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c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:55 PM Page 101

PROBLEMS 101

curves indicate that Julie has a diminishing marginal rate f ) On the same graph draw a second indifference curve
of substitution of food for clothing? (Make sure your an- U 2 , with U 2 U 1 .
swers to parts (b) and (c) are consistent!)
3.16. Answer all parts of Problem 3.15 for the utility
3.11. Sandy consumes only hamburgers (H ) and milk- function U(x, y) 1xy . The marginal utilities are
shakes (M ). At basket A, containing 2 hamburgers and MU x 1y (21x) and MU y 1x (21y) .
10 milkshakes, his MRS H,M is 8. At basket B, containing
6 hamburgers and 4 milkshakes, his MRS H,M is 1 2 . Both 3.17. Answer all parts of Problem 3.15 for the utility
baskets A and B are on the same indifference curve. Draw function U(x, y) xy x . The marginal utilities are
the indifference curve, using information about the MU x y 1 and MU y x.
MRS H,M to make sure that the curvature of the indiffer- 3.18. Answer all parts of Problem 3.15 for the utility
0.4 0.6
ence curve is accurately depicted. function U(x, y) x y . The marginal utilities are
0.6
0.4
MU x 0.4( y x 0.6 ) and MU y 0.6(x y 0.4 . )
3.12. Adam likes his caffé latte prepared to contain
exactly 1 4 espresso and 3 4 steamed milk by volume. On 3.19. Answer all parts of Problem 3.15 for the utility
a graph with the volume of steamed milk on the horizon- function U 1x 21y . The marginal utilities for x
tal axis and the volume of espresso on the vertical axis, and y are, respectively, MU x 1 (21x) and
draw two of his indifference curves, U 1 and U 2 , with MU y 1 1y .
U 1 U 2 .
3.20. Answer all parts of Problem 3.15 for the utility
2
3.13. Draw indifference curves to represent the follow- function U(x, y) x y 2 . The marginal utilities are
ing types of consumer preferences. MU x 2x and MU y 2y.
a) I like both peanut butter and jelly, and always get the
same additional satisfaction from an ounce of peanut 3.21. Suppose a consumer’s preferences for two goods
butter as I do from 2 ounces of jelly. can be represented by the Cobb–Douglas utility function

U Ax y , where A, , and are positive constants.
b) I like peanut butter, but neither like nor dislike jelly. 1
The marginal utilities are MU x
Ax y and MU y
c) I like peanut butter, but dislike jelly. Ax y . Answer all parts of Problem 3.15 for this
1
d) I like peanut butter and jelly, but I only want 2 ounces utility function.
of peanut butter for every ounce of jelly.
3.22. Suppose a consumer has preferences over two
3.14. Dr. Strangetaste buys only food (F) and clothing goods that can be represented by the quasilinear utility
(C) out of his income. He has positive marginal utilities function U(x, y) 21x y The marginal utilities are
for both goods, and his MRS F,C is increasing. Draw two of MU x 1 1x and MU y 1.
Dr. Strangetaste’s indifference curves, U 1 and U 2 , with a) Is the assumption that more is better satisfied for both
U 2 U 1 . goods?
The following exercises will give you practice in working with b) Does the marginal utility of x diminish, remain con-
a variety of utility functions and marginal utilities and will stant, or increase as the consumer buys more x? Explain.
help you understand how to graph indifference curves. c) What is the expression for MRS x,y ?

3.15. Consider the utility function U(x, y) 3x y, d) Is the MRS x,y diminishing, constant, or increasing as
with MU x 3 and MU y 1. the consumer substitutes more x for y along an indiffer-
ence curve?
a) Is the assumption that more is better satisfied for both
goods? e) On a graph with x on the horizontal axis and y on the
vertical axis, draw a typical indifference curve (it need not
b) Does the marginal utility of x diminish, remain con-
stant, or increase as the consumer buys more x? Explain. be exactly to scale, but it should accurately reflect
whether there is a diminishing MRS x,y ). Indicate on your
c) What is MRS x,y ?
graph whether the indifference curve will intersect either
d) Is MRS x,y diminishing, constant, or increasing as the or both axes.
consumer substitutes x for y along an indifference curve?
f ) Show that the slope of every indifference curve will be
e) On a graph with x on the horizontal axis and y on the the same when x 4. What is the value of that slope?
vertical axis, draw a typical indifference curve (it need not
be exactly to scale, but it needs to reflect accurately 3.23. Daniel and Will each consume two goods. When
whether there is a diminishing MRS x,y ). Also indicate on they consume the same basket, Daniel’s marginal utility
your graph whether the indifference curve will intersect of each good is higher than Will’s. But at any basket they
either or both axes. Label the curve U 1 . both have the same marginal rate of substitution of one