CHAPTER 4 Consumer Choice

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1 CHAPTER 4 Consumer Choice CHAPTER OUTLINE 4.1 Preferences Properties of Consumer Preferences Preference Maps 4.2 Utility Utility Function Ordinal Preference Utility and Indifference Curves Utility and Marginal Utility Utility and Marginal Rates of Substitution 4.3 Budget Constraint Slope of the Budget Constraint Purchasing Fractional Quantities Effect of a Change in Price on Consumption Effect of a Change in Income on Consumption 4.4 Consumer s Constrained Choice The Consumer s Optimum Bundle Optimal Bundles Are on Convex Sections of Indifference Curves Buy Where More Is Better Food Stamps TEACHING TIPS The material in this chapter represents a challenge to students for a number of reasons. Most will not have learned constrained optimization before. Also, utility analysis may seem quite abstract to them. Finally, if, you are using calculus, this is the point where the calculus techniques go beyond the methods they are likely to have learned in their calculus prerequisite. To overcome these stumbling blocks, you may want to devote one class to presenting and discussing the technical points of utility theory and budget lines using one or two running examples, and another class to applications. Whether you decide to present applications as you go, or after presenting the technical material, I find that this is one area where the students have to see the theory applied in order for it to make sense. You might also consider breaking the class into small groups to work on problems that test their understanding (additional problems are provided below). This way, you can migrate among the groups to get a better idea of how many are struggling and who needs help. The methods presented in this chapter will be needed in Chapter 5 for the derivation of demand curves. If they are not confident with the material in this chapter, they are almost certain to struggle not only in Chapter 5, but also when it comes to production and cost, since the methodology there is similar. When discussing the marginal rate of substitution, note that in this text, MRS is left as a negative number (MRS = / Y, which, of course, is negative, rather than MRS = - / Y as in some texts). When presenting the MRS, it is worth the time spent to try to get the class to think hard about the special cases of perfect complements and substitutes. Not only does it help them with commodities that fall into this category, but it also helps them to better understand the more typical case of convexity. The class should see that the notion of a measurable util is nonsense and will find the model more believable given that only ordinal measurability is needed. Section 4.3 on budget lines should not require much class time, though it is worth putting an equation and graph on the board and briefly discussing how parameter changes affect the constraint. 21

2 22 Part One\Teaching Aids For Section 4.4, Constrained Choice, the technical portion of the material should come fairly easily if the class has a solid grasp of indifference curves. Using plenty of examples and applications will make the process of utility maximization seem less abstract. There are several in the text as well as the additional application below. You may want to spend some time contrasting interior solutions and corner solutions. Corner solutions are an effective way to show that the model makes predictions that are intuitively accurate. For example, you can choose combinations of goods such that they are perfect substitutes to demonstrate that the model predicts that the consumer will always buy the one that is cheaper. You can also choose two goods such that one is a good and the other is neutral to show that the consumer will only purchase the good that provides utility. I also spend some time on the notion of bads (see discussion question #2 below). I try to get the class to realize that if the bad is turned on its head, it becomes a good (e.g., less air pollution is more clean air) and that clean air has a positive price that can be measured as the price of abatement. Once transformed, standard maximization methods apply. Finally, I show, using graphs, simple examples of cases where the standard maximization solution does not exist, such as nonlinear budget lines. For example, a consumer may be shown to be indifferent between fewer goods at a higher price, and a larger quantity at a lower price in the case of declining marginal prices. If you are using calculus to present utility theory, indifference curves, and maximization, students should read Appendices 4A and 4B. The Lagrange Multiplier method is covered in Appendix 4B. Note that endof-chapter problems can be assigned as calculus problems. There are also additional calculus-based problems below. Even if you are teaching students who are very strong mathematically, it is probably not worth the time it would take to use functions more complex than Cobb-Douglas or quadratic. Instead, If you have extra time, I recommend that you devote time to the interpretation of the Lagrange multiplier as the marginal value of the constraint. ADDITIONAL APPLICATIONS Transitivity of Preferences How realistic is the assumption of transitive preferences? A number of studies of both humans and animals show that preferences usually are transitive. For example, Wienstein (1968) uses an experiment to determine how frequently people give intransitive responses. 1 None of the subjects knew the purpose of the experiment. They were given choices between ten goods, offered in pairs, in every possible combination. The goods included $3 in cash, an 8-cup Wearever aluminum coffee percolator, and a free pass to the next four Saturday matinees at the subject s favorite movie theatre. They were told that all of the goods had a value of $3, to ensure that monetary value would not affect their calculations. Weinstein found that 93.5% of the responses of adults (at least 18 years old) were transitive. Of children aged 9-12, however, only 79.2% of the responses were transitive. He reasoned that this difference may be a justification for political and economic restrictions and protections placed on youths. Psychologists have also tested for transitivity using preferences for colors, photos of faces, and so forth. For example, Bradbury and Ross (1990) found that, given a choice of three colors, nearly half of 4-5 year olds are intransitive, compared to 15% for year olds, and 5% for adults. 2 Bradbury and Ross show that a novelty (preference for a new color) is responsible for most intransitive responses, and that this effect is especially strong in children. 1. Show using indifference curves and a budget line that if preferences are intransitive, standard utility maximization solutions may not result. 1 Weinstein, Arnold A., Transitivity of Preferences: A Comparison Among Age Groups, Journal of Political Economy, 76(2), March/April 1968: Bradbury, Hinton and Karen Ross, The Effects of Novelty and Choice Materials on the Intransitivity of Preferences of Children and Adults, Annals of Operations Research. 23(1 4) June 1990:

