CHAPTER SEVEN UTILITY MAXIMIZATION

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1 CHAPTER SEVEN UTILITY MAXIMIZATION CHAPTER OVERVIEW This microeconomics chapter explains consumer decision-making on a more sophisticated level, enhancing understanding of demand concepts introduced in Chapter 3. The law of diminishing marginal utility is developed further, leading to a detailed discussion of the theory of consumer choice. The discussion of the diamond-water paradox helps students look beyond what may be their first conclusions about the importance and value of products. The chapter concludes with an examination of several examples of consumer behavior. The Appendix to this chapter introduces indifference curve analysis for those intending to pursue further study in economics, or for those who desire a more rigorous explanation of consumer choice. This material is linked to the coverage provided in the rest of the chapter by using indifference curve analysis to develop an individual s demand curve for a product. The material in the Appendix is not included in the AP exam. AP EMPHASIS This section is provided to help instructors identify material that is emphasized in the AP curriculum. The Learning Objectives (LO) for each chapter will be rated on a scale of zero to three stars. The ratings are based on a two semester course consisting of micro- and macro- economics. While all the material in the text is open to examination by the College Board, the material with stars should be given higher emphasis. No stars low priority 1 star important, but spend limited time 2 stars important; students seeking a score of 3 or better must master 3 stars very important; fundamental to AP economics course LO7.1 *** Total compared to marginal LO7.2 ** -to-price ratios LO7.3 Deriving a demand curve LO7.4 * Income and substitution effects LO7.5 LO7.6 Real world examples Indifference curves and budget lines PACING GUIDE FOR MICROECONOMICS Assign this chapter during study of the law of demand. Micro Students will need at least two class periods for this chapter s concepts, formulas, and practice. Students tend to enjoy the topic and will learn at different paces, so group work is helpful. The equations are part of the AP Microeconomics curriculum, and the utility-maximizing decision has appeared in free response questions on the AP Microeconomics exam. Indifferences curves, found in the Appendix, has never been part of microeconomics for AP. This chapter should be omitted in a semester course on macroeconomics.

2 INSTRUCTIONAL OBJECTIVES After completing this chapter, students should be able to: 1. Define marginal utility and state the law of diminishing marginal utility. 2. Explain and graph the relationship between marginal utility and total utility. 3. List four assumptions made in the theory of consumer behavior. 4. State the utility-maximizing rule. 5. Use the utility-maximizing rule to determine a consumer s spending (and demand curve) when given income, utility, and price data. 6. Use the theory of consumer behavior to define the market shift from portable CD players to ipods since their introduction in Explain the diamond-water paradox. 8. Explain why a cash gift will give the receiver more utility than a noncash gift costing the same amount. 9. Define and identify terms and concepts listed at the end of the chapter and appendix. COMMENTS AND TEACHING SUGGESTIONS 1. Students sometimes have difficulty with the notion that the income and substitution effects happen simultaneously. You may want to use the following concept illustration to clarify the point. Concept Illustration--Income and Substitution Effects Students sometimes mistakenly think that the substitution effect and the income effect are part of a sequential process that changes quantity demanded when price changes. First, one effect occurs, and then the other occurs. In reality, the substitution effect and income effect work simultaneously. The following analogy may help clarify this point. Picture a boat that is adrift on the ocean with the tide moving it directly south. Also suppose that the wind is from the north such that it, too, is pushing the boat directly south. The two forces, the tide and the wind, work simultaneously. We would need to measure their strengths to sort out their relative importance. But it is evident that, over some time period, the boat will drift farther to the south than if only the tide or the wind was singularly at work. The substitution effect of a price change is analogous to the tide and the income effect is analogous to the wind. For a normal good, the two effects increase quantity demanded when price falls, and decrease quantity demanded when price increases. The substitution effect (the tide) works through a change in the relative price of the good; the income effect (the wind) works through the change in the real income that results from the price change. The two effects work simultaneously to change quantity demanded (move the boat) when price changes. 2. Demonstrate the law of diminishing marginal utility, and perhaps even the idea of a saturation point (MU = 0), by having a student consume jelly beans or something similar while you review the law of demand and the income and substitution effects. Normally, the student will stop eating in a few minutes, even though the candy is free. Be careful not to use something that will cause an adverse reaction (e.g., caffeinated beverages) if the student, in an effort to prove the teacher wrong, consumes too much.

