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These problems provide some practice in examining utility functions by looking at
indifference curve maps and at a few functional forms. The primary focus is on
illustrating the notion of quasi-concavity (a diminishing MRS) in various contexts. The
concepts of the budget constraint and utility maximization are not used until the next
chapter.
Comments on Problems
3.1 This problem requires students to graph indifference curves for a variety of
functions, some of which are not quasi-concave.
3.2 This problem introduces the formal definition of quasi-concavity (from Chapter
2) to be applied to the functions studied graphically in Problem 3.1.
3.3 This problem shows that diminishing marginal utility is not required to obtain a
diminishing MRS. All of the functions are monotonic transformations of one
another, so this problem illustrates that diminishing MRS is preserved by
monotonic transformations but diminishing marginal utility is not.
3.4 This problem focuses on whether some simple utility functions exhibit convex
indifference curves.
3.5 This problem is an exploration of the fixed-proportions utility function. The
problem also shows how the goods in such problems can be treated as a
composite commodity.
3.6 This problem asks students to use their imaginations to explain how advertising
slogans might be captured in the form of a utility function.
3.7 This problem shows how utility functions can be inferred from MRS segments. It
is a very simple example of “integrability.”
3.8 This problem offers some practice in deriving utility functions from indifference
curve specifications.
CHAPTER 3:
Preferences and Utility
Chapter 3: Preferences and Utility
14
Analytical Problems
3.9 Initial endowments. This problem shows how initial endowments can be treated
in simple indifference curve analysis.
3.10 Cobb–Douglas utility. This problem provides some exercises with the Cobb–
Douglas function, including how to integrate subsistence levels of consumption
into the functional form.
3.11 Independent marginal utilities. This problem shows how analysis can be
simplified if the cross-partials of the utility function are zero.
3.12 CES utility. This problem shows how distributional weights can be incorporated
into the CES form introduced in the chapter without changing the basic
conclusions about the function.
3.13 The quasi-linear function. This problem provides a brief introduction to the
quasi-linear form, which (in later chapters) will be used to illustrate a number of
interesting outcomes.
3.14 Preference relations. This problem provides a very brief introduction to how
preferences can be treated formally with set-theoretic concepts.
3.15 The benefit function. This problem introduces Luenberger’s notion of reducing
preferences to a cardinal number of replications of a basic bundle of goods.
Solutions
3.1 Here we calculate the MRS for each of these functions:
Chapter 3: Preferences and Utility
15
3.2 Because all of the first-order partials are positive, we must only check the second-
order partials.
y
yy
shows that monotonic transformations may affect diminishing marginal
utility, but not the
.MRS
3.4 a. In the range in which the same good is limiting, the
Chapter 3: Preferences and Utility
16
b. Again, in the range in which the same good is maximum, the indifference
curve can be shown to be linear. Consider a range in which different goods
c. Here,
Hence, the indifference curve is linear.
f. Raise prices so that a fully condimented hot dog rises in price to $2.60.
Chapter 3: Preferences and Utility
17
3.6 For all the suggested utility functions, let x represent some other good and the
good in question is represented by the appropriate letter:
b. We know that for a Cobb–Douglas utility function,
c. Yes, there was a redundancy. We never used the information about the
Chapter 3: Preferences and Utility
18
Chapter 3: Preferences and Utility
19
Analytical Problems:
3.9 Initial endowments
a.
b. Any trading opportunities that differ from the MRS at
,xy
will provide
c. A preference for the initial endowment will require that trading
3.10 Cobb–Douglas utility
Chapter 3: Preferences and Utility
20
3.11 Independent marginal utilities
From Problem 3.2,
0
xy yx
UU==
implies diminishing MRS providing
3.12 CES utility with weights
d. Follows from part (a). If
3.13 The quasi-linear function
a.
Chapter 3: Preferences and Utility
21
b. Check
22
2 0.
xx y xy x y yy x
U U U U U U U− +
We have
d. Since the marginal utility of
x
is a constant at 1 while that of
y
is
decreasing as
y
increases (as it is of the form
1y
), we would expect
3.14 Preference relations
All of the suggested preference relations are complete, transitive, and continuous.
a. Summation:
Complete: Clearly all bundles are ranked by the sum of items
b. Lexicographic:
Complete: All bundles can be ranked in this ordered way.
Chapter 3: Preferences and Utility
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c. Bliss
Complete: Clearly all bundles are ranked by the distance metric.
Transitive: The distance metric itself imposes a cardinal ranking,
3.15 The benefit function
c. In the graph below, the benefit associated with any initial endowment is
d. In the graph below, two initial endowments are shown
12
( and )EE
. The
Chapter 3: Preferences and Utility
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y