is nevertheless reasonable to assume that more is better for amounts of a good that a consumer might actually purchase. For example, in Figure 3.3 we would normally only need to draw the utility function for the first seven hamburgers. The consumer would never consider buying more than seven hamburgers because it would make no sense for her to spend money on ham- burgers that reduce her satisfaction. PREFERENCES WITH MULTIPLE GOODS: MARGINAL UTILITY, INDIFFERENCE CURVES, AND THE MARGINAL RATE OF SUBSTITUTION Let\u2019s look at how the concepts of total utility and marginal utility might apply to a more realistic scenario. In real life, consumers can choose among myriad goods and services. To study the trade-offs a consumer must make in choosing his optimal bas- ket, we must examine the nature of consumer utility with multiple products. We can illustrate many of the most important aspects of consumer choice among multiple products with a relatively simple scenario in which a consumer, Brandon, must decide how much food and how much clothing to purchase in a given month. Let x measure the number of units of food and y measure the number of units of clothing purchased each month. Further, suppose that Brandon\u2019s utility for any basket (x, y) is measured by A graph of this consumer\u2019s utility function is shown in Figure 3.4. Because we now have two goods, a graph of Brandon\u2019s utility function must have three axes. In Figure 3.4 the number of units of food consumed, x, is shown on the right axis, and the number of units of cloth- ing consumed, y, is represented on the left axis. The vertical axis measures Brandon\u2019s level of satisfaction from purchasing any basket of goods. For example, U \ufffd 1xy. FIGURE 3.4 Graph of the Utility Function U \ufffd The level of utility is shown on the vertical axis, and the amounts of food (x) and clothing (y) are shown, respectively, on the right and left axes. Contours representing lines of constant utility are also shown. For example, the consumer is indifferent between baskets A, B, and C because they all yield the same level of utility (U \ufffd 4). 1xyA B C U = 2 2 2 2 4 4 4 6 6 6 8 8 8 10 10 10 12 12 12 0 U = 10 U = 8 U = 6 U = 4 x, units of fo od y , units of clothing U , le ve l o f u tili ty c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 81 82 CHAPTER 3 CONSUMER PREFERENCES AND THE CONCEPT OF UTIL ITY basket A contains two units of food (x \ufffd 2) and eight units of clothing ( y \ufffd 8). Thus, Brandon realizes a level of utility of with basket A. As the graph indicates, Brandon can achieve the same level of utility by choosing other bas- kets, such as basket B and basket C. The concept of marginal utility is easily extended to the case of multiple goods. The marginal utility of any one good is the rate at which total utility changes as the level of consumption of that good rises, holding constant the levels of consumption of all other goods. For example, in the case in which only two goods are consumed and the utility function is U(x, y), the marginal utility of food (MUx) measures how the level of satisfaction will change (\ufffdU ) in response to a change in the consumption of food (\ufffdx), holding the level of y constant: (3.2) Similarly, the marginal utility of clothing (MUy) measures how the level of satis- faction will change (\ufffdU ) in response to a small change in the consumption of clothing (\ufffdy), holding constant the level of food (x): (3.3) One could use equations (3.2) and (3.3) to derive the algebraic expressions for MUx and MUy from U(x, y).3 When the total utility from consuming a bundle (x, y) is the marginal utilities are and So, at basket A (with x \ufffd 2 and y \ufffd 8), and Learning-By-Doing Exercise 3.1 shows that the utility function satisfies the assumptions that more is better and that marginal utilities are diminishing. Because these are widely regarded as reasonable characteristics of consumer preferences, we will often use this utility function to illustrate concepts in the theory of consumer choice. U \ufffd 1xy MUy \ufffd 12\ufffd(218) \ufffd 1\ufffd4.MUx \ufffd 18\ufffd(212) \ufffd 1 MUy \ufffd 1x\ufffd(21y).MUx \ufffd 1y\ufffd(21x) U \ufffd 1xy, MUy \ufffd ¢U ¢y 2 x is held constant MUx \ufffd ¢U ¢x 2 y is held constant U \ufffd 1(2)(8) \ufffd 4 3Learning-By-Doing Exercise A.7 in the Mathematical Appendix shows how to derive the equations of MUx and MUy in this case. Let\u2019s look at a utility function that satisfies the assumptions that more is better and that marginal utilities are diminishing. Suppose a consumer\u2019s prefer- ences between food and clothing can be represented by the utility function where x measures the number of units of food and y the number of units of clothing, and the marginal utilities for x and y