3 3. Numerical illustrations of the utility-maximizing rule should be viewed as a pedagogical technique, rather than an attempt to portray the actual choice-making process of consumers. When this illustration is explained by order of purchase, the brief algebraic summary of consumer equilibrium should pose no great difficulty for most students. Ask students to recall the best and the worst purchases they have made. Can they explain each using this concept of utilitymaximization? 4. Imagine a case in which marginal utility rises, rather than diminishes, with increased consumption. (Drug addiction and gambling come to mind.) Reason with students how this would ultimately result in a consumer spending all of his/her income on this one good. Emphasize that this is fortunately not consistent with most consumer behavior. This helps to support the law of diminishing marginal utility and the process by which consumers allocate their incomes among many different goods and services. 5. Ask students to identify people who would be likely to value time very highly and others who would place a low value on time. Have them discuss how those two groups of people would react to such situations as: (a) waiting in line to buy a product; (b) shopping for hours to find the best prices on products such as cars, stereo equipment, and soft drinks; or (c) hiring cleaning, laundry, and gardening services. 6. The marginal-total relationship presented in this chapter is the first of many the students will be asked to learn as they go through the microeconomics chapters. Spend a little extra time on this relationship and explain that they will be using the concept in somewhat altered forms later in the course. 7. What consumers decide to purchase depends on their personal preferences and priorities. No two people will make exactly the same decisions. Test this hypothesis by handing out monopoly money in different amounts and asking students to write down the purchases they would make with the sum given. Then have the students compare their choices. (This would be a good group activity.) Were there any identical lists? Some students got much more money than others. Was this fair? This exercise can be used to demonstrate the problem of making interpersonal comparisons and the problem of income inequality. Explain that the concept of consumer equilibrium puts each person in charge of his/her own satisfaction. The objective for each consumer is to maximize his/her satisfaction while staying within a budget. Students will need to practice this model and method. The same type of methodology will be used later in the course when producers combine resources with the goal of greatest output of cost. 8. The final objective of the chapter is to illustrate the concept of a constrained maximum. Without the use of calculus, this demonstration can be time consuming and confusing. Yet the equimarginal principle is a powerful tool and is central to the concept of efficiency with many applications. Consumer equilibrium, where total utility is maximized, occurs when marginal utility is the same for every good consumed. This means that the consumer s dollars are doing equal work at the margin. A dollar s worth of taffy apples is giving the same satisfaction as a dollar s worth of peanut butter cookies. Students may be able to grasp the concept intuitively by using physical examples such as the Consider This box on fast food lines in Chapter 1. Fast food customers try to join the shortest line and, in the process, they make the food servers do equal work. The same thing happens in the grocery store as shoppers jockey for position at the check-out stand. A foreman of an assembly line achieves efficiency when he assigns workers tasks requiring equal time. 9. Ask students to think about the silliest, most undesirable, ugliest, or useless gift they ever received. Assuming the giver had good intentions, why was there a loss of efficiency? 10. Debate: Be it resolved that welfare recipients should be provided income in-kind (food, clothing, or shelter), rather than cash.

4 11. The AP exam has never asked students to derive a demand curve using utility curves--nor has the exam used utility curves graphically. STUDENT STUMBLING BLOCKS 1. Students frequently attempt to compare the marginal utility of two products, rather than the marginal utility of the products, in determining the utility-maximizing combination of products. Emphasize that the price is important in the relative comparisons of products. For example, a car holds significantly higher utility than a candy bar, but we must consider the relative prices of the products in determining how many of each product the consumer will buy to maximize utility. 2. Students will find the marginal utility computations cumbersome at first. Model the procedure first, and then have students practice with a few exercises in class. Teachers will find that it will be helpful to use one as a model and another as an assessment. 3. Recent student problem areas on the FRQ section of AP exams, The 2012 Microeconomics exam provided a chart of marginal utilities for two items. Students were asked to determine the maximum utility attainable with an income constraint of $11. The 2010 exam asked about Diminishing returns, which is covered in this chapter using Diminishing. LECTURE NOTES I. Learning Objectives In this chapter students will learn: A. How to define and explain the relationship between total utility, marginal utility, and the law of diminishing marginal utility. B. How rational consumers maximize utility by comparing the marginal utility-to-price ratios of all of the products they could possibly purchase. C. How to derive a demand curve by observing the outcomes of price changes in the utilitymaximization model. D. How the utility-maximization model helps highlight the income and substitution effects of a price change. E. About real world examples of consumer behavior. F. How the indifference curves and budget lines lead to utility maximization II. Law of Diminishing A. Although consumer wants in general are insatiable, wants for specific commodities can be fulfilled. The more of a specific product that consumers obtain, the less likely they are to desire more units of that product. This can be illustrated with almost any item. The text uses the automobile example, but houses, clothing, and even food items work just as well. 1. and usefulness are not the same, but both have value. 2. is a subjective notion in economics, referring to the amount of satisfaction a person gets from consumption of a certain item. The utility of a product can and does vary person to person. 3. is difficult to measure. We will use a theoretical unit of measure called utils.

