g) Does Ann have a diminishing marginal rate of substitution of food for clothing? Show this graphically and algebraically.

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1 PROBLEMS 141 $1.50 per month. Each movie he attends costs $10. Let x denote the square feet of housing, and let y denote the number of movies he attends per month. a) What is the expression for Pedro s budget constraint? b) Draw a graph of Pedro s budget line. c) What is the maximum number of square feet of housing he can purchase given his monthly stipend? d) What is the maximum number of movies he could attend given his monthly stipend? e) Suppose Pedro s parents increase his stipend by 10 percent. t the same time, suppose that in the college town where he lives, all prices, including housing rental rates and movie ticket prices, increase by 10 percent. What happens to the graph of Pedro s budget line? 4.2. Sarah consumes apples and oranges (these are the only fruits she eats). She has decided that her monthly budget for fruit will be $50. Suppose that one apple costs $0.25, while one orange costs $0.50. a) What is the expression for Sarah s budget constraint? b) Draw a graph of Sarah s budget line. c) Show graphically how Sarah s budget line changes if the price of apples increases to $0.50. d) Show graphically how Sarah s budget line changes if the price of oranges decreases to $0.25. e) Suppose Sarah decides to cut her monthly budget for fruit in half. Coincidentally, the next time she goes to the grocery store, she learns that oranges and apples are on sale for half price, and will remain so for the next month; that is, the price of apples falls from $0.25 per apple to to $0.125 per apple, and the price of oranges falls from $0.50 per orange to $0.25 per orange. What happens to the graph of Sarah s budget line? 4.3. Julie has preferences for food F and clothing C described by a utility function U(F,C ) FC. Her marginal utilities are MU F Cand MU C F. Suppose that food costs $1 a unit and that clothing costs $2 a unit. Julie has $12 to spend on food and clothing. a) On a graph draw indifference curves corresponding to u 12,u 18, and u 24. Using the graph (and no algebra), find the optimal (utility-maximizing) choice of food and clothing. Let the amount of food be on the horizontal axis and the amount of clothing be on the vertical axis. b) Using algebra (the tangency condition and the budget line), find the optimal choice of food and clothing. c) What is the marginal rate of substitution of food for clothing at her optimal basket? Show this graphically and algebraically. d) Suppose Julie decides to buy 4 units of food and 4 units of clothing with her $12 budget (instead of the optimal basket). Would her marginal utility per dollar spent on food be greater than or less than her marginal utility per dollar spent on clothing? What does this tell you about how she should substitute food for clothing if she wanted to increase her utility without spending any more money? 4.4. The utility that nn receives by consuming food F and clothing C is given by U(F,C ) FC F. The marginal utilities of food and clothing are MU F C 1 and MU C F. Food costs $1 a unit, and clothing costs $2 a unit. nn s income is $22. a) nn is currently spending all of her income. She is buying 8 units of food. How many units of clothing is she consuming? b) Graph her budget line. Place the number of units of clothing on the vertical axis and the number of units of food on the horizontal axis. Plot her current consumption basket. c) Draw the indifference curve associated with a utility level of 36 and the indifference curve associated with a utility level of 72. re the indifference curves bowed in toward the origin? d) Using a graph (and no algebra), find the utilitymaximizing choice of food and clothing. e) Using algebra, find the utility-maximizing choice of food and clothing. f ) What is the marginal rate of substitution of food for clothing when utility is maximized? Show this graphically and algebraically. g) Does nn have a diminishing marginal rate of substitution of food for clothing? Show this graphically and algebraically Consider a consumer with the utility function U(x,y) min(3x, 5y); that is, the two goods are perfect complements in the ratio 3:5. The prices of the two goods are P x $5 and P y $10, and the consumer s income is $220. Determine the optimum consumption basket Jane likes hamburgers (H) and milkshakes (M). Her indifference curves are bowed in toward the origin and do not intersect the axes. The price of a milkshake is $1 and the price of a hamburger is $3. She is spending all her income at the basket she is currently consuming, and her marginal rate of substitution of hamburgers for milkshakes is 2. Is she at an optimum? If so, show why. If not, should she buy fewer hamburgers and more milkshakes, or the reverse? 4.7. Ray buys only hamburgers and bottles of root beer out of a weekly income of $100. He currently consumes 20 bottles of root beer per week, and his marginal utility of root beer is 6. The price of root beer is $2 per bottle. Currently, he also consumes 15 hamburgers per week, and his marginal utility of a hamburger is 8. Is Ray maximizing utility at his current consumption basket? If not, should he buy more hamburgers each week, or fewer?

