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Chapter 3
A Consumer’s Constrained Choice
Chapter Outline
3.1 Preferences
Properties of Consumer Preferences
Completeness
Transitivity
More is Better
Application: You Can’t Have Too Much Money
Preference Maps
Indifference Curves
Solved Problem 3.1
3.2 Utility
Utility Function
Ordinal Preferences
Utility and Indifference Curves
Willingness to Substitute Between Goods
The Relationship Between the Marginal Rate of Substitution and Marginal Utility
Solved Problem 3.2
Application: MRS Between Recorded Tracks and Live Music
Diminishing Marginal Rate of Substitution
Curvature of Indifference Curves
Straight Line Indifference Curve
Right Angle Indifference Curve
Convex Indifference Curve
Solved Problem 3.3
Application: Indifference Curves Between Food and Clothing
3.3 Budget Constraint
3.4 Constrained Consumer Choice
Finding an Interior Solution Using Graphs
Solved Problem 3.4
Finding an Interior Solution Using Calculus
Substitution Method
40 Perloff • Microeconomics: Theory and Applications with Calculus, Third Edition
©2014 Pearson Education, Inc.
Solved Problem 3.5
Lagrangian Method
Solved Problem 3.6
Solved Problem 3.7
Application: Utility Maximization for Recorded Tracks and Live Music
Finding Corner Solutions
Perfect Substitutes Utility Function
Quasilinear Utility Function
Optimal Bundles on Convex Sections of Indifference Curves
Minimizing Expenditure
Solved Problem 3.8
3.5 Behavioral Economics
Tests of Transitivity
Endowment Effect
Application: How You Ask the Question Matters
Salience
Teaching Tips
The material in this chapter represents a challenge to students for a number of reasons. Most will not have
learned constrained optimization before. Also, utility analysis may seem quite abstract to them. Some take
issue with the assumption of rational behavior on the part of individuals and it needs to be stressed that the
models require only rational behavior on average. Finally, this is the point where the calculus techniques
go beyond the methods they are likely to have learned in their calculus prerequisite.
To overcome these stumbling blocks, you may want to devote one class to presenting and discussing the
technical points of utility theory and budget lines using one or two running examples, and another class to
applications. Whether you decide to present applications as you go, or after presenting the technical material,
this is one area where the students have to see the theory applied in order for it to make sense. You might
also consider breaking the class into small groups to work on problems that test their understanding (additional
problems are provided below). This way, you can migrate among the groups to get a better idea of how
many are struggling and who needs help. The methods presented in this chapter will be needed in Chapter 4
for the derivation of demand curves. If they are not confident with the material in this chapter, they are
almost certain to struggle not only in Chapter 4, but also when it comes to production and cost, since the
methodology there is similar.
The class should see that the notion of a measurable util is nonsense and will find the model more believable
given that only ordinal measurability is needed. In Section 3.3 on budget lines, more difficult
manipulations of the budget constraint can be introduced using taxes, subsidies, and binding quotas with or
without transferability, and this is good background for the problems that follow.
For Section 3.4, Constrained Choice, the technical portion of the material should come fairly easily if the
class has a solid grasp of indifference curves. Using plenty of examples and applications will make the
process of utility maximization seem less abstract. There are several in the text as well as in the additional
applications below. You may want to spend some time contrasting interior solutions and corner solutions.
Corner solutions are an effective way to show that the model makes predictions that are intuitively accurate.
For example, you can choose combinations of goods such that they are perfect substitutes to demonstrate
Chapter 3 A Consumer’s Constrained Choice 41
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that the model predicts that the consumer will always buy the one that is cheaper. You can also choose two
goods such that one is a good and the other is neutral to show that the consumer will only purchase the good
that provides utility. Also spend some time on the notion of “bads” (see discussion question 2 below).
Try to get the class to realize that if the “bad” is turned on its head, it becomes a good (e.g., less air pollution
is more clean air) and that clean air has a positive price that can be measured as the price of abatement.
Once transformed, standard maximization methods apply. Finally, show, using graphs, simple examples of
cases where the standard maximization solution does not exist, such as nonlinear budget lines. For example,
a consumer may be shown to be indifferent between fewer goods at a higher price, and a larger quantity at
a lower price in the case of declining marginal prices.