5 III. B. Figure 7.1 (Key Graph) and the accompanying table illustrate the relationship between total and marginal utility. 1. Total utility increases as each additional taco is purchased through the first five, but utility rises at a diminishing rate since each taco adds less and less to the consumer s satisfaction. 2. At some point, marginal utility becomes zero and then even negative at the seventh unit and beyond. If more than six tacos were purchased, total utility would begin to fall. This illustrates the law of diminishing marginal utility. C. Diminishing marginal utility explains why the demand curve slopes downward. Because each successive unit of a product yields less and less marginal utility, the price of the next unit must fall in order to entice the consumer to purchase it. D. CONSIDER THIS Vending Machines and 1. Newspaper vending machines normally allow one to take multiple papers; publishers allow this because they believe that people rarely take more than one paper because the marginal utility of the second paper is often zero, and it has little shelf life. 2. Soft drink vending machines distribute one can or bottle at a time. Even if the marginal utility of the second unit of soda is low in the short run, the long shelf life would allow people to keep sodas for later consumption. Theory of consumer behavior uses the law of diminishing marginal utility to explain how consumers allocate their income. A. Consumer Choice and the Budget Constraint 1. Consumers are assumed to be rational, trying to get the most value for their money. 2. Consumers have clear-cut preferences for various goods and services and can judge the utility they receive from successive units of various purchases. 3. Consumer incomes are limited because their individual resources are limited. Thus, consumers face a budget constraint, as we saw in Chapter Goods and services have prices and are scarce relative to the demand for them. Consumers must choose among alternative goods with their limited money incomes. B. The utility maximizing rule: To maximize satisfaction, the consumer should allocate income so that the last dollar spent on each product purchased yields the same amount of extra (marginal) utility. 1. A consumer is in equilibrium when utility is balanced () at the margin. When this is true, there is no incentive to alter the expenditure pattern unless tastes, income, or prices change. 2. Table 7.1 provides a numerical example for Holly, who has $10 to spend. Follow the reasoning process to see why 2 apples and 4 oranges will maximize Holly s utility, given the $10 spending limit. 3. It is marginal utility spent that is equalized; that is, consumers compare the extra utility from each product with its cost. 4. As long as one good provides more utility than another, the consumer will buy more of the first good; as more of the first product is bought, its marginal utility diminishes until the amount of utility just equals that of the other product.

6 IV. 5. Table 7.1 summarizes the step-by-step decision-making process the rational consumer will pursue to reach the utility-maximizing combination of goods and services attainable. 6. The algebraic statement of this utility-maximizing state is that the consumer will allocate income in such a way that: MU of product A (Apples) / Price of A (Apples) = MU of product B (Oranges) / Price of B (Oranges) Maximization and the Demand Curve A. Determinants of an individual s demand are tastes, income, and prices of other goods. B. The demand curve can be derived using product B (oranges) in Table 7.1 and considering alternative prices at which product B (oranges) might be sold. At lower prices, using the utility-maximizing rule, we see that more will be purchased as the price falls.. V. Income and Substitution Effects are explained by the utility-maximizing rule. VI. A. When the price of an item declines, the consumer will no longer be in equilibrium until more of the item is purchased and the marginal utility of the item declines to match the decline in price. More of this item is purchased rather than another relatively more expensive substitute. B. The income effect is shown by the fact that a decline in price expands the consumer s real income and the consumer must purchase more of this and other products until equilibrium is once again attained for the new level of real income. Applications and Extensions A. The ipod 1. The ipod came on the market in November Less than six years later, Apple sold its 100 millionth unit. Furthermore, those units enabled Apple to sell more than 2.5 billion songs through its online itunes store. 2. The improved portability and storage significantly increased consumer satisfaction, increasing the marginal utility compared to the price. Thus demand for the ipod increased as demand decreased for the portable CD player it replaced. 3. This example demonstrates a simple but important point: New products succeed by enhancing consumers total utility. B. The Diamond-Water Paradox 1. Before marginal analysis, economists were puzzled by the fact that some essential goods like water had lower prices than luxuries like diamonds. 2. The paradox is resolved when we look at the abundance of water relative to diamonds. 3. Theory tells us that consumers should purchase any good until the ratio of its marginal utility to price is the same as that ratio for all other goods. a. The marginal utility of an extra unit of water may be low, as is its price, but the total utility derived from water is very large. b. The total utility of all water consumed is much larger than the total utility of all diamonds purchased. c. However, society prefers an additional diamond to an additional drop of water, because of the abundant stock of water available.

7 VII. C. Opportunity Cost and the Value of Time 1. Time also has a value in the opportunity cost of forgone income, so this must be considered in decision-making and utility maximization. The total price of an item must include the value of the time spent in consuming the product (i.e., the wage value of an hour of time). When time is considered, consumer behavior appears to be much more rational. 2. Highly paid doctors may not spend hours hunting for bargains because their time is more valuable than the money to be saved from finding the best buy. 3. Foreigners observe that Americans waste material goods but conserve time. This could be because our high productivity makes our time more valuable than many of the goods we waste. D. Buying Medical Care or Eating at a Buffet 1. Most Americans have health insurance for which they pay a fixed monthly premium, which covers, say, 80 percent of their health care costs. Therefore, the cost of obtaining care is only 20 percent of its stated price for the insured patient. 2. Following the law of demand, people purchase a larger quantity of medical care than if they had to pay the full price for each visit. 3. If you buy a meal at an all-you-can-eat buffet, you eat more than if you paid separately for each item. 4. Because the marginal utility is positive and the price is low or zero, consumers buy more in each of these situations. E. Cash and Noncash Gifts 1. Noncash gifts may yield less utility to the receiver than a cash gift of equal monetary value, because the noncash gift may not match the receiver s preferences. 2. Individuals know their own preferences better than the gift giver. 3. Look back at Table 7.1. If Holly had no income and was given a $2 gift, she would rather have the cash transfer to spend on B than to be given 2 units of A. (She gets more utility or satisfaction by spending her $2 on B.) LAST WORD: Criminal Behavior A. Now that this chapter has explained much of consumer behavior, this Last Word uses our new concepts to explore crime and punishment. B. The benefit of crime (burglary or theft) is the value of the object stolen. The cost of the crime has several sources, but society can impose fines and imprisonment. A potential burglar or thief must weigh these additional societal imposed costs against the chance of being apprehended and successfully prosecuted. ANSWERS TO END-OF-CHAPTER QUESTIONS AND PROBLEMS Chapter 07 - Maximization DISCUSSION QUESTIONS 1. Complete the following table and answer the questions below: LO1