2 142 CHPTER 4 CONSUMER CHOICE 4.8. Dave currently consumes 10 hot dogs and 6 sodas each week. t his current consumption basket, his marginal utility for hot dogs is 5 and his marginal utility for sodas is 3. If the price of one hot dog is $1 and the price of one soda is $0.50, is Dave currently maximizing his utility? If not, how should he reallocate his spending in order to increase his utility? 4.9. Helen s preferences over CDs (C ) and sandwiches (S) are given by U(S,C) SC 10(S C), with MU C S 10 and MU S C 10. If the price of a CD is $9 and the price of a sandwich is $3, and Helen can spend a combined total of $30 each day on these goods, find Helen s optimal consumption basket The utility that Corey obtains by consuming hamburgers (H) and hot dogs (S) is given by U(H, S) 1H 1S 4. The marginal utility of hamburgers is and the marginal utility of steaks is equal to # 1H 1S 4 a) Sketch the indifference curve corresponding to the utility level U 12. b) Suppose that the price of hamburgers is $1 per hamburger and the price of steak is $8 per steak. Moreover, suppose that Corey can spend $100 per month on these two foods. Sketch Corey s budget line for hamburgers and steak given this budget. c) Based on your answer to parts (a) and (b), what is Corey s optimal consumption basket given his budget? This problem will help you understand what happens if the marginal rate of substitution is not diminishing. Dr. Strangetaste buys only french fries (F) and hot dogs (H) out of his income. He has positive marginal utilities for both goods, and his MRS H,F is increasing. The price of hot dogs is P H, and the price of french fries is P F. a) Draw several of Dr. Strangetaste s indifference curves, including one that is tangent to his budget line. b) Show that the point of tangency does not represent a basket at which utility is maximized, given the budget constraint. Using the indifference curves you have drawn, indicate on your graph where the optimal basket is located Julie consumes two goods, food and clothing, and always has a positive marginal utility for each good. Her income is 24. Initially, the price of food is 2 and the price of clothing is 2. fter new government policies are implemented, the price of food falls to 1 and the price of clothing rises to 4. Suppose, under the initial budget constraint, her optimal choice is 10 units of food and 2 units of clothing. a) fter the prices change, can you predict whether her utility will be higher, lower, or the same as under the initial prices? b) Does your answer require that there be a diminishing marginal rate of substitution of food for clothing? Explain Toni likes to purchase round trips between the cities of Pulmonia and Castoria and other goods out of her income of $10,000. Fortunately, Pulmonian irways provides air service and has a frequent flyer program. round trip between the two cities normally costs $500, but any customer who makes more than 10 trips a year gets to make additional trips during the year for only $200 per round trip. a) On a graph with round trips on the horizontal axis and other goods on the vertical axis, draw Toni s budget line. (Hint: This problem demonstrates that a budget line need not always be a straight line.) b) On the graph you drew in part (a), draw a set of indifference curves that illustrates why Toni may be better off with the frequent flyer program. c) On a new graph draw the same budget line you found in part (a). Now draw a set of indifference curves that illustrates why Toni might not be better off with the frequent flyer program consumer has preferences between two goods, hamburgers (measured by H) and milkshakes (measured by M ). His preferences over the two goods are represented by the utility function U 1H 1M. For this utility function MU H 1/(21H) and MU M 1/(2/1M). a) Determine if there is a diminishing MRS H,M for this utility function. b) Draw a graph to illustrate the shape of a typical indifference curve. Label the curve U 1. Does the indifference curve intersect either axis? On the same graph, draw a second indifference curve U 2, with U 2 U 1. c) The consumer has an income of $24 per week. The price of a hamburger is $2 and the price of a milkshake is $1. How many milkshakes and hamburgers will he buy each week if he maximizes utility? Illustrate your answer on a graph Justin has the utility function U xy, with the marginal utilities MU x yand MU y x. The price of x is 2, the price of y is p y, and his income is 40. When he maximizes utility subject to his budget constraint, he purchases 5 units of y. What must be the price of y and the amount of x consumed? student consumes root beer and a composite good whose price is $1. Currently, the government imposes an excise tax of $0.50 per six-pack of root beer. The student now purchases 20 six-packs of root beer per month. (Think of the excise tax as increasing the price of root beer by $0.50 per six-pack over what the price would