When discussing the marginal rate of substitution, note that in this text, MRS is left as a negative number
(MRS =
∂
X/
∂
Y, which, of course, is negative, rather than MRS = −
∂
X/
∂
Y as in some texts). When presenting
the MRS, it is worth the time spent to try to get the class to think hard about the special cases of perfect
complements and substitutes. This is especially true when using the Lagrangian technique where the method
fails to find an optimal bundle. Not only does it help them with commodities that fall into this category,
but it also helps them to better understand the more typical case of convexity. Several examples are normally
required before students become comfortable with solving these types of problems.
Introduction of the duality at this time makes it easier when discussing profit maximization and cost
minimization later on.
Section 3.5, Behavioral Economics, is largely conceptual, but very interesting. It is harder to do explicit
problems with this material, but it is wonderful fodder for class discussions, especially after the
mathematical rigor of the previous section. The increasing literature on this subject gives room for a
variety of applications.
Additional Applications
Transitivity of Preferences
How realistic is the assumption of transitive preferences? A number of studies of both humans and animals
show that preferences usually are transitive. For example, Wienstein (1968) uses an experiment to determine
how frequently people give intransitive responses.1 None of the subjects knew the purpose of the experiment.
They were given choices between ten goods, offered in pairs, in every possible combination. The goods
included $3 in cash, an 8-cup “Wearever” aluminum coffee percolator, and a free pass to the next four
Saturday matinees at the subject’s favorite movie theatre. They were told that all of the goods had a value
of $3, to ensure that monetary value would not affect their calculations. Weinstein found that 93.5% of the
responses of adults (at least 18 years old) were transitive. Of children aged 9–12, however, only 79.2%
of the responses were transitive. He reasoned that this difference may be a justification for political and
economic restrictions and protections placed on youths.
Psychologists have also tested for transitivity using preferences for colors, photos of faces, and so forth.
For example, Bradbury and Ross (1990) found that, given a choice of three colors, nearly half of 4–5 year
olds are intransitive, compared to 15% for 11–13 year olds, and 5% for adults.2 Bradbury and Ross show
that a novelty (preference for a new color) is responsible for most intransitive responses, and that this effect
is especially strong in children.
1Arnold A. Weinstein, “Transitivity of Preferences: A Comparison Among Age Groups,” Journal of Political Economy, 76(2),
March/April 1968:307–11.
2Hinton Bradbury and Ross Karen, The Effects of Novelty and Choice Materials on the Intransitivity of Preferences of Children
and Adults,” Annals of Operations Research, 23(1–4) June 1990:141–59.
42 Perloff • Microeconomics: Theory and Applications with Calculus, Third Edition
©2014 Pearson Education, Inc.
1. Show using indifference curves and a budget line that if preferences are intransitive, standard utility
maximization solutions may not result.
2. Suppose three of the subjects meet on the street after the above experiment. Each is carrying one item
(the money, the tickets, or the coffee percolator). Two of these individuals have intransitive preferences;
the other does not. How many exchanges could take place involving all three people before one would
be unwilling to trade? What would your answer be if only one had intransitive preferences?
Additional Questions and Problems
1. Maximizing behavior in the context of school performance (not utility maximization) would imply
trying to get straight A’s. Is maximizing behavior a good assumption in this case? Can you think of
another assumption that may be more appropriate for some individuals when attempting to model
school performance behavior?
2. Consider two goods that are perfect substitutes. What is likely to be true about their relative prices?
Can you confirm your hypothesis with examples?
3. From the total utility schedule shown below, calculate the marginal utility of each additional unit of
X consumed.
Units of
X 1 2 3 4 5 6 7 8 9
Utility
15 35 50 62 70 74 76 77 77
4. Suppose a consumer’s utility derived from consuming bananas is described by the function
U = 10X + 3X 2
− (1/3)X3
. Compute marginal utility.
a. Make a table showing total and marginal utility for X from 0 to 7 units.
b. Would this individual ever choose to consume more than 7 units? Explain.
5. What information is contained in the slope of an indifference curve? Why are these curves typically
convex to the origin?
6. For each of the utility functions below, draw a set of indifference curves showing utility levels
U = 12, U = 16, and U = 24.
a. U = XY
b. U = X + Y
c. U = X − Y
d. What is true about the commodities in (b)?
e. What about the commodities in (c)?
7. Suppose a consumer has an income of $500 and faces prices pX = 5 and pZ = 10.
a. Write the equation for the budget constraint.
b. Draw the budget constraint, placing good X on the horizontal axis. Label it BC.
c. What is the slope of BC?
d. Suppose income decreases to $300. Draw the new budget constraint and label it RS.
8. Larry and Teri allocate their consumption between two goods: hats and bats. The price of hats is $4