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relationship between the indifference curve and the marginal rate of substitution of hamburgers for Cokes. c) Suppose the marginal rate of substitution of ham- burgers for Cokes is constant. In this case, are ham- burgers and Cokes perfect substitutes or perfect complements? d) Suppose that Michael always wants two hamburgers along with every Coke. Draw a typical indifference curve. In this case, are hamburgers and Cokes perfect substi- tutes or perfect complements? 11. Suppose a consumer is currently purchasing 47 different goods, one of which is housing. The quantity of housing is measured by H. Explain why, if you wanted to measure the consumer\u2019s marginal utility of housing (MUH) at the current basket, the levels of the other 46 goods consumed would be held fixed. c03consumerpreferencesandtheconceptofutility.qxd 7/14/10 2:13 PM Page 99 100 CHAPTER 3 CONSUMER PREFERENCES AND THE CONCEPT OF UTIL ITY P R O B L E M S 3.1. Bill has a utility function over food and gasoline with the equation U \ufffd x2y, where x measures the quan- tity of food consumed and y measures the quantity of gasoline. Show that a consumer with this utility function believes that more is better for each good. 3.2. Consider the single-good utility function U(x) \ufffd 3x2, with a marginal utility given by MUx \ufffd 6x. Plot the utility and marginal utility functions on two separate graphs. Does this utility function satisfy the principle of diminishing marginal utility? Explain. 3.3. Jimmy has the following utility function for hot dogs: U(H) \ufffd 10H \ufffd H2, with MUH \ufffd 10 \ufffd 2H. a) Plot the utility and marginal utility functions on two separate graphs. b) Suppose that Jimmy is allowed to consume as many hot dogs as he likes and that hot dogs cost him nothing. Show, both algebraically and graphically, the value of H at which he would stop consuming hot dogs. 3.4. Consider the utility function with the marginal utilities and . a) Does the consumer believe that more is better for each good? b) Do the consumer\u2019s preferences exhibit a diminish- ing marginal utility of x? Is the marginal utility of y diminishing? 3.5. Carlos has a utility function that depends on the number of musicals and the number of operas seen each month. His utility function is given by U \ufffd xy2, where x is the number of movies seen per month and y is the number of operas seen per month. The corresponding marginal utilities are given by: MUx \ufffd y2 and MUy \ufffd 2xy. a) Does Carlos believe that more is better for each good? b) Does Carlos have a diminishing marginal utility for each good? 3.6. For the following sets of goods draw two indiffer- ence curves, U1 and U2, with U2 \ufffd U1. Draw each graph placing the amount of the first good on the horizontal axis. a) Hot dogs and chili (the consumer likes both and has a diminishing marginal rate of substitution of hot dogs for chili) b) Sugar and Sweet\u2019N Low (the consumer likes both and will accept an ounce of Sweet\u2019N Low or an ounce of sugar with equal satisfaction) c) Peanut butter and jelly (the consumer likes exactly 2 ounces of peanut butter for every ounce of jelly) MUy \ufffd 1xMUx \ufffd y\ufffd(21x) U(x, y) \ufffd y1x d) Nuts (which the consumer neither likes nor dislikes) and ice cream (which the consumer likes) e) Apples (which the consumer likes) and liver (which the consumer dislikes) 3.7. Alexa likes ice cream, but dislikes yogurt. If you make her eat another gram of yogurt, she always requires two extra grams of ice cream to maintain a constant level of satisfaction. On a graph with grams of yogurt on the vertical axis and grams of ice cream on the horizontal axis, graph some typical indifference curves and show the directions of increasing utility. 3.8. Joe has a utility function over hamburgers and hot dogs given by where x is the quantity of hamburgers and y is the quantity of hot dogs. The mar- ginal utilities for this utility function are MUx \ufffd 1 and Does this utility function have the property that MRSx,y is diminishing? 