3 PROBLEMS 143 be without the tax.) The government is considering eliminating the excise tax on root beer and, instead, requiring consumers to pay $10.00 per month as a lump-sum tax (i.e., the student pays a tax of $10.00 per month, regardless of how much root beer is consumed). If the new proposal is adopted, how will the student s consumption pattern (in particular, the amount of root beer consumed) and welfare be affected? (ssume that the student s marginal rate of substitution of root beer for other goods is diminishing.) When the price of gasoline is $2.00 per gallon, Joe consumes 1,000 gallons per year. The price increases to $2.50, and to offset the harm to Joe, the government gives him a cash transfer of $500 per year. Will Joe be better off or worse off after the price increase and cash transfer than he was before? What will happen to his gasoline consumption? (ssume that Joe s marginal rate of substitution of gasoline for other goods is diminishing.) Paul consumes only two goods, pizza (P) and hamburgers (H), and considers them to be perfect substitutes, as shown by his utility function: U(P,H) P 4H. The price of pizza is $3 and the price of hamburgers is $6, and Paul s monthly income is $300. Knowing that he likes pizza, Paul s grandmother gives him a birthday gift certificate of $60 redeemable only at Pizza Hut. Though Paul is happy to get this gift, his grandmother did not realize that she could have made him exactly as happy by spending far less than she did. How much would she have needed to give him in cash to make him just as well off as with the gift certificate? Jack makes his consumption and saving decisions two months at a time. His income this month is $1,000, and he knows that he will get a raise next month, making his income $1,050. The current interest rate (at which he is free to borrow or lend) is 5 percent. Denoting this month s consumption by x and next month s by y, for each of the following utility functions state whether Jack would choose to borrow, lend, or do neither in the first month. (Hint: In each case, start by assuming that Jack would simply spend his income in each month without borrowing or lending money. Would doing so be optimal?) a) U(x,y) xy 2,MU x y 2,MU y 2xy b) U(x,y) x 2 y,mu x 2xy,MU y x 2 c) U(x,y) xy,mu x y,mu y x The figure in this problem shows a budget set for a consumer over two time periods, with a borrowing rate r B and a lending rate r L, with r L r B. The consumer purchases C 1 units of a composite good in period 1 and C 2 units in period 2. The following is a general fact about consumers making consumption decisions over two time periods: Let denote the basket at which a consumer spends exactly his income each period (the point at the kink of the budget line). Then a consumer with a diminishing MRS C1, C 2 will choose to borrow in the first period if at basket MRS C1, C r B and will choose to lend if at basket MRS C1, C r L. If the MRS lies between these two values, then he will neither borrow nor lend. (You can try to prove this if you like. Keep in mind that diminishing MRS plays an important role in the proof.) Using this rule, consider the decision of Meg, who earns $2,000 this month and $2,200 the next with a utility function given by U(C 1,C 2 ) C 1 C 2, where the C s denote the value of consumption in each month. For this utility function MU C1 C 2 and MU C2 C 1. Suppose r L 0.05 (the lending rate is 5 percent) and r B 0.12 (the borrowing rate is 12 percent). Would Meg borrow, lend, or do neither this month? What if the borrowing rate fell to 8 percent? C 2 2, ,000(1.05) 2,200 Slope = ,000 Slope = , ,200/ Sally consumes housing (denote the number of units of housing by h) and other goods (a composite good whose units are measured by y), both of which she likes. Initially she has an income of $100, and the price of a unit of housing (P h ) is $10. t her initial basket she consumes 2 units of housing. few months later her income rises to $120; unfortunately, the price of housing in her city also rises, to $15. The price of the composite good does not change. t her later basket she consumes 1 unit of housing. Using revealed preference analysis (without drawing indifference curves), what can you say about how she ranks her initial and later baskets? Samantha purchases food (F ) and other goods (Y ) with the utility function U FY, with MU F Y and MU y F. Her income is 12. The price of a food is 2 and the price of other goods 1. a) How many units of food does she consume when she maximizes utility? b) The government has recently completed a study suggesting that, for a healthy diet, every consumer should consume at least F 8 units of food. The government is considering giving a consumer like Samantha a cash subsidy that would induce her to buy F 8. How large would the cash subsidy need to be? Show her optimal basket with the cash subsidy on an optimal choice diagram with F on the horizontal axis and Y on the vertical axis. C 1

