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100 CHAPTER 3 CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY

PROBLEMS

3.1. Bill has a utility function over food and gasoline d) Nuts (which the consumer neither likes nor dislikes)
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with the equation U x y, where x measures the quan- and ice cream (which the consumer likes)
tity of food consumed and y measures the quantity of e) Apples (which the consumer likes) and liver (which
gasoline. Show that a consumer with this utility function the consumer dislikes)
believes that more is better for each good.
3.7. Alexa likes ice cream, but dislikes yogurt. If you
3.2. Consider the single-good utility function make her eat another gram of yogurt, she always requires
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U(x) 3x , with a marginal utility given by MU x 6x. two extra grams of ice cream to maintain a constant level
Plot the utility and marginal utility functions on two of satisfaction. On a graph with grams of yogurt on the
separate graphs. Does this utility function satisfy the vertical axis and grams of ice cream on the horizontal
principle of diminishing marginal utility? Explain.
axis, graph some typical indifference curves and show the
3.3. Jimmy has the following utility function for hot directions of increasing utility.
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dogs: U(H) 10H H , with MU H 10 2H.
3.8. Joe has a utility function over hamburgers and hot
a) Plot the utility and marginal utility functions on two dogs given by U x 1y, where x is the quantity of
separate graphs.
hamburgers and y is the quantity of hot dogs. The mar-
b) Suppose that Jimmy is allowed to consume as many ginal utilities for this utility function are MU x 1 and
hot dogs as he likes and that hot dogs cost him nothing. MU y 1 (21y ).
Show, both algebraically and graphically, the value of H
at which he would stop consuming hot dogs. Does this utility function have the property that MRS x,y
is diminishing?
3.4. Consider the utility function U(x, y) y1x with
the marginal utilities MU x y (21x) and MU y 1x .
3.9. Julie and Toni consume two goods with the follow-
a) Does the consumer believe that more is better for ing utility functions:
each good?
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b) Do the consumer’s preferences exhibit a diminish- U Julie (x y) , MU x Julie 2(x y), MU y Julie 2(x y)
ing marginal utility of x? Is the marginal utility of y U Toni x y, MU x Toni 1, MU y Toni 1
diminishing?
a) Graph an indifference curve for each of these utility
3.5. Carlos has a utility function that depends on the functions.
number of musicals and the number of operas seen each b) Julie and Toni will have the same ordinal ranking of
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month. His utility function is given by U xy , where x different baskets if, when basket A is preferred to basket
is the number of movies seen per month and y is the B by one of the functions, it is also preferred by the other.
number of operas seen per month. The corresponding Do Julie and Toni have the same ordinal ranking of dif-
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marginal utilities are given by: MU x y and MU y 2xy. ferent baskets of x and y? Explain.
a) Does Carlos believe that more is better for each good?
b) Does Carlos have a diminishing marginal utility for 3.10. The utility that Julie receives by consuming food
each good? F and clothing C is given by U(F, C) FC. For this
utility function, the marginal utilities are MU F C and
3.6. For the following sets of goods draw two indiffer- MU C F.
ence curves, U 1 and U 2 , with U 2 U 1 . Draw each graph a) On a graph with F on the horizontal axis and C on the
placing the amount of the first good on the horizontal vertical axis, draw indifference curves for U 12, U 18,
axis. and U 24.
a) Hot dogs and chili (the consumer likes both and has a b) Do the shapes of these indifference curves suggest
diminishing marginal rate of substitution of hot dogs for that Julie has a diminishing marginal rate of substitution
chili) of food for clothing? Explain.
b) Sugar and Sweet’N Low (the consumer likes both and c) Using the marginal utilities, show that the MRS F,C
will accept an ounce of Sweet’N Low or an ounce of C F . What is the slope of the indifference curve U 12
sugar with equal satisfaction) at the basket with 2 units of food and 6 units of clothing?
c) Peanut butter and jelly (the consumer likes exactly What is the slope at the basket with 4 units of food and
2 ounces of peanut butter for every ounce of jelly) 3 units of clothing? Do the slopes of the indifference