3.9. Julie and Toni consume two goods with the follow- ing utility functions: a) Graph an indifference curve for each of these utility functions. b) Julie and Toni will have the same ordinal ranking of different baskets if, when basket A is preferred to basket B by one of the functions, it is also preferred by the other. Do Julie and Toni have the same ordinal ranking of dif- ferent baskets of x and y? Explain. 3.10. The utility that Julie receives by consuming food F and clothing C is given by U(F, C) \ufffd FC. For this utility function, the marginal utilities are MUF \ufffd C and MUC \ufffd F. a) On a graph with F on the horizontal axis and C on the vertical axis, draw indifference curves for U \ufffd 12, U \ufffd 18, and U \ufffd 24. b) Do the shapes of these indifference curves suggest that Julie has a diminishing marginal rate of substitution of food for clothing? Explain. c) Using the marginal utilities, show that the MRSF,C \ufffd . What is the slope of the indifference curve U \ufffd 12 at the basket with 2 units of food and 6 units of clothing? What is the slope at the basket with 4 units of food and 3 units of clothing? Do the slopes of the indifference C\ufffdF MUToniy \ufffd 1MUTonix \ufffd 1,UToni \ufffd x \ufffd y, MUJuliey \ufffd 2(x \ufffd y)MUJuliex \ufffd 2(x \ufffd y),UJulie \ufffd (x \ufffd y)2, MUy \ufffd 1\ufffd(21y ). U \ufffd x \ufffd 1y, c03consumerpreferencesandtheconceptofutility.qxd 7/14/10 12:08 PM Page 100 PROBLEMS 101 curves indicate that Julie has a diminishing marginal rate of substitution of food for clothing? (Make sure your an- swers to parts (b) and (c) are consistent!) 3.11. Sandy consumes only hamburgers (H ) and milk- shakes (M ). At basket A, containing 2 hamburgers and 10 milkshakes, his MRSH,M is 8. At basket B, containing 6 hamburgers and 4 milkshakes, his MRSH,M is . Both baskets A and B are on the same indifference curve. Draw the indifference curve, using information about the MRSH,M to make sure that the curvature of the indiffer- ence curve is accurately depicted. 3.12. Adam likes his caffé latte prepared to contain exactly espresso and steamed milk by volume. On a graph with the volume of steamed milk on the horizon- tal axis and the volume of espresso on the vertical axis, draw two of his indifference curves, U1 and U2, with U1 \ufffd U2. 3.13. Draw indifference curves to represent the follow- ing types of consumer preferences. a) I like both peanut butter and jelly, and always get the same additional satisfaction from an ounce of peanut butter as I do from 2 ounces of jelly. b) I like peanut butter, but neither like nor dislike jelly. c) I like peanut butter, but dislike jelly. d) I like peanut butter and jelly, but I only want 2 ounces of peanut butter for every ounce of jelly. 3.14. Dr. Strangetaste buys only food (F ) and clothing (C) out of his income. He has positive marginal utilities for both goods, and his MRSF,C is increasing. Draw two of Dr. Strangetaste\u2019s indifference curves, U1 and U2, with U2 \ufffd U1. The following exercises will give you practice in working with a variety of utility functions and marginal utilities and will help you understand how to graph indifference curves. 3.15. Consider the utility function U(x, y) \ufffd 3x \ufffd y, with MUx \ufffd 3 and MUy \ufffd 1. a) Is the assumption that more is better satisfied for both goods? b) Does the marginal utility of x diminish, remain con- stant, or increase as the consumer buys more x? Explain. c) What is MRSx,y? d) Is MRSx,y diminishing, constant, or increasing as the consumer substitutes x for y along an indifference curve? e) On a graph with x on the horizontal axis and y on the vertical axis, draw a typical indifference curve (it need not be exactly to scale, but it needs to reflect accurately whether there is a diminishing MRSx,y). Also indicate on your graph whether the indifference curve will intersect either or both axes. Label the curve U1. 3\ufffd41\ufffd4 1\ufffd2 f ) On the same graph