4 144 CHPTER 4 CONSUMER CHOICE c) s an alternative to the cash subsidy in part (b), the government is also considering giving consumers like Samantha food stamps, that is, vouchers with a cash value that can only be redeemed to purchase food. Verify that if the government gives her vouchers worth $16, she will choose F 8. Illustrate her optimal choice on an optimal choice diagram. (You may use the same graph you drew in part (b).) s shown in the following figure, a consumer buys two goods, food and housing, and likes both goods. When she has budget line BL 1, her optimal choice is basket. Given budget line BL 2, she chooses basket B, and with BL 3, she chooses basket C. Housing B C BL 1 BL 2 BL 3 Food a) What can you infer about how the consumer ranks baskets,b, and C? If you can infer a ranking, explain how. If you cannot infer a ranking, explain why not. b) On the graph, shade in (and clearly label) the areas that are revealed to be less preferred to basket B, and explain why you indicated these areas. c) On the graph, shade in (and clearly label) the areas that are revealed to be (more) preferred to basket B, and explain why you indicated these areas The following graph shows the consumption decisions of a consumer over bundles of x and y, both of which he likes. When faced with budget line BL 1,he chose basket, and when faced with budget line BL 2,he chose basket B. If he were to face budget line BL 3, what possible set of baskets could he choose in order for his behavior to be consistent with utility maximization? y B BL 3 BL 1 BL x Darrell has a monthly income of $60. He spends this money making telephone calls home (measured in minutes of calls) and on other goods. His mobile phone company offers him two plans: Plan : Pay no monthly fee and make calls for $0.50 per minute. Plan B: Pay a $20 monthly fee and make calls for $0.20 per minute. Graph Darrell s budget constraint under each of the two plans. If Plan is better for him, what is the set of baskets he may purchase if his behavior is consistent with utility maximization? What baskets might he purchase if Plan B is better for him? Figure 4.17 illustrates the case in which a consumer is better off with a quantity discount. Can you draw an indifference map for a consumer who would not be better off with the quantity discount? ngela has a monthly income of $120, which she spends on MP3s and a composite good (whose price you may assume is $1 throughout this problem). Currently, she does not belong to an MP3 club, so she pays the retail price of an MP3 of $2; her optimal basket includes 20 MP3s monthly. For the past several months steroid, a media company, has offered her the chance to join their Premium Club ; to join the club she would need to pay a membership fee of $60 per month, but then she could buy all the MP3s she wants at a price of $0.50. She has decided not to join the club. steroid has now introduced an Economy Club ; to join, ngela would need to pay a membership fee of $30 per month, but then she could buy all the MP3s she wants at a price of $1. Draw a graph illustrating (1) ngela s budget line and optimal basket when she joins no club, (2) the budget line she would have faced had she joined the Premium Club, and (3) her budget line if she joins the Economy Club. Will ngela surely want to join the Economy Club? If she were to join the club, how many MP3s per month might she buy? Show how you arrive at your answers using a revealed preference argument lex buys two goods, food (F) and clothing (C). He likes both goods. His preferences for the goods do not change from month to month. The following table shows his income, the baskets he selected, and the prices of the goods over a two-month period. Month P F P C Income Basket Chosen F 16,C F 14,C 5