8 a. At which rate is total utility increasing: a constant rate, a decreasing rate, or an increasing rate? How do you know? b. A rational consumer will purchase only 1 unit of the product represented by these data since that amount maximizes marginal utility. Do you agree? Explain why or why not. c. It is possible that a rational consumer will not purchase any units of the product represented by these data. Do you agree? Explain why or why not. Answer: Missing total utility data, top bottom: 18; 33. The missing total utility for the second unity can be found by adding the marginal utility (change in utility) to the total utility for the first unit. By consuming the second unit, 8 more units of utils are added; thus total utility is 18 (= ). Missing marginal utility data, top bottom: 7; 5; 1. The missing marginal utility values are found by subtracting the total utility for the previous unit consumed from the total utility of the unit with the missing value (the change in utility). The marginal utility for the third unit is 7, which equals 25 (total utility for the third unit) minus 18 (total utility for the second unit). a. A decreasing rate; because marginal utility is declining. b. Disagree. The marginal utility of a unit beyond the first may be sufficiently great (relative to product price) to make it a worthwhile purchase. Consumers are interested in maximizing total utility, not marginal utility. c. Agree. This product s price could be so high relative to the first unit s marginal utility that the consumer would buy none of it. 2. Mrs. Simpson buys loaves of bread and quarts of milk each week at prices of $1 and 80 cents, respectively. At present she is buying these products in amounts such that the marginal utilities from the last units purchased of the two products are 80 and 70 utils, respectively. Is she buying the utilitymaximizing combination of bread and milk? If not, how should she reallocate her expenditures between the two goods? LO2 Answer: Mrs. Wilson is not buying the utility-maximizing combination of bread and milk because the marginal utility per cent spent on each good is not equal. The marginal utility per cent of bread is 0.8 (= 80 utils/100 cents); the utility per cent of milk is (= 70 utils/80 cents). Mrs. Wilson should buy more milk and less bread. 3. How can time be incorporated into the theory of consumer behavior? Explain the following comment: Want to make millions of dollars? Devise a product that saves Americans lots of time. LO2

9 Answer: Time is money. This expression is a time-saving way of making the point that for a person who can make so much per hour, every hour spent not working is so much money not made. A person can be said to consume a ball game or an evening at the theater. If the ball game costs $10 and the theater $20, at first sight one could say the ball game is a better deal. But if the person makes $20 an hour and is forgoing this in taking the time off, then we must take into account the time spent at the ball game or at the theater. If the ball game goes into extra innings and takes 4 hours, then its total cost is $90 (= $10 + $80). If the theater takes 3 hours, its total cost is $80 (= $20 + $60). Assuming the marginal utility of the ball game and attending the theater are the same, the theory of consumer behavior (with time taken into account) would therefore have this consumer going to the theater. 4. Explain: LO2 a. Before economic growth, there were too few goods; after growth, there is too little time. b. It is irrational for an individual to take the time to be completely rational in economic decision making. c. Telling your spouse where you would like to go out to eat for your birthday makes sense in terms of utility maximization. Answer: a. Before economic growth, most people lived at the subsistence level. By practically anyone s definition, this implies too few goods. After economic growth, goods are in relative abundance. To make (or consume) more takes time, but the relative abundance of goods means that there are already many goods to enjoy. So, now there is a clash between the use of time to make more goods and the use of time to relax and enjoy the goods one already has. There just isn t enough time. b. To be completely rational in economic decision making, provided one does not take time into consideration, one has to take account of every factor. This would take a great deal of time. One could not, for example, make any purchase without first searching the classifieds to see whether a better deal could be had, rather than simply heading for the nearest store. However, this would be most irrational, for time does have value. While making an extensive search before making any deal, one would be forgoing the income to make this or any deal. For every penny saved to make the perfect deal, one would be losing dollars in income because of the time spent in making the perfect deal. c. There is little time sacrificed in making a request to your spouse for the restaurant where you eat on your birthday. If you eat there, the benefit will likely exceed the cost. It also reduces the probability of eating at a restaurant where the market value (purchase price) exceeds the utility to the recipient. 5. In the last decade or so there has been a dramatic expansion of small retail convenience stores (such as 7 Eleven, Kwik Shop, and Circle K), although their prices are generally much higher than prices in large supermarkets. What explains the success of the convenience stores? LO2 Answer: These stores are selling convenience as well as the goods that are purchased there. Because of their small size and convenient locations, they save busy consumers time. In an era when most consumers are working at least 40 hours per week, their time is valuable, and when only a few items are needed, the time saved must be worth the additional cost one pays for shopping at these convenience stores. (You seldom, if ever, see anyone buying a week s worth of groceries at such shops.)

