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the worst tasting of the five products. The bad news is that this result was not statistically significant! These kinds of studies do not suggest that all consumers regard all beer, wine, or paté style prod- ucts to be perfect substitutes. However, when a con- sumer does not have a strong preference for one brand over another, the marginal rate of substitution of brand A for brand B might be nearly constant, and probably near 1, since a consumer would probably be willing to give up one unit of one brand for one unit of another. A recent study of wine drinkers came to a similar conclusion.8 The food and wine publishing firm Fearless Critic Media organized 17 blind tastings of wine by 506 participants in 2007\u20132008. Wines ranged from $1.65 to $150 per bottle. Tasters were asked to as- sign a rating to each wine. The data were then statisti- cally analyzed by economists. They found a small and negative correlation between price and rated quality. They did find a positive correlation between price and quality among tasters with wine training, but the cor- relation was small and had low statistical significance. Two members of that research team recently col- laborated on a similar study that is perhaps a bit more troubling than the wine research.9 Noting that canned dog food and paté are both made at least partially from small pieces of ground meat, they studied whether (human) tasters could distinguish the two products in a 8R. Goldstein et al., \u201cDo More Expensive Wines Taste Better? Evidence from a Large Sample of Blind Tastings,\u201d Journal of Wine Economics (Spring 2008). 9J. Bohannon et al., \u201cCan People Distinguish Paté from Dog Food?\u201d American Association of Wine Economists\u2019 Working Paper #36, April 2009. 10Spam is an inexpensive canned food made out of precooked chopped pork and gelatin. FIGURE 3.13 Indifference Curves with Perfect Substitutes A consumer with the utility function U \ufffd P \ufffd 2W always views two pancakes as a perfect substitute for one waffle. , and so indifference curves are straight lines with a slope of .\ufffd1\ufffd2 MRSP,W \ufffd 1\ufffd2 U = 8 U = 4 P, pancakes W , w a ffl es Preference directions 4 2 4 8 PERFECT COMPLEMENTS In some cases, consumers might be completely unwilling to substitute one good for an- other. Consider a typical consumer\u2019s preferences for left shoes and right shoes, depicted in Figure 3.14. The consumer wants shoes to come in pairs, with exactly one left shoe for every right shoe. The consumer derives satisfaction from complete pairs of shoes, but gets no added utility from extra right shoes or extra left shoes. The indifference curves in this case comprise straight-line segments at right angles, as shown in Figure 3.14. c03consumerpreferencesandtheconceptofutility.qxd 7/14/10 12:07 PM Page 93 94 CHAPTER 3 CONSUMER PREFERENCES AND THE CONCEPT OF UTIL ITY The consumer with the preferences illustrated in Figure 3.14 regards left shoes and right shoes as perfect complements in consumption. Perfect complements are goods the consumer always wants in fixed proportion to each other; in this case, the desired proportion of left shoes to right shoes is 1:1.11 A utility function for perfect complements\u2014in this case, left shoes (L) and right shoes (R)\u2014is U(R, L) \ufffd 10min(R, L), where the notation \u201cmin\u201d means \u201ctake the min- imum value of the two numbers in parentheses.\u201d For example, at basket G, R \ufffd 2 and L \ufffd 2; so the minimum of R and L is 2, and U \ufffd 10(2) \ufffd 20. At basket H, R \ufffd 3 and L \ufffd 2; so the minimum of R and L is still 2, and U \ufffd 10(2) \ufffd 20. This shows that bas- kets G and H are on the same indifference curve, U2 (where U2 \ufffd 20). THE COBB\u2013DOUGLAS UTILITY FUNCTION The utility functions and U \ufffd xy are examples of the Cobb\u2013Douglas utility function. For two goods, the Cobb\u2013Douglas utility function is more generally repre- sented as U \ufffd Ax\ufffdy\ufffd, where A, \ufffd, and \ufffd are positive constants.12 The Cobb\u2013Douglas utility function has three properties that make it of interest in the study of consumer choice. \u2022 The marginal utilities are positive for both goods. The marginal utilities are MUx \ufffd \ufffdAx\ufffd 1y\ufffd and MUy \ufffd \ufffdAx\ufffdy\ufffd 1; thus, both MUx and MUy are positive when A, \ufffd, and \ufffd are positive constants. This means that \u201cthe more is better\u201d assumption is satisfied. \u2022 Since the marginal utilities are both positive, the indifference curves will be downward sloping. \u2022 The Cobb\u2013Douglas utility function also exhibits a diminishing marginal rate of substitution. The indifference curves will therefore be bowed in toward the U \ufffd 1xy G H U3, the utility from three pairs of shoes U2, the utility from two pairs of shoes U1, the utility from one pair of shoes R, Right shoes L, Le ft sh oe s 0 1 2 3 1 2 3 FIGURE 3.14 Indifference Curves with Perfect Complements The consumer wants exactly one left shoe for every right shoe. For example, his utility at basket G, with 2 left shoes and 2 right shoes, is not increased by moving to basket H, containing 2 left shoes and 3 right shoes. perfect complements (in consumption) Two goods that the consumer always wants to consume in fixed proportion to each other. 11The fixed-proportions utility function is sometimes called a Leontief utility function, after the economist Wassily Leontief, who employed fixed-proportion production functions to model relationships between sectors in a national economy. We shall examine Leontief production functions in Chapter 6. 12This type of function is named for Charles Cobb, a mathematician at Amherst College, and Paul Douglas, a professor of economics at the University of Chicago (and later a U.S. senator from Illinois). It has often been used to characterize production functions, as we shall see in Chapter 6 when we study the theory of production. The Cobb\u2013Douglas utility function can easily be extended to cover more than two goods. For example, with three goods the utility function might be represented as U\ufffdAx\ufffdy\ufffdz\ufffd, where z measures the quantity of the third commodity, and A, \ufffd, \ufffd, and \ufffd are all positive constants. Cobb\u2013Douglas utility function A function of the form U \ufffd Ax \ufffdy \ufffd, where U measures the con- sumer\u2019s utility from x units of one good and y units of another good and where A, \ufffd, and \ufffd are positive constants. c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 94 origin, as in Figure 3.11. Problem 3.21 at the end of the chapter asks you to verify that the marginal rate of substitution is diminishing. QUASILINEAR UTILITY FUNCTIONS The properties of a quasilinear utility function often simplify analysis. Further, eco- nomic studies suggest that such functions may reasonably approximate consumer preferences in many settings. For example, as we shall see in Chapter 5, a quasilinear utility function can describe preferences for a consumer who purchases the same amount of a commodity (such as toothpaste or coffee) regardless of his income. Figure 3.15 shows the indifference curves for a quasilinear utility function. The distinguishing characteristic of a quasilinear utility function is that, as we move due north on the indifference map, the marginal rate of substitution of x for y remains the same. That is, at any value of x, the slopes of all of the indifference curves will be the same, so the indifference curves are parallel to each other. The equation for a quasilinear utility function is U(x, y) \ufffd v(x) \ufffd by, where b is a positive constant and v(x) is a function that increases in x\u2014that is, the value of v(x) in- creases as x increases This utility function is linear in y, but generally not linear in x. That is why it is called quasilinear. [e.g., v(x) \ufffd x2or v(x) \ufffd 1x ]. 3.3 SPECIAL PREFERENCES 95 quasilinear utility function A utility func- tion that is linear in at least one of the goods consumed, but may be a nonlinear function