5 PPENDIX 1: THE MTHEMTICS OF CONSUMER CHOICE 145 a) On the graph with F on the horizontal axis and C on the vertical axis, plot and clearly label the budget lines and consumption baskets during these two weeks. Label the consumption bundle in week 1 by point on the graph and the consumption basket in week 2 by point B. Using revealed preference analysis, what can you say about lex s preferences for baskets and B (i.e., how does he rank them)? b) In month 3 lex s income rises to 57. The prices of food and clothing are both 3. ssuming his preferences do not change, describe the set of baskets he might consume in month 3 if he continues to maximize utility. Show this set of baskets in the graph Brian consumes units of electricity (E) and a composite good (Y ), whose price is always 1. He likes both goods. In period 1 the power company sets the price of electricity at $7 per unit, for all units of electricity consumed. Brian consumes his optimal basket, 20 units of electricity and 70 units of the composite good. In period 2 the power company then revises its pricing plan, charging $10 per unit for the first 5 units and $4 per unit for each additional unit. Brian s income is unchanged. Brian s optimal basket with this plan includes 30 units of electricity and 60 units of the composite good. In period 3 the power company allows the consumer to choose either the pricing plan in period 1 or the plan in period 2. Brian s income is unchanged. Which pricing plan will he choose? Illustrate your answer with a clearly labeled graph Carina consumes two goods, X and Y, both of which she likes. In month 1 she chooses basket given budget line BL 1. In month 2 she chooses B given budget line BL 2, and in month 3 she chooses C given budget line BL 3. ssume her indifference map is unchanged over the three months. Use the theory of revealed preference to show whether her choices are consistent with utilitymaximizing behavior. If so, show how she ranks the three baskets. If it is not possible to infer how she ranks the baskets, explain why not. Y B C BL 2 BL 3 BL 1 X PPENDIX 1: The Mathematics of Consumer Choice In this section we solve the consumer choice problem using the calculus technique of Lagrange multipliers. Suppose the consumer buys two goods, where x measures the amount of the first good and y the amount of the second good. The price of the first good is P x and the price of the second is P y. The consumer has an income I. Let s assume that the marginal utilities of both goods are positive, so we know that he will expend all of his income at his optimal basket. The consumer choice problem is then: maxu(x, y) (x,y) subject to: P x x P y y I (4.1) We define the Lagrangian ( ) as (x, y, l) U(x, y) l(i P x x P y y), where is a Lagrange multiplier. The first-order necessary conditions for an interior optimum (with x 0 and y 0) are 0 0x 010U (x, y) 0x lp x (4.2) 0 0y 010U (x, y) 0y lp y 0 0l 01I P xx P y y 0 (4.3) (4.4)

6 200 CHPTER 5 THE THEORY OF DEMND PROBLEMS 5.1. Figure 5.2(a) shows a consumer s optimal choices of food and clothing for three values of weekly income: I 1 $40, I 2 $68, and I 3 $92. Figure 5.2(b) illustrates how the consumer s demand curve for food shifts as income changes. Draw three demand curves for clothing (one for each level of income) to illustrate how changes in income affect the consumer s purchases of clothing Use the income consumption curve in Figure 5.2(a) to draw the Engel curve for clothing, assuming the price of food is $2 and the price of clothing is $ Show that the following statements are true: a) n inferior good has a negative income elasticity of demand. b) good whose income elasticity of demand is negative will be an inferior good If the demand for a product is perfectly price inelastic, what does the corresponding price consumption curve look like? Draw a graph to show the price consumption curve nn consumes five goods. The prices of all goods are fixed. The price of good x is p x. She spends 25 percent of her income on good x, regardless of the size of her income. a) Show that her income elasticity of demand of good x is the same for any level of income, and determine its value. b) Would the value of the income elasticity of demand for x be different if nn always spends 60 percent of her income on good x? 5.6. Suzie purchases two goods, food and clothing. She has the utility function U(x, y) xy, where x denotes the amount of food consumed and y the amount of clothing. The marginal utilities for this utility function are MU x y and MU y x. a) Show that the equation for her demand curve for clothing is y I (2P y ). b) Is clothing a normal good? Draw her demand curve for clothing when the level of income is I 200. Label this demand curve D 1. Draw the demand curve when I 300 and label this demand curve D 2. c) What can be said about the cross-price elasticity of demand of food with respect to the price of clothing? 