10 6. Many apartment-complex owners are installing water meters for each apartment and billing the occupants according to the amount of water they use. This is in contrast to the former procedure of having a central meter for the entire complex and dividing up the collective water expense as part of the rent. Where individual meters have been installed, water usage has declined 10 to 40 percent. Explain that drop, referring to price and marginal utility. LO3 Answer: The way we pay for a good or service can significantly alter the amount purchased. An individual living in an apartment complex who paid a share of the water expense measured by a central meter would have little incentive to conserve. Individual restraint would not have much impact on the total amount of water used. Suppose there were 10 apartments in the complex; each apartment would be billed for one-tenth of the cost of the water. A single gallon of water would carry a price equal to one-tenth the amount charged by the water district. The very low price per gallon would encourage the use of water until the marginal utility of an additional gallon was correspondingly low. If the tenants paid separately for their own water, the full market price of water would be considered when making their consumption choices. 7. Using the utility-maximization rule as your point of reference, explain the income and substitution effects of an increase in the price of product B, with no change in the price of product A. LO4 Answer: The utility-maximization rule compares the marginal utilities of goods under consideration (in this case A and B). An increase in the price of product B would reduce the marginal utility of B. This would discourage consumption of B, and with a pure income effect, would not alter consumption of A. If the increased price of B caused the marginal utility of the last unit of B to fall below the MU/$ of the next unit of A, we would expect the consumer to substitute A for B in consumption (substitution effect). 8. ADVANCED ANAYLSIS A mathematically fair bet is one in which the amount won will on average equal the amount bet, for example when a gambler bets, say, $100 for a 10 percent chance to win $1,000 ($100 =.10 x $1,000). Assuming diminishing marginal utility of dollars, explain why this is not a fair bet in terms of utility. Why is it even a less fair bet when the house takes a cut of each dollar bet? So is gambling irrational? LO4 Answer: Because the marginal utility of money diminishes the more you have, the utility of the $100 used to make the bet is greater than the $900 that you might gain ($1,000 - $100) if you win the bet. It is even less of a fair bet when the house takes its cut, because the $100 bet has the possibility of yielding less than $900 in winnings. Is gambling irrational? Maybe. The activity of gambling may provide enough extra utility to offset the poor utility odds of winning. 9. LAST WORD In what way is criminal behavior similar to consumer behavior? Why do most people obtain goods via legal behavior as opposed to illegal behavior? What are society's main options for reducing illegal behavior?

11 Answer: Both the lawful consumer and the criminal try to maximize their total utility. (Answers may vary.) Most people choose legal behavior because they see the price of illegal behavior to be greater than the marginal benefit. The costs include guilt, direct costs (supplies and tools), loss of freedom, and the forgone income earned through legal behavior. Society's main option to reduce illegal behavior is to increase the cost of engaging in it. They can do this many ways. Guilt costs can be increased through family, educational, and religious efforts. They can get better locks and alarms forcing the criminal to spend more money on better tools. They can provide higher incomes for legal behavior, increasing the opportunity cost. They can increase policing to make it more likely the criminal will be caught. AP REVIEW QUESTIONS 1. True or false. The law of diminishing marginal utility predicts the consumption behavior of addicts quite well. LO1 Answer: false. The law of diminishing marginal utility does not predict the consumption behavior of addicts well at all. That is because the law predicts that, other things equal, successive units will be less and less valuable to a person. Thus, people should not keep on consuming additional units relentlessly. But an alcoholic, for instance, will often purse a 15 th drink on a given day with as much intensity as he pursued his first drink that day. And a gambling addict will often bet his final dollar with as much alacrity as he bet his first. Those suffering from addiction appear to behave as if the marginal utility of the drug or behavior to which they are addicted remains high no matter how much of the drug or behavior they engage in. 2. Frank spends $75 on 10 magazines and 25 newspapers. The magazines cost $5 each and the newspapers cost $2.50 each. Suppose that his MU from the final magazine is 10 utils while his MU from the final newspaper is also 10 utils. According to the utility-maximizing rule, Frank should: LO2 a. Reallocate spending from magazines to newspapers. b. Reallocate spending from newspapers to magazines. c. Be satisfied because he is already maximizing his total utility. d. None of the above. Answer: a, reallocate spending from magazines to newspapers Frank is not currently maximizing the utility that he could get from his $75 budget and should reallocate spending from magazines to newspaper in order to increase the total utility that he receives from his budget. This can be seen by examining the MU that he is currently getting from the last magazine and the last newspaper. The MU from the last magazine is 10/$5 = 2 utils. The MU from the last newspaper is 10/$2.50 = 4 utils. Thus, Frank could gain more total utils if he spent less on newspapers and more on dollars. As an example, imagine that Frank cut his purchases of magazines by one, reducing them from 10 purchased to only 9 purchased. That would free up $5 that he could spend on newspapers. Since newspapers cost $2.50 each, he could purchase two newspapers. Now, look at what happens to his total utility as he makes that change in consumption. He will lose 10 utils by