3 Chapter 4\Consumer Choice Suppose three of the subjects meet on the street after the above experiment. Each is carrying one item (the money, the tickets, or the coffee percolator). Two of these individuals have intransitive preferences; the other does not. How many exchanges could take place involving all three people before one would be unwilling to trade? What would your answer be if only one had intransitive preferences? DISCUSSION QUESTIONS 1. Our analysis of consumer behavior focuses on how to maximize the well-being of individual consumers. What alternative objectives might we consider? 2. How can we analyze commodities that are bads (garbage, water pollution)? 3. Name pairs of goods that you consume that are perfect substitutes. 4. Name pairs of goods that you consume that are perfect complements. 5. Can you think of a person who might be satiated in all goods (does not want more of anything)? 6. Discuss the democratic model of one person one vote as a method for the determination of social policy. ADDITIONAL QUESTIONS AND MATH PROBLEMS 1. Maximizing behavior in the context of school performance (not utility maximization) would imply trying to get straight A s. Is maximizing behavior a good assumption in this case? Can you think of another assumption that may be more appropriate for some individuals when attempting to model school performance behavior? 2. Consider two goods that are perfect substitutes. What is likely to be true about their relative prices? Can you confirm your hypothesis with examples? 3. From the total utility schedule shown below, calculate the marginal utility of each additional unit of consumed. Units of : Utility: Suppose a consumer s utility derived from consuming bananas is described by the function U= (1/3) 3. Use the derivative formula found in Equation 4A.1 to compute marginal utility. a) Make a table showing total and marginal utility for from 0 to 7 units. b) Would this individual ever choose to consume more than 7 units? Explain. 5. What information is contained in the slope of an indifference curve? Why are these curves typically convex to the origin? 6. For each of the utility functions below, draw a set of indifference curves showing utility levels U = 12, U = 16, and U = 24. a) U = Y b) U = + Y c) U = - Y d) What is true about the commodities in (b)? e) What about the commodities in (c)?

4 24 Part One\Teaching Aids 7. Suppose a consumer has an income of $500 and faces prices p = 5 and p Z = 10. a) Write the equation for the budget constraint. b) Draw the budget constraint, placing good on the horizontal axis. Label it BC. c) What is the slope of BC? d) Suppose income decreases to $300. Draw the new budget constraint and label it RS. 8. Larry and Teri allocate their consumption between two goods: hats and bats. The price of hats is $4 each and the price of bats is $8 each. For Larry, the marginal utility of the last hat consumed was 8 and the marginal utility of the last bat was 24. For Teri the marginal utility of the last hat was 6 and the marginal utility of the last bat was 12. Which consumer is not maximizing his/her utility? How can you tell? How should he/she change their allocation? 9. Suppose a consumer has income of $120 per period, and faces prices p = 2 and p Z = 3. Her goal is to maximize her utility, described by the function U = 10.5 Z.5. Calculate the utility maximizing bundle (*, Z*) using the methods described in Appendix 4B. 10. Confirm that if a consumer s utility function is described by U = 2 + Z, and prices are p = 2 and p Z = 1, there is no unique utility maximizing solution regardless of income level. What does this tell you about and Zas commodities? (Hint: draw a graph showing a budget constraint and indifference curve using the information provided.) ANSWERS TO ADDITIONAL QUESTIONS AND MATH PROBLEMS 1. Some individuals may practice what is known as satisficing behavior. In the context of school performance, this would mean performing to some minimum acceptable standard, such as C quality work. In the work environment, it implies working just hard enough to not get fired. However, in cases where the individual does not have good information about what minimal performance standards are, or the standards cannot be achieved with certainty, it becomes a very risky strategy. 2. Goods that are perfect substitutes are generally priced equally, or nearly so. For example, stores often price different brands of golf or tennis balls all at one level, or at very similar prices because they must be produced to professional association standards, and so are by definition good substitutes. Different brands of basic food commodities such as bread and milk are also usually priced in a very narrow range at a given store. 3. Units of : MU: MU = a) Units of : MU: (10) b) No. Marginal utility of the 8 th unit would be negative. 5. The slope of an indifference curve indicates the rate at which the consumer is willing to exchange one commodity for the other. This is the Marginal Rate of Substitution (MRS). Indifference curves are typically convex because the consumer places greater value on the last unit of the commodity that is more scarce.

5 Chapter 4\Consumer Choice a) Figure 4.1 b) Figure 4.2 Y Y U = 24 U = 16 U = 12 U = 24 U = 16 U = 12 c) Figure 4.3 Y U = 12 U = 16 U = 24 d) They are perfect substitutes. e) is a good, and Y is a bad, providing exactly the opposite of in utility per unit.

6 26 Part One\Teaching Aids 7. a) Z = 500 b) See Figure 4.4 c) Slope = -1/2 d) See Figure 4.4 Figure 4.4 Z 50 B 30 R 8. Based on Equation 4.6, the ratio of marginal utility per dollar should be equal for the last unit of all commodities. For Teri, the marginal utilities per dollar are equal, for Larry they are not. Larry needs to increase his expenditures on bats, and decrease expenditures on hats, which will cause the marginal utility of hats to rise, and marginal utility of bats to fall. 9. Setting up the Lagrangian as in 4B.2 results in S C L = 10.5 Z.5 - (2+3Z-120) Taking the partial derivative of this function with respect to, Z and λ, and setting equal to zero, results in three equations and three unknowns: L/ = Z.5-2 = 0 L/ Z = 5.5 Z -5-3 = 0 L/ = Z = 0 The easiest way to solve is to take the ratio of the first two equations, and plug the resulting /Z ratio into the third. In this case, * = 30, Z* = Because the commodities are perfect substitutes, and the price ratio is equal to the MRS, any budget line drawn would be coincident with some indifference curve, resulting in no unique solution. The consumer is equally happy choosing any point on the budget line.

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