5.7. Karl s preferences over hamburgers (H ) and beer (B) are described by the utility function: U(H, B) min(2h, 3B). His monthly income is I dollars, and he only buys these two goods out of his income. Denote the price of hamburgers by P H and of beer by P B. a) Derive Karl s demand curve for beer as a function of the exogenous variables. b) Which affects Karl s consumption of beer more: a one dollar increase in P H or a one dollar increase in P B? 5.8. David has a quasilinear utility function of the form U(x, y) 1x y, with associated marginal utility functions MU x 1/(21x ) and MU y 1. a) Derive David s demand curve for x as a function of the prices, P x and P y. Verify that the demand for x is independent of the level of income at an interior optimum. b) Derive David s demand curve for y. Is y a normal good? What happens to the demand for y as P x increases? 5.9. Rick purchases two goods, food and clothing. He has a diminishing marginal rate of substitution of food for clothing. Let x denote the amount of food consumed and y the amount of clothing. Suppose the price of food increases from P x1 to P x2. On a clearly labeled graph, illustrate the income and substitution effects of the price change on the consumption of food. Do so for each of the following cases: a) Case 1: Food is a normal good. b) Case 2: The income elasticity of demand for food is zero. c) Case 3: Food is an inferior good, but not a Giffen good. d) Case 4: Food is a Giffen good Reggie consumes only two goods: food and shelter. On a graph with shelter on the horizontal axis and food on the vertical axis, his price consumption curve for shelter is a vertical line. Draw a pair of budget lines and indifference curves that are consistent with this description of his preferences. What must always be true about Reggie s income and substitution effects as the result of a change in the price of shelter? Ginger s utility function is U(x, y) x 2 y, with associated marginal utility functions MU x 2xy and MU y x 2. She has income I 240 and faces prices P x $8 and P y $2. a) Determine Ginger s optimal basket given these prices and her income. b) If the price of y increases to $8 and Ginger s income is unchanged, what must the price of x fall to in order for her to be exactly as well off as before the change in P y? nn s utility function is U(x, y) x y, with associated marginal utility functions MU x 1 and MU y 1. nn has income I 4. a) Determine all optimal baskets given that she faces prices P x 1 and P y 1.

7 PROBLEMS 201 b) Determine all optimal baskets given that she faces prices P x 1 and P y 2. c) What is demand for y when P x 1 and P y 1? What is demand for y when P x 1 and P y 1? What is demand for y when P x 1 and P y 1? Plot nn s demand for y as a function of P y. d) Repeat the exercises in (a), (b) and (c) for U(x, y) 2x y, with associated marginal utility functions MU x 2 and MU y 1, and with the same level of income Some texts define a luxury good as a good for which the income elasticity of demand is greater than 1. Suppose that a consumer purchases only two goods. Can both goods be luxury goods? Explain Scott consumes only two goods, steak and ale. When the price of steak falls, he buys more steak and more ale. On an optimal choice diagram (with budget lines and indifference curves), illustrate this pattern of consumption Dave consumes only two goods, coffee and doughnuts. When the price of coffee falls, he buys the same amount of coffee and more doughnuts. a) On an optimal choice diagram (with budget lines and indifference curves), illustrate this pattern of consumption. b) Is this purchasing behavior consistent with a quasilinear utility function? Explain (This problem shows that an optimal consumption choice need not be interior and may be at a corner point.) Suppose that a consumer s utility function is U(x, y) xy 10y. The marginal utilities for this utility function are MU x y and MU y x 10. The price of x is P x and the price of y is P y, with both prices positive. The consumer has income I. a) ssume first that we are at an interior optimum. Show that the demand schedule for x can be written as x I (2P x ) 5. b) Suppose now that I 100. Since x must never be negative, what is the maximum value of P x for which this consumer would ever purchase any x? c) Suppose P y 20 and P x 20. On a graph illustrating the optimal consumption bundle of x and y, show that since P x exceeds the value you calculated in part (b), this corresponds to a corner point at which the consumer purchases only y. (In fact, the consumer would purchase y I P y 5 units of y and no units of x.) d) Compare the marginal rate of substitution of x for y with the ratio (P x P y ) at the optimum in part (c). Does this verify that the consumer would reduce utility if she purchased a positive amount of x? e) ssuming income remains