12 consuming one fewer magazine (because the MU of magazines is 10). But he will gain more than 10 utils by consuming two additional newspapers (because the MU of an additional newspaper is 10 utils and he will be consuming two additional newspapers). Thus, by reallocating his spending from the product with the lower MU to the product with the higher MU, Frank will be able to increase the total utility that he receives from his budget. 3. A consumer s demand curve for a product is downsloping because: LO3 a. Total utility falls below marginal utility as more of a product is consumed. b. utility diminishes as more of a product is consumed. c. Time becomes less valuable as more of a product is consumed. d. The income and substitution effects precisely offset each other. e. The income effect would cause a decrease in quantity when the price goes down. Answer: utility diminishes as more of a product is consumed. 4. Jermaine spends his money on cucumbers and lettuce. If the price of cucumbers falls, the MU per dollar of cucumbers will and Jermaine will cucumbers for lettuce. LO4 a. Fall; substitute b. Rise; substitute c. Fall; supply d. Rise; demand Answer: b, rise; substitute When the price of cucumbers falls, the MU of cucumbers will rise. As a result, each dollar spent on cucumbers will now bring more marginal utility than it did previously. Cucumbers thus become relatively more attractive than they were previously and so Jermaine will switch, or substitute, some of his spending away from lettuce and towards cucumbers in order to maximize his utility given the new, lower price of cucumbers. 5. Tammy spends her money on lemonade and iced tea. If the price of lemonade falls, it is as though her income. LO4 a. increases b. decreases c. stays the same Answer: a, increases. A person s income is not same thing as their budget. Rather, their income is the amount of goods and services that they can purchase with their budget. Thus, when the price of something that they purchase falls, their income rises because they can now purchase more goods and services than they did before. Consider the following example. Suppose that Tammy originally has a $6 budget and maximizes utility by purchasing 2 glasses of lemonade for $2 each and 2 glasses of iced tea for $1 each. So her income of goods and services is 2 glasses of lemonade and 2 glasses of iced tea. Then suppose that the price of iced tea falls in half, to only $1 per glass. Her budget of $6 remains unchanged, but she will now be able to purchase more total items and thereby enjoy a higher income due to the lower price of lemonade.

13 This can be seen most directly by noting that her old income of 2 glasses of lemonade and 2 glasses of iced tea will now only cost $4 which means that she has $2 left over to increase her consumption of either iced tea or lemonade. The price reduction allows her to consume more overall, which is the same as saying that her income of goods and services has increased. PROBLEMS 1. Mylie s total utility from singing the same song over and over is 50 utils after one repetition, 90 utils after two repetitions, 70 utils after three repetitions, 20 utils after four repetitions, -50 utils after five repetitions, and -200 utils after six repetitions. Write down her marginal utility for each repetition. Once Mylie s total utility begins to decrease, does each additional singing of the song hurt more than the previous one or less than the previous one? LO1 Answers: 50, 40, -20, -50, -70, -150; More than the previous one. Feedback: Consider the following values: Mylie s total utility from singing the same song over and over is 50 utils after one repetition, 90 utils after two repetitions, 70 utils after three repetitions, 20 utils after four repetitions, -50 utils after five repetitions, and -200 utils after six repetitions. Mylie's marginal utility can be found by calculating the change in total utility as she sings the song one more time. The utility from singing the song the first time is 50, which is the total utility of singing the song the first time, 50, minus the total utility of not singing the song at all, 0. The utility from singing the song the second time is 40, which is the total utility from singing the song the second time, 90, minus the total utility from singing the song the first time, 50. Apply the same procedure to following repetitions of the song. The marginal utility of the third time is -20 (= 70-90). The fourth time, -50 (= 20-70), the fifth time, -70 (= ). The sixth time, -150 (= (-50) = ). Once Mylie's total utility begins to decrease, each additional song hurts more than the previous one because she is becoming worse in terms of utility (which is a measure of individual welfare). As we can see above, after the second repetition, Mylie is actually worse off. Also, the decline in utility increases after the second repetition. 2. John likes Coca-Cola. After consuming one Coke, John has a total utility of 10 utils. After two Cokes, he has a total utility of 25 utils. After three Cokes, he has a total utility of 50 utils. Does John show diminishing marginal utility for Coke or does he show increasing marginal utility for Coke? Suppose that John has $3 in his pocket. If Cokes cost $1 each and John is willing to spend one of his dollars on purchasing a first can of Coke, would he spend his second dollar on a Coke, too? What about the third dollar? If John s marginal utility for Coke keeps on increasing no matter how many Cokes he drinks, would it be fair to say that he is addicted to Coke? LO1 Answers: Increasing; Yes; Yes. Feedback: Consider the following values: After consuming one Coke, John has a total utility of 10 utils. After two Cokes, he has a total utility of 25 utils. After three Cokes, he has a total utility of 50 utils. Also, assume John has $3 in his pocket and Cokes cost $1 each. John's marginal utility equals the change in total utility as he consumes an additional unit. Thus, his marginal utility from the first Coke is 10 (= 10-0), his second Coke is 15 (= 25-10), and his third Coke is 25 (= 50-25). Since his marginal utility is increasing at an increasing rate he shows increasing marginal utility for Coke.