at 100, draw the demand schedule for x for all values of P x. Does its location depend on the value of P y? The accompanying figure illustrates the change in consumer surplus, given by rea BEC, when the price decreases from P 1 to P 2. This area can be divided into the rectangle BDC and the triangle BDE. Briefly describe what each area represents, separately, keeping in mind the fact that consumer surplus is a measure of how well off consumers are (therefore the change in consumer surplus represents how much better off consumers are). (Hint: Note that a price decrease also induces an increase in the quantity consumed.) P 1 P 2 C Demand B D q 1 q The demand function for widgets is given by D(P) 16 2P. Compute the change in consumer surplus when the price of a widget increases from $1 to $3. Illustrate your result graphically Jim s preferences over cookies (x) and other goods ( y) are given by U(x, y) xy with associated marginal utility functions MU x y and MU y x. His income is $20. a) Find Jim s demand schedule for x when the price of y is P y $1. b) Illustrate graphically the change in consumer surplus when the price of x increases from $1 to $ Lou s preferences over pizza (x) and other goods ( y) are given by U(x, y) xy, with associated marginal utilities MU x y and MU y x. His income is $120. a) Calculate his optimal basket when P x 4 and P y 1. b) Calculate his income and substitution effects of a decrease in the price of food to $3. c) Calculate the compensating variation of the price change. d) Calculate the equivalent variation of the price change Carina buys two goods, food F and clothing C, with the utility function U FC F. Her marginal utility of food is MU F C 1 and her marginal utility of E

8 202 CHPTER 5 THE THEORY OF DEMND clothing is MU C F. She has an income of 20. The price of clothing is 4. a) Derive the equation representing Carina s demand for food, and draw this demand curve for prices of food ranging between 1 and 6. b) Calculate the income and substitution effects on Carina s consumption of food when the price of food rises from 1 to 4, and draw a graph illustrating these effects. Your graph need not be exactly to scale, but it should be consistent with the data. c) Determine the numerical size of the compensating variation (in monetary terms) associated with the increase in the price of food from 1 to Suppose the market for rental cars has two segments, business travelers and vacation travelers. The demand curve for rental cars by business travelers is Q b P, where Q b is the quantity demanded by business travelers (in thousands of cars) when the rental price is P dollars per day. No business customers will rent cars if the price exceeds $140 per day. The demand curve for rental cars by vacation travelers is Q v P, where Q v is the quantity demanded by vacation travelers (in thousands of cars) when the rental price is P dollars per day. No vacation customers will rent cars if the price exceeds $80 per day. a) Fill in the table to find the quantities demanded in the market at each price. Market Business Vacation Demand Price (thousands of (thousands of (thousands of ($/day) cars/day) cars/day) cars/day) b) Graph the demand curves for each segment, and draw the market demand curve for rental cars. c) Describe the market demand curve algebraically. In other words, show how the quantity demanded in the market Q m depends on P. Make sure that your algebraic equation for the market demand is consistent with your answers to parts (a) and (b). d) If the price of a rental car is $60, what is the consumer surplus in each market segment? There are two types of consumers in a market for sheet metal. Let P represent the market price. The total quantity demanded by Type I consumers is Q P, for 0 P 50. The total quantity demanded by Type II consumers is Q 2 40 P, for 0 P 40. Draw the total market demand on a clearly labeled graph There are two consumers on the market: Jim and Donna. Jim s utility function is U(x, y) xy, with associated marginal utility functions MU x y and MU y x. Donna s utility function is U(x, y) x 2 y, with associated marginal utility functions MU x 2xy and MU y x 2. Income of Jim is I J 100 and income of Donna is I D 150. a) Find optimal baskets of Jim and Donna when price of y is P y 1 and price of x is P. b) On separate graphs plot Jim s and Donna s demand schedule for x for all values of P. c) Compute and plot aggregate demand when Jim and Donna are the only consumers. d) Plot aggregate demand when there is one more consumer that has identical utility function and income as Donna One million consumers like to rent movie videos in Pulmonia. Each has an identical demand curve for