14 If John consumes the first can of Coke, we know that his marginal utility was sufficient to induce the purchase. Since his marginal utility increases as he consumes more Coke, his marginal utility will also increase (price does not change). This implies that he will purchase the second Coke because it provides even more marginal utility than the first Coke did. The same is true for the third can of Coke. He will spend his entire income on Coke, thus it is fair to say he is addicted to Coke. 3. Suppose that Omar s marginal utility for cups of coffee is constant at 1.5 utils per cup, no matter how many cups he drinks. On the other hand, his marginal utility per doughnut is 10 for the first doughnut he eats, 9 for the second he eats, 8 for the third he eats, and so on (that is, declining by 1 util per additional doughnut). In addition, suppose that coffee costs $1 per cup, doughnuts cost $1 each, and Omar has a budget that he can spend only on doughnuts, coffee, or both. How big would that budget have to be before he would spend a dollar buying a first cup of coffee? LO2 Answer: $10. Feedback: Consider the following example: Suppose that Omar s marginal utility for cups of coffee is constant at 1.5 utils per cup, no matter how many cups he drinks. On the other hand, his marginal utility per doughnut is 10 for the first doughnut he eats, 9 for the second he eats, 8 for the third he eats, and so on (that is, declining by 1 util per additional doughnut). In addition, suppose that coffee costs $1 per cup, doughnuts cost $1 each, and Omar has a budget that he can spend only on doughnuts and/or coffee. To answer this question, we first need to calculate the marginal utility for doughnuts. Recall that the marginal utility for a good is the marginal utility divided by the price of the good (=MU/P). For the first doughnut we have 10 (=10/$1), the second doughnut 9 (=9/$1), third 8, fourth 7, fifth 6, sixth 5, seventh 4, eighth 3, ninth 2, and tenth 1. The marginal utility for every cup of coffee is 1.5 (=1.5/$1). To determine how big the budget would have to be before John would spend a dollar buying his first cup of coffee, we compare the marginal utility values. John will purchase the first doughnut before he buys a cup of coffee because the marginal utility for the doughnut is greater than the marginal utility for the cup of coffee (10>1.5). The same is true for the second through the ninth doughnut. This implies John will buy 9 doughnuts at the price of $1 before he buys his first cup of coffee. Therefore his budget will need to $10 before he buys his first cup of coffee, $9 on the doughnuts and $1 for the cup of coffee. 4. Columns 1 through 4 in the table below show the marginal utility, measured in utils, that Ricardo would get by purchasing various amounts of products A, B, C, and D. Column 5 shows the marginal utility Ricardo gets from saving. Assume that the prices of A, B, C, and D are, respectively, $18, $6, $4, and $24 and that Ricardo has an income of $106. LO2

15 a. What quantities of A, B, C, and D will Ricardo purchase in maximizing his utility? b. How many dollars will Ricardo choose to save? c. Check your answers by substituting them into the algebraic statement of the utility-maximizing rule (verify that all of the income has been exhausted between the various goods and savings). Answer: (a) 4 units of good A, 3 units of good B, 3 units of good C, zero units of good D; (b) $4; (c) $106 (= $18x4 + $6x3 +$4x3 + $24x0 + $4) Feedback: Consider the following information and use the table above: the prices of A, B, C, and D are, respectively, $18, $6, $4, and $24 and Ricardo has an income of $106. The first step is to convert the marginal utility values into marginal utility values. Recall this is MU/P. Column 1 Good A Column 2 Good B Column 3 Good C Column 4 Good D Column 5 Saving unit 1 unit 2 unit 3 unit 4 unit 5 unit 6 unit 7 unit units

16 The table above reads across rows (for example, the first row labeled 'Column 1-Good A' is marginal utility for good A). The columns represent the marginal utility for the different goods for each unit consumed and saving (for example, column 1 ' unit 1' tells us the marginal utility for the first unit of every good and savings). This will makes the comparison a little easier. To determine the optimal amount of goods that Ricardo will purchase, we choose the goods with the highest marginal utility as we move from left to right. We continue to do this until income is exhausted. We see that the first unit of savings ($1) provides $5.00 worth of utility per dollar, so Ricardo makes this choice. Next he purchases one unit of good A and good B with a marginal utility of 4 He will also save the second dollar (an additional $1) because the marginal utility is 4. The next decision leads Ricardo to purchase the first unit of good C with a marginal utility of 3. Then he moves on to purchase the second unit of good A, the second unit of good B, and he saves an additional dollar because the marginal utility per dollar is 3for these goods. After these purchases, he has an income left of $53. Since he has income left, he continues to buy more goods using the approach above. In conclusion, Ricardo will consume 4 units of good A, 3 units of good B, 3 units of good C, zero units of good D, and saves $4. This will exhaust his income $106 (= $18 x 4 + $6 x 3 +$4 x 3 + $24 x 0 + $4). 5. You are choosing between two goods, X and Y, and your marginal utility from each is as shown in the table below. If your income is $9 and the prices of X and Y are $2 and $1, respectively, what quantities of each will you purchase to maximize utility? What total utility will you realize? Assume that, other things remaining unchanged, the price of X falls to $1. What quantities of X and Y will you now purchase? Using the two prices and quantities for X, derive a demand schedule (prices and quantities demanded table) for X. LO3 Answer: X = 2 units, Y = 5 units; total utility = 48; X = 4, Y = 5; Demand schedule = Price Demanded Quantity Demanded