movies. The price of a rental is $P. t a given price, will the market demand be more elastic or less elastic than the demand curve for any individual? (ssume there are no network externalities.) Suppose that Bart and Homer are the only people in Springfield who drink 7-UP. Moreover their inverse demand curves for 7-UP are, respectively, P 10 4Q B and P 25 2Q H, and, of course, neither one can consume a negative amount. Write down the market demand curve for 7-UP in Springfield, as a function of all possible prices Joe s income consumption curve for tea is a vertical line on an optimal choice diagram, with tea on the horizontal axis and other goods on the vertical axis. a) Show that Joe s demand curve for tea must be downward sloping. b) When the price of tea drops from $9 to $8 per pound, the change in Joe s consumer surplus (i.e., the change in the area under the demand curve) is $30 per month. Would you expect the compensating variation and the equivalent variation resulting from the price decrease to be near $30? Explain Consider the optimal choice of labor and leisure discussed in the text. Suppose a consumer works the first

9 8 hours of the day at a wage rate of $10 per hour, but receives an overtime wage rate of $20 for additional time worked. a) On an optimal choice diagram, draw the budget constraint. (Hint: It is not a straight line.) b) Draw a set of indifference curves that would make it optimal for him to work 4 hours of overtime each day Terry s utility function over leisure (L) and other goods (Y ) is U(L, Y ) Y LY. The associated marginal utilities are MU Y 1 L and MU L Y. He purchases other goods at a price of $1, out of the income he earns from working. Show that, no matter what Terry s wage rate, the optimal number of hours of leisure that he consumes is always the same. What is the number of hours he would like to have for leisure? Consider Noah s preferences for leisure (L) and other goods (Y ), U(L, Y ) 1L 1Y. The associated marginal utilities are MU L 1/(21L) and MU Y 1/(21Y ). Suppose that P Y $1. Is Noah s supply of labor backward bending? Raymond consumes leisure (L hours per day) and other goods (Y units per day), with preferences described by U(L, Y ) L 21Y. The associated marginal utilities are MU Y 1 and MU L 1/1L. The price of other goods is 1 euro per unit. The wage rate is w euros per hour. a) Show how the number of units of leisure Raymond chooses depends on the wage rate. b) How does Raymond s daily income depend on the wage rate? c) Does Raymond work more when the wage rate rises? Julie buys food and other goods. She has an income of $400 per month. The price of food is initially $1.00 per unit. It then rises to $1.20 per unit. The prices of other goods do not change. To help Julie out, her mother offers to send her a check each month to supplement her income. Julie tells her mother, Thanks, Mom. If you would send me a check for $50 per month, I would be exactly as happy paying $1.20 per unit as I would have been paying $1.00 per unit and not receiving the $50 from you. Which of the following statements is true? Explain. The increased price of food has: a) an income effect of $50 per month b) an income effect of $50 per month PROBLEMS 203 c) a compensating variation of $50 per month d) a compensating variation of $50 per month e) an equivalent variation of $50 per month f ) an equivalent variation of $50 per month Gina lives in Chicago and very much enjoys traveling by air to see her mother in Italy. On the accompanying graph, x denotes her number of round trips to Italy each year. The composite good y measures her annual consumption of other goods; the price of the composite good is p y, which is constant in this problem. Several indifference curves from her preference map are drawn, with levels of utility U 1 U 2 U 3 U 4 U 5. If she spends all her income on the composite good, she can purchase y* units, as shown in the graph. When the initial price of air travel is $1,000, she can purchase as many as 18 round trips if she spends all her income on air travel to Italy. a) Make a copy of the graph, and use it to determine the income and substitution effects on the number of round trips Gina makes as the price of a round trip increases from $1,000 to $3,000. Clearly label these effects on the graph. b) Using the graph, estimate the numerical size of the compensating variation associated with the price increase. You may refer to the graph to explain your answer. c) Will the consumer surplus measured using Gina s demand for air travel to Italy provide an exact measure of the monetary value she associates with the price increase? In a sentence, explain why or why not. y* y 0 U 2 U1 U 5 U 4 U x