17 Feedback: Consider the following table and information as an example: Your income is $9 and the prices of X and Y are $2 and $1, respectively. The first step is to convert the marginal utility values into marginal utility values. Recall this is MU/P. See the table below. Good Y (first price) Good X (first price) Good Y (second price) Good X (second price) unit 1 unit 2 unit 3 unit 4 per dollar unit 5 per dollar unit Units The first two rows define the marginal utility for goods X and Y when the price of good X is $2 and the price of Good Y is $1. Each column is the marginal utility for the unit consumed ( utility for unit 1 is for the first unit consumed, utility for unit 2 is for the second unit consumed, etc...). To determine the optimal number of goods to purchase we choose the goods with the highest marginal utility as we move from left to right. We continue to do this until income is exhausted. First, we only consider the first two rows for Goods X and Y for the first price of good X (price of X is $2). We purchase the first unit of good Y with a marginal utility of 8. Next we purchase the second unit of good Y with a marginal utility of 7. Followed by the third unit of good Y with a marginal utility of 6. Continuing on, we purchase the fourth and fifth unit of Good Y and the first and second unit of good X 5and4marginal utility ). Thus, we purchase 5 units of good Y and 2 units of good X. The total cost is $9 (= $5 x 5 + $2 x 2), which exhausts income. The total utility is then found by adding up the marginal utility values for the goods consumed using the values in the original table. The total utility from good Y is 30 (= ) and

18 the total utility from good X is 18 (=10 + 8). The sum of these two values is the total utility, which equals 48 (= ). Now we consider the case where the price of good X falls to $1. Here, we only consider the last two rows for Goods X and Y for the second price of good X. Here our first purchase is the first unit of good X with a marginal utility of 10. Next we purchase the second unit of good X and the first unit of good Y with a marginal utility of We continue this process until income is exhausted. The optimal consumption level for good X is 4 and for good Y is 5. This gives us the ordered pairs for good X: (P = 2, Q = 2) and (P = 1, Q = 4). We can use these ordered pairs to graph the demand schedule (not required). 6. ADVANCED ANAYLSIS Let MU A = z = 10 - x and MU B = z = 21-2y, where z is marginal utility measured in utils, x is the amount spent on product A, and y is the amount spent on product B. Assume that the consumer has $10 to spend on A and B that is, x + y = 10. How is the $10 best allocated between A and B? How much utility will the marginal dollar yield? LO3 Answer: $7 on good B; $3 on good A; MU A = 7, MU B = 7. Feedback: To solve this system of equations we set the MU A =MU B. This is required if we are maximizing utility. This leaves with only x and y to solve for using the individual's budget constraint. Consider the following equations: Let MU A = z = 10 - x and MU B = z = 21-2y, where z is marginal utility measured in utils, x is the amount spent on product A, and y is the amount spent on product B. Also, assume that the consumer has $10 to spend on A and B that is, x + y = 10. We know from our optimization rule that MU A =MU B, which implies 10 - x = 21-2y. Or, after some rearranging, 2y - x = 11. Combining this equation with our budget constraint, x + y =10, we have two equations and two unknowns (x and y). Use the budget constraint to solve x, or x = 10 - y. Substitute x from this equation into our optimality rule (2y - x = 11), which gives us 2y - (10 -y) = 11. Again after some rearranging, we have 2y y = 11, or 3y = 21. Thus, we spend $7 on good B. Using our budget constraint 9x + y =10), this implies we spend $3 on good A. To check our answer, substitute these values into the marginal utility equations: MU A = 10 - x, or MU A = 10-3 = 7 and MU B = 21-2y = 21-2x7 = 7. Thus, MU A =MU B., and the marginal dollar will yield 7 utils. 7. Suppose that with a budget of $100, Deborah spends $60 on sushi and $40 on bagels when sushi costs $2 per piece and bagels cost $2 per bagel. But then, after the price of bagels falls to $1 per bagel, she spends $50 on sushi and $50 on bagels. How many pieces of sushi and how many bagels did Deborah consume before the price change? At the new prices, how much money would it have cost Deborah to buy those same quantities (the ones that she consumed before the price change)? Given that it used to take Deborah s entire $100 to buy those quantities, how big is the income effect caused by the reduction in the price of bagels? LO4 Answers: 30 pieces of sushi and 20 bagels; $80 (= (30 x $2) + (20 x $1)); $20. Feedback: Consider the following values as an example. Suppose that with a budget of $100, Deborah spends $60 on sushi and $40 on bagels when sushi costs $2 per